SENSORLESS STATOR WINDING TEMPERATURE ESTIMATION FOR INDUCTION MACHINES
A Dissertation Presented to The Academic Faculty
by
Zhi Gao
In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Electrical and Computer Engineering
Georgia Institute of Technology December 2006
SENSORLESS STATOR WINDING TEMPERATURE ESTIMATION FOR INDUCTION MACHINES
Approved by: Dr. Thomas G. Habetler, Advisor School of Electrical and Computer Engineering Georgia Institute of Technology
Dr. Thomas E. Michaels School of Electrical and Computer Engineering Georgia Institute of Technology
Dr. Ronald G. Harley School of Electrical and Computer Engineering Georgia Institute of Technology
Dr. Roy S. Colby Center for Innovation and Technology Schneider Electric
Dr. Deepakraj M. Divan School of Electrical and Computer Engineering Georgia Institute of Technology Date Approved: October 13, 2006
To my mother, Qihui Zhao, and my father, Yongsheng Gao, for their love and support.
ACKNOWLEDGEMENTS
A doctoral dissertation is usually considered to be a personal accomplishment. However, it would not have been possible for me to finish this work without the inspiration, encouragement and support from many people. Dr. Thomas Habetler has been a wise and trusted advisor throughout the entire process. It is due to his constant inspiration and encouragement that I have gained a deeper understanding of engineering and made progress toward solving problems and improving my communication skills as a research engineer. Had it not been for his vision, encouragement and his confidence in my ability, much of this work would not have been completed. I am deeply grateful for his guidance. I would also like to express my gratitude to Dr. Ronald Harley. Throughout the whole project, his patient guidance, constant encouragement and meticulous attention to detail provide me with tremendous motivation. I am also indebted to Dr. Deepak Divan, Dr. Thomas Michaels for their time and invaluable input into my research. This work has been funded through a Georgia Tech research contract with the Schneider Electric North America Operating Division / Square D Company. It would not have been possible to come to a fruitful and mutually beneficial conclusion without the wise guidance and suggestions from Dr. Roy S. Colby. I was fortunate to work with many exceptional fellow colleagues in my research group. I would like to thank Dr. Sang-Bin Lee, Dr. Jason Stack, Dr. Wiehan le Roux, Dr. Ramzy Obaid, Dr. Jung-Wook Park, Dr. Vinod Rajasekaran, Dr. Dong-Myung Lee, Dr. Xianghui Huang, Dr. Salman Mohagheghi, Dr. Satish Rajagopalan, Joy Mazumdar, Afroz Imam, Young-Kook Lee, Wei Qiao, Long Wu, Bin Lu, Yi Yang, Ari Zachas and Wei Zhou for their help on various aspects of this work, and other fellow graduate
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students in the research group for their friendship and support over the past four years of my endeavor. In addition, I would also like to thank the machine shop technicians: Mr. Lorand Csizar and Mr. Louis Boulanger, for their help and assistance to my experimental work. There are numerous names of faculty, family and friends that should be mentioned here, who have supported me directly or indirectly during my stay at Georgia Tech. I express my gratitude to all of the people I have known. Most of all, I would like to thank my parents for being a constant source of encouragement and motivation throughout my pursuit for the doctoral degree. I could never fully express my love and gratitude to them.
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TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ............................................................................................ iv LIST OF TABLES .............................................................................................................x LIST OF FIGURES ......................................................................................................... xi SUMMARY .................................................................................................................. xvi CHAPTER 1 INTRODUCTION ......................................................................................1 1.1
Stator Winding Insulation Failure.........................................................................2
1.2
Temperature Monitoring.......................................................................................5
1.3
Problem Statement ................................................................................................8
1.4
Dissertation Outline ............................................................................................10
CHAPTER 2 SUMMARY OF PREVIOUS WORK ON STATOR TEMPERATURE ESTIMATION .........................................................12 2.1
Temperature Estimation Based on Thermal Models ..........................................13
2.1.1
Dual-element Time-delay Fuses ................................................................14
2.1.2
Thermal Models with a Single Time Constant ..........................................16
2.1.3
Complex Thermal Networks......................................................................20
2.2
Temperature Estimation Based on Motor Parameters ........................................22
2.2.1
Induction Machine Model-based Resistance Estimation...........................24
2.2.2
Resistance Estimation Using dc Injection..................................................26
2.3
Comparison of Different Temperature Estimation Schemes ..............................28
2.4
Chapter Summary ...............................................................................................29
CHAPTER 3 INDUCTION MACHINE THERMAL ANALYSIS .............................31 3.1
Analysis of Induction Machine Thermal Behaviors ...........................................32
3.1.1
State-space Representation of Induction Machine Thermal Models .........32 vi
3.1.2 3.2
Induction Machine Thermal Behaviors under Different Duty Types........34
Hybrid Thermal Models of Induction Machines ................................................37
3.2.1
Full Order Hybrid Thermal Model ............................................................38
3.2.2
Reduced Order Hybrid Thermal Model.....................................................49
3.3
Chapter Summary ...............................................................................................51
CHAPTER 4 INDUCTION MACHINE ONLINE PARAMETER ESTIMATION..........................................................................................53 4.1
The Overall Architecture of the Sensorless Parameter Estimation Algorithm ...54
4.2
Complex Space Vector Modeling of Induction Machines..................................54
4.2.1
Complex Space Vector Representation of Three Phase Variables ............55
4.2.2
Complex Space Vector Representation of Induction Machines ................62
4.3
Online Inductance Estimation Algorithm ...........................................................64
4.3.1
Derivation of the Inductance Estimation Algorithm..................................64
4.3.2
Criterion for Good Estimates of Inductances.............................................70
4.3.3
Influences from Numerical Precision and A/D Resolution on the Inductance Estimation Algorithm ..............................................................79
4.4
Online Rotor Resistance Estimation Algorithm .................................................87
4.5
Fast and Efficient Extraction of Positive and Negative Sequence Components 88
4.5.1
Estimation Error from Negative Sequence Fundamental Frequency and Other Frequency Components ...................................................................89
4.5.2
Goertzel Algorithm ....................................................................................93
4.6
Sensorless Rotor Speed Detection from Current Harmonic Spectral Estimation98
4.6.1
Rotor Slot Harmonics ................................................................................98
4.6.2
Rotor Dynamic Eccentricity Harmonics..................................................100
4.6.3
Sensorless Rotor Speed Detection ...........................................................102
4.6.4
Experimental Validation ..........................................................................104
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4.7
Chapter Summary .............................................................................................107
Chapter 5 INDUCTION MACHINE SENSORLESS STATOR WINDING TEMPERATURE ESTIMATION....................................110 5.1
Online Calculation of Rotor Temperature ........................................................110
5.1.1
Rotor Temperature Calculation................................................................110
5.1.2
Experimental Validation ..........................................................................111
5.2
Online Adaptation of Reduced Order Hybrid Thermal Model.........................114
5.2.1
State-Space Representation of the Reduced Order Hybrid Thermal Model .......................................................................................................115
5.2.2
Online Parameter Tuning.........................................................................117
5.2.3
Experimental Validation ..........................................................................124
5.3
Chapter Summary .............................................................................................126
CHAPTER 6 EXPERIMENTAL SETUP AND IMPLEMENTATION OF VARIOUS TESTS ...........................................................................128 6.1
Experimental Setup...........................................................................................128
6.1.1
Motor and Load........................................................................................130
6.1.2
Current and Voltage Measurements.........................................................134
6.1.3
Speed Measurement .................................................................................143
6.1.4
Temperature Measurement ......................................................................145
6.2
Implementation of Various Tests......................................................................151
6.2.1
Motor Operation with Unbalanced Voltage Supply ................................152
6.2.2
Motor Operation with Impaired Cooling .................................................153
6.2.3
Motor Operation with Continuous-operation Periodic Duty Cycles .......155
6.3
Chapter Summary .............................................................................................159
CHAPTER 7 INDUCTION MACHINE ONLINE THERMAL CONDITION MONITORING .............................................................161 7.1
Induction Machine Thermal Monitoring under Impaired Cooling Condition ..161 viii
7.2
Induction Machine Thermal Monitoring under Continuous-operation Periodic Duty Cycles.......................................................................................................164
7.2.1
Proportional Integral Observer ................................................................165
7.2.2
Operation of the Proportional Integral Observer .....................................167
7.2.3
Experimental Results ...............................................................................168
7.3
Chapter Summary .............................................................................................173
CHAPTER 8 CONCLUSIONS, CONTRIBUTIONS AND RECOMMENDATIONS.......................................................................175 8.1
Conclusions.......................................................................................................175
8.2
Contributions.....................................................................................................179
8.3
Recommendations for Future Work..................................................................182
APPENDIX A MOTOR PARAMETERS ...................................................................186 APPENDIX B RELATIONSHIP BETWEEN TEMPERATURE AND RESISTIVITY........................................................................................188 APPENDIX C SINGULAR VALUE DECOMPOSITION AND MOORE-PENROSE INVERSE ...........................................................190 REFERENCES...............................................................................................................196 VITA
..................................................................................................................204
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LIST OF TABLES Page Table 1.1: Temperature limits for different insulation classes ........................................... 2 Table 2.1: Relationship between τth and t6X at different service factors............................ 19 Table 2.2: Comparison of different temperature estimation techniques........................... 30 Table 4.1: Online inductance estimation results for the 5 hp TEFC motor ...................... 69 Table 4.2: Online inductance estimation results for the 5 hp ODP motor........................ 69 Table 4.3: Online inductance estimation results for the 7.5 hp TEFC motor ................... 70 Table 4.4: Approximate constants for 3-phase induction motors [50] ............................. 74 Table 4.5: Online inductance estimation results for the 5 hp TEFC test motor at same load level......................................................................................................... 86 Table 4.6: Total computations for each algorithm to extract positive sequence fundamental frequency component................................................................. 96 Table 4.7: Rotor slot harmonic frequencies from Figure 4.23.......................................... 99 Table 6.1: Nameplate data of motors used in the experiments ....................................... 130 Table 6.2: Nameplate data of the dc machine................................................................. 133 Table A.1: Parameters of the 5 hp TEFC Motor............................................................. 186 Table A.2: Parameters of the 5 hp ODP Motor .............................................................. 186 Table A.3: Parameters of the 7.5 hp TEFC Motor.......................................................... 187 Table B.1: Relationship between temperature and resistivity ........................................ 188
