THE DYNAMIC ANALYSIS AND CONTROL OF A SELF-EXCITED INDUCTION GENERATOR DRIVEN BY A

by

Dawit Seyoum

A thesis submitted to The University of New South Wales for the Degree of Doctor of Philosophy

School of Electrical Engineering and Telecommunications March, 2003

CERTIFICATE OF ORIGINALITY

I hereby declare that this submission is my own work and that, to the best of my knowledge and belief, it contains no material previously published or written by another person nor material which to a substantial extent has been accepted for the award of any other degree or diploma of a university or other institute of higher learning, except where due acknowledgment is made in the text.

______Dawit Seyoum

ii ACKNOWLEDGEMENTS

First, thanks be to God who gave me the physical and spiritual health to pursue my Ph.D. study.

I would like to thank my supervisors Associate Professor M.F. Rahman and Associate Professor Colin Grantham for their guidance and financial assistance throughout this study.

Special acknowledgement is due to Mr. Doug McKinnon for proof reading the thesis and for sharing ideas. I thank Daniel Indyk from Energy Australia for his assistance to visit a site. Thank you to the laboratory staff for their logistical support. Thanks also go to my colleagues in the Energy Systems Research Group for their suggestions, Mr. Baburaj Karanayil, Mr. Chathura Mudannayake, Dr. Enamul Haque, Mr. Lixin Tang, Mr. Phuc Huu To and Mr. Phop Chancharoensook.

I thank my late father who encouraged me to go to school when I was a little boy and my mother who raised me and helped me to go to school as a single mother.

Last, but foremost, thanks go to my family. To my wife Abeba, thank you for your patience, understanding, encouragement and help, especially when I was spending most of the time doing research. And thanks to my little daughter, Lwam, for your patience in enjoying the little time that I had to spend with you.

iii ABSTRACT This thesis covers the analysis, dynamic modelling and control of an isolated self- excited induction generator (SEIG) driven by a variable speed wind turbine. The voltage build up process of an isolated induction generator excited by AC capacitors starts from charge in the capacitors or from a remnant magnetic field in the core. A similar voltage build up is obtained when the isolated induction generator is excited using an inverter/rectifier system with a single DC capacitor on the DC link of the converter. In this type of excitation the voltage build up starts from a small DC voltage in the DC link and is implemented using vector control.

The dynamic voltage, current, power and frequency developed by the induction generator have been analysed, simulated and verified experimentally for the loaded and unloaded conditions while the speed was varied or kept constant. Results which are inaccessible in the experimental setup have been predicted using the simulation algorithm.

To model the self excited induction generator accurate values of the parameters of the induction machine are required. A detailed analysis for the parameter determination of induction machines using a fast data acquisition technique and a DSP system has been investigated. A novel analysis and model of a self-excited induction generator that takes iron loss into account is presented in a simplified and understandable way.

The use of the variation in magnetising inductance with voltage leads to an accurate prediction of whether or not self-excitation will occur in a SEIG for various capacitance values and speeds in both the loaded and unloaded cases. The characteristics of magnetising inductance, Lm, with respect to the rms induced voltage or magnetising current determines the regions of stable operation as well as the minimum generated voltage without loss of self-excitation.

In the SEIG, the frequency of the generated voltage depends on the speed of the prime mover as well as the condition of the load. With the speed of the prime mover of an isolated SEIG constant, an increased load causes the magnitude of the generated voltage

iv and frequency to decrease. This is due to a drop in the speed of the rotating magnetic field. When the speed of the prime mover drops with load then the decrease in voltage and frequency will be greater than for the case where the speed is held constant. Dynamic simulation studies shows that increasing the capacitance value can compensate for the voltage drop due to loading, but the drop in frequency can be compensated only by increasing the speed of the rotor.

In vector control of the SEIG, the reference flux linkage varies according to the variation in rotor speed. The problems associated with the estimation of stator flux linkage using integration are investigated and an improved estimation of flux linkage is developed that compensates for the integration error.

Analysis of the three-axes to two-axes transformation and its application in the measurement of rms current, rms voltage, active power and from data obtained in only one set of measurements taken at a single instant of time is discussed. It is also shown that from measurements taken at two consecutive instants in time the frequency of the three-phase AC power supply can be evaluated. The three-axes to two- axes transformation tool simplifies the calculation of the electrical quantities.

v CONTENTS

ACKNOWLEDGEMENTS...... iii

ABSTRACT ...... iv

CONTENTS ...... vi

LIST OF FIGURES ...... xii

LIST OF TABLES ...... xix

LIST OF SYMBOLS ...... xx

1 INTRODUCTION ...... …1 1.1 General...... …1 1.2 Thesis outline...... …4 1.3 Literature review...... …8 1.3.1 Self-excited induction generator...... …8 1.3.2 Capacitance and rotor speed for self-excitation ...... 11 1.3.3 Representation of magnetising inductance...... 11 1.3.4 Control of generated voltage and frequency...... 13 1.3.5 Wind powered generators...... 13 1.3.6 Cross saturation...... 15 1.4 References...... 16

2 WIND POWER...... 21 2.1 Source of wind...... 21 2.2 Wind Turbine...... 22 2.2.1 Vertical axis wind turbine...... 22

vi 2.2.2 Horizontal axis wind turbine ...... 23 2.3 Power extracted from wind...... 24 2.4 Torque developed by a wind turbine ...... 31 2.5 Tip-Speed Ratio...... 35 2.6 Power control in wind turbines...... 36 2.6.1 Pitch control...... 38 2.6.2 Yaw control ...... 38 2.6.3 Stall control...... 39 2.7 Wind powered electric generation...... 40 2.8 Economics of wind powered electric generation...... 41 2.9 Summary...... 42 2.10 References...... 43

3 THREE AXES TO TWO AXES TRANSFORMATION AND ITS APPLICATION ...... 44 3.1 Introduction...... 44 3.2 General change of variables in transformation...... 45 3.2.1 Transformation into a stationary reference frame ...... 46 3.2.2 Transformation into a rotating reference frame...... 51 3.3 Voltage measurement ...... 53 3.4 Current measurement...... 55 3.5 Power measurement...... 58 3.6 Power factor measurement ...... 60 3.7 Frequency measurement...... 61 3.8 Measurement in a balanced non sinusoidal three phase system...... 63 3.9 Summary...... 64 3.10 References...... 64

4 INDUCTION MACHINE MODELING ...... 66 4.1 Introduction ...... 66 4.2 Conventional induction machine mode ...... 67 4.3 D-Q axes induction machine model ...... 70

vii 4.4 Simulation of induction machine...... 74 4.5 D-Q axes induction machine model in rotating reference frame ...... 86

4.6 Development of D-Q axes induction machine model with Rm ...... 87 4.7 Summary...... 93 4.8 References ...... 93

5 DATA ACQUISITION AND DIGITAL SIGNAL PROCESSING...... 95 5.1 Introduction ...... 95 5.2 DS1102 DSP board...... 96 5.3 Data acquisition...... 98 5.3.1 Voltage and Current measurement ...... 98 5.3.1.1 Anti-aliasing filter...... 99 5.3.1.2 Voltage measurement ...... 101 5.3.1.3 Current measurement...... 102 5.4 Speed and angle measurement...... 103 5.4.1 Angle measurement ...... 105 5.4.2 Speed measurement ...... 107 5.5 Digital signal processing ...... 108 5.5.1 Digital filter...... 108 5.5.1.1 Infinite Impulse Response (IIR) filter ...... 109 5.5.1.2 Finite Impulse Response (FIR) filter...... 110 5.5.1.3 Comparison of IIR and FIR filters...... 111

5.5.2 Digital filter design from analog filter...... 111

5.5.3 Implementation of a digital filter by approximating analog filter circuits...... 112

5.6 Summary...... 113

5.7 References ...... 114

6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE.... 115 6.1 Introduction ...... 115 6.2 Open-circuit and short-circuit test...... 117 6.2.1 Open-circuit test...... 117

viii 6.2.2 Short-circuit test...... 118 6.2.3 Induction machine with constant rotor parameters...... 119 6.2.4 Induction machine with variable rotor parameters...... 120 6.2.5 Results for DSP based parameter determination ...... 125 6.3 Sensitivity study on variable rotor parameters ...... 137 6.3.1 The effect of combining measurement errors ...... 138 6.3.1.1 Percentage errors ...... 138 6.3.1.2 Combining errors...... 139 6.3.2 Induction machine parameters for analysis of measurement error..... 139 6.3.3 Statistical tools...... 140 6.3.4 Simulation of parameter determination with measurement error...... 142 6.4 Summary...... 146 6.5 References ...... 147

7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS...... 149 7.1 Introduction...... 149 7.2 Model of self-excited induction generator ...... 151 7.3 Analysis of self-excitation process...... 153 7.3.1 RLC circuit characteristics ...... 154 7.3.2 Conditions for self-excitation in induction generator...... 156 7.3.2.1 Using matrix partition...... 158 7.3.2.2 Direct matrix inversion...... 162 7.4 Characteristics of magnetising inductance in induction machine ...... 164 7.5 Minimum speed and capacitance for self-excitation...... 166 7.6 Magnetising inductance and its effect on stability of generated voltage .... 170 7.7 Onset of self-excitation when the SEIG is loaded...... 173 7.8 Simulation of self-excited induction generator ...... 175 7.8.1 The modelling of self-excitation process...... 175 7.8.1.1 Determination of initial conditions...... 175 7.8.1.2 The dynamic representation of self-excitation at no load .... 176 7.8.2 The dynamic representation of a loaded SEIG...... 186 7.9 Characteristics of wind turbine and its effect on generator output...... 194

ix 7.10 Effect of rotor parameters variation on self-excitation...... 199 7.11 Summary...... 205 7.12 References...... 207

8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT...... 208 8.1 Introduction...... 208

8.2 SEIG dynamic model including Rm ...... 209

8.3 Characteristics of Lm and Rm...... 210

8.4 Analysis of SEIG including Rm ...... 211

8.5 Simulation of dynamic self-excitation including Rm ...... 213 8.5.1 Simulation of dynamic self-excitation at no load...... 213 8.5.2 Dynamics of SEIG during loading ...... 216 8.6 Summary...... 220 8.7 References...... 221

9 INVERTER/RECTIFIER EXCITATION OF A THREE-PHASE INDUCTION GENERATOR...... 222 9.1 Introduction...... 222 9.2 Vector control...... 224 9.2.1 Rotor flux oriented vector control ...... 225 9.2.1.1 Direct (feedback) flux oriented vector control ...... 227 9.2.1.2 Indirect (feed forward) flux oriented vector control...... 231 9.2.2 Rotor flux oriented control with voltage as the controlled variable.. 232 9.2.3 Stator flux oriented vector control...... 234 9.3 System description...... 239 9.4 Establishment of reference flux linkage ...... 241 9.5 Details for the implementation of vector control ...... 243 9.5.1 Implementation of direct rotor flux oriented vector control...... 244 9.5.2 Implementation of indirect rotor flux oriented vector control...... 245 9.5.3 Implementation of rotor flux oriented vector control with voltage as a control variable...... 246

x 9.5.4 Implementation of stator flux oriented vector control...... 247 9.6 Results...... 248 9.7 Summary...... 254 9.8 References...... 256

10 FLUX LINKAGE ESTIMATION AND COMPENSATION IN INDUCTION MACHINES ...... 258 10.1 Introduction...... 258 10.2 Theory of Integrator ...... 259 10.3 Numerical integrator...... 263 10.4 Proposed integration offset adjustment ...... 263 10.4.1 Strategy I - without input offset minimization ...... 264 10.4.2 Strategy II - with input offset minimization ...... 265 10.5 Stator flux linkage estimation with the proposed method...... 265 10.6 Summary...... 267 10.7 References...... 268

11 CONCLUSIONS AND SUGGESTION FOR FUTURE WORK...... 269 11.1 Conclusions ...... 269 11.2 Suggestions for future work ...... 277

APPENDICES A DETERMINATION OF INERTIA AND FRICTION COEFIENT OF THE INDUCTION GENERATOR SYSTEM ...... 278 B MEASUREMENT AND CONTROL SYTEMS HARDWARE...... 283 C DETAILS IN INDUCTION MACHINE MODELLING ...... 289 C.1 Introduction ...... 289 C.2 Relationship of parameters in steady state model and d-q model of induction machines ...... 289

C.3 Expanded equations for induction machine modelling including Rm.. 292 D LIST OF PUBLICATIONS ...... 296

xi LIST OF FIGURES

Fig. 1.1 Kooragang wind turbine generator, Newcastle, NSW, Australia ...... …2 Fig. 1.2 Wind farm around San Francisco, California, USA (Photo 2002) ...... …3 Fig. 2.1 Vertical axis wind turbine...... …23 Fig. 2.2 Horizontal axis wind turbine (a) upwind machine (b) downwind machine…24 Fig. 2.3 Detail of a wind turbine driven power generation system ...... …25 Fig. 2.4 Change of wind speed and wind pressure around the wind turbine ...... 27

Fig. 2.5 Power coefficient versus V2/V1 ...... 30 Fig. 2.6 Wind turbine output power to shaft speed characteristic curve...... 31 Fig. 2.7 Air flow around cross section of a blade of a wind turbine ...... 32 Fig. 2.8 Air flow around cross section of a blade during stall condition ...... 32 Fig. 2.9 Wind turbine output torque to shaft speed characteristic curve...... 33 Fig. 2.10 Detail of a twisted rotor blade...... 34 Fig. 2.11 Cross section of a twisted rotor blade from tip to base...... 35 Fig. 2.12 Typical power coefficient versus tip speed ratio ...... 36 Fig. 2.13 Histogram and Weibull function for the probability of a given wind speed (data measured in 1m/s intervals) ...... 37 Fig. 2.14 Wind turbine control regions ...... 38 Fig. 2.15 Power coefficient verses tip speed ratio under yaw control ...... 39 Fig. 2.16 Growth of wind energy capacity worldwide...... 41 Fig. 2.17 Trend in the cost of electricity generated from wind energy...... 42 Fig. 3.1 Three-axes and two-axes in the stationary reference frame...... 46 Fig. 3.2 Three-axes and two-axes in the stationary reference frame with d-axis and a-axis aligned ...... 49 Fig. 3.3 Steps of the abc to rotating dq axes transformation...... 52 Fig. 3.4 Voltage vector and its component in dq axes ...... 54 Fig. 3.5 Current vector and its component in stationary dq axes...... 58 Fig. 3.6 Voltage and current vectors with their components in the stationary dq-axes ...... 59

xii Fig. 4.1 Stator side of the per-phase equivalent circuit of a three-phase induction machine...... 67 Fig. 4.2a Rotor side of the per-phase equivalent circuit of a three-phase induction machine...... 68 Fig. 4.2b Rotor side of the induction machine with adjustment ...... 68 Fig. 4.3 Per-phase equivalent circuit of three-phase induction machine neglecting core loss...... 69 Fig. 4.4 Per-phase equivalent circuit of three-phase induction machine including core loss...... 69 Fig. 4.5 D-Q representation of induction machine...... 71 Fig. 4.6 Detailed d-q representation of induction machine in stationary reference frame (a) d-axis circuit (b) q-axis circuit ...... 72 Fig. 4.7 Experimental setup to find the characteristics of induction machine in the motoring and generating regions ...... 75 Fig. 4.8 Variation of stator phase current for constant supply voltage and frequency (a) Current and voltage when the rotor speed is varied from standstill to twice the synchronous speed (b) detail of motoring region (c) detail around the synchronous speed (d) detail in the generating region...... 77 Fig. 4.9 Relationship between phase voltage vector and phase current vector (a) in the motoring region (b) between motoring and generating (at synchronous speed) (c) in the generating region...... 78 Fig. 4.10 Induction machine torque, power and efficiency characteristics (a) torque

(b) electrical power (c) mechanical power (Pm=ZmTe) (d) efficiency ...... 79 Fig. 4.11 Space vector angles measured with respect to the stator voltage space

vector angle for (a) stator current Is (b) stator flux linkage Os

(c) rotor current Ir (d) magnetising current Im...... 80

Fig. 4.12 Magnitude of space vector for (a) stator voltage (b) stator current Is

(c) stator flux linkage Os (d) rotor current Ir (e) magnetising current Im ...... 81 Fig. 4.13 Space vector diagram for stator voltage, stator current, rotor current, magnetising current and stator flux linkage (a) during motoring mode (b) during generating mode...... 82 Fig. 4.14 Stator current in the de-qe axes of the excitation reference frame

xiii (a) qe-axis current (b) de-axis current...... 84 Fig. 4.15 Stator voltage in the de-qe axes of the excitation reference frame (a) de-axis voltage (b) qe-axis voltage ...... 84 Fig. 4.16 Magnetising current in the de-qe axes of the excitation reference frame (a) de-axis magnetising current (b) qe-axis magnetising current...... 85 Fig. 4.17 Rotor current in the de-qe axes of the excitation reference frame (a) de-axis rotor current (b) qe-axis rotor current ...... 85 Fig. 4.18 Rotor current in different reference frames (a) rotor current in a rotating reference frame that is rotating at the rotor speed (b) rotor current in the stator (stationary) reference frame ...... 86

Fig. 4.19 D-Q representation of induction machine in the excitation (Ze) reference frame (a) d-axis circuit (b) q-axis circuit ...... 87

Fig. 4.20 D-Q model of induction machine including core loss represented by Rm (a) d-axis (b) q-axis...... 90 Fig. 5.1 Block diagram for data acquisition and signal processing...... 95 Fig. 5.2 Hardware and software system configuration...... 96 Fig. 5.3 Block Diagram of the DS1102...... 97 Fig. 5.4 Voltage measurement system (a) voltage sensor (b) signal conditioning for the sensed voltage...... 101 Fig. 5.5 Current measurement system (a) current transducer (b) signal conditioning for the sensed current in terms of voltage signal ...... 103 Fig. 5.6 Output signals of and incremental angle encoder ...... 104 Fig. 5.7 Block diagram of an incremental encoder interface ...... 105 Fig. 5.8 Block diagram for FIR filter ...... 110 Fig. 5.9 Simple first order analog low pass filter ...... 112 Fig. 6.1 The per-phase equivalent circuit with shunt magnetising branch impedance represented in parallel...... 115 Fig. 6.2 Per-phase equivalent circuit with shunt magnetising branch impedance represented in series form ...... 116 Fig. 6.3 Per-phase equivalent circuit of three-phase induction machine under no load test ...... 117 Fig. 6.4 Per-phase equivalent circuit at standstill (short-circuit test)...... 118 Fig. 6.5 Current displacement with rotor speed a) zero speed b) intermediate

xiv speed c) close to synchronous speed...... 121 Fig. 6.6 Rotor parameter variations with slip for deep bar induction machine...... 123 Fig. 6.7 Per-phase equivalent circuit with variable rotor parameters ...... 123 Fig. 6.8 Monitoring system for parameter determination ...... 126 Fig. 6.9 Three-phase input quantities as a function of time (a) measured line voltage (b) measured line current (c) measured input power...... 128 Fig. 6.10 Three-phase induction motor input quantities as a function of speed (a) measured line voltage (b) measured line current (c) measured input power...... 130 Fig. 6.11 Variation of rotor parameters for machine single-cage rotor...... 131 Fig. 6.12 Variation of rotor parameters with slip and supply line to line voltage ...... 132 Fig. 6.13 Effect of temperature on rotor parameters ...... 134 Fig. 6.14 Variation of (a) magnetizing reactance with voltage at 95oC (b) iron loss resistance with voltage at 95oC (c) magnetizing reactance with temperature and voltage (d) iron loss resistance with temperature and voltage ...... 137

Fig. 6.15 Values of rotor resistance, Rr, and rotor leakage reactance, Xlr ...... 140 Fig. 6.16 Measurement error with a normal distribution ...... 141 Fig. 6.17 Data generated for simulation of measurement error...... 143 Fig. 6.18 Error in rotor parameters due to ±0.5% error in voltage current and/or power...... 144 Fig. 6.19 Error in rotor parameters due to ±1% error in voltage current and/or power...... 144 Fig. 6.20Error in rotor parameters due to ±1.5% error in voltage current and/or power...... 145 Fig. 6.21 Simulated shaft torque for variable and constant rotor parameters ...... 146 Fig. 7.1 SEIG with a capacitor excitation system driven by a wind turbine...... 150 Fig. 7.2 D-Q representation of self-excited induction generator...... 151 Fig. 7.3 Detailed d-q model of SEIG in stationary reference frame (a) q-axis circuit (b) d-axis circuit...... 152 Fig. 7.4 RLC circuit...... 154 Fig. 7.5 Current in series RLC circuit (a) for R = 1.2: and (b) for R = -1.2:...... 156G

xv Fig. 7.6 Variation of magnetising inductance with phase voltage at rated frequency 165 Fig. 7.7 Flow chart to determine the minimum speed and minimum capacitance for SEIG at no load ...... 167 Fig. 7.8 Values of minimum capacitance and rotor speed for self-excitation at no load...... 169 Fig. 7.9 Error in capacitance when calculated using the approximate method...... 170 Fig. 7.10 Measured unsuccessful self-excitation at C=60PF (a) generated phase voltage (b) speed ...... 171 Fig. 7.11 Measured self-excitation at C = 60PF and lower speed (a) generated phase voltage (b)speed...... 172 Fig. 7.12 Measured self-excitation at C = 60PF with speed and generated voltage close to rated values (a) generated phase voltage (b)speed ...... 173G

Fig. 7.13 Required capacitance and speed for self-excitation with load, RL...... 174 Fig. 7.14 Relationship between capacitance value, rotor speed and generated voltage at no load ...... 178 Fig. 7.15 Variation of magnetising inductance with phase voltage at different frequencies ...... 179 Fig. 7.16 Variation of magnetising inductance with magnetising current ...... 180 Fig. 7.17 DC motor speed regulator...... 181 Fig. 7.18 Measured self-excitation at C = 60PF and with regulated speed (a) generated phase voltage (b) speed (c) stator current ...... 182G Fig. 7.19 Simulated self-excitation at C = 60PF and with regulated speed (a) generated phase voltage (b) speed (c) stator current ...... 183 Fig. 7.20 Simulated self-excitation at C = 60PF and with regulated speed (a) magnetising inductance (b) rms magnetising current (c) peak stator flux-linkage ...... 184G Fig. 7.21 Three dimensional d-axis flux-linkage and q-axis flux-linkage as a function of time during self-excitation process ...... 185 Fig. 7.22 Self-excitation process initiated by a charged capacitor of 60PF and rotor speed of 1480rpm (a) experimental result (b) simulated result...... 186G Fig. 7.23 d-q model of a loaded SEIG in a stationary reference frame (a) q-axis

xvi circuit (b) d-axis circuit...... 187 Fig. 7.24 Relationship between rotor speed and synchronous speed in a SEIG ...... 189 Fig. 7.25 Experimental loading of SEIG after the voltage has developed to its steady state value (a) phase voltage (b) speed (c) frequency (d) rms phase voltage (e) generated power (f) rms stator current ...... 190 Fig. 7.26 Simulated loading of SEIG after the voltage has developed to its steady state value (a) phase voltage (b) speed (c) frequency (d) rms phase voltage (e) generated power (f) rms stator current ...... 191 Fig. 7.27 Simulated loading of SEIG (a) rms stator current (b) rms capacitor current (c) rms load current...... 192 Fig. 7.28 Simulated loading of SEIG (a) Lm (b) peak flux-linkage (c) rms magnetising current...... 192 Fig. 7.29 Measured variation of generated voltage with load for a 60PF capacitance 193G Fig. 7.30 Measured variation of generated frequency with load for a 60PF capacitance...... 193G Fig. 7.31 Wind turbine output torque as a function of rotor speed ...... 195 Fig. 7.32 Simulated results for wind turbine with variable rotor speed (a) load resistance (b) capacitance (c) rotor speed (d) phase voltage (e) frequency as a function of time...... 197 Fig. 7.33 Simulated results for wind turbine with variable rotor speed (a) rms stator current (b) rms capacitor current (c) rms load current (d) electromagnetic torque (e) output power as a function of time...... 198 Fig. 7.34 Input to the hypothetical SEIG (a) capacitance, PF (b) load resistance, :G (c) speed, rpm...... 200 Fig. 7.35 Comparison of constant and variable rotor parameters performance in SEIG (a) rms phase voltage (b) rms stator current (c) rms capacitor current (d) rms load current (e) rms magnetising current (f) magnetising inductance ...... 203 Fig. 7.36 Comparison of constant and variable rotor parameters performance in SEIG (a) generated frequency (b) slip (c) electromagnetic torque (d) electrical generated output power (e) mechanical input power (f) efficiency...... 204

xvii Fig. 8.1 No load D-Q model of a SEIG including core loss represented by Rm (a) d-axis (b) q-axis...... 210

Fig. 8.2 Values of capacitance and speed for self-excitation with and without Rm at no load...... 213

Fig. 8.3 No load RMS phase voltage during self-excitation with and without Rm ..... 216 Fig. 8.4 Variation of connected capacitor and resistor...... 218 Fig. 8.5 The dynamic rms generated voltage with variation of load and capacitance 218 Fig. 8.6 Dynamic currents in the load, capacitor and stator with variation in load and capacitance...... 219 Fig. 8.7 The dynamic output power with variation in load and capacitance ...... 219 Fig. 8.8 The dynamic electromagnetic torque with variation in load and capacitance220 Fig. 9.1 Electrical and mechanical connections ...... 224 Fig. 9.2 Vector diagram for rotor flux oriented vector control ...... 226 Fig. 9.3 Vector diagram for stator flux oriented vector control ...... 235 Fig. 9.4 System description ...... 240 Fig. 9.5 Relationship between generator rotor speed and flux linkage...... 242 Fig. 9.6 Implementation of direct rotor flux oriented vector control with current controlled PWM VSI ...... 244 Fig. 9.7 Implementation of indirect rotor flux oriented vector control with current. 246 Fig. 9.8 Implementation of direct rotor flux oriented vector control with stator voltage as a control variable ...... 247 Fig. 9.9 Implementation of stator flux oriented vector control with current controlled PWM VSI...... 248 Fig. 9.10 Generated DC voltage for different capacitance value ...... 250 Fig. 9.11 Rotor speed and angular frequency of the generated voltage for different capacitance value...... 250 Fig. 9.12 Flux linkage at different rotor speeds of the induction generator for 1000PF ...... 251 Fig. 9.13 Generated line to line voltage at the terminals of the induction generator.. 251

Fig. 9.14 Loading of the induction generator (a) RL (b) rotor speed (c) VDC (d) flux

e e linkage (e) ids (f) iqs (g) Idc (h) Output power (i) Slip (j) Electromagnetic torque ...... 254

xviii Fig. 10.1 Offset error equal to Am as a result of the integration initial condition...... 261 Fig. 10.2 No integrator error ...... 261 Fig. 10.3 Error produced due to measurement offset...... 262 Fig. 10.4 Error produced due to measurement offset and integration initial condition262 Fig. 10.5 Numerical integrator representation ...... 263 Fig. 10.6 Proposed offset adjustment in a numerical integrator...... 265 Fig. 10.7 Proposed integrator with input offset adjustment ...... 265 Fig. 10.8 Detail for integration error compensation ...... 266 Fig. 10.9 Stator flux linkage estimation using the proposed method...... 267 Fig. A.1 Electromagnetic torque versus motor speed at steady state...... 280 Fig. A.2 Variation of speed with time (a) DC motor field supply on (b) DC motor field supply off ...... 280 Fig. B.1 Interconnection of hardware system ...... 283 Fig. B.2 DSPACE DS1102 DSP controller board ...... 283 Fig. B.3 Multiplexer board control to dSPACE DS1102 DSP card connection...... 283 Fig. B.4 DAC output for DC motor speed control...... 284 Fig. B.5 Dead time Generator board and DS1102 DSP card connection ...... 284 Fig. B.6 Incremental encoder DS1102 DSP card connection...... 284 Fig. B.7 Four isolated 15V Power supply for optocoupler circuit...... 285 Fig. B.8 Optocoupler to Mitsubishi PM50RVA120 IPM ...... 286 Fig. B.9 8 to 4 multiplexer with Sample and Hold ...... 287 Fig. B.10 Cross over protection board (dead time generator)...... 288 Fig. D.1 Student award...... 300

LIST OF TABLES

Table 2.1 Rough Categories of Wind Generator Sizes ...... 39

xix LIST OF SYMBOLS

Generally symbols are defined locally. The list of principal symbols is given below

V1 - Upwind velocity, m/s

V2 - Downwind velocity, m/s

VT - wind velocity at the wind turbine, m/s U - density of air, Kg/m3 m - mass of air, Kg V - velocity of air, m/s F - force applied on rotor blades, N

PT - power extracted by the wind turbine, Watt A - area swept by the blades of the wind turbine, m2

ZT - angular velocity of the wind turbine, rad/s

Vtn - tangential speed of the blades at the tips

TT - torque produced by the wind turbine, Nm

Vw - the undisturbed wind speed in the site, m/s

Ve - the maximum fraction of the undisturbed wind that can be absorbed by the rotor

blade for maximum capture of wind power, Ve = 2/3*Vw, m/s

Va - is the wind created due to rotation of the wind turbine and increases with radius (Va

is perpendicular to Ve and Vw), m/s

Vres - the resultant incident wind speed due to Va and Ve, m/s r - total radius the rotor blade respectively, m r1, r2 and r3 - radiuses at points 1, 2 and 3 of the rotor blade respectively, m

TSR - Tip-Speed Ratio (dimensionless ratio of tip linear speed of blades to Vw) 2 Prf - Steady state wind pressure, which is equal to atmospheric air pressure, N/m Pr  - wind pressure just after the wind turbine, N/m2 f Pr  - wind pressure just before the wind turbine, N/m2 f m - Mass flow rate of air per unit time, Kg/s Q - Volume flow rate of air per unit time, m3/s

Cp - Dimensionless power coefficient s s s fa , fb , and fc a b c axes instantaneous quantities in stationary reference frame

xx s s s fq , fd , and fo dq axes instantaneous quantities in stationary reference frame e e fq , and fd dq axes DC quantities in excitation reference frame va, vb and vc phase voltages in three axes system (stationary reference frame), V ia, ib and ic phase currents in three axes system (stationary reference frame), A s s vq , and vd phase voltages in two axes system (stationary reference frame), V s s iq , and id phase currents in two axes system (stationary reference frame), A e e iq , and id phase currents in two axes system (excitation reference frame), A ds-qs stationary dq axes de-qe dq axes in rotating reference frame (rotating at excitation frequency) vds d-axis stator voltage, V vqs q-axis stator voltage, V vdr d-axis rotor voltage, V vqr q-axis rotor voltage, V ids d-axis stator current, A iqs q-axis stator current, A idr d-axis rotor current, A iqr q-axis rotor current, A imd d-axis magnetising current, A imq q-axis magnetising current, A

Ods d-axis stator flux linkage, web-turn

Oqs q-axis stator flux linkage, web-turn

Odr d-axis rotor flux linkage, web-turn

Oqr q-axis rotor flux linkage, web-turn

Odm d-axis air gap flux linkage, web-turn

Oqm q-axis air gap flux linkage, web-turn

Vm peak phase voltage, V

Im peak phase current, A

Vrms rms phase voltage, V

Irms rms phase current, A

Vdq phase voltage space vector, V

Idq phase current space vector, A

Ts sampling time(period), seconds

xxi T angle between the two axes and three axes, rad I phase shift between current and voltage Z angular speed of the space vector, speed of the general reference frame, rad/s

Ze angular speed of the excitation reference frame, synchronous speed, rad/s

Zr electrical rotor angular speed, rad/sec

Zm mechanical rotor (shaft) angular speed(Zm = Zr /Pp ), rad/sec fe excitation frequency, Hz s the slip of the rotor with respect to the stator magnetic field

Pp number of pole pairs of the induction machine

Ne synchronous speed in revolutions per minute (rpm)

Vs rms stator voltage, V

Is rms stator current, A

Ir rms rotor current, A

Rs stator winding resistance, :

Rr rotor winding resistance, :

Rm equivalent resistance representing iron loss or core loss, :

Lls stator leakage inductance, H

Llr rotor leakage inductance, H

Lm magnetising inductance, H

Ls stator leakage inductance (Lls) + magnetising inductance (Lm) , H

Lr rotor leakage inductance (Llr) + magnetising inductance (Lm), H p d/dt, the differential operator

Es rms induced emf in the stator winding due to the rotating magnetic field that links the stator and rotor windings, V

Er rms induced voltage in the rotor when the rotor is stationary, V sZe rotor current angular frequency

Te electromagnetic torque, Nm

Tm mechanical torque JJG Om air gap flux linkage JG Ir rotor current space vector D friction coefficient, Nm/rad/sec J inertia, Kg-m2

xxii Aincr incremental count of the position counter, incremental steps

Iincr incremental position, radians

Zres speed measurement resolution, rad/s

Tres angle measurement resolution, rad

VO the measured open-circuit phase voltage, V

IO the measured open-circuit phase current, A

PO the measured open-circuit three-phase power, W

Vsh the measured short-circuit input phase voltage, V

Ish the measured short-circuit input phase current, A

Psh the measured short-circuit three-phase input power, W

Superscript * commanded variables

Abbreviations SEIG Self-Excited Induction Generator emf Electromotive force PWM Pulse Width Modulation IGBT Insulated Gate Bipolar Transistor RMS root mean square DSP Digital Signal Processor (Processing) ADC Analog to digital converter IIR Infinite impulse response FIR Finite impulse response PI Proportional and integral (PI controller) VSI Voltage source inverter IPM Intelligent power module VAR Volt ampere reactive

xxiii CHAPTER 1

INTRODUCTION

1.1 General Today, most of the electricity generated comes from fossil fuels (, oil, and ). These fossil fuels have finite reserves and will run out in the future. The negative effect of these fossil fuels is that they produce pollutant gases when they are burned in the process to generate electricity. Fossil fuels are a non- source. However, renewable energy resources (solar, wind, hydro, , geothermal and ocean) are constantly replaced, hence will not run out, and are usually less polluting [1].

Due to an increase in greenhouse gas emissions more attention is being given to renewable energy. As wind is a renewable energy it is a clean and abundant resource that can produce electricity with virtually no pollutant gas emission. Induction generators are widely used for wind powered electric generation, especially in remote and isolated areas, because they do not need an external power supply to produce the excitation magnetic field. Furthermore, induction generators have more advantages such as cost, reduced maintenance, rugged and simple construction, brushless rotor (squirrel cage) and so on.

In the literature, starting in the 1930s, it is well known that a three-phase induction machine can be made to work as a self-excited induction generator (SEIG) [2, 3]. In an isolated application a three-phase induction generator operates in the self-excited mode by connecting three AC capacitors to the stator terminals [2-4] or using a converter and a single DC link capacitor [5]. The dynamic performance of an isolated induction generator excited by three AC capacitors or a single DC capacitor with a converter is discussed in detail in this work.

1 CHAPTER 1 INTRODUCTION

Induction machines are more robust and cheaper than other electrical machines for the same rating. They need less maintenance when manufactured with a squirrel cage rotor. Depending on the condition of operation the induction machine can be used as a motor or generator. Induction machines are available in single-phase or three-phase constructions. In this work the modelling and analysis given is only for the three-phase induction machine and the induction machine is operated as a generator. The definition of slip in this study is the usual one and is the same for the induction generator and induction motor.

In a grid connected induction generator driven by a wind turbine the magnetic field is produced by excitation current drawn from the grid. In different countries there are many induction generators with high power ratings that use wind power as their prime mover. These export to the grid. The Kooragang wind turbine generator, shown in Fig. 1.1, which is owned and operated by Energy Australia, in Newcastle, NSW, Australia, is connected to the grid and has rated power of 600KW and the turbine is a Vestas V44-600KW machine [6].

Fig. 1.1 Kooragang wind turbine generator, Newcastle, NSW, Australia (Photo 2002)

2 CHAPTER 1 INTRODUCTION

For this generating system the angular speed of the wind turbine rotor measured on the wind turbine side is 28rpm. A gear box steps up the shaft speed and on the generator side the angular speed of the generator rotor is approximately 1500rpm [6].

Multiple wind turbine generators can be installed at a given site to form a wind farm. Fig. 1.2 shows part of a wind farm around San Francisco, California, USA.

Fig. 1.2 Wind farm around San Francisco, California, USA (Photo 2002)

The output voltage and frequency of an isolated induction generator vary depending on the speed of the rotor and the load connected to the generator. This is due to a drop in the speed of the rotating magnetic field [7]. The wind turbine can be designed to operate

3 CHAPTER 1 INTRODUCTION at constant speed or variable speed. When the speed of the prime mover of the isolated induction generator drops with load, then the decrease in voltage and frequency will be greater than for the case where the speed is held constant. The AC voltage can be compensated by varying the exciting AC capacitors or using a controlled inverter and a DC capacitor. However the frequency can be compensated only if there is a change in the rotor speed. Because the frequency of the three-phase isolated induction generator varies with loading its application should be for the supply of equipment insensitive to frequency deviations, such as heaters, water pumps, lighting, battery charging etc.

For applications that require constant voltage and frequency the rectified DC voltage of the isolated induction generator should be controlled to remain at a given reference value. Then the constant DC voltage can be converted to constant AC voltage and frequency using an output inverter. In this way a control mechanism is implemented to regulate the output voltage and frequency from an induction generator.

1.2 Thesis outline There are eleven chapters and four appendices in this thesis. The thesis presents the modelling of the dynamic characteristics of an isolated self-excited induction generator driven by a wind turbine. To have a good understanding of the prime mover an overview of the characteristics of wind turbines is presented. Analysis of an induction generator is discussed using modelling and the theory of induction machines.

In Section 1.3 of this chapter the literature related to isolated induction generators and wind turbines is reviewed. This involves clarifying the strengths and limitations of the previous works and highlighting the advantages of the research covered in the thesis.

In Chapter 2 a detailed explanation about wind as a power source and the mechanism of conversion of wind power to mechanical power is presented. The variation of output power and output torque with rotor angular speed and wind speed is discussed. The economics and growth of wind powered electric generation is given and the projection for the future is also discussed.

4 CHAPTER 1 INTRODUCTION

The three-axes to two-axes transformation presented in Chapter 3 is applicable for any balanced three-phase system. In electrical machines analysis a three-axes to two-axes transformation is applied to produce simpler expressions that provide more insight into the interaction of the different parameters. The D-Q model for dynamic analysis is obtained using this transformation. It is shown that the three-axes to two-axes transformation simplifies the calculation of dynamic rms current, rms voltage, active power and power factor in a three-phase system and more specifically for this application, the three-phase induction machine. Traditional methods of measuring these quantities are unable to obtain peak values of current and voltage in less than one quarter of a cycle. However using the three-axes to two-axes transformation in the manner described in Chapter 3, it is possible to evaluate the rms or peak magnitudes of three-phase AC currents and voltages from one set of measurements taken at a single instant of time. Furthermore from measurements taken at two consecutive instants in time the frequency of the three-phase AC power supply can be evaluated.

In Chapter 4 the modelling of an induction machine using the conventional or steady state model and the D-Q or dynamic model are explained. The voltage, current and flux linkage in the rotating reference frame and their phase relationships in the motoring region and generating region are presented. Chapter 4 gives the fundamentals of induction machine modelling and characteristics as a preparation of the modelling and analysis of an isolated induction generator. The induction machine model in D-Q axes has been improved to include the equivalent iron loss resistance, Rm. This improved model is presented in a simple and understandable way. Using this model the dynamic current, torque and power can be calculated more accurately.

In Chapter 5 the data acquisition system and signal processing are discussed. The measurement of voltages, currents, rotor angle and angular speed with their appropriate sensors is explained. The detail of the digital signal processing (DSP) card and transducer board used in the experimental setup is given. The sensors for current and voltage are Hall-Effect devices. Rotor speed and angle measurements are taken using an optical incremental encoder. The resolution of angle and speed for a given encoder is derived. Anti-aliasing filters are introduced in the analog signals of the sensor outputs to prevent the high frequencies appearing as a low frequency when the analog signal is

5 CHAPTER 1 INTRODUCTION digitised in the A/D converter. The advantage of digital signal processing is discussed and different types of filter design are presented which are used in the simulation and experimental procedures.

Machine modelling requires knowledge of the parameters of the machine. Whether the three-phase induction machine is modelled using the conventional per-phase equivalent circuit or the D-Q method the parameters of the machine are required. Chapter 6 discusses a rapid way of determining the parameters that is fast enough to determine the parameters at rated voltage of the induction machine without damaging it due to overheating. The error in the values of induction motor parameters arising from measurement error in voltage, current and power have been presented. Rotor parameter variations in squirrel cage induction machines and the cause of this variation is examined. The variation of induction machine parameters with temperature is also presented.

Chapter 7 deals with the modelling, analysis and dynamic performance of an isolated three-phase induction generator excited by three AC capacitors connected at the stator terminals. The mathematical model of a self-excited induction generator including the representation of the remnant magnetic flux in the iron core and the initial charge in the capacitor is given. The initiation and process of self-excitation is presented, starting from a simple RLC circuit as an analogy to a complete dynamic representation of a self- excited induction generator, i.e. the complete representation includes both steady state and transient conditions. The variation of magnetising inductance of the induction machine is important in the voltage build up and stabilisation of the generated voltage. It is shown that the characteristics of magnetising inductance with respect to the rms induced stator voltage or magnetising current determines the regions of stable operation as well as the minimum generated voltage without loss of self-excitation. The variation of the generated voltage and frequency for a self excited induction generator driven by a wind turbine at constant and variable speeds has been investigated. Using simulation algorithms more results which are not accessible in an experimental setup have been predicted.

6 CHAPTER 1 INTRODUCTION

In Chapter 8 the modelling of an isolated self-excited induction generator taking iron loss into account is discussed. Iron loss or core loss is represented in the induction machine model using Rm, a resistance value which has the same power loss as the total iron loss in the induction machine. The method presented here is a novel analysis and modelling for the dynamics of the self-excited induction generator driven by a variable speed prime mover taking iron loss into account. It is noted that this method is easily understood, having drawn on many familiar concepts and using the standard terminology and nomenclature of D-Q unified machine theory. This improved model takes into consideration the variations of Rm with air gap voltage and, as in Chapter 7, the variation of magnetising inductance. This model is then coupled to the characteristics of a variable speed prime mover and the analysis of this system is produced and discussed.

In Chapter 9 the voltage build up process and terminal voltage control in an isolated wind powered induction generator using an inverter/rectifier excitation with a single capacitor on the DC link is discussed. A vector control technique is developed to control the excitation and the active power producing currents independently. That is, the current control scheme causes the currents to act in the same way as in a DC generator where the field current and the current are decoupled. When the speed of the prime mover is varied the flux linkage in the induction generator is made to vary inversely proportional to the rotor speed so that the generated voltage will remain constant. Since the torque produced by a wind turbine drops at high turbine rotor speed the induction generator will run at high generator rotor speed when loaded with a small load and the rotor speeds decrease with an increase in load. As the turbine rotor shaft and the generator rotor shaft are connected via a gear box, both rotor speeds will increase and decrease proportionally at constant gear ratio. The flux linkage of the induction generator is controlled by controlling the d-axis current in the synchronously rotating reference frame. Two vector control strategies: rotor flux oriented vector control and stator flux oriented vector control are presented. It is shown that the estimation of rotor flux linkage is more dependent on the induction machine parameters whereas estimation of stator flux linkage is dependent only on the stator resistance.

7 CHAPTER 1 INTRODUCTION

Chapter 10 investigates the problems and the solutions in the estimation of stator flux linkage using integration of the voltage behind the stator resistance. This voltage is calculated from the measured voltages and currents. Accurate flux estimation is very crucial in the control of induction motor drives and induction generators using vector control. The method of flux linkage estimation proposed in this chapter is new and effective. It eliminates the error produced by the measurement offset error and integrator output error due to initial integration in a continuous time integrator or numerical/discrete time integrator. It is shown that if the integration ramp output due to the existence of measurement offset error is large then subtracting the output of a low pass filter of the signal from the signal to be integrated minimizes the offset. A signal with small input offset will have a small increment of ramp that will appear at the output of the integrator. As the time increases the ramp keeps on increasing and eventually the distortion in flux will be unacceptable. However, if the ramp is eliminated every cycle, the flux distortion due to the offset correction at the output is insignificant.

In Chapter 11 conclusions and suggestions for future work are given.

1.3 Literature review In this section previous work carried out in the area of self-excited induction generators that are driven by variable speed prime movers and in particular by wind turbines are reviewed. If there is a controller to regulate the output voltage and frequency, then an isolated induction generator can be driven by a variable speed prime mover. However, for loads which are insensitive to frequency, then the controller needs only to regulate the generated voltage.

1.3.1 Self-excited induction generator The early work on three-phase SEIGs excited by three capacitors was mainly experimental analysis [2, 3]. The main methods of representing a SEIG are the steady state model and the dynamic model. The steady state analysis of SEIG is based on the steady state per-phase equivalent circuit of an induction machine with the slip and angular frequency expressed in terms of per unit frequency and per unit angular speed. The steady state analysis includes the loop-impedance method [8-13] and the nodal

8 CHAPTER 1 INTRODUCTION admittance method [14-15]. The loop-impedance method is based on setting the total impedance of the SEIG, i.e. including the exciting capacitance, equal to zero and then to find the steady state operating voltage and frequency using an iteration process. In the nodal admittance method the real and imaginary parts of the overall admittance of the SEIG are equated to zero. The equations are formulated based on the steady state conditions of the SEIG.

The main draw back of using the per-phase steady state equivalent circuit model is that it cannot be used to solve transient dynamics because the model was derived from the steady state conditions of the induction machine.

The dynamic model of a SEIG is based on the D-Q axes equivalent circuit or unified machine theory. For analysis the induction machine in three axes is transformed to two axes, D and Q, and all the analysis is done in the D-Q axes model. The results are then transformed back to the actual three axes representation. In the D-Q axes if the time varying terms are ignored the equations represent only the steady state conditions. The SEIG represented in D-Q axes and analysed under steady state conditions are reported in [16-17]. In [18-21] the dynamic equations for the representation of SEIG conditions are given. In these papers the initial conditions that take into account the initial charge in the exciting capacitors and the remnant magnetic flux linkage in the iron core are not given and in some of the papers the complete dynamic equations are not presented.

The D-Q axes model of SEIG given in [20] reported that the dynamic generated voltage varies with the applied load, but there are no results that show what happens to the dynamic speed of the rotor when the generator is loaded. Hence it cannot be proven whether the variation in voltage is exaggerated due to a change in speed or not. To investigate this, the characteristic of the dynamic voltage is simulated and measured keeping the speed at a constant value by applying a speed regulator to a DC motor which is used as a prime mover for the SEIG. For the constant speed drive test a PI (proportional and integrator) speed controller and an inner loop PI current controller is used. The dynamic frequency of the generated voltage, during loading conditions, is calculated from measured voltages or from measured voltages and currents. A three- axes to two-axes transformation is used in the calculation of the dynamic frequency

9 CHAPTER 1 INTRODUCTION value. Here the transformation is used to simplify the calculation. The measured and simulated dynamic currents, active power and electromagnetic torque generated by the SEIG are also given in this paper.

The normal connection of a SEIG is that the three exciting capacitors are connected across the stator terminals and there is no electrical connection between the stator and rotor windings. However, in the literature a SEIG with electrical connection between rotor and stator windings is reported [22]. This paper deals with the steady state performance of a SEIG realised by a series connection of stator and rotor windings of a slip-ring type induction machine and solved using D-Q analysis. In this type of connection it has been claimed that it has the advantage of operating at a frequency independent of load conditions for a fixed rotor speed, however the angular frequency of the output voltage is equal to half of the rotor electrical angular speed, which means the prime mover should rotate at twice the normal speed to generate voltage with standard frequency. There is also concern regarding the current carrying capability of the rotor and stator windings because both of them are carrying the same current. Whether any wound rotor induction machine can be used in this way or not is not specified.

Shridhar et al reported that if a single valued capacitor bank is connected, i.e. without voltage regulator, a SEIG can safely supply an induction motor rated up to 50% of its own rating and with a voltage regulator that maintains the rated terminal voltage the SEIG can safely feed an induction motor rated up to 75% of its own rating [23]. In this case the SEIG can sustain the starting transients of the induction motor without losing self-excitation.

Since a SEIG operates in the saturation region, it has been shown that to saturate the core, the width of the stator yoke is reduced so that the volume and the weight of the induction generator will be less than the corresponding induction motor [24]. The voltage drop for a constant capacitor induction motor used as a generator was 30% while the voltage drop of the corresponding designed induction generator was 6% [24].

10 CHAPTER 1 INTRODUCTION

A three-phase SEIG can be used as a single-phase generator with excitation capacitors connected in C-2C mode where capacitors C and 2C connected across two phases respectively and nil across the third phase [25].The steady state performance of an isolated SEIG when a single capacitor is connected across one phase or between two lines supplying one or two loads is presented in [26]. However in these applications the capacity of the three-phase induction generator cannot be fully used.

1.3.2 Capacitance and rotor speed for self-excitation The minimum and maximum values of capacitance required for self-excitation of a three-phase induction generator have been analysed previously using a current model [9, 11, 20]. Calculation of the minimum capacitance required for self excitation using a flux model has also been reported [27].

In the calculation of capacitance required for self-excitation, economically and technically, it is not advisable to choose the maximum value of capacitance. This is due to the fact that for the same voltage rating the higher capacitance value will cost more. In addition, if the higher capacitance value is chosen then there is a possibility that the current flowing in the capacitor might exceed the rated current of the stator due to the fact that the capacitive reactance reduces as the capacitance value increases.

It has been shown that a de-excited induction generator can re-excite even if the load is already connected to it [30], but the relationship between the value of the load, capacitance and speed has not been given. In this thesis the relationship between speed, capacitance and load is given so that the characteristics of the induction generator for self-excitation with a load can be established. This relationship is also important to find the region where the induction generator can continue to operate without loss of self- excitation.

Wind speed can change from the minimum set point to the maximum set point randomly and the SEIG can be started at any point within the range of speed. It is essential to find the minimum and maximum speed required for self-excitation, when the generator is loaded. In this thesis the author has developed the analysis and

11 CHAPTER 1 INTRODUCTION calculation of the minimum and maximum speeds for self-excitation to occur and for a particular value of capacitance. 1.3.3 Representation of magnetising inductance In the SEIG the variation of magnetising inductance is the main factor in the dynamics of voltage build up and stabilisation. Several papers have reported on the representation of the variation of magnetising inductance (Lm) or magnetising reactance (Xm) during voltage build up.

One of the ways of representation is Xm as a function of Vg/f (V/Hz to relate to flux) [8-

9, 11-13, 15, 21], where Vg is the voltage across Xm and f is the frequency of excitation, or Lm as a function of Vg [14, 26] for a known frequency of operation. In these papers it has been shown that the value of Xm, as the value of Vg/f or Vg increases from zero, starts at a given unsaturated value, remains constant at the unsaturated value for low values of air-gap voltage or ratio of air gap voltage to frequency, and then starts to decrease up to its rated value, which is a saturated value. In fact, in [9] the measured values show the actual variation of magnetising reactance. This is the magnetizing reactance as the air gap voltage increases from zero. It starts at a given value, increases until it reaches its maximum value and then starts to decrease down to its rated value, which is a saturated value. However, in the analysis of the SEIG the magnetising reactance for values of air gap voltage close to zero were ignored. Since Xm is dependent on frequency it is not good for transient dynamic analysis, rather Lm should be used.

The other representation is Xm as a function of magnetising current [20, 28] or Lm as a function of magnetising current [16, 29, 30]. In these papers it has been illustrated that the magnetising inductance or magnetising reactance starts at a maximum unsaturated value and then decreases when the iron core saturates, however in [16] the authors have indicated that the value of magnetising inductance starts at a given unsaturated value, increases and then finally decreases as the magnetising current increases from zero. Although this representation depicts the actual variation of magnetising inductance, the significance of this characteristic has not been presented.

12 CHAPTER 1 INTRODUCTION

The reason for this variation in magnetising reactance and the effect on self-excitation is discussed in this thesis. As the magnetizing reactance is dependent on frequency, magnetizing inductance is used in the analysis and its effect on the initiation of self- excitation and stabilisation is discussed in detail and confirmed experimentally.

1.3.4 Control of generated voltage and frequency The main problem in using a SEIG is the control of the generated voltage because the voltage amplitude and frequency drops with loading as well as with a decrease in the generator rotor speed [7]. The magnitude and frequency of the output voltage of a stand alone induction generator driven by a variable speed rotor can be controlled by employing the rotor excitation of a wound-rotor induction machine [31]. In a similar way it can be controlled by varying the rotor resistance of a self-excited slip-ring induction generator [32]. However a self-excited slip-ring induction generator will require more maintenance than a squirrel cage rotor due to the slip-rings and brush gear.

The rms value of the generated voltage, irrespective of its frequency, can be controlled using variable capacitance values [33], or a fixed capacitor thyristor controlled reactor static VAR compensator [34], or continuously controlled shunt capacitors using antiparallel IGBT across the fixed excitation capacitor [35].

It has been shown that copper loss decreases in the stator and increases in the rotor in the generating mode when compared to the motoring mode [36]. In a SEIG, a squirrel cage rotor is preferable to a wound rotor because the squirrel cage rotor has a higher thermal withstand capability and requires less maintenance. Due to the higher thermal withstand capability of the squirrel cage rotor, a higher copper loss in the rotor is acceptable.

1.3.5 Wind powered generators For a fixed speed wind turbine system that can be connected to the grid, maintaining a constant frequency is not a problem, irrespective of whether an induction or synchronous generator is used. Such systems typically employ induction machines connected directly to the grid. In grid connected systems there are two generating schemes for variable speed wind turbine systems [37-43]. The first scheme employs

13 CHAPTER 1 INTRODUCTION machine control using feeding the rotor circuit (wound rotor induction machine) or a second winding in the stator of an induction machine (squirrel cage rotor or wound rotor) to adjust the frequency and generated voltage when the generator rotor speed is varied. The second scheme applies to single stator winding fed induction generators which produce a constant DC output voltage that is then inverted to have an output of constant rms voltage and frequency. The generation of constant DC voltage is implemented using scalar or vector control [44-45] or using a DC-DC converter to produce constant DC voltage from the variable rectified DC voltage [46]. In a variable speed wind turbine system the mechanical stresses caused in the structural elements by gusts and varying wind speed are diminished by letting the rotor follow the wind. Also when the rotor speed is allowed to vary with the wind the turbine can be operated at peak efficiency. However, the necessary power electronics can be expensive.

Brushless doubly-fed induction machines have two stator windings of different pole number [39-42]. Although the system has reduced size and cost of the power electronics, the induction machine is expensive because it is specially made. A double output induction generator is a wound rotor induction machine with the control power electronics connected on the rotor circuit [43, 45]. In this arrangement the induction generator gives more than its rated power without being overheated. The power generation can be realised for a wide range of wind speed. They have a rotor inverter and front end converter while the stator is linked directly to the grid.

The methods discussed above can also be used to control the output voltage from a stand alone induction generator. In the literature it is reported that a stand alone induction generator excited by a single DC capacitor and inverter/rectifier system can be used instead of the AC capacitor excited system. If a constant DC voltage is achieved then a load side inverter is used to produce a constant rms voltage and frequency. For this application an inverter/rectifier can be shunt connected so that it carries only the exciting current [47-49] or a converter can be connected in series so that it carries the full current [50-51], i.e. the exciting and load current. In both cases the initiation of voltage build up is the same. However in these papers the details of the control mechanism and the generation of reference currents are not given. The minimum DC

14 CHAPTER 1 INTRODUCTION capacitance required for the initiation of voltage build up has been discussed [50]. When the converter carries only the exciting current an additional rectifier is required to produce the DC voltage that supplies the load.

Artificial Intelligence is the branch of science that concentrates on making computers or computer-based technology to function like humans. Advanced intelligent control of a variable speed wind generation system has been reported in the literature [52-57]. Artificial intelligence techniques include fuzzy logic, neural network, and genetic algorithm, etc [56-57]. The evolving adaptive and elastic versions of fuzzy logic control in combination with the artificial neural network algorithms promise to revolutionize the applicability of fuzzy logic control in reference trajectory tracking, state estimation and parameter adaptation of control strategies [52]. It has been shown that fuzzy control algorithms are universal, give fast convergence, are parameter insensitive, and accept noisy and inaccurate signals [57]

It has been reported that artificial intelligent has been used extensively to optimize efficiency and enhance performance in a grid connected variable speed wind generation system [52-57]. A fuzzy controller tracks the generator speed with the wind velocity to extract the maximum power. For a grid connected variable speed wind generation system any power generated by the induction generator is absorbed by the grid. The draw back of using Artificial Intelligence control is, at the current price of microprocessors, it is expensive to implement in induction generators with small power rating. Of course the general drawback of Artificial Intelligence is the inability to untangle the complicated web of human intelligence.

1.3.6 Cross saturation The idea of cross saturation is to have an interest focused on the so called cross coupling effect by which two windings with their magnetic axes in space quadrature exhibit a specific magnetic interaction due to saturation of the main flux paths. With the concept of cross saturation the flux produced by the current flowing in the d-axis magnetising inductance will have effect on the q-axis circuit. With the consideration of cross saturation some authors have argued that the model describes physically sound phenomenon in the smooth air gap machine that are known from the theory of salient

15 CHAPTER 1 INTRODUCTION pole machines [58-60]. However others have argued that cross coupling in smooth air gap machines is not possible and that the existence of cross coupling terms and the inequality of mutual inductances along the d-axis and q-axis is purely a consequence of the mathematical derivation and it has been shown that the concept of cross saturation has no physical significance; it is an erroneous conclusion only, obtained from otherwise correct mathematical formulation [61].

The motivation for transforming the actual three-axes model into a fictitious two-axes or D-Q axes model is to avoid cross coupling between the D and Q phases and simplify the analysis of induction machines. Therefore, since the contribution of cross saturation in the area of induction machine analysis has not been resolved and since the intention of the D-Q analysis is to decouple the axes, cross saturation will not be considered in this thesis. Any variation due to saturation will be represented by the variation of magnetising inductance along the q-axis and the d-axis. In this way, the basic essence of transforming the three-axes model to the D-Q axes model is maintained.

1.4 References [1] M. R. Patel, “Wind and Systems”, CRC Press LLC, Boca Raton, 1999. [2] E. D. Basset and F.M. Potter, “Capacitive excitation of induction generators”, Trans. of the Amer. Inst. Electr, Eng., Vol. 54, No. 5, May 1935, pp. 540-545. [3] C. F. Wagner , “Self-excitation of induction Motors”, Trans. of the Amer. Inst. Electr, Eng., Vol. 58, Feb. 1939, pp. 47-51. [4] J. M. Elder, J.T. Boys and J.L. Woodward, “Self-excited induction machine as a small low-cost generator”, IEE Proc. C, Vol.131, No. 2, March 1984, pp. 33-41. [5] D. W. Novotny, D. J. Gritter and G. H. Studtmann, “Self-excitation in inverter driven induction machines”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-96, no.4, July/August 1977, pp. 1117-1125. [6] Energy Australia “Kooragang wind turbine generator fact sheet”, 2001. [7] D. Seyoum, C. Grantham and F. Rahman, "The dynamic characteristics of an isolated self-excited induction generator driven by a wind turbine", Proceedings IEEE- IAS 2002 Annual Meeting Pittsburgh, USA, October 13-18, 2002, pp 731-738. [8] S. S. Murthy, O. P Malik and A. K. Tandon, “Analysis of self-excited induction generators”, IEE Proc., Vol. 129, No. 6, Nov. 1982, pp. 260-265. [9] N. H. Malik, and A. H. Al-Bahrani, A.H., “Influence of the terminal capacitor on the performance characteristics of a self-excited induction generator ”, IEE Proc C., Vol. 137, No. 2, March 1990, pp. 168-173.

16 CHAPTER 1 INTRODUCTION

[10] A. K. Al Jabri and A. I. Alolah, "Capacitance requirements for isolated self-excited induction generators", IEE Proc., B, Vol. 137, No. 3, 1990, pp.154-159. [11] A. K. Al Jabri and A.I. Alolah, “Limits on the performance of the three-phase self excited induction generators” IEEE Trans. on Energy Conversion, Vol. 5, No. 2, June 1990, pp. 350-356. [12] A.K. Tandon S. S. Murthy and G. J. Berg, “Steady state analysis of capacitor self-excited induction generators”, IEEE Transaction on Power Apparatus and Systems, Vol. 103, No.3, March 1984, pp. 612-618. [13] L. Shridhar, B. Singh and C. S. Jha, “A step towards improvements in the charactersitics of self- excited induction generator”, IEEE Transactions. on Energy Conversion, vol. 8, No .1, March 1993, pp. 40-46. [14] T. F. Chan, “Capacitance requirements of self-excited induction generators”, IEEE Trans. on EC, Vol.8, No.2, June 1993, pp. 304-311. [15] S. Rajakaruna and R. Bonert, “A technique for the steady-state analysis of a self-excited induction generator with variable speed”, IEEE Transactions on Energy Conversion, Vol. 8, No. 4, December 1993, pp. 757-761. [16] M. Elder, J. T. Boys and J. L. Woodward, “Self-excited induction machine as a small low-cost generator”, IEE Proc. C, Vol. 131, No. 2, March 1984, pp. 33-41. [17] Y. Uctug and M. Demirekler, “Modelling, analysis and control of a wind-turbine driven self-excited induction generator”, IEE Proceedings, Vol. 135, Pt. C, No. 4, July 1988, pp. 268 -275. [18] C. Grantham, D. Sutanto and B. Mismail, “Steady-state and transient analysis of self-excited induction generators”, IEE Proc. B, Vol. 136, No. 2, pp. 61-68, March 1989. [19] M. H. Salama, and P.G. Holmes, “Transient and steady-state load performance of stand-alone self- excited induction generator”, IEE Proc. -Electr. Power Appl., Vol. 143, No. 1, pp. 50-58, January 1996. [20] L. Wang and L. Ching-Huei, “A novel analysis on the performance of an isolated self-excited induction generator”, IEEE Trans. on Energy Conversion, Vol. 12, No. 2, pp. 109-117, June 1997. [21] L. Wang and Jian-Yi Su, “Effect of long-shunt and short-shunt connection on voltage variations of a self-excited induction generator”, IEEE Trans. on Energy Conversion, vol. 12, No .4, pp. 368- 374, December 1997. [22] A. S. Mostafa, A. L. Mohamadein, E. M. Rashad, “Analysis of series-connected wound-rotor self- excited induction generator”, IEE Proceedings-B, Vol. 140, No. 5, September 1993, pp. 329-336. [23] L. Shridhar, B. Singh, C. S. Jha and B. P. Singh, “Analysis of self excited induction generator feeding induction motor”, IEEE Transactions on Energy Conversion, vol. 9, No .2, June 1994, pp. 390-396. [24] J. Faiz, A. A Dadgari, S. Horning and A. Keyhani, “Design of a three-phase self-excited induction generator”, IEEE Transactions on Energy Conversion, Vol. 10, No. 3, September 1995, pp. 516- 523. [25] J. L. Bhattacharya and J. L. Woodward, “Excitation balancing of a self-excited induction generator for maximum power output”, IEE Proceedings, Vol. 135, Pt. C, No. 2, March 1988, pp. 88-97.

17 CHAPTER 1 INTRODUCTION

[26] Y.H.A. Rahim, “Excitation of isolated three-phase induction generator by a single capacitor”, IEE Proceedings-B, Vol. 140, No. 1, January 1993, pp. 44-50. [27] R. J. Harrington and F. M. M. Bassiouny, “New approach to determine the critical capacitance for self-excited induction generators”, IEEE Transactions on Energy Conversion, Vol. 13, No. 3, September 1998, pp. 244-249. [28] C. H. Lee and L. Wang, “A novel analysis of parallel operated self-excited induction generators”, IEEE Transactions on Energy Conversion, Vol. 13, No. 2, June 1998, pp. 117-123. [29] Li Wang and Jian-Yi Su, “Dynamic performances of an isolated self-excited induction generator under various loading conditions”, IEEE Transactions on Energy Conversion, Vol. 14, No. 1, March 1999, pp. 93-100. [30] L. Shridhar, B. Singh and S. S. Jha, “Transient performance of the self regulated short shunt self excited induction generator”, IEEE Transactions on Energy Conversion, Vol. 10 No. 2, June 1995, pp. 261-267. [31] Y. Kawabata Y. Morine, T. Oka, E. C. Ejiogu and T. Kawabata, “New stand-alone power generating system using wound-rotor induction machine”, IEEE-Power Electronics and Drive Systems Conference, Vol. 1, October 2001, pp. 335-341. [32] T. F. Chan, K. Nigim and L. L. Lai, “Voltage and frequency control of self-excited slip ring induction generators”, IEEE IEMDC Conference, 2001, pp. 410-414. [33] P. G. Casielles, L. Zarauza and J. Sanz, “Analysis and design of wind turbine driven self-excited induction generator”, IEEE-IAS Conference, 1988, pp. 116-123. [34] E. S. Abdin and W. Xu, “Control design and dynamic performance analysis of a wind turbine- induction generator unit”, IEEE Transactions on Energy Conversion, Vol. 15, No. 1, March 2000, pp. 91-96. [35] M. A, Al-Saffar E.-C. Nho and T. A. Lipo, “Controlled shunt capacitor self-excited induction generator”, IEEE - Industry Applications Conference, 12-15 October 1998, pp. 1486-1490. [36] S. S.; Murthy, C. S. Jha and P. S. N. Rao, “Analysis of grid connected induction generators driven by hydro/wind turbines under realistic system constraints”, IEEE Transactions on Energy Conversion, Vol. 5, No. 1, March 1990, pp. 1-7. [37] I. Schiemenz and M. Stiebler, “Control of a permanent magnet synchronous generator used in a variable speed wind energy system”, IEEE- IEMDC 2001 Conference, 2001, pp. 872-877. [38] J. Jayadev, “Harnessing the wind”, IEEE Spectrum, Vol. 32, No. 11, Nov. 1995, pp. 78-83. [39] C. S Brune, R. Spee and A. K. Wallace, “Experimental evaluation of a variable-speed, doubly-fed wind-power generation system”, IEEE Transactions on Industry Applications, Vol. 30 No. 3, May/June 1994, pp. 648-655. [40] R. Spee, S. Bhowmik and J. H. R. Enslin, “Adaptive control strategies for variable-speed doubly- fed wind power generation systems”, IEEE-Industry Applications Society Conference, 2-6 Oct. 1994, pp. 545-552.

18 CHAPTER 1 INTRODUCTION

[41] S. Bhowmik, R. Spee and J. H. R Enslin, “Performance optimization for doubly fed wind power generation systems”, IEEE Transactions on Industry Applications, Vol. 35, No. 4, July/August 1999, pp. 949-958. [42] M. Y. Uctug, I. Eskandarzadeh and H. Ince, “Modelling and output power optimisation of a wind turbine driven double output induction generator”, IEE Proc. Electr. Power Appl., Vol. 141, No. 2, March 1994, pp. 33-38. [43] L. Zhang and C. Watthanasarn “A matrix converter excited doubly-fed induction machine as a wind power generator”, “IEE - Power Electronics and Variable Speed Drives Conference, 21-23 September 1998, pp. 532 -537. [44] A. Miller, E. Muljadi and D. S Zinger, “IEEE Transactions on Energy Conversion, Vol. 12, No. 2, June 1997, pp. 181-186. [45] R. S Pena, R. J. Cardenas, G. M. Asher and J. C. Clare, “Vector controlled induction machines for stand-alone wind energy applications”, IEEE - Industry Applications Conference, 2000, pp. 1409- 1415. [46] S. Jiao, G. Hunter, V. Ramsden and D. Patterson, “Control System Design for a 20KW wind Turbine Generator with a Boost Converter and Battery Bank Load”, IEEE -Power Electronics Specialists Conference, Vancouver, 2001, pp. 2203-2206. [47] S. R. Silva, R. O. C Lyra, “PWM converter for excitation of induction generators”, Fifth European Power Electronics Conference, 1993, pp 174-178. [48] M. S. Miranda, R. O. C. Lyra, and S. R. Silva, “ An Alternative Isolated Wind Erlecric Pumping System Using Induction Machines”, IEEE Transactions on Energy Conversion, Vol. 14, No. 4, Dec. 1999, pp 1611-1616. [49] L. A. C. Lopes and R. G. Almeida, “Operation aspects of an isolated wind driven induction generator regulated by a shunt voltage source inverter”, IEEE - Industry Applications Conference, Oct 2000, pp. 2277-2282. [50] S. N. Bhadra, K. V. Ratnam and A. Manjunath, “Study of voltage build up in a self-excited, variable speed induction generator/static inverter system with DC side capacitor”, “International Conference on Power Electronics, Drives and Energy Systems for Industrial Growth, 1996, 8-11 Jane 1996, pp. 964 -970. [51] R. Cardenas, R. Pena, G. Asher and J. Clare, “Control strategies for enhanced power smoothing in wind energy systems using a flywheel driven by a vector-controlled induction machine”, IEEE Transactions on Industrial Electronics, Vol. 48, No. 3, June 2001, pp. 625-635. [52] H. M. Mashaly; A. M. Sharaf, A. A. El-Sattar and M. M. Mansour, “Implementation of a fuzzy logic controller for wind energy induction generator DC link scheme” Proc. of the Fuzzy Systems IEEE Conference on Computational Intelligence, 26-29 June 1994, pp. 978 -982. [53] R. M. Hilloowala and A. M. Sharaf, “A rule-based fuzzy logic controller for a PWM inverter in a stand alone wind energy conversion scheme”, Proc. of the IEEE- IAS 1993 Annual Meeting Pittsburgh, 2-8 Oct. 1993, pp. 2066 -2073.

19 CHAPTER 1 INTRODUCTION

[54] R. M. Hilloowala and A. M. Sharaf, “A rule-based fuzzy logic controller for a PWM inverter in a stand alone wind energy conversion scheme”, IEEE Transactions on Industry Applications, Vol. 32, No. 1, January/February 1996, pp. 57 –65. [55] H. M. Mashaly; A. M. Sharaf, A. A. El-Sattar and M. M. Mansour, “A fuzzy logic controller for wind energy utilization”, Proc. of the IEEE Conference on Control Applications, 24-26 Aug. 1994, pp. 221 -226. [56] M. G. Simoes, B. K. Bose and R. J. Spiegel, “Fuzzy logic based intelligent control of a variable speed cage machine wind generation system”, IEEE Transactions on Power Electronics, Vol. 12, No. 1, Jan. 1997, pp. 87 -95. [57] M. G. Simoes, B. K. Bose and R. J. Spiegel, “Design and performance evaluation of a fuzzy-logic- based variable-speed wind generation system”, IEEE Transactions on Industry Applications, Vol. 33, No. 4, July-Aug. 1997, pp. 956 –965. [58] J. E. Brown, K. P. Kovacs, P. Vas, “A method of including the effects of main flux path saturation in the generalised equations of AC machines”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-102, No. 1, January 1983, pp. 96-106. [59] K. E. Hallenius, P. Vas, J. E. Brown, “The analysis of a saturated self-excited asynchronous generator”, IEEE Transactions on Energy Conversion, Vol. 6, No. 2, June 1991, pp. 336-345. [60] E. Levi, “Impact of cross saturation on accuracy of saturated induction machine models”, IEEE Transactions on Energy Conversion, Vol. 12, No. 3, September 1997, pp. 211-216. [61] K. P. Kovacs, “On the theory of cylindrical rotor AC machines including main flux saturation”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-103, No. 4, April 1984, pp. 754- 761.

20 CHAPTER 2

WIND POWER

2.1 Source of wind Wind is a result of the movement of atmospheric air. Wind comes from the fact that the regions around the equator, at 0° latitude, are heated more by the sun than the polar region. The hot air from the tropical regions rises and moves in the upper atmosphere toward the poles, while cool surface winds from the poles replace the warmer tropical air. These winds are also affected by the earth’s rotation about its own axis and the sun. The moving colder air from the poles tends to twist toward the west because of its own inertia and the warm air from the equator tends to shift toward the east because of this inertia. The result is a large counterclockwise circulation of air streams about low- pressure regions in the northern hemisphere and clockwise circulation in the southern hemisphere [1]. The seasonal changes in strength and direction of these winds result from the inclination of the earth’s axis of rotation at an angle of 23.5o to the axis of rotation about the sun, causing variations of heat radiating to different areas of the planet.

Local winds are also created by the variation in temperature between the sea and land. During the daytime, the sun heats landmasses more quickly than the sea. The warmed air rises and creates a low pressure at ground level, which attracts the cool air from the sea. This is called a sea breeze. At night the wind blows in the opposite direction, since water cools at a lower rate than land. The land breeze at night generally has lower wind speeds, because the temperature difference between land and sea is smaller at night. Similar breezes are generated in valleys and on mountains as warmer air rises along the heated slopes. At night the cooler air descends into the valleys. Although global winds, due to temperature variation between the poles and the equator, are important in

21 CHAPTER 2 WIND POWER determining the main winds in a given area, local winds have also influence on the larger scale wind system.

Meteorologists estimate that about 1% of the incoming solar radiation in converted to wind energy. Since the solar energy received by the earth in just ten days has an energy content equal to the world’s entire fossil fuel reserves (coal, oil and gas), this means that the wind resource is extremely large. As of 1990 estimation, one percent of the daily wind energy input, i.e. 0.01% of the incoming solar energy, is equivalent to the world daily energy consumption [2]. It is encouraging to know that the global wind resource is so large and that it can be used to generate more electrical energy than what is currently being used.

2.2 Wind Turbine A wind turbine is a turbine driven by wind. Modern wind turbines are technological advances of the traditional windmills which were used for centuries in the history of mankind in applications like water pumps, crushing seeds to extract oil, grinding grains, etc. In contrast to the windmills of the past, modern wind turbines used for generating electricity have relatively fast running rotors [1].

In principle there are two different types of wind turbines: those which depend mainly on aerodynamic lift and those which use mainly aerodynamic drag. High speed wind turbines rely on lift forces to move the blades, and the linear speed of the blades is usually several times faster than the wind speed. However with wind turbines which use aerodynamic drag the linear speed can not exceed the wind speed as a result they are low speed wind turbines. In general wind turbines are divided by structure into horizontal axis and vertical axis.

2.2.1 Vertical axis wind turbine The axis of rotation for this type of turbine is vertical. It is the oldest reported wind turbine. The modern vertical axis wind turbine design was devised in 1920s by a French electrical engineer G.J.M. Darrieus [2]. It is normally built with two or three blades. A typical vertical axis wind turbine is shown in Fig. 2.1. Note that the C-shaped rotor blade is formally called a 'troposkien'.

22 CHAPTER 2 WIND POWER

Guy wires C-shaped rotor

Tower support Generator and gearbox

Fig. 2.1 Vertical axis wind turbine

The primary aerodynamic advantage of the vertical axis Darrieus machine is that the turbine can receive the wind from any direction without the need of a yaw mechanism to continuously orient the blades toward the wind direction. The other advantage is that its vertical drive shaft simplifies the installation of gearbox and electrical generator on the ground, making the structure much simpler. On the disadvantage side, it normally requires guy wires attached to the top for support. This could limit its applications, particularly for offshore sites. Wind speeds are very low close to ground level, so although it might save the need for a tower, the wind speed will be very low on the lower part of the rotor. Overall, the vertical axis machine has not been widely used because its output power can not be easily controlled in high winds simply by changing the pitch. Also Darrieus wind turbines are not self-starting, however straight-bladed vertical axis wind turbines with variable-pitch blades are able to overcome this problem [3].

2.2.2 Horizontal axis wind turbine Horizontal axis wind turbines are those machines in which the axis of rotation is parallel to the direction of the wind. At present most wind turbines are of the horizontal axis type. Depending on the position of the blades wind turbines are classified into upwind machines and down wind machines as shown in Fig. 2.2. Most of the horizontal axis wind turbines are of the upwind machine type. In this study only the upwind machine design is considered.

23 CHAPTER 2 WIND POWER

Wind Wind direction direction

Fig. 2.2 Horizontal axis wind turbine (a) upwind machine (b) downwind machine

Wind turbines for electric generation application are in general of three blades, two blades or a single blade. The single blade wind turbine consists of one blade and a counterweight. The three blades wind turbine has 5% more energy capture than the two blades and in turn the two blades has 10% more energy capture than the single blade. [2]. These figures are valid for a given set of turbine parameters and might not be universally applicable.

The three blade wind turbine has greater dynamic stability in free yaw than two blades, minimising the vibrations associated with normal operation, resulting in longer life of all components [4].

2.3 Power extracted from wind Air has a mass. As wind is the movement of air, wind has a kinetic energy. To convert this kinetic energy of the wind to electrical energy, in a wind energy conversion system, the wind turbine captures the kinetic energy of the wind and drives the rotor of an electrical generator.

The kinetic energy (KE) in wind is given by 1 KEmV 2 (2.1) 2 where m- is the mass of air, in Kg V-is the speed of air, in m/s

24 CHAPTER 2 WIND POWER

Generator Gear box Controller Wind direction Rotor brake High speed shaft Anemometer Wind vane

Pitch

Wind direction

Nacelle Rotor hub

Yaw motor

Yaw drive Wind direction

Rotor blade

Tower

Foundation

Fig. 2.3 Detail of a wind turbine driven power generation system

25 CHAPTER 2 WIND POWER

The power in wind is calculated as the flux of kinetic energy per unit area in a given time, and can be written as

dKE 1dm22 1 P VmV  (2.2) dt 2 dt 2 where m is the mass flow rate of air per second, in kg/s, and it can be expresses in terms of the density of air (U in kg/m3) and air volume flow rate per second (Q in m3/s) as given below mQAV UU (2.3) where A-is the area swept by the blades of the wind turbine, in m2.

Substituting equation (2.3) in (2.2), we get 1 P U AV 3 (2.4) 2 This is the total wind power entering the wind turbine. Remember that for this to be true V must be the wind velocity at the rotor, which is lower than the undisturbed or free stream velocity. This calculation of power developed from a wind turbine is an idealised one-dimensional analysis where the flow velocity is assumed to be uniform across the rotor blades, the air is incompressible and there is no turbulence where flow is inviscid (having zero viscosity).

The volume of air entering the wind turbine should be equal to the volume of air leaving the wind turbine because there is no storage of air in the wind turbine. As a result volume flow rate per second, Q , remains constant, which means the product AV remains constant. Hence when the wind leaves the wind turbine, its speed decreases and expands to cover more area [5]. This is illustrated in Fig. 2.4.

Fig. 2.4 shows the idealised case where the speed of wind continues to flow at a value of V2 downstream of the rotor. In reality the slow air in the wake 'diffuses' into the surrounding air through turbulence, so that further down stream the velocity of air will be equal to the undisturbed up stream wind speed because of the gain of energy from the surrounding wind.

26 CHAPTER 2 WIND POWER

Wind turbine

VT V1 V2

A1 AT A Wind speed 2

V1 VT

V2

Distance in the direction of wind

Wind pressure

+ PrT Prf Prf

_ PrT

Distance in the direction of wind

Fig. 2.4 Change of wind speed and wind pressure around the wind turbine

As shown in Fig. 2.4 Prf is the wind at atmospheric pressure. The turbine first causes the approaching wind to slow down gradually, which results in a rise in wind pressure. Applying Bernoulli’s equation the wind has highest pressure, Pr  , just before the wind f turbine and the wind has lowest pressure (lower than atmospheric pressure), Pr  , just f after the wind turbine. As the wind proceeds down stream, the pressure climbs back to atmospheric value, causing a further slowing down of the wind speed.. The pressures immediately upwind and downwind of the rotor are related to the far upwind and downwind velocities V1 and V2 by applying Bernoulli's equation separately upwind and downwind. Using momentum theory the downwind force on the rotor is equal to the pressure drop across it times the rotor blade area [2].

The force F on the rotor blades can be given by the rate of change of momentum,

27 CHAPTER 2 WIND POWER

F mV 12 V (2.5)

Using equation (2.2), the power extracted by the wind turbine PT is the difference between the upstream wind power, at A1, and the downstream wind power, at A2 given by , 1 P mV 22 V (2.6a) T 2 12 1 PT1212 mV  V V V (2.6b) 2 This power is calculated assuming that all the power lost by the wind has been extracted by the wind turbine and none has been lost through turbulence.

The force F on the rotor blades multiplied by the wind speed at the rotor blades, VT produces power given by

PTT FV (2.7) Substituting equation (2.5) in (2.7) gives

PT12T mV  V V (2.8) Equating equations (2.6b) and (2.8) gives VV V 12 (2.9) T 2

Therefore the wind speed at the rotor blades, VT is the average of the undisturbed up stream wind speed, V1, and the down stream wind speed, V2.

Using equation (2.3) the mass flow rate of air through the rotating blades of the wind turbine is

mAV U TT (2.10)

Substituting equation (2.9) in (2.10) the mass flow rate of air at the wind turbine is given by VV mA U 12 (2.11) T 2

Substituting equation (2.11) in (2.6a) gives the power absorbed by the wind turbine, which is the mechanical power at the shaft of the wind turbine, as 28 CHAPTER 2 WIND POWER

1 22§·VV12 PAVVTT12 U ¨¸ (2.12) 22©¹ This power is calculated assuming that all the power lost by the wind has been extracted by the wind turbine and none has been lost through turbulence. If all the power in the wind were extracted, the wind speed V2 would be zero and the air could not leave the wind turbine. However, if there is no wind leaving the wind turbine the power extracted is zero because air has to exit the wind turbine in order to make the rotor blades rotate.

Rearranging the equation (2.12) to express the mechanical power developed in the wind turbine in terms of the upstream wind speed at A1, shown in Fig. 2.2, gives

§·§·2 §·§VV22 · ¨¸¨¸11¨¸¨ ¸ ¨¸¨¸VV11 1 3 ©¹©¹© ¹ PAVTT1 U ¨ ¸ (2.13) 22¨¸ ¨¸ ¨¸ ©¹

From equation (2.4) the total wind power P1 at area A1 is 1 P U AV3 (2.14) 1T12 Then the ratio of wind power extracted by the wind turbine to the total wind power at area A1 is the dimensionless power coefficient Cp, where §·2 §·§VV22 · ¨¸11¨¸¨ ¸ ¨¸©¹©VV11 ¹ C ©¹ (2.15) p 2 Substituting equation (2.15) into equation (2.14) the wind power extracted by the wind turbine can be written as 1 P U AV3 C (2.16) TT1p2 or 1 P US DV23 C (2.17) TT1p2

where DT is the sweep diameter of the wind turbine.

29 CHAPTER 2 WIND POWER

In equation (2.17) it is clearly shown that the power output of a given wind turbine depends on the square of the rotor blade diameter and the cube of the wind speed. For a given turbine if the wind speed is doubled the output power will be multiplied by 8.

The maximum value of power coefficient Cp gives the maximum power absorbed by the wind turbine. To simplify the calculation of Cp, the substitution of x=V2/V1 in Equation (2.15) is made.

The maximum Cp is then obtained from dC p 0 dx which gives a solution of x = 1/3 or x = -1 and in terms of V1 and V2, V1 = 3V2 or

V1 = -V2. The solution is V1=3V2 because V1=-V2 shows equal and opposite winds coming from upstream and down stream towards the wind turbine, which is not realistic.

Power coefficient Cp 0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 0.2 0.4 0.6 0.8 1 V2/V1

Fig. 2.5 Power coefficient versus V2/V1

Substituting V1=3V2 in equation (2.11), gives the maximum value of Cp as 16 C (2.18) p 27

30 CHAPTER 2 WIND POWER

Here maximum Cp is about 0.59. This is called the Betz limit [2]. In practical designs, the maximum achievable Cp is below 0.5 for high speed, two blade wind turbines, and between 0.2 and 0.4 for slow speed turbines with more blades [6].

From equations (2.14) and (2.16) the power extracted by the wind turbine is given by

PT1p PC (2.19) Then the theoretical maximum extracted power by the wind turbine is given by 16 P P (2.20) T127 This shows that the maximum theoretical efficiency of a wind turbine is about 59%.

For the same wind speed the output power of the wind turbine varies with the shaft speed. Fig. 2.6 shows a typical set of wind turbine output power versus shaft speed characteristics for a 7.5kW machine at fixed pitch angle.

Fig. 2.6 Wind turbine output power to shaft speed characteristic curve [7]

2.4 Torque developed by a wind turbine The torque in a wind turbine is produced due to the force created as a result of pressure difference on the two sides of each blade of the wind turbine. From fluid mechanics it is known that the pressure in fast moving air is less than in stationary or slow moving air. This principle helps to produce force in an aeroplane or in a wind turbine. To explain the detail of the force created due to the wind, a cross sectional area of the blade of a 31 CHAPTER 2 WIND POWER wind turbine is given in Fig. 2.7. The air travelling from A to B follows two paths. The shape of the upper surface (path 1) results in higher velocity than the lower surface (path 2). This will create a low pressure on path 1 side of the blade. Hence force F, at 90o to the air flow, will be produced and pushes the blade upwards. This force F multiplied by the radial distance from the hub at which the force is created gives the torque. F

1 B A

2

Fig. 2.7 Air flow around cross section of a blade of a wind turbine

In reality the angle of incidence of the incoming wind can be different from the one given in Fig. 2.7 but the principle remains the same.

If the angle between the incoming wind and blade increases for some small angle, the force produced increases. However, if the angle increases above a given value then the air flow on path 1 stops sticking to the surface of the blade. Instead the air whirls around in an irregular path and creates turbulence as shown in Fig. 2.8. Then the force that was pulling upward on the low pressure side of the blade disappears. This phenomenon is known as stall [8].

1 A

B 2

Fig. 2.8 Air flow around cross section of a blade during stall condition

On a wind turbine rotor the blades are at some angle to the plane of rotation. At low shaft speeds, the angle of incidence on a blade element at some radius from the hub is large, the blades are stalled and only a small amount of driving force will be created. As a result small torque will be produced at low shaft speed. As the shaft speed increases 32 CHAPTER 2 WIND POWER the velocity of the wind hitting the blade element increases, because of the additional component of wind due to the blade's rotational speed. In addition, the angle of incidence decreases. If this angle is below the blade's stall angle, lift increases and drag decreases, resulting in higher torque. As the shaft speed increases further, the angle of incidence on the blade element decreases towards zero as the free wind speed becomes insignificant relative to the blade's own velocity. Since lift generated by a blade is proportional to the angle of incidence below stall, the torque reduces towards zero at very high shaft speeds. This variation in torque produced by the wind turbine is shown in the typical wind turbine torque versus shaft speed characteristics for a 7.5kW machine given in Fig. 2.9 [9]. In Fig. 2.9, at zero shaft speed, the wind turbine produces a small starting torque otherwise it will not self start.

Fig. 2.9 Wind turbine output torque to shaft speed characteristic curve

For a horizontal axis wind turbine, operating at fixed pitch angle, the torque developed by the wind turbine, TT, can be expressed as

PT TT (2.21) ZT where ZT - angular velocity of the wind turbine, rad/s.

The wind which hits the rotor blades of a wind turbine will not come from the direction in which the wind is blowing at the site, i.e. from the front of the turbine. This is because the rotor blades themselves are moving. As the rotor blade rotates it will see

33 CHAPTER 2 WIND POWER different wind speed along its length from its base to its tip. The wind will be coming from a much steeper angle (more than from the general wind direction at the site) as you move towards the base of the blade. Therefore, the rotor blade has to be twisted, so as to achieve an optimal angle of incidence throughout the length of the blade and follow the change in direction of the resultant wind as shown in Fig. 2.10. Otherwise, as discussed above, if the blade is hit by wind at an angle of incidence which is too steep, the rotor blade will stop producing the turning force, causing the blade to stall [2]. Wind direction viewed from blade cross sections

Vres3 3 Ve=(2/3)Vw Plane of rotation Va3 = ZTr3

Vres2 Direction of rotation at this 2 Ve=(2/3)Vw Plane of rotation instant of time r3 Va2 = ZTr2

Vres1 r2 1 Ve = (2/3)Vw Plane of rotation

Va1 = ZTr1 r1

Blade Cross sections Fig. 2.10 Detail of a twisted rotor blade

Fig. 2.10 shows the detail of the twisting of the rotor blade at different radius from the center of rotation where

Vw - undisturbed wind speed in the site

Ve - maximum fraction of the undisturbed wind that can be absorbed by the rotor

blade for maximum capture of wind power, Ve = (2/3)Vw

Va - is the wind created due to rotation of the wind turbine and increases with

radius, Va1 = ZT r1 (Va at blade radius of r1 is Va1). Va is perpendicular to Ve

and Vw

Vres - the resultant wind speed of Va and Ve (Vres at blade radius of r1 is Vres1)

r1, r2 and r3 - radiuses at points 1, 2 and 3 of the rotor blade respectively The twisting of the rotor blade when viewed from its tip is given in Fig. 2.11 [10].

34 CHAPTER 2 WIND POWER

Fig. 2.11 Cross section of a twisted rotor blade from tip to base

2.5 Tip-Speed Ratio A tip speed ratio TSR is simply the rate at which the ends of the blades of the wind turbine turn (tangential speed) in comparison to how fast the wind is blowing. The tip speed ratio TSR is expressed as: V Z r TSR tn T (2.22) VVww

Where Vtn - tangential speed of the blades at the tips

ZT - angular velocity of the wind turbine r - radius of the wind turbine

Vw - undisturbed wind speed in the site.

The tip speed ratio dictates the operating condition of a turbine as it takes into account the wind created by the rotation of the rotor blades. A typical power coefficient Cp versus tip speed ratio TSR is given in Fig. 2.12 [11]. The tip speed ratio shows tangential speed at which the rotor blade is rotating compared with the undisturbed wind speed.

As the wind speed changes, the tip speed ratio and the power coefficient will vary. The power coefficient characteristic has single maximum at a specific value of tip speed ratio. Therefore if the wind turbine is operating at constant speed then the power coefficient will be maximum only at one wind speed.

35 CHAPTER 2 WIND POWER

Power Coefficient Cp

Tip speed ratio TSR

Fig. 2.12 Typical power coefficient versus tip speed ratio [11]

Usually, wind turbines are designed to start running at wind speeds somewhere around 4 to 5 m/s. This is called the cut in wind speed. The wind turbine will be programmed to stop at high wind speeds of 25 m/s, in order to avoid damaging the turbine. The stop wind speed is called the cut out wind speed [6].

2.6 Power control in wind turbines The output power of a wind turbine is a function of the wind speed. The determination of the range of wind speed at which the wind turbine is required to operate depends on the probability of wind speed obtained from wind statistics for the site where the wind turbine is to be located. A typical histogram of the wind speed is shown in Fig.2.13 [12]. This histogram is derived from long term wind data covering several years. The histogram indicates the probability, or the fraction of time, where the wind speed is within the interval given by the width of the columns. The sum of the height of the columns is 1 or 100%, since the probability that the wind will be blowing at some wind speed including zero must be 100%. Fig. 2.13 shows the probability of wind speed being in a 1m/s interval centred on a given value. For example the probability of the wind speed being between 4.5 and 5.5 m/s is 0.104 or (0.104u8760) = 910 hours per year. When the width of the columns becomes smaller, the histogram becomes a continuous function called a probability density function, which can be fitted to a Weibull distribution or function as shown in Fig. 2.13. 36 CHAPTER 2 WIND POWER

Probability (%)

Fig. 2.13 Histogram and Weibull function for the probability of a given wind speed (data measured in 1m/s intervals)

In a wind power system typically the wind turbine starts operating (cut in speed) when the wind speed exceeds 4-5m/s, and is shut off at speeds exceeding 25 to 30m/s. In between, it can operate in the optimum constant Cp region, the speed-limited region or the power limited region as shown in Fig. 2.14 [6]. This design choice was made in order to limit the strength and therefore the weight and cost of the components of the wind turbine. Over the year some energy will be lost because of this operating decision. However, considering the typical wind speed distribution of Fig. 2.13, the number of hours per year is quite small when the wind speed exceeds 15m/s.

As discussed in Section 2.3 the power absorbed by the wind turbine is proportional to the cube of the wind speed. Hence there should be a way of limiting the peak absorbed power. Wind turbines are therefore generally designed so that they yield maximum output at wind speeds around 15 metres per second. In case of stronger winds it is necessary to waste part of the excess energy of the wind in order to avoid damaging the wind turbine. All wind turbines are therefore designed with some sort of power control to protect the machine. There are different ways of doing this safely on modern wind turbines.

37 CHAPTER 2 WIND POWER

Fig. 2.14 Wind turbine control regions

2.6.1 Pitch control In pitch controlled wind turbines the power sensor senses the output power of the turbine. When the output power goes above the maximum rating of the machine, the output power sensor sends a signal to the blade pitch mechanism which immediately pitches (turns) the rotor blades slightly out of the wind. Conversely, the blades are turned back into the wind whenever the wind speed drops again. On a pitch controlled wind turbine, in order to keep the rotor blades at the optimum angle and maximise output for all wind speeds, the pitch controller will generally pitch the blades by a small angle every time the wind changes. The pitch mechanism is usually operated using hydraulics [5].

2.6.2 Yaw control Yaw control is a mechanism of yawing or tilting the plane of rotation out of the wind direction when the wind speed exceeds the design limit. In this way the effective flow cross section of the rotor is reduced and the flow incident on each blade considerably modified. The effect of yawing on the power coefficient is given in Fig. 2.15 [5].

Fig. 2.15 shows a drastic drop in the power coefficient resulting from turning wind turbines out of the wind, with consequent blade stall.

38 CHAPTER 2 WIND POWER

Power coefficient Cp

Tip speed ratio O

Fig. 2.15 Power coefficient verses tip speed ratio under yaw control

2.6.3 Stall control Under normal operating conditions, a stalled rotor blade is unacceptable. This is because the power absorbed by the wind turbine will decrease, even to the point where no power is absorbed. However, during high wind speeds, the stall condition can be used to protect the wind turbine. The stall characteristic can be designed in to the rotor blades so that when a certain wind speed is exceeded, the power absorbed will fall to zero, hence protecting the equipment from exceeding its mechanical and electrical ratings. In stall controlled wind turbines the angle of the rotor blades is fixed. The cross sectional area of the rotor blade has been aerodynamically designed to ensure that the moment the wind speed becomes too high, it creates turbulence on the side of the rotor blade which is not facing the wind, similar to that shown in Fig. 2.8. The stall prevents the creation of a tangential force which pulls the rotor blade to rotate. The rotor blade has been designed with a slight twist along its length, from its base to the tip, which helps to ensure that the wind turbine stalls gradually, rather than abruptly, when the wind speed reaches its critical value.

The main advantage of stall control is that it avoids moving parts in the rotor blade itself, and a complex control system. However, stall control represents a very complex aerodynamic design problem, and related design challenges in the structural dynamics of the whole wind turbine, e.g. to avoid stall-induced vibrations. Around two thirds of the wind turbines currently being installed in the world are stall controlled machines [13].

39 CHAPTER 2 WIND POWER

2.7 Wind powered electric generation In general electrical generators for wind powered application include induction machines and synchronous machines. The generation of electricity using wind energy systems has found application for grid connected and stand alone systems. The capacity of wind powered electric generation has been growing. The category of electricity produced from wind power with respect to size is given in Table 2.1.

Type Rotor Size Electricity Produced Micro 0.5 – 1.25 m 20 – 300 W Mini 1.25 – 2.75 m 300 – 850W Household 2.75 – 7 m 0.85 – 10KW Industrial 7 – 30 m 10 – 100KW Utility 30 – 90 m 0.1 – 4 MW

Table 2.1 Rough Categories of Wind Generator Sizes [14]

Wind projects are relatively easy to site and expand, have low environmental impacts (including no carbon emissions) and are highly desirable to buyers of “green” power

[15].

Wind power has emerged as the world's fastest growing electricity generating technology, growing by more than 40 percent annually since 1993. Fig. 2.16 shows the worldwide growth of wind energy capacity [17]. Total world wide wind power capacity is now estimated at more than 24,000MW [16].

Harnessing the power of the wind has a rich tradition that is enjoying resurgence due to recent political, economic and technological developments. The strategic plan in USA is to provide 5 percent of the nation's electricity from wind power by the year 2020. This plan equates to 80,000 MW of new USA wind power capacity (compared to today's less than 3,000 MW) [18, 19].

40 CHAPTER 2 WIND POWER

Fig. 2.16 Growth of wind energy capacity worldwide

One major trend driving the development of wind power is concern over global climate change. In Europe, countries have established specific target dates for reducing carbon dioxide (CO2), a pollutant which is linked to rising global temperatures. Wind power, which is projected to reach 40,000 MW on the European continent by 2010, has become the technology of choice to reduce CO2 emissions in Europe and much of the rest of the world [18].

2.8 Economics of wind powered electric generation The increasingly attractive economics of wind energy is perhaps the most important factor in the expansion of wind powered electric generation. The costs of wind power have declined almost 90% since the early 1980s [19]. Between 1990 and 2000, the average cost to produce electricity from wind turbines has decreased from around ten cents to less than five cents per kWh in regions with very favourable wind resources. Some projects are selling power under long-term contracts at 3.5 cents per kWh [20].

The cost depends upon the particular wind turbine, the nature of the local wind resource, and economies of scale associated with the size of the wind farm. Prices are projected to drop another 20 to 40 percent over the next ten years. The goal of the USA Department of Energy (DOE) program is to get the cost down to 2.5 cents/kWh (at 6.7 m/s wind sites). However, wind turbines, particularly off-grid models, are already cost effective throughout much of the developing world where the cost of dirty diesel-fired 41 CHAPTER 2 WIND POWER power is often as high as 30 cents/kWh. The dramatic cost reductions are a result of increased size and an increase in efficiency when converting kinetic wind energy into electricity. The graphical decrease in the cost of electricity generated from wind energy is given in Fig. 2.17 [21].

Fig.2.17 Trend in the cost of electricity generated from wind energy

2.9 Summary The general definition of wind and the source of wind have been presented in this chapter. The analysis of power absorbed by a wind turbine is based on the horizontal axis wind turbine. The mechanism of production of force from wind that causes the rotor blades to rotate in a plane perpendicular to the general wind direction at the site has been discussed in detail. The importance of having twisted rotor blades along the length from the base to the tip is given. The variation of the torque produced by the wind turbine with respect to the rotor angular speed has been presented.

Power absorbed by a wind turbine is proportional to the cube of the wind speed. Wind turbines are designed to yield maximum output power at a given wind speed. In case of stronger winds it is necessary to waste part of the excess energy of the wind in order to avoid damaging the wind turbine. Different ways of power control to protect the

42 CHAPTER 2 WIND POWER machine have been presented. The economics and growth of wind powered electric generation has been discussed and the projection for the future has been presented.

2.10 References [1] N. P. Cheremisnoff, “Fundamentals of Wind Energy”, Ann Arbor Science Publishers, Michigan, 1978. [2] L.L, Freris, “Wind Conversion Systems”, Prentice Hall International, London, 1990. [3] Nick Pawsey, “Private conversation with Nick Pawsey”, School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, Australia, 2002. [4] WASTWIND Turbines, Australia (http://www.westwind.com.au/tubines.htm). [5] S. Heier, “Grid Integration of Wind Energy Conversion Systems”, John Wiley & Sons Ltd, Chichester, 1998, pp. 23. [6] M. R. Patel, “Wind and Solar Power Systems”, CRC Press LLC, Boca Raton, 1999, pp.37. [7] Z. Zhang, C. Watthanasarn and W. Shepherd, “Application of a Matrix Converter for the Power Control of a Variable-Speed Wind-Turbine Driving a Doubly-Fed Induction Generator”, in Proc. 1997 IEEE IECON’97 Conference, pp. 906-911. [8] Danish wind Industry Association (http://www.windpower.dk/tour/wtrb/rotor.htm) [9] Z. Zhang and C. Watthanasarn, “A Matrix Converter Excited Doubly-Fed Induction Machine as a Wind Power Generator”, IEE- Power Electronics and Variable Speed Drives conference, 21-23 September 1998, pp. 532-537. [10] http://www.tuat.ac.jp/~akilab/renewables/wind.html. [11] R. Spee, S. Bhowmik and J. HR Enslin, “Adaptive Control Strategies for Variable-Speed Doubly- Fed Wind Power Generation System”, IEEE IAS Annual Meeting Conf., 1994, pp. 545-552. [12] J. F. Walker and N. Jenkins, “Wind Energy Technology”, John Wiley & Sons, New York, Inc, 1997. [13] http://www.windpower.org/tour/wtrb/powerreg.htm. [14] http://www.aocwind.net/wbasics.htm. [15] M. Begovic, A. Pregelj, A. Rohatgi and C. Honsberg, “Green power: status and perspectives”, Proceedings of the IEEE, Vol. 89 Issue: 12, Dec. 2001, pp. 1734 -1743. [16] IEEE “The Institute” Vol.26, No.5, May 2002, pp. 1-8. [17] National Renewable Energy Laboratory, USA Department of Energy “Wind energy Information report”, May 2001. [18] http://www.ceert.org/ip/wind.html. [19] R.; Swisher, C.R.; De Azua and J. Clendenin, “Strong winds on the horizon: wind power comes of age”, Proceedings of the IEEE, Vol. 89 Issue: 12, Dec. 2001, pp. 1757 -1764. [20] http://www.eren.doe.gov/consumerinfo/refbriefs/ad2.html. [21] National Renewable Energy Laboratory, USA Department of Energy “Wind energy Information report”, 1997.

43 CHAPTER 3

THREE-AXES TO TWO-AXES TRANSFORMATION AND ITS APPLICATION

3.1 Introduction Mathematical transformations are tools which make complex systems simple to analyse and solutions easy to find. For instance, the differential equation is transformed into its corresponding Laplace transform representation. The Laplace equation is then simply analyzed using algebra. Once the solution is obtained in algebraic form the inverse Laplace transform is applied to yield the time domain solution.

In electrical machines analysis a three-axes to two-axes transformation is applied to produce simpler expressions that provide more insight into the interaction of the different parameters.

The different transformations studied in the past are [1]: -In the late 1920s, R.H.Park formulated a change of variables, which, in effect, replaced the variables (Voltages, currents, and flux linkages) associated with the stator windings of a synchronous machine with variables associated with fictitious windings rotating with the rotor. He transformed, or referred, the stator variables to a frame of reference fixed in the rotor. -In the late 1930s, H.C Stanley employed a change of variables in the analysis of induction machines. He transformed the rotor variables to a frame of reference fixed in the stator. -G. Kron introduced a change of variables which eliminated the time-varying inductances of a symmetrical induction machine by transforming both the stator

44 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION variables and the rotor variables to a reference frame rotating in synchronism with the rotating magnetic field. This reference frame is commonly referred to as the synchronously rotating reference frame. -D. S. Brereton et al, employed a change of variables which also eliminated the time- varying inductances of a symmetrical induction machine by transforming the stator variables to a reference frame fixed in the rotor. This is essentially Park’s transformation applied to induction machines.

Park, Stanley, Kron and Brereton et al. developed changes of variables each of which appeared to be uniquely suited for a particular application. Consequently, each transformation was derived and treated separately in literature until it was noted in 1965[2] that all known transformations used in induction machine analysis are contained in one general transformation which eliminates all time-varying inductances by referring the stator and rotor variables to a frame of reference which may rotate at any angular velocity or remain stationary. All known real reference transformations may then be obtained by simply assigning the appropriate speed of rotation to this so- called arbitrary reference frame.

3.2 General change of variables in transformation Mathematical transformations are often used to decouple variables. Transformations help to facilitate the solution of difficult equations with time–varying coefficients, or to refer all variables to a common reference frame [3].

The three axes are representing the real three phase supply system. However, the two axes are fictitious axes representing two fictitious phases perpendicular, displaced by 90o, to each other. The transformation of three-axes to two-axes can be done in such a way that the two-axes are in a stationary reference frame, or in rotating reference frame. The transformation actually achieves a change of variable, creating the new reference frame. Transformation into a rotating reference frame is more general and can include the transformation to a stationary reference frame. Transformation to a stationary reference frame is one particular condition of transformation to a rotating reference

45 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION frame. If the speed of rotation of the reference frame is zero it becomes a stationary reference frame.

If the reference frame is rotating at the same angular speed as the excitation frequency, when the variables are transformed into this rotating reference frame, they will appear as a constant value instead of time-varying values.

3.2.1 Transformation into a stationary reference frame Here the assumption taken is that the three-axes and the two-axes are in a stationary reference frame. It can be rephrased as a transformation between abc and stationary dq0 axes.

To visualize the transformation from three-axes to two-axes, the trigonometric relationship between three-axes and two-axes is given below.

s fb Z

s fq

T

 s fa

s s fc fd

Fig. 3.1 Three-axes and two-axes in the stationary reference frame

In the above diagram, Fig. 3.1, f can represent voltage, current, flux linkage, or electric charge. The subscript s indicates the variables, parameters, and transformation

46 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

associated with stationary circuits. The angular displacement T shows the displacement of the two-axes, dq-axes, from the three-axes, abc-axes.

s s s s s fq and fd variables are directed along paths orthogonal to each other. fa , fb , and fc may be considered as variables directed along stationary paths each displaced by 120 electrical degrees.

A change of variables which formulates a transformation of the three-phase variables of stationary circuit elements to an arbitrary reference frame can be expressed as [1]:

ªº§·§·22SS «»cosT cos¨¸¨¸TT- cos  s ©¹©¹33s ªºfq «»ªºfa «» s 2 «»§·§·22SS«»s «»f sinT sinTT sin fb (3.1) d 3 «»¨¸¨¸«» «» ©¹©¹33s f s «»«»f ¬¼o 11 1¬¼c «» ¬¼«»22 2

This transformation is a special case of the classical symmetrical components s transformation [3]. In Equation (3.1) fo is a variable that takes care of the unbalance in the variables of the three-axes system and is the same as the zero-sequence component in three phase system. It is important to note that the zero-sequence variables are not associated with the arbitrary reference frame. Instead, the zero-sequence variables are related arithmetically to the abc variables, independent of T.

s s s It is essential not to confuse fa , fb , and fc with phasors, because phasors are used in s s s steady state expressions. Instead fa , fb , and fc are instantaneous quantities, which may be any function of time. Portraying the transformation as shown in Fig. 3.1 is s s particularly convenient when applying it to ac machines where the direction of fa , fb , s and fc may also be thought of as the direction of the magnetic axes of the stator windings. They can also represent space vectors or the axes of distribution of the phase s s windings. The direction of fq and fd can be considered as the direction of the magnetic axes of the new windings created by the change of variables.

47 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

The inverse of Equation (3.1), which can be derived directly from the relationship given in Fig. 3.1, is

ªºcosTTsin 1 ªºf s «»ªºf s a §·§·22SSq «»«»cosTT sin 1 «» s ¨¸¨¸ s (3.2) «»fb «»©¹©¹33«»fd s «»«» «»f §·§·22SSf s ¬¼c «»cos¨¸¨¸TT sin 1 ¬¼o ¬¼«»©¹©¹33

The change of variables may be applied to variables of any waveform and time sequence. However, the transformation given above is particularly appropriate for an abc sequence. That is, as the magnetic field rotates at the speed Z in the direction shown in Fig. 3.1, it will induce an abc phase sequence in the abc axes, i.e. voltage will be induced first in axis-a then b then in c. This shows voltage in phase a leads the voltage in phase b and so on. And for the dq-axes, voltage will be induced first in the d- axis then in the q-axis. That is the voltage in the d-axis will lead the voltage in the q- axis.

In Fig. 3.1, if the q-axis is aligned with the a-axis, i.e. T = 0, Equation (3.1) will be written as:

ªº11 1  s «»22s ªºfq «»ª fa º «» s 2 «»33« s » «»f 0  fb (3.3) d 3 «»« » «»s 22s f «»« f » ¬¼o 11 1¬ c ¼ «» ¬¼«»22 2 and Equation (3.2) will be simplified to: ªº10 1 ªºf s «»ªºf s a 13 q «»«»1 «» s  s (3.4) «»fb «»22 «»fd s «»«» «»f 13 f s ¬¼c «» 1 ¬¼o ¬¼«»22

48 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

In Equation (3.3) and (3.4) the magnitude of the phase quantities, voltages and currents, in the three (abc) axes and two (dq) axes remain the same. This transformation is based on the assumption that the number of turns of the windings in each phase of the three axes and the two axes are the same. Here the advantage is the peak values of phase voltages and phase currents before and after transformation remain the same. The detail of transformation for voltages and currents is given in Sections 3.3 and 3.4 respectively. However, for the power, to have the same magnitude in the three axes and two axes, there should be a multiplying factor 3/2 in the two axes power calculation as given in Section 3.5.

Another way of transforming the three axes to two axes, with the magnitude of per phase voltages and per phase currents remaining unchanged in both axes systems, is to align the d-axis along the a-axis, T = 0 in Fig. 3.2. Fig. 3.2 is similar to Fig. 3.1 except that the d-q axes are rotated by 90o while the abc axes remain the same. s fq

s fb Z

s fd T s fa

s fc

Fig. 3.2 Three-axes and two-axes in the stationary reference frame with d-axis and a-axis aligned

49 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

From Fig. 3.2, for T = 0 and taking the zero sequence current into account, the transformation equation can be written as [4, 5]

ªº11 1  s «»22s ªºfd «»ª fa º «»s 2 «»33« s » «»fq 0  fb (3.3a) 3 «»« » «»s 22s f «»« f » ¬¼o 11 1¬ c ¼ «» ¬¼«»22 2 and the inverse transform will be ªº10 1 ªºf s «»ªºf s a «»13 d «»s  1 «»s «»fb «»22 «»fq (3.4b) s «»«» «»f 13 f s ¬¼c «» 1 ¬¼o ¬¼«»22

Whether the q-axis or d-axis is aligned along the a-axis is a matter of choice. The difference is only a rotational translation of the d-q axes by 90o. In this thesis the option of aligning the q-axis along the a-axis has been used.