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LIST OF FIGURES Page Figure 1.1: Stator winding damage [Courtesy of Electrical Apparatus Service Association (EASA) Inc., St. Louis, USA]....................................................... 4 Figure 2.1: Typical thermal limit curves from reference [21]. ......................................... 13 Figure 2.2: The structure of a dual-element time-delay fuse from reference [8].............. 14 Figure 2.3: Dual-element fuse operating mechanisms under different conditions [8]. .... 15 Figure 2.4: Motor starting and running curves and the dual-element time-delay fuse thermal limit curve [8]. ................................................................................... 15 Figure 2.5: Thermal model with a single thermal time constant. ..................................... 16 Figure 2.6: Equivalent electrical circuit of the temperature estimator with a single thermal time constant...................................................................................... 18 Figure 2.7: Thermal model by a complex thermal network for induction machine #1. ... 21 Figure 2.8: Thermal model by a complex thermal network for induction machine #2. ... 22 Figure 2.9: Overall structure of the induction machine model-based Rr and Rs estimator.25 Figure 2.10: DC equivalent circuit of dc injection circuit and the motor. ........................ 26 Figure 2.11: Equivalent circuit during dc injection mode. ............................................... 27 Figure 2.12: Equivalent circuit during normal mode........................................................ 28 Figure 3.1: Thermal Network and Parameters [25]. ......................................................... 33 Figure 3.2: Stator winding temperature rise for the 7.5 hp TEFC motor.......................... 36 Figure 3.3: Full order hybrid thermal model for an induction motor. .............................. 38 Figure 3.4: Identification of the dominant component in the rotor thermal transient....... 44 Figure 3.5: Different components in a typical rotor thermal transient. ............................ 45 Figure 3.6: Normalized steady-state stator winding temperature. .................................... 47 Figure 3.7: Effects of variations in R1, R2 and R3 on τth.................................................... 48 Figure 3.8: Reduced order hybrid thermal model for an induction motor........................ 50 xi
Figure 4.1: Overall architecture for the induction machine sensorless parameter estimation algorithm. ...................................................................................... 54 Figure 4.2: Trajectories of the complex current and phase-to-neutral voltage space vectors in a stationary reference frame. .......................................................... 57 Figure 4.3: Positive and negative sequence fundamental frequency components in a complex current space vector in the stationary reference frame..................... 59 Figure 4.4: Current and voltage spectra from the complex current and phase-to-neutral voltage space vectors in the range of −300~300 Hz. ...................................... 60 Figure 4.5: Positive and negative sequence complex current and voltage space vectors in a synchronous reference frame. .................................................................. 61 Figure 4.6: Steady-state positive sequence equivalent circuit using complex vectors. .... 63 Figure 4.7: Steady-state positive sequence motor equivalent circuit using phasors......... 64 Figure 4.8: Phasor diagram of the equivalent circuit........................................................ 65 Figure 4.9: Flowchart of the online inductance estimation algorithm. ............................. 68 Figure 4.10: The relationship between K=Vsy/Is and the motor’s load levels. .................. 73 Figure 4.11: K=Vsy/Is versus the motor’s load levels for various motors with different parameters - variations in Rs (Xls=Xlr=0.0675 p.u.; Xm=2.4 p.u.; Rr=0.045 p.u.). ................................................................................................................ 75 Figure 4.12: K=Vsy/Is versus the motor’s load levels for various motors with different parameters - variations in Xls and Xlr (Rs=0.045 p.u.; Xm=2.4 p.u.; Rr=0.045 p.u.). ................................................................................................................ 76 Figure 4.13: K=Vsy/Is versus the motor’s load levels for various motors with different parameters - variations in Xm (Rs=0.045 p.u.; Xls=Xlr=0.0675 p.u.; Rr=0.045 p.u.). ................................................................................................................ 77 Figure 4.14: K=Vsy/Is versus the motor’s load levels for various motors with different parameters - variations in Rr (Rs=0.045 p.u.; Xls=Xlr=0.0675 p.u.; Xm=2.4 p.u.). ................................................................................................................ 78 Figure 4.15: Block diagram of the measurement and data acquisition system for the inductance estimation algorithm. .................................................................... 81 Figure 4.16: The phasor diagram to illustrate the load spread in inductance estimation algorithm. ........................................................................................................ 84 Figure 4.17: Positive and negative sequence complex current and voltage space vectors in a synchronous reference frame. .................................................................. 90 xii
Figure 4.18: Simulation results from the online rotor temperature estimation algorithm e are used............................................................................. 91 when iqdse and vqds Figure 4.19: Experimental results from the online rotor temperature estimation e algorithm when iqdse and vqds are used. ........................................................... 92 Figure 4.20: Experimental results from the online rotor temperature estimation algorithm when i1 e and v1e are used............................................................... 93
Figure 4.21: Flowchart of the Goertzel algorithm to extract positive sequence fundamental frequency components from the complex space vectors. .......... 95 Figure 4.22: The performance of various algorithms in extracting positive sequence fundamental frequency components from complex space vectors. ................ 97 Figure 4.23: Rotor slot harmonics in the current harmonic spectrum. ........................... 100 Figure 4.24: Rotor eccentricity harmonics in the current harmonic spectrum. .............. 101 Figure 4.25: Relationship between the estimated and the measured speeds................... 104 Figure 4.26: Result from the sensorless rotor speed detection algorithm for the test motor - 5 hp TEFC motor, Is=10.7 A (85% FLC). ....................................... 105 Figure 4.27: Result from the sensorless rotor speed detection algorithm for the test motor - 5 hp ODP motor, Is=13.0 A (100% FLC). ....................................... 106 Figure 4.28: Result from the sensorless rotor speed detection algorithm for the test motor - 7.5 hp TEFC motor, Is=19.7 A (101% FLC). .................................. 107 Figure 5.1: Results from the rotor resistance estimation and the rotor temperature calculation algorithm for the test motors. ..................................................... 113 Figure 5.2: Bode diagram for the frequency-response characteristics of the reduced order hybrid thermal model (τth=534 sec, R1=0.5 ºC/W)............................... 116 Figure 5.3: Two-stage approach to the online parameter tuning algorithm.................... 117 Figure 5.4: Kaiser window in the time- and frequency- domain. ................................... 120 Figure 5.5: Flowchart of the online parameter tuning algorithm.................................... 124 Figure 5.6: Stator winding temperature predicted by the the reduced order hybrid thermal model at Is=22.5 A, with the thermal parameters identified from heat run at Is=19.7 A. .................................................................................... 125
xiii
Figure 6.1: Overall experimental setup to validate the proposed stator winding temperature estimation algorithm. ................................................................ 129 Figure 6.2: The motor-load configuration. ..................................................................... 131 Figure 6.3: Baker D12R digital motor tester [70]........................................................... 132 Figure 6.4: DC machine equivalent circuit. .................................................................... 133 Figure 6.5: The Hall effect transducers [71]................................................................... 136 Figure 6.6: Schematic of the current and voltage transducers on PCB. ......................... 137 Figure 6.7: SCXI-1000 4-slot chassis. ............................................................................ 138 Figure 6.8: SCXI-1305 AC/DC coupling BNC terminal block [72]. ............................. 138 Figure 6.9: SCXI-1141 8-channel lowpass elliptical filter module [73]. ....................... 139 Figure 6.10: PCI-6036E data acquisition scheme........................................................... 141 Figure 6.11: The LabView data acquisition program for current and voltage measurements................................................................................................ 142 Figure 6.12: Measuring the rotor speed with a non-contact photo tachometer. ............. 144 Figure 6.13: The locations of the thermocouples............................................................ 146 Figure 6.14: Temperature measurements for slot and end windings (Is=150% FLC). ... 147 Figure 6.15: SR630 thermocouple monitor. ................................................................... 148 Figure 6.16: LabView program for data acquisition of temperature measurements. ..... 150 Figure 6.17: Experimental setup to create unbalanced voltage supply........................... 152 Figure 6.18: Experimental setup to create impaired cooling conditions. ....................... 154 Figure 6.19: Continuous operation periodic duty – duty type S6 [40]. .......................... 156 Figure 6.20: Experimental setup to create continuous-operation periodic duty cycles. . 158 Figure 7.1: Rotor temperatures estimated for motors with impaired cooling conditions and unbalanced supply.................................................................................. 162 Figure 7.2: Block diagram of the sensorless stator winding temperature estimator based on a proportional integral observer. .............................................................. 165
xiv
Figure 7.3: Performance of the sensorless adaptive stator winding temperature estimator (Tc=60 min, cyclic duration factor 50%). ..................................................... 170 Figure 7.4: Performance of the sensorless stator winding temperature estimator (Tc=30 min, cyclic duration factor 50%). ................................................................. 172 Figure 8.1: Stator winding temperature rise for the 7.5 hp TEFC motor........................ 184 Figure B.1: Relationship between resistivity and temperature. ...................................... 189 Figure C.1: The calculation of the singular value decomposition and the Moore-Penrose inverse. .......................................................................................................... 191
xv
SUMMARY
The organic materials used for stator winding insulation are subject to deterioration from thermal, electrical, and mechanical stresses. Stator winding insulation breakdown due to excessive thermal stress is one of the major causes of electric machine failures; therefore, prevention of such a failure is crucial for increasing machine reliability and minimizing financial loss due to motor failure. This work focuses on the development of an efficient and reliable stator winding temperature estimation scheme for small to medium size mains-fed induction machines. The motivation for the stator winding temperature estimation is to develop a sensorless temperature monitoring scheme and provide an accurate temperature estimate that is capable of responding to the changes in the motor’s cooling capability. A discussion on the two major types of temperature estimation techniques, thermal model-based and parameter-based temperature techniques, reveals that neither method can protect motors without sacrificing the estimation accuracy or motor performance. Based on the evaluation of the advantages and disadvantages of these two types of temperature estimation techniques, a new online stator winding temperature estimation scheme for small to medium size mains-fed induction machines is proposed in this work. The new stator winding temperature estimation scheme is based on a hybrid thermal model. By correlating the rotor temperature with the stator temperature, the hybrid thermal model unifies the thermal model-based and the parameter-based temperature estimation techniques. Experimental results validate the proposed scheme for stator winding temperature monitoring. The entire algorithm is fast, efficient and reliable, making it suitable for implementation in real time stator winding temperature monitoring.
xvi
1
CHAPTER 1
INTRODUCTION
Three-phase induction machines are used extensively in modern industry due to their cost effectiveness, ruggedness and low maintenance requirements. A single industrial facility may have thousands of induction motors operating along its assembly lines. As a result of this coordinated operation, malfunction of an induction motor may incur financial losses not only associated with the individual motor’s repair or replacement, but also losses associated with the down time of the entire assembly line and the loss of productivity. For this reason, reliable motor operation is crucial in many industrial processes. To ensure reliable motor operation, protection devices, such as thermal relays, are widely used in modern industry. Condition monitoring of induction machines, the underlying technology in motor protective devices, has experienced rapid growth in recent years. A major task of induction machine condition monitoring is to provide accurate and reliable overload protection for motors.
According to IEEE Industry Applications
Society (IAS) and Electric Power Research Institute (EPRI) surveys, 35–40% of motor failures are related to the stator winding insulation and iron core [1]-[3]. These failures are primarily caused by severe operating conditions, such as cyclic overload operation; or harsh operating environments, such as those in the mining or petrochemical industries [4]. Although induction motors are rugged and reliable, the stator winding insulation failure is potentially destructive. It often leads to stator winding burnout and even total motor failure. Protection of the stator winding from insulation failure is the main theme of this work.
1
1.1
Stator Winding Insulation Failure
The organic material used for insulation in stator windings of an induction motor must work below a certain temperature limit. Operating above this temperature limit for short durations does not seriously affect the life of the motor, but prolonged operations beyond the permissible temperature limit will produce accelerated and irreversible deterioration of the stator winding insulation material. Such deterioration often expedites the motor’s aging process and eventually reduces the motor’s life. As a rule of thumb, the motor’s life is reduced by 50% for every 10°C increase above the stator winding temperature limit. Since excessive thermal stress is identified from industry practice as the primary cause of stator winding insulation degradation, especially for small-size mains-fed induction machines, the National Electrical Manufacturers Association (NEMA) has established permissible temperature limits for the stator windings of an induction machine based on its insulation class to ensure its continuous and reliable operation. Typical temperature limits for the stator windings are given in Table 1.1 [5].
Table 1.1: Temperature limits for different insulation classes Insulation
Ambient
Rated Temperature
Hot Spot
Hot Spot
Class
Temperature (ºC)
Rise (ºC)
A
40
60
5
105
B
40
80
10
130
F
40
105
10
155
H
40
125
15
180
Allowance (ºC) Temperature (ºC)
There are several conditions under which the temperature limit can be exceeded, resulting in acceleration of stator winding insulation degradation: transient overloads, running overloads and abnormal cooling conditions [4]-[6].
2
Transient overloads and running overloads are related to two regions of motor operation [6]. The first region of motor operation is the transient overloads with 250 to 1000% full-load current. These overloads include motor starting, wherein the motor draws up to 6 times its rated current during acceleration; motor stall, wherein the motor fails to accelerate the load to the desired speed during its starting phase; and motor jam, wherein the motor is stopped during its normal operation due to a sudden mechanical lock. In each of these scenarios, a significant amount of heat is generated by the large amount of inrush current in the stator winding due to the stator I2R loss. The lack of ventilation, caused by the slow or even complete halt of rotor movement, makes it difficult for the heat to be dissipated [7]. Therefore, the transient overloads can be regarded as adiabatic processes with very fast thermal transients. Normally it takes between 25 to 30 seconds for a typical motor stator winding to reach 140°C rise above its ambient temperature during a locked rotor condition [6]. In most applications, general and special-purpose NEMA T-frame motors may be considered to be protected at transient overloads when NEMA Class 20 overload relays are used. These relays allow 6 times full-load current to pass through the motor for 20 seconds [8]. The second region of motor operation is the running overload with 1 to 2 times the full load current. In this region the motor is continuously running, thereby providing a certain degree of heat dissipation for the internal losses, and resulting in a gradual increase of the stator winding temperature. Unlike the motor operation in transient overloads, the internal heat is transferred to the motor ambient by means of conduction and convection during running overloads. Therefore, the thermal time constant under this type of motor operation is far larger than that under the transient overloads. This thermal time constant is determined by a number of factors, such as the motor design, the rotor speed and the temperature of the surrounding air. As a result, while a definite time relay can be used to protect the motor 3
from transient overloads, a more sophisticated scheme is needed to protect the motor from running overloads. This defines the scope of the research presented in this work. Abnormal cooling conditions are another possible cause of stator winding temperature rising beyond its limit. Typically the cooling ability of a motor is reduced due to a defect or fault in any of the components in the motor’s cooling system. This often leads to an abnormal motor temperature rise. For instance, when the fins or casing of the motor is clogged with dust or other particles, transfer of motor internal heat to its ambient is obstructed, and as the result the motor temperature increases.
Another
example is when the cooling of the motor is compromised due to high ambient temperature. Standard motors are designed to operate at an ambient temperature below 40°C, therefore the insulation life decreases significantly as the motor ambient temperature increases. There are even more serious situations in motor cooling, caused either by a broken cooling fan or accidentally blocked air vents or ducts. All of them decrease the motor’s cooling ability and lead to possible motor failure.
(a) locked rotor (b) running overload Figure 1.1: Stator winding damage [Courtesy of Electrical Apparatus Service Association (EASA) Inc., St. Louis, USA].
4
Two examples of the damage in the stator winding due to excessive thermal stress are shown in Figure 1.1. Figure 1.1(a) shows the damage in the stator and rotor caused by a locked rotor condition, which is one type of transient overloads. Figure 1.1(b) shows the stator insulation damage due to excessive motor running overloads. 1.2
Temperature Monitoring
To safeguard the stator winding from insulation failure and extend the motor life, the stator winding temperature must be continuously monitored.
Whenever the stator
winding temperature exceeds the permissible limit, the motor should be shut down to avoid damage to its stator winding insulation materials. Many techniques have been developed for induction motor protection under overload conditions to guarantee reliable motor operation. These techniques can be classified into 3 major categories: 1)
Direct temperature measurement
2)
Thermal model-based temperature estimation
3)
Parameter-based temperature estimation
Direct temperature measurement of the stator winding temperature is performed using embedded
thermocouples,
thermally
sensitive
resistors
(thermistors),
resistive
temperature detectors (RTDs) or infrared cameras [9]. Such thermal sensors are capable of providing reliable temperature readings at their installed locations. However, since most thermal stresses lead to localized failures inside the stator winding, where these thermal sensors are not installed, the direct temperature measurement may not provide complete overload protection for the whole stator winding.