There is another way of transformation from three-axes into two-axes where the magnitude of the power remains the same but the magnitude of phase voltages and currents in the two axes are higher than that of the three-axes [6, 7, 8]. In these references the d-q axes are given as D-E axes. Replacing the D-E with d-q and making some rearrangement, it can be written as

ªº11 1  s «»22s ªºfq «»ªºfa «» 2 33«» s «»s (3.5) «»fd 0  «»fb 3 «»22 «»s «»«»f s ¬¼fo 11 1¬¼c «» ¬¼«»22 2

Here the q-axis is aligned along the a-axis but their magnitude is different. The inverse transformation will be

50 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

ªº1 «»10 s 2 s ªºfa «»ª fq º «» 2 «»131« » s s (3.6) «»fb «» « fd » 3 222 s «»« » «»f f s ¬¼c «»131¬ o ¼ «» ¬¼«»222

In this type of transformation the transformation matrix from three-axes to two-axes is the transpose of the inverse transformation matrix and it can be shown that s 2 s 2 s 2 s 2 s 2 s 2 (fa ) +(fb ) +(fc ) = (fq ) +(fd ) +(fo ) . The phase voltages and currents in the two-axes are 32times the quantities in the three-axes. As a result when the power is calculated in the two-axes the 1.5 multiplying factor is not required.

In this thesis the transformation that gives constant magnitude of voltages and currents in the two-axes and three-axes is used. In addition the q-axis of the two axes is aligned along the a-axis of the three axes.

3.2.2 Transformation into a rotating reference frame The rotating reference frame can have any speed of rotation depending on the choice of the user. Selecting the excitation frequency as the speed of the rotating reference frame gives the advantage that the transformed variables, which had instantaneous values, appear as constant (DC) values. In other words, an observer moving along at that same speed will see the space vector as a constant spatial distribution, unlike the time-varying values in the stationary abc axes.

In the previous section the transformation from abc axes to a stationary qd axes is given. Here the stationary qd axes will be transformed into a rotating qd reference frame, which is rotating at Ze, excitation frequency.

To see through the eyes of the so-called moving observer is equivalent, mathematically to resolving whatever variables that we want to see onto a rotating reference frame. This reference frame moves at the same speed as the observer. Since we are dealing with two-dimensional variables, the rotating reference can be any two independent basic

51 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION space vectors, which for convenience another pair of orthogonal qd axes will be used [3]. The zero-sequence component remains the same as before. Fig. 3.3 shows the abc to rotating dq transformation in two steps, ie, first transforming to stationary dq axes and then to rotating dq axes.

s fb Z e fq

s T fa s s at T= 0 fq fq

s e fc fd T=Z t s s fd fd (a) (b)

Fig. 3.3 Steps of the abc to rotating dq axes transformation (a) abc to stationary dq axes b) stationary ds-qs to rotating de-qe axes

The equation for the abc to stationary dq-axes transformation is given in Equation (3.7). Using geometry, it can be shown that the relation between the stationary ds-qs axes and rotating de-qe axes is expressed as:

es ªºfqqªºcosTT sin ªºf «» «» (3.7) es«»sinTT cos ¬¼«»fdd¬¼¬¼«»f

The angle,T, is the angle between the q-axis of the rotating and stationary ds-qs axes. T is a function of the angular speed, Z(t), of the rotating de-qe axes and the initial value, that is

t TZT()ttdt () (0) (3.8) ³0

If the angular speed of the rotating reference frame is the same as the excitation frequency then the transformed variables in the rotating reference frame will appear constant (DC).

52 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

3.3 Voltage measurement For a balanced, pure sinusoidal three phase supply the sum of the three phase voltages is zero, as a result the zero-sequence voltage will be zero. The voltage transformation for this type of source can be shown to be

ªº11v s «»1 ªºa ªºvq 2 22«» «» «»vb (3.9) vs 3 «»33«» ¬¼«»d 0  «»v ¬¼«»22¬¼c

Assume that the three phase supply voltages are given by

vVam cosZ e t (3.10)

vVbm cos(ZS e t 2 3) (3.11)

vVcm cos(ZS e t 2 3) (3.12)

Applying Equation (3.9) to the three phase voltages given above

s vv v 2 v bc (3.13) q 3 a 22

vs 2 3 []vv (3.14) d 3 2 bc

For a balanced system since vvvabc   then (3.13) becomes

s vvVqam cosZ e t (3.15) Simplifying Equation (3.14)

s 1 vvvdcb  3 1 Vtmecos(ZS 2 3) Vt me cos( ZS  2 3) 3 Applying Euler’s identity,

jj wtee2SS /3  wt2 /3 jj wt ee 2 SS /3 wt 2 /3 Vm §·()ee() ee ¨¸ (3.16) 3 ©¹22 Simplifying Equation (3.16)

s vVdme  sinZ t (3.17)

s s For a balanced three phase system the dq voltages,vd and vq , are orthogonal and they have the same peak values as the abc phase voltages. The explanation given in Section

53 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

3.2.1 indicates that the voltage in the d-axis leads the voltage in the q-axis. This is confirmed by the results given in Equations (3.15) and (3.17).

Based on the discussion given above the dq component voltages can be written as

ss VvjvVdq q d m cosZZ e tj  sin e t

Zet VVedq m (3.18)

So Vdq , the resultant voltage vector in the stationary reference frame, rotates counter clockwise at a speed of Ze , as shown in Fig. 3.4, from an initial position at t=0, which

s s is co-phase with the a-phase axis. At t=0 vq = Vdq and vd = 0.

s vd Vdq

Zet s vq q- axis

d- axis

Fig. 3.4 Voltage vector and its component in dq axes

s s For a direct on line measurement vd and vq will be calculated from the measured instantaneous values of va , vb and vc . For a balanced three-phase supply the magnitude of the peak phase voltage can be calculated as:

ss22 2 2 2 VvvVdq d q m[(sinZZ e t )  (cos e t ) ]

VVdq m (3.19) Then the rms voltage will be V V dq (3.20) rms 2 From Equation (3.20) it is shown that the rms voltage can be calculated from the measured instantaneous values of va and vb . vc is not included because for a balanced only system vvvcab   . The rms value is readily available by taking one set of measurements, i.e. va and vb , at one instant of time [9].

54 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

If the measured voltages are line-to-line values, like in a delta connected induction machine, then two line-to-line voltages will be measured. The third line-to-line voltage vca can be calculated from the other two measured values, vab and vbc , as it is assumed the supply is a balanced three-phase system.

If the instantaneous measured line to line voltages are vab and vbc , then for a balanced system

vvvca  ab  bc (3.21a) and based on the fact that

va + vb + vc =0 (3.21b)

vvvab b a (3.21c)

vvvbc b c (3.21a)

vvvca c a (3.21a) the phase quantities can be calculated from line quantities using the above equations to give

v 1 vv (3.22) a 3 ab ca

v 1 vv (3.23) b 3 bc ab

v 1 vv (3.24) c 3 ca bc

3.4 Current measurement When the general transformation given in Equation (3.3) is applied to three phase currents the equation becomes

ªº11 «»1  ªºis 22 q «»ªia º «» s 2 «»33« » «»i 0  ib (3.25) d 3 «»22« » «»s i «»«ic » ¬¼o 11 1¬ ¼ «» ¬¼«»22 2

55 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

If the three-phase currents are balanced sinusoids which are 120 electrical degrees apart, the sum of the three currents flowing in the phases of the system is zero. Therefore the zero-sequence current in the system, provided there is a path for the zero-sequence current, will be zero and Equation (3.25) is simplified to

ªº11i s «»1 ª a º ªºiq 2 22« » «» «»ib (3.26) is 3 «»33« » ¬¼«»d 0  «i » ¬¼«»22¬ c ¼

Then,

s ii i 2 i bc q 3 a 22

is 2 3 ii d 3 2 >@bc

Since ia+ib+ic=0, the above equations can be reduced to

s iiqa (3.27)

s 1 iiidcb  (3.28) 3

Corresponding to the three phase voltages given in equations (3.10) to (3.12), the currents flowing in the system may be described as

iIam cos(ZI e t ) (3.29)

iIbm cos(ZSI e t 2 3 ) (3.30)

iIcm cos(ZSI e t 2 3 ) (3.31) Where I is the phase shift between the voltage and the current for a lagging power factor. Substituting the instantaneous values of the currents in the abc axes into the dq axes

s iIqm cos(ZI e t ) (3.32)

iIts 1 cos(ZSI 2 3 ) It cos( ZSI 2 3 ) dme3>@ me

j wtee2/3SI j wt 2/3SI j wt ee 2/3SI j wt 2/3SI s Im §·()ee() ee id ¨¸ (3.33) 3 ©¹22 Simplifying the above equation results in,

56 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

s iIdm sin(ZI e t  ) (3.34)

s s s To compare the phase of id with the phase of iq , id can be expressed as

s iIdm cos(ZS e t 2 I ) . This shows that the current in the d-axis leads that in the q-

s s axis by S 2 . The peak value of iq and id is the same as the peak value of ia , ib and ic .

The resultant current in the stationary dq axes, Idq , can be written in terms of the components in the d-axis and the q-axis as,

ss IijiIdq q d m cos(ZI e t  )  j sin( ZI e t  )

jt()ZIe  IIedq m (3.35)

Equation (3.35) highlights the fact that Idq is a vector with a magnitude of Im and rotates at the excitation angular frequency Ze . The components of Idq along the d-axis and q- axis vary with time. It can also be stated that the magnitudes of the components along the d-axis and the q-axis are instantaneous values, similar to the three-phase currents in the abc axes. The magnitude of Idq can be calculated as

s22s Iiidq d q

22 2 IIdq m[(sin(ZI e t )) (cos( ZIe t ))

IIdq m (3.36)

The currents in the stationary dq-axes can be shown as in Fig. 3.5. s id Idq Zet-I s iq q-axis

d-axis Fig. 3.5 Current vector and its component in stationary dq axes

57 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

Since the magnitude of Idq is equal to Im and is the same as the peak magnitude of phase current in the abc-axes, the rms current can be evaluated from the instantaneous values in the dq-axes. Therefore using Equation (3.36) I I dq (3.37) rms 2

If the system is a balanced three-phase system, then only two phase currents (ia and ib) are required to be measured, the third one (ic) can be derived from the assumption that the three-phase currents are 120 electrical degrees apart and instantaneously add to zero. Taking one sample of instantaneous values of currents flowing in any two phases of a three-phase system, the rms and the peak currents of the three-phase system can be obtained instantaneously.

3.5 Power measurement The transformation from three axes to two axes is done based on the concept that the peak values of the phases in three axes as well as two axes are the same. The total power in the system under consideration should remain the same regardless of the choice of reference frame and choice of axes.

Since the voltages and currents in the three axes have the same peak values to the currents and voltages in the two axes, the power in the two-axes system should be multiplied by a factor 3/2 so that the transformation will keep the value of total power the same.

Fig. 3.6 shows the voltage and current vectors with their components in the stationary dq-axes. Once the components of the currents and voltages are calculated in the d and q axes then power is evaluated as

3 P ()ivs sss iv (3.38a) 2 dd qq

58 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

If the currents and voltages are substituted in Equation (3.38a) with the expressions given in the voltage and current measurement sections then the classical power expression becomes 3 PIV cosI (3.38b) 2 mm

s Vdq vd

Zet I i s dq d I

Zet-I s s q-axis iq vq

d-axis

Fig. 3.6 Voltage and current vectors with their components in the stationary dq-axes

The two axes can be visualized as a machine having two windings. The power in the two axes system is related to that of the three axes system by the factor 3/2. Therefore, with the currents and voltages in the dq-axes, calculated from the instantaneous values of two-phase currents and two-phase voltages, the value of the active power (average power) can be computed instantaneously.

This is quite a remarkable result. Traditionally to calculate the average power, measured values are averaged over one cycle. With the method described above, the average power can be calculated from a single set of instantaneous voltages and currents measured at a single instant in time.

59 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

3.6 Power factor measurement Any circuit with effective resistive and reactance will have a phase shift, I, between the voltage vector and the current vector. To transform the voltages and currents into a rotating dq axes reference frame Equation (3.7) is applied in the following way

es ªºvvqqªºcosTT sin ªº «» «» (3.39) es«»sinTT cos ¬¼«»vvdd¬¼¬¼«»

es ªºiiqqªºcosTT sin ªº «» «» (3.40) es«»sinTT cos «»¬¼iidd¬¼¬¼«»

If it is assumed that the reference frame is rotating at the excitation frequency, Ze, then

T =Zet and for voltages

es s vvqq cosTT v d sin (3.41)

es s vvdq sinTT v d cos (3.42)

s s With T =Zet and substituting the expressions vVqdqe cosZ tand vVddqe  sinZ t into Equations (3.41) and (3.42), then simplifying gives

e vVqdq

e vd 0

Hence with the given arrangement and analysis of the rotating reference frame, the q-

e axis voltage in the rotating reference frame, vq , and the rotating voltage space vector,

Vdq , are in the same axis and they are equal. That is why the d-axis component of the voltage vector is always zero in the rotating reference frame rotating at the excitation frequency Ze. The AC voltage in the stationary reference frame appears as if it is DC or a constant value in the rotating reference frame.

For the currents, if it is assumed that the reference frame is rotating at the excitation frequency, Ze, then Equation (3.40) will become

es s iiqq cosZZ e ti d sin e t (3.43)

es s iidq sinZZ e ti d cos e t (3.44)

s s Substituting iIqdqe cos(ZI t ) and iIddqe sin(ZI t  ) in Equation (3.43)

60 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

e iIqdqe cos(ZI t )cos Z e t [ I dqe sin( ZI t )]sin Z e t

e iIqdqe [cosZIZIZ t cos sin e t sin ]cos e tI  dqe [sin ZIIZZ t cos sin cos et ]sin e t

e 22 iIqdqe [cosZIZIZ t cos sin e t sin cos e t  sin ZIIZZ e t cos sin cos et sin e t ]

e iIqdq cosI (3.45) Applying the same analysis to Equation (3.44) gives

e iIddq sinI (3.46) From Equation (3.45) it can be seen that the component of the current in the q-axis contains the term cosI which is the power factor of the system

s s The expressions vVqdqe cosZ tand vVddqe  sinZ t lead to

s vq cosZet (3.47) Vdq

s vd sinZet (3.48) Vdq

e Substituting Equations (3.47) and (3.48) in Equation (3.43) iq can be rewritten as,

s s esvq svd iiqq  i d (3.49) VVdq dq and from Equation (3.45) ie cosI q (3.50) Idq

Applying the above equation, the power factor of the system can be calculated instantaneously from data measured at any single instant of time.

3.7 Frequency measurement For a balanced three-phase system it is enough to have measured the instantaneous values of voltages and currents from two phases to determine all the required values including frequency. If the measured instantaneous voltages and currents are given by

vVam cosZ e t

vVbm cos(ZS e t 2 3)

iIam cos(ZI e t )

iIbm cos(ZSI e t 2 3 )

61 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

The equations given above have four known values va , vb , ia and ib , which are measured at a single instant of time, and four unknowns Vm ,Zet , Im andI . Having four equations and four unknowns there is a solution for all the unknowns. The three axes to two axes transformation helps to simplify the mathematics in attaining this objective.

The information for the excitation frequency Ze is concealed in the Zet term.

Differentiating Zet with respect to time give Ze from which the frequency can be obtained as fe=Ze/2S.

Differentiation can be implemented by having two measurements at a small interval of time and experimental differentiation is implemented as the difference between two consecutive measurements divided by the sampling time.

Applying Equation (3.47) at two instants of time with a small interval of time between the two measurements vs §·vs cosZ t q1 ===>Z t cos1 q1 (3.51) e 1 V e 1 ¨¸V dq ©¹dq

vs §·vs cosZ t q2 ===>Z t cos1 q2 (3.52) e 2 V e 2 ¨¸V dq ©¹dq Then

()ZZeett21 Ze (3.53) ()tt21

Here t2-t1 can be replaced by a sampling time Ts of the measurement and Ts should be less than one fourth of the period of the frequency to be measured. For power frequency it is more than adequate to have a sampling time less than one millisecond.

With a sampling time of Ts,

§·vvss §· cos11qq21 cos ¨¸VV ¨¸ ©¹dq ©¹ dq Ze (3.54) Ts

62 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

3.8 Measurement in a balanced non sinusoidal three phase system It has been explained above that for a sinusoidal (free of harmonics) balanced three phase system the electrical quantities can be measured at a single instant of time from only one set of measurements. However, most systems contain harmonics.

Fourier analysis states that any periodic waveform consists of a summation of sinusoidal waveforms that are integral multiples of the fundamental. These integral multiples of the fundamental are harmonics. A signal with harmonics will have a distorted sinusoidal waveform [10]. To identify and isolate each harmonic the original signal can be filtered using a digital filter. The digital filter technique will be discussed in Chapter 5. The harmonic filters are easily instituted in software. These digital filters are applied to the data produced by a Digital Signal Processor (DSP). This can be done on-line in real-time using filters that are generally supplied with most DSP cards or off- line using filters generated in some mathematical analysis software, such as the MATLAB software package.

Using different digital filters all harmonics with magnitude greater than a given value can be separated. Then each three phase harmonic will be a pure sinusoidal waveform. To transform the three axes to two axes Equations (3.9) and (3.26) are applied for voltages and currents respectively. In the two axes the rms quantities of each harmonic can be calculated individually.

The rms quantity for each harmonic of hth order is expressed as

fh_dq fh_rms (3.55) 2 th where fh_rms -rms value of the h harmonic

fh_dq -the peak value

f - represents either voltage or current

The total rms value of a distorted sinusoidal waveform is calculated from each harmonic and is given by [11]

222 2 frms f 1_ rms f 2 _ rms f 3 _ rms ... f h _ rms (3.55)

63 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

And the total power measured from the distorted waveform is[12]

P P123 P P ... P h (3.56) th where Ph - power due to h harmonic.

The space vector of the fundamental signal of the distorted waveform rotates at angular th speed Ze. Consequently the angular frequency of the space vector of the h harmonic will be hZe.

3.9 Summary The three-axes to two-axes transformation presented in this chapter is applicable for any balanced three-phase system. It has been discussed that the three-axes to two-axes transformation simplifies the calculation of rms current, rms voltage, active power and power factor in a three-phase system. Only one set of measurements taken at a single instant of time is required when using the method described to obtain rms current, rms voltage, active power and power factor. And from measurements taken at two consecutive instants in time the frequency of the three-phase AC power supply can be evaluated. Existing electrical measuring methods, such as the Fast Fourier Transform, require many samples from a significant period of the measured waveform’s cycle to be processed by elaborate computation techniques in order to evaluate rms or peak magnitudes of AC currents and voltages. These traditional methods are unable to obtain peak values in less than one quarter of a cycle. However using the three-axes to two- axes transformation it is possible to evaluate the rms or peak magnitudes of three-phase AC currents and voltages from one set of measurements taken at a single instant of time.

3.10 References [1] P.C. Krause, O. Wasynczuk and S. D Sudhoff, “Analysis of Electric Machinery”, IEEE press, New York, 1995. [2] P.C. Krause and C. H. Thomas, “Simulation of Symmetrical Induction Machinery,” IEEE Trans. Power Apparatus and Systems, Vol. 84, November 1965, pp.1038-1053. [3] Chee-Mun Ong, “Dynamic Simulation of Electric Machinery –using Matlab/Simulink”, Prentice- Hall Inc., Upper Saddle River, New Jersey, 1998.

64 CHAPTER 3 THREE-AXIS TO TWO-AXIS TRANSFORMATION AND ITS APPLICATION

[4] Peter Vas, “Sensorless Vector and Direct Torque Control”, Oxford University Press, New York, 1998. [5] Andrzej M. Trzynadlowski, “The field Orientation Principle in Control of Induction Motors”, Kluwer Academic Publisher, Boston, 1994. [6] H. Akagi, Y. Kanazawa and A Nabae, “Generalized Theory of the Instantaneous Reactive Power in Three-Phase Circuits, IPEC’83-Int. Power Electronics Conf., Tokyo, Japan, 1983, pp. 1375-1386. [7] H. Akagi, Y. Kanazawa and A Nabae, “Instantaneous Reactive Power Compensator Comprising Switching Devices without Components”, IEEE Trans. Industry Application, vol.20, May/June 1984, pp. 625-630. [8] J. Afonso, C. Couto and J. Martins, “Active Filters with Control Based on the p-q Theory, IEEE Industrial Electronics Society Newsletter, September 2000, pp. 5-11. [9] C. Grantham, D. Seyoum and H. Tabatabaei-Yazdi, “Very fast and accurate Electrical Measurements”, AUPEC/EECON’99 Conference Proceedings, 26-29 September 1999, Darwin, Australia, pp 99-103. [10] D. McKinnon, D. Seyoum and C. Grantham, "Rapid Determination of Fundamental and Harmonic RMS Quantities in a 3-Phase System”, Proc. AUPEC 2001, Perth, 23 - 26 September. 2001, pp. 73 - 78. [11] Muhammad H. Rashid, “Power Electronics Circuits, Devices and Applications”, Prentice-Hall Inc, New Jersey, 1993. [12] L. Peretto, R. Sasdelli and G. Serra, “Measurement of harmonic losses in supplying nonsinusoidal load currents”, IEEE Transactions on Instrumentation and Measurement, Vol. 49, No. 2, April 2000, pp. 315 – 319.

65 CHAPTER 4

INDUCTION MACHINE MODELLING

4.1 Introduction The main aspect which distinguishes the induction machine from other types of electric machines is that the secondary currents are created solely by induction, as in a , instead of being supplied by a DC exciter or other external power source, through slip rings or a commutator, as in synchronous and DC machines. Depending on the condition of operation, the induction machine can be used as a motor or generator. Induction machines are available in single-phase or three-phase winding configurations. In this thesis the modelling and investigation is given only for the three-phase induction machine.

When the stator is excited from a balanced three-phase supply, the three phases together create a constant magnitude, synchronously revolving mmf or field in the air gap with a crest value 3/2 times the peak value of the alternating field due to one phase alone [1].

This field rotates around the air-gap at synchronous speed Ne, which can be calculated as

60 fe Ne (4.1) Pp

Where fe - excitation frequency in cycles per second (Hz)

Pp - number of pole pairs

Ne - synchronous speed in revolutions per minute (rpm)

Ne is also expressed as the rotational speed of the stator magnetic field, or mmf.

The slip of a motor, s, which is defined as the slip of the rotor with respect to the stator magnetic field, can be given as

66 CHAPTER 4 INDUCTION MACHINE MODELLING

NN s er (4.2a) Ne where Nr - the rotational speed of the rotor in rpm. If the speeds are expressed in radians per second the slip is given by ZZ s er (4.2b) Ze where Ze - synchronous speed in radians per second (rad/sec)

Zr - rotor speed in rad/sec.

4.2 Conventional induction machine model The relative speed between the synchronous speed and the rotor speed is expressed in its equivalent electrical speed as Ze-Zr or sZe, where the electrical rotor speed is the product of the mechanical speed and the number of pole pairs.

Rotation of the rotor changes the relationships between stator and rotor emfs. However, it does not directly change the inductance and resistance parameters. The angular frequency of the induced current in the rotor is sZe and the induced voltage in the rotor will be sEr, where Er is the induced voltage in the rotor when the rotor is stationary. This is based on the assumption that the induction machine is only supplied from the stator terminals.

Assuming that the winding is distributed sinusoidally in angular space around the stator to produce a sinusoidally distributed magnetic field [2] and the rotor winding is similar in form to the stator winding, then the per-phase equivalent circuit of the stator side of the three-phase induction machine can be represented as follows

Rs jZeLls

Is

Vs Es

Fig. 4.1 Stator side of the per-phase equivalent circuit of a three-phase induction machine

where Vs - stator voltage, V

67 CHAPTER 4 INDUCTION MACHINE MODELLING

Is - stator current, A

Rs - stator winding resistance, :

Lls - stator leakage inductance, H

Es - induced emf in the stator winding due to the rotating magnetic field that links the stator and rotor windings, V

Ze - stator current angular frequency, rad/sec

For constant stator flux the voltage induced in the rotor depends solely on the slip, which is the relative speed between the stator flux rotating at synchronous speed and rotor speed. Maximum induced voltage occurs in the rotor when the rotor is stationary. Without any external input on the rotor side, the rotor circuit is given by

Rr jsZeLlr

Ir

sEr

Fig. 4.2a Rotor side of the per-phase equivalent circuit of a three-phase induction machine

where sEr - induced voltage in the rotor, V

Ir - rotor current, A

Rr - rotor winding resistance, :

Llr - rotor leakage inductance, H

sZe - rotor current angular frequency, rad/sec.

If all the terms in the rotor side are divided by the slip, s, a modified circuit is obtained as shown in Fig. 4.2b.

jZeLlr

Ir Rr/s Er

Fig. 4.2b Rotor side of the induction machine with adjustment

68 CHAPTER 4 INDUCTION MACHINE MODELLING

Using the appropriate voltage transformation ratio between the stator and rotor, the rotor voltage, Er, referred to the stator is then equal to Es, in Fig. 4.1. The stator and rotor circuits are linked because of the mutual inductance Lm. When all circuit parameters are referred to the stator, the stator and rotor circuits can be combined to give the circuit shown in Fig. 4.3.

Rs jZeLls jZeLlr

Is Ir Vs jZeLm Rr/s

Fig. 4.3 Per-phase equivalent circuit of three-phase induction machine neglecting core loss

In Fig. 4.3 the core loss, which is due to hysteresis and eddy current losses, is neglected. It can be compensated by deducting the core loss from the internal mechanical power at the same time as the friction and windage losses are subtracted [3]. The no load current in three-phase induction machines consists of the iron loss or core loss component and the magnetizing component. From the iron loss current component and from the applied voltage the equivalent resistance for the excitation loss can easily be calculated. There is also some core loss in the rotor. Under operating conditions, however, the rotor frequency is so low that it may reasonably be assumed that all core losses occur in the stator only [4].

The core loss can be accounted for by a resistance Rm in the equivalent circuit of the induction machine [18]. Rm is dependent on the flux in the core and frequency of excitation. For constant flux and frequency Rm remains unchanged. As Rm is independent of load current it is connected in parallel with the magnetising inductance

Lm. The equivalent circuit including Rm is shown in Fig. 4.4.

Rs jZeLls jZeLlr

Is Ir

Ic Im jZeLm Rr/s Vs Rm

69 CHAPTER 4 INDUCTION MACHINE MODELLING

Fig. 4.4 Per-phase equivalent circuit of three-phase induction machine including core loss 4.3 D-Q axes induction machine model Using the D-Q representation, the induction machine can be modelled as shown in Fig.4.5. This representation is a general model based on the assumption that the supply voltage can be applied to both the stator and/or rotor terminals. In squirrel cage induction machines voltage is supplied only to the stator terminals. In general power can be supplied to the induction machine (induction motor) or power can be extracted from the induction machine (induction generator). It all depends on the precise operation of the induction machine. If electrical power is applied to the stator of the induction machine then the machine will convert electrical power to mechanical power. As a result the rotor will start to rotate and the machine is operating as a motor. On the other hand, if mechanical power is applied to the rotor of the induction machine then the machine will convert mechanical power to electrical power. In this case the machine is operating as an induction generator. When the induction machine operating as a generator is connected to the grid or supplying an isolated load, driven by an external prime mover, then the rotor should be driven above synchronous speed.

When the machine is operated as a motor, power flows from the stator to the rotor, crossing the air gap. However, in the generating mode of operation, power flows from the rotor to the stator. Only these two modes of operation are dealt with in this investigation. The braking region, where the rotor rotates opposite to the direction of the rotating magnetic field, is not dealt with here.

The conventional model and the d-q (or D-Q) axes model are the same for steady state analysis. The advantage of the d-q axes model is that it is powerful for analysing the transient and steady state conditions, giving the complete solution of any dynamics. The general equations for the d-q representation of an induction machine, in the stationary stator reference frame, are given as [5]:

70 CHAPTER 4 INDUCTION MACHINE MODELLING

ªºviqs ªºRpLss 00 pL m ªºqs «» «» v«»00 R pL pL i «»ds «» s s m«» ds (4.3) «» «» vpLLRpLLiqr«» m--ZZ r m r r r r qr «»«»«» ZZLpLLRpL «»¬¼vidr «»¬¼rm m rr r r «»¬¼dr where Rs - stator winding resistance, :

Rr - rotor winding resistance, :

Lm - magnetising inductance, H

Ls - stator leakage inductance (Lls) + magnetising inductance (Lm), H

Lr - rotor leakage inductance (Llr) + magnetising inductance (Lm), H

Zr - electrical rotor angular speed in rad/sec and p=d/dt, the differential operator.

Q-axis

iqs

vqs

Zr iqr

vqr

D-axis idr ids vdr vds

Fig. 4.5 D-Q representation of induction machine

Equation (4.3) can be written in a first order differential equation form as given in the following matrix equation

71 CHAPTER 4 INDUCTION MACHINE MODELLING

L0 L 0 2 ªºpiqsªºrm ªº v qs ªºLRrs L mrZZ L mr R  L mrr L ªºiqs «»«» «» « 2 » «» pids110Lrm 0 L v ds LLRLLLRZZ ids «» «» «»« mr ss mrr mr » «» (4.4) «»«» «» « » «» piqrLLL0 L 0 v qr LR LZZ L LR L L iqr «»«»ms «» « ms srm sr srr» «» 0L0L ¬¼«»pidr«»¬¼mm ¬¼«» v dr ¬«LLLRsrmZZ ms LL srr LR sr¼» ¬¼«»idr

2 where LLLL s rm.

Using the matrix shown in Equation (4.3), the d-q representation given in Fig. 4.5 can be redrawn in detail, in a stationary stator reference frame, with separate direct and quadrature circuits as shown in Fig. 4.6.

-Zr Oqr Rs Lls Llr R r + -

ids idr imd Lm vds vdr Ods Odr

(a)

Zr Odr Rs Lls Llr R r + -

iqs iqr imq Lm vqs vqr Oqs Oqr

(b) Fig. 4.6 Detailed d-q representation of induction machine in stationary reference frame (a) d-axis circuit (b) q-axis circuit

From the stator side (for simplicity the superscript “s” which indicates stationary reference frame is not included with the currents, voltages and fluxlinkages)

Ods L siLi ds m dr (4.5)

Oqs Li s qs L m i qr (4.6) dO vRi ds (4.7) ds s ds dt dO vRi qs (4.8) qs s qs dt

72 CHAPTER 4 INDUCTION MACHINE MODELLING

From the rotor side

Odr L miLi ds r dr (4.9)

Oqr Li m qs Li r qr (4.10) dO vRi dr ZO (4.11) dr r drdt r qr dO vRi qr ZO (4.12) qr r qrdt r dr

For the air gap flux linkage

Odm L miLiLi md m ds m dr (4.13)

Oqm L miLiLi mq m qs m qr (4.14)

The stator electrical input power to the induction machine during motoring operation or the stator electrical output power in generating mode is given by 3 P iv iv (4.15) edsdsqsqs2

The electromagnetic torque Te generated by the induction machine is given by [6] 3 JJG JG TPI O u (4.16) epmr2 JJG where Om - air gap flux linkage JG Ir - rotor current space vector

Pp - number of pole pairs of the induction machine.

Solving the cross product in Equation (4.15) gives 3 TPLiiii  (4.17) e2 p m qsdr dsqr The mechanical equation in the motoring region is dZ TJ m  DZ T (4.18) emmdt and in the generating region it is given as dZ TJ m  DTZ (4.19) medt 73 CHAPTER 4 INDUCTION MACHINE MODELLING

where Tm - mechanical torque in the shaft, Nm Te - electromagnetic torque, Nm

Zm - mechanical shaft speed (Zm = Zr /Pp ), rad/sec D - friction coefficient, Nm/rad/sec J - inertia, Kg-m2.

The mechanical power generated during motoring or the mechanical power required to drive the induction generator is given by

PTmem Z (4.20)

4.4 Simulation of induction machine The simulation of the induction machine shows its characteristics in the motoring and generating regions without the need to use a hardware experimental setup, which is time consuming, space demanding, noisy and expensive. Also, at low rotor speeds, away from synchronous speed, the induction machine will draw high current compared to its rated value. If the data acquisition takes a long time the induction machine may be damaged due to excessive temperature rise. The simulation is useful for finding the characteristics of the induction machine from standstill to twice the synchronous speed or for any rotor speed. In order to find the characteristics of a real induction machine in the motoring and generating region, an experimental setup such as shown in Fig. 4.7 is used. With a fast data acquisition system the conventional meters can be replaced with sensors so that all the data is measured in a personal computer, quickly and accurately, using a Digital Signal Processing (DSP) interface.

The induction machine used in this investigation is a three-phase wound rotor induction motor with specification: 415V, 7.8A, 3.6kW, 50Hz, and 4 poles. The simulation of the induction machine is done using the d-q equivalent circuit because it provides the complete solution, comprising of both the transient and the steady state. The parameters of the induction machine obtained using the standstill test and the no-load test data at rated values of voltage, current and frequency are Lls = Llr = 12mH, Lm = 181mH,

Rs = 1.6:, Rr = 2.75:.

74 CHAPTER 4 INDUCTION MACHINE MODELLING

Ammeter, Voltmeter and DC Supply Watmeter

Three-phase Ammeter and supply Voltmeter d

Speed meter

DC machine Induction machine Fig. 4.7 Experimental setup to find the characteristics of induction machine in the motoring and generating regions

The simulation is carried out based on the fact that when the induction machine is operating as a motor it drives the DC machine so that the DC machine is operating as a generator. When the induction machine is operating as a generator it is driven by the DC machine, operating as a DC motor. Since the area of interest is in the characteristics of the induction machine the investigation is done only for the induction machine. The DC machine is used as a prime mover and as a load for the induction machine. In the simulation, to avoid the transients in the motoring region, the induction machine is started by rotating in the reverse direction. However, the region of operation investigated here is between zero speed and twice the synchronous speed. A constant full voltage is applied and the induction machine is allowed to accelerate at constant acceleration. The system takes 10 seconds to accelerate from standstill (zero speed) to twice the synchronous speed. In this case the synchronous speed is 1500rpm (4-pole, 50Hz machine).

Using Equations (4.4) and (4.16) a model is developed in Matlab/Simulink. The simulation model has two inputs namely stator voltage and rotor speed. As the rotor terminals of the induction machine are shorted there is no electrical input from the rotor side. With the help of the developed model the necessary characteristics of the induction machine are obtained. 75 CHAPTER 4 INDUCTION MACHINE MODELLING

Whether the induction machine is operating as a motor or a generator, it needs an exciting current (reactive current) to magnetise the core and produce the required amount of magnetomotive force (mmf). This mmf will produce an equivalent amount of magnetising flux linkage determined by the air gap electromotive force (emf), which is dependent on the stator terminal voltage and the voltage drop in the stator resistance and stator leakage inductance.

76 CHAPTER 4 INDUCTION MACHINE MODELLING

Phase voltage(V) and 10*Phase current(A) in stationary reference frame 500 10*Current 400 Voltage 300 200 100 0 -100 -200 -300 -400 -500 0 1 2 3 4 5 6 7 8 9 10 A B C time(sec)

(a)

Detail of A Detail of B 500 500 Voltage Voltage

0 0 (b) (c) 10*Current 10*Current

-500 -500 2.62 2.63 2.64 5 5.01 5.02 Detail of C 500 10*Current

0 (d)

Voltage -500 6.64 6.65 6.66 Fig. 4.8 Variation of stator phase current for constant supply voltage and frequency (a) Current and voltage when the rotor speed is varied from standstill to twice the synchronous speed (b) detail of motoring region (c) detail around the synchronous speed (d) detail in the generating region

From Fig. 4.8 it can be seen that the phase current lags the voltage by an angle less than 90o in the motoring region. Close to synchronous speed the current lags by about 90o

77 CHAPTER 4 INDUCTION MACHINE MODELLING and in the generating region the lag angle is greater than 90o. The time varying phase current shown in Fig. 4.8 can be iqs or ids and for the voltage it can be vqs or vds. The currents iqs and ids correspond to the voltages vqs and vds respectively. The angle between the phase current and the phase voltage remains the same in the q-axis and the d-axis as well as in the three axis representation a, b and c.

T V T V I I

(a) (b)

V T

I

(c)

Fig. 4.9 Relationship between phase voltage vector and phase current vector (a) in the motoring region (b) between motoring and generating (at synchronous speed) (c) in the generating region

In Fig. 4.9, V is the stator voltage and I is the stator current. In the two axis representation V can be Vds or Vqs and in three axis representation V is the phase voltage and can be Va, Vb or Vc and I represents phase current, i.e. Ids, Iqs, Ia, Ib or Ic.

The stator electrical input/output power, the mechanical output/input power, the electromagnetic torque and efficiency are given in Fig. 4.10. The electrical input power and the mechanical output power are considered as positive during motoring operation. Hence during generating operation the electrical output power and the mechanical input power are negative because the flow of power is reversed. Positive slip is motoring and negative slip is generating.

78 CHAPTER 4 INDUCTION MACHINE MODELLING

100

0 Torque (Nm) -100 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (a) 20

10 (KW)

e 0 P

-10 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (b) 10

0 (KW) m P -10

-20 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (c) 100 75 50 25

Efficiency (%) 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (d) Slip Fig. 4.10 Induction machine torque, power and efficiency characteristics (a) torque (b) electrical power (c) mechanical power (Pm=ZmTe) (d) efficiency

The efficiency calculation is only taking into account the losses in the induction machine and is related to the electrical power, Pe, and mechanical power, Pm, in the equations following. It does not include the friction and windage loss in the system nor does it include stray losses. The efficiency in the motoring region is calculated by P Efficiency m u100 Pe and for the generating region P Efficiency e u100 Pm

79 CHAPTER 4 INDUCTION MACHINE MODELLING

The instantaneous space vector angles and magnitudes of the stator voltage Vs, stator current Is, magnetising current Im, rotor current Ir, and stator flux linkage Os are calculated using

22 1 §·fq T fdq f f and ftan ¨¸ ©¹fd

Where f can be Vs, Is, Im, Ir, or Os.

Stator current space vector angle 0 T Is (deg) -30 (a) -60 -90 -120 -150 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Stator flux likage space vector angle s (deg) 0 TO

-30 (b) -60

-90 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Rotor current space vector angle T Ir (deg) 180 150 120 (c) 90 60 30 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Magnetising current space vector angle 0 T Im (deg) -30 (d) -60 -90 -120 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Slip Fig. 4.11 Space vector angles measured with respect to the stator voltage space vector angle for (a) stator current Is (b) stator flux linkage Os (c) rotor current Ir (d) magnetising current Im

80 CHAPTER 4 INDUCTION MACHINE MODELLING

To have a comparison between the different space vectors, all the space vector angles are calculated with respect to the stator voltage space vector angle. Fig. 4.11 shows the space vector angle for stator current, stator flux linkage, rotor current and magnetising current.

340

(a) (V) 200 s V

0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 60

40

(b) (A) s I 20

0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

1 (c) (web-Turn)

s 0.5 O

0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

60

40 (A) r

(d) I 20

0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 6

4

(e) (A) 2 m I 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Slip

Fig. 4.12 Magnitude of space vector for (a) stator voltage (b) stator current Is (c) stator flux linkage Os (d) rotor current Ir (e) magnetising current Im

The magnitude of stator voltage, stator current, stator flux linkage, rotor current and magnetising current are shown in Fig. 4.12. Using the angles given in Fig. 4.11 and the 81 CHAPTER 4 INDUCTION MACHINE MODELLING magnitudes given in Fig. 4.12, the space vectors for stator voltage, stator current, stator flux linkage, rotor current and magnetising current are shown in Fig. 4.13.

I e r q

TIr

qs TIm Vs Im TIs

TOs

Is Te Os

de ds

Ze (a)

Ir qe

TIr qs

Vs TIm

Im

TOs TIs

Is Te Os

de ds Ze (b) Fig. 4.13 Space vector diagram for stator voltage, stator current, rotor current, magnetising current and stator flux linkage (a) during motoring mode (b) during generating mode

From Fig. 4.10d maximum efficiency occurs close to synchronous speed and efficiency is zero at synchronous speed (s = 0) and at standstill when the rotor is stationary (s = 1).

82 CHAPTER 4 INDUCTION MACHINE MODELLING

In Fig. 4.11 it can be observed that compared to the angle of the stator voltage space vector, the stator current space vector always lags. Because of the stator winding resistance, the maximum magnitude of angle for the stator flux linkage is close to 90o lag but not exactly 90o. Close to synchronous speed the rotor current is more resistive, and there is a reversal of current when it goes from the generating mode to the motoring mode. The magnetising space vector, which is the sum of the stator current space vector and the rotor current space vector, has an angle of 90o lag close to synchronous speed.

As can be seen in Fig. 4.12, the magnitude and frequency of the terminal voltage space vector are kept constant during the motoring and generating modes. There is a minimum stator current space vector at synchronous speed. The magnitude of the stator flux linkage space vector shows that there is a small increase as the machine goes from motoring to generating. The variation in stator flux linkage is a reflection of the variation in the emf of the induction machine. There is no closed-loop control to control the exciting current and the stator flux linkage. As will be discussed in the vector control of induction machines in Chapter 8 the magnitude of the terminal voltage space vector can be controlled by varying the stator flux linkage. At synchronous speed the rotor current is zero and the magnetising current approaches its maximum value.

Fig. 4.13 shows the diagram of all the magnitudes and angles of the space vectors discussed above. The space vectors shown in Fig. 4.13 are helpful in analysing and designing stator oriented vector control of the induction machine in the generating and motoring regions.

To transform the signals into the rotating reference frame the d-axis of the rotating reference frame (de) is aligned with the stator flux linkage. Hence at T e = 0 the d-axis component of the stator flux linkage in the stationary reference frame is equal to the total stator flux linkage. At a given operating slip the space vectors are rotating at the same speed Ze. Therefore, all the components of the space vectors appear as DC values in the rotating reference frame.

83 CHAPTER 4 INDUCTION MACHINE MODELLING

Iqs in exictation reference frame (A) 20

0

-20

-40 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (a)

Ids in exictation reference frame (A) 60

40

20

0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Slip (b) Fig. 4.14 Stator current in the de-qe axes of the excitation reference frame (a) qe-axis current (b) de-axis current

Vds in the exitaion reference frame (V)

100

50

0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (a) Vqs in the exitaion reference frame (V)

300

200

100

0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 slip (b)

Fig. 4.15 Stator voltage in the de-qe axes of the excitation reference frame (a) de-axis voltage (b) qe-axis voltage

84 CHAPTER 4 INDUCTION MACHINE MODELLING

Imd in the exitaion reference frame (A) 6

4

2

0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (a) Imq in the exitaion reference frame (A) 2

1

0

-1

-2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 slip (b) Fig. 4.16 Magnetising current in the de-qe axes of the excitation reference frame (a) de-axis magnetising current (b) qe-axis magnetising current

Idr in the exitaion reference frame (A) 0

-10

-20

-30

-40 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 (a) Iqr in the exitaion reference frame (A) 40

20

0

-20 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 slip (b)

Fig. 4.17 Rotor current in the de-qe axes of the excitation reference frame (a) de-axis rotor current (b) qe-axis rotor current

85 CHAPTER 4 INDUCTION MACHINE MODELLING

As shown in Figures 4.14 – 4.17, all time varying signals transformed to the excitation reference frame look like DC quantities. In the excitation reference frame it is easier to see the components of currents that produce the electromagnetic torque.

Transforming the rotor current to a reference frame fixed on the rotor gives an with a frequency equal to the slip frequency Ze-Zr or sZe. Fig. 4.18 shows the rotor current in a reference frame fixed to the rotor and in a reference frame fixed to the stator, which is a stationary reference frame. The transformation angle to the rotor reference frame is calculated using the rotor speed as

TZT dt  (4.21) rrr³ to The magnitude of the rotor current given is when it is referred to the stator side.

rotor current in the rotor reference frame (A) 50

25

0

-25

-50 0 1 2 3 4 5 6 7 8 9 10 (a) rotor current in the stationary reference frame (A) 50

25

0

-25

-50 0 1 2 3 4 5 6 7 8 9 10 time (sec) (b)

Fig. 4.18 Rotor current in different reference frames (a) rotor current in a rotating reference frame that is rotating at the rotor speed (b) rotor current in the stator (stationary) reference frame

4.5 D-Q axes induction machine model in rotating reference frame The transformation of currents and voltages to a rotating reference frame gives a characteristic from a different perspective. The speed of the rotating reference frame can

86 CHAPTER 4 INDUCTION MACHINE MODELLING have any value. If the reference frame is rotating exactly at the excitation frequency then the difference between the speed of the rotating reference frame, Ze, and the rotor speed, Zr, gives the slip frequency Zsl.

Assuming the induction machine is only supplied from the stator side the equivalent circuit in the excitation reference frame of the d and q axes is shown in Fig. 4.19.

e e R -ZeOqs L L R (Ze-Zr)Oqr s +  ls lr r + 

e e ids idr Lm e e e Vds Ods Odr

(a)

e e ZeOds -(Ze -Zr )Odr Rs +  Lls Llr Rr + 

e e iqs iqr Lm e e e Vqs Oqs Oqr

(b)

Fig. 4.19 D-Q representation of induction machine in the excitation (Ze) reference frame (a) d-axis circuit (b) q-axis circuit

Unlike the stationary reference frame, in the excitation or synchronous reference frame the reference frame is rotating at the same speed as the excitation frequency or the synchronous speed. Since the voltages and currents have the excitation frequency they will appear as DC values.

4.6 Development of D-Q axes induction machine model with Rm The conventional way of three-phase induction machine steady state modelling is to use the per-phase equivalent circuit, including Rm, as given in Fig. 4.4. The equivalent resistance for core loss or iron loss is incorporated into the circuit by adding Rm in

87 CHAPTER 4 INDUCTION MACHINE MODELLING

parallel with the magnetizing reactance, Xm. To simplify the analysis of the three-phase induction machine, Rm is often neglected from the per-phase equivalent circuit [7].

To date the three-phase induction machine d-q axes model has been represented neglecting Rm [8, 9, 19]. Neglecting Rm will definitely simplify the analysis of the induction motor, but it will introduce some error in the results that are obtained from the d-q axes model. The error, which is introduced by neglecting Rm, will have more effect especially if the application of the analysis is to compute efficiency or analyse losses in the machine [10, 11].

There is an interest in modelling the iron loss in the d-q model of an induction machine. In the literature different ways of representing iron loss have been presented that attempt to treat iron loss in a unified way [12, 13, 14, 15, 16]. These representations do not give the relationship between the conventional model and the d-q model of a three- phase induction machine. The d-q model of a three-phase induction machine including iron loss needs to be presented in an easily understandable way.

In this analysis a novel d-q model of an induction machine for analysing the effects of iron loss under steady-state and transient conditions is provided. In collaboration with D. McKinnon, a fellow research student, the d-q model including iron loss is presented in a simple, understandable way [17]. It should be noted that Mr McKinnon contributed more to the inclusion of iron loss into the traditional d-q model. The author, Mr McKinnon and C. Grantham exchanged ideas frequently in this area of the investigation.

Using Fig. 4.3, the conventional steady state per-phase equivalent circuit model of a three-phase induction machine with Rm neglected and all rotor quantities referred to the primary/stator side, the following substitution is made (Ze-Zr)/ Ze = s and p=jZe, and with power being supplied only on the stator side, the voltages and currents can be related as:

ªºvs ªºR+Lpss Lm p ªºis «»= «»«» (4.22) ¬¼0 ¬¼Lp-jmrmrrrrȦ LR+Lp-jȦ L ¬¼ir

88 CHAPTER 4 INDUCTION MACHINE MODELLING

Where Ls =Lls+Lm, Lr =Llr+Lm, p=d/dt and Zr is the rotor speed in electrical rad/sec.

From the conventional per-phase equivalent circuit of a three-phase induction machine given in Fig. 4.4 with Rm included in the model, the relation between the input stator voltage, and the stator and rotor currents can be derived as:

ªºv s ªºR+LssNEW p LN p ªºis «»= «»«» (4.23) ¬¼0 ¬¼Lp-jNrNrrNEWrrNEWȦ LR+Lp-jȦ L ¬¼ir Where

RmmL L=N Rmm+L p

L=L+LsNEW ls N

L=L+LrNEW lr N

From Equation (4.3), neglecting Rm and with vdr = vqr = 0, the following matrix equation can be obtained.

ªºvqs ªºR+Lpss00 L m p ªºiqs «»«»«» v 00R +L p L p i «»ds = «»s sm«»ds «» (4.24) «»0 «»Lpmr -Ȧ LR+Lp-mrrrȦ Lirqr «»«»«» 0 Ȧ LLpȦ LR+Lp ¬¼«»¬¼«»rm m rr r r ¬¼«»idr

To represent Rm in the d-q model, Equation (4.24) needs to be modified. Using the comparison between Equation (4.22) and (4.24), Equation (4.23) gives the matrix

Equation in the d-q model, i.e. including Rm, as:

ªºvqs ªºR+LssNEW p00 L N p ªºiqs «»«»«» v 00R +L p L p i «»ds = «»s sNEW N «»ds «» (4.25) «»0 «»LpNr -Ȧ LR+Lp-mrrNEWrȦ LirNEWqr «»«»«» 0 Ȧ LLpȦ LR+Lp ¬¼«»¬¼«»rN N rrNEW r rNEW ¬¼«»idr

89 CHAPTER 4 INDUCTION MACHINE MODELLING

This is a new form of matrix expression, which takes into consideration the effect of Rm in the model. Equation (4.25) is the matrix form for the relationship between voltages and currents and it can be used for dynamic analysis of induction machines with Rm included.

With only a stator supplied induction machine, i.e. vdr = vqr = 0, the d-q model, including Rm, is derived from Equation (4.25) and is given in Fig. 4.20. The d-q model of the induction machine, including Rm, given in Fig. 4.20 is the same as the model given in Fig. 4.6 except that Rm is now added in parallel with Lm.

-Zr Oqr Rs Lls Llr R r + -

ids idr imd Rm Lm vds Ods Odr

(a)

Zr Odr Rs Lls Llr R r + -

iqs iqr imq Rm Lm vqs Oqs Oqr

(b) Fig. 4.20 D-Q model of induction machine in the stationary reference frame including

core loss represented by Rm (a) d-axis (b) q-axis

From Fig. 4.20 it is clear that iiqsz qr i mq and iids drz i md . There is current flowing in

Rm, which is the representation of the core loss.

Expanding and rearranging Equation (4.25) gives 2 P I = AopV + A1V + BopI + B1I (4.26)

90 CHAPTER 4 INDUCTION MACHINE MODELLING

ªºiqs ªº1 ªºR «» 0 m 0 i ªºv «»L «»LL «»ds qs «»ls «»mls Where I , V «», Ao , A1 «»i v «»1 «»R «»qr ¬¼ds 0 0 m «»L «»LL ¬¼«»idr ¬¼ls ¬¼mls

ªº§·RRRsmm Rm «»¨¸00 «»©¹LLLls m ls Lls «» §·RRsmm R Rm «»00¨¸  «»©¹LLLls m ls Lls Bo «» «»RRmm§·Rr Rm 0 ¨¸Zr «»LLLL «»lr ©¹lr m lr «» RRmm§·Rr Rm «»0 Zr ¨¸ ¬¼«»LLlr ©¹lrL mL lr

ªºRR  ms 00 0 «»LL «»mls «»RR 000 ms «»LL «»mls B 1 «» ZrmRRRRR mr§· m m «»0 Zr ¨¸ «»LLLLLlr m lr©¹ lr m «» «»ZrmRRRRR§· m m mr 0 Zr ¨¸ «»LLLLL ¬¼«»lr©¹ lr mr m lr

From the magnetising inductance branch circuit of Fig. 4.20 the q-axis magnetising current is

Rm iiimq qs qr (4.27a) RLpmm and in integral form it can be written as R iiiidt m  (4.27b) mq³ qs qr mq Lm

For the d-axis magnetising current

Rm iiimd ds dr (4.28a) RLpmm and in integral form it is written as

91 CHAPTER 4 INDUCTION MACHINE MODELLING

R iiiidt m  (4.28b) md³ ds dr md Lm

The q-axis air gap flux linkage is

RLmm Omq L miii mq qs  qr (4.29) RLpmm and the d-axis air gap flux linkage is

RLmm Omd Li m md i ds  i dr (4.30) RLpmm

From the stator side the q-axis stator flux linkage is given as

RLmm Oqs Li ls qs Li m mq Li ls qs i qs  i qr (4.31) RLpmm and the d-axis stator flux linkage is given as

RLmm Ods Li ls ds Li m md Li ls ds i ds  i dr (4.32) RLpmm

From the rotor side the q-axis stator flux linkage is given as

RLmm Oqr Li lr qr Li m mq Li lr qr i qs  i qr (4.33) RLpmm and the d-axis stator flux linkage is given as

RLmm Odr Li lr dr Li m md Li lr dr i ds  i dr (4.34) RLpmm

Based on the definition of electromagnetic torque [6] 3 JJG JG TPI O u (4.35) epmr2 JJG JG where Om and Ir are the space vectors for the air gap flux linkage and for the rotor current respectively. JJG JG Substituting Om and Ir in Equation (4.35) and rearranging using vector manipulation gives:

3 RLmm Te P p ii qs dr ii ds qr (4.36) 2 RLpmm

92 CHAPTER 4 INDUCTION MACHINE MODELLING and in integral form

3 §·R TPRiiiiTdt m (4.37) e p m³¨¸ qsdr dsqr e 2 ©¹Lm

4.7 Summary The modelling of an induction machine using the conventional or steady state model and the d-q or dynamic model are explained in detail. The voltage, current and flux linkage in the rotating reference and their phase relationships in the motoring region and generating region are presented. For the same stator terminal voltage the magnitude of the electromagnetic torque in the generating region is higher than the electromagnetic torque in the motoring region. The reason for the difference in electromagnetic torques is that during the motoring region all the electrical losses in the induction machine are supplied by an external electrical power source and the electromagnetic torque is the output of the system. However, in the generating region the electromagnetic torque is equivalent to the external mechanical input torque and all the electrical power losses in the induction machine are indirectly supplied by the external mechanical power source and the terminal voltage is the output of the system. Hence to overcome all the internal power losses in the induction machine and have the same terminal voltage as in the motoring region the electromagnetic torque in the generating region is higher than the motoring region.

D. McKinnon and C. Grantham, in collaboration with the author, has improved the induction machine model in D-Q axes including the equivalent iron loss resistance, Rm, is presented in a simple and understandable way. Using this model the dynamic current, torque and power can be calculated.