In addition, direct
temperature measurement is only considered a cost-effective method for large machines. The installation of thermal sensors in small machines is extremely difficult and costly. Thermal model-based temperature estimation is the most commonly used technique in motor overload protection. Dual-element time-delay fuses, eutectic alloy overload 5
relays and microprocessor-based motor protective relays are 3 major types of protective devices based on the thermal models of induction machines. The dual-element time-delay fuse, which is the most extensively used device for motor overload protection due to its low cost, consists of a short-circuit element and an overload element [8].
A properly sized dual-element time-delay fuse can provide
protection for both short-circuit and running overload conditions. However, each time the motor is overloaded, the fuse needs to be replaced. The eutectic alloy overload relays are another type of motor protective relays based on the emulation of the thermal characteristics of the stator winding. When coordinated with the proper short-circuit protection, this type of overload relays is intended to protect the motor against overheating due to excessive over currents. Nevertheless, the thermal discrepancy between the eutectic alloy overload relays and the motors makes it difficult to match both heating and cooling characteristics of the motor under all thermal conditions.
As a result, the device often trips the motor based on an approximate
estimate of the stator winding temperature, and spurious trips are common with these devices [10]. Among all devices using thermal model-based temperature estimation techniques, the microprocessor-based motor protective relays represent the state-of-the-art in motor protection [6]. To provide an estimate of the motor’s stator winding temperature, the microprocessor-based overload relay first calculates the power losses from the current measurements at motor terminals based on the induction motor equivalent circuit. The relay then derives the stator winding temperature from a thermal model for the induction motor. Thermal model-based temperature estimation provides an accurate and reliable temperature estimate when compared to fuses or eutectic alloy overload relays, thus ensuring complete motor overload protection. In addition, it can be adjusted easily for different classes of motors due to its flexible software-based algorithm. However, similar 6
to fuses and eutectic alloy overload relays, it cannot respond to changes in the cooling capability of a motor, which are often caused by either a clogged motor casing or a broken ventilation fan. Parameter-based temperature estimation technique presents an alternative method in estimating the stator winding temperature.
Since resistance is a direct indicator of
temperature, this type of method provides superior performance over the thermal modelbased temperature estimation. Besides the high accuracy associated with the estimated stator winding temperature in this method, it is capable of responding to the changes in the motor cooling condition because the temperature variation is reflected immediately on the stator resistance estimate. Compared with the direct temperature measurements from either thermocouples or RTDs, this method requires no temperature detectors, and is therefore non-intrusive in nature and inexpensive. Reference [11] presents a detailed method of calculating the stator resistance, Rs, and the rotor resistance, Rr, from the induction machine equivalent circuit. However, as indicated in [12], a direct estimate of stator resistance at high speed operation is extremely difficult and susceptible to parametric errors from rotor resistance and motor inductances. To avoid the large error in the estimated stator resistance, one method assumes a fixed ratio between Rs and Rr [13]. Since Rr is strongly dependent on the rotor frequency due to skin effect, while Rs is uncorrelated to rotor frequency, the stator resistance estimate obtained in this manner is not the ‘true’ stator resistance, and consequently it is not a direct indicator of stator temperature. Other researchers propose dc injection method for line-connected and soft-started induction machines for parameter-based temperature estimation [14]-[16]. However, the major problem with using dc injection for Rs estimation is the torque pulsation and the negative torque induced by the dc current component [15].
7
In addition to the aforementioned variety of devices used for overload protection, bimetallic thermal protectors are also a popular type of temperature monitoring device. They are typically used on fractional to small integral-horsepower (up to 5 hp) ac induction motors to provide built-in overheating protection. Detecting abnormal cooling conditions during motor operation is also one important aspect of induction machine temperature monitoring. In case of a cooling system fault, the motor may operate at a higher temperature under the same load or thermal condition compared to when the cooling system is healthy. This results in accelerated stator winding insulation degradation. In references [17]-[18], methods for detecting abnormal cooling situations are proposed. By comparing the difference in temperature estimated from the thermal model and the temperature estimated from the resistance, the motor cooling system is monitored. If the difference is beyond a predetermined threshold value, a fault signal is generated to indicate a malfunction in the motor’s cooling system. The implementation of this scheme requires complete knowledge of the motor electrical and thermal models. Sophisticated signal processing techniques are necessary to unify these two models and produce a reliable estimate. 1.3
Problem Statement
It was shown in the previous sections that temperature monitoring of the stator winding is crucial to protecting not only an individual motor but also the whole industrial process driven by motors. This work focuses on the development and implementation of a fast, efficient and reliable algorithm to estimate the stator winding temperature online with only voltage and current measurements from the terminals of small to medium size mains-fed induction machines. In addition, motor cooling system condition monitoring is also explored for complete stator winding protection. The ultimate goal of this work is to
8
provide a comprehensive set of algorithms for motor overload protection to the next generation microprocessor-based protective relays. The development of a thermal monitoring tool begins with a thorough investigation of state-of-the-art techniques for stator temperature estimation. The thermal model-based temperature estimation technique, though simple and reliable, suffers from inaccuracies in the thermal model parameters. These inaccuracies often lead to conservative estimates of stator winding temperature, resulting in spurious trips and unnecessary interruption of the whole manufacturing process. On the other hand, the parameter-based temperature estimation technique, though accurate, is highly susceptible to the errors in the induction machine electrical parameters. Theoretically, estimation of Rs using the negative or zero sequence model is insensitive to motor parameter errors; however, continuous monitoring of Rs in practice is virtually impossible since small negative sequence or zero sequence current often causes singularity problems in signal processing. If the negative or zero sequence currents are intentionally injected into the machine to obtain an estimate of Rs, the inherent motor asymmetry in different phases may also cause large errors in the Rs estimate. Other problems associated with the current injection method include the deterioration of motor performance due to torque pulsations and motor internal heating. For example, the dc injection technique proposed in references [15]-[16] usually introduces undesired torque pulsations and motor performance deterioration. Based on the analysis of the pros and cons of both the thermal model-based temperature estimation techniques and the parameter-based techniques, a new method is suggested in this proposal. First, a hybrid thermal model (HTM) is proposed to correlate the stator temperature with the rotor temperature. This model also accounts for the disparities in thermal operating conditions for different motors of the same rating. Then the rotor temperature, obtained from the rotor resistance estimation, is regarded as an indicator of the motor’s thermal characteristics. The rotor temperature is used to tune the 9
parameters in the HTM to reflect the specific motor’s cooling capability. Finally the HTM is run independently after the tuning process to provide an accurate and reliable estimate of the stator winding temperature. An abnormal cooling condition in motor operation, such as a clogged motor casing or a broken ventilation fan is also considered in this work. The entire algorithm is fast, efficient and reliable, making it suitable for implementation in real time for protection purposes. 1.4
Dissertation Outline
A brief overview of the results of previous research related to stator winding temperature estimation is given in Chapter 2. Chapter 3 analyzes the induction machine thermal behavior via networks consisting of thermal resistors and thermal capacitors. Hybrid thermal models are also proposed in this chapter based on the analysis of design and thermal behavior of small to medium size mains-fed induction machines. As the first step in implementing the stator winding temperature estimation scheme via the hybrid thermal model, Chapter 4 gives detailed procedures to obtain the induction machine rotor resistance from only current and voltage measurements. A detailed analysis of the algorithm requirement for the motor operating points is also covered in this chapter. Chapter 5 derives the rotor temperature from the estimated rotor resistance and then gives the general rules to tune the thermal parameters in the hybrid thermal model, so that the true motor cooling capability is reflected by the tuned thermal model. To validate the proposed stator winding temperature estimation scheme, Chapter 6 shows the detailed experimental setup, including the hardware platform and the software used for data acquisition. The experimental results for the induction machine online thermal condition monitoring are given in Chapter 7. Chapter 8 summarizes this work with conclusions and contributions.
Recommendations for future work on the algorithms for the online
adaptive stator winding temperature estimator are also described to provide a more 10
accurate and reliable estimation of the stator winding temperature for small to medium size mains-fed induction machines.
11
2
CHAPTER 2
SUMMARY OF PREVIOUS WORK ON STATOR TEMPERATURE ESTIMATION
The temperature rise inside an induction machine is caused by the accumulation of heat on both the stator and the rotor. The heat is produced from the motor losses. The motor losses are made up of following losses [19]: •
Losses dependent on the motor current o Stator I2R loss o Rotor I2R loss o Stray-load loss
•
Losses independent of the motor current o Core loss due to eddy current and hysteresis o Friction and windage loss
During the conversion from electrical energy to mechanical energy by the induction machine, these losses are generated inside the machine and are dissipated in the form of heat by means of conduction and convection. For most modern small-size mains-fed induction machines, the major portion of the heat comes from the I2R losses. A complete overload protection scheme needs to provide protection to the induction machine at all times [20]. However, the thermal characteristics of an induction machine during its starting phase are vastly different from that during the running phase. Therefore, a good stator winding temperature estimator should be able to distinguish between these two different motor operating modes and adjust the temperature estimator accordingly. There are currently two major types of stator winding temperature estimation techniques available: the thermal model-based temperature estimation technique and the 12
parameter-based temperature estimation technique. Their basic concepts are summarized and evaluated in this chapter. In addition, a comparison is made between these two types of techniques. 2.1
Temperature Estimation Based on Thermal Models
Thermal limit curves are typically used to provide knowledge of the safe operating time for an induction machine under locked rotor conditions, acceleration condition and running overload conditions [21].
The thermal model-based temperature estimation
techniques normally emulate the thermal limit curves to achieve complete motor overload protection. Figure 2.1 shows the typical thermal limit curves for an induction machine. To insure proper motor operation, the protective devices must trip the motor once it goes beyond its normal starting or running condition and reaches its thermal limits.
Figure 2.1: Typical thermal limit curves from reference [21].
13
Both the dual-element time-delay fuses and the microprocessor-based overload relays simulate the motor internal heating based on the given thermal limit curves. 2.1.1
Dual-element Time-delay Fuses
Dual-element time-delay fuses consist of a short-circuit element and an overload element, as shown in Figure 2.2, providing complete protection to the motor at both the starting phase and the running phase.
Figure 2.2: The structure of a dual-element time-delay fuse from reference [8]. As shown in Figure 2.3(a), when a short circuit occurs in the induction machine, the inrush current cause the restricted portion in the short-circuit element to melt. After the arc is suppressed by the arc quenching material and the increased arc resistance, a gap is produced inside the short-circuit element of the fuse, indicated by Figure 2.3(b). The power supply is therefore cut off from the motor. Under sustained overload condition, the trigger spring fractures the calibrated fusing alloy and releases the connector in Figure 2.3(c). The release of the connector produces a break inside the overload element of the fuse, as illustrated in Figure 2.3(d). Figure 2.4 shows the time-current diagram for a properly sized dual-element timedelay fuse in protecting a motor. The thermal limit curve emulated by the fuse lies on the right side of the normal motor starting and running condition, therefore, the motor is protected at both the starting phase and the running phase.
14
(a) During short-circuit condition
(b) After short-circuit condition
(c) During overload condition
(d) After overload condition
Figure 2.3: Dual-element fuse operating mechanisms under different conditions [8]. The dual-element time-delay fuse is not only suitable for standalone motor overload protection, but also an economical means in providing backup protection to an overload relay. It is inexpensive and virtually free from maintenance. However, this type of device trips the motor based only on a crude estimate of the stator winding temperature and is subject to spurious trips as well as under-protection [6], [10].
Figure 2.4: Motor starting and running curves and the dual-element time-delay fuse thermal limit curve [8].
15
2.1.2
Thermal Models with a Single Time Constant
Most microprocessor-based motor overload protective relays, representing the stateof-the-art in motor protection, rely on the motor heat transfer models to predict the stator winding temperature. Thermal models with a single thermal capacitor and a single thermal resistor are widely adopted in the industry. The thermal capacitance and thermal resistance are normally predetermined by a set of parameters for a given class of motors, classified by their full load current (FLC), service factor (SF) and trip class (TC) according to reference [22]. Thermal models with a single thermal capacitor and a single thermal resistor are derived from the heat transfer of a uniform object, as shown in Figure 2.5,
Figure 2.5: Thermal model with a single thermal time constant. The quantities, θ and θA, in ºC, are temperatures of the uniform object and its ambient, respectively. The power input into this uniform object is determined by the power losses from the current, I (unit: A), on the resistor, R (unit: Ω). Heat is dissipated through the boundary of the uniform object (the shaded region in Figure 2.5) to the ambient. The thermal resistance, Rth, in ºC/W, models this heat transfer. The thermal capacitance, Cth, in J/ºC, is defined to be the energy needed to elevate temperature by one degree Celsius for the object. It represents the total thermal capacity of the object. 16
The difference between the input power and the output power is used to elevate the temperature of the uniform object, Pin − Pout = Cth
d (θ − θ A ) dt
(2.1)
The input power is the heat, I2R, generated by the current on the resistor. The output power is the heat transfer,
θ −θ A Rth
, across the boundary of the object to its ambient.
Therefore, Equation (2.1) is rewritten as, I 2R −
θ −θ A Rth
= Cth
d (θ − θ A ) dt
(2.2)
By solving Equation (2.2) as a first order differential equation, a closed form solution is obtained, t − ⎛ τ th θ (t ) = I R ⋅ Rth ⎜1 − e ⎜ ⎝ 2
⎞ ⎟ +θ A ⎟ ⎠
(2.3)
where τth=RthCth is the thermal time constant of the uniform object. If the constant current, I, flows in this uniform object for a sufficiently long time, i.e.: t → ∞ , the final temperature of this uniform object is θ (∞ ) = I 2 R ⋅ Rth + θ A . For a specific motor, it is designed to work under some maximum permissible temperature, θmax, determined by its stator winding insulation material [5].
This
maximum permissible temperature determines the maximum permissible current through the stator winding,
I max =
θ max − θ A
(2.4)
R ⋅ Rth
For a motor, if its stator current exceeds a predetermined value for certain time, the stator winding temperature will rise above its maximum permissible value.
The
microprocessor-based overload relay monitors the stator current and calculates the time to trip the motor and ensures proper motor operation below its stator winding maximum 17
permissible temperature.
Figure 2.6 shows the one possible implementation of the
temperature estimation scheme by an equivalent electrical circuit, which consists of an RC circuit and an Op-Amp.
Figure 2.6: Equivalent electrical circuit of the temperature estimator with a single thermal time constant. While Equations (2.3) and (2.4) are sufficient in predicting the stator winding temperature, it is usually difficult to obtain the thermal resistance and the thermal capacitance for a particular motor. Therefore, the full load current (FLC), service factor (SF) and trip class (TC) are used instead to calculate the time to trip. From Equation (2.3), for a motor with its stator winding initially at the ambient temperature, under given constant current, I, and with a known stator winding maximum permissible temperature θmax, the time needed to trip this motor is, ⎡ ⎤ I 2 R ⋅ Rth t = τ th ln ⎢ 2 ⎥ ⎣ I R ⋅ Rth − (θ max − θ A ) ⎦
(2.5)
2 Substituting (θ max − θ A ) in Equation (2.5) with I max R ⋅ Rth according to Equation (2.4),
gives,
18
⎛ ⎞ ⎛ I2 ⎞ I 2 R ⋅ Rth τ ln t = τ th ln ⎜ 2 = ⎟ th ⎜ 2 2 ⎟ 2 ⎝ I R ⋅ Rth − I max R ⋅ Rth ⎠ ⎝ I − I max ⎠
(2.6)
By defining service factor to be, SF =
I max I rated
(2.7)
Equation (2.6) is further simplified, ⎛ ⎞ I2 t = τ th ln ⎜ 2 pu 2 ⎟ ⎜ I − SF ⎟ ⎝ pu ⎠
where I pu =
I I rated
(2.8)
.
Trip class, often denoted as t6X, is defined to be the maximum time [seconds] for an overload trip to occur when a cold motor’s operating current is six times its rated current. From Equation (2.8),
⎛ 62 ⎞ t6 X = Tth ln ⎜ 2 2 ⎟ ⎝ 6 − SF ⎠
(2.9)
Therefore, the relationship between t6X and τth can be established once the service factor of that specific motor is known. Table 2.1 shows the relationship between the thermal time constant and the trip class at different service factors according to Equation (2.9).
Table 2.1: Relationship between τth and t6X at different service factors Service Factor Thermal Time Constant 1.00
35.5 t6X
1.05
32.0 t6X
1.10
29.2 t6X
1.15
26.7 t6X
19
As a brief conclusion, given the full load current, the service factor and the trip class of a motor, the time to trip can be calculated from Table 2.1 and Equation (2.8). For example, for the test motor given in Table A.1, the service factor is 1.15, and its trip class is 20. Therefore, for an overload with I = 1.5Irated, the time to trip is,
⎛ ⎞ 1.52 t = 26.7 ⋅ t6 X ln ⎜ 2 = 26.7 × 20 × 0.8862 = 473.23 (sec) 2 ⎟ ⎝ 1.5 − 1.15 ⎠
(2.10)
2.1.3 Complex Thermal Networks The thermal model of a single thermal time constant is derived from the thermal behavior of a uniform object. However, the motor is not thermally homogeneous, the temperature rise at various parts of the motor, such as the stator, the rotor, or the iron core, is different. Even areas in the same part, such as the stator slot winding and the end winding, have different thermal characteristics. Therefore, complex thermal networks have been proposed as one type of thermal model-based temperature estimation techniques [6], [23]-[26]. Figure 2.7 illustrates one type of the complex thermal networks proposed in reference [23].
Figure 2.7(a) shows the structure of the totally enclosed fan-cooled (TEFC)
induction machine along with the specific locations where the temperature is estimated. Figure 2.7(b) shows how each of the ten components are linked to form a network of an induction machine thermal model and the actual heat flow between them.
The
temperature of the stator components, such as stator winding, stator end winding, stator core and stator teeth can be estimated using this model. However, the thermal resistors and capacitors need to be evaluated in advance from the physical dimensions and construction materials of the motor.