4.8 References [1] P. L. Alger, “The nature of induction machines”, Gordon and Breach Inc., New York, 1965. [2] G. R. Slemon, “Electric Machines and Drives”, Addison-Wesley Publishing Company, New York, 1992. [3] A. E. Fitzgerald and JR. Charles Kingsley, “Electric Machinery”, McGraw-Hill Book Company, New York, 1961. [4] J. Rosenblatt, and M. H. Friedman, “Direct and Alternating Current machinery”, CBS Publishers and Distributors, New Delhi, 2000. 93 CHAPTER 4 INDUCTION MACHINE MODELLING

[5] N. N. Hancock, “Matrix Analysis of Electrical Machinery”, Pergamon Press, New York, 1974. [6] B. K. Bose, “Modern Power Electronics and AC Drives”, Printice-Hall, New Jersey, 2002. [7] B. K. Bose, “Power Electronics and AC Drives”, Printice-Hall, New Jersey, 1968. [8] Chee-Mun Ong, “Dynamic Simulation of Electric Machinery using Matlab/Simulink” Printice-hall, New Jersey, 1998. [9] C. Grantham, D. Seyoum, H.Tabatabaei-Yazdi, “Very Fast and Accurate Electrical Measurements”, AUPEC ’99, The Northern Territory University, Darwin, Australia, 26th-29th Sept. 1999, pp. 99 - 103. [10] H. Auinger, “Efficiency of electric motors under practical conditions”, IEE Journal, June 2001, pp. 163 - 167. [11] C. Grantham and H. Tabatabaei-Yazdi, “Rapid Parameter Determination for use in the Control High Performance Induction Motor Drives”, IEEE 1999 International Conference on Power Electronics and Drive Systems, PEDS’99, July 1999, Hong Kong, pp. 267 - 272 . [12] E. Levi, “Impact of iron loss on behavior of vector controlled induction machines”, IEEE Trans. On Industry Applications, Vol. 31, No. 6, November/December 1995, pp. 1287 - 1296. [13] E. Levi, M. Sokola, A. Boglietti and M. Pastorelli, “Iron loss in rotor-flux-oriented induction machines: identification, assessment of detuning, and compensation”, IEEE Transactions on Power Electronics, Vol. 11 No. 5, Sept. 1996, pp. 698 - 709. [14] G.O. Garcia, J.A. Santisteban and S.D. Brignone, “Iron losses influence on a field-oriented controller”, In Proc. IECON '94, Vol. 1, 1994, pp. 633 - 638. [15] Sung-Don Wee, Myoung-Ho Shin and Dong-Seok Hyun, “Stator-flux-oriented control of induction motor considering iron loss”, IEEE Trans. On Industrial Electronics, Vol. 48, No. 3, June 2001, pp. 602 - 608. [16] J.W. Choi, D. W. Chung, and S. K. Sul, “Implementation of Field Oriented Induction Machine Considering Iron Losses”, IEEE- APEC '96. Conference Proceedings, 1996, pp. 375-379. [17] D. McKinnon, D. Seyoum and C. Grantham, "Novel Dynamic Model for a Three-Phase Induction Motor With Iron Loss and Variable Rotor Parameter Considerations", Proc. AUPEC 2002, Melbourne, 29 Sept. - 2 Oct. 2002. ISBN 0-7326-2206-9. [18] M.G. Say, “Alternating current machines” Pitman, London, 1983, pp. 260-262. [19] P. C., Krause, O. Wasynczukand and S. D. Sudhoff, “Analysis of Electric Machinery”, IEEE press, New York, 1995.

94 CHAPTER 5

DATA ACQUISITION AND DIGITAL SIGNAL PROCESSING

5.1 Introduction The remarkable increase in the speed and power of digital computers and special purpose hardware over recent years has ensured the continued growth of interest in data acquisition and Digital Signal Processing (DSP) for monitoring and control of electrical machines. A signal is any variable that contains information from a transducer or from a controller that can be manipulated for a given application. Digital Signal Processing, as the term suggests, is the processing of signals by digital means.

Each measurement system in the experimental set up involves sensors, signal conditioning, transmission, sampling, manipulation and interpretation of the resultant data. The output of the measurement process is presented to be displayed or processed for the control of physical parameters. The over all interconnection of a data acquisition and signal processing system is shown in Fig. 5.1.

Digital Parameter to A/D Digital Digital D/A Analog sensor Analog signal converter signal converter signal be measured signal signal processor

Fig. 5.1 Block diagram for data acquisition and signal processing

Filtering reduces noise errors in the signal. This can be done with either an analog or digital filter. For most applications a low-pass filter is used. This allows the lower frequency components through but attenuates the higher frequencies. The cut-off frequency must be compatible with the frequencies present in the actual signal (as opposed to possible contamination by noise) and the sampling rate used for the A/D

95 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING conversion. A high pass filter is used to remove DC component or signals with low frequency.

Fast data acquisition and Digital Signal Processing systems are used to acquire thousands of samples of a measured signal in a second. This was impossible to implement a few years ago. Data acquisition deals with the way a signal is sensed and conditioned, whilst Digital Signal Processing deals with the manipulation of a signal in digital form. a

b Sensor Induction c board machine

ia ib vab vbc

Anti-aliasing filter

Speed measurement DS1102 DSP board Hardware

Software 3 axes to 2 axes transformation

id iq vd vq

Fig. 5.2 Hardware and software system configuration

5.2 DS1102 DSP board The fast data acquisition and control system in the experimental set up is based on a DS1102 controller board produced by dSPACE GmbH. The DS1102 board is built around a Texas Instruments TMS320C31 floating point Digital Signal Processor (DSP) running at 40MHz clock rate and the slave-DSP is a Texas Instruments TMS320P14 DSP running at 25MHz clock rate. The processor can be accessed from the host computer through graphically oriented tools for on-line visualization (TRACE) and on- line parameter tuning (COCKPIT). Control desk combines the cockpit and trace options together in the same window. The most significant advantage in using the DS1102 is the

96 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING relative ease of use, freeing the designer from the burden of manually programming the DSP. The operation of the DS1102 DSP board requires the Texas Instrument C compiler, assembler, linker and loader. The LD31 utility program performs the actual loading of the TMS320C31 DSP object code modules and optionally the TMS320P14 object code modules. The C environment of the DS1102 controller board contains a library with predefined functions, header files and other application files.

The TMS320P14 can be used for numerical calculation through loading the program into its memory. The TMS320C31 communicates with the slave TMS320P14 through a 32 bit communication port and programs can be executed in both processors concurrently by two separate timers. The block diagram of the DS1102 is shown in Fig. 5.3 [1].

128Ku32 TMS 320P14 26 static RAM Digital I/O zero wait states 16-bit ADC 1 16-bit ADC 2

serial 12-bit ADC 3 interface TMS320C31 12-bit ADC 4 12-bit DAC 1 12-bit DAC 2

JTAG 12-bit DAC 3

connector JTAG Interface 12-bit DAC 4 Incr. encoder 1 Noise filter Analog/Digital I/O connector I/O Analog/Digital Host Incr. encoder 2 Noise filter DS1102 DSP-board Interface

PC PC/AT Expansion Bus

Fig. 5.3 Block Diagram of the DS1102

From Fig. 5.3 the signal groups used in the experimental set up are: x Four ADC (analog to digital conversion) inputs: The two ADC inputs are of 16 bit with A/D conversion time of 10Ps and two ADC inputs are of 12 bit with

97 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING

A/D conversion time of 3Ps. All ADCs have single ended bipolar inputs with ±10V input span. ±10V appears as ±1V in the software environment. x Four DAC outputs: All DAC outputs are of 12 bit with programmable output voltage range. The DACs have single ended voltage outputs with ±10V span. x Two channels of incremental encoder interfaces. x Digital I/O subsystem: This is based on the TMS320P14 DSP-microcontroller. It contains 16 pins bit selectable parallel I/O port, six PWM circuits and other lines which are not used in the experimental setup.

5.3 Data acquisition Taking measurements using conventional meters is very slow and is able to show only the steady state values. For a fast data acquisition system fast sensors are required to track the change in the measured quantities. The voltage, current and speed of the induction machine are measured using appropriate sensors to be processed in the Digital Signal Processing part for machine analysis and control. The temperature of the induction machine is sensed and displayed on a dedicated display system as a conventional thermometer.

5.3.1 Voltage and Current measurement In a balanced three-phase system with the measurement of only two phase voltages and two line currents it is possible to calculate the rms voltage, rms current, three-phase power, power factor and frequency of the three-phase system at an instant of time. This was illustrated in Chapter 3. To implement a fast current and voltage measurement, fast responding (high bandwidth) and high accuracy sensors are required.

The sensors used for voltage and current measurement are based on the Hall-Effect principle and are able to measure voltage and current signals in the frequency range of DC up to 100 KHz. The principle of measurement for the Hall-Effect voltage measurement and current measurement are the same. Hall-Effect produces Hall-Voltage based on a magnetic flux created by current. If a conducting material is placed in a magnetic field perpendicular to the direction of current flow, a Hall-Voltage is

98 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING developed across the material in a direction perpendicular to both the initial current direction and the magnetic field.

The sensors for current and voltage are isolated from the circuit being measured and use a separate power supply to avoid any effect on the voltages and currents being measured. To avoid ground loops between the sensor boards and the DSP card separate return lines are used for the connected sensors and the signal line is shielded to avoid any external interference. The voltage and current sensors and their associated amplification circuits are located on one PCB. This is referred to as the Sensor Board.

The DS1102 has only 4 analog ADC inputs, however to measure 6 analog inputs (2 AC voltages, 2 AC currents, 1 DC voltage and 1 DC current) an 8 to 4 multiplexer is used and the multiplexer circuit is controlled by 2 lines from the 16 I/O pins. The function ds1102_ad(channel) returns the ADC input value of the converter specified by the parameter channel which must be set to 1, 2, 3 or 4. The ADC data is read subsequently and scaled to its floating-point value in the range -1.0 … +1.0. Since the ADC input value is a 16 or 12-bit signed integer left aligned within a 32-bit data word, the factor 2-31 is required for scaling. The conversion of the ADCs must be started by a preceding call to the function ds1102_ad_start( ).

5.3.1.1 Anti-aliasing filter According to the Sampling Theorem [2], any signal can be accurately reconstructed from values sampled at uniform intervals as long as it is sampled at a rate that is at least twice the highest frequency present in the signal. Failure to satisfy this requirement will result in aliasing of higher-frequency components, meaning that these components will appear to have frequencies lower than their true values.

One way of avoiding the problem of aliasing is to apply a low-pass filter to the signal prior to the sampling stage to remove any frequency components above the "folding" or Nyquist frequency (half the sampling frequency) [3]. The anti-aliasing filters are introduced at the end of the analog signal conditioning and just before the analog to digital converter. These anti-aliasing filters are implemented using conventional analog

99 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING circuitry with cut off frequency below half of the minimum sampling frequency (maximum sampling time) used in the DS1102 application programs.

The anti-aliasing filter also helps to have smooth voltage and current signals in the PWM controlled inverter source. Each analog signal from voltage and current sources must have an anti-aliasing filter.

5.3.1.2 Voltage measurement For voltage measurement the sensor circuit is designed in such a way that it will be able to measure the full range of the applied line to line voltage at the terminals of the induction machine and the maximum DC voltage available in the DC link of the inverter. The peak voltage to be measured is 590V (415u1.414). LV 25-P, produced by LEM components, is the transducer used for the electronic measurement of DC and AC voltages. The detail of the voltage measurement system is shown in Fig. 5.4.

When the voltage required for measurement is connected across terminals a-b, a proportional current flows through resistor RV (47K). This current produces a magnetic flux, which in turn is proportional to the voltage. The magnetic flux is constantly controlled at zero by a compensating current flowing through the secondary coil using the Hall-Effect device and associated electronic circuit. The amount of secondary (compensating) current required to hold the zero flux is a measure of the primary current flowing multiplied by the ratio of the secondary winding. The secondary current multiplied by RM is the output voltage of the sensor. The signal conditioning part of the circuit is calibrated by adjusting the potentiometer P2 in such a way that the maximum voltage to be measured will give an output voltage signal of peak value ±10V. To protect the ADC input of the DS1102 board from over voltage, two zener diodes (BZX55-C9V1), with breakdown voltage of 9.1V, are connected back to back. The zener voltage of 9.1V is chosen so that when the back to back diode combination is conducting, the voltage drop will not exceed 10V (9.1V + forward biased voltage drop).

The output of the voltage measurement circuits are connected to ADC3 and ADC4 of the DS1102 board. ADC3 and ADC4 contain 12 bit successive approximation analog to

100 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING digital converters with integrated sample and hold units. Each converter achieves a conversion time of 3Ps.

Turns ratio 1:2.5 a RV +15V + + + 47K IM Sensed voltage Voltage to be LV 25-P M measured - RM - - -15V 100R b (a)

R6 P2 9K1 20K

C1 0.01PF +15V 13 _ 14 R7 To DS1102 From voltage R1 10 _ 4 sensor 8 12 DSP board 5K6 + 2K2 R2 9 + LT1058 +15V 5K6 11 LT1058 9.1VZ

-15V R3 10K 9.1VZ

Jumper P1 2K

R5 270R R4 10K

-15V

(b) Fig. 5.4 Voltage measurement system (a) voltage sensor (b) signal conditioning for the sensed voltage

The connection of resistor R5 to ground in the signal conditioning part of the voltage measurement system reduces the effect of offset error to a negligible value as compared to the measured signal of interest. The voltage sensor given here is good for measuring voltages above 10V. For input voltages below 10V there is more error because the accuracy is given as absolute error relative to the nominal current. The voltage sensor can go up to its rated insulation capability by increasing the value of the resistance RV.

101 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING

The combined resistance of R6 and potentiometer P2 in parallel with capacitor C1 provides the anti-aliasing filter with a cut off frequency fc = 1/(2SRC).

5.3.1.3 Current measurement The current transducer used in the current sensor circuit is an LTA 50P/SP1 produced by LEM Components. The current transducer LTA 50P/SP1, similar to the voltage transducer, employs Hall-Effect to measure DC, AC and impulse signals. The LTA 50P/SP1 is connected in its open-loop mode to give an output voltage proportional to the current under measurement. This arrangement gives a bandwidth from DC up to 25KHz. The principle of operation of the LTA 50P/SP1 is that the magnetic flux created by the primary current (current to be measured) is connected in a magnetic circuit and measured using the Hall-Effect device. The output voltage from the Hall-Effect device and associated electronic circuit provide an exact representation of the primary current. The output voltage is 100mV/Amp, i.e. 10A primary current gives 1V output voltage. The detail of the current measurement system is shown in Fig. 5.5.

In Fig 5.5(a) Vout is the output voltage signal of the current transducer and this has to be signal conditioned to be fed to the DS1102 board. The signal conditioning part of the circuit is calibrated by adjusting the potentiometer P2 in such a way that the maximum current to be measured will give an output voltage signal of peak value ±10V.

The output of the current measurement circuits are connected to ADC1 and ADC2 of the DS1102 board. ADC1 and ADC2 contain 16 bit successive approximation analog to digital converters with integrated sample and hold units. Each converter achieves a conversion time of 10Ps.

The purpose of the resistor R5, capacitor C1, zener diode and other resistors is similar to that of the voltage measurement set up described in the previous Section.

102 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING

1 2 -15V

4 Vout 5 6 +15V

Current to be measured (primary current)

(a)

R6 P2 9K1 20K

C1 0.01PF +15V 13 _ 14 R7 To DS1102 From current R1 10 _ 4 sensor 8 12 DSP board 5K6 + 2K2 R2 9 + LT1058 +15V 5K6 11 LT1058 9.1VZ

-15V R3 10K 9.1VZ

Jumper P1 2K

R5 270R R4 10K

-15V

(b) Fig. 5.5 Current measurement system (a) current transducer (b) signal conditioning for the sensed current in terms of voltage signal

5.4 Speed and angle measurement Speed and angular position can be measured using different types of sensors. An optical incremental encoder is the most frequently used. Incremental measurement means measurement by counting. That is, the output signals of incremental rotary encoders are supplied to an electronic counter in which the measured value is derived by counting individual “increments”. The information for speed and angle calculation is mainly determined by counting the encoder pulses during a given sampling interval.

103 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING

In the experimental set up, the position measuring system is based on a ROD 426 incremental rotary encoder produced by Heidenhain GmbH with 5,000 lines (or increments) per revolution [4]. The ROD 426 operates on the principle of photo electrically scanning very fine gratings forming an incremental track. Output signals are incremental with TTL square wave pulse trains Ua1, Ua2, and their inverted pulse trains

Ua1 and Ua2 produced by four-field scanning. A reference signal of a single square wave pulse Ua0 is produced every revolution and there is a Ua0 , which is the inverted o pulse of Ua0. Ua2 lags Ua1 by 90 (1/4 of a cycle) with clockwise rotation (viewed from shaft side) as shown in Fig. 5.6. One signal cycle of 360o electrical corresponds to the angle of rotation for one pitch of the radial grating (i.e. one line and one space).

Signal period 360oelec.

Ua1

90oelec. phase shift

Ua2

edge separation a 90oelec.

Ua0

td td

Fig. 5.6 Output signals of and incremental angle encoder

The output Ua2 and Ua1 pulses produce four states represented by 2 bits 00, 01, 11 and 10 for each line space of the encoder, as shown in Fig. 5.6. Hence the minimum resolution of the increment encoder is one fourth of the grating pitch of the radial grating. The value of the edge separation, a, is greater than or equal to 0.45Ps at the maximum scanning frequency 300 KHz. The lag time of pulse Ua0 to signals Ua1 and

Ua2 is less than 50ns. In this application with a sampling time around 200Ps, equivalent

104 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING to a scanning frequency of about 5KHz, the requirements for edge separation is satisfied and the effect of lag time is negligible.

5.4.1 Angle measurement The 5,000 line count per revolution of the ROD 426 encoder with the four states per line space gives 20,000 measuring steps (counts) per revolution. Then the value of the minimum angle measurement step is 360o/20,000 = 0.018o. Hence the angle resolution is 0.018o or as a fraction of a revolution is 5u10-5 revolution.

The DS1102 DSP card contains two incremental sensor interfaces (Channel 1 and Channel 2) to support optical incremental sensors commonly used in position control. Each interface contains line receivers for the input signals, a digital noise pulse filter eliminating spikes on the phase lines, a quadrature decoder which converts the sensor’s phase information to count-up and count-down pulses, a 24-bit counter which holds the current position of the sensor and a 24-bit output latch [1]. Using a C program the current value of the position counter is read from the output latch. Fig. 5.7 shows a block diagram for an incremental sensor interface.

25MHz RESET STROBE

Ua1 Phi0 1 24-bit 24-bit Line Noise Quadrature position output Ua2 Phi90 1 receiver filter decoder counter latch

Ua0 index Line receiver

Fig. 5.7 Block diagram of an incremental encoder interface

In the experimental set up Channel 1 incremental encoder position counter register address 024000H of the DS1102 is used. The incremental position can be determined from the 24 bit position counter that counts up when Ua1 is the leading sequence and

105 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING

counts down when Ua2 is the leading sequence. When digitized, both edges of Ua1 and

Ua2 are counted, thus the incremental position Iincr in radians is given by 2S I A I (5.1) incr 4Nu incr o

where Aincr - incremental count of the position counter in incremental steps N - the line count of the encoder (here it is 5000 lines)

Io - the initial position

One incremental step is equivalent to a 90° electrical phase shift of the signals Ua1 and

Ua2.

To express the incremental position between 0 and 2S, when the angle exceeds 2S it resets to 0. To have this option the angle measurement can be implemented using 2S II AA  (5.2) incr incr 4Nu incr__ new incr old

where Aincr_new and Aincr_old are the new and old incremental count of the position counter respectively. At start Iincr is equal to Io.

A function read_inc(1) is derived from the original dSPACE function ds1102_inc from the DS1102 software environment to read Channel 1 of the 24 bit position counter register and return the output as a 32 bit left aligned data word. The left alignment is done by shifting the 24 bits by 8 bits towards the left or it is multiplied by 28 = 256. In the DS1102 software environment the angle measurement is implemented as 2S II  AA (5.3) incr incr4uu N 256 incr__ new incr old

where Aincr_new and Aincr_old are the new and old values of the function read_inc(Channel1) incremental count of the position counter respectively. The 256 is included to cancel the multiplication factor of 256 introduced due to the conversion from 24 bit to the left aligned 32 bit.

106 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING

5.4.2 Speed measurement The speed information is determined by counting the number of pulses, which represent an increment in angular position, during a specified sampling period. The 5000 line encoder gives 20000 measuring steps per revolution. In general the speed resolution is expressed as

Tres Zres (5.4) Ts

-5 where Tres - incremental angular position resolution equal to 5u10 of a revolution = 3.142u10-4radians

Ts - sampling period (seconds) Hence the speed resolution expressed in revolutions per minute (rpm) is

5u 105 0. 003 Zres u60 rpm (5.6) TTs s

With a sampling time of 200Psec (Ts = 200Ps) the speed resolution will be 15rpm. This means that the speed measurement is expressed as a discrete value in 15rpm steps. The maximum error, irrespective of the speed, will be 15rpm. At high speed the relative error in speed measurement is small however at low speed the relative error is large. If the sampling time is 500Ps then the speed resolution is 6rpm. For a sampling time of 1ms the speed resolution is 3rpm.

The accuracy of speed measurement is dependent on the sampling time. The sampling time has to be large enough so that the accuracy of the speed measurement will be improved. However the sampling time should not be increased to an extent where it will not be able to follow the change in speed quickly and affect a control system dependent on rotor speed.

A function ds1102_inc(1) updates the Channel 1 incremental encoder counter output register and returns the position counter value. The 24-bit position counter value is scaled to a floating point value in the range -1.0 to 1.0 by the factor 2-31 because the data word is a 24 bit signed integer left aligned within the 32 bit data word [5]. The speed is calculated as

107 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING

count count speed __new old uk (5.7) Ts where k is the multiplication factor, k = 25165.824. count_new and count_old are the new and old value of the function ds1102_inc(Channel1) respectively. The value of count_new and count_old is between -1.0 and 1.0 floating point value.

The multiplying factor k is derived from the way the counted value is stored in the register and returned from the function ds1102_inc(Channel1) to the software environment. Based on the encoder used 1rpm = 333.333counts/sec. The 333.333 is stored in the 24-bit signed integer left aligned position counter register. When the 24-bit is converted to a left aligned 32-bit word, it has effectively been multiplied by 256 or 28. Finally the function ds1102_inc(Channel1) returns the counted value in 32-bit form divided by 231 to have a scaling in the range -1.0 … +1.0 floating point value so that 1rpm will be returned from the function as 3.974u10-5/sec. To express the speed measurement in rpm the incremental counted value per second should be multiplied by a factor k calculated as 60 231 k u 25165. 824 (5.8) 20000 256

5.5 Digital Signal Processing The currents and voltages used in the three axes to two axes transformation should be a sinusoidal waveform. Due to the resolution in the incremental encoder the speed is filtered. The filters used to smooth the currents and voltages are designed using digital signal processing tools. The filters used are first order and second order low pass filters. A high pass filter is also used to remove offset from the signal. For band pass filtering the combination of the low pass filter and high pass filter is used.

5.5.1 Digital filter Digital filters are applied to sampled data in the discrete time domain. There are two types of digital filters: infinite impulse response (IIR) filters which are recursive and finite impulse response (FIR) filters which are non recursive. For the recursive type there is at least one feed back path, however for the non recursive type there is no feed back path. Either type of filter, in its basic form, can be represented by its impulse

108 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING response sequence, h(k) (where k = 0, 1, 2, …). The input and output signals to the filter are related by the convolution sum, which is given in Equation (5.9) for the IIR and Equation (5.10) for the FIR filter [2, 6].

f yn() ¦ hkxn ()( k ) (5.9) k0

M1 yn() ¦ hkxn ()( k ) (5.10) k0 From Equations (5.9) and Equations (5.10) it is clear that for the IIR filter the impulse response is of infinite duration, whereas for the FIR it has finite duration.

5.5.1.1 Infinite Impulse Response (IIR) filter An IIR filter can not be realized using Equation (5.9) because of the theoretical infinite length of its impulse response. Instead, the IIR filtering equation is expressed in a recursive form given by

f ML yn()()() hkxn k axn () k byn () k ¦¦¦ kk (5.11) k0 k0 k1 where x(n) is the input signal, y(n) is the output signal and the constants a0, a1, a2 … aM, b1, b2 … bL are filter coefficients.

The current output sample y(n), is a function of past outputs as well as present and past input samples. That is, an IIR filter is some sort of feedback system. The transfer function for the IIR filter consists of poles and zeroes and the generalized form is:

M azk Yz() ¦ k Hz() k0 (5.12) Xz() §·L 1bzk ¨¸ ¦ k ©¹k1

IIR filters are implemented in a recursive fashion. An important part of the IIR filter design process is to find suitable values for the coefficients ak and bk such that some aspect of the filter characteristic behaves in a desired manner. The transfer function of the IIR filter, H(z), given in Equation (5.10) can be factorised as K()()()zzzz" zz  HZ() 12 N (5.13) ()()()zpzp12" zp  L

109 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING

where z1, z2, …, zN are the zeros of H(z), and p1, p2, …, pL are the poles of H(z). Instability can occur if any of the magnitudes of the roots of the poles are greater than 1 (outside the unit circle in the z plane).

5.5.1.2 Finite Impulse Response (FIR) filter Finite Impulse Response (FIR) Filters are generally non recursive digital filters and they have a finite duration. From Equation (5.10) the current output sample y(n), is a function of present and past input samples. This means that the FIR filter appears as a moving average filter where the nth output is a weighted average of the most recent M inputs. The block diagram realization for FIR filters is relatively straight forward. For a FIR polynomial, the realization is simply a combination of multipliers and delay elements. A generalized version in block diagram format is presented in Fig. 5.8.

x(n) z-1 z-1 z-1

h(0) h(1) h(2) h(M-2) h(M-1)

¦ y(n)

Fig 5.8 Block diagram for FIR filter

Using Equation (5.10) the transfer function for FIR filters is expressed as Yz() M1 Hz() ¦ hkz ()k (5.14) Xz() k0 Upon expansion the equation yields M-1 poles at the origin, meaning that FIR filters are unconditionally stable as the poles cannot ever lie outside the unit circle in the z-plane [7]. Yz() 1 M1 Hz() hkz ()()M1 k (5.15) M1 ¦ Xz() z k0

In Equation (5.11) when bk = 0 the filter is of non recursive type and becomes an FIR filter where h(k) = ak, (k= 0,1, … ).

110 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING

5.5.1.3 Comparison of IIR and FIR filters The comparison of IIR and FIR filters is summarised as follows x FIR filters are realised non recursively and are always stable. The stability of IIR filters cannot always be guaranteed. Round off noise and quantisation errors are much less severe in FIR filters than in IIR filters. x FIR filters can have linear phase response however IIR filters have non linear phase response, especially at the band edges. x IIR filters require fewer coefficients than FIR filters for sharp cut off filters. Fewer coefficients mean less processing time and less storage requirement. x In IIR filters there is no direct relationship between complexity and the length of the impulse response, which is infinite by definition. Filters with high selectivity can be realised with relatively low complexity. However, in FIR filters complexity is proportional to the length of the impulse response. x Analog filters can be readily transferred into equivalent IIR digital filters meeting similar specifications, but FIR filters have no analog counter part. x IIR filters can be designed using design formulae. FIR filter design procedures are normally iterative procedures and design equations do not exist.

5.5.2 Digital filter design from analog filter In the literature there are many ways of converting analog filters to digital filters [2, 6, 8]. The Matlab software provides a function, c2dm, to change a transfer function from continuous time to discrete time. One of the ways of converting an analog filter (s- domain) to a digital filter (z-domain) is to use a binomial Bilinear Transformation.

A bilinear transformation is implemented by substituting 21 z1 2z 1 s (5.16) T1 z1 Tz 1 in the analog filter in the s-domain. With the bilinear transformation, the entire left half s-plane maps to the interior of the unit circle in the z-plane. Hence all stable analog filters will result in stable digital ones. Also, the bilinear transformation maps the entire imaginary axis in the s-plane onto the unit circle in the z-plane.

111 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING

It can be shown that the digital frequencies, (ZD), are related to frequencies in the s- plane, (ZA), by a mapping of the frequency in the s-plane to the frequency in the z- plane, that is,

2 1 §·ZAT ZD tan ¨¸ (5.17) T2©¹ where T is the sampling time. Correction for this frequency scale wrapping may be accomplished by redesigning (pre- warping) the critical frequencies of the desired transfer function G(s) before applying the bilinear z-transform. If the digital filter cut-off frequency is Zk then the equivalent analog filter cut-off frequency (Zke) is found using the relation (pre-warping) given by

2 1 §·ZkT Zke tan ¨¸ (5.18) T2©¹

5.5.3 Implementation of a digital filter by approximating analog filter circuits Continuous time filters can be converted to discrete time filters by approximating the derivative in a differential equation representing the input output relationship of a dv circuit. For example the derivative can be approximated by using a backward dt vn() vn ( 1 ) vn() vn ( 1 ) difference or ,where Ts is the sampling time. The +t Ts sampling rate is the time between two samples and its inverse is the sampling frequency.

A simple analog RC low pass filter is given in Fig. 5.9.

R

x C y

Fig. 5.9 Simple first order analog low pass filter

112 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING

In Fig. 5.9 y is the filtered output and x is the unfiltered input. The differential equation relating the input and output is dy RCyx (5.19) dt The Laplace transform, which is the analog transfer function in the frequency domain, is Ys() Z c (5.20) Xs() sZc 1 where Zc = is the cut-off frequency. RC

Using the backward Euler method the differential equation given in Equation (5.19) can be approximated by a finite difference equation given by yn() yn ( 1 ) RC y() n x () n (5.21) +t 1 substituting Zc = and Ts = 't and rearranging leads to RC 1 yn() yn ( 1 )Zc Txn () (5.22) 1TZcs Here y(n) is the filtered value and x(n) is the unfiltered value.

A second order low pass filter and high pass filter can be derived in a similar way as given in the first order low pass filter.

5.6 Summary The data acquisition system which is used for the measurement of voltages, currents, angle and speed with their sensors is explained. The outputs of the voltage and current measurements are calibrated so that the voltage signals of the sensor outputs will not exceed the ADC input rating of the DS1102 DSP board. The sensors for current and voltage are Hall-Effect devices. Anti-aliasing filters are introduced in the analog signals of the sensor outputs to prevent the high frequencies appearing as a low frequency when the analog signal is digitised in the A/D converter.

113 CHAPTER 5 DATA ACQUISITION and DIGITAL SIGNAL PROCESSING

Speed and angle measurements are taken using an optical incremental encoder. The resolution of angle and speed for a given encoder is derived. For a given encoder the resolution of the angle measurement is constant; however the resolution of speed measurement is dependent on the sampling period used. High sampling period (low sampling frequency) gives small resolution in speed (discrete steps) producing less error in the measurement of the speed.

The advantage of digital signal processing is discussed. Different types of filter design are presented. Digital filters are used in the simulation and experimental results presented in the following chapters.

5.7 References [1] DSPACE, “DS1102 User’s Guide - Document Version 2.0”, Paderborn, Germany. [2] Alan V. Oppenheim and Ronald W. Schafer, “Discrete time Signal Processing”, Prentice-Hall Inc., Upper Saddler River, N.J., 1999. [3] Paul A. Lynn and Wolfgang Fuerst, “Introductory Digital and Signal Processing with Computer Applications”, John Wiley and Sons, New York, 1989. [4] Heidenhain, “Rotary Encoders” Catalogue 1999. [5] DSPACE, “DSP-CITeco DS1102 Software Environment - Document Version 2.0”, Paderborn, Germany. [6] Emmanuel C. Ifeachor and Barrie W. Jervis, “Digital Signal Processing a Practical Approach”, Addison-Wesley Publishers Ltd., Wokingham, England, 1993. [7] John G. Proakis and Dimitris G. Manolakis, “Digital Signal Processing Principles, Algorithms, and Applications”, Prentice-Hall Inc., Upper Saddler River, N.J., 1996. [8] Charles L. Philips and Troy H. Nagle, “Digital Control System Analysis and Design, Prentice-Hall, Englewood Cliffs, N.J., 1995.

114 CHAPTER 6

PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

6.1 Introduction Machine modelling requires knowledge of the parameters of the machine. Whether the three-phase induction machine is modelled using the conventional equivalent circuit or dq method, the parameters of the machine are required. To have an accurate model of the machine, which represents all the characteristics of the physical machine, the parameters need to be determined accurately. An in depth analysis and simulation of an induction machine can be carried out only with accurate parameters that represent the actual machine. Consequently to accurately model a three-phase induction machine, accurate parameter values which represent the actual operating conditions being modelled should be known.

There are different ways to determine the parameters of an induction machine modelled by the conventional or steady state method. In this work the parameters are obtained by taking measurements of input voltage, current and power over a wide speed range [1, 2]. For a three-phase induction machine with variable rotor parameters or constant rotor parameters, the determination of parameters is dependent on phase voltage, phase current, phase power and rotor speed.

The parameter determination method is based on the well known equivalent circuit shown in Fig. 6.1. In this equivalent circuit the arrow through Xlr and Rr/s indicates that these two parameters may be treated as variables for the case where rotor parameter variations are taken into account

115 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

Rs jXls jXlr

Rm jXm Rr/s

Fig. 6.1 The per-phase equivalent circuit with shunt magnetising branch impedance represented in parallel

To convert the shunt magnetising branch impedance from parallel to series form

2 RXmm RM 22 (6.1) Rmm X

2 RXmm X M 22 (6.2) Rmm X

Where RM and XM are the resistive and the reactive equivalent components, respectively, of the shunt magnetising branch represented in series form.

And to revert, to the parallel branch parameters:

22 RMM X Rm (6.3) RM

22 RMM X X m (6.4) X M

The modified form of the per-phase equivalent circuit is given in Fig. 6.2. Here the shunt magnetising branch elements are connected in series.

Rs jXls jXlr

RM

Rr/s jXM

Fig. 6.2 Per-phase equivalent circuit with shunt magnetising branch impedance represented in series form

116 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

6.2 Open-circuit and short-circuit test Open-circuit and short-circuit tests are conducted to find the parameters of the induction machine. At each test all the parameter values were taken into consideration. Even when two parameters having a large ratio in their values were compared, the smaller value was not neglected. It should be noted that the rotor leakage reactance is referred to the stator frequency and from the usual assumption Xls = Xlr .

Rs is obtained from a DC measurement of stator resistance taking some consideration for skin effect. Alternatively a Ware test [3] can be used, where Rs is measured by removing the rotor and supplying the stator with AC voltage at 50Hz .The difference between the Ware test and the simple DC test can be about 5%. The details of the tests are given below.

6.2.1 Open-circuit test The open-circuit test is conducted by supplying rated voltage to the stator while driving the induction motor at its synchronous speed using an external prime mover. When the motor runs at synchronous speed the slip, s, will be zero and as a result the current flowing in the rotor becomes zero. Then for the open-circuit test, the conventional equivalent circuit model can be reduced to the one shown in Fig. 6.3.

Rs Xls

IO RM

VO XM

Fig. 6.3 Per-phase equivalent circuit of three-phase induction machine under no load test

With VO- the measured open-circuit phase voltage

IO- the measured open-circuit phase current

PO- the measured open-circuit three-phase power So at slip s = 0:

117 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

PO Total input resistance under open-circuit condition RO 2 (6.5) 3IO

VO Total input impedance under open-circuit condition ZO (6.6) IO

22 Total input reactance under open-circuit condition XZROOO  (6.7)

Then RMOs RR (6.8)

XXXMOls  (6.9)

6.2.2 Short-circuit test The short-circuit test (or locked rotor or standstill test) is conducted by blocking the motor using a locking mechanism or using another prime mover to hold the induction motor at zero speed. At standstill, rated current is supplied to the stator. When the speed of the rotor is zero, the slip will be unity. At this slip, the resistive value on the rotor side will be Rr, which is the referred rotor winding resistance. Fig. 6.4 shows the per- phase equivalent circuit for the short-circuit or standstill test condition.

Rs jXls jXlr

Ish RM

Vsh Rr jXM

Fig. 6.4 Per-phase equivalent circuit at standstill (short-circuit test)

At slip s = 1

Psh Total input resistance under short-circuit condition Rsh 2 (6.10) 3Ish

Vsh Total input impedance under short-circuit condition Zshs RjXhsh (6.11) Ish

22 Total input reactance under short-circuit condition XZRshshsh  (6.12)

118 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

Where Vsh - the measured short-circuit input phase voltage

Ish - the measured short-circuit input phase current

Psh - the measured short-circuit three-phase input power

6.2.3 Induction machine with constant rotor parameters For a wound rotor induction machine the rotor winding arrangement is similar to that of the stator winding. The rotor parameters for the wound rotor induction machine do not change with speed or slip.

From Equation (6.11) and Fig. 6.4 the exact expression for the short-circuit test is given as

()()RjXRjXMMrlr RjXRjXsh sh s ls (6.13) ()()RRrM jXX lrM 

Using the assumption that Xls = Xlr and substituting the expressions for RM and XM, Equations (6.8) and (6.9) respectively, into Equation (6.13) gives

(()()ROsRjXXRjX O  lsr  lr RjXRjXsh sh s lr (6.14) ()RRrOs R  jX O

Multiplying both sides of Equation (6.14) by ()RrOsRR  jX O and then simplify by equating the real and imaginary parts on both sides of the equation, gives an expression for the rotor winding resistance and the rotor leakage reactance.

From the real part, the expression for Rr is

22 RshORRRXXRRR shs shO  sO s2 XXX lsO ls Rr (6.15) RROsh and from the imaginary part the expression for X lr is

RshXXRXRXRRXXR O sh r sh O  sh s s O O r X lr (6.16) 2(RROs ) Simplifying Equation (6.16), gives

2 aX111lr bX lr c 0 (6.17)

Where aX1 Osh X

bXabc110 2(O  0 )

119 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

22 ccXdXXXa10 sh 0 O O sh 1

bRR0 Osh

cRR0 Os

dRR0 shs

Solving Equation (6.17) gives the solution for X lr or X ls as follows

2 rbbac11 4 11 X lr (6.18a) 2a1 Here there are two mathematical solutions for the rotor leakage reactance. To determine the realistic solution the terms can be partially substituted to obtain

2 RROshOs RR bac111 4 XXlr O  r (6.18b) 22 XXOsh a1

In Equation (6.18b), if the + sign is chosen the rotor leakage reactance, Xlr, will be more than the total input reactance during the Open-circuit test, XO. This is not a realistic solution. Hence the solution must be

2 bbac11 4 11 X lr (6.18c) 2a1

Minimising Equation (6.15) using the coefficients given in Equation (6.17) gives

2 dc00 XshO X2 X lrO X  X lr Rr (6.19) b0

Hence Rr is calculated by substituting the value of Xlr in Equation (6.19).

6.2.4 Induction machine with variable rotor parameters The rotor parameters of induction machines with a deep bar rotor or double-cage squirrel-cage rotor vary due to rotor current displacement effect, or skin effect [4]. The ideal machine for motoring application will have varying rotor resistance, large at standstill and decreasing as the speed rises. The referred rotor leakage reactance will increase as the speed rises. The explanation for the variation in rotor parameters is illustrated in Fig. 6.5.

120 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

Stator Mutual magnetic flux

Air gap

Rotor

Current

Rotor leakage magnetic flux Rotor bar (a) (b) (c)

Fig. 6.5 Current displacement with rotor speed a) zero speed b) intermediate speed c) close to synchronous speed

The variation of rotor resistance and rotor leakage reactance can be explained as follows. At zero rotor speed, slip equal to one, there is maximum relative speed between the rotor and the rotating magnetic flux set by the stator current. As a result, at zero rotor speed the frequency of the induced rotor current is the same as the stator excitation frequency. The leakage flux produced by the rotor current will pass through the iron below the bottom of the rotor bar, as shown in Fig. 6.5, because of high permeability of the iron core. Now imagine the bar to consist of an infinite number of layers of differential depth. The layers at the lower part of the bar will be linked by all rotor leakage flux produced by the rotor current flowing in all layers of the bar. However the layers at the upper part of the bar will be linked only by the flux produced due to the current flowing in these layers. As inductance is flux linkage per unit current, the leakage inductance at the bottom of the rotor bar is greater than that of the upper part of the rotor bar. With alternating current flowing in the bars this leakage inductance effect produces leakage reactance, which is higher at the bottom of the bar than at the top of the bar. This is significant when the frequency of the rotor current is the same as the frequency of the stator excitation current, i.e. at slip s = 1. The leakage reactance decreases along the bar from the bottom to the top. Hence at slip equal to one current will be displaced upwards and almost all the current will flow in the upper part of the bar, as shown in Fig. 6.5a. This phenomenon is basically the same as skin and proximity effect in any system of conductors with alternating current in them. Since the upper part

121 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE is close to the air gap the possibility of the flux produced by the rotor current crossing the air gap and linking the stator core to become part of the mutual flux increases. As the flux produced by the rotor current crossing the air gap to link the stator increases the leakage flux in the rotor will decrease. A decrease in rotor leakage flux per unit current decreases the rotor leakage inductance and thereby the rotor leakage reactance when referred to the stator side.

However the resistance will increase because the effective area utilized by the current decreases. Hence, referred to the stator, at high slip, the rotor resistance is high but the rotor leakage inductance is at its lowest value. The frequency of the rotor current decreases with an increase in rotor speed (a decrease in slip) because of the decrease in relative motion between the rotating magnetic field and the rotor bars. For this reason at a speed half way between standstill and synchronous speed, about s = 0.5, the cross- sectional area of the bar used by the current has increased, as shown in Fig. 6.5b. At a rotor speed close to synchronous speed, s = 0, the rotor current frequency is low and there is almost no rotor current displacement, as shown in Fig. 6.5c. Close to synchronous speed the magnitude of the induced current is low and this small current flows over almost all of the cross-sectional area of the rotor bar. The spread of the current over almost the whole cross-sectional area of the bar increases the effective cross-sectional area and consequently the resistance to the flow of current decreases. As a result the rotor resistance will have a lower value at higher speeds. In this condition the rotor flux linking only the rotor increases and the rotor leakage inductance also increases. A typical example of rotor parameter variation for an induction motor with double-cage rotor is given in Fig. 6.6 [5].

122 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

9

8 Rotor leakage reactance

7

6

5

4

3

2 Rotor resistance and reactance (ohm) Rotor resistance 1

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Slip Fig. 6.6 Rotor parameter variations with slip for deep bar induction machine

For an induction machine with variable rotor parameters, once the open-circuit test and short-circuit tests are carried out the rotor parameters at slip equal to one can be calculated in the same way as in the induction machine with constant rotor parameters. The variation of rotor parameters at different slip can be obtained by measuring the stator phase voltage (VS(S)), phase current (IS(S)) and the three-phase power (PS(S)) supplied to the induction machine at a given slip. At slip equal to one Xlr(1) = Xls. Fig. 6.7 shows the variable rotor parameters and associated variation in input quantities which are functions of slip.

Rs jXls jXlr(S)

IS (S ) RM RrS() V S (S ) jXM s

Fig.6.7 Per-phase equivalent circuit with variable rotor parameters

123 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

Xlr(S) and Rr(S) are the rotor leakage reactance and rotor resistance respectively, referred to the stator. They vary as functions of slip (rotor frequency) or rotor speed. Viewed from the stator input side

VSS() Total input impedance at any slip, s, Z SS() (6.20) I SS()

PSS() Total input resistance at any slip, s, R SS() 2 (6.21) 3I SS()

22 Total input reactance at any slip, s, XZRSS() sS () sS () (6.22)

Then using Fig. 6.7 R RjX§·rS() MM ¨¸ jX lr() S s RjXRX  ©¹ (6.23) SS() SS () s ls R §·rS() j XX ¨¸ RM  lr() S M ©¹s

§·R Multiplying both sides of Equation (6.23) by rS() j XX and then ¨¸ RM  lr() S M ©¹s equating the real parts leads to

RrS() RSS()RR s MXRlr() S X ls  X SS ()  X M M RR s  SS ()  XX M( SS ()  X ls s (6.24) and equating the imaginary parts gives

RrS() XSS() X ls X MXRXlr() S R SS ()  RR s M M X ls  X SS () M RR s  SS () s (6.25) Equations (6.24) and (6.25) can be written as

ARBXErS() lrS () (6.26)

CRrS() DX lrS () F (6.27)

Solving the above two equations simultaneously, the rotor resistance and the rotor leakage reactance at a given slip can be calculated from: BF ED R (6.28) rS() BC AD

124 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

CE AF X (6.29) lr() S BCAD R RR Where A SS() s M s

B XXls S() S  X M

XXX C SS() ls M s

B RRRSS() s M

ER MM Rs RXXSS()  X SS() ls

FR MM XXls S() S  X RRs S() S

Brown & Grantham[6] devised a method whereby the parameters are determined using a variable speed test, where a speed control method, such as Ward-Leonard, is used to vary the speed, and the values of voltage, current and power are measured at each value of speed. However, the test time can be quite long for this method so the temperature can change during the test, hence changing the resistance values. Also, because of the period of time spent at each speed to record the data, it is not possible to determine the parameters under rated conditions. The motor temperature would become excessive at high slips due to the increased current (up to ten times the full-load current at standstill) causing overheating and damage to the induction machine.

To avoid the temperature change during the test, the data can be collected rapidly using a DSP data acquisition system. The DSP data acquisition can be done so quickly that the required data can be collected as the induction machine runs up to speed without any variation in temperature and it allows the standstill test to be implemented at full voltage.

6.2.5 Results for DSP based parameter determination Variation of rotor parameters exists in squirrel-cage induction machines. Even in induction machines with a single-cage rotor the variation in rotor parameters is significant [1, 2]. The monitoring system used for the parameter determination is as shown in Fig. 6.8. The monitoring system for this part of the experimental setup is based

125 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE on a 603 PowerPC floating-point processor running at 250MHz with a TMS320F240 slave DSP. The testing and measurements described below were carried out in collaboration with D. McKinnon [2].

C B Induction Motor A

RRR Thermocouple

Analogue Anti-Aliasing Filters

Temperature display Digital Digital Filters Filters Incremental Encoder Incremental

3 3 VI 2 2

Digital Real Time Filters

Digital Signal Processing System

Fig. 6.8 Monitoring system for parameter determination

The parameter determination algorithm discussed in the previous section together with the above monitoring system were used for the parameter evaluation of a 3-phase, 7.5kW, 4-pole, 13.8A, 415V, 50Hz delta connected induction motor with a single-cage rotor and fully enclosed 215T frame. The speed was monitored by a very accurate incremental encoder with 10000 lines/revolution, together with a four-fold pulse multiplication mechanism i.e. 40000 pulses/revolution and a 24-bit counter at the incremental encoder input.

In any induction machine parameter determination technique the most important data to be measured are the ones at zero and synchronous speed, which for on-line systems, are the most difficult values to sample. The difficulty at zero speed is that when a machine

126 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE is switched on with full voltage supply, there exists a transient current, the peak value of which can be substantially higher than the steady state current. Furthermore, as soon as voltage is applied to the machine, the rotor will start to rotate. Consequently a separate test is normally needed to measure the necessary data at zero speed.

Both of the above problems can be overcome by simply rotating the motor in the reverse direction, reversing the phase sequence of the supply and then start sampling as soon as the machine reaches zero speed. It has also been established that a convenient method of reducing the severity of the speed reversal transients is to ramp up the voltage in the negative speed region such that the voltage is at rated value as the machine approaches zero speed.

Initially 50% of the rated voltage was applied while the induction machine was rotating in the reverse direction. The motor decelerated in the reverse direction to zero and then accelerated in the forward direction. All the transients are settled when the machine crosses zero speed. The measured voltage, current and power as a function of time is shown in Fig. 6.9 and Fig. 6.10 shows the input quantities as a function of speed.

220

200

180

160 Line Voltage (V) 140

120

100

80

60

40

20

0 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 Time (Sec)

(a)

127 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

30

25

Line Current (A) 20

15

10

5

0 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 Time (Sec)

(b)

4000

3500

3000

2500

Total Input Power (W) 2000

1500

1000

500

0

-500

-1000 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 Time (Sec)

(c) Fig. 6.9 Three-phase induction motor input quantities as a function of time (a) measured line voltage (b) measured line current (c) measured input power

128 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

200

180

160

Line Voltage(V) 140

120

100

80

60

40

20

0 0 300 600 900 1200 1500 Speed (rpm)

(a)

30

25

Line(A) Current 20

15

10

5

0 0 300 600 900 1200 1500 Speed (rpm)

(b)

129 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

4000

3500

3000

2500

Total Input Power (W) Power Input Total 2000

1500

1000

500

0

-500

-1000 0 300 600 900 1200 1500 Speed (rpm)

(c) Fig. 6.10 Three-phase induction motor input quantities as a function of speed (a) measured line voltage (b) measured line current (c) measured input power

The negative power close to synchronous speed clearly indicates that the speed actually passes through synchronous speed and consequently synchronous speed data can be evaluated from this test to calculate Xm and Rm. The value of power measured when the rotor speed crosses the synchronous speed is still in the transient condition. It is difficult to calculate the steady state parameters from this transient condition. However, the measured power when the rotor speed crosses the synchronous speed for the last time or when it just reaches the synchronous speed, gives parameters close to the actual values. Even though the values are not exact, this method of parameter determination does not suffer from several problems identified for the existing method. That is, there is no need to carry out a separate open-circuit and short-circuit tests, there is no need for a variable speed drive and the problem associated with temperature rise and voltage levels is eliminated due to the very much shorter period of time required for the test. Fig. 6.11 shows the variation of Xlr and Rr with rotor frequency for the test machine with single- cage rotor. The test was carried out at room temperature and the value of stator winding resistance measured was 1.8:. The small dip in power Fig. 6.10(c) is related to the 5th

130 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE harmonic torque [7] and it is suspected also that the natural resonance frequency of the mechanical system has contribution to this slight dip. 8

: ) 7 Rotor Leakage Reactance 6

5

4

3

2 Rotor Resistance 1 Rotor Resistance and Reactance ( and Reactance Resistance Rotor

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Slip Fig. 6.11 Variation of rotor parameters for machine single-cage rotor

In Fig. 6.11 it is important to notice that even for the single-cage type of rotor there is a variation of rotor parameters due to rotor current displacement effect i.e. the skin and/or proximity effect. The variation in rotor parameters cannot be identified by the conventional open-circuit and short-circuit tests.

At rated voltage, the free-running motor suffers under severe transient conditions and the steady-state model can no longer be applied. Therefore, in order to capitalize on the benefits of the fast data acquisition system and the parameter determination method, an inertial load was coupled to the shaft of the machine to provide damping of the transient conditions. Whilst this treatment diminishes the transient effect, it has the disadvantage of preventing the machine from accelerating up through synchronous speed, thus inhibiting the calculation of the effective iron loss resistance, Rm, and the magnetization reactance, Xm. Therefore, a synchronous machine was coupled to the induction motor’s shaft, providing both the inertial load and the means for driving the induction motor at synchronous speed. When the induction motor neared synchronous speed the synchronous motor was activated to draw the machine up to exactly synchronous speed. With this method the parameters at synchronous speed are obtained very accurately,

131 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE because there is no transient condition and the synchronous motor is locked at precisely synchronous speed.

Fig. 6.12 shows the results of this test method for various supply voltages at a constant motor temperature of 40oC. The run up to speed tests were conducted over an appropriate time period for the applied voltage, that is, a long duration (30 seconds) for low voltage and a short duration (5 seconds) for high voltage. 8 Rotor Leakage Reactance

: 7

6 lowest Voltage 5

200VL-L 4 250VL-L 300VL-L 350V 3 highest Voltage L-L

Rotor Resistance Reactance and Resistance Rotor 2 lowest Voltage

highest Voltage 1 Rotor Resistance

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Slip Fig. 6.12 Variation of rotor parameters with slip and supply line to line voltage

The set of results in Fig. 6.12 shows that both the rotor leakage reactance and rotor resistance decrease with increasing supply voltage at each slip. This can be attributed to saturation of the iron core. Due to saturation, for the same slip or rotor current frequency, if the voltage is increased the rotor leakage inductance (rotor leakage flux per unit rotor current) will decrease. The decrease in rotor leakage reactance with increased voltage is as expected due to saturation of the leakage paths. The rotor leakage inductance decreases and there will be less shifting of the rotor current towards the upper part of the rotor bars due to current displacement effect. When current flows in an increased effective cross-sectional area, the resistance of the current path drops and therefore rotor resistance decreases.

132 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

The induction machine can operate at different temperature values depending on the loading condition and surrounding air temperature. The temperature of the induction machine was varied to see the effect of temperature on the rotor parameters. The data acquisition system and the parameter determination algorithm were used to identify the effect of temperature variation on induction machine parameters. Because the test is so rapid the temperature remains almost the same even as the speed of the induction machine varies. It is not possible to achieve the result with the conventional way of manual data recording.

Fig. 6.13 shows the variation of Xlr and Rr with slip at 50% of the rated voltage. The test was carried out under the same operating conditions, but at two different temperatures: 40 and 75oC. A thermocouple is embedded in the stator winding for temperature measurement. When the temperature of the stator is changed the temperature of the rotor will also change. In this experiment the aim is to see the variation of rotor parameters with temperature. The rotor temperature will follow the stator temperature variation. If necessary the rotor temperature can be measured by inserting a thermocouple in the rotor when the induction machine stops and compare with the stator temperature. This will give the gradient of temperature between the stator core and rotor core and be able to determine the exact temperature of the rotor while the temperature sensor is in the stator.

From Fig. 6.13 it can be seen that, as expected, a rise in temperature causes a rise in resistance at each slip or rotor current frequency. This rise in resistance results in a lower induced current in the rotor and therefore causes a small rise in the leakage inductance because the rotor leakage flux linkage per unit of rotor current slightly increases. That is, since at each rotor current frequency the rotor flux linkage is almost constant due to saturation for a particular supply voltage, the leakage inductance must increase as current decreases. However, it is clear that temperature rise affects the rotor resistance significantly more than the rotor reactance.

133 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

8

40 oC 7 75 oC ) :

6 Rotor Leakage Reactance

5

4

3 Rotor Resistance and Reactance ( Reactance and Resistance Rotor

2

Rotor Resistance 1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Slip Fig. 6.13 Effect of temperature on rotor parameters

Fig. 6.14 shows the variation for magnetising reactance Xm and iron loss resistance Rm with supply voltage. These results were derived from measurements made as the induction machine was driven at synchronous speed (open-circuit test) by a synchronous machine and the supply voltage varied accordingly. Two techniques were used to confirm the synchronous speed test data. In the first technique discrete data is stored for 30 seconds for each temperature at each fixed voltage level. This method is referred to as the discrete method. The average of each quantity during this period is then used to evaluate Xm and Rm. The second technique varies the supply voltage continuously during the test, from 50V to rated voltage, for each temperature. This method is referred to as the continuous method. The duration of this test is 60 seconds for each temperature.