20
(a) Detailed construction of motor #1
(b) Heat flow inside motor #1
1. Motor frame 3. Stator teeth 5. Air gap 7. End cap air 9. Rotor back iron
2. Stator back iron 4. Stator slot winding 6. Stator end winding 8. Rotor winding 10. Motor shaft
Figure 2.7: Thermal model by a complex thermal network for induction machine #1. To avoid calculation of the thermal resistances and capacitances from the physical dimensions and construction materials of an induction machine, some researchers propose a complex thermal network based on parameter estimation, as shown in Figure 2.8 [26].
First, the embedded thermal sensors measure the temperatures at various
locations inside the induction machine, as indicated in Figure 2.8(a).
The thermal
resistances and capacitances are then identified online by applying a recursive least square method on the thermal network illustrated in Figure 2.8(b). Once the thermal resistances and capacitances are identified, the thermal network is capable of predicting the temperatures at various locations inside the motor.
21
(a) Detailed construction of motor #2
(b) Thermal network for motor #2
1. Rotor cage center 3. Stator end winding center 5. Frame and end brackets
2. Stator embedded winding center 4,6,7. Stator core
Figure 2.8: Thermal model by a complex thermal network for induction machine #2. Although the parameter estimation technique used here eliminates the need to calculate the thermal resistances and capacitances from physical dimensions and construction materials of an induction machine, it requires high-precision temperature measurements from the embedded thermal sensors. This is often impractical for smallsize mains-fed induction machines due to economic reasons. 2.2
Temperature Estimation Based on Motor Parameters
The thermal model-based temperature estimation techniques give an estimate of the stator winding temperature based on the thermal model, and the knowledge of the thermal parameters in the model is very crucial in giving an accurate temperature estimate. In practice, the thermal capacitance and resistance are normally predetermined by a set of parameters for a class of motors of the same ratings, such as full load current, service factor and trip class. Consequently, the thermal model is incapable of giving an accurate stator winding temperature estimate tailored to a specific motor’s thermal capacity. The parameter-based temperature estimation techniques, on the other hand, derive average 22
stator and rotor temperatures from the stator and rotor resistances, respectively. Compared with the temperatures estimated from the thermal models, the temperatures estimated from the stator resistance, Rs, and the rotor resistance, Rr, are more direct measures of the stator temperature, θs, and the rotor temperature, θr, respectively. According to reference [19], the temperature and the resistance have the following relationship, R2 = R1 ⋅
θ2 + k θ1 + k
(2.11)
where θ1 represents the reference temperature [°C]; R1 and R2 are the resistances [Ω] at temperature θ1 and θ2, respectively; k is the inferred temperature coefficient for zero resistance and varies for different materials: for 100% International Annealed Copper Standard (IACS) conductivity copper
1
, it is 234.5, for aluminum with a volume
conductivity of 62%, it is 225. In parameter-based temperature estimation techniques, once the resistance is estimated, the temperature is calculated from Equation (2.11), which is rewritten as,
θ2 =
R2 ⋅ (θ1 + k ) − k R1
(2.12)
As indicated by Equation (2.12), the key issue in parameter-based temperature estimation is an accurate estimate of the resistance. Since the purpose of the temperature monitoring is to protect motors from overheating in its stator winding, direct estimation of the stator resistance now becomes the focus of this review of previous research. There are two major approaches to estimating the stator resistance: 1) Resistance estimation based on the induction machine model 2) Resistance estimation using dc injection.
1
IACS: a measure of conductivity used to compare electrical conductors to a traditional copper-wire standard. Conductivity is expressed as a percentage of the standard. 100% IACS represents a conductivity of 58 mega-siemens per meter (MS/m); this is equivalent to a resistivity of 1/58 ohm per meter for a wire one square millimeter in cross section.
23
These two approaches are summarized in this section. 2.2.1
Induction Machine Model-based Resistance Estimation
Estimation schemes for Rs have been proposed mainly for improving the performance of field oriented drives at low speed [27]-[30], or for obtaining a better estimate of shaft speed for speed sensorless control at low speed [31]-[34]. This section reviews the basic principles used in Rs estimation techniques. Using a synchronously rotating reference frame (ω=ωe) and aligning the current vector with the d-axis ( iqse = piqse = 0 ) in the q-d coordinates, the operation of a e e = pλqdr = 0 ) is described by symmetrical induction machine under steady state ( pλqds
[35],
vqse = ωe λdse , vdse = Rs idse − ωe λqse
(2.13)
0 = Rr iqre + sωe λdre , 0 = Rr idre − sωe λqre
(2.14)
λqse = Lmiqre , λdse = Ls idse + Lmidre
(2.15)
λqre = Lr iqre , λdre = Lr idre + Lmidse
(2.16)
where Rs, Rr are the stator and rotor resistance respectively; Ls, Lr, Lm are the stator, rotor and mutual inductance respectively; ωe is the angular speed [rad/s] of the synchronous reference frame; s is the slip; and p is the differential operator,
d ; sωe is the slip dt
frequency defined as the difference between the synchronous speed, ωe, and the rotor speed, ωr; λqds and λqdr are the stator and rotor flux linkage, respectively. e e and λqds , Equations (2.13)-(2.16) are further reduced to, By eliminating iqdr
⎛ ⎞ L vqses = ω e ⎜ σ Ls idses + m λdres ⎟ Lr ⎝ ⎠
24
(2.17)
vdses = Rs idses − ω e
(2.18)
Rr es λqr + sω e λdres Lr
(2.19)
Rr es λdr − Lmidses ) − sω e λqres ( Lr
(2.20)
0= 0=
Lm es λqr Lr
Equations (2.17), (2.19) and (2.20) can be used to calculate Rr. By eliminating λqre , expressions for estimates of λdre and Rr, which are independent of Rs, are, Lr e σ Ls Lr e v + ids ω e Lm qs Lm
(2.21)
sω e ⋅ Lr Rˆ r = Lmidse −1 λˆ e
(2.22)
λˆdre =
dr
Similarly, Rs is calculated from (2.18), (2.20)-(2.22) as,
v e sω 2 L λˆ e Rˆ s = eds − e me dr ids Rˆr ids
(2.23)
The overall structure of the Rr and Rs estimator is shown in Figure 2.9.
Figure 2.9: Overall structure of the induction machine model-based Rr and Rs estimator.
25
2.2.2
Resistance Estimation Using dc Injection
The dc injection circuit proposed in references [15]-[16] and [36]-[37] consists of a power MOSFET and an external resistor connected in parallel. Figure 2.10 shows the dc equivalent circuit of the motor, source, and dc injection circuit from the source to the motor terminals in one phase,. The dc injection circuit operates in two modes: dc injection mode (DIM) and normal mode (NM: no injection of dc), for intermittent injection of a dc bias into the motor.
+ Vsw,dc _
a'
a
Ias,dc
Rs
n
s
Rs
Rs c'
b'
b
c
Figure 2.10: DC equivalent circuit of dc injection circuit and the motor. During the dc injection mode, the FET is turned off when ias > 0, and turned on when ias < 0; the equivalent circuits for each case is shown in Figure 2.11(a) and (b), respectively, and the v-i characteristics under DIM is shown in Figure 2.11(c). The asymmetrical resistance causes the voltage drop across the circuit to be asymmetrical (dc component in vsw), resulting in the injection of a dc current component into the motor. Under DIM, Rs is updated using,
26
v 2 ⋅ vab ,dc 2 ⋅ vsw, dc Rˆ s = as ,dc = =− ias ,dc 3 ⋅ ias , dc 3 ⋅ ias ,dc
(2.24)
The dc injection scheme causes torque pulsation inside the motor and power dissipation in both the stator winding and Rext. To adjust the torque distortion and power dissipation to be within an acceptable level, the value of Rext is adjusted depending on the nominal Rs and the rated ias. In addition, the variation of the stator winding temperature is slow, therefore it is not necessary to inject a constant dc bias into the motor and estimate Rs continuously. As a result, the dc injection circuit can be operated under NM in between DIMs, by setting vgs to Vgs,on (FET on). The equivalent circuit and v-i characteristics of the dc injection circuit under NM are shown in Figure 2.12(a) and (b), respectively.
(a) operation at positive haversine in DIM
iasx2, then, Ls = x1
(4.33)
σ Ls = x2
(4.34)
Figure 4.9 illustrates the flowchart of the online inductance estimation algorithm. By taking voltage and current measurements as well as the stator winding resistance as its input, the online inductance estimation algorithm produces Ls and σLs as its outputs. Since Equations (4.19)-(4.34) are derived from the steady-state positive sequence motor equivalent circuit (Figure 4.7), the positive sequence voltage and current components should be used in the inductance estimation algorithm, as indicated by the first block in
67
Figure 4.9. A detailed discussion on how to extract the positive sequence voltage and current components from online measurements will be discussed in Section 4.5 [49].
Figure 4.9: Flowchart of the online inductance estimation algorithm.
4.3.1.2 Experimental Validation To validate the proposed scheme, experiments have been performed on three induction machines: one 5 hp motor with TEFC enclosure, one 5 hp motor with open drip proof (ODP) enclosure and one 7.5 hp motor with TEFC enclosure.