Fig. 6.14(a) and Fig. 6.14(b) show the comparison of the results from the test methods carried out at synchronous speed. The result shows that there is no difference between discretely varying and continuously varying the supply voltage. The results were taken

134 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE when the motor temperature was 95oC. A low pass filter is applied during the continuous method to aid in identifying Rm. It is clear that the two test methods are in excellent agreement.

Fig. 6.14(c) and Fig. 6.14(d) present the effects that variations in supply voltage and temperature have on the effective magnetizing reactance, and iron loss resistance respectively. Only the results from the discrete voltage method have been shown for clarity in these figures.

As shown in Fig 6.14 the magnetizing reactance rises to a peak value and then, due to saturation, decreases steadily as the voltage rises. The iron loss resistance curve exhibits the predicted linear rise with voltage up to close to the rated supply voltage. It is noted that the magnetizing reactance rises slightly with temperature, whereas the iron loss resistance appears to have no consistent relationship with temperature.

200 Discrete voltages 180 Continuous voltages

160 ) :

( 140 m X 120

100

80

60

40

Magnetizing reactance - reactance Magnetizing 20

0 0 50 100 150 200 250 300 350 400 450 Line voltage (V)

(a)

135 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

6000 Discrete voltages Continuous voltages

5000

) 4000 : ( m R 3000

2000

1000 Iron loss resistance – resistance loss Iron

0 0 50 100 150 200 250 300 350 400 450 Line voltage (V) (b)

200

) 180 : ( m

X 160

140

120

100

80 o Xm 40 C Magnetizing reactance - reactance Magnetizing 60 o Xm 55 C o Xm 65 C 40 o Xm 75 C X 95oC 20 m

0 0 50 100 150 200 250 300 350 400 450 Line voltage (c)

136 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

6000

5000 ) :

( 4000 m R

3000

o 2000 Xm 40 C o Xm 55 C o Xm 65 C o Iron lossresistance – 1000 Xm 75 C o Xm 95 C

0 0 50 100 150 200 250 300 350 400 450 Line Voltage (V) (d) Fig. 6.14 Variation of (a) magnetizing reactance with voltage at 95oC (b) iron loss resistance with voltage at 95oC (c) magnetizing reactance with temperature and voltage (d) iron loss resistance with temperature and voltage

6.3 Sensitivity study on variable rotor parameters To effectively model a three-phase induction motor the parameter values should be accurately known. These parameters are obtained from the measured values of voltage, current and power as discussed in Section 6.2. Any practical meter will have discrepancy between the actual value and the measured value. The accuracy of measurement influences the accuracy of the calculated parameters. Consequently any measurement error in obtaining the values of current, voltage and power would be directly reflected as an error in the calculated parameters. The extent of the error reflected as a deviation from the actual values of parameters was studied [8].

To achieve this analysis a hypothetical model of a three-phase induction motor was considered and a MATLAB program was used. Rotor parameter variations due to skin or proximity effect are taken into account. MATLAB functions can generate any data with a given standard deviation and percentage error of the measured sample values

137 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE randomly, which follow the normal curve in statistics. All the randomly generated data are above and below the true value within a predetermined range.

6.3.1 The effect of combining measurement errors The parameters of a three-phase induction machine can be determined using the most well known method that uses no load and locked rotor tests or the fast DSP based parameter determination discussed previously. The parameters can be calculated taking into consideration the effect of temperature, speed, or any factor that affects the values of the parameters and can be achieved in different ways. All the procedures used in calculating the parameters could be correct, but that does not show anything related to the measurement errors in the conventional meters and sensors used to measure voltage, current and power. There is a need to calculate the magnitude of error in the parameters for a given percentage of error, or range of error, in the measurement devices, i.e. a sensitivity study on the effect of the parameters due to measurement data error.

6.3.1.1 Percentage errors In practice, it is often useful to express an error in relation to the magnitude of the quantity being measured. Expressing error as a percentage is important because it helps predict the accuracy needed for individual measurements in order to achieve a given percentage error in the final result [9]. Manufacturers mostly give the accuracy of a given measuring device or sensor in percentage error.

The percentage error in a given measurement is given by mt e100 u (6.30) t where e - percentage error m - measured value t - true value

Here e is the size of the error in percent when the measured value of a quantity is compared with its true value.

138 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

Rearranging Equation (6.30), can give a measured quantity expressed in terms of the percentage error and the true value as follows. m1et  (6.31)

6.3.1.2 Combining errors The errors in two measured values are combined when the two measured values are added, subtracted, multiplied or divided. Assume that the measured values for two true quantities of a and b are am and bm respectively. And assume the measurement errors for the two measured quantities are ea and eb. Then the measured quantities can be expressed as

aaema r (6.32)

bbemb r (6.33) When the two measured quantities are added, subtracted, multiplied or divided the combined error in the result will be the sum of the individual errors [9].

6.3.2 Induction machine parameters for analysis of measurement error For the purpose of analysis and simulation a hypothetical three-phase induction machine with double-cage or deep-bar rotor is used. The values of the parameters were chosen to have similarity with a real motor in the laboratory. To have a wider operating region the parameters were selected for speeds starting from –1500rpm to 1500rpm, where 1500rpm is the synchronous speed of a 4-pole, 50Hz induction machine.

The hypothetical three-phase induction motor that was selected for this simulation has the following parameters:

X1s=2:

Rs=1.5:

Rm=800:

Xm=100: and the values of rotor leakage reactance and rotor resistance with respect to speed were as shown in Fig. 6.15 below.

139 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

6

Ohms 5

Rr 4

3

2 Xlr

1

0 -1500 -1000 -500 0 500 1000 1500 Speed(rpm)

Fig. 6.15 Values of rotor resistance, Rr, and rotor leakage reactance, Xlr

The values of Rr, and Xlr are dependent on speed, as shown in Fig. 6.15. All parameters in the per-phase equivalent circuit of the three-phase induction motor were substituted with the parameter values given above. The advantage of using a hypothetical machine for the sensitivity study is that the value of the parameters and parameter variations are precisely known. These exact values can then be compared precisely with the new values calculated as a result of the known error in the simulated measured data.

6.3.3 Statistical tools The main problem in experimentally investigating parameters with the aid of measuring systems is the properties of the systems on the measurement results. The source of measuring errors could be an internal problem of the meter or the influence of external disturbances. Whatever the cause of the disturbance or error in the measuring system a statistical tool is needed to analyse the measurement error.

The preferred statistical tool for the analysis of this measurement error is the normal (Gaussian) distributions. Normal distributions are continuous probability distributions.

140 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

They are frequently used as a population model because they provide a reasonable approximation to the distribution of many different variables [10]. It is an assumption that the distribution of error as measured from the true value follows the normal curve. According to an Empirical Rule [10] if the distribution of error can be reasonably well approximated by a normal curve, as shown in Fig. 6.16, then: x Approximately 68% of the measured data are within one standard deviation of the actual measured value x Approximately 95% of the measured data are within two standard deviations of the actual measured value x Approximately 99.7% of the measured data are within three standard deviations of the actual measured value.

error error

2V 2V

34% 34%

2.35% 2.35% 13.5% 1V 1V 13.5%

3V 1V 1V 1V Actual measurement Fig. 6.16 Measurement error with a normal distribution

Any deviation from the actual measurement (mean), P, is an error of measurement. The measurement error can be expressed in terms of standard deviation (V) or percentage of error. Using the normal curve approximately 99.7% of all the observations are within three standard deviations of the mean. If the actual value is 100 and the standard deviation is 0.333 then one standard deviation is 0.333 in the unit of measured value. Therefore, a value three standard deviations from the mean will give an error of 0.999 which is about 1%. To simulate 1.5% error for an actual value of 100 the standard deviation is 0.5 in the unit of measured value.

141 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

6.3.4 Simulation of parameter determination with measurement error The hypothetical induction motor with the equivalent circuit and parameter values specified above was considered when connected in a star configuration to a 50Hz supply with line to line voltage of 415V.

The speed was allowed to vary from –1500rpm to 1500rpm. The corresponding values of slip were 2 to 0. The synchronous speed was assumed to be 1500rpm (i.e. 4-pole). The simulation was carried out at a constant supply voltage and then finding the actual values of current and power for a given speed or slip. The given value of voltage and the calculated values of current and power from the equivalent circuit model of the induction machine represent the actual values of measurement. Any deviation from these actual values would signify an error of measurement in a practical application of the parameter determination technique. Parameters calculated from the actual values of voltage, current and power correspond to the actual parameters of the motor and any calculation with the introduction of some error in voltage, current and/or power gives the values of parameters with a given percentage of error from the actual values.

Random generated “measured” values using MATLAB functions that fit to a normal (Gaussian) distribution were used in the investigation. The measurement error was expressed in terms of percentage error. The standard deviation of the true (actual) measured value can be calculated from the percentage measurement error.

For the implementation of the measurement error analysis expressed in terms of the standard deviation of the true value, a MATLAB program was developed with the possibility of adjusting measurement error in the voltage, current, and/or power. The percentage of error was selected in such a way that it would represent the practical error, which could occur in a practical measurement setup. To simulate the different measurement errors that can occur in the conventional meters and sensors for measuring the values of current, voltage and power a MATLAB function given by normrnd(mean, standard deviation, N o of columns, N o of rows) was used to generated the hypothetical measured values with distribution as shown in Fig. 6.17.

142 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE Density

actual value

Hypotetical measured value

Fig. 6.17 Data generated for simulation of measurement error

In Fig. 6.17 since the values were generated randomly, at any point there is any possibility of minimum or maximum error to represent the error in voltage, current or power measurement. In general the possibility of measuring the precise value or a value with maximum error is a matter of probability. Of course the extent of the deviation from the true value is proportional to the percentage of measurement error or the standard deviation. At each speed there are 200 hypothetical measured values generated by the MATLAB program to represent the possibility of measurement error within a given percentage error of the conventional meter reading.

As shown in Fig. 6.18 to Fig. 6.20 the maximum error in the rotor parameters increases as the percentage measurement error in voltage, current and/or power increases. Close to synchronous speed, when the slip approaches zero, there is greater error, in the calculated values of Xlr, but reduced error in the calculated values of Rr.

143 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

8

7 Ohms

6 RRr2 (actual value) 5

4

3 Values with error

2

XXlr2 (actual value) 1

0 -1500 -1000 -500 0 500 1000 1500 Speed (rpm)

Fig. 6.18 Error in rotor parameters due to ±0.5% error in voltage current and/or power

8

7 Ohms 6 RRr2 (actual value) 5

4

3 Values with error

2

XXlr2 (actual value) 1

0 -1500 -1000 -500 0 500 1000 1500 Speed (rpm)

Fig. 6.19 Error in rotor parameters due to ±1% error in voltage current and/or power

144 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

8

7 Ohms

6 RRr2 (actual value) 5

4

3 Values with error

2

XXlr2 (actual value) 1

0 -1500 -1000 -500 0 500 1000 1500 Speed (rpm)

Fig. 6.20 Error in rotor parameters due to ±1.5% error in voltage current and/or power

The reason for the error close to synchronous speed is the magnitude of Xlr is very small compared to Rr /s. The result shows that induction motor performance is very insensitive to large errors in rotor leakage reactance close to synchronous speed.

From the result of the simulation, i.e. for 1% measurement error, the maximum error in

Xls is 6.4%, in Xm 1.5% and in Rm 2.2%. For the rotor parameters, as can be seen in

Fig.19, the maximum error in the motoring region for Rlr is 3.3% and 11.4% for that of

Xlr. Except near to synchronous speed, the absolute error of all values of Xlr and Rr is almost the same.

The developed shaft torque for the hypothetical induction motor discussed above, using constant rotor parameters and variable rotor parameters, is shown in Fig. 6.21. When the rotor parameter variations are ignored there is a maximum error of 61%. Similar percentage error is obtained for current and power [1, 2]. These differences in the calculated results are enormous when compared with the errors introduced by the inaccuracy in measurement. This vividly illustrates the need to take rotor parameter

145 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE variations into account for squirrel-cage machines which exhibit any significant degree of rotor current displacement effect.

120

For variable Rr and Xlr 100 Torque (Nm) 80

Assuming constant Rr and Xlr 60

40

20

0 -1500 -1000 -500 0 500 1000 1500 Speed (rpm)

Fig. 6.21 Simulated shaft torque for variable and constant rotor parameters

6.4 Summary The results of an investigation into the variation of magnetizing reactance, iron loss resistance and rotor parameters with temperature and supply voltage have been presented. A monitoring system employing digital signal processing techniques was used to observe these effects.

To date parameter identification methods over a large speed range have typically required reduced voltages to prevent the motor overheating from excessive current at high slips. The run up to speed test described enables the parameters to be determined at supply voltages up to and including the rated voltage without damaging the motor under test. A separate test was used to determine the iron loss resistance and magnetizing reactance. This test incorporated two methods. Both methods were conducted at synchronous speed, one using continuously varied supply voltage, the other at discrete voltages. Agreement between the results was excellent. All tests were performed at corresponding temperatures.

146 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

Parameter identification is especially important in the design of electrical drive systems. It is obvious from these results that parameter variations cannot be ignored. It has been shown that rotor parameter variations exist even for a single-cage induction motor.

The error in the values of induction motor parameters arising from measurement error in voltage, current and power used to determine the parameters have been presented. These three measurement quantities are essential for accurate parameter determination. Even with measurement errors of r1.5% the determined parameters are still very acceptable and considerably more accurate than if rotor parameter variations are ignored. Only very close to synchronous speed is there substantial error in the value of

Xlr determined, but this is almost irrelevant in any meaningful analysis of the inductive machine, because close to synchronous speed the effect of Xlr is swamped by the effect of Rr /s. When rotor parameter variations are ignored the percentage errors in the current, power and torque are substantial for machines which exhibit a significant degree of rotor current displacement effect.

6.5 References [1] C. Grantham and H. Tabatabaei-Yazdi, "Rapid Parameter Determination for use in the Control High Performance Induction Motor Drives" IEEE 1999 International Conference on Power Electronics and Drive Systems, PEDS’99, July 1999, Hong Kong, pp. 267-272. [2] D. McKinnon, D. Seyoum and C. Grantham “Investigation of the effects of supply voltage and temperature on parameters in a 3-phase induction motor including iron loss”, Proc. AUPEC’02, Melbourne, Sep29-Oct 2, 2002, ISBN0-7326-2206-9. [3] D. H. Ware, “Measurement of stray load losses in induction machines”, Trans., AIEE, 1945, pp. 194-196. [4] P. L Alger, “The nature of Induction Machines”, Gordon and Breach Inc., New York, 1965, pp. 111. [5] C. Grantham and H. Tabatabaei_Yazdi, "Rapid Parameter Determination for use in the Control of High Performance Induction Motor Drives”, Proc. AUPEC 2001, Perth, 23 - 26 September. 2001, pp. 31 - 36. [6] J. E. Brown and C. Grantham, "Determination of the parameters and parameter variations of a 3- phase induction motor having a current displacement rotor", Sept.1975, Proc. IEE, 122, No. 9, pp. 919-921. [7] G. R. Slemon, “Electric Machines and Drives”, Addison-Wesley Publishing Inc., Ontario, 1992.

147 CHAPTER 6 PARAMETER DETERMINATION FOR AN INDUCTION MACHINE

[8] C. Grantham, D. Seyoum D. Indyk and D. McKinnon, “Calculation of the Parameters and Parameter Variations of an Induction motor and the effect of measurement error”, Proc. AUPEC’00, Brisbane, Australia, 2000, pp. 225-228. [9] Donald T. Graham, “Principles of radiological Physics”, Edinburgh, UK, 1996, pp. 86-88. [10] J. Devore and R. Peck, “STATISTICS The Exploration and Analysis of Data”, Duxbury press, Belmont 1997, pp. 187 - 232.

148 CHAPTER 7

EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

7.1 Introduction Any induction machine requires excitation current to magnetise the core and produce a rotating magnetic field. The excitation current for an induction generator connected to an external source, such as the grid, is supplied from that external source. If this induction generator is driven by a prime mover above the synchronous speed, electrical power will be generated and supplied to the external source. An isolated induction generator without any excitation will not generate voltage and will not be able to supply electric power irrespective of the rotor speed.

In general an induction generator requires reactive power for its operation. Three charged capacitors connected to the stator terminals of the induction generator can supply the reactive power required by the induction generator. Provided that the conditions for self-excitation are satisfied the charged capacitors cause the terminal voltage to build up at the stator terminals of the induction generator. When the charged capacitors are connected to the terminals a transient exciting current will flow and produce a magnetic flux. This magnetic flux will generate voltage and the generated voltage will be able to build the charge in the capacitors. As the charge increases, more exciting current is supplied to the induction generator. The magnetic flux continues to increase hence producing a higher generated voltage. In this way voltage is built up.

However, if the capacitors are not charged, and a remnant magnetic flux in the core exists, then a small voltage will be generated at the terminals of the induction generator due to that remnant flux. This small voltage will charge the capacitor. The charged

149 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS capacitor will now be able to produce a small exciting current. With time the exciting current grows and produces magnetic flux more than the remnant magnetic flux and voltage will be built up. This is similar to the way that current and voltage interact in a resonant circuit. For the voltage to build up across the terminals of the induction generator, there are certain requirements for minimum rotor speed and capacitance value that must be met. When capacitors are connected across the stator terminals of an induction machine, driven by an external prime mover, voltage will be induced at its terminals. The induced emf and current in the stator windings will continue to rise until steady state is attained. At this operating point the voltage and current will continue to oscillate at a given peak value and frequency. The rise of the voltage and current is influenced by the magnetic saturation of the machine. In order for self-excitation to occur with a particular capacitance value there is a corresponding minimum speed.

Self-excited induction generators are good candidates for wind powered , especially in remote areas, because they do not need an external power supply to produce the excitation magnetic field. Permanent magnet generators can also be used for wind energy applications; however the generated voltage increases linearly with wind turbine speed. An induction generator can cope with a small increase in speed from its rated value because, due to saturation, the rate of increase of generated voltage is not linear with speed. Furthermore when there is a short circuit at the terminals of the self-excited induction generator (SEIG) the voltage collapses providing a self-protection mechanism. Additional advantages of SEIGs include lower cost, reduced maintenance, they are rugged with simple construction, and they have a brush- less rotor (squirrel cage). Fig. 7.1 shows the SEIG driven by a wind turbine.

Rotor blade

Gear box Wind direction

C C C

Induction generator

Fig. 7.1 SEIG with a capacitor excitation system driven by a wind turbine

150 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

7.2 Model of self-excited induction generator Basically the model of a SEIG is similar to an induction motor. The only difference is that the self-excited induction generator has capacitors connected across the stator terminals for excitation. The conventional steady state per-phase equivalent circuit representation of an induction machine is convenient to use for steady state analysis. However, the d-q representation is used to model the self-excited induction generator under dynamic conditions. The d-q representation of a self-excited induction generator with capacitors connected at the terminals of the stator windings and without any electrical input from the rotor side is shown in Fig. 7.2 below.

The representation shown in Fig. 7.2 can be redrawn in detail, in a stationary stator reference frame, with direct and quadrature circuits separately represented as given in Fig. 7.3. The capacitance is labelled C in Fig. 7.3.

The capacitor voltages in Fig. 7.3 can be represented as 1 VidtV  (7.1) cqC ³ qs cqo

1 VidtV  (7.2) cdC ³ ds cdo

Where VV and VV are the initial voltage along the q-axis and d-axis cqo cq t0 cdo cd t0 capacitors, respectively.

With Ls = Lls+Lm and Lr = Llr+Lm the rotor flux linkage is given by

OOqr Li m qs Li r qr qro (7.3)

OOdr Li m ds Li r dr dro (7.4)

Where OO and OO are the remnant or residual rotor flux linkages qro qr t0 dro dr t0 along the q-axis and d-axis, respectively.

Then, with an electrical rotor speed of Zr, the rotational voltage in the rotor circuit along the q-axis is

ZOrdr Z r Li mds Li rdrZO rdro

ZOr dr Z r Li m ds Li r dr K qr (7.5)

151 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

Q-axis

iqs

Zr iqr

Q-axis idr ids

Fig. 7.2 D-Q representation of self-excited induction generator

Zr Odr S Rs Lls Llr R r + -

iqs iqr C imq Vcq Lm Oqs Oqr

(a)

-Zr Oqr S Rs Lls Llr R r + -

ids idr C imd Vcd Lm Ods Odr

(b) Fig. 7.3 Detailed d-q model of SEIG in stationary reference frame (a) q-axis circuit (b) d-axis circuit

152 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS and the rotational voltage in the d axis of the rotor circuit is

ZOrqr Z r Li mqs Li rqrZO rqro

ZOr qr Z r Li m qs Li r qr K dr (7.6) where Kdr ZO r qro and Kqr ZO r dro are constants, which represent the initial induced voltages along the d-axis and q-axis, respectively. The constants Kdr and Kqr are due to the remnant or residual magnetic flux in the core. And Zr is the equivalent electrical rotor speed in radians per second. That is, Electrical speed = number of pole pairs u mechanical speed

The matrix equation for the d-q model of a self-excited induction generator, in the stationary stator reference frame, using the SEIG model given in Fig. 7.3 and from Equations (7.5) and (7.6), is given as:

iV ªº0 ªºRpLpCss10 pL m 0ª qsºª cqo º « »« » «»0 «»010RpLpC pLi V «» «»ss mdscdo« »« » (7.7) « »« » «»0 «»pLmrZZ Lmrrr R pL Lrq ir Kqr «» «»« »« » 0 ZZLpLLRpL ¬¼ «»¬¼rm m rr r r ¬«iKdr ¼¬»«dr ¼»

Z IV VV

Where Z is the impedance matrix, IV is the stator and rotor currents vector and VV is the voltage vector due to initial conditions.

7.3 Analysis of self-excitation process Whether it is a wound rotor induction machine or a squirrel cage induction machine voltage will develop across the capacitors connected to the stator terminals when the rotor of the induction machine is driven by an external prime mover. The voltage developed across the capacitors is the terminal voltage of the self-excited induction generator.

153 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

7.3.1 RLC circuit characteristics The behaviour and analysis of the self-excited induction generator is similar to an RLC circuit. Since the equations in the induction generator are complex, the principle of self- excitation process will be explained first using a simple RLC circuit. For analysis purpose the RLC circuit given in Fig. 7.4 will be considered. The plus sign at the capacitor is the polarity for the initial capacitor voltage.

RLS

i(t) + C Vco

Fig. 7.4 RLC circuit

Energy may be stored in an inductor or in a capacitor. A resistor is incapable of energy storage. S in Fig. 7.4 is closed at t = 0. In general the two initial conditions at t = 0- are: current might have been flowing in the inductor (provided that the inductor was part of another circuit, not shown in Fig. 7.4, before t = 0-) or initial voltage exists in the capacitor. If all initial conditions are zero then there will not be any transient or steady state current flow.

In Fig. 7.4 assume that at the instant the switch is closed, the current is zero and the voltage across the capacitor is vc = -Vco. When the switch is closed, the voltage equation in the RLC circuit is given by di 1 Ri L idt  V 0 (7.8) dt C ³ co Introducing the p operator for d/dt Equation (7.8) can be rewritten as §·1 ¨¸R pL i() t Vco (7.9) ©¹pC Then i(t) can be expressed as pV it() co (7.10) pL2  pR1 C If the denominator of Equation (7.10) is equated to zero, then pL2  pR1 C 0 (7.11)

154 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

Equation 7.11 is called the characteristic equation because it contains the information about the behaviour of the resulting current. The roots of the characteristic equation are

2 2 RR§· 1 RR§· 1 p1  ¨¸  and p2  ¨¸  (7.12) 22L ©¹LLC 22L ©¹LLC

Using the roots given in Equation (7.12) the complete solution for the current expression in Equation (7.10) is

pt1 p2 t it() Ae12 Ae (7.13) where A1 and A2 are determined from the initial conditions and circuit parameters, and p1 and p2 are determined from the values of the circuit parameters R, L, and C. If the voltage across the capacitor vc(t) is the output voltage of interest then 1 vt() idtV. ccC ³ o

2 §·R 1 In Equation (7.12), if ¨¸ then the roots p1 and p2 of Equation (7.13) are ©¹2L LC complex quantities which can be expressed as p1=V +jZ and p2=V -jZ. Relating these expressions with Equation (7.12), the real part of the roots, V, is always negative provided the resistance R is positive. As a result with positive R there will be a decaying oscillation. V represents the rate at which the transient decays and Z, the imaginary part of the roots, represents the frequency of oscillation. In passive circuits, like the RLC circuit mentioned above, all transient solutions have negative V, meaning that the transient is reducing in magnitude with the progression of time and finally decays to zero. However, if V is positive, this implies that the transient is growing with the progression of time, and in theory would increase to infinity. V can be positive only if the resistance R is negative. Negative resistance implies a power source whereas positive resistance implies a power sink. Fig. 7.5 shows the current in the RLC circuit when L = 0.1H, C = 100PF, Vco = -10V and the magnitude of resistance R is 1.2: with positive value in Fig. 7.5(a) and negative value in Fig. 7.5(b). Close to t = 0 the magnitude of the instantaneous current flowing in the RLC circuit in both cases is the same.

155 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

0.3

detail of A 0

-0.3

0 0.05 0.1

0.3 150

0.2 100

0.1 50 Current (A) Current (A) 0 A 0 -0.1

-50 -0.2

-0.3 -100

-0.4 -150 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 time (sec) time (sec) (a) (b) Fig. 7.5 Current in series RLC circuit (a) for R = 1.2: and (b) for R = -1.2:

Transients which grow in magnitude as shown in Fig. 7.5(b), with a positive value of V, are very rare. There is no variation in any of the values of R, L or C and as a result the current keeps on growing. Any current flowing in a circuit dissipates power in the circuit resistance. If there is an increasing current that dissipates increasing power, there must be some energy source available to supply the increasing power. This is in fact the case in the self-excited induction generator. The example above of a very rare transient is characteristic of a SEIG where the power source is a prime mover.

7.3.2 Conditions for self-excitation in induction generator Basically an induction machine is modelled using RLC circuit parameters. Self- excitation in an induction generator is the growth of current and the associated increase in the voltage across the capacitor without an external excitation system. As in Fig. 7.5(b), transients that grow in magnitude (self-excitation), with a positive real part of the root, can only happen if there is an external energy source that is able to supply all the power losses associated with the increasing current. The self-excited induction generator is able to have a growing transient because of the external source that is driving the induction generator. The process of terminal voltage build up continues in the manner described until the iron circuit saturates and the voltage therefore stabilises. In terms of the transient solution considered above, the effect of this

156 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

saturation is to modify the magnetisation inductance Lm, such that the real part of the roots becomes zero; the transient then neither increases nor decreases and becomes a steady-state quantity giving a continuous self-excitation.

The energy source, referred to above, which is necessary for this type of unusual transient to occur, is provided by the kinetic energy (KE) of the rotor. If the rotor is driven by an external prime mover, the KE of the rotor is maintained and self-excitation and energy transfer continues permanently. The initiation of the process of self- excitation is therefore a transient phenomenon and is better understood if analysed using instantaneous values of currents and voltages.

Unlike the simple RLC circuit that has been discussed, the roots for the self-excited induction generator which can be derived from Equation (7.7), are dependent on the induction machine parameters, the capacitor connected at the stator terminals of the induction generator and the rotor speed when coupled to an external prime mover. Determination of the roots of the characteristic equation of the currents in the induction generator is the key to finding out whether the induction generator will self-excite or not.

Equation (7.7) can be re written as

ªºRpLpCss10 pLm 0ªºªiVqs  cqo º «»« » «»010RpLpC pLi  V «»s sm«»«ds cdo » (7.14) «»« » «»pLmrZZ Lmrrr R pL Lrq ir Kqr «»«»« » ZZLpLLRpL «»¬¼rm m rr r r ¬¼¬«»«iKdr  dr ¼»

The characteristic equation of the currents can be obtained from the expression for the current vector in Equation (7.14). This characteristic equation for the currents can be solved using a matrix partitioning, which gives a compact polynomial expression or a direct matrix inversion, which produces an expression with a lower order polynomial characteristic equation.

157 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

7.3.2.1 Using matrix partition Equation (7.14) can be rewritten as

ªºªZsssrZVªºIs osº «»««» » (7.15) ¬¼¬ZrsZV rr¬¼Ir or ¼ Where

ªºVcqo Vos «», initial conditions on the stator side, ¬¼Vcdo

ªºKqr Vor «», initial conditions on the rotor side ¬¼Kdr

ªºiqs ªºiqr Is «», and Ir «», ¬¼ids ¬¼idr

ªºRpLpCss10 ªºpLm 0 Zss «», Z sr «», ¬¼01RsspL pC ¬¼0 pLm

ªºpLLmrZ m ªRpLrrZ rr Lº Zrs «», Zrr « » ¬¼ZrmLpL m ¬ ZrrLRp r L¼ From Equation (7.15)

VZIZIos ss s sr r (7.16)

VZIZIor rs s rr r (7.17) From Equation (7.17)

11 Irrrorrrrss ZV ZZI (7.18)

1 ªºR  pLZ L Where Z 1 rrrr rr 222«» ()RpLrrZ rr L¬¼ZrrLRpL r Substituting Equation (7.18) in Equation (7.16)

1 1 VZZVZZZZIos sr rr or ( ss sr rr rs) s (7.19) Equation (7.19) can be rewritten as VK ªºcqo pLm ªºRpLLrrrr Z ªºqr «»«» «»LRpL ¬¼VKcdo ' ¬¼Zrr r r ¬¼dr (7.20) ªºR pL1() pC p22 L R pL pL2 2 L pL2 R ss  mrr   mrrZZ ' mrr 'ªºiqs «»«» «»pL22 R R pL1() pC p L2 R pL pL22 L i ¬¼''mrrZZs s m r r mrr ¬¼ds

2 22 2 22 22 where ' ()Rr pL r ZZ rr L R r  2 pR rr L  p L r  rr L

158 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

From Equation (7.19) the stator currents can be calculated as

11 1 Is ()ZZZZVZZVss sr rr rs os  sr rr or (7.21) and in detail

§·ªº22 2 2 2 RpLpCpLRpLpLLss1()  mrr   mrrZZ ' pLR mrr ' ¨¸«» ¨¸«»pL22 R R pL1() pC p L2 R pL pL22 L ªºiqs mrrZZ' s s m r  r mrr ' ¨¸¬¼u «»¨¸ ¬¼ids 4 ¨¸ ¨¸ ©¹ §·VK ªºcqo pLm ªºRpLLrrrr Z ªºqr (7.22) ¨¸«» «» ¨¸«»Z LR pL ©¹¬¼VKcdo ' ¬¼rr r r ¬¼d

2 2 Where R pL1() pC p22 L R pL pL 2 2 L pL 2 R 4 ss    mrr   mrrZZ '  mrr '

When a balanced three phase system is transformed to a two axis system the stator o currents iqs and ids have similar waveforms. The difference is that iqs lags ids by 90 .

Hence it suffices to analyse only the expression for ids. The numerator of Equation (7.22) helps to determine the multiplying constants for the solution of the current in the time domain and these constants are dependent on the machine parameters, capacitance value, rotor speed and initial conditions. However, setting the denominator of Equation (7.22) equal to zero, 4 = 0, gives the characteristic equation because it contains the information about the behaviour of the resulting current. If any of the roots of the characteristic equation has a positive real part then there will be a growing transient indicating that there will be self-excitation. If there is no positive real root then there will not be any self-excitation.

Using Equation (7.22) the expression for ids can be given as

U i (7.23) ds 2 2 ªºR pL1() pC p22 L R pL pL 2 2 L pL 2 R ¬¼ss  mrr   mrrZZ ' mrr ' where

159 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

ªºp222C() R pL pC2 p 42 C L 2 () R pL p32 C L 2 R 2 L V ''''¬¼s smrrmrZrr cdo 322 2 32 32 'pC Lm R rZZ r V cqo ' pC Lm r LK r qr ' pC Lm() R s  pL s K dr U 2 pC'

In Equation (7.23) U represents all the terms on the numerator and is dependent on the initial charge in the capacitors, the remnant flux in the core, capacitance, rotor speed and the machine parameters. U only has an effect on the coefficients of the partial fraction expansion of (2), which determine the constants that will be multiplied with the exponential current expression in the time domain, and does not affect the behaviour of the current. The detail of the expression of U is long and it is not necessary to consider when determining whether there is self-excitation or not. The analysis here is to identify if there is self-excitation or not which is solely dependent on the expression in the denominator of Equation (7.23).

Setting the denominator of Equation (7.23) to zero gives the characteristic equation expressed by

2 2 ªºR pL1() pC  p22 L R  pL  pL 2ZZ 2 L ' pL 2 R ' 0 ¬¼ss mrr mrr mrr

2 2 ªº(1pCR p23222222 CL)() p CL R pL p CL L p CL R ¬¼'ss mrrmrrmrrZZ  2 0 pC'

2 ªº()RpL222 LpCRpCL ( 2 1)() pCLRpLpCLL 32 222 ¬¼ r rZZ rr s  s m r r mrr 22 2  pCLRmrrZ 0 (7.24) 222222 pC(2 Rrrrrrr pR L p LZ L ) Hence

2 ªº222 2 32 222 ()RrpL rZZ rr L ( pCR s  p CL s  1)() p CLm R r pL r p CLmrr L ¬¼ (7.25) 22 2  pCLRmrrZ 0

2222222 ªº(2RpRLpLLpCRpCLZ )(  1) 2 rrrrrrs s pCLR22Z 0 (7.26) «»32 222 mrr ¬¼«»pCLmr() R pL r pCL mrrZ L

Equation (7.26) can be written as

160 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

22 Ap43 Bp Dp 2  Ep F Gp 2 0 (7.27) where

22 A CLL()rs LL mr

22 B CLR(2rs RLL rrs  LL mr)

22222 2 DRLRR 2(rrs rZZ rr LLLLC ) s mrr L r

222 E 2(RLrr CR s R r Z r L r)

222 F RLrrZ r

2 GCLR mrrZ The expanded form of Equation (7.27) is

8765432 ap876543210 ap ap ap ap ap ap ap a 0 (7.28) where

2 aA8

aA7 2 B

2 aAD6 2 B

aBDA5 22E

22 aAFBEDG4 22

aBFD3 22E

2 aDFE2 2 

aE1 2 F

2 aF0 It is easy to find the roots of Equation (7.28), the characteristic equation, using MATLAB or any mathematical equation solving program.

When Equation (7.28) is factorized it gives (-p VZjp )(-- VZ jp )(- VZ jp )(-- VZ jp )(- VZ  jp )(-- VZ j ) 11 11 2 2 22 33 33 (7.29) (-pjpjVZ44 )(- VZ 44 - ) 0

161 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

If any of the roots in Equation (7.29) has a positive real value then there is self- excitation. To determine the required capacitance value for an induction generator running at a given rotor speed, the roots in Equation (7.29) are evaluated by increasing the capacitance value until one of the real parts in the roots becomes positive.

7.3.2.2 Direct matrix inversion The current expression for the equation representing a self-excited induction generator, given in Equation (7.14) can be solved by applying Cramer’s rule or by finding the inverse of the impedance matrix.

Cramer’s rule [1] is a mathematical tool for finding one of the variables in an unknown vector in a matrix equation based on the calculation of determinants. Applying Cramer’s rule to Equation (7.14) results in

RpLpCVss10 cqom pL

00Vpcdo Lm

pLmq Kr Rr pLrZr Lr

ZZrmLKLRpdr rr rL r ids (7.30) RpLpCss10 pLm 0

010RpLpCss pLm

pLmrZZLmrrr R pL Lr

ZZrmLpLLRm rr r pL r

Since the characteristic equation of the d-axis stator current is the determinant of the denominator, only the denominator part of ids will be expanded. The determinant of the numerator will be represented by a variable U, which is dependent on the machine parameters, initial conditions, capacitance and electrical rotor speed. U affects only the magnitude of the current ids and does not contain any information on the behaviour of the resulting current. The determinants in Equation (7.30) can be evaluated to give U ids 65432 22 (7.31) ()Ap6543210 Ap Ap Ap Ap Ap A/( C p ) where

222 2 4 A6 CLLLLLL s rsrmm-2 

162 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

22 22 2 A5 2C RLLs sr LRL rsm LLR rsr LRL srm

2222 2 22 22 42 2 A4 C Ls ()4ZZrr L R r LR rs LRRL rsrs 2 LLL rsmr L mrZ 2 RRL srm 22 2CLL sr LL rm A 22ªº C222 LLRZZ LL 22 R LRR 2 RLR 2 C RL 2 LLR RL2 3 ¬¼ srsr rmsr rrs ssr sr rsr rm

22 2 2 2 22 2 2 2 2 A2 2(LCs ZZrr L R r ) R s C ( rr L R r )2 LCL r mr Z L r4 LRCR rs r

22 2 A1 2(RCs Zrr L R r )2 RL rr

22 2 A0 ZrrLR r

Analysing the denominator of Equation (7.31) is sufficient to determine whether initiation of self-excitation will occur. To determine if there is an onset of self- excitation, or not, the denominator of Equation (7.31) is set to zero. That is

65432 Ap6543210 Ap Ap Ap Ap Ap A 0 (7.32)

Equation (7.32) is a sixth order characteristic equation and it has six distinct roots which are first order complex roots in the form of

(-pjpjpjpjpjpjVZ11 )(- VZ 11 - )(- VZ 2 2 )(-- VZ 22 )(- VZ 33  )(- VZ 33 - ) 0 (7.33) If any of the roots has a positive real part, then at that given specific operating point there will be self-excitation.

The current and voltage will grow until the magnetising inductance saturates and makes the real part of the roots zero, which shows that there is a continuous oscillation (Alternating Current and Voltage) as long as the prime mover is driving the induction generator. The transient and steady state solution due to each of the roots can be obtained by using partial fraction expansion.

The eighth order and the sixth order expressions given in Equations (7.28) and (7.32), respectively, are solutions for the same variable in the same machine. The eighth order ended up as a higher order because it has a term with a second order expressed as

22 22 2 Lprrrrr2 RLpZ L Rr that can be factorized in the numerator of Equation (7.24)

163 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS and cancelled with one of the expressions in the denominator. As the eighth order polynomial is very long it is difficult to factorise and see the terms that can be readily cancelled. At a specific operating point, i.e., for a given speed and capacitance two of the roots in the eighth order polynomial due to the terms that would be cancelled can be determined as

22 22 2 Lprrrrrr20 RLpZ L R

pR())())r jLLpRZZ rrr  r jLL rrr 0 (7.34)

For any speed and capacitance values, the real part of the roots for the expression given in Equation (7.34) is always negative and is given by -Rr/Lr. This means that whether the 8th order or the 6th order characteristic equation is used the result is exactly the same. The roots with positive real parts in both characteristic equations are exactly the same. The additional roots in the 8thorder have a negative real part which gives a damped transient response.

7.4 Characteristics of magnetising inductance in induction machine In the modelling of an induction machine used for motoring applications, it is important to determine the magnetising inductance, Lm, at rated voltage. In the SEIG the variation of magnetising inductance is the main factor in the dynamics of voltage build up and stabilisation. In this investigation the magnetising inductance is determined by driving the induction machine at synchronous speed and taking measurements when the applied voltage was varied from zero to 120% of the rated voltage with rated frequency. The magnetising inductance is calculated, without approximation, using the parameter determination method discussed in Chapter 6. Here, conventional high accuracy meters are used for measurements of voltage, current and power, because the accuracy of the voltage and current sensors in the fast measurement system, discussed in Chapter 5, are not good for low values (close to zero) of voltages and currents. The computed power will be erroneous if the accuracy of voltage and current measurements is poor. This is especially important because the magnetising inductance for voltages and currents close to zero is used in the calculation for the initiation of self-excitation process.

164 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

The variation of the magnetising inductance, measured at rated frequency, for the induction machine used in this investigation is given in Fig. 7.6, where the dots are experimental results and the curve is a fourth order curve fit given by:

11 4 8 3 5 2 3 LVVVVmp uuuu1.62 10hp 2.67 10hp 1.381 10hp 1.76 10h 0.23 (7.35)

Where Vph is the phase voltage.

B

(H) 0.3 m L

0.25 A 0.2 C 0.15 - Experimental

Magnetising inductance - Fourth order curve fit 0.1

0.05

0 0 50 100 150 200 250 300

Phase voltage Vph (V)

Fig. 7.6 Variation of magnetising inductance with phase voltage at rated frequency

As can be observed in Fig. 7.6, Lm starts from a small value then increases to reach its peak value and finally starts to drop. This change in Lm is due to the characteristics of the magnetising curve and the fact that: O L m (7.36) m I

Where Om - rms flux linkage, in wb-turn I - rms magnetising current, in A

165 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

It was found by conducting a test in the laboratory that the variation of magnetising inductance in a transformer follows a similar pattern to the one shown in Fig. 7.6.

7.5 Minimum speed and capacitance for self-excitation The induction machine used as the SEIG in this investigation is a three-phase wound rotor induction motor with specification: 4 pole, 415V star connected, 7.8A, 3.6kW, 50Hz. The d-q model, shown in Fig. 7.3 is used because it provides the complete solution, transient and steady state, of the self-excitation process. The parameters obtained from parameter determination tests at rated values of voltage and frequency are Lls = Llr = 11.4mH, Lm = 181mH, Rs = 1.6:, Rr = 2.75:.

When the three capacitors are connected in star the voltage rating of each capacitor is equal to the rated phase voltage. However, if the capacitors are connected in delta the voltage rating of each capacitor should be equal to the line-to-line voltage. In delta connected capacitors, even though the voltage rating of each capacitor is higher than the rating of the capacitors in star connection by a factor of 1.73, the magnitude of the capacitance is lower by a factor 3, i.e. 1/3 of the capacitance in the star connection.

When the induction machine, as shown in Fig. 7.3, with switch S closed, is driven by a prime mover, voltage will start to develop at a corresponding minimum speed. The minimum speed for the onset of self-excitation can be obtained by solving the roots of the 8th order polynomial equation given in Equation (7.28) or the 6th order polynomial equation given in Equation (7.32) and then searching if there is a positive real part in the roots. The minimum capacitance required for a given rotor speed of the induction generator can be found by fixing the rotor speed and then increasing the value of the capacitance until one of the real parts of the roots changes from negative to positive, passing through zero. The value of capacitance that makes the real part of one of the complex roots greater than zero is the minimum value of capacitance required for self- excitation. To have a smooth plot of the minimum rotor speed versus minimum capacitance requirement, the capacitance was incremented by a small value. The detail of this procedure, theoretical determination of the minimum speed and minimum

166 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS capacitance for a fixed speed by incrementing the capacitance, is given in the flow chart of Fig. 7.7.

Another way of finding the minimum rotor speed and corresponding minimum capacitance required for self-excitation is first to set the capacitance at a given value and then increase the rotor speed until one of the real parts of the complex roots becomes positive. This is a good way to find the minimum capacitance and its corresponding minimum rotor speed in the experimental setup.

start

Read machine parameters

Find roots from C = 0 Equations (7.32) or Minimum (7.28) Z r =0 Increment C

No Any positive real root? Increment Yes Z r Reset C to zero Save values of

C and Z r

No Maximum No

Z r?

Yes

Stop

Fig. 7.7 Flow chart to determine the minimum speed and minimum capacitance for SEIG at no load

In the experimental setup a DC motor was used as a prime mover. The rotor speed of the SEIG was varied by varying the speed of the DC motor while the capacitance was kept at a given value. It is possible to increase the capacitance for a given speed,

167 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS however it is not as convenient as varying the speed and it is difficult to find capacitor values that will give a smooth variation of capacitance.

There are two important rotor speeds; the first is the point at which self-excitation occurs and the second is where self-excitation is lost. For a given capacitance value the speed of the test machine was increased until the SEIG started to generate voltage, this is the normal way of achieving self-excitation. The capacitance value and the rotor speed at which the self-excitation started were recorded. Another test was conducted where the self-excited induction generator is already generating voltage and the speed is reduced until the SEIG loses its self-excitation. For a particular capacitance value, the minimum rotor speed for self excitation, determined by increasing the rotor speed from zero, is greater than the minimum rotor speed obtained by decreasing the rotor speed until the SEIG losses its self-excitation. Because the SEIG is always started in the unexcited mode, that is, from zero speed, the correct minimum rotor speed is chosen from the first test, even though the second test produces a lower minimum speed.

The minimum rotor speed and minimum capacitance for self-excitation calculated and measured are given in Fig. 7.8. For the theoretical determination of minimum rotor speed and minimum capacitance, the effect of magnetising inductance on the onset of self-excitation using the rated value of Lm (i.e. 0.18H), as used in motoring analysis, and the unsaturated value of 0.23H is shown in Fig. 7.8. The different values of Lm are obtained from Fig. 7.6. If the rated Lm, 0.18H, is used for determining the onset of self- excitation there will be an error as shown in Fig. 7.8. It was found out that the minimum rotor speed and minimum capacitance required for self-excitation are dependent on magnetising inductance but not on rotor parameters variation.

The principle of finding the minimum capacitance and the minimum rotor speed for self-excitation at no load can be approximated by neglecting the stator winding resistance and stator leakage inductance so that the capacitive reactance and the inductive reactance will be equal. Since the induction generator starts without load the rotor speed is almost the same as the synchronous speed of the induction machine. Hence the approximate minimum capacitance required for self-excitation can be calculated using

168 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

1 Cmin 2 (7.37) ZrmL

Where Z r - the electrical rotor speed, in rad/s

Lm - the value of magnetising inductance close to zero voltage, in H.

SpeedSpeed (rpm) (rpm) 1500

1000

500 - measured values

- using Lm =0.18H (saturated value) - using Lm =0.23H (unsaturated value)

0 0 50 100 150 200 250 300 350 CapacitanceCapacitance ( microPF) F Fig. 7.8 Values of minimum capacitance and rotor speed for self-excitation at no load

A given induction machine can operate as a motor and as a generator depending on whether the power source is electrical or mechanical. Considering motoring operation the capacitance given in Equation (7.37) is equivalent to the capacitance required to have a unity power factor when the induction motor is operating at no load, neglecting friction, supplied from a source with angular frequency equal to Zr. In no load motoring or generating operation the synchronous speed is almost equal to the electrical rotor speed.

The approximate minimum capacitance calculated using Equation (7.37) is slightly more than the actual minimum capacitance calculated using the exact solution. The

169 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS percentage error of capacitance calculated using the approximate method, given in Equation (7.37), and the exact method, discussed in Section 7.3.2, is shown in Fig. 7.9. The error introduced by using the approximate calculation reaches up to 4.5% which is acceptable to find an approximate minimum capacitance value required for an onset of self-excitation in the SEIG using a simple calculation.

error in capacitance value ( %) 5

4

3

2

1

0 600 700 800 900 1000 1100 1200 1300 1400 1500 Speed (rpm)

Fig. 7.9 Error in capacitance when calculated using the approximate method

The SEIG needs to be started at no load. Hence it can be generalised that the capacitance required for the onset of self-excitation in the SEIG rotating at a rotor speed of Z r, is almost equal to the capacitance required to have a unity power factor in a motoring application, as the induction motor operates at no load and with an angular supply frequency of Z r.

7.6 Magnetising inductance and its effect on stability of generated voltage The magnetising inductance, measured at 50Hz frequency of excitation, varies with voltage as shown in Fig. 7.6. At the start of self-excitation (point A), where the voltage is close to zero, Lm is close to 0.23H. Once self-excitation starts the generated voltage will grow and then Lm also increases up to point B. When there is an increase in magnetising inductance it increases the value of the positive real root of the

170 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS characteristic equation and consequently the generated voltage grows faster. Beyond point B, up to point C, Lm decreases while the voltage continues to grow until it reaches its steady state value determined by the Lm value, capacitance and the rotor speed.

Referring to Fig. 7.6, the unstable region is between points A and B. If the SEIG starts to generate in this region, a small decrease in speed will cause a decrease in voltage and this will bring a decrease in Lm, which in turn decreases the voltage, and finally the voltage will collapse to zero. Once the voltage collapses there is no transient phenomenon and there will not be voltage build up even if the speed increases once again to its initial value as shown in Fig. 7.10. This condition can cause demagnetisation of the core. When the core is demagnetised there will not be self- excitation. In order to initiate self-excitation in the demagnetised core, the core should be magnetised by running the generator as a motor or exciting the windings from a DC supply. The other arrangement is to charge the exciting capacitors from a DC supply.

200

(V) 100 ph 0

-100 Generated V -200 0 2 4 6 8 10 12 (a)

1000 800 600 400 Speed (rpm) 200 0 0 2 4 6 8 10 12 time (sec) (b) Fig. 7.10 Measured unsuccessful self-excitation at C=60PF (a) generated phase voltage (b) speed Between points B and C is a stable operating region. When the speed of the prime mover decreases voltage will decrease and Lm increases which enables the self-excited

171 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS machine to continue to operate at a lower voltage as shown in Fig. 7.11. An increase in

Lm means an increase in the positive real roots of the characteristic equation which is good for stable operation of the SEIG. The results given in Fig. 7.10 and Fig. 7.11 reflect the condition of a SEIG driven by a wind turbine when the wind turbine is operating at low wind speed. This condition is mapped by the curves for low torque versus rotor speed as given in Fig. 2.9 of Chapter 2. From the characteristic of a wind turbine, the gradient of the torque versus rotor speed curve is gentle, so that a large change in rotor speed brings a small change in output torque.

400

200 (V) ph 0

-200 Generated V Generated -400 0 2 4 6 8 10 12 (a) 1200 1000 800 600 400 Speed (rpm) Speed 200 0 0 2 4 6 8 10 12 time (sec) (b)

Fig. 7.11 Measured self-excitation at C=60PF and lower speed (a) generated phase voltage (b)speed

For the same capacitance value, if the rotor speed is increased the generated voltage also increases. The rated voltage given in Fig. 7.12 can be generated by increasing the rotor speed above the speed given in Fig. 7.11(b) to a speed close to the synchronous speed. The result given in Fig. 7.12 shows the voltage build up for the case where the speed is close to the synchronous speed and is typically of the result to be expected

172 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS from a SEIG driven by a wind turbine. The small drop in speed is due to the power loss in the self-excited induction generator associated with the generation of voltage.

400

200 (V) ph 0

-200 Generated V Generated -400 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (a)

1500

1000

Speed (rpm) Speed 500

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time (sec) (b)

Fig. 7.12 Measured self-excitation at C=60PF with speed and generated voltage close to rated values (a) generated phase voltage (b)speed

7.7 Onset of self-excitation when the SEIG is loaded The SEIG equivalent circuit shown in Fig. 7.3 can be loaded with a resistive load by connecting a resistance RL across the capacitor, C. With resistive load Equation (7.7) is modified to the following equation ªºR RpL L 00 pL «»ss m iV ªº0 1 RpCL ªºªqs cqo º «»«»« » «»0 «»RiV «» Lds«»«cdo » «»00RpLss pL m  (7.38) «»0 1 RpC «»«iK » «»L «»«qr q » «» «» ¬¼0 pLmrmrrrrZZ L R pL L «»«iK » «»¬¼¬dr d ¼ ¬¼«»ZZrmLpLLRpL m rr r r

173 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

Equation (7.38) can be solved to obtain its characteristic equation using a similar procedure to that used to solve Equation (7.7). Analysing in a similar way, as discussed in Section 7.3.2, the curves in Fig. 7.13 are obtained for different load resistors.

Speed (rpm) 1500

RL = 25: RL = 11:

RL = 9.5: RL = 13:

1000

RL > 1000:

500

0 0 100 200 300 400 500 600 Capacitance (PF)

Fig. 7.13 Required capacitance and speed for self-excitation with load, RL

For a given capacitance value the wind speed can vary without warning. Without load the SEIG requires only a minimum speed for self-excitation, but a loaded SEIG has a minimum and maximum speed for self-excitation as shown in Fig. 7.13. When RL is large the characteristic is similar to the no load self-excitation case. If RL is small, larger load, there is a minimum and maximum speed to produce self-excitation at a particular capacitance value.

The characteristics in Fig. 7.13 help to find the minimum and maximum speed set points for a given capacitance value. Once the minimum and maximum speed points are obtained, the speed range for a safe generating range can be identified. It is clear that a loaded generator has different onset of self-excitation characteristics for different values of load resistance. This also helps to determine the speed range for the steady-state generating characteristic of the SEIG. At high load resistance or at no load the maximum speed limit is so high that it is not necessary to take that into consideration. 7.8 Simulation of self-excited induction generator

174 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

In conducting an experimental analysis of a self-excited induction generator, it is difficult to get continuous values of a capacitance and determine the corresponding speed. Mostly the components are in discreet values. It is also hard to see the condition of the self-excitation beyond the rated values of the machine as it can damage the machine. Simulation is extremely useful in predicting the condition of self-excitation within the rated values of the machine and/or beyond these rated values.

7.8.1 The modelling of self-excitation process To find the dynamics of a self-excited induction generator a mathematical model is developed. The solution of this mathematical model gives the complete characteristics comprising of transient and steady state for the voltage, current, power and frequency of a self-excited induction generator. With the help of this mathematical model the dynamic values of voltage, current and power of the SEIG at any given time can be evaluated. The mathematical model takes into account the initial conditions in the induction generator, namely the initial voltage in the exciting capacitors and the initial induced voltage due to remnant magnetic flux in the magnetic core.

7.8.1.1 Determination of initial conditions The initial conditions required in the equation for the simulation of self-excited induction generator given in Equation (7.7) can be determined from measurements performed on the induction machine and the capacitors.

Using the self-excited induction generator model shown in Fig. 7.3 the remnant or residual flux linkages along the d-axis Odro and along the q-axis Oqro can be estimated from the stator terminal voltage of the induction machine measured without the exciting capacitors, i.e. when the switch S is open. Then the rotational initial voltages in the rotor are calculated from the remnant flux linkages and the rotor speed using

Kqr ZO r dro and Kdr ZO r qro .

The initial voltage in the exciting capacitor can be measured using a conventional voltmeter while the switch S in Fig. 7.3 is open. The initial voltage in the capacitor

175 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS decreases with time because of the internal leakage and discharge through the internal resistance of the voltmeter.

7.8.1.2 The dynamic representation of self-excitation at no load Based on the general equation of the self-excited induction generator a mathematical model is developed to represent the dynamic characteristic involved in the voltage build up of the SEIG. The dynamic analysis has to demonstrate the transient and steady state values of voltage and frequency developed by the induction generator at the no load condition.

Using Equations (7.1) and (7.2) Equation (7.7) can be written as iV ªº0 ªºRpLss 00 pLm ª qs ºªcq º « »« » «»0 «»00RpL pL i V «» «»ss m« dscd»« » (7.39) « »« » «»0 «»pLmrmrrrrqZZ L R pL L ir K qr «» «»« »« » 0 ZZLpLLRpL ¬¼ ¬¼«»rm m rr r r ¬«iKdr ¼¬»«dr ¼»

Equation (7.39) can be rearranged to give pi i V ªºªLLsm00ªºqs Rs 000 ºªºªºqs cq ªº0 «»««» »«»«» 00LLpiR 0 00 i V«»0 «»«smd«»s s »«»«»dscd «» (7.40) «»««» »«»«» LLmr00 piLRLiK qr 0ZZ rmrrrqr  qr«»0 «»««» »«»«»«» 00LLZZ L 0 LR 0 ¬¼¬«»«m r «»¬¼pidr rm rr r ¼»«»«»¬¼¬¼idr Kdr ¬¼

The above equation can be simplified as

ApIV + BIV + VV =0 (7.41) where

ªºLLsm00 ªºRs 000 «»00L L «»00R 0 A «»sm B «»s «»LL00 «»0  ZZL RL «»mr «»rm r rm ¬¼«»00LmrL ¬¼«»ZZrmLL0 rrR r

ªºiqs ªºVcq «» «» i V I «»ds V «»cd V «»i V «»K «»qr «»qr ¬¼«»idr ¬¼«»Kdr

176 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

Using Matrix inversion, Equation (7.41) can be written as -1 -1 pIV = -A BIV -A VV (7.42) Where

ªºL L r 00m «»LL «» L L «»00r m 1 «»LL 2 A «» and L LLs rm L LL «»ms00 «»LL «»LL «»00ms ¬¼«»LL

Then evaluating Equation (7.40) in a state space form gives 2 LV L K ªºLRrs L mrZZ L mr R  L mrr L ªº rcq m qr «»«» «»LLLL«» LL ªºpi2 ªº i qs «»LLRLLLRqs «»L VLK «»mrZZ ss mrr mr «» rcd m dr pi«» i «» «»ds LLLL «»ds LL (7.43) «» «» «»«» piqr «»LR LLZZ LRLL iqr «»LVmcq LK s qr «»ms srm sr srr «»  «»pi«»LLLL «» i «»LL ¬¼dr «» ¬¼dr «» LLZZ LR LL LR LV LK «»srm ms srr sr «»mcd s dr ¬¼«»LL L L ¬¼«»LL

Expanding Equation (7.43) gives four sets of first order differential equations as given below 1 pi L Ri L2 ZZ i  L Ri  L Li  LV L K (7.44) qs L r s qs m r ds m r qr m r r dr r cq m qr 1 pi L2 ZZiLRiLLiLRiLVLK (7.45) ds L mrqs rsds mrrqr mrdr rcd m dr 1 pi L RiLLiLRiLLiLVLKZZ (7.46) qr L m s qs s r m ds s r qr s r r dr m cq s qr 1 pi L ZZLi LRi L Li LRi LV LK (7.47) dr L s r m qs m s ds s r r qr s r dr m cd s dr and from the equations of the capacitors i pV qs (7.48) cq C i pV ds (7.49) cd C

177 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

Vcd and Vcq are the no load output generated voltages and the three phase voltages can be obtained by using the three-phase to two-axes transformation discussed in Chapter 3.

Since the mechanical time constant is much larger than the electrical time constant, in equations 7.44 - 7.47 the rotor speed is assumed constant for small changes in the voltages and currents.

The relationship between capacitance, rotor speed and generated voltage at the no load condition is given in Fig. 7.14. Generated voltage

C1 C2 C3 Z r3

Z r2

Z r1

O

C3 > C2 > C1

Z r3 > Z r2 > Z r1

Exciting current

Fig. 7.14 Relationship between capacitance value, rotor speed and generated voltage at no load

The capacitor provides the exciting current required by the induction generator and the induction generator charges the capacitor to increase the terminal voltage. An increase in capacitor voltage provides an increase in exciting current to the induction generator. In this way the voltage build up continues until the magnetising inductance decreases to its saturated value and an equilibrium point is attained. The process of voltage build up

178 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS in a SEIG is similar to a shunt connected DC generator [2]. It can be assumed that point O is the rated voltage in Fig. 7.14. The rated voltage can be generated using capacitance

C2 at rotor speed Zr2. To generate the same amount of voltage at lower rotor speed, Zr1, the capacitance increases to C3 and the exciting current can exceed the rating of the stator current of the induction machine. And at higher speed, Zr3, the rated voltage can be generated with a small capacitance value C1.

The magnetising inductance given in Fig. 7.6 applies only at rated frequency 50Hz. The characteristic of magnetising inductance as a function of voltage at other frequencies is shown in Fig. 7.15. The magnetising inductance can be expressed as a function of magnetising current as shown in Fig. 7.16. However, since voltage is the output parameter of interest in a SEIG, the magnetising inductance as a function of voltage clearly indicates the generated voltage at a given operating point and also shows the minimum voltage that can be generated without loss of self-excitation.

LLm (H) m

0.25

55Hz 0.2

0.15 50Hz 35Hz 0.1

0.05 40Hz 45Hz

0 0 50 100 150 200 250 300 350 400 V (V) Vphph (V)

Fig. 7.15 Variation of magnetising inductance with phase voltage at different frequencies

179 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

From Fig. 7.15 the minimum stable generated voltage operating at low frequency, corresponding to low speed (large capacitance value), is less than the minimum stable generated voltage operating at high frequency corresponding to high speed (small capacitance value). For example the minimum stable generated voltage at 35Hz is lass than that of 50Hz.

In the simulation, the polynomial equations for the curves in Fig. 7.15, derived from the polynomial curve for 50Hz, can be used. The current in a capacitor is calculated from the voltage across the capacitor by applying differentiation and differentiation can be represented by a high pass filter. Hence the rms current might need a low pass filter to avoid unnecessary noise if a polynomial curve fit for Lm as a function of the magnetising current is used.

L (H) Lmm (H) 0.3

0.25

0.2

0.15

0.1

0.05

0 0 1 2 3 4 5 6 7 8 9 10 Magnetising currentI (A) I (A) m m Fig. 7.16 Variation of magnetising inductance with magnetising current

The components of the magnetising current Im are calculated as

imq = iqs + iqr (7.50)

180 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

imd = ids + idr (7.51) and the rms value of the magnetising current Im is

22 iimq md Im (7.52) 2

In Fig. 7.16 the magnetising inductance as a function of rms magnetising current Im is represented by two polynomial curve fits.

For Im < 1.157A

43 2 Lmmmm 0.I 063 0 .I 14 0 .I 017 0 .I 125 m 0 . 23 (7.53a) and for Im t 1.157A

64 43 32 L.mmmmm 3 98u 10 I u 2 . 4 10 I u 5 . 48 10 I  0 .I. 0605  0 3552 (7.53b)

In order to have a constant speed before and after self-excitation of the induction generator the rotor was driven at constant speed using a DC motor with speed regulator as shown in Fig. 7.17. 3 phase supply DC motor Speed Current controller controller Controlled Reference PI PI 3 phase speed rectifier

KI

KS

Fig. 7.17 DC motor speed regulator

When the induction generator is driven using the DC motor with speed regulator, as illustrated in Fig. 7.17, it is easier to compare the experimental results with the results obtained from the simulation of the dynamic model. SIMNON [3] simulation software was used to predict the generated voltage of a three-phase SEIG rotating at a given speed with appropriate capacitors connected at the stator terminals.

181 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

With zero initial charge in the capacitors the experimental dynamic self-excitation process with per-phase capacitance of 60PF is given in Fig. 7.18. Since the initial charge in the capacitors is zero, the initial condition for self-excitation is coming from the remnant or residual flux in the iron core. Fig. 7.19 is the dynamic simulated result with the same initial conditions, capacitance values and speed as the experimental results shown in Fig. 7.18. In the experimental result of Fig. 7.18b there is a small transient dip in speed during the self-excitation process because of the associated electrical power loss in the induction generator. However this small dip in speed is compensated for by the speed regulator.

Experimental result for capacitance 60micrF 400 RMS phase voltage 200

0 Va (V)

-200

-400 0 0.5 1 1.5 2 2.5 3 3.5 (a)

1500

1450 Speed (rpm) Speed

1400 0 0.5 1 1.5 2 2.5 3 3.5 (b)

4

Is (A) 2

0 0 0.5 1 1.5 2 2.5 3 3.5 (c) time(sec)

Fig. 7.18 Measured self-excitation at C=60PF and with regulated speed (a) generated phase voltage (b) speed (c) stator current

182 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

The experimental results in Fig. 7.18 and the simulated results in Fig 7.19 are in good agreement. The rms current and voltage are calculated at each sampling time using the algorithm developed in Chapter 3. The rms and instantaneous voltages are captured during the transient condition. The algorithm for calculating rms values is based upon instantaneous measurements, and therefore is effective for both steady state and transient conditions.

Simulation result for capacitance 60micrF 400 RMS phase voltage 200

0 Va (V)

-200

-400 0 0.5 1 1.5 2 2.5 3 3.5 (a)

1500

1450 Speed (rpm) Speed

1400 0 0.5 1 1.5 2 2.5 3 3.5 (b)

4

Is (A) 2

0 0 0.5 1 1.5 2 2.5 3 3.5 (c) time(sec)

Fig. 7.19 Simulated self-excitation at C=60PF and with regulated speed (a) generated phase voltage (b) speed (c) stator current

183 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

Since the experimental and simulated results given in Fig. 7.18 and Fig. 7.19, respectively, are in good agreement it is possible to predict with confidence other parameters which are not convenient to measure in the real induction generator. The magnetising current, magnetising inductance and the stator flux-linkage are given in Fig. 7.20, where the speed of the induction generator and capacitance values are the same as that of Fig. 7.19. The dynamic magnetising inductance varies with the generated voltage and/or exciting current during the self-excitation process. The magnetising current and stator flux linkage grow with the generated voltage.

Simulated result for capacitance 60micrF 0.3

0.2 Lm (H) Lm

0.1 0 0.5 1 1.5 2 2.5 3 3.5 6

4

Im (A) 2

0 0 0.5 1 1.5 2 2.5 3 3.5 1.5

1

0.5 flux (web-turn) flux 0 0 0.5 1 1.5 2 2.5 3 3.5 time(sec)

Fig. 7.20 Simulated self-excitation at C=60PF and with regulated speed (a) magnetising inductance (b) rms magnetising current (c) peak stator flux-linkage

The simulation results in Fig. 7.20 indicate that as the magnetising current increases from zero the magnetising inductance increases, reaches its peak value, then starts to decrease and finally reaches its saturated value.

184 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

Fig. 7.21 shows the build-up of d-axis stator flux-linkage and q-axis stator flux-linkage during the self-excitation process as a function of time in a three dimensional plot. The flux-linkages continue to grow until they reach their steady state values which are the saturated flux-linkages. The peak values of the d-axis flux-linkage and the q-axis flux linkage are equal.

1.5

1

time (sec) time 0.5

0

1 1.5 1 0 0.5 flux-q (web-turn) 0 -0.5 flux-d (web-turn) -1 -1 -1.5

Fig. 7.21 Three dimensional d-axis flux-linkage and q-axis flux-linkage as a function of time during self-excitation process

The results of the dynamic self-excitation process given in the previous figures are based on remnant or residual flux in the iron core providing the initial condition required by the self-excited induction generator. When the initial conditions for self- excitation are satisfied the flux grows and associated with the growth of flux linkage the generated voltage also grows.

As discussed previously the self-excitation process can be also initiated with a charged capacitor. A charged capacitor will provide magnetising current to the induction generator and the flux and the terminal voltage will grow. When the self-excitation process is started from a charged capacitor there is a step voltage at the moment the

185 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS capacitors are connected to the terminals of the induction generator and provides a transient exciting current. The experimental and simulated results for the self-excitation process initiated by a charged capacitor for a capacitance of 60PF and rotor speed of 1480rpm are given in Fig. 7.22 below.

For capacitance 60micrFof 60PF and speed 1480rpm 400

200

0 Va (V)

-200

-400 0 0.5 1 1.5 2 2.5 3 3.5 (a)

400

200

0 Va (V) -200

-400 0 0.5 1 1.5 2 2.5 3 3.5 (b)

Fig. 7.22 Self-excitation process initiated by a charged capacitor of 60PF and rotor speed of 1480rpm (a) experimental result (b) simulated result

7.8.2 The dynamic representation of a loaded SEIG The generated voltage in a self-excited induction generator grows and reaches a steady state value where the peak of the generated voltage remains at a constant peak value. Once steady state is attained a load can be connected to the SEIG. In a SEIG operating without load the stator current and the capacitor current are equal. However, in a loaded SEIG the stator current is divided into capacitor current and load current. The equations used to analyse the loaded SEIG are modified from the equations representing the unloaded SEIG, discussed in the previous section, and additional equations are developed to take into account the relationship between the stator, capacitor and load

186 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS currents of the loaded SEIG. With a good model and implementing the model using simulation software the dynamic characteristics of generated voltage, stator current, capacitor current, load current, magnetising current, power, and electromagnetic torque can be studied with confidence.

The model for the loaded SEIG is shown in Fig. 7.23.

Zr Odr Rs Lls Llr R r + - i Lq iqr icq iqs imq Vcq Lm RL C Oqs Oqr

(a)

-Zr Oqr Rs Lls Llr R r + - i Ld idr icd ids imd Vcd Lm RL C Ods Odr

(b) Fig. 7.23 d-q model of a loaded SEIG in a stationary reference frame (a) q-axis circuit (b) d-axis circuit

The differential equations given in Equations (7.44) to (7.47) are also used for the loaded condition of the SEIG. For a resistive load the additional equations needed are:

Vcq iLq (7.54) RL

Vcd iLd (7.55) RL

iiicq qs Lq (7.56)

iiicd ds Ld (7.57) i pV cq (7.58) cq C

187 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

i pV cd (7.59) cd C

The dynamic equations developed for the loaded SEIG are simulated using SIMNON [3] simulation software. Using this simulation algorithm it is possible to determine the characteristic of a loaded SEIG and the dynamic change in the parameters during loading and unloading of a SEIG. The no load condition can be represented by introducing a very large value of load resistance.

When the induction machine operates as a motor at a constant frequency, the speed of the air gap rotating magnetic field is fixed. When this motor is loaded the rotor speed will be varied relative to the synchronous speed to produce an output power equivalent to the mechanical power demand. For a SEIG, with constant rotor speed, the speed of the rotating magnetic field lags behind the rotor speed. When the load of the SEIG is increased the magnitude of the negative slip also increases. In this case, as the rotor speed is the input and is constant, the increase in slip is only due to a decrease in the speed of the rotating magnetic field. The generated frequency and voltage are proportional to the speed of the rotating magnetic field. For the same capacitance value a decrease in the speed of the rotating magnetic field will inevitably decrease the generated voltage and its frequency. This explanation is illustrated in Fig. 7.24.

When the SEIG is operating at no load, point A in Fig. 7.24, the slip is almost zero and the generated voltage is equal to its rated value, curve 1. When the SEIG is loaded the rotor speed remains at point A. However, the synchronous speed has to decrease to point B in order to operate at a negative slip equivalent to the electrical load demand which is equivalent to the electromagnetic torque at point O, curve 2. The speed of the rotating magnetic field at point B is less than that of point A. Since the generated voltage is proportional to the speed of the rotating magnetic field a decrease in the synchronous speed decreases the generated voltage. Hence loading will decrease the generated voltage and frequency of the SEIG. The simulation and experimental results given in Figs. 7.25 to 7.30 confirm this.

188 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

The instantaneous phase voltage is measured using a data acquisition board. The instantaneous frequency is calculated by applying a d-q transformation to two voltage samples as discussed in Chapter 3 or it can be calculated from flux-linkages of the induction machine using the differentiation technique [4].

60 1 40

20

Torque (Nm) 2 A 0 B

-20 O

-40

-60

-80

-100 0 500 1000 1500 2000 2500 3000 Speed (rpm)

Fig. 7.24 Relationship between rotor speed and synchronous speed in a SEIG

The steady state variation of voltage and frequency are given in Fig. 7.29 and Fig. 7.30, respectively. The voltage and frequency are dependent on the amount of loading. In Fig. 7.30 the frequency of the generated voltage, due to remnant magnetic flux in the core, of the free running induction generator without capacitance and at no load, is given by: nP f P (7.60) 60 where Pp - the number of pole pair n - rotor speed, rpm

189 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

ExperimentalExperimental result for capacitancecapacitance 60 60micrFPF and andRL = RL=55ohm 55: 500

0 Va (V)

-500 0 0.5 1 (a) 1.5 2 2.5 3 3.5 1500

1450 speed(rpm) 1400 0 0.5 1 (b) 1.5 2 2.5 3 3.5 60

40

20 frequency(Hz) 0 0 0.5 1 (c) 1.5 2 2.5 3 3.5

300

200

Vrms (V) 100

0 0 0.5 1 (d) 1.5 2 2.5 3 3.5 3000

2000

1000 Power (W) 0 0 0.5 1 (e) 1.5 2 2.5 3 3.5

4

2 Is-rms(A)

0 0 0.5 1 1.5 2 2.5 3 3.5 (f) time (sec) Fig. 7.25 Experimental loading of SEIG after the voltage has developed to its steady state value (a) phase voltage (b) speed (c) frequency (d) rms phase voltage (e) generated power (f) rms stator current

190 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

SimulationSimulation result result for for capacitance capacitance 60 60micrFPF and R andL = 55RL=55ohm: 500

0 Va (V)

-500 0 0.5 1 (a) 1.5 2 2.5 3 3.5 1500

1450 speed(rpm) 1400 0 0.5 1 (b) 1.5 2 2.5 3 3.5 60

40

20 frequency(Hz) 0 0 0.5 1 (c) 1.5 2 2.5 3 3.5

300

200

Vrms (V)Vrms 100

0 0 0.5 1 (d) 1.5 2 2.5 3 3.5 3000

2000

1000 Power (W) 0 0 0.5 1 (e) 1.5 2 2.5 3 3.5 6

4

2 Is-rms(A)

0 0 0.5 1 1.5 2 2.5 3 3.5 (f) time (sec) Fig. 7.26 Simulated loading of SEIG after the voltage has developed to its steady state value (a) phase voltage (b) speed (c) frequency (d) rms phase voltage (e) generated power (f) rms stator current

191 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

For 60micrF and RL=55ohm For 60PF and RL = 55: 6

4

Is (A) 2

0 0 0.5 1 1.5 2 2.5 3 3.5 (a)

4

2 Ic (A)

0 0 0.5 1 1.5 (b) 2 2.5 3 3.5 6

4

IL (A) 2

0 0 0.5 1 1.5 (c) 2 2.5 3 3.5 tim e (s ec)

Fig. 7.27 Simulated loading of SEIG (a) rms stator current (b) rms capacitor current (c) rms load current

For 60micrFFor 60PF and and RL=55ohm RL = 55:

0.2

0.1 Lm (H) Lm

0 0 0.5 1 1.5 2 2.5 3 3.5 (a) 1.5

1

0.5 flux (web-turn) flux 0 0 0.5 1 1.5 (b) 2 2.5 3 3.5

4

2 Im (A)

0 0 0.5 1 1.5 (c) 2 2.5 3 3.5 tim e (sec)

Fig. 7.28 Simulated loading of SEIG (a) Lm (b) peak flux-linkage (c) rms magnetising current

192 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

Generated phase voltage (V) 300 No load 280 RL=139ohm RL=70ohm 260 RL=55ohm

240

220

200

180

160

140

120

100

80 1150 1200 1250 1300 1350 1400 1450 1500 1550 Speedspeed (rpm)

Fig. 7.29 Measured variation of generated voltage with load for a 60PF capacitance

55 f=np/60- Vgen due to remnant flux (no C) no load (C=60microF) RL=139ohm (C=60microF) RL=70ohm (C=60microF) 50 RL=55ohm (C=60microF)

Frequency (Hz) 45

40

35

30 1000 1100 1200 1300 1400 1500 1600 speed(rpm)

Fig. 7.30 Measured variation of generated frequency with load for a 60PF capacitance

193 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

When the capacitors are connected to the induction machine, voltages and currents build up and power is dissipated in the machine. The induction generator has to absorb an equivalent amount of power from the prime mover, which makes it operate at a synchronous speed which is marginally lower than the rotor speed. When the load of the induction generator increases, its synchronous speed keeps on decreasing to produce the required amount of slip at each operating point.

7.9 Characteristics of wind turbine and its effect on generator output As discussed in Chapter 2, when the angular rotor speed of the wind turbine increases from zero the torque and power produced by the wind turbine increases from zero value, reach their peaks and then decrease. At zero rotor speed there is a small torque which is enough to start and rotate the wind turbine. A typical wind turbine torque-rotor speed characteristic is given in Fig. 7.31. This characteristic is modified from the original characteristic given in Reference [5] to have an increased torque and decreased rotor speed by changing the gear ratio.

There are different torque-rotor speed curves representing different wind speeds. The torque produced by the wind turbine and measured at the shaft of the induction generator can be represented by a function dependent on the linear wind speed and angular rotor speed at the induction generator. However to simplify the representation and dynamics of the torque produced by the wind turbine, it is assumed that the wind speed is constant at 9m/s during the operation of the self-excitation and the torque produced by the wind turbine will follow a single characteristic curve as shown in Fig. 7.31 represented by the solid line. The torque-speed characteristic is approximated using a fifth order polynomial curve fit given by:

13 5 9 4 6 3 3 2 T.t 4 123 u 10 n.  1 993 u 10 n.  3 428 u 10 n.  2 395 u 10 n.n  0 54  36 (7.61) where Tt - wind turbine torque measured at the generator shaft, Nm n - angular rotor speed, rpm

194 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

100 12m/s Wind speed 80 11m/s

60 10m/s Wind turbine torqye (Nm) 9m/s 40 8m/s

7m/s

20 6m/s 5m/s

0 0 200 400 600 800 1000 1200 1400 1600 1800 Rotor speed (rpm)

Fig. 7.31 Wind turbine output torque as a function of rotor speed

Here the aim is not to operate at maximum output power or maximise the output torque of the wind turbine but is simply to analyse the dynamic effect of variation in load and variation in angular rotor speed on the characteristic of the SEIG. The analysis is simplified when the wind speed is assumed constant. At constant wind speed variation in the load connected to the SEIG will change the electromagnetic torque developed by the induction generator. A change in the electromagnetic torque will vary the mechanical torque demand from the wind turbine. The torque output from the wind turbine is adjusted by changing the rotor speed. The equation that relates the mechanical output torque produced by the wind turbine measured at the shaft of the induction generator and the electromagnetic torque developed by the induction generator is given by

dZ TJ m  DZ T (7.62) t dt me

where Tt - turbine output torque measured at the generator shaft, Nm

195 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

Te - induced electromagnetic torque in the induction generator, Nm

Zm - the angular mechanical rotor speed, rad/s D - friction coefficient referred to the generator shaft, Nm/rad/s J - effective inertia referred to the generator shaft, Kg-m2

Fig. 7.32 and Fig. 7.33 show the dynamic characteristics of different parameters of the induction generator when driven by the wind turbine given in Fig. 7.31. At t = 0 sec a capacitor with capacitance of 60PF is connected at the stator terminals of the induction generator without any load, i.e. RL = f, and voltage is generated because of the available rotor speed at t = 0. The generated voltage Vph, shown in Fig. 7.32d, is expressed as an instantaneous value and rms value. At t = 2sec a 55: load resistor is connected. The generated voltage, frequency, capacitor current (ic) and stator current

(is) decrease. However, the load current (iL) and induced electromagnetic torque increase.

At t = 6sec the capacitance is increased to 201PF to compensate for the voltage drop. The voltage rises to its no load value but the frequency of the generated voltage decreases further. All currents, torque and output power increase.

At t = 10sec the resistance value is increased to 90:, which demands less power, and the generated voltage rises to a value higher than the no load voltage. The stator current, capacitor current, frequency, and speed increase. However the load current, output power and torque decrease.

At t = 14sec the capacitance value is decreased to 126PF to reduce the generated voltage to its no load value. All currents, electromagnetic torque and output power decrease. However the speed and frequency increase. As can be observed from the results the variation of the frequency follows the variation of the rotor speed, which is dependent on the characteristics of the wind turbine.

196 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

100

80

60 (ohm)

L 40 R 20

0 0 2 4 6 8 10 12 14 16 18 20 (a)

250

200

150

(micro F) 100 C 50

0 0 2 4 6 8 10 12 14 16 18 20 (b)

1500

1000 (rpm)

500 Speed

0 0 2 4 6 8 10 12 14 16 18 20 (c)

500 Vrms 250

(V) 0 ph V -250

-500 0 2 4 6 8 10 12 14 16 18 20 (d)

60

40 (Hz) f 20

0 0 2 4 6 8 10 12 14 16 18 20 (e) time (sec) Fig. 7.32 Simulated results for wind turbine with variable rotor speed (a) load resistance (b) capacitance (c) rotor speed (d) phase voltage (e) frequency as a function of time

197 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

15

10 (A) S i

5

0 0 2 4 6 8 10 12 14 16 18 20 (a)

15

10 (A) C i 5

0 0 2 4 6 8 10 12 14 16 18 20 (b)

6

4 (A) L i 2

0 0 2 4 6 8 10 12 14 16 18 20 (c)

60

40 (Nm) e T 20

0 0 2 4 6 8 10 12 14 16 18 20 (d)

6000

4000 (W)

2000 Power

0 0 2 4 6 8 10 12 14 16 18 20 (e) time (sec) Fig. 7.33 Simulated results for wind turbine with variable rotor speed (a) rms stator current (b) rms capacitor current (c) rms load current (d) electromagnetic torque (e) output power as a function of time

198 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

At a given speed, adjusting the excitation capacitance might compensate the variation in voltage caused by loading. However the frequency is dependent on the rotor speed and amount of connected load. Adjusting the capacitance does not have direct effect on the frequency.

7.10 Effect of rotor parameters variation on self-excitation The hypothetical three-phase induction machine discussed in Section 6.3.2 is used for the purpose of analysis and simulation of a SEIG with rotor parameter variations. This hypothetical induction machine is typical of a double-cage or deep-bar rotor where rotor parameter variation is significant. The values of the parameters were chosen to have similarity with a real induction machine in the laboratory with specifications 415V, 4- pole, 50Hz.

The hypothetical three-phase induction machine that was selected for this simulation has parameters L1s = 6mH, Rs = 1.5: and Lm = 318mH. The variation of magnetising inductance with voltage at rated frequency, which was derived from Equation 7.35, is given by

11 4 8 3 5 2 3 LVVVmp 1.757(uuu 1.62 10h 2.67 10ph 1.381 10ph u 1.76 10Vph 0.23)

(7.63) where Vph is the voltage across the magnetising inductance.

The multiplying factor 1.757 was chosen from the ratio of that for Lm at rated values so that the variation of magnetising inductance will be similar to the real machine. When the magnetising inductance is expressed as a function of rms magnetising current Im, it is represented by two polynomial curve fits given by:

for Im < 0.6345 A

43 2 Lmm 1.I 06 1 .I 34 m 0 . 0894 I m  0 . 387 I m  0 . 404 (7.64a) and for Im t 0.6345 A

54 33 2 Lmm 7.I.I.I.I. 01u 10  2 357 u 10 mmm  0 0303  0 189  0 624 (7.64b)

199 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

The values of rotor leakage inductance Llr and rotor resistance Rr with respect to the magnitude of slip were determined from Fig. 6.15 and are expressed as

432 Rr 0. 969 s 4 .s 26 4 .s 9 1 .s 52 1 . 48 (7.65)

432 L.s.s.s.s.lr 0 0032 0 0144 0 0252 0 027 0 0189 (7.66)

In the calculation, the magnitude of the slip was taken because the rotor current frequency is the same whether the slip is positive or negative.

If the rotor parameters are considered constant then Llr = 6mH and Rr = 4.6:, which are the values at s = 1. For variable rotor parameters the parameters start close to Llr =

20mH and Rr = 1.4:, which are the values at s = 0, no load operation. The variations of load resistance and excitation capacitance at the input of the SEIG, and the rotor speed are given in Fig. 7.34.

100

50 C (microC F) 0 0 2 4 6 8 10 12 14 16 (a) 100

(ohm) 50 L R

0 0 2 4 6 (b) 8 10 12 14 16 2000

1000 speed (rpm) speed

0 0 2 4 6 8 10 12 14 16 (c) time (sec)

Fig. 7.34 Input to the hypothetical SEIG (a) capacitance, PF (b) load resistance, : (c) speed, rpm

200 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

The capacitance is 40PF for time t less than 10sec and then increased to 68PF to make the generated voltage equal to the no load voltage. The SEIG is operating at no load (infinity load resistance) for time t less than 5sec and then loaded with load resistance of 45:. The rotor speed is fixed at 1500rpm during the operation.

The results for the hypothetical SEIG with both constant and variable rotor parameters are given in Fig. 7.35 and Fig. 7.36 for the variation of excitation capacitance, load resistance and rotor speed shown in Fig. 7.34. The simulation results for the variable rotor parameters are indicated with black lines and for the constant rotor parameters with grey lines.

It has been previously established that the minimum capacitance and minimum rotor speed required for the initiation of self-excitation are not dependent on the variation of rotor parameters. The onset of self-excitation is mainly dependent on the magnitude of magnetising inductance.

The total rotor inductance is the sum of the magnetising inductance plus the leakage inductance. The leakage inductance is very small compared to the magnetising inductance and so any change in leakage inductance will have a negligible effect on the rotor circuit time constant. However, the rotor circuit time constant is inversely proportional to the rotor resistance. And as discussed in Chapter 6, the rotor resistance of an induction machine with variable rotor parameters increases with an increase in the magnitude of slip. Consequently, variation in the rotor resistance causes the rotor circuit time constant to change. For an induction machine with constant rotor parameters, however, the value of the rotor resistance which is obtained at slip equal to one (locked rotor test), remains constant and hence the rotor circuit time constant does not change.

During the initiation of self-excitation the induction generator operates at slip close to zero. An induction machine with variable rotor parameters has minimum rotor resistance at slip close to zero. Hence if the rotor resistance of an induction machine with variable rotor parameters is calculated only from the locked rotor test then this rotor resistance will be higher than the value of rotor resistance that occurs at low slips (close to zero).

201 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

The simulation of an induction machine with variable rotor parameters is contrasted with the same machine considering only constant rotor parameters and is shown in Fig. 7.35 and 7.36. The results related to the variable rotor parameters are shown with a dark line and indicated with “variable”. In Fig. 7.35a, the voltage build up of a SEIG when rotor parameters are assumed constant is faster than the voltage build up when rotor parameter variations are taken into account. When the SEIG with variable rotor parameters is loaded at t = 5sec the dynamic performance of the generated voltage is slow (large time constant) because of the small rotor resistance close to slip equal to zero. However, if the rotor parameter variations are neglected, that is, the rotor parameters are taken as constant, then the dynamic performance of the generated voltage is fast (small time constant). This is because of the large rotor resistance close to slip equal to one. When the excitation capacitance is increased at t = 10sec the variation of rotor parameters affects the dynamics of the generated voltage as explained for the case during loading conditions.

The dynamic performances of stator current (Fig. 7.35b), capacitor current (Fig. 7.35c) and magnetising current (Fig. 7.35e) are affected by variation in rotor parameters during the process of self-excitation, loading conditions and change in excitation capacitance. The dynamics of the load current (Fig. 7.35d) during loading and changes in excitation capacitance is similar to the dynamics of the stator and capacitor currents. The dynamic variation of magnetising inductance is shown in Fig. 7.35f.

202 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

400

200

Vrms (V) Vrms variable 0 0 2 4 6 8 10 12 14 16 (a) 10

5 Is (A) Is

0 0 2 4 6 8 10 12 14 16 (b) 6

4

Ic (A) 2

0 0 2 4 6 (c) 8 10 12 14 16

6

4 (A) L I 2

0 0 2 4 6 8 10 12 14 16 (d)

4 (A) m I 2 variable 0 0 2 4 6 8 10 12 14 16 (e) 0.6

0.4

Lm (H)Lm 0.2

0 0 2 4 6 (f) 8 10 12 14 16 time (sec)

Fig. 7.35 Comparison of constant and variable rotor parameters performance in SEIG (a) rms phase voltage (b) rms stator current (c) rms capacitor current (d) rms load current (e) rms magnetising current (f) magnetising inductance

203 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

60

40

20 frequency (Hz) 0 0 2 4 6 8 10 12 14 16 (a) 0.1 variable 0

slip s slip -0.1

-0.2 0 2 4 6 8 10 12 14 16 (b) 300

200

100 Torque (Nm) Torque

0 0 2 4 6 (c) 8 10 12 14 16

variable 4

2

0 0 2 4 6 8 10 12 14 16 (d) 6

4

Pmech (KW)2 (KW) Pelec

0 0 2 4 6 8 10 12 14 16 (e) 100

75

50

25 Efficiency % Efficiency 0 0 2 4 6 (f) 8 10 12 14 16 time (sec)

Fig. 7.36 Comparison of constant and variable rotor parameters performance in SEIG (a) generated frequency (b) slip (c) electromagnetic torque (d) electrical generated output power (e) mechanical input power (f) efficiency

204 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

The operating point of an induction generator is close to synchronous speed (slip equal to zero). Taking the rotor parameter variations into consideration the rotor resistance will have its minimum value in its operating region. However, when the rotor parameter variations are ignored the value of rotor resistance will be high, because the rotor resistance corresponds to a slip equal to one. As shown in Fig. 7.36 the effect of using constant rotor parameters for an induction generator exhibiting variable rotor parameters is the same as if external rotor resistance is added to the rotor circuit. When constant rotor parameters are used, the frequency (Fig. 7.36a) and efficiency (Fig. 7.36f) drop. Also, the magnitude of the operating slip (Fig. 7.36b) increases. In general, using constant rotor parameters means a larger rotor resistance than if the variable rotor resistance is used, resulting in the synchronous speed and efficiency both decreasing. A decrease in frequency reduces the generated voltage for the same magnitude of magnetising current. For electromagnetic torque, generated power and required mechanical power, they are dependent on the generated voltage and load current; however their dynamic performance is slower when modelled with variable rotor parameters.

It is noted that when the generated output power, calculated using constant rotor parameters, is almost the same as that when calculated using variable rotor parameters, the former requires greater mechanical power than the latter.

7.11 Summary The use of the variation in magnetising inductance with voltage leads to an accurate prediction of whether or not self-excitation will occur in a SEIG for various capacitance values and speeds in both the loaded and unloaded cases. The characteristics of magnetising inductance, Lm, with respect to the rms induced stator voltage or magnetising current determines the regions of stable operation as well as the minimum generated voltage without loss of self-excitation. Once self-excitation has been initiated and a steady state condition has been reached, the speed at which self-excitation ceases is always lower than the speed to initiate self-excitation.

205 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

At a particular speed the capacitance required for self-excitation, when the machine operates at no load, is less than the capacitance required for self-excitation when it is loaded.

When an induction machine operates as a motor the speed of the rotating air gap magnetic field is totally dependent on the excitation frequency. In synchronous generators the frequency of the generated voltage is dependent only on the speed of the prime mover, for a given number of poles. However, in the SEIG the frequency of the generated voltage depends on the speed of the prime mover as well as the condition of the load. Keeping the speed of the prime mover constant with increased load causes the magnitude of generated voltage and frequency of an isolated SEIG to decrease. This is due to a drop in the speed of the rotating magnetic field. When the speed of the prime mover drops with load, then the decrease in voltage and frequency will be greater than for the case where the speed is held constant.

The dynamic voltage, current, power and frequency developed by the induction generator have been analysed, simulated and verified experimentally for the loaded and unloaded conditions while the speed was varied or kept constant. Using the simulation algorithm more results which are not accessible in an experimental setup have been predicted. Increasing the capacitance value can compensate for the voltage drop due to loading but the drop in frequency can be compensated only by increasing the speed of the rotor. The variation of magnetising inductance follows the variation in terminal voltage or magnetising current. Increasing the capacitance can compensate the generated voltage, however it increases stator current. Hence care should be taken not to exceed the stator rated current.

The dynamic calculated performance comparison between the machine with constant and variable rotor parameters has been simulated and discussed considering a hypothetical induction machine that represents a typical double-cage or deep-bar rotor induction machine where rotor parameter variation is significant.

206 CHAPTER 7 EXCITATION OF THREE PHASE INDUCTION GENERATOR USING THREE AC CAPACITORS

All of these characteristics are the basic tools required to develop a control system, using power electronics, which will regulate the generated voltage and frequency for a SEIG over a wide variation in speed.

7.12 References [1] G. Strang, “Linear Algebra and its Application”, Harcourt Brace Jovanovich Publishers, San Diego, 1988. [2] A. E. Fitzgerald, C. Kingsley and S. D. Umans, “Electric machinery”, McGraw-Hill, London, 1992. [3] SIMNON-Simulation of nonlinear systems, SSPA Systems, Gothenburg, Sweden, 1993. [4] B. K. Bose, “Modern Power Electronics and AC Drives”, Printice-Hall, New Jersey, 2002. [5] Z. Zhang, C. Watthanasarn and W. Shepherd, “Application of a matrix converter for the power control of a variable-speed wind-turbine driving a doubly-fed induction generator”, in Proc. 1997 IEEE IECON’97 Conference, pp. 906-911.

207 CHAPTER 8

MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT

8.1 Introduction As discussed in Chapter 4, the iron loss in induction machines arises due to hysteresis and eddy current losses. Iron loss or core loss is represented in the induction machine model using Rm, a resistance value which has the same power loss as the total iron loss in the induction machine. For induction generators with small iron loss, i.e. large Rm, neglecting Rm in the machine model will make negligible difference to predicting the performance of the induction machine.

The steady state analysis of the SEIG including iron loss has already been reported [1, 2]. However, the steady state analysis is not able to show the dynamics of the SEIG. In all analyses reported in the literature based on the generalized machine theory and using the D-Q axes model of the SEIG, iron loss has been neglected. It is important to note that for stable operation of the self-excited induction generator, the machine has to operate in the region of magnetic saturation. Therefore, iron loss should be included in any accurate analysis. For small induction machines, the current associated with iron loss has almost the same per-unit value as the magnetising current [3]. Neglecting the iron loss in this case will cause a large error in the analysis. Few works have been reported which include iron loss in the D-Q axes model of an induction motor [4-6].

In this chapter a novel analysis (to the best knowledge of the author) for the dynamics of the self-excited induction generator driven by a variable speed prime mover and taking iron loss into account is given and establishes the error introduced if iron loss is

208 CHAPTER 8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT

neglected. Iron loss is represented as resistance Rm in the standard D-Q axes equivalent circuits. The D-Q model of the induction generator discussed in this chapter takes into account the actual non-linear variation of magnetising inductance Lm as a function of air gap voltage or magnetising current, and Rm as functions of air gap voltage and predicts the dynamics of the self-excitation process in the time domain. The complete study of the stability behaviour of the SEIG was previously not possible; however, this new model that includes the iron loss and variation in magnetising inductance provides the tool to do so and is presented below in a simplified way.

8.2 SEIG dynamic model including Rm To model the SEIG effectively, the parameters of the machine should be measured accurately. The parameters used in the SEIG can be obtained by conducting tests on the induction generator when it is used as a motor. The traditional tests used to determine the parameters are the open circuit (no load) test and the short circuit (locked rotor) test as discussed in Chapter 6. The induction machine used as the SEIG in this investigation is a three-phase wound rotor induction motor with specification: 415V, 7.8A, 3.6kW, 50Hz, and 4 poles. The parameters given in the D-Q equivalent circuit shown in Fig. 8.1 are obtained by conducting parameter determination tests on the above mentioned induction machine. As it is a wound rotor induction machine there is no variation of rotor parameters with speed. The parameters obtained from the test at rated values of voltage and frequency are Lls = Llr=11.4mH, Lm = 181mH, Rm = 1600:, Rs = 1.6:,

Rr = 2.75:.

For motoring application these parameters can be used directly. However, for self- excited induction generator application the variation of Lm with the induced stator voltage or magnetising current, and variation of Rm with the induced stator voltage should be taken into consideration to find the correct voltage build up. Using the correct parameters the dynamic currents, output power and induced electromagnetic torque can be predicted accurately during no load and loaded conditions.

209 CHAPTER 8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT

-Zr Oqr Rs Lls Llr R S r + - iLd i i ids dr C Cd imd Rm Lm RL Vcd Ods Odr

(a)

Zr Odr Rs Lls Llr R S r + - iLq i i iqs qr C Cq imq Rm Lm RL VCq Oqs Oqr

(b)

Fig. 8.1 No load D-Q model of a SEIG including core loss represented by Rm (a) d-axis (b) q-axis

8.3 Characteristics of Lm and Rm

As discussed in Chapter 7 Lm starts from a small value then increases to reach its peak value and finally starts to drop [7]. The characteristic of Lm is helpful for the stability of generated voltage and to determine the minimum generated voltage without loss of self- excitation. The value of Rm also exhibits variation and in general increases with generated voltage [2]. When the per-unit value of Rm is almost the same as Xm, neglecting Rm will give rise to an error in the analysis. However if the magnitude of Rm is much larger than Xm neglecting Rm will not have any significant effect on the results.

The variation of magnetising inductance, Lm, used in this investigation is the same as the one used in Chapter 7. The machine investigated in Chapter 7 is not a good example to illustrate the effect of iron loss because Rm is very large with that machine and its effect is minimal. However, this is not the case for all practical machines and an example to illustrate this is the machine investigated by Grantham [3]. The variation in

Rm is modeled by the following curve fit [2]

Rm = 3Vph+50 (8.1)

210 CHAPTER 8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT

where Vph is the phase rms voltage across Rm in parallel with Lm.

Vph can be calculated by subtracting the sum of the voltage drop in the stator winding resistance and stator leakage inductance from the stator terminal voltage of the induction machine while the induction machine is motoring. For generator application

Vph is the sum of the voltage drop in the stator impedance and the voltage at the terminals of the stator.

8.4 Analysis of SEIG including Rm Using the D-Q equivalent circuit model in Fig. 8.1, the equations for a SEIG including

Rm can be derived and used to simulate the machine dynamics. These equations for a self-excited induction generator, in the stationary stator reference frame are given as: iV ªº0 ªºRpLssNEW10 pC pL N 0 ªºªqs cqo º «»« » «»0 «»010RpL pC pL i V «» «»ssNEW N«»« dscdo » (8.2) «»« » «»0 «»pLNrmrrNEWrrNEWqrqrZZ L R pL L i K «» «»«»« » 0 ZZLpLLRpL ¬¼ «»¬¼rN N rrNEW r rNEW ¬¼¬«»«iKdr dr ¼»

Z IV VV Where

RmmL L=N Rmm+L p

L=L+LsNEW ls N

L=L+LrNEW lr N

VV and VV , are the initial voltages along the q-axis and d-axis cqo cq t0 cdo cd t0 capacitors respectively.

Kqr ZO r dro and Kdr ZO r qro are constants, which represent the initial induced voltages along the d-axis and q-axis respectively, and are due to remnant or

residual d-axis magnetic flux (Odro) and q-axis magnetic flux (Oqro) in the core.

From Equation (8.2) it is given that

0 = [Z][I]+[Vo] (8.3) The self-excitation currents are obtained from Equation (8.3) in the normal way, i.e. -1 [I] = –[Z] [Vo] (8.4)

211 CHAPTER 8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT from which, following a similar procedure as in Section 7.3.2.1, the stator self- excitation current along the d-axis is given by: U i (8.5) d ApBpDpEpFpGpHpJpM87 6543 2 Here, U is a function of the machine parameters, capacitance C, rotor speed and initial conditions. As U is in the numerator its detail is not important at this stage. However, the terms in the denominator contain the roots which are important in the determination of a growing or decreasing transient solution. In the denominator A, B, D, E, F, G, H, J, and M are functions of the machine parameters. The full expression for each of these coefficients is included in the Appendix C.

If one of the roots of the denominator in the expression for id has a positive real part then there is self-excitation. A positive real part of the root produces a growing transient response until saturation of the magnetising inductance is reached. Hence to find the roots, the denominator of Equation (8.5) is set to zero as in Equation (8.6) below.

Ap87 Bp Dp 6543 Ep Fp Gp Hp 2 Jp M 0 (8.6)

When there is self-excitation at least one of the eight roots will have a positive real part. During the initiation of self-excitation, as the generated voltage is close to zero, the values of Rm and Lm should be selected corresponding to a phase voltage close to zero. The curve for the onset of self-excitation is given in Fig. 8.2.

As can be seen in Fig. 8.2, when the magnitude of Rm is almost the same as Xm, for the same rotor speed, the SEIG requires more capacitance to have an onset of self- excitation. Or it can be said that to have self-excitation for a given capacitance value the model including Rm requires a higher speed than the model with Rm neglected. If the magnitude of Rm is very much larger than the magnitude of Xm there is no difference whether Rm is included or neglected.

212 CHAPTER 8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT

1500

|R |=|X | m m

1000 neglecting R m Speed (rpm)

500

0 0 50 100 150 200 250 300 Capacitance (micro F)

Fig. 8.2 Values of capacitance and speed for self-excitation

with and without Rm at no load

8.5 Simulation of dynamic self-excitation including Rm The matrix equation given in Equation (8.2) represents a SEIG in the D-Q axes model. Differential equations derived from this matrix equation are used to simulate the SEIG including Rm during loaded and unloaded conditions.

8.5.1 Simulation of dynamic self-excitation at no load Rearranging Equation (8.2) of the SEIG gives a 2nd order differential equation represented by

2 p IApIAIBpVBV o1o1 (8.7)

Where

ªºiqs ªºVcq «» «» i V I «»ds V «»cd «»i «»K «»qr «»qr ¬¼«»idr ¬¼«»Kdr

213 CHAPTER 8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT

ªº§·RRsmm R Rm «»¨¸00 «»©¹LLLls m ls Lls «» §·RRsmm R Rm «»00¨¸  «»©¹LLLls m ls Lls Ao «» «»RRmm§·Rr Rm 0 ¨¸Zr «»LLLL «»lr ©¹lr m lr «» RRmm§·Rr Rm «»0 Zr ¨¸ ¬¼«»LLlr ©¹lrL mL lr

ªºRR  ms 00 0 «»LL «»mls «»RR 000 ms «»LL «»mls A1 «» ZrmRRRRR mr§· m m «»0 Zr ¨¸ «»LLLLLlr m lr©¹ lr m «» «»ZrmRRRRR§· m m mr 0 Zr ¨¸ «»LLLLL ¬¼«»lr©¹ lr mr m lr

ªºR  m 000 «»LL «»mls «»R 000 m «»LL B «»mls o «»R «»00 m 0 «»LLmlr «»R «»000 m ¬¼«»LLmlr ªº1  000 «»L «»ls «»1 000 «»L B «»ls 1 «»1 00 0 «»L «»lr «»1 «»000 ¬¼«»Llr 1 1 VidtV , VidtV  cq C ³ qs cqo cd C ³ ds cdo

214 CHAPTER 8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT

Expanding Equation (8.7) gives four sets of second order differential equations which are given below

2 RRms() LR ms RL mls RL mm R m R m pVcq piqs i qs pi qs  pi qr V cq (8.8) LLm ls LL m ls L ls LL m ls L ls

2 RmsRLRRLRLR() ms mls mm m Rm pVcd pids ids  pids  pidr Vcd (8.9) LLmls LLmls Lls LLmls Lls RZ R RR() LR RL RL p2ipiii mrmmrmrm lrmmpi qr Lqs Lds LL qr LL qr lr lr m lr m lr (8.10) Zrmmlrm()RL LR Rm pKqr ipiKdrZ r dr qr LLm lr LLm lr Llr ZZR R() RL LR RR p2iipiipii rm m r mm lrm Z mr dr LLqs ds LLqr r qr LLdr lr lr m lr m lr (8.11) ()LRmr RL mlr RL mm Rm pKdr pidr Kdr LLmlr LLmlr Llr

Since the SEIG is operating at no load the stator current is equal to the capacitor current. Hence the equations relating the capacitor currents and capacitor voltages are i pV qs (8.12) cq C i pV ds (8.13) cd C

The simulation was carried out using SIMNON [8] to solve the 1st order and 2nd order differential equations given in Equations (8.8) to (8.13). The dynamic simulation of the SEIG operating at no load is given in Fig. 8.3.

The error in the steady state no load developed voltage, shown in Fig. 8.3, is very small such that it can not be seen with the given scale. From the simulation it was discovered that if the SEIG is operating at no load or if the magnitude of Rm is much greater than

Xm then there is no significant difference in the analysed results whether Rm is included or neglected.

215 CHAPTER 8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT

300

Without RmRm

250

200

With RmRm (V) ph V 150

100

50

0 0 0.5 1 1.5 2 2.5 3 3.5 4 time(sec)

Fig. 8.3 No load RMS phase voltage during self-excitation with and without Rm

8.5.2 Dynamics of SEIG during loading Power will flow to the load when the switch S in Fig. 8.1 is closed. The load is connected across the capacitors and the terminal voltage is the voltage developed across the capacitors. Equations (8.8) to (8.11) used in the no load condition will be used in the loaded case. The equations that relate the stator current, load current, capacitor current and terminal voltage or capacitor voltage are the same as the equations given in Equations (7.54) to (7.59).

Since Rm is included, the magnetising current is not the sum of the stator current and the rotor current. Some of the current is bypassed through Rm. Hence from the magnetising inductance branch circuit of Fig. 8.1 the q-axis magnetising current is

Rm iiimq qs qr (8.14a) RLpmm and in integral form it can be written as

216 CHAPTER 8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT

R iiiidt m  (8.14b) mq³ qs qr mq Lm

For the d-axis magnetising current

Rm iiimd ds dr (8.15a) RLpmm and in integral form it is written as R iiiidt m  (8.15b) md³ ds dr md Lm

As discussed in the induction machine model including Rm in Section 4.6, the induced electromagnetic torque or the mechanical torque required to drive the SEIG is given by [9]: 3 JJG JG TPI O u (8.16) epmr2 JJG JG Substituting Om and Ir in Equation (8.16) and rearranging using vector manipulation gives:

3 RLmm Te P p ii qs dr ii ds qr (8.17) 2 RLpmm

The dynamic induced electromagnetic torque with Rm included can be expressed in the integral form as

3 §·R TPRiiiiTdt m (8.18) e p m³¨¸ qsdr dsqr e 2 ©¹Lm

The results given below are from the simulation of the SEIG when it is driven at constant speed (1500rpm) and variable capacitance and resistive load are connected at the stator terminals. The variations of capacitance and resistance are given in Fig. 8.4.

The dynamic variation of voltage current, power and torque, for variations in resistive load and capacitance, are shown in Figs. 8.5 to. 8.8. At no load the effect of Rm is insignificant. However, when the induction generator is loaded, neglecting Rm will result in an error. When Rm is included, which depicts the actual situation, the generated voltage, currents, and output power, are lower than for the case when Rm is neglected.

217 CHAPTER 8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT

However, due to additional losses the electromagnetic torque developed in the model with Rm included is higher than that without Rm.

80

70

60 C (micro Farad)

50 0 2 4 6 8 10 12 14 16 18 20

100

80 (ohm) L R 60

40 0 2 4 6 8 10 12 14 16 18 20 tim e (s ec)

Fig. 8.4 Variation of connected capacitor and resistor

300

without Rm

250

200 with Rm (V) ph V 150

100

50

0 0 2 4 6 8 10 12 14 16 18 20 tim e (s ec) Fig. 8.5 The dynamic rms generated voltage with variation of load and capacitance

218 CHAPTER 8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT

6

without Rm 4 (A) L i 2 w ith Rm 0 0 2 4 6 8 10 12 14 16 18 20 10

w ithout Rm

5 (A) c i with Rm

0 0 2 4 6 8 10 12 14 16 18 20 10 w ithout Rm

5 (A) s i w ith Rm

0 0 2 4 6 8 10 12 14 16 18 20 time (sec)

Fig. 8.6 Dynamic currents in the load, capacitor and stator with variation in load and capacitance

3500

w ithout Rm 3000

2500

2000

Power (W) 1500 w ith Rm

1000

500

0 0 2 4 6 8 10 12 14 16 18 20 time (sec)

Fig. 8.7 The dynamic output power with variation in load and capacitance

219 CHAPTER 8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT

30

25

w ith Rm (Nm)

e 20 T

15 w ithout Rm

10

5

0 0 2 4 6 8 10 12 14 16 18 20 time (sec)

Fig. 8.8 The dynamic electromagnetic torque with variation in load and capacitance

8.6 Summary In this chapter a novel D-Q axes model that includes the iron loss equivalent resistance

Rm in the dynamic analysis and simulation of the SEIG has been described in a simple and understandable way. In some instances it is possible to neglect the iron loss; however its effect has to be shown to be negligible. This paper provides the tool to reach such a decision. When Rm is included, which depicts the actual situation, the generated voltage, currents, and output power, are lower than that when Rm is neglected. However, due to additional losses the electromagnetic torque necessary to drive the machine is higher when Rm is included than when it is neglected.

Additionally, the method presented here is the first to include the effects of iron loss in the more general dynamic analysis of induction machines and their high performance drives. It is noted that this method is easily understood, having drawn on many familiar concepts and using the standard terminology and nomenclature of D-Q unified machine theory.

220 CHAPTER 8 MODELLNG OF AN ISOLATED SELF-EXCITED INDUCTION GENERATOR TAKING IRON LOSS INTO ACCOUNT

8.7 References [1] N. H. Malik and S. E. Haque, “Steady state analysis and performance of an isolated self-excited induction generator”, IEEE Transaction on EC, Vol. 1, No.3, 1986, pp.134-139. [2] T. F. Chan, “Analysis of Self-Excited induction generators using an iterative method”, IEEE Transaction on EC, Vol. 10, No.3, 1995, pp.502-507. [3] C. Grantham, “Determination of Induction Motor Parameter Variations From a Variable Frequency Standstill test”, Electric Machines and Power Systems, Vol.10, No.2-3, 1985, pp.239-248. [4] S. D. Wee M. H. Shin and D. S. Hyun, “Stator-flux-oriented control of induction motor considering iron loss”, IEEE Trans. on Industrial Electronics, Vol.48, No.3, June 2001, pp. 602-608. [5] J. W. Choi D. W. Chung and S. K. Sul, “Implementation of field oriented induction machine considering iron losses”, IEEE- APEC '96. Conference Proceedings, 1996, pp. 375-379. [6] E. Levi M. Sokola A. Boglietti and M. Pastorelli, “Iron loss in rotor-flux-oriented induction machines: identification, assessment of detuning, and compensation”, IEEE Trans. on Power Electronics, Vol.11, 1996, pp. 698 –709. [7] D. Seyoum C. Grantham and F. Rahman, “Analysis of an isolated self-excited induction generator driven by variable speed prime mover”, Proc. AUPEC’01, 2001, pp.49-54. [8] Simnon-Simulation of nonlinear systems, SSPA Systems, Gothenburg, Sweden, 1993. [9] B. K. Bose “Modern Power Electronics and AC Drives”, Prentice-Hall, NJ, 2002.