The motor
parameters obtained from the standard no load and locked rotor tests according to [19] are shown in Tables A.1-A.3 in Appendix A. All experiments are performed at an ambient temperature of 25 °C, and the experimental data are captured by a National Instrument data acquisition system with PC interfaces. The 5 hp TEFC motor is supplied with rated voltage and runs at different load levels: one experimental set at no load with Is= 4.64 A, which is approximately 37% of the full load current (FLC); one set at light load with Is=7.71 A (62% FLC); one set at heavy load with Is=9.24 A (74% FLC); and one set at almost full load with Is=11.71 A (94% FLC). At a sampling frequency of 5 kHz, a “snapshot” with 5000 samples is taken at 10 seconds after the rated voltage is applied to the motor terminals; another two snapshots
68
with the same length of data are taken at 15 seconds and 30 seconds respectively after the motor has started. This procedure is repeated for each load level. The proposed inductance estimation algorithm is applied to the experimental data, assuming only Rs is known. The results are shown in Table 4.1.
Table 4.1: Online inductance estimation results for the 5 hp TEFC motor Parameters
Standard test
Ls (mH)
Test run t=10 sec
t=15 sec
t=30 sec
74.3
78.1
78.2
78.3
σLs (mH)
7.24
7.02
7.01
7.01
ε (%)
—
5.15
5.16
5.18
The relative error, ε, in Table 4.1, is defined as,
ε = max
Lˆ − L* × 100% L*
(4.35)
where L* is the inductance value obtained from the standard no load and locked rotor tests, as shown in Appendix A; Lˆ is the inductance value estimated from the proposed inductance estimation algorithm.
Table 4.2: Online inductance estimation results for the 5 hp ODP motor Parameters
Standard test
Ls (mH)
Test run t=10 sec
t=15 sec
t=30 sec
61.6
66.5
66.6
66.5
σLs (mH)
6.26
5.70
5.67
5.66
ε (%)
—
8.99
9.43
9.58
Similar experiments are performed on the 5 hp ODP motor. The motor is again supplied with rated voltages and runs at different load levels: no load with Is=5.46 A 69
(42% FLC); a light load with Is=7.20 A (55% FLC); a heavy load with Is=9.60 A (74% FLC); and full load with Is=13.09 A (100% FLC). Following the same procedure as that
for the 5 hp TEFC motor, the final estimated inductance values are shown in Table 4.2. The 7.5 hp TEFC motor is supplied with rated voltage and is tested at 2 different load levels: one test at Is=14.19 A (72% FLC); and the other test at Is=16.46 A (84% FLC). The results are shown in Table 4.3.
Table 4.3: Online inductance estimation results for the 7.5 hp TEFC motor Parameters
Standard test
Ls (mH)
Test run t=10 sec
t=15 sec
t=30 sec
43.8
40.2
40.3
40.2
σLs (mH)
2.62
2.69
2.73
2.67
ε (%)
—
8.18
7.96
8.18
The only a priori information needed in the inductance estimation algorithm is Rs at 25 °C. The large amount of inrush current during the motor start causes a rapid increase in its stator winding temperature.
However, this phenomenon is ignored in the
inductance estimation algorithm. The same Rs is used in all three cases (t=10, 15 and 30 seconds) to calculate the inductance values for each machine. Consequently, there are small variations in the accuracies of the estimated inductance values over time. To minimize the influence of the stator winding temperature drift, the proposed inductance estimation algorithm should be applied as soon as possible when the motor enters its steady-state operation after being energized. 4.3.2 Criterion for Good Estimates of Inductances
The inductance estimation algorithm, described in the previous section, requires at least two different load levels to achieve an estimate of a motor’s inductance values.
70
Suppose that two experiments are performed and voltage and current measurements are acquired, the inductance estimation algorithm, described in the matrix form, is, ⎡ω I V U = ⎢ e s ,1 sy ,1 ⎢⎣ω e I s ,2Vsy ,2
−ω e2 I s2,1 ⎤ ⎥ −ω e2 I s2,2 ⎥⎦
⎡ Vs2,1 + I s2,1 Rs2 − 2 Rs I s ,1Vsx ,1 ⎤ y=⎢ 2 ⎥ 2 2 ⎣Vs ,2 + I s ,2 Rs − 2 Rs I s ,2Vsx ,2 ⎦
(4.36)
(4.37)
where ωe is the synchronous speed, for a mains-fed induction machine with 60 Hz power supply, it is 377 rad/s; Vs,1, Vsx,1, Vsy,1 and Is,1 are phasor quantities obtained from the voltage and current measurements during the first experiment; Vs,2, Vsx,2, Vsy,2 and Is,2 are phasor quantities obtained during the second experiment. If the load level of the first experiment is different from that of the second experiment, Vs,1, Vsx,1, Vsy,1 and Is,1 are different from Vs,2, Vsx,2, Vsy,2 and Is,2. Consequently, ωeIs,1Vsy,1
is normally different from ωeIs,2Vsy,2, so is ωe2 I s2,1 from ωe2 I s2,2 . Therefore, the first row of U in Equation (4.36) is linearly independent of the second row, as a result, U is full rank,
and Equation (4.31) has a unique solution. The condition number is often used in this case to indicate the linear independence among different rows in U, and consequently to quantify the difference in load levels. For a U with “reasonably different” load levels, the condition number should be small. However, the computation of the condition number for U involves complicated singular value decomposition, and this may not be desirable for real-time implementation in a low-cost hardware platform. In addition, the condition number is a purely mathematical index and does not carry sufficient information associated with the operation of the induction machine. Therefore, a simpler and more direct criterion is desired. This criterion should not only carries with it useful information on the motor load levels, but also indicates whether U is ill-conditioned or not.
71
For a mains-fed machine with fixed supply frequency, the ratio between Vsy and Is, designated by K, can be used as a criterion to determine whether different rows in U are linearly independent from one another, and subsequently determine the goodness of the estimate. The detailed derivation is as follows. Assume that the rows of U in Equation (4.36) are linearly dependent on each other. Therefore, ⎡ −ω e2 I s2,1 ⎤ ⎡ ω e I s ,1Vsy ,1 ⎤ + =0 α1 ⎢ α ⎥ 2 ⎢ 2 2 ⎥ I V ω I − ω ,2 ,2 e s sy ,2 e s ⎣ ⎦ ⎣ ⎦
(4.38)
with α1≠0, and α2≠0. Simplification of Equation (4.38) yields, ⎡ Vsy ,1 ⎤ ⎢ ⎥ ⎢ I s ,1 ⎥ = α 2 ⋅ ω = K ⎢ Vsy ,2 ⎥ α1 e ⎢ ⎥ ⎢⎣ I s ,2 ⎥⎦
(4.39)
Therefore, in order to have linearly independent rows in U, Vsy,1/Is,1 must be different from Vsy,2/Is,2. During the steady-state operation of a motor, K=Vsy/Is versus the per phase shaft output power is plotted in Figure 4.10(a) for the 5 hp TEFC motor, whose parameters are specified in Table A.1. Figure 4.10(b) shows the relationship between K and the per phase input real power. From Figure 4.10(a) and (b), the relationship between K and the motor’s load level, described by either the shaft output power or the input real power, is monotonic. Therefore, suppose K1=Vsy,1/Is,1 and K2=Vsy,2/Is,2 are two indices computed from two experiments, the larger the difference between K1 and K2, the larger the difference between the corresponding load levels for these two experiments, and consequently the more reliable and accurate the estimates of the inductance values are.
72
Vsy/Is vs. Pshaft 30
Vsy/Is (Ω)
25
20
15
10
5 0
500 1000 Shaft output power (W)
1500
(a) K=Vsy/Is versus per phase shaft output power Vsy/Is vs. Pin 30
Vsy/Is (Ω)
25
20
15
10
5 0
200
400
600 800 1000 Input power (w)
1200
1400
1600
(b) K=Vsy/Is versus per phase input power Figure 4.10: The relationship between K=Vsy/Is and the motor’s load levels.
73
To confirm that the relationship between K and the motor’s load levels is monotonic for various small to medium size mains-fed induction machines, and consequently that the above conclusions are valid, simulations were carried out for various motors with parameter ranges shown in Table 4.4 according to [50].
Table 4.4: Approximate constants for 3-phase induction motors [50] Rating
Full Load efficiency
(hp) ≤5 5-25 25-200 200-1000 ≥1000
(%) 75-80 80-88 86-92 91-93 93-94
Full Load Full Load Power Slip Factor (%) (%) 75-85 3.0-5.0 82-90 2.5-4.0 84-91 2.0-3.0 85-92 1.5-2.5 88-93 ~1.0
R and X in per unit c Xls+Xlr d
Xm
Rs
Rr
(p.u.) 0.10-0.14 0.12-0.16 0.15-0.17 0.15-0.17 0.15-0.17
(p.u.) 1.6-2.2 2.0-2.8 2.2-3.2 2.4-3.6 2.6-4.0
(p.u.) 0.040-0.06 0.035-0.05 0.030-0.04 0.025-0.03 0.015-0.02
(p.u.) 0.040-0.06 0.035-0.05 0.030-0.04 0.020-0.03 0.015-0.025
c: Sbase is the motor’s full load VA rating; Vbase is the motor’s rated voltage. d: Assume Xls=Xlr for constructing the IEEE-recommended equivalent circuit.