221 CHAPTER 9

INVERTER/RECTIFIER EXCITATION OF A THREE- PHASE INDUCTION GENERATOR

9.1 Introduction The main drawback of using induction generators excited by three AC capacitors is their inherently poor voltage regulation and uncontrollable frequency of operation. The output voltage of a SEIG can be controlled by introducing an appropriate voltage regulating scheme. A number of schemes have been suggested for this purpose [1-5]. However, the variation of the frequency of the SEIG with load and speed cannot be regulated by static means. As a result the equipment supplied by the three-phase SEIG discussed in Chapter 7 should be frequency insensitive (e.g. heater, water pump, lighting, battery charging etc).

The scheme based on switched capacitors [1] finds limited application because it regulates the terminal voltage in discrete steps. A saturable reactor scheme of voltage regulation [3, 5] involves a potentially large size and weight, due to the necessity of a large saturating inductor. In the short/long shunt configuration [4] the series capacitor used causes the problem of resonance while supplying power to an inductive load.

In a three phase capacitor excited induction generator the value of capacitance should be varied so that the terminal voltage remains constant at different rotor speeds. It is also shown that the value of capacitance is influenced by the load as well as by the load power factor [1]. The problem is further aggravated by the uncertainty of the machine to re-excite after a short circuit unless some charge is provided [6]. Loss of self-excitation could be disastrous in applications like aircraft power supplies. There should be a way to avoid this problem. An isolated induction generator with an excitation system

222 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR provided by a single capacitor on the DC link side of the inverter can re-excite even after a short circuit. Since a battery is required to control the switching of the IGBTs of the inverter, the same battery can be used for the initiation of voltage build up using vector control.

The excitation provided by a single capacitor on the DC link side of the inverter is reported in [7-9]. In these papers the mechanism of maintaining constant output DC link voltage for variable rotor speed is not clearly indicated. When the excitation comes from the DC side capacitor of the converter, as covered in this chapter, then varying the current flowing to the generator by controlling the switching of the IGBTs varies the flux in the generator. Due to the switching of the inverter/rectifier the single DC side capacitor acts like a three-phase capacitor. When the fundamental switching frequency of the converter is varied the reactive capacitance of the DC side capacitor will be varied as seen from the induction machine side. Overall the single DC side capacitor provides all the reactive current or the VAR required by the induction generator.

In a grid connected induction generator, the grid acts as a stiff voltage source so that the generator control structure is similar to a standard drive with sinusoidal front-end converter, i.e. by varying the modulation index the terminal voltage at the induction generator can be varied with the rotor speed while the DC bus is maintained at constant voltage.

For an induction generator operating in stand-alone mode there should be a system that regulates the output voltage. The output voltage is the DC voltage and the control system, which is implemented using vector control, is required to keep this DC voltage at a constant level. The frequency of the AC voltage can vary with speed but the aim is to have constant peak voltage and as a result to have constant DC voltage. Once a constant DC voltage is achieved a DC load can use it directly or, if required, it is a matter of having an inverter to produce a constant voltage and frequency AC output. The electrical and mechanical connections for an isolated induction generator driven by a wind turbine are shown in Fig. 9.1. To simplify the diagram the control system is not included in Fig. 9.1.

223 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

In Fig. 9.1 the turbine rotor speed will be varied depending on the wind speed. The system is loaded by a DC load connected at the terminals of the DC capacitor or an AC load can be connected via a second inverter that adjusts the frequency and peak voltage of the generated AC power supply. Induction generators can be excited from a single DC capacitor by using an inverter/rectifier arrangement. The voltage build up process is started from a small voltage in a charged DC capacitor or from a battery. During the voltage build up process the DC capacitor gets its charge from the induction generator via the rectifier and the capacitor in turn supplies the excitation current or the reactive VA (VAR) required by the induction generator for its operation. This is implemented using the well established field oriented vector control technique.

Rotor blade

Gear box

Wind direction

Cdc

Induction generator

Fig. 9.1 Electrical and mechanical connections

9.2 Vector control Since the introduction of the basic principles of vector control in the 1970s, vector control of induction machines has been the standard means of achieving high performance in induction motor drives. Vector control is a mathematical control method based on the space vector theory implemented by controlling the phase angle and modules of the current in the synchronous reference frame [10-14]. In these papers it has been shown that with the help of field oriented vector control, induction motors can have performance similar to separately excited DC motors, where the torque and flux

224 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR are controlled independently by the naturally decoupled armature current and field current, respectively.

The technique of vector control established to control induction motor drives will be used in the control of an induction generator to control the generated voltage for different loading conditions and variable rotor speeds. In a DC generator the main flux (due to field current) and the load current can be controlled independently because they are fixed in space. The field current is fixed, corresponding to a given speed, to generate rated voltage. The armature current increases without affecting the field current when the DC generator is loaded. For an increase in armature speed the flux should be decreased by decreasing the field current to maintain the generated voltage at the rated value. The generated voltage in the DC machine is given by [15]

Ekaa IZ (9.1) where k – dependent on the armature winding

Ea – armature generated voltage I – flux per pole

Za – angular speed of the armature

The DC generator will continue to operate based on a control system that adjusts the field current inversely proportional to the change in armature angular speed. The principle of decoupling the flux producing current and power producing current in an induction generator will be used to produce performance similar to the DC generator. The principle of field oriented vector control will be discussed briefly before its use in the induction generator. Based on the orientation of the flux, the three field-oriented control schemes are, rotor flux oriented, magnetizing flux oriented and stator flux oriented. The magnetizing flux oriented vector control will not be discussed here as its analysis is similar to the principles discussed in rotor oriented and stator oriented vector controls schemes.

9.2.1 Rotor flux oriented vector control As its name states, the rotor flux oriented vector control is produced when the alignment of the d-axis current in the synchronously rotating reference frame is along the rotor

225 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR flux linkage space vector. This technique makes the q-axis rotor flux linkage to be equal to zero and the d-axis flux linkage to be equal to the total rotor flux linkage in the induction machine.

The vector diagram for the rotor flux oriented vector control of an induction machine is shown in Fig. 9.2. In Fig 9.2 the rotor flux linkage, which is aligned along the d-axis of the rotating reference frame, is rotating at synchronous speed, however the rotor speed is lagging behind the rotor flux linkage space vector when the induction machine is motoring and it is leading when the machine is generating, as discussed in Chapter 4.

 e = O qr

e iqs s Oqr q s e ids

e Tsl Odr = Or s Odr Tr Ro Te to r a Zsl x i s d e Ze Zr s d Fig. 9.2 Vector diagram for rotor flux oriented vector control

s s If the d-axis rotor flux linkage, Odr , and the q-axis rotor flux linkage, Oqr , in the stationary reference frame, are calculated, then the modulus or magnitude of the rotor flux linkage will be

ss22 OOrdrqr ()() O (9.2)

and the space angle of Or is given as §·O s T tan1 qr (9.3) e ¨¸s ©¹Odr

226 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

If the magnitude of the rotor flux linkage, Or, is calculated in the rotating reference frame the space angle of Or cannot be evaluated from components of the rotor flux linkage. In this case the space angle Te is obtained from the slip angle Tsl and rotor angle

Tr as illustrated in Fig. 9.2, and is given by

TTTeslr  (9.4)

The slip angle Tsl is negative for induction generators.

With rotor flux oriented vector control the space vector of the rotor flux linkage is aligned along the d-axis of the synchronously rotating reference frame so that all the rotor flux linkage will be produced by the d-axis current in the rotating reference frame.

The space angle Te is used for transformation from stationary reference frame to the rotating reference frame and vice versa. The rotor flux oriented vector control is implemented using two schemes, direct and indirect, to obtain the magnitude and space angle of the rotor flux linkage space vector.

9.2.1.1 Direct (feedback) flux oriented vector control With the direct rotor flux oriented control the modulus and space angle of the rotor flux linkage are calculated from the measured currents, voltages and/or rotor speed using different types of flux models, or the rotor flux linkage can be directly measured using hall-effect sensors, search coils, etc. In this investigation the direct estimation of rotor flux linkage from the measured currents, voltages and/or rotor speed was used. a) Rotor flux linkage calculation utilising monitored stator voltages and currents in the stationary reference frame Once the stator flux linkage is calculated from the measured stator voltages and currents then using Equation (4.5) the d-axis rotor current is expressed as

s s s Ods Li s ds idr (9.5) Lm and from Equation (4.6) the q-axis rotor current is expressed as

s s s Oqs Li s qs iqr (9.6) Lm

227 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

Substituting Equation (9.5) in Equation (4.9)

s Lr ss OOVdr dsLi s ds (9.7) Lm L 2 where V 1  m LLrs And Substituting Equation (9.6) in Equation (4.10)

s Lr ss OOVqr qsLi s qs (9.8) Lm Here the estimation of rotor flux linkage is dependent on the parameters of the induction machine Ls, Lm, Lr, and Rs, where the effect of stator resistance is coming from the estimation of stator flux linkage. b) Rotor flux linkage calculation utilising monitored stator currents and rotor speed evaluated in the stationary reference frame With a shorted rotor circuit Equations (4.11) and (4.12), the rotor side voltage equations, are rewritten as dO s dr Ri ssZO 0 (9.9) dt rdr r qr dO s qr Ri ssZO 0 (9.10) dt rqr r dr

From Equation (4.9) and (4.10) the rotor currents are expressed in terms of rotor flux linkages and stator currents as

s s s Odr Li m ds idr (9.11) Lr

s s s Oqr Li m qs iqr (9.12) Lr The rotor flux linkages can be expressed in terms of the stator currents by substituting Equation (9.11) in (9.9) and Equation (9.12) in (9.10) and after simplifying gives

§·L O s OZOss mdridt s (9.13) dr³¨¸ ds r qr ©¹TTrr

228 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

s ss§·Lm Oqr s OZOqr ¨¸idt qs r dr (9.14) ³¨¸TT ©¹rr

Lr Where Tr is the rotor circuit time constant. Rr From Equations (9.13) and (9.14) the modules and space angle of the rotor flux linkage can be estimated using Equations (9.2) and (9.3), respectively. However, the accuracy of the estimation is dependent on the rotor inductance and rotor resistance. Hence any variation in rotor parameters will affect the estimation procedure. c) Rotor flux linkage calculation utilising monitored stator currents and rotor speed evaluated in the rotor flux linkage oriented rotating reference frame

Using the D-Q representation of an induction machine in the reference frame rotating at synchronous speed, as illustrated in Fig. 4.19, the voltage equations from the rotor side are given by

e dOdr ee Rirdr ZZO e r qr 0 (9.15) dt

e dOqr ee Rirqr ZZO e r dr 0 (9.16) dt and the rotor flux linkages are expressed as

eee Odr Li m ds Li r dr (9.17)

eee Oqr L miLi qs r qr (9.18) From Equations (9.17) and (9.18) the rotor currents are given as eeLi e Odr m ds idr (9.19) Lr

eeLi e Oqr m qs iqr (9.20) Lr To express the Equations (9.15) and (9.16) in terms of accessible and measurable stator currents, Equations (9.19) and (9.20) are used to give

e dROdr reee L m OZOdrRi r ds  sl qr 0 (9.21) dt Lrr L

229 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

e dOqr RLrmeee OZOqrRi r qs  sl dr 0 (9.22) dt Lrr L

where ZZZsler  is the slip frequency as illustrated in Fig. 9.2.

With rotor flux oriented vector control, shown in Fig. 9.2, the total rotor flux is aligned along the d-axis of the rotating reference frame so that

e OOdr r (9.23) and

e Oqr 0 (9.24) Then dO e qr 0 (9.25) dt

Substituting Equations (9.23) to (9.25) in Equation (9.21) and utilising Tr = Lr/Rr gives dO TLir O e (9.26) rrmdsdt Equation (9.26) shows that the rotor flux linkage can be evaluated from the d-axis measured stator current in the rotating reference frame. It is implemented by applying a

e low pass filter to ids , with a cut off frequency of 1/Tr and a gain of Lm. Since the flux linkage is dependent only on the d-axis current and not on the q-axis current the rotor flux oriented vector control scheme gives decoupled control when implemented with a current regulated source.

And substituting Equations (9.23) to (9.25) in Equation (9.22) will give

Lm e Zslq i s (9.27) OrrT At steady state the rotor flux linkage is constant. Hence Equation (9.26) can be rewritten as

e Ormds Li (9.28) and

230 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

e 1 iqs Zsl e (9.29) Tirds

Here also, the accuracy of the rotor flux linkage estimation is dependent on the rotor inductance and rotor resistance. Any variation in rotor parameters will affect the estimated rotor flux linkage and slip speed. The rotor resistance can vary with temperature and the rotor inductance can be varied with the magnetising saturation level of the iron core.

9.2.1.2 Indirect (feed forward) flux oriented vector control In this scheme the reference d-axis current is computed from reference values in a feed forward manner not from measured values. In the implementation of the indirect rotor flux oriented control of an induction machine the space angle of the rotor flux linkage space vector is obtained from the rotor angle and the computed slip angle.

The derivation is the same as given in Part c of Section 9.2.1.1. The main difference is that in the indirect flux oriented scheme the rotor flux linkage is given as a reference value and then the reference current is evaluated from Equation (9.26) given by

* e* 1 TprrO ids (9.30) Lm At steady state the derivative term in Equation (9.30) is eliminated because at a given operating point the magnitude of the rotor flux linkage is constant.

The slip speed, from Equation (9.27) will be given as

e* Lm iqs Zsl * (9.31) TrrO

This scheme is a simple one; however it needs a good prior knowledge of the characteristics of the rotor flux linkage because there is no flux controller. As in Part c of Section 9.2.1.1 the calculation of reference current and the slip speed is dependent on rotor parameters.

231 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

9.2.2 Rotor flux oriented control with voltage as the controlled variable The implementation of the rotor flux oriented vector control method discussed in Section 9.2.1 is based on a current regulated source with the current controllers in the stator stationary reference frame. However it is also possible to utilise stator voltage equations in the synchronously rotating reference frame and express the stator voltage equations using rotor flux linkages and stator currents. In this case the current controllers are implemented in the synchronously rotating reference frame to produce the voltage reference signal.

Using the D-Q representation of an induction machine in the reference frame rotating at synchronous speed illustrated in Fig. 4.19 the voltage equations from the stator side are given by dO e VRiee ZO e ds (9.32) ds s ds e qs dt

dO e VRiee ZO e qs (9.33) qs s qs e ds dt and the stator flux linkages are given by

ee e Ods Li s ds L m i dr (9.34)

ee e Oqs Li s qs L m i qr (9.35)

Substituting Equation (9.19) into (9.34) and Equation (9.20) into (9.35) and then simplifying gives

eeLm e OVds Li s ds O dr (9.36) Lr

eeLm e OVqs Li s qs O qr (9.37) Lr L 2 where V 1  m LLrs Substituting Equations (9.36) and (9.37) into Equations (9.32) and (9.33) and then rearranging to give

232 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

ee ee eZOemLdiLd e ds m dr VRiLids sdsZV e sqs  O qr  V L s  (9.38) LdtLdtrr

ee ee eZemLL ediqs m dO qr VRiLiqs sqsZV e sds  O dr  V L s  (9.39) LdtLdtrr Since the stator voltage equations are to be expressed in the reference frame fixed to the rotor flux linkage space vector, the total rotor flux linkage space vector is aligned along the d-axis of the reference frame rotating at synchronous speed Ze. Hence the conditions given in Equations (9.23) to (9.25) will be substituted into Equations (9.38) and (9.39) to give

e ee edids L m dO r VRiLiLds sdsZV e sqs  V s  (9.40) dt Lr dt

e ee eZemL diqs VRiLiqs sqsZV e sds  O r  V L s (9.41) Ldtr

e In Equation (9.40) the q-axis term iqs appears in the d-axis voltage expression and in

e e Equation (9.41) the d-axis terms ids and Or (equal to Odr ) appear in the q-axis voltage expression. These give rise to unwanted coupling and do not produce the ideal DC

e machine like characteristics, where the d-axis stator current, ids , is the rotor flux

e linkage producing component and the q-axis stator current, iqs , is the active power

e e producing component. The stator currents ids and iqs can only be independently controlled if the stator voltage Equations (9.40) and (9.41) are decoupled. Then controlling the stator voltages of the induction machine will indirectly control the stator

e e currents ids and iqs .

Taking the rotor flux linkage as constant at a given rotor angular speed the stator current components can be independently controlled if the decoupling rotational voltage components are arranged in such a way that they will cancel the effect of the coupling. The decoupling technique is implemented by adding the coupling terms to the voltage signals obtained from the output of the controllers for the d-axis and q-axis stator currents in the synchronously rotating reference frame. The decoupling term to be added in the d-axis voltage is

233 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

e VLiddeco ZV e s qs (9.42) and the decoupling term to be added in the q-axis voltage is

e ZemL VLiqdeco ZV e s ds O r (9.43) Lr

Using the decoupling terms in the calculation of d-axis reference voltage and q-axis reference voltages the currents that control the flux and the active power generated by induction generator will be controlled indirectly.

9.2.3 Stator flux oriented vector control In the stator oriented reference frame all variables are expressed in a reference frame aligned with the stator flux linkage space vector. In stator oriented vector control the stator flux linkages are determined from the measured phase voltages and phase currents in the stationary (stator) reference frame, as illustrated in Fig. 4.6, and given by

O sss ()vRidt (9.44) ds³ ds s ds O sss ()vRidt (9.45) ds³ ds s ds Then the modulus or magnitude of the stator flux linkage will be calculated using

ss22 OOsdsqs ()() O (9.46) and the space angle of Os is given as §·O s T tan1 qs (9.47) e ¨¸s ©¹Ods In the estimation of the stator flux linkage space vector the stator resistance is the only machine parameter involved. Hence the accuracy of the estimated stator flux linkage space vector is dependent on the stator resistance. Because of accessibility the stator resistance can be measured with good accuracy and it is possible to predict its variation with temperature. In the rotor flux oriented vector control, with the involvement of rotor resistance, the rotor resistance variation becomes dominant due to temperature and skin effect in squirrel cage induction machines as discussed in Chapter 6. Compensation of this parameter is difficult because of inaccessibility, but it is easier to compensate stator resistance [16].

234 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

The induction generator is controlled to generate almost constant terminal voltage which is the rated voltage. It is reasonable to consider that the stator flux linkage can be estimated accurately. However, the small effect of variation of stator resistance creates a small error in the estimated flux, though this is seen as a reasonable compromise.

It can be shown that using trigonometric identities and derivative rules for tanT x then ddT 1 dx tan1 x (9.48) dt dt1 x2 dt Using Equations (9.44), (9.45), (9.47) and (9.48) the synchronous angular frequency can be estimated as

s s dOqs dOds s §·OOds qs ddTe 1 Oqs dt dt Ze ¨¸tan dt dt ¨¸s 22 ©¹OOOds qs ds

OOds viR qs qs s qs viR ds  ds s Ze 22 (9.49) OOqs ds

The vector diagram for stator flux oriented vector control is illustrated in Fig. 9.3

 e = O qs

e i qs s Oqs q s e ids

e Ods = Os s Ods

Te

d e Ze

s d Fig. 9.3 Vector diagram for stator flux oriented vector control

235 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

Since in the stator oriented reference frame all variables are expressed in a reference frame oriented to the stator flux linkage space vector, a mathematical model needs to be developed to find the relationship between the stator flux linkage and the stator currents. The rotor flux linkages in the synchronously rotating reference frame can be expressed in terms of stator flux linkages and stator currents from Equations (9.36) and (9.37) as

eeeLr OOVdr dsLi s ds (9.50) Lm

eeeLr OOVqr qsL si qs (9.51) Lm L 2 where V 1  m LLrs

Using the induction machine model in a reference frame rotating at synchronous speed, as shown in Fig. 4.19, the stator flux linkages are expressed as

ee e Ods Li s ds L m i ds (9.52)

ee e Oqs Li s qs L m i qs (9.53) From Equations (9.52) and (9.53) the rotor currents can be expressed as

ee e Ods Li s ds idr (9.54) Lm

ee e Oqs Li s qs iqr (9.55) Lm

Equations (9.50), (9.51), (9.54) and (9.55) are substituted in the voltage equations (9.15) and (9.16) to eliminate the rotor flux linkages and rotor currents from the induction machine equations and to express the machine equations in the reference frame fixed to the stator flux linkage space vector. The new expressions are given by ddO eei RLOVZOZVeeds RLiLLL ds  ee  LLi 0 (9.56) rds rdt rsds r sdt slrqs slr sqs ddO eei RLOVZOZVeeqs RLiLLL qs  ee  LLi 0 (9.57) rqs rdt rsqs r sdt slrds slr sds

236 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

With stator flux oriented vector control, illustrated in Fig. 9.3, the total stator flux is aligned along the d-axis of the synchronously rotating reference frame so that

e OOds s (9.58) and

e Oqs 0 (9.59) Then dO e qs 0 (9.60) dt

Using Equations (9.58) to (9.60) and utilising Tr = Lr/Rr, Equations (9.56) and (9.57) can be rewritten as

ee 11TprOV s  Tp r Li s ds ZV sl T r Li s qs (9.61)

ee ZOVslrTLiLTpi s sds s 1 V r qs (9.62) where p is the differential operator, i.e. p = d/dt

In rotor flux oriented vector control with current as the controlled variable, the total rotor flux linkage, as derived in Equation (9.26), is controlled by the d-axis current in the synchronously rotating reference frame. That is the rotor flux oriented vector control with current as the controlled variable gives a natural decoupling. However, as shown in Equation (9.61), with stator flux oriented vector control, which is based on an impressed stator current controller, the total stator flux linkage, Os , and the q-axis stator current

e e e iqs , are coupled. This means that any change in iqs without changing ids will cause unwanted transients to occur in the stator flux linkage.

The undesirable coupling can be eliminated by utilising a decoupling circuit in the flux linkage control loop. The decoupling circuit is implemented at the output of the stator flux linkage controller. The stator flux linkage controller is a PI controller and its output is designed to be the d-axis current required to produce the reference stator flux linkage

e say ids 1 given by

e* iGds1 OO s s (9.63) where G is the transfer function of a PI controller.

237 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

e However, due to the coupling problem, ids 1 does not have total control over the stator

e* flux linkage. The reference d-axis current, ids , with full control of the stator flux linkage is expressed as

e* e iids ds1  i deco

e* * iGds OOs s i deco (9.64) Substituting Equation (9.64) in (9.61) gives

*e 11TprsOV  Tp rss LG OOV  s 1Tp rsdecoslrsqs Li ZV T Li (9.65)

For decoupled control with the help of ideco the last two terms of Equation (9.65) should be cancelled, i.e.

e 10VZVTprsdecoslrsqs Li T Li (9.66) so that

e ZVslrTi qs ideco (9.67) 1 VTpr The angular slip frequency is available from Equation (9.62) given by

e LTpis 1 V rqs (9.68) Zsl e TLirs OV sds

A differentiator is very sensitive to noise and the differentiator in the numerator of Equation (9.68) can introduce noise in the current regulator. A low pass filter with appropriate cut off frequency can be introduced to overcome the noise problem. In the steady state, Equations (9.67) and (9.68) can be written as

e iTideco ZV sl r qs (9.69)

Li e sqs (9.68) Zsl e TLirs OV sds

In stator flux oriented vector control the rotor speed can be estimated from the estimated synchronous angular frequency and the estimated angular slip frequency given by

ZZZresl 

238 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

e In induction generators the value of Zsl andiqs are negative because power is being converted from mechanical to electrical power.

The advantage of the stator flux oriented vector control is that the stator flux linkage vector is derived by integration of the voltage behind the stator resistance and it is sensitive to stator resistance only, which can be compensated somewhat easily because of accessibility. The drawback of the stator flux oriented vector control is the coupling effect which requires the introduction of the decoupling compensation ideco . In the stator flux oriented vector control the induction generator behaves like a separately excited DC generator by aligning the total stator flux linkage of the induction generator along the d-axis of the rotating reference frame. Hence the stator flux linkage in the induction machine is controlled by the d-axis stator current in the rotating (excitation) reference frame.

9.3 System description The system description for the implementation of vector control in an isolated induction generator is shown in Fig. 9.4. The proposed system starts its excitation process from an external battery Vb or it can be started from a charged capacitor. The external battery Vb helps to charge the capacitor and also to start the build up of flux in the core. When the generated DC voltage rises to a value higher than Vb then the diode blocks the back flow of current to the battery. The diode can also be replaced with a switch that is operated by comparing the value of the battery voltage and the magnitude of the generated DC voltage in the capacitor.

239 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

S Inverter/ Rectifier Induction Generator Load Voltage Wind C V Turbine and current b sensors

Speed sensor PWM generator

Iabc Vabc VDC

Zr

PC with dSPACE DS1102 Fig. 9.4 System description

As discussed in Chapter 7 a self-excited induction generator using three AC capacitors and without any voltage control can start its voltage build up only from a remnant magnetic flux in the core. The voltage build up starts when the induction generator is driven at a given speed and an appropriate capacitance is connected at its terminals. However, for a system with a single DC capacitor, as proposed in this Section, the voltage build up process cannot start from the remnant flux in the core due to the following reasons: a) There is no way of initiating the injection of exciting current into the induction generator via the inverter using vector control. b) The terminal voltage is not continuous because of the PWM switching in the inverter and the current is not sinusoidal. c) The power loss in the machine, the switching power losses in the inverter and the characteristic of all the instantaneous losses due to PWM switching. d) The losses in the generator increase because of the harmonics.

As a result an initial voltage is required in the DC capacitor to start the vector control and to allow the voltage build up process. The initial voltage can be obtained from a previously charged DC capacitor or from a battery connected to the DC capacitor. The easiest method is to utilise the battery that supplies power to the IGBT drivers. The minimum initial voltage required in the DC capacitor is dependent on the components used in the inverter/rectifier, their combined forward voltage drop in the converter

240 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR arrangement and the parameters of the induction generator. The simulation of the system is implemented using Matlab/Simulink. From the simulation, the voltage build up process can start for voltage as low as 10V. However, to have a good control and faster voltage build up process 48V is used. 48V is appropriate to implement the proposed scheme at the lowest speed and the 48V can also be obtained from connecting several standard commercially available 12V batteries in series.

As discussed in Chapter 7 during the voltage build up process the magnetizing inductance, Lm, in an induction machine varies with the air gap voltage across it. The implementation of the variation of magnetizing inductance in the model of induction generator depicts the actual variation in the real system.

9.4 Establishment of reference flux linkage Wind speed can vary at any time. Variation in wind speed affects the variation in rotor speed of the induction generator. Then variation in rotor speed affects the generated voltage unless there is a well-designed control system to regulate the generated voltage.

In motoring applications, all control schemes use constant flux for rotor speeds lower than the rated speed. The flux will be reduced inversely proportional to the speed when the induction motor is operated above its rated value. That is, the motor is being operated in the flux weakening mode. For all speeds less than the rated value the control scheme adjusts the voltage and frequency to regulate the desired reference speed or torque while keeping the flux at a constant value. With an induction generator the aim is to have a constant generated voltage. Of course the frequency of the generated voltage is dependent on the rotor speed but once it is rectified the DC voltage depends only on the magnitude of the peak AC voltage.

In general the no load generated terminal voltage, E, in an induction generator can be approximated by

EK ZOr (9.69) where K - constant

Zr - rotor speed in rad/s

241 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

O - flux linkage in web-turn

From Equation (9.69) the product of the rotor speed and the flux linkage should remain constant so that the terminal generated voltage does not change. Ideally this is perfect for all variations of rotor speed. However, in practical applications there is magnetic flux saturation of the core of the induction generator. Hence the minimum rotor speed corresponds to the maximum allowable flux linkage, which is the saturated flux linkage.

The maximum speed is limited by the mechanical rating of the mechanical system of the induction generator. The minimum flux linkage corresponds to the maximum generator rotor speed.

Once the allowable variation between maximum and minimum rotor speeds is defined it is required to find the corresponding minimum and maximum (saturated) flux linkages respectively. Graphically the variation of flux linkage with generator rotor speed is shown in Fig. 9.5. O

Omax

Omin

Zr_min Z r_max Z r Fig. 9.5 Relationship between generator rotor speed and flux linkage

Since the maximum value of flux linkage is determined by the saturation level of the core, the flux linkage required at any speed is calculated based on this maximum flux linkage, the minimum set speed and the speed of operation. Hence,

Zr_min OO max (9.70) Zr

242 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

When the rotor speed decreases to a value lower than Zr_min, theoretically the flux linkage should increase to a value higher that Omax. However, in an induction machine once the saturation level is reached the controller forces more direct axis current in the synchronously rotating reference frame to produce more flux linkage. The magnitude of this exciting current can exceed the rated current of the machine without approaching the required reference flux linkage. As a result the magnitude of the generated voltage drops for speeds lower than the minimum set value which is Zr_min.

From Fig. 9.5, the maximum speed, Zr_max, is determined by the mechanical limitation of the induction generator and the wind turbine. A minimum exciting d-axis current in the synchronously rotating reference frame that will be enough to produce the minimum flux linkage, Omin, corresponding to the maximum speed, Zr_max, is provided by the controller.

9.5 Details for the implementation of vector control The detail for the generation of d-axis current and q-axis current will be discussed in this Section. The d-axis current in the synchronously rotating reference frame is generated from the flux linkage controller or from the reference flux linkage as in the indirect method. The reference flux linkage is varied according to the change in rotor speed. The q-axis current in the synchronously rotating reference frame is generated from the DC voltage controller. When there is an increase in loading, the DC voltage will decrease and this will be interpreted with an increase in q-axis current in the synchronous reference frame. This q-axis reference current in the synchronously rotating reference frame is negative in induction generators and was discussed in Chapter 4.

In Chapter 2 it was shown that at a particular wind speed the output power and the output torque of a wind turbine decreases at high turbine rotor speeds. Hence at no load the induction generator operates at high speed. When the induction generator driven by the wind turbine is loaded, the generator rotor speed decreases. The operating points of the output power and output torque characteristic of the wind turbine follow the output power and the induced electromagnetic torque of the induction generator. The reference

243 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR flux linkage of the induction generator varies inversely proportional to the variation in angular rotor speed. For different loading conditions the generator rotor speed can remain constant provided that there is variation in wind speed. In this case the reference flux linkage remains constant and any change in loading will only affect the DC voltage controller in which its output is the q-axis reference current in the rotating reference frame.

9.5.1 Implementation of direct rotor flux oriented vector control The detail of the direct rotor flux oriented vector control implemented in an isolated induction generator is shown in Fig. 9.6. The flux linkage estimation can be achieved using one of the schemes for direct rotor flux oriented vector control.

S

Induction Inverter / Rectifier Generator Load Wind Voltage and current C V Turbine b sensors

Speed sensor PWM generator

Vabc Iabc Current controllers

ic ib - + ia - + - * + * * ia ib ic Zr Flux linkage Te e e estimation d q / abc * e* e* VDC ids i Or qs * - + Or + Zr_min .Or_max PI PI VDC Numerator y -

Fig. 9.6 Implementation of direct rotor flux oriented vector control with current controlled PWM VSI

In Fig. 9.6 the current controllers are implemented using hysteresis controllers or proportional integral (PI) controllers. With the utilization of hysteresis controllers the actual current continuously tracks the reference or command current within a given hysteresis band. For a hysteresis band of h, if the measured current is less than the reference current minus the hysteresis band then an output signal “1” is generated to increase the actual current by turning on the upper power switch of the inverter. When the measured current is less than the reference current plus hysteresis band then an

244 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR output signal “-1” is generated to decrease the actual current by turning off the upper power switch of the inverter. The implementation of a hysteresis controller is very simple. However, the drawbacks are uncontrolled and high switching frequencies when a small hysteresis band is used or high ripple when the hysteresis band is larger. The selection of the hysteresis band needs some compromise for the magnitude of the ripple and switching frequency. The output signals “-1” and “1” are chosen instead of “0” and “1” because the range of the value -1.0 to 1.0, when using the function for PWM generation in the DSPACE DS1102 DSP card, represents a duty cycle of 0 to 100%. Three similar hysteresis controllers are used for the three phases.

When the current controller is implemented using a PI controller the error between the reference current and the actual current will be fed to the PI controller. The PI controller converts the error signal to the sinusoidal reference voltage. This sinusoidal reference voltage signal is then fed to the converter as a PWM signal. The PI controller is simple and the switching frequency can be predicted, however due to its limited bandwidth there will be a phase lag and a magnitude error between the reference current and the actual current [12]. The phase lag and magnitude error are detrimental and they will increase with rotor speed because at high rotor speed the frequency of the current will be high.

As shown in Fig. 9.6 and which will be also given in the implementation of other schemes of vector control, the control mechanism involves cascaded control structures with inner and outer closed loop systems. From control theory the cascaded control structure can only work under the assumption that the bandwidth of the control increases towards the inner closed loops [14, 17]. The most inner loop of the cascaded control structure should be the fastest and the most outer loop the slowest. The inner loop can perform well if there is enough time to execute the command given by the next outer loop. The fact that each feedback variable can be limited by limiting the reference signal of importance is a major advantage of cascade control.

9.5.2 Implementation of indirect rotor flux oriented vector control The detail of the indirect rotor flux oriented vector control implemented in an isolated induction generator is shown in Fig. 9.7. The command d-axis current in the

245 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

e* synchronously rotating reference frame, ids , is calculated from the magnitude of the rotor flux linkage divided by the magnetising inductance as given in Equation (9.30).

S Inverter/ Rectifier Induction Generator Load Wind C Turbine Vb

Speed sensor PWM generator

Current controllers

ic ib - + ia - + - + * * ia ib ic Zr Te e e Te estimation d q / abc e* e* i iqs e* e* ds ids i qs * VDC

* + Or 1Tpr Zr_min .Or_max PI VDC Numerator y - Lm

Fig. 9.7 Implementation of indirect rotor flux oriented vector control with current controlled PWM VSI

The previous explanation for the types of current controllers discussed in the direct rotor flux oriented vector control also applies in this indirect rotor flux oriented vector control scheme.

9.5.3 Implementation of rotor flux oriented vector control with voltage as a control variable The implementation of rotor flux oriented vector control discussed in Sections 9.5.1 and 9.5.2 utilises a current controlled PWM voltage source inverter with fast closed loop current control of the stator currents and with sinusoidal reference currents in the stationary reference frame [11]. However if the stator currents cannot be impressed by fast control loops, it is necessary to utilise a scheme with stator voltage as a control variable as discussed in Section 9.2.2. This method is more complicated than the current controlled rotor flux oriented vector control scheme because it involves decoupling terms and there are more PI controllers and more magnitude limiters at the output of the PI controllers.

246 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

In this section the outputs of the d-axis and q-axis current controllers in the synchronously rotating reference frame are added with their respective decoupling signals to produce d-axis and q-axis reference voltages in the synchronously rotating reference frame. Then using the transformation from two-axes in the synchronously rotating reference frame to three-axes in the stationary reference frame the three-phase stator voltages in the stationary reference frame are obtained. The overall implementation is shown in Fig. 9.8.

Fig. 9.8 shows only the direct rotor flux oriented voltage control scheme. However the same figure can be used to implement the indirect rotor oriented vector control provided that the evaluation for the d-axis current is changed.

S Inverter/ Rectifier Induction Generator Load Voltage and Wind C V Turbine current sensors b

Speed sensor

Vabc PWM generator

Iabc * * * v v vc Zr Flux linkage, Ze and Te a b estimation Te abc / deqe e e Te d q / abc e e* Or Ze ids iqs e* e* Vds Vqs

VL s + Lm /Lr + + Zr_min .Or_max +

VDC L Numerator Or V s e* Vddeco * i - e* - * Or + - ds + - - iqs VDC PI PI + PI PI + Zr y + Fig. 9.8 Implementation of direct rotor flux oriented vector control with stator voltage as a control variable

9.5.4 Implementation of stator flux oriented vector control The stator flux linkage is estimated in a direct way from the monitored stator terminal voltages and stator currents using integration of the back emf along the d-axis and q- axis in the stationary reference frame as discussed in Section 9.2.3. The implementation of stator flux oriented vector control is illustrated in Fig. 9.9.

247 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

S Inverter/ Induction Rectifier Generator Load Wind Voltage Turbine and current C Vb sensors

Speed sensor Vabc Iabc PWM generator

Current controllers

ic ib - + - + ia - i * + i * i * Te a b c Zr Flux linkage, Z and e e e d q / abc Te estimation e e iqs ids e* e* decoupling ids iqs * Ze VDC Os i - dcou + + + + Zr_min .Or_max PI PI VDC Numerator y * - Os

Fig. 9.9 Implementation of stator flux oriented vector control with current controlled PWM VSI

The current controllers in Fig. 9.9 can be hysteresis or PI controllers. The detail in using these two types of controllers is discussed in Section 9.5.1.

The stator flux oriented vector control can be implemented with stator voltage as a control variable; however it requires additional decoupling in the stator voltage equations expressed in the synchronously rotating reference frame and is not studied in this work.

9.6 Results The simulation for rotor oriented vector control and stator oriented vector control are implemented using MATLAB/SIMULINK. The features in the Power Systems Blockset are used to model an inverter, rectifier and all circuit components. The induction machine model in the Power Systems Blockset is modified to include speed as an input and to update the variation of magnetising inductance as the voltage builds up during self-excitation.

248 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

The results obtained in using direct rotor oriented vector control and the stator oriented vector control are similar. It is simply a matter of tuning the PI controllers in the DC voltage controller and flux linkage controller.

The dynamics of the DC voltage build up process, at a rotor speed of 1500rpm for a capacitance value of 1000PF and 1500PF, is shown in Fig. 9.10. When the capacitance is large it takes a longer time to reach its steady state value. If the capacitance is too small there will not be enough exciting current and as a result there will not be any voltage build up.

In Fig. 9.11 the frequency of the generated voltage is estimated using Equation (9.49). For a rotor speed of 314rad/s the AC voltage build up process starts with a low frequency and then rises until it reaches its steady state value of 311 rad/sec. The small slip is required to overcome all power loses in the induction machine and in the inverter that supplies the exciting current and the load current.

From Fig. 9.12 it can be observed that the value of the flux linkage in the machine varies inversely proportional to the rotor speed of the induction generator. Fig. 9.13 shows the generated voltage.

The loading for the induction generator is shown in Fig. 9.14. The no load resistance is simulated by a large resistance. The load resistance is decreased from its large value at no load to an effective resistance of about 120: and the rotor speed is varied from 1500rpm (157rad/sec) to about 2400rpm (250rad/sec). In response to these variations in load and rotor speed the variations in DC voltage and induction generator flux linkage are shown in Fig. 9.14c and Fig. 9.14d. Fig. 9.14e shows the variation in d-axis stator current in the rotating reference frame and Fig. 9.14f shows the variation in q-axis stator current in the rotating reference frame.

DC current in the load output electrical power, operating slip of the induction generator and the induced electromagnetic torque in the induction generator are also given in in Fig. 9.14

249 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

DC voltage @ n=1500rpm 700

600

C=1000micr F 500

C=2000micr F 400 (V) dc V 300

200

100

0 0 0.5 1 1.5 time (sec)

Fig. 9.10 Generated DC voltage for different capacitance value

estimated excitation freqency w @ w = 314rad/s(electrical) e r 350 w r 300

C=1000 micro F 250

C=2000 micro F 200 (rad/s) e

w 150

100

50

0 0.5 1 1.5 time (sec)

Fig. 9.11 Rotor speed and angular frequency of the generated voltage for different capacitance value

250 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

Estimated Fluxlinkage for C=1000micro F 1.4 Rotor speed=1500rpm

1.2

1 Rotor speed=1900rpm

0.8

0.6

Flux linkage(web-turn) 0.4

0.2

0 0 0.5 1 1.5 time (sec)

Fig. 9.12 Flux linkage at different rotor speeds of the induction generator for 1000PF

Generated voltage V @ n=1500rpm and C=1000micr F ab 800

600

400

200

0 Vab (V) Vab -200

-400

-600

-800 0 0.5 1 1.5 time (sec)

Fig. 9.13 Generated line to line voltage at the terminals of the induction generator

251 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

150

100

RL (ohm) RL 50

0 0 2 4 6 8 10 12 14 (a) 3000

2000

Speed (rpm) 1000

0 0 2 4 6 8 10 12 14 (b)

800

600

400 Vdc (V) Vdc 200

0 0 2 4 6 8 10 12 14 (c) 1.5

1

0.5 Flux (web-turn)

0 0 2 4 6 8 10 12 14 (d) time (sec)

252 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

8

6

4 Ide (A) Ide

2

0 0 2 4 6 8 10 12 14 (e) 0

-2

-4

Iqe (A) Iqe -6

-8

-10 0 2 4 6 8 10 12 14 (f) time (sec)

6

(A) 4 dc I

2

0 0 2 4 6 8 10 12 14 (g) 4

3

2 Power (KW) 1

0 0 2 4 6 8 10 12 14 (h) time (sec)

253 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

Slip -0.03

-0.06 0 2 4 6 8 10 12 14 (i) 0

-10

Torque (Nm) Torque -20

-30 0 2 4 6 8 10 12 14 (j) time (sec)

Fig. 9.14 Loading of the induction generator (a) RL (b) rotor speed (c) VDC (d) flux

e e linkage (e) ids (f) iqs (g) Idc (h) Output power (i) Slip (j) Electromagnetic torque

In Fig 9.14 when the speed increases the flux linkage decreases and the d-axis current in the rotating reference frame decreases also. This d-axis current is not affected by the load. Loading affects the magnitude of q-axis current in the rotating reference frame. At no load there is a small slip and induced electromagnetic torque to overcome the power loses in the induction generator and in the converter.

The vector control can be implemented in real time using commercially available DSPs, microcontrollers and PLCs. The versatile commercially available DSP from Texas Instruments is TMS320LF2407 and an advanced one is the TMS320LF2812. This project is, however, not concerned with the implementation of the proposed techniques in appropriate real time hardware.

9.7 Summary The voltage build up process of an induction generator with a single capacitor on the DC side of the inverter using vector control is discussed. Since the induction generator

254 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR operates at a frequency away from the DC frequency, any integration offset error is easily removed by offset adjustment, as discussed in Chapter 10, when stator flux linkage estimation is used.

With stator flux oriented vector control the total flux linkage is aligned to the d-axis of the stator flux linkage and with rotor flux oriented vector control the total flux linkage is aligned to the d-axis of the rotor flux linkage in the excitation reference frame. A decoupling signal is generated to cancel the effect of the q-axis current on the d-axis flux for the stator oriented vector control. The main advantage of stator flux oriented vector control is that the error in the estimated flux depends only on the stator resistance. Unlike the rotor resistance the variation of stator resistance depends mainly on temperature. If the variation of stator resistance causes a significant error in the estimated flux then a compensation block can be added in the model.

The advantage of rotor oriented vector control is that there is no need to generate a decoupling signal because the flux linkage is dependent only on the d-axis current and not on the q-axis current. The rotor flux oriented vector control scheme gives a decoupled control when implemented with a current regulated source.

e When the load is varied iqs varies to provide the current demand of the load. When the

e rotor speed is varied ids is varied to provide the required amount of flux linkage corresponding to the rotor speed in a similar way as in flux weakening operation of induction motors. In this way the DC voltage will remain constant as the speed of the wind turbine varies.

In a similar way the different flux oriented vector control schemes discussed here can be used in an automotive combined starter/generator system which is an area likely to be very typical in the future due to the expected change from 12V to 36V electrical systems.

255 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

9.8 References [1] N. H. Malik and A. H. Al-Bahrani, “Influence of the terminal capacitor on the performance characteristics of a self-excited induction generator ”, IEE Proc C., Vol. 137, No. 2, March 1990, pp. 168-173. [2] V. N. Nandakumar, K. Yadukumar, T. Sureshkumar, S. Ragupathi and R. K. Hegde, “A wind driven self-excited induction generator with terminal voltage controller and protection circuits”, IEEE Power Conversion Conference, 1993, pp. 484-489. [3] H. C. Rai and A. K. Tandan, “Voltage regulation of self-excited induction generator using passive elements”, 6th International Conf. on Electrical Machines and Drives, 1993, pp. 240-245. [4] S. S. Murthy, C. Parabhu, A. K. Tandon and M. O. Vaishya, “Analysis of series compensated self- excited induction generators for autonomous power generation”, IEEE Conference on Power Electronics, Drives and Energy Systems for Industrial Growth, 1996, pp.687-693. [5] S. M. Alghuwainew, “Steady-state analysis of a self-excited induction generator self-regulated by shunt saturable reactor”, IEEE International Conf. on Electrical Machines and Drives, 1997, pp.101-103. [6] L. Shridhar, B. Singh and C. C. Jha, “Transient performance of the self regulated short shunt self excited induction generator”, IEEE Transactions on Energy Conversion, Vol.10, No.2, June 1995, pp. 261-267. [7] S.N., Bhadra, K.V., Ratnam, and A, Manjunath, “Study of Voltage Build up in a Self-Excited, Variable Speed Induction Generator/ Static Inverter System with D.C. Side Capacitor”, International Conference on Power Electronics, Drives and Energy System, Vol. 2, 1996, pp. 964- 970. [8] M. S., Miranda, R.O, Lyra, and S.R., Silva, “An Alternative Isolated Wind Electric Pumping System Using Induction Machines”, IEEE Transactions on Energy Conversion, Vol. 14, No. 4, December 1999, pp. 1611-1616. [9] R., Cardenas, R., Pena, G., Asher, and , J. Clare., “Control Strategies for Enhanced Power Smoothing in Wind Energy Systems Using a Flywheel Drive by a Vector-Controlled Induction Machine”, IEEE Transaction on Industrial Electronics, Vol.48, No. 3, June 2001, pp. 625 – 635. [10] F. Blaschke, “The principle of field orientation as applied to the new Transvektor closed loop control system for rotating field machines”, Siemens Review, Vol. 34, 1972, pp. 217 – 220. [11] Peter Vas, “Sensorless Vector and Direct Torque Control”, Oxsford University Press, New York, 1998. [12] B. K. Bose, “Modern Power Electronics and AC Drives”, Prentice-Hall, NJ, 2002. [13] T. Matsuo, V. Blasko, J. C. Moreira and T. A. Lipo, “Field oriented control of induction machines employing rotor end ring current detection”, IEEE Trans. on Power Electronics, Vol. 9, No. 6, November 1994. [14] Werner Leonhard, “Control of Electrical Drives”, Springer-Verlag Berlin Heidelberg, New York, 1996.

256 CHAPTER 9 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

[15] G. R. Slemon, “Electric Machines and Drives”, Addison-Wesley Publishing Company, New York, 1992. [16] E. D. Mitronikas, A. N. Safacas and E. C. Tatakis, “A new stator resistance tuning method for stator-flux-oriented vector-controlled induction motor drive”, IEEE Transactions on Industrial Electronics, Vol. 48 Issue: 6, Dec. 2001, pp. 1148 –1157. [17] W. S. Levine, “The Control Handbook”, CRC press with IEEE press, New York 1996.

257 CHAPTER 10

FLUX LINKAGE ESTIMATION AND COMPENSATION IN INDUCTION MACHINES

10.1 Introduction Flux information is needed in induction machine control for the purpose of synchronous angle and synchronous speed estimation, flux regulation and torque regulation. Accurate flux estimation is very crucial in the control of induction motor drives and induction generators if vector control or direct torque control (DTC) is used.

The stator flux is calculated by integrating the back electromotive force (emf), which is the terminal voltage minus the voltage drop in the stator resistance. This method is preferred because it requires knowledge of only one parameter of the induction machine, namely the stator resistance. The stator resistance can be easily obtained from measurement. If there is variation of stator resistance due to change in temperature it can be readily compensated [1]. The back emf is calculated from the stator terminal phase voltage and from stator phase current. When a pure integrator is implemented in discrete form such as that used when controlling an induction machine from a DSP system, an error can arise. This error comprises a drift produced by the discrete integrator and also a drift produced by the offset error in the back emf. The offset error in the back emf is due to the use of analog components in the sensor and amplifier circuits for the measurement of voltages and currents. A small DC offset in the measured signal, no mater how small it is, can drive the pure integrator into saturation. The integration error associated with the implementation of the integrator is constant and appears as an offset in the initial integrated value. From the signal at the input of the integrator it is not easy to know whether the integrated signal will have an offset or not.

258 CHAPTER 10 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

The constant offset available at the output of the integrator can be explained as follows. Generally, when a sine signal (sinZt) is applied to the input of an integrator with an integrator gain of Z then a negative cosine (-cosZt) signal is obtained at the output of the integrator. In a discrete integrator this is only true when the input sine wave is applied at its positive peak, sin(Zt+S/2), or negative peak, sin(Zt-S/2). If the sine signal is applied at any other initial condition, a constant DC offset will appear at the output of the integrator. This DC offset is an undesirable error and it distorts the output of the integrated signal. When a sine input signal (sinZt) is applied to the integrator, the maximum constant DC offset that can occur is equal to the peak value of the sine signal.

Attempts have been made to modify the pure integrator by implementing it using a low pass filter [2]. A low pass filter will produce errors in magnitude and phase angle, especially if the excitation frequency is lower than the cut off frequency of the low pass filter. It has also been reported that the pure integrator can be replaced by a programmable low pass filter which is implemented by tuning the angle and changing the magnitude of the output vector of the low pass filter according to the calculated error [2-7]. In all these applications the cut off frequency of the low pass filter is decided from the estimated excitation frequency. However the accuracy of the calculated excitation frequency depends on the estimated flux. As a result the magnitude and phase of the estimated flux is dependent on the calculated cut off frequency of the low pass filter.

The measurement-offset error will appear as a ramp signal at the output of the integrator. The modified integrator proposed in this chapter has feedback to cancel the DC offset value due to the integrator output initial condition and the ramp due to measurement offset error.

10.2 Theory of Integrator As discussed in Chapter 9 the stator flux linkage is estimated by integrating the terminal voltage minus the voltage drop across the stator winding resistance. To analyse the principle of integration and the practical difficulty in the implementation of an integrator, a theoretical signal is applied and the output is analysed. If a signal X is fed

259 CHAPTER 10 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR to an integrator then an integrated output signal Y will be obtained. This can be written as: YXdt ³ (10.1) for XA m sin Z t, then 1 YAtAt mmcos ZZ  cos Z t 0 1 YAtA mmcos Z  (10.2) Z

However, if XAsin tS Acos t then mm ZZ2 A Yt m sinZ (10.3) Z

If the signal X has an offset where X is represented by:

X AtAmdsin Z c then 1 YAtAtAtAt m cos ZZ m cos dc  dc Z t 0 t 0 1 YAtAAt mmcos Z  dc (10.4) Z

However, if X Atsin S AAtAcos then md ZZ2 c mdc A YtAt m sinZ  (10.5) Z dc

In Equation (10.2) there is a constant offset error Am/Z at the output of the integrator due to the initial value of the limit of integration, i.e. taking t= 0 as the starting point. The input and output signals are displayed in Fig. 10.1. However in Equation (10.3) there is no error in the integrated output, which is the expected type of signal in the determination of flux linkage. These signals are shown in Fig. 10.2. Depending on the phase shift of the sinusoidal function the value of the integration error can vary from zero to ±Am/Z.

From Equation (10.5) the ramp error Adct at the output of the integrator is due to a DC signal or offset in the signal that is fed to the input of the integrator and is shown in Fig.

260 CHAPTER 10 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

10.3. However, in Equation (10.4) there are two types of errors involved: the first one is similar to that expressed in Equation (10.2) and the second one is similar to the error expressed in Equation (10.5). The ramp error signal Adct keeps on increasing with time. This type of error is displayed in Fig. 10.4.

X sin Z t

YXdt Z ³

Fig. 10.1 Offset error equal to Am as a result of the integration initial condition

Xtsin S Z 2

YXdt Z³

Fig. 10.2 No integrator error

261 CHAPTER 10 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

XsintS 008 . Z 2

YXdt Z³

Fig. 10.3 Error produced due to measurement offset

Xt sin Z  0.08

YXdt Z ³

Fig. 10.4 Error produced due to measurement offset and integration initial condition

In order to obtain a sinusoidal integrated signal a technique is needed to eliminate the effect of Am/Z, appearing as a constant value and Adct, appearing as a ramp signal at the output of the integrator. With time the ramp signal will easily drive the integrator into

262 CHAPTER 10 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

saturation irrespective of the non-zero magnitude of offset Adc occurring in the input signal.

Figs. 10.1 to 10.4 show the different conditions that the input AC signals can have and the different conditions the integrator output can exhibit. These results represent the conditions that can happen to the back emf at the input of the integrator and the computed flux linkage at the output of the integrator in any experimental flux linkage estimation. The output of the integrator is multiplied by a constant, which is the angular frequency of the signal to be integrated, so that the peak to peak value of the AC signal at the input and output remain constant. Then the input and output signals can be compared easily.

10.3 Numerical integrator The integrator discussed in Section 10.2 can be implemented as a numerical integrator for application in a discrete time system. With a sampling time of Ts, the numerical integrator is given by:

k YXT kns ¦ (10.6) n0 This can be represented by the following figure:

Z-1

X + Y Ts + Fig. 10.5 Numerical integrator representation

10.4 Proposed integration offset adjustment In general the dc offset in the voltage and current sensors must be minimized to prevent saturation of the integrator. In the experimental setup, calibrating the sensors every time the system is started minimizes the dc offset in the measured signals.

The average of a pure sinusoidal signal in a full cycle is zero. This information is used to correct the offset present at the output of the integrator. Every cycle the average value is calculated and then subtracted from the integrator output. This method is dealt with in

263 CHAPTER 10 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

Strategy I below. The measurement offset error is almost constant for a given duration of time. However the integration error due to the offset at the input of the integrator keeps on increasing with time. If the measurement offset error present in the signal of interest is small then the ramp build up within a cycle is insignificant compared with the integrated signal. Strategy II deals with a technique of minimizing the offset of the signal at the input of the integrator.

Since the mechanical time constant of induction machines is large, the change of peak current within a given cycle is small. The main advantage of the proposed technique is that a pure integrator is used, as it is, without any approximations.

10.4.1 Strategy I - without input offset minimization This strategy, as illustrated in Fig. 10.6, is applicable when the offset generated in the hardware sensor and signal conditioning error is small. If the offset at the input of the integrator is small then the increase of the ramp at the output of the integrator in one cycle will also be small. Hence adjustment of the ramp every cycle will not distort the signal of interest. Using a moving average makes it possible to track changes that occur before the end of a full cycle.

Z-1

X + Y Ts + -

Output error adjustment Average in a period

Once in a cycle Fig. 10.6 Proposed offset adjustment in a numerical integrator

In Fig 10.6 the period of one cycle used in the proposed offset adjustment can be obtained from the measured voltage or current using the zero crossing points. The time between two zero crossing points is equal to half the period of the signal. The period can be also calculated from the frequency of the signal to be integrated.

264 CHAPTER 10 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

10.4.2 Strategy II - with input offset minimization This strategy is applicable when the offset error in the hardware sensor and signal conditioning is large. The measurement offset error at the input of the integrator can be reduced to an acceptable value by introducing an offset adjustment at the input of the integrator. This can be realized using a low pass filter as illustrated in Fig. 10.7.

The offset at the output of the integrator generated due to the initial condition, as given in Equation 10.2 and Fig. 10.1, cannot be removed using any adjustment at the input of the integrator. The offset generated from the integrator due to integration initial condition is constant and once it is adjusted it will not appear again during the rest of the procedure.

Low Pass Filter

Z-1

X - + Y Ts + + -

Output error adjustment Average in a period

Once in a cycle Fig. 10.7 Proposed integrator with input offset adjustment

A sinusoidal signal, shown in Fig. 10.8a, with an offset of 0.008 and a peak value of 1 unit is applied to the proposed integration error correction system. The integrator output without any integration error compensation and with the proposed integrator error compensation is shown in Figs. 10.8b and 10.8d respectively. The magnitude of the error adjustment signal generated every cycle to compensate the integration error at the output of the integrator is shown in Fig. 10.8c.

10.5 Stator flux linkage estimation with the proposed method Flux estimation is the important part in induction machine control. The stator flux linkage can be estimated from the measured terminal voltage and current [8]. As discussed in Chapter 9, once the stator flux linkage is available it is easy to calculate the rotor flux.

265 CHAPTER 10 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

2

1

(a) 0

-1

-2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0.01

0.008

0.006 (b) 0.004

0.002

0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

-3 x 10 4

3

(c) 2

1

0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0.01

(d) 0

-0.01 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 time (sec) Fig. 10.8 Detail for integration error compensation (a) input X = sin(Zt)+0.008 (b) output YXdt ³ without output error adjustment (c) output error adjustment signal value (d) output with output error adjustment

266 CHAPTER 10 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR

In the estimation of stator flux linkage the voltages and currents in the three axes model are transformed to voltages and currents in the two axes model using the method discussed in Chapter 3 and then all the analysis is done in the two axes model [9].

The integration given in Equations (9.44) and (9.45) is implemented in the numerical integrator using the proposed integrator described in Section 10.4. The implementation of the d-axis flux linkage estimation and q-axis flux linkage estimation using the proposed scheme are illustrated in Fig. 10.9.

ids Rs

Vds - Proposed Ods + integrator

(a)

iqs Rs

Vqs - Proposed Oqs + integrator

(b) Fig. 10.9 Stator flux linkage estimation using the proposed method (a) d-axis flux estimation (b) q-axis flux estimation

10.6 Summary The method of flux estimation proposed in this chapter is new and effective. It eliminates the error produced by the measurement offset error and integrator output error due to initial integration in a continuous time integrator or numerical/discrete time integrator. The numerical integrator can be used in a discrete system such as in digital signal processing applications. The integrator output offset due to the initial condition is not dependent on frequency.

If the integration ramp output due to the existence of measurement offset error is large then subtracting the output of a low pass filter of the signal from the signal to be

267 CHAPTER 10 INVERTER/RECTIFIER EXCITATION OF THREE-PHASE INDUCTION GENERATOR integrated minimizes the offset. A signal with small input offset will have a small increment of ramp that will appear at the output of the integrator. As the time increases the ramp keeps on increasing and eventually the distortion in flux will be unacceptable. However, if the ramp is eliminated every cycle, the flux distortion due to the offset correction at the output is insignificant.

Overall a simple method is proposed to compensate the error produced due to the initial limit of integration as well as measurement error at the input of the integrator.

10.7 References [1] E. D. Mitronikas, A. N. Safacas and E. C. Tatakis, “A new stator resistance tuning method for stator-flux-oriented vector-controlled induction motor drive”, IEEE Transactions on Industrial Electronics, Vol. 48, Issue: 6, Dec. 2001, pp. 1148 –1157. [2] K. D. Hurst, T. G. Habetler, G. Griva and F. Profumo, “Zero speed tacholess IM torque control: simply a matter of stator voltage integration”, IEEE Transactions on Industry Applications, Vol. 34, No. 4, July/August 1998, pp. 790-795. [3] B. K. Bose and N. R. Patel, “A programmable cascaded low-pass filter-based flux synthesis for stator flux-oriented vector-controlled induction motor drive”, IEEE Transaction on Industrial Electronics, Vol.44, No.1, February 1997, pp.140-143. [4] J. Hu and B. Wu, “New integration algorithms for estimating motor flux over a wide speed range”, IEEE Transaction on Power Electronics, Vol. 13, No. 5 September 1998, pp.969-977. [5] M. –H. Shin, D. –S. Hyun, S. –B. Cho and S. –Y. Choe, “An improved stator flux estimation for speed sensorless stator flux oriented control of induction motors”, IEEE Transaction on Power Electronics, Vol. 15, No. 2, March 2000, pp.312-318. [6] N. R. N. Idris, and A. H. M. Yatim, “An improved stator flux estimation in steady-state operation for direct torque control of induction machines”, IEEE Transactions on Industry Applications, Vol. 38, No. 1, January/February 2002, pp. 110-116. [7] M. Hinkkanen, and L. Luomi, “Modified integrator for voltage model flux estimation of induction motors”, IECON’01: The 27th Annual Conference of the IEEE Industrial Electronics Society, 2001, pp.1339-1343. [8] X. Xu, R. D. Doncker and D. Novotny “A stator flux oriented induction machine”, in Conf. Rec. IEEE-PESC, 1988, pp. 870-876. [9] P. Vas, “Sensorless Vector and Direct Torque Control”, Oxford University Press, New York, 1998.

268 CHAPTER 11

CONCLUSIONS AND SUGGESTION FOR FUTURE WORK

11.1 Conclusions Many important and interesting aspects of an isolated self-excited induction generator have been discussed and presented in this thesis. The study comprises theoretical analysis, simulation and experimental results related to induction generators. The modelling and characteristics of induction machines in general has also been presented to provide an overall perspective of induction generators.

For an isolated self-excited induction generator driven by a wind turbine the characteristics of the output power and torque of the wind turbine are important. The output power and torque of a wind turbine drop at high turbine angular rotor speed. Any change of electrical load connected to the induction generator will be transferred to the wind turbine and as a result the wind turbine operating point will be adjusted by a change in angular rotor speed.

For a grid connected induction generator all the power generated can be supplied to the grid as the grid can absorb theoretically an infinite amount of power. However for an isolated induction generator the maximum power available in the wind might not be utilised by a constant electrical load connected to the induction generator. If an isolated self-excited induction generator is supplying a constant load then the operating points of the output power and output torque of the wind turbine will be adjusted as per the operation of the induction generator.

269 CHAPTER 11 CONCLUSIONS AND SUGGESTION FOR FUTURE WORK

The literature related to isolated induction generators and wind turbines is reviewed in Chapter 1. This has involved clarifying the strengths and limitations of the previous works and highlighted the advantages of the research covered in this thesis. A detailed explanation about wind as a power source and the mechanism of conversion of wind power to mechanical power is presented in Chapter 2. The general definitions of wind and wind as a source of power have been presented in this chapter. The analysis of power absorbed by a wind turbine is based on the horizontal axis wind turbine. The mechanism of production of force from wind that causes the rotor blades to rotate in a plane perpendicular to the general wind direction at the site has been discussed in detail. The importance of having twisted rotor blades along the length from the base to the tip is given. The variation of the torque produced by the wind turbine with respect to the rotor angular speed has been presented and associated with this output torque the power output from a wind turbine is evaluated.

Power absorbed by a wind turbine is proportional to the cube of the wind speed. Wind turbines are designed to yield maximum output power at a given wind speed. In case of stronger winds it is necessary to waste part of the excess energy of the wind in order to avoid damaging the wind turbine. Different ways of power control to protect the machine have been presented. The economics and growth of wind powered electric generation is given and the projection for the future is also discussed.

The three-axes to two-axes transformation presented in Chapter 3 is applicable for any balanced three-phase system. It has been discussed that the three-axes to two-axes transformation simplifies the calculation of rms current, rms voltage, active power and power factor in a three-phase system. Only one set of measurements taken at a single instant of time is required when using the method described to obtain rms current, rms voltage, active power and power factor. Furthermore from measurements taken at two consecutive instants in time the frequency of the three-phase AC power supply can be evaluated.

Existing electrical measuring methods, such as the Fast Fourier Transform, require many samples from a significant period of the measured waveform’s cycle to be processed using elaborate computation techniques in order to evaluate rms or peak

270 CHAPTER 11 CONCLUSIONS AND SUGGESTION FOR FUTURE WORK magnitudes of AC currents and voltages. These traditional methods are unable to obtain peak values in less than one quarter of a cycle. Therefore, the measurement system presented here supersedes the traditional methods for the monitoring of dynamically changing quantities in a three-phase system.

As preparation for the modelling and analysis of an isolated induction generator a detailed model of an induction machine using the conventional or steady state model and the D-Q or dynamic model are explained and analysed in Chapter 4. The voltage, current and flux linkage in the rotating reference frame and their phase relationships in the motoring region and generating region are presented.

For the same stator terminal voltage of an induction machine the magnitude of the electromagnetic torque in the generating region is higher than the electromagnetic torque in the motoring region. The reason for the difference in electromagnetic torques is that during motoring all the electrical losses in the induction machine are supplied by an external electrical power source and the electromagnetic torque is the output of the system. However, in the generating region the electromagnetic torque is equivalent to the external mechanical input torque and all the electrical power losses in the induction machine are indirectly supplied by the external mechanical power source and the terminal voltage is the output of the system. Hence to overcome all the internal power losses in the induction machine and have the same terminal voltage as in the motoring region the electromagnetic torque in the generating region must be higher than the electromagnetic torque for motoring.

The D-Q axes induction machine model has been improved to include the equivalent iron loss resistance, Rm. This improved model was developed by D. McKinnon and C. Grantham in collaboration with the author and is presented here in a simple and understandable way. Using this model the dynamic current, torque and power can be calculated more accurately.

The data acquisition system which is used for the measurement of voltages, currents, angle and speed with their appropriate sensors is explained in Chapter 5. The outputs of the voltage and current measurement circuits are calibrated so that the voltage signals of

271 CHAPTER 11 CONCLUSIONS AND SUGGESTION FOR FUTURE WORK the sensor outputs will not exceed the ADC input rating of the DS1102 DSP board. The sensors for current and voltage are Hall-Effect devices. Anti-aliasing filters are introduced in the analog signals of the sensor outputs to prevent the high frequencies appearing as a low frequency when the analog signal is digitised in the A/D converter. Speed and angle measurements for the induction machine are taken using an optical incremental encoder. The resolution of angle and speed for a given encoder is derived. For a given encoder the resolution of the angle measurement is constant; however the resolution of speed measurement is dependent on the sampling period used. High sampling period (low sampling frequency) gives small resolution in speed (discrete steps) producing less error in the measurement of the speed. The advantage of digital signal processing is discussed. Different types of filter design are presented. Digital filters are used in the simulation and experimental set up with the results presented in the thesis.

Induction machine parameter determination and the results of an investigation into the variation of magnetizing reactance, iron loss resistance and rotor parameters with temperature and supply voltage have been presented in Chapter 6. Most of the experimentation was carried out by D. McKinnon. A monitoring system employing digital signal processing techniques was used to observe these effects. To date, parameter identification methods over a large speed range have typically required reduced voltages to prevent the motor overheating from excessive current at high slips. The run up to speed test described enables the parameters to be determined at supply voltages up to and including the rated voltage without damaging the motor under test.

A separate test was used to determine the iron loss resistance and magnetizing reactance. This test incorporated two methods. Both methods were conducted at synchronous speed, one using continuously varied supply voltage, the other at discrete voltages. Agreement between the results was excellent. All tests were performed at corresponding temperatures.

Parameter identification is important in the modelling of and control of induction machines. It is obvious from the results in this thesis that parameter variations cannot be

272 CHAPTER 11 CONCLUSIONS AND SUGGESTION FOR FUTURE WORK ignored. It has been shown that rotor parameter variations exist even for a single-cage induction motor.

The error in the values of induction motor parameters arising from measurement error in voltage, current and power used to determine the parameters have been presented. These three measurement quantities are essential for accurate parameter determination. Even with measurement errors of r1.5% the determined parameters are still very acceptable and considerably more accurate than if rotor parameter variations are ignored. Only very close to synchronous speed is there substantial error in the determined value of Xlr, but this is almost irrelevant in any meaningful analysis of the inductive machine, because close to synchronous speed the effect of Xlr is swamped by the effect of Rr /s.

When rotor parameter variations are ignored the percentage errors in the current, power and torque are substantial for machines which exhibit a significant degree of rotor current displacement effect.

The modelling, analysis and dynamic performance of an isolated three-phase induction generator excited by three AC capacitors connected at the stator terminals is presented in Chapter 7. The use of the variation in magnetising inductance with voltage leads to an accurate prediction of whether or not self-excitation will occur in a SEIG for various capacitance values and speeds in both the loaded and unloaded cases. The characteristics of magnetising inductance, Lm, with respect to the rms induced stator voltage or magnetising current determines the regions of stable operation as well as the minimum generated voltage without loss of self-excitation.

Once self-excitation has been initiated and a steady state condition has been reached, the speed at which self-excitation ceases is always lower than the speed to initiate self- excitation. At a particular speed the capacitance required for self-excitation, when the machine operates at no load, is less than the capacitance required for self-excitation when it is loaded.

273 CHAPTER 11 CONCLUSIONS AND SUGGESTION FOR FUTURE WORK

When an induction machine operates as a motor the speed of the rotating air gap magnetic field is totally dependent on the excitation frequency. However, in the SEIG it is shown that the frequency of the generated voltage depends on the speed of the prime mover as well as the condition of the load. Keeping the speed of the prime mover constant with increased load causes the magnitude of generated voltage and frequency of an isolated SEIG to decrease. Modelling and experimental results show that this is due to a drop in the speed of the rotating magnetic field. When the speed of the prime mover drops with load, then the decrease in voltage and frequency will be greater than for the case where the speed is held constant.

The dynamic voltage, current, power and frequency developed by the induction generator have been analysed, simulated and verified experimentally for the loaded and unloaded conditions while the speed was varied or kept constant. Using the simulation algorithm more results which are not accessible in an experimental setup have been predicted.

Increasing the capacitance value can compensate for the voltage drop due to loading but the drop in frequency can be compensated only by increasing the speed of the rotor. The variation of magnetising inductance follows the variation in terminal voltage or magnetising current. Increasing the capacitance can compensate the generated voltage, however it increases stator current. Hence care should be taken not to exceed the stator rated current.

The variation of the generated voltage and frequency for a self excited induction generator driven by a wind turbine at constant and variable speeds has been investigated. The dynamic calculated performance comparison between the machine with constant and variable rotor parameters has been simulated and discussed considering a hypothetical induction machine that represents a typical double-cage or deep-bar rotor induction machine where rotor parameter variations are significant.

A novel D-Q axes model that includes the iron loss equivalent resistance Rm in the dynamic analysis and simulation of the SEIG has been described in Chapter 8 in a simple and understandable way. It is noted that this method is easily understood, having

274 CHAPTER 11 CONCLUSIONS AND SUGGESTION FOR FUTURE WORK drawn on many familiar concepts and using the standard terminology and nomenclature of D-Q unified machine theory. In some instances it is possible to neglect the iron loss; however its effect has to be shown to be negligible. This study provides the tool to reach such a decision. When Rm is included, which depicts the actual situation, it has been shown that the generated voltage, currents, and output power, are lower than that when Rm is neglected. However, due to additional losses the electromagnetic torque necessary to drive the induction generator is higher when Rm is included than when it is neglected.

Using vector control the voltage build up process and terminal voltage control in an isolated wind powered induction generator using an inverter/rectifier excitation with a single capacitor on the DC link is discussed in Chapter 9. Due to the fundamental switching of the IGBTs of the inverter, controlled by the vector control principle, the single DC capacitor appears as if three capacitors were connected across the three stator terminals of the induction generator.

Different types of vector control techniques are developed to control the excitation and the active power producing currents independently. That is, the current control scheme causes the currents to act in the same way as in a DC generator where the field current and the armature current are decoupled. When the speed of the prime mover is varied the flux linkage in the induction generator is varied inversely proportional to the rotor speed so that the generated voltage will remain constant.

Since the torque produced by a wind turbine drops at high turbine rotor speed the induction generator will run at high generator rotor speed when loaded with a small load and the rotor speeds decrease with an increase in load. As the turbine rotor shaft and the generator rotor shaft are connected via a gear box, both rotor speeds will increase and decrease proportionally for a constant gear ratio. The flux linkage of the induction generator is controlled by controlling the d-axis current in the synchronously rotating reference frame. All vector control strategies presented are either rotor flux oriented vector control or stator flux oriented vector control.

275 CHAPTER 11 CONCLUSIONS AND SUGGESTION FOR FUTURE WORK

It is shown that the estimation of rotor flux linkage is more dependent on the induction machine parameters whereas estimation of stator flux linkage is dependent only on the stator resistance. If the variation of stator resistance is causing a significant error in the estimated flux then a compensation block can be added in the model, however it is difficult to compensate the variation in rotor parameters, especially rotor resistance. The dynamic results for generated voltage, current, power, torque and slip for different rotor speed and load have been given.

When the load is varied the q-axis current in the synchronously rotating reference frame varies to provide the current demand of the load. Hence the q-axis current in the synchronously rotating reference frame is varied when there is variation in load. When the rotor speed is varied the d-axis current in the synchronously rotating reference frame is varied to provide the required amount of flux linkage corresponding to the rotor speed in a similar way as in flux weakening operation of induction motors. In this way the DC voltage will remain constant as the speed of the wind turbine varies and also when the electrical load changes. Once a constant DC voltage is achieved a DC load can use it directly or, if required, it is a matter of having an inverter to produce a constant voltage and frequency AC output.

The estimation of stator flux linkage using integration of the voltage behind the stator resistance is the easiest way of estimating the flux linkage. However, there are problems associated with the integration. The method of flux estimation proposed in Chapter 10 is an effective method to compensate the error produced by the integrator. The cause of the error in an integrator is illustrated in detail by using a sinusoidal signal as an input to the integrator and then solutions are proposed in a simple and understandable way.

The compensation method eliminates the error produced by the measurement offset error and integrator output error due to the initial integration in a continuous time integrator or numerical/discrete time integrator. The numerical integrator can be used in a discrete system such as in digital signal processing applications. The integrator output offset due to the initial condition is not dependent on frequency.

276 CHAPTER 11 CONCLUSIONS AND SUGGESTION FOR FUTURE WORK

If the integration ramp output due to the existence of measurement offset error is large then subtracting the output of a low pass filter of the signal from the signal to be integrated minimizes the offset. A signal with small input offset will have a small increment of ramp that will appear at the output of the integrator. As the time increases the ramp keeps on increasing and eventually the distortion in flux will be unacceptable. However, if the ramp is eliminated every cycle, the flux distortion due to the offset correction at the output is insignificant. Overall a simple method is proposed to compensate the error produced due to the initial limit of integration as well as measurement error at the input of the integrator.

11.2 Suggestions for future work The analysis and explanations presented in this thesis provide a good foundation for further research in the area of isolated induction generators driven by a wind turbine. Some of the topics recommended for future work are: x Implementation of vector control to regulate the output voltage and frequency of an isolated self-excited induction generator and compare with the simulation results. x Implementation of the vector control technique for an isolated induction generator with variable rotor parameters. As the contribution of rotor leakage inductance is small compared to the total rotor inductance the focus is mainly in the updating of the rotor resistance variation. This variation in rotor resistance can also be explored and extended to include variation of rotor resistance due to the operating temperature of the rotor conductors. x Development of a model that takes into account the variation in stator resistance or a way of compensating the stator resistance in the rotor resistance. Stator resistance is important in the estimation of stator flux linkage. x Technical and economic comparison between series connected inverter/rectifier (discussed in Chapter 9 of this thesis) and shunt connected inverter/rectifier associated with a separate rectifier to supply the load for an isolated induction generator. x Use of Direct Torque Control instead of Vector control in an isolated induction generator. The development of a control system based on the balance of power

277 CHAPTER 11 CONCLUSIONS AND SUGGESTION FOR FUTURE WORK

between the output power and the available power in the prime mover is to be envisaged.

278 APPENDICES

APPENDIX A

DETERMINATION OF INERTIA AND FRICTION COEFIENT OF THE INDUCTION GENERATOR SYSTEM

The induction generator system consists of an induction machine and a DC motor. Here the DC motor is the prime mover and is always coupled to the induction machine. Therefore the total inertia and friction coefficient determined here consists of the sum of the inertia in the induction machine and in the DC motor.

The dynamics of the motor is given by dZ TJ r  DZ T (A.1) erLdt where

Te – electromagnetic torque (Nm),

TL – load torque (Nm),

Zr –rotor or shaft speed (rad/sec), D – friction coefficient (Nm/rad/sec) and J – rotor inertia (Kg-m2).

To determine J and D the induction generator system was driven by the DC motor. During the test the terminals of the induction machine were open. Hence no load torque, i.e TL =0.

From the voltage equation of the DC motor, the back emf in the armature can be calculated as:

EVIRaaa  (A.2)

278 APPENDIX A DETERMINATION OF INERTIA AND FRICTION COEFFIENT where

Ea – back-emf (V), V – applied voltage (V),

Ia – motor current (A) and

Ra – Armature winding resistance (ohm)

The output power, amount of power converted to mechanical power, is given by

P EIaa (A.3) and the electromagnetic torque is the mechanical power divided by the shaft speed, which is calculated as:

EIaa Te (A.4) Zr

At steady state the electromagnetic torque developed by the DC motor, for TL = 0, is given by:

TDer Z (A.5)

Te is determined from (A.5) and then D is calculated as. T D e (A.6) Zr For the determination of D, the voltage and current measurements at a given speed were taken by giving some time until steady state speed was attained. In this part the measurement was taken manually making sure that there was no change of speed at a particular value of speed. The torque versus speed graph to determine D is given in Fig. A.1. A line is drawn using Matlab linear curve fit and the slop of this line is the friction coffiecient..

From (A.6) D is the slope of the line in the plot Te versus Zr. From the curve fit the slop of the line is 0.0027, which is the average gradient.

Therefore D=0.0027 Nm/(rad/sec)

279 APPENDIX A DETERMINATION OF INERTIA AND FRICTION COEFFIENT

Torque, Te (Nm)

Speed or Zr (rad/sec) Fig. A.1 Electromagnetic torque versus motor speed at steady state

The effect of J appears when there is change in speed. To calculate J the variation of speed with time should be recorded accurately. In this part of the test a fast data acquisition system (dSPACE DS1102 DSP card) was used. The speed of the DC motor reaches at a given point and then the DC supply to the motor is switched off.

If the supply to the DC motor is switched off, while the motor is running at a specific speed, then the dynamics of the motor is given by: dZ 0 J r DZ (A.7) dt r The transient solution of (A.7) is given by

D t J ZZrro e (A.8)

where Zro is the initial speed, which is the speed of the motor when the DC supply was switched off.

D The magnitude of can be calculated by solving the equation in (A.8) using curve J fitting or based on the value of speed and time obtained from measurement.

280 APPENDIX A DETERMINATION OF INERTIA AND FRICTION COEFFIENT

Or J can be calculated by rearranging the basic equation given A.7 [1] as DZ J  r (A.9) dZr dt

The DC supply to the armature of the motor and to the field winding of the motor should be switched off at the same time. The variation of motor speed recorded when the DC motor field supply was on and off is shown in Fig. A.2. Of course the armature supply of the DC motor was switched off.

Speed or Zr (rad/sec)

(b) (a)

Fig. A.2 Variation of speed with time (a) DC motor field supply on (b) DC motor field supply off

When the supply is switched off at t = 0 there is transient due to the collapsing of the field and this transient filed will enable the machine to generate a transient condition Close to t = 0 the curves with field supply on and with field supply off appear as if they overlap. Hence the inertia for the test with field supply off appears as if there is smaller inertia than the rest of the time.

Using curve (a) in Fig. A.2 can lead to a wrong answer because the DC machine field supply was on during the test. If the DC machine field supply is on, while the motor armature supply is off, the DC motor will act as a generator and it will generate an open

281 APPENDIX A DETERMINATION OF INERTIA AND FRICTION COEFFIENT circuit voltage. Due to the generation of open circuit voltage it will have some internal electrical losses. These losses will decelerate the speed faster. The effect of faster deceleration is reflected on the value of the J in the system and it will appear as if it has a smaller value than the J obtained from the test with the DC motor field supply off. J calculated from curve (a) in Fig. A.2 is 0.025Kg-m2 and J calculated from Fig. A.2 curve (b) is 0.045Kg-m2/rad.

The correct value of the inertia J is the one calculated from the speed versus time variation when the supplies to the field winding as well as the supply to the armature are switched off at the same time. The curve that satisfies this condition is curve (b) given in Fig. A.2. Hence the correct value of inertia, J, for the induction generator system used in the lab is 0.045Kg-m2/rad.

Reference Werner Leonhard, “Control of Electrical Drives”, Springer-Verlag Berlin Heidelberg, New York, 1996

282 APPENDIX B

MEASUREMENT AND CONTROL HARDWARE SYTEMS

The interconnection between the different measurement and control hardware systems is given in this appendix.

Voltages and MITSBUSHI MITSBUSHI currents sensors PM50RSA120 PM50RVA120 IPM

Incremental 8 – to – 4 Optocoupler Optocoupler encoder multiplexer board board

Encoder interface Cross over protection board and 5V encoder DS1102 power supply board DC motor speed DSP card in PC reference Fig. B.1 Interconnection of hardware system

Fig. B.2 DSPACE DS1102 DSP controller board

IOP0 Red IOP0 5 28 IOP1 Orange IOP1 1 7 GND Black GND 3 60 Single core coaxial cable CON1 Output1 ADC1 DSP Multiplexer 1 dSPACE DS1102 board Output2 Single core coaxial cable ADC2 card CON2 2

Output3 Single core coaxial cable ADC3 CON3 3 Output4 Single core coaxial cable CON4 ADC3 4

Fig. B.3 Multiplexer board control to dSPACE DS1102 DSP card connection

283 APPENDIX B MEASUREMENT AND CONTROL HARDWARE SYTEMS

DSP dSPACE DAC4 BNC socket 46 DS1102 card Coaxial cable for speed GND 47 feed back

Fig. B.4 DAC output for DC motor speed control

PWM1 blue PWM1 CMP0 3 54

PWM2 green PWM2 CMP1 2 55

PWM3 yellow PWM3 CMP2 1 56 PWM4 Orange PWM4 CMP3 Dead time generator 6 57 dSPACE for IGBTs gate DS1102 DSP PWM5 Red PWM5 CAP3 drivers in the 5 52 card Inverter (Cross over PWM6 Brown PWM6 CAP4 protection board) 4 53

Break White Break-to reduce DC bus voltage IOP4 7 30

0V digitalGND 8 Black GND 15

Fig. B.5 Dead time Generator board and DS1102 DSP card connection

U brown Ua1 Ua1 blue Phi0 1 a1 1 1 20

Ua1 green Ua1 2 Ua1 yellow Phi0 1 Ua2 gray U 2 41 a2 3 U pink U Ua2 gray Phi90 1 a2 a2 4 3 19 Incremental Encoder Uao red Uao rotary encoder interface Ua2 orange Phi90 1 5 40 dSPACE HEIDENHAIN U board with 4 Ua0 black a0 DS1102 6 5V supply ROD 426 Uao green Index1 DSP card 5V(sensor) 5V 5 21 5000line blue 7 0V(sensor) White 0V U 8 ao Red Index1 42 5V(U )Brown/green 5V 6 p 14 0V(UN)White/green 0V 0V Black Digital GND 60 15 8

Fig. B.6 Incremental encoder DS1102 DSP card connection

284 APPENDIX B MEASUREMENT AND CONTROL HARDWARE SYTEMS

+15V 8 + 33PF, VN1 50V 22K: 7 0V

5 20V 4 5 6 10 +15V 11 12 Mitsubishi + 10PF, V Mitsubishi M57140-01 22K: UP1 Diode + 330 F, 50V + 220PF, M57120L-01 P 0V a240V bridge 50V 4u15V isolated 9 GBU6K 400V DC-DC DC output converter +15V 2 1 2 3 12 6 1 + 10PF, VUP1 + 50V 22K: 0V 11 +15V 14 + 10PF, 22K: VWP1 50V 13 0V

Fig. B.7 Four isolated 15V Power supply for optocoupler circuit

285 APPENDIX B MEASUREMENT AND CONTROL HARDWARE SYTEMS

Fig. B.8 Optocoupler to Mitsubishi PM50RVA120 IPM

286 APPENDIX B MEASUREMENT AND CONTROL HARDWARE SYTEMS

V +1 + 1 V -15 - 4 LF398 Output Input Input 5 3 C h 6 Logic 10nF 8 L. Ref 7 2 Offset adjust +15 V 1K 24K +15 -15V +5 +1 + 1 V V V VL -15 - 4 LF398 Output + - Input Input 5 3 11 14 12 C D1 2 h 6 Logic S1 16 1 10nF 8 1K : L. Ref 7 S3 D3 ADC1 22 4 Analog 3 1 Offset adjust 9.1V +15 S2 Switch D1 1K 24K 9 8 +1 V+ 1 DG403 D3 1K : ADC2 V S4 6 2 -15 - 4 LF398 Output 5 To Input Input 5 GND 9.1V 3 13 DS1102 Ch 15 10 6 Logic ADC3 3 10nF 8 DSP card L. Ref 7 2 IN1 IN2 ADC Offset adjust 1K 24K +15 inputs V +1 + 1 ADC4 V 4 -15 - 4 LF398 Output Input Input 5 9.1V 3 +5V Ch 1K : 6 Logic 1K : 10nF 8 L. Ref 7 2 Offset adjust 1K 24K +15 9.1V V 1K : +1 + 1 V -15 - 4 LF398 Output Input Input 5 3 IOP0 C 1 h 6 Logic +5V 10nF 8 L. Ref 7 2 1K : 2 IOP1 Offset adjust 5

+15 1K 24K V+ +15V -15V +5V +1 1 V- 9 pin socket -15 4 LF398 Output V+ V- VL Input Input 5 3 11 14 12 Connector C h 6 Logic S1 1 D1 to DS1102 10nF 8 16 L. Ref 7 DSP card 2 S3 3 D3 Offset adjust 4 Analog +15 1K 24K S2 Switch 8 D1 +1 V+ 9 1 DG403 V S4 D3 -15 - 4 LF398 Output 5 6 Input Input 5 GND 3 13 C 3 h 6 Logic 15 10 10nF 8 L. Ref 7 2 IN1 IN2 Offset adjust +15 V 1K 24K +1 + 1 V -15 - 4 LF398 Output Input Input 5 3 C 8 h 6 Logic 10nF 8 L. Ref 7 2 Offset adjust 1K 24K +15 Fig. B.9 8 to 4 multiplexer with Sample and Hold

287 +5V 18 Vcc +5 Break 8 Br 2 ENAR 7 4 ENAS +5V 1K: 11 +5V 6 ENAT 1 14 +5 10 4 3 PWM4 T 8 RESET 9 6 INHIBIT 14 6 5 12 74LS00 1K 8 U 2 : 17 RU 2 NAND 11 P To optocoupler Dead time +5V 16 RL 5 UN 5 Generator 10 7 IGBT gate 15 SU VP 3 5 PWM5 S IXYS 13 driver 3 SL VN 6 1K: IXDP630 14 1 +5 14 W 4 13 TU 4 74LS00 3 P +5V 12 TL 2 NAND 6 WN 7 PWM6 R 5 7 4 1 11 1K 15 : 47K: 20K: 9 10 +5V To 7 4.7pF AND 1 OUTEN SN74LS08N DSP card 3 To Single IGBT for OUTEN 1 GND DC voltage control 7 18 Vcc +5 2 8 2 ENAR 7 +5V +5V 4 ENAS 11 PWM1 T 6 ENAT 1 1 +5V 10 3 5 4 3 1K: 8 RESET 9 74LS00 6 INHIBIT 14 Dead time 17 RU 1 NAND 8 UP 2 +5V Generator 2 1 To optocoupler 16 RL 5 UN 5 IXYS 1 7 IGBT gate PWM2 S 15 SU V 3 2 3 IXDP630 1 P 1K: 14 SL +5V VN 6 1 1 W 4 13 TU 4 74LS00 P +5V NAND 3 12 TL 2 6 WN 7 PWM3 R 5 1 1 11 7 1K: 9 47K: 20K: 15 10 4.7pF

Fig. B.10 Cross over protection board (dead time generator) 288 APPENDIX C

DETAILS IN INDUCTION MACHINE MODELLING

C.1 Introduction In this part of the appendix the relationship between the induction machine parameters in the conventional steady state equivalent circuit model and in the dynamic equivalent d-q model is explained. The expanded form of the equations representing the dynamic d-q model of an induction machine including Rm are given.

C.2 Relationship of parameters in steady state model and d-q model of induction machines The parameters of an induction machine are obtained with the help of parameter determination method using the conventional steady state equivalent circuit. The conventional steady state model gives only steady state solution however the d-q model gives the complete solution which includes the transient and steady state solutions based on the steady state parameters. To find the relationship between the parameters in the conventional steady state model and d-q model, the d-q model will be analysed based on its steady state conditions. From Equation (4.24)

ªºvqs ªºR+Lpss00 L m p ªºiqs «»«»«» v 00R +L p L p i «»ds = «»s sm«»ds (C.1) «» «»0 «»Lpmrmrrrrq -Ȧ LR+Lp-Ȧ Lir «»«»«» 0 Ȧ LLpȦ LR+Lp ¬¼«»¬¼«»rm m rr r r ¬¼«»idr

The steady state form is obtained by substituting p by jZe and Zr by (1-s)Ze where Ze and s are the excitation angular frequency and rotor slip respectively

ªºvqs ªºRss +jX00 jX m ªºiqs «»«»«» v 00R +jX jX i «»ds = «»s sm«»ds (C.2) «» «»0 «»jXmmrrrq X(1 s ) R +jX -X(1 s ) i r «»«»«» 0 X(1 s ) jX X(1 s ) R +jX ¬¼«»¬¼«»mmrrr¬¼«»idr 289 APPENDIX C DETAILS IN INDUCTION MACHINE MODELLING

The rotor current terms can be eliminated easily by partitioning the matrix equation and rewriting as

ªºV1 ªºªºZ11ZI 12 1 «» «»«» (C.3) ¬¼0 ¬¼¬¼Z21ZI 22 2 where

ªºvqs ªºªºªºRss+jX 00jX mjX m X m (1 s) VZ11 «» 1 ZZ12 21 «»«»«»00R +jX jX X (1 s) jX ¬¼vds ¬¼¬¼¬¼ss m m m

ªºR+jX -X(1 s ) ªºiiqs ªº qr ZI rr r I 22 «»XsR+jX(1 ) 1 «»2 «» ¬¼rrr ¬¼iids ¬¼ dr

1 ªºR+jX X(1 s ) Z 1 rr r 22 22«»XsR(1 ) +jX R+jXrr X r(1 s ) ¬¼r rr From Equation (C.3)

1 IZZI222211  (C.4)

1 VZZZZI1111222211  (C.5) Expanding Equation (C.5)

ªºvqs §ªºªºRss +jX00 jX m1 ªºR+jXrr X r(1 s ) ª jX m X m(1) s º·ªºiqs ¨¸ «» «»«»00R +jX jX«» X (1)( s R +jX«» X1) s jX «» ¬¼vds ©¹¬¼¬¼ss m' ¬¼ r rr¬ m m ¼¬¼ids (C.6) Where

22 ' R+jXrr  X r(1  s ) Using mathematical equation a2 + b2 = (a + jb) (a – jb)

' Rrr+jX  jX r(1  s ) R rr +jX  jX r(1  s )

' Rrr jX(2  s ) R rr +jsX (C.7) Simplifying Equation (C.6) vi2 ªºqs §·ªºR+jXss0 jX m ªº10ªºR+jXrr X r(1 s ) ªjs (1) º ªºqs ¨¸ «»«»00R +jX«»1«» X(1 s)( R +jX «1 s) j » «» ¬¼vids ©¹¬¼ss' ¬¼¬¼ r rr ¬  ¼¬¼ds

2 2 ªºviqs §·ªºR+jXss0 jX ªºjR s X R(1 s ) ªºqs ¨¸m rr r «»¨¸«»0 R+jX «» «» ¬¼vids ©¹¬¼ss ' ¬¼Rsrrr(1 ) sXsjR (2  ) ¬¼ds 290 APPENDIX C DETAILS IN INDUCTION MACHINE MODELLING

22 2 ªºjXmr() jR s X r jX mr R (1) s «»R+jXss ªºviqs '' ªºqs «» «» «» (C.8) vi2 2 ¬¼ds «»jXmr R(1 s ) jXmr  sX(2 s ) jR r ¬¼ds «»R+jXss ¬' ' ¼

The d-axis stator voltage vds in Equation (C.8) can be expressed as

2 2 jX R(1 s ) §·jXmr  sX(2 s ) jR r viR+jXi mr  (C.9) ds qs¨¸ s s ds ''©¹ As discussed in Chapter 3 the q-axis current lags the d-axis current by 90o.Hence substituting ijiqs  ds in Equation (C.9) gives

2 §·XjsXssRmr (2 ) r vR+jX  i (C.10) ds¨¸ s s ds ©¹' Substituting Equation (C.7) in (C.10)

2 §·XjsXssRmr (2 ) r vR+jXds ¨¸ s s i ds ¨¸RjXsR+jsX(2 ) ©¹ rr rr

§·sX 2 vR+jX m i (C.11) ds¨¸ s s ds ©¹ R+jsXrr

Bur Xs and Xr are the total stator reactance and rotor reactance respectively. Expressing the reactances with the mutual and their leakage reactances

XXXs ls m

XXXrlrm  Then Equation (C.11) can be rewritten as

§·§·Rr ¨¸jXmlr¨¸ jX ¨¸©¹s vR+jXds s ls i ds (C.12) ¨¸§·Rr ¨¸¨¸+jXlr jX m ©¹©¹s

Equation (C.12) represents the steady state per-phase equivalent circuit which can be derived from Fig. 4.3. Hence at steady state the D-Q model converges to the conventional steady state model.

C.3 Expanded equations for induction machine modelling including Rm

291 APPENDIX C DETAILS IN INDUCTION MACHINE MODELLING

The matrix equation representing a stator supplied induction machine, from Equation

4.25, including Rm is

ªºvqs ªºR+LssNEW p00 L N p ªºiqs «»«»«» v 00R +L p L p i «»ds = «»s sNEW N «»ds «» (C.13) «»0 «»LpNr -Ȧ LR+Lp-mrrNEWrȦ LirNEWqr «»«»«» 0 Ȧ LLpȦ LR+Lp ¬¼«»¬¼«»rN N rrNEW r rNEW ¬¼«»idr where

RmmL L=N Rmm+L p

L=L+LsNEW ls N

LrNEW=L lr +L N The expanded form of Equation (C.13) is given in Equation (C.15).

For the D-Q model of a self-excited induction generator the matrix equation including

Rm, as given in Equation 8.1, is, iV ªº0 ªºRpLssNEW10 pC pL N 0 ªºªqs cqo º «»« » «»0 «»010RpL pC pL i V «» «»ssNEW N«»« dscdo » (C.14) «»« » «»0 «»pLNrmrrNEWrrNEWqrqrZZL R pL L i K «» «»«»« » 0 ZZLpLLRpL ¬¼ «»¬¼rN N rrNEW r rNEW «»«¬¼¬iKdr dr »¼

The expanded form of Equation (C.14) is given in Equation (C.16).

292 ª §·RLmm RLpmm º «RLsls¨¸ p 00» RLp RLp « ©¹mmmm » v « §·RL RLp » ªºi ªºqs « mm mm » qs «» 00RLsls¨¸ p «» v « ©¹RLpmmRLpmm » ids «»ds «» (C.15 « » «» «»0 RLp Z RLp §·§·RL RLp iqr « mm r mm mm mm » «» «» -Rrlr¨¸¨¸Lp-ZrlrL  «»0 « RLpRLp RLp RLp » «»i ¬¼« mm mm ©¹©¹mm mm » ¬¼dr « » ZrmmRLp RLpmm §·§·RLpmmRL mm « Zrlr¨¸¨¸L  RLrlr p» «¬ RLpmm RLpmm ©¹©¹RLpmmRLp mm ¼»

ª §·RLmm 1 RLpmm º «RLsls¨¸ p  00» « ©¹RLpmm pC RLpmm » « §·RL 1 RLp » ªºªiV º ªº0 « mm mm » qs cqo 00RLsls¨¸ p  «»« » «» « RLp pC RLp » 0 ©¹mm mm «»«iVds cdo » «» « »  «»0 «»«iK » « RLpmmZ rRLp mm§·§·RL mmRLp mm » qr q «» -Rrlr¨¸¨¸Lp-Z rlrL  «»« » 0 « RLpRLp RLp RLp » «»«iK » ¬¼ « mm mm©¹©¹ mm mm » ¬¼¬dr d ¼ « » ZrmmRLp RLp mm §·§·RLpmm RLmm « Zrl¨¸¨¸L rrRLlr p» ¬« RLpmmRLp mm ©¹©¹RLpmm RLpmm ¼» (C.16)

293 From Equation (8.5) the d-axis current for the SEIG including Rm is given by U i (C.17) d ApBpDpEpFpGpHpJpM87 654  3 2 

The roots in the denominator (poles) contain the behaviour of the solution for the d-axis current. To find the roots of the denominator ApBpDpEpFpGpHpJpM087 6  54 3 2  (C.18) A, B, D, E, F, G, H, J, and M are coefficients which are functions of the machine parameters, exciting capacitance and rotor speed. The detail these coefficients are given as follows.

2222 A CLls L lr L m

22 2 22 2 2 22 2 22 222 B 2(CLlsrmlrm RL L RCL lslrm LL RCLL s lslrm L RCLL m lslrm L2 RCL m lslrm L L)

2222 22 22 2 22 22 2 D CLRLls r m 22 CLLLls lr m RCLLL m ls lr m  4 RCLRLL m ls r lr m  2 RCLRL m ls r m 22222 2 2 2222 2222 222 CLlsrlrmZ LL42 RCLRLL s lsrmlrm  RCLL lsm RCLL m mlr RRCLL ms lrm 22 2222222 2 2 44RRCLLLms lslrm RRCLL ms lslrm L  4 RCLRL m lsrm Llr RCL s lr L m 22 2 22 2 2 22 2 22RCLLLm lslrm RCLL m lslr RCLLL m lslrm

22 2 22 2 2 ECRLLCLRLLRCLLRCLLLRCLLL 24mlrm lsrmlr  24 s lrm m lslrm  4 m lslrm 22 2 2 2 2 2 2 2 2 2 2 24Rs C RL rmlr L R ms RC RL rmlr L  22 RC s L lsrm R L RC s L lsrlrmZ L L 2222 222 2 2 22 2 2 2 2222RmCL lsZZ r LL lr m RCL m ls r LL lr m RRCLL m s lr m RRCLL m s lr m 22 2 22 2 222 222 2222RRCLLm s ls lr RRCLL m s ls m RRCLL m s lr m RRCLL m s lr m 222222222 8422RRCLRLLms lsrlrm RRCLRL ms lsrm RCL m lsr R L m RCLR m lsr L m 222222 222 222 2422RCLmlsrlrmZ L L R mslslrmm RCLLL R CL mrlrm RL R CLRL lsrm 22 22 2 22 2  4RCLmlsRLL r lr m22 R m C L ls RL r lr R m C L ls RL r m

22222 222222 J 2(RRLrmmZZ rlrmm LLR rlrmm L RL R ms RC Z rm L  RRL+2RRC rmlrmsZ rlrm LL 222222 +Rrm R L m +R m R s CZ r L lr +R m R s CR r)

22 222 222 2 2 MRR rmZZ rmm RL  rlrm LR 2 Z rlrmm LRL

294 22 2 2222 2222222 222 F Llr L m 42222242 CRRLm r m L lr  CRLm lr L m  CRLLm lr m  CLRLls r m  R m CLL ls lr  R m CLL ls m  R m CLLL ls lr m  CL lsZ r L lr L m 2 2 2 2 22 2 2 22 222 222 48RCRLLs r m lr RCLRLL m ls r lr m  4 RCLRL m ls r m  4 RRCLL m s lr m  4 RRCLL m s lr m  RCRL s r m RC sZ r LLlr m 2 RRCRL m s r m 2 2 2 2 22 22 2 2 22 2 2 2 2 2 2 22 2 2 242RmsRCZ r L lrm L  RRCRLL ms rlrm  RRCRL ms rm R m RCL s m 242 R m RCRL s rm R m RCRLL s rlrm R m CL lsrZ LL lrm 222 222 222 222 22 22222 4444Rms RCL lsrrlrZZZ RL R ms RCL lsrrm RL RRCL ms lsrlrm L L RRCL ms lsZZ rLL lr m RRCLR m s ls r L m R m CL ls r L lr 22 2 2 22 2 2 22 2 2 2 2 22 2 22 2 22 2 2 2 22 2 2 22Rm CL lsZZ r LL lr m R m CL ls R r R m CL ls r L m R m RCLL s lr m R m RCL s lr R m CL mZ r L lr R m CL m R r 22 2 2 22 2 22RCLm lsZ r LL lr m RCLRL m ls r m

G =2*Rr*Lm^2*Llr+2*Llr^2*Rm*Lm+2*Llr*Lm^2*Rm+2*c*Rm*Rr^2*Lm^2+2*c*Rm^2*Rr*Lm^2 +2*c*Rm*wr^2*Llr^2*Lm^2 +2*Rs*c*wr^2*Llr^2*Lm^2+2*Rm^2*Rs*c*Lm^2+4*Rm^2*c*Lls*Rr*Llr+4*Rm^2*c*Lls*Rr*Lm+2*Rs*c*Rr^2*Lm^2 +4*c*Rm^2*Rr*Llr*Lm+2*Rm^2*Rs*c*Llr^2+4*Rm*c*Lls*wr^2*Llr^2*Lm+4*Rm*c*Lls*wr^2*Llr*Lm^2+4*Rm^2*Rs*c*Llr*Lm +4*Rm*c*Lls*Rr^2*Lm+8*Rm*Rs*c*Rr*Llr*Lm+4*Rm*Rs*c*Rr*Lm^2+2*Rm^2*Rs*c^2*wr^2*Llr^2*Lm+2*Rm^2*Rs*c^2*Rr^2*Lm +2*Rm^2*Rs*c^2*wr^2*Llr*Lm^2+2*Rm*Rs^2*c^2*wr^2*Llr^2*Lm+2*Rm*Rs^2*c^2*Rr^2*Lm+2*Rm*Rs^2*c^2*wr^2*Llr*Lm^2 +2*Rm^2*Rs*c^2*Lls*wr^2*Llr^2+4*Rm^2*Rs*c^2*Lls*wr^2*Llr*Lm+2*Rm^2*Rs*c^2*Lls*Rr^2+2*Rm^2*Rs^2*c^2*Rr*Llr +2*Rm^2*Rs^2*c^2*Rr*Lm+2*Rm^2*Rs*c^2*Lls*wr^2*Lm^2

H =wr^2*Llr^2*Lm^2+4*Rm^2*Rs*c*Rr*Lm+4*Rr*Rm*Llr*Lm+Rm^2*Lm^2+2*Rr*Lm^2*Rm+Llr^2*Rm^2+2*Llr*Rm^2*Lm +2*c*Rm^2*wr^2*Llr*Lm^2+Rm^2*Rs^2*c^2*wr^2*Lm^2+4*Rm*Rs*c*Rr^2*Lm+2*Rm^2*c*Lls*Rr^2+4*Rm^2*Rs*c*Rr*Llr +2*c*Rm^2*Rr^2*Lm+Rm^2*Rs^2*c^2*Rr^2+2*c*Rm^2*wr^2*Llr^2*Lm+4*Rm*Rs*c*wr^2*Llr^2*Lm+4*Rm^2*c*Lls*wr^2*Llr*Lm +4*Rm*Rs*c*wr^2*Llr*Lm^2+2*Rm^2*c*Lls*wr^2*Llr^2+2*Rm^2*Rs^2*c^2*wr^2*Llr*Lm+2*Rm^2*c*Lls*wr^2*Lm^2+Rr^2*Lm^2 +Rm^2*Rs^2*c^2*wr^2*Llr^2

295 APPENDIX D

LIST OF PUBLICATIONS

This appendix contains the lists of author’s publications during the course of this study.

Journal Publications [1] D. Seyoum, C. Grantham and M. F. Rahman, “The dynamic characteristics of an isolated self-excited induction generator driven by a wind turbine”, scheduled for publication in the IEEE Transactions on Industry Applications, Vol. 39, No. 4, July-Aug. 2003, pp. 936 -944. [2] D. Seyoum, M. F. Rahman and C. Grantham, “Improved flux estimation in induction machines for control application”, accepted for publication in the special issue of Journal of Electrical and Electronic Engineering Australia (JEEEA), Institute of Engineers Australia (IEAust), Vol. 22, No. 23, 2003, pp. 243-248. [3] 3 D. McKinnon, D. Seyoum and C. Grantham, “Novel dynamic model for a three- phase induction motor with iron loss and variable rotor parameter considerations”, accepted for publication in the special issue of Journal of Electrical and Electronic Engineering Australia (JEEEA), Institute of Engineers Australia (IEAust), Vol. 22, No. 23, 2003, pp. 219-225. [4] C. Grantham, M.F. Rahman and D. Seyoum, “A regulated self-excited induction generator for use in a remote area power supply”, International Journal of Renewable Energy Engineering, Vol. 2, Curtin University of Technology, 2000, pp. 135-140.

Conference Publications [5] D. Seyoum, M. F. Rahman and C. Grantham “Inverter Supplied Voltage Control System for an Isolated Induction Generator Driven by Wind Turbine”, accepted in IEEE Industry Applications Society 38th Annual Meeting, The Grand America Hotel, Salt Lake City, Utah, USA, October 12 - 16, 2003.

296 APPENDIX D LIST OF PUBLICATION

[6] D. Seyoum, D. McKinnon, M. F. Rahman, C. Grantham and H. P. To, “A novel account of iron loss in the analysis and modeling of an isolated self-excited induction generator”, accepted in the 6th International Power Engineering Conference, Singapore, 22-24 May, 2003.(due to the spread of SARES rescheduled to Nov. 27-29, 2003). [7] D. Seyoum, M. F. Rahman and C. Grantham, “Simplified flux estimation for control application in induction machines”, IEEE- International Electric Machines and Drives Conference (IEMDC), June 1st - 4th, 2003, Madison, WI, USA, pp. 691 -695. [8] D. Seyoum, M. F. Rahman and C. Grantham, “Terminal voltage control of a wind turbine driven isolated induction generator using stator oriented field control”, IEEE-Applied Power Electronics Conference and Exposition, Miami Beach, Florida, USA, February 9-13 2003, pp. 846 -852. [9] D. Seyoum, C. Grantham and F. Rahman F., "The Dynamic Characteristics of an Isolated Self-Excited Induction Generator Driven by a Wind Turbine", Proceedings IEEE- IAS 2002 Annual Meeting Pittsburgh, USA, 2002, October 13-18, pp 731- 738. [10] D. Seyoum, F. Rahman and C. Grantham, “Terminal voltage control of a wind turbine driven isolated induction generator”, Proc. AUPEC’02, Melbourne, Australia, Sep29-Oct 2, 2002, ISBN 0-7326-2206-9. [11] D. Seyoum, C. Grantham and F. Rahman, “A Novel analysis and modelling of an isolated self - excited induction generator taking iron loss into account”, Proc. of Australasian Universities Power Engineering Conference (AUPEC 2002), Melbourne, Australia, Sep29-Oct 2, 2002, ISBN 0-7326-2206-9. [12] D. Seyoum, F. Rahman and C. Grantham, “An improved flux estimation in induction machine for control application”, Proc. AUPEC’02, Melbourne, Australia, Sep29-Oct 2, 2002, ISBN 0-7326-2206-9. [13] D. McKinnon, D. Seyoum and C. Grantham “Investigation of the effects of supply voltage and temperature on parameters in a 3-phase induction motor including iron loss”, Proc. AUPEC’02, Melbourne, Australia, Sep29-Oct 2, 2002; ISBN 0-7326- 2206-9. [14] D. McKinnon, D. Seyoum and C. Grantham, "Novel Dynamic Model for a Three- Phase Induction Motor With Iron Loss and Variable Rotor Parameter

297 APPENDIX D LIST OF PUBLICATION

Considerations", Proc. of Australasian Universities Power Engineering Conference (AUPEC 2002), Melbourne, Australia, 29 Sept.-2 Oct. 2002, ISBN 0-7326-2206-9. [15] D. Seyoum, C. Grantham and F. Rahman, “An Insight into the Dynamics of Loaded and Free Running Isolated Self-Excited Induction Generators”, Proceedings IEE-PEMD 2002, University of Bath, UK, 2002, pp. 580-585.

[16] D. Seyoum, C. Grantham and F. Rahman, “Analysis of an Isolated Self-Excited Induction Generator Driven by Variable Speed Prime Mover”, Proc. AUPEC’01, Perth, Australia, 2001, pp. 49-54. [17] D. Seyoum, C. Grantham and F. Rahman, “The Dynamics of an Isolated Self- Excited Induction Generator Driven by a Wind Turbine”, Industrial Electronics Society, IECON2001, the 27th Annual Conference of the IEEE, Denver, USA, 2001 pp. 1364-1369. (received student award) [18] D. McKinnon, D. Seyoum and C. Grantham, “Fast Algorithm for Rapid Determination of RMS Values in a 3-Phase System”, Proc. AUPEC’01, Perth, Australia, 2001, pp. 360-365. [19] D. McKinnon D. Seyoum and C. Grantham, “Rapid Determination of Fundamental and Harmonic RMS Quantities in a 3-Phase System”, Proc. AUPEC’01, Perth, Australia, 2001, pp. 73-78. [20] C. Grantham F. Rahman, D. Seyoum, “A self-excited induction generator with voltage regulator for use in remote area power supply”, IEEE Power Electronics and Motion Control Conference, PIEMC 2000, China, Vol.2, 2000, pp. 710 - 715. [21] C. Grantham, D. Seyoum, D. Indyk and D. McKinnon, “Calculation of the parameters and parameter variations of an induction motor and the effect of measurement error”, Proc. AUPEC’00, Brisbane, Australia, 2000, pp. 225-228. [22] C. Grantham, D. Seyoum. and H. Tabatabaei-Yazdi, “Very fast and accurate electrical measurements”, Proc. AUPEC /EECON ’99, Darwin, Australia, 1999, pp. 99-103. [23] C. Grantham, M. F. Rahman and D. Seyoum, “A regulated self-excited induction generator for use in a remote area power supply”, Proc. AUPEC /EECON ’99, Darwin, Australia, 1999, pp. 438-443.

298 APPENDIX D LIST OF PUBLICATION

299 APPENDIX D LIST OF PUBLICATION

Fig. D.1 Student award

300