The simulation results are shown in Figure 4.11-Figure 4.14. From these figures, for any induction machine with parameter ranges specified in Table 4.4, the relationship between K and the motor’s load levels, expressed either in input power, or in shaft output power, is strictly monotonic. Therefore, given K1 and K2, and K1≠K2, their corresponding load levels, P1 and P2, are different from each other. Consequently, the rows in matrix U are linearly independent on each other, and the inverse of U exists. In addition, it can be seen from Figure 4.11-Figure 4.14 that the largest slopes of the curves occur in the middle of the plots. These regions correspond to normal motor operation around 20-80% of its full load. It means that changes in the motor’s load levels can be reflected by changes in the magnitudes of K.
74
Vsy/Is vs. Pshaft
2.5 Rs = 0.030 p.u. Rs = 0.045 p.u.
Vsy/Is (p.u.)
2
Rs = 0.060 p.u.
1.5
1
0.5
0 0
0.1
0.2
0.3
0.4 0.5 0.6 0.7 Shaft output power (p.u.)
0.8
0.9
1
(a) K=Vsy/Is versus per phase shaft output power Vsy/Is vs. Pin
2.5 Rs = 0.030 p.u. Rs = 0.045 p.u.
Vsy/Is (p.u.)
2
Rs = 0.060 p.u.
1.5
1
0.5
0 0
0.1
0.2
0.3
0.4 0.5 0.6 Input power (p.u.)
0.7
0.8
0.9
1
(b) K=Vsy/Is versus per phase input power Figure 4.11: K=Vsy/Is versus the motor’s load levels for various motors with different parameters - variations in Rs (Xls=Xlr=0.0675 p.u.; Xm=2.4 p.u.; Rr=0.045 p.u.). 75
Vsy/Is vs. Pshaft
2.5 Xls = Xlr = 0.05 p.u. Xls = Xlr = 0.0675 p.u.
Vsy/Is (p.u.)
2
Xls = Xlr = 0.085 p.u.
1.5
1
0.5
0 0
0.1
0.2
0.3
0.4 0.5 0.6 0.7 Shaft output power (p.u.)
0.8
0.9
1
(a) K=Vsy/Is versus per phase shaft output power Vsy/Is vs. Pin
2.5 Xls = Xlr = 0.05 p.u. Xls = Xlr = 0.0675 p.u.
Vsy/Is (p.u.)
2
Xls = Xlr = 0.085 p.u.
1.5
1
0.5
0 0
0.1
0.2
0.3
0.4 0.5 0.6 Input power (p.u.)
0.7
0.8
0.9
1
(b) K=Vsy/Is versus per phase input power Figure 4.12: K=Vsy/Is versus the motor’s load levels for various motors with different parameters - variations in Xls and Xlr (Rs=0.045 p.u.; Xm=2.4 p.u.; Rr=0.045 p.u.). 76
Vsy/Is vs. Pshaft
3.5 Xm = 1.6 p.u. Xm = 2.4 p.u.
3
Xm = 3.2 p.u.
Vsy/Is (p.u.)
2.5
2
1.5
1
0.5
0 0
0.1
0.2
0.3
0.4 0.5 0.6 0.7 Shaft output power (p.u.)
0.8
0.9
1
(a) K=Vsy/Is versus per phase shaft output power Vsy/Is vs. Pin
3.5 Xm = 1.6 p.u. Xm = 2.4 p.u.
3
Xm = 3.2 p.u.
Vsy/Is (p.u.)
2.5
2
1.5
1
0.5
0 0
0.1
0.2
0.3
0.4 0.5 0.6 Input power (p.u.)
0.7
0.8
0.9
1
(b) K=Vsy/Is versus per phase input power Figure 4.13: K=Vsy/Is versus the motor’s load levels for various motors with different parameters - variations in Xm (Rs=0.045 p.u.; Xls=Xlr=0.0675 p.u.; Rr=0.045 p.u.). 77
Vsy/Is vs. Pshaft
2.5 Rr = 0.030 p.u. Rr = 0.045 p.u.
Vsy/Is (p.u.)
2
Rr = 0.060 p.u.
1.5
1
0.5
0 0
0.2
0.4 0.6 0.8 Shaft output power (p.u.)
1
1.2
(a) K=Vsy/Is versus per phase shaft output power Vsy/Is vs. Pin
2.5 Rr = 0.030 p.u. Rr = 0.045 p.u.
Vsy/Is (p.u.)
2
Rr = 0.060 p.u.
1.5
1
0.5
0 0
0.5
1
1.5
Input power (p.u.)
(b) K=Vsy/Is versus per phase input power Figure 4.14: K=Vsy/Is versus the motor’s load levels for various motors with different parameters - variations in Rr (Rs=0.045 p.u.; Xls=Xlr=0.0675 p.u.; Xm=2.4 p.u.). 78
From Figure 4.13, for a motor with relatively small mutual reactance running close to no load, the change in K with respect to the change in load is small. Consequently, different load levels, such as Pin,1=0.05 p.u. and Pin,2=0.1 p.u., may produce almost the same K (K1=1.662 p.u. and K2=1.632 p.u. in Figure 4.13), and this may lead to linearly dependent rows in the matrix U, and subsequently an ill-conditioned U. The inversion of this ill-conditioned U produces solutions to a nearby problem, thus invalidating the inductance estimation algorithm. In conclusion, K can be used as an index to indicate the linear independence of different rows in the matrix U. In order to have good estimates of the inductance values, each row’s K must be sufficiently different from each other. However, if the motor’s true mutual inductance is small, the motor must be operated at load levels much larger than its no load condition to yield sufficiently different K for a successful matrix inversion. 4.3.3 Influences from Numerical Precision and A/D Resolution on the Inductance Estimation Algorithm
The effects of A/D resolution and numerical precision are notoriously known as the “finite word length effects” in the DSP world. It involves A/D conversion, number representation, fixed- or floating- point quantization error, round-off noise, limit cycles and overflow oscillations. In some cases, even though the load levels are different from each other, the “finite word length effects” of DSP chips may have adverse effects on the matrix inversion in Equation (4.31). Suppose that the load level of one experiment is only slightly different from that of the other experiment, ωeIs,1Vsy,1 might be close to or even same as ωeIs,2Vsy,2 due to the fixed-point number representation, and so are ωe2 I s2,1 and ωe2 I s2,2 . This leads to an ill-conditioned U in Equation (4.36), and subsequently a solution ξ to a so-called “nearby” problem in Equation (4.31), which is usually far from the true solution according to Wilkinson’s research in matrix inverse [47]. 79
The above discussion indicates that the load levels must be “reasonably different” from each other, so that ωeIs,1Vsy,1 can be distinguished from ωeIs,2Vsy,2, and ωe2 I s2,1 be distinguished from ωe2 I s2,2 , by the DSP chips, and a well-conditioned U can therefore be obtained. According to reference [51], the analysis of the round-off error alone can be further divided into 3 categories: 1) Worst-case error bound 2) Steady-state worst-case error bound 3) Stochastic estimate of the error bound Considering the specific nature of this research – motor overload protection, the most conservative error bound, the worst-case error bound, is preferred to the other two error bounds. Usually, a target hardware platform needs to be specified before proper investigation can be carried out. This report assumes that specifications for such a hardware platform are known and quantified. Based on this assumption, a method is provided to analyze the performance of the proposed inductance estimation algorithm, and the minimum spread in load condition is quantified based on the assumed specifications. 4.3.3.1 Theoretical Analysis Assume that the signal flow in the measurement and data acquisition system is similar to the one shown in Figure 4.15.
80
Figure 4.15: Block diagram of the measurement and data acquisition system for the inductance estimation algorithm. For a given input current, i, to the measurement and data acquisition system, the output is assumed to be i+∆i, where ∆i (∆i≥0) is the error bound of the overall measurement and data acquisition system. This error is caused by the combined effects of the A/D resolution and the numerical precision of the measurement system. This includes items such as the ratio of the primary and secondary windings in a current transducer, the number of bits used to represent the scaled current signal from the secondary side of the transducer, and the fixed- or floating-point number representation. Generally speaking, it indicates that no error greater than ±∆i can be expected for any reading, i, that might be taken from this assumed measurement and data acquisition system [52]-[53]. Similar assumptions can be made for the voltage measurement and data acquisition system, as well. For the convenience of the subsequent analysis, the voltage of a mains-fed induction machine is assumed to be fixed.
81
Designating the measured current as i′ ( t ) . It is related to the true current, i ( t ) , by,
i′ ( t ) = i ( t ) + ∆i
(4.40)
for each phase. In Equation (4.40), ∆i is the aforementioned measurement error bound, introduced by the measurement and data acquisition system. According to Equation (4.4), the complex current space vector then becomes, iqdss ′ ( t ) =
D ⎡ 2 1 ⎤ ⎡ia ( t ) + ∆i ⎤ 2⎡ 1 e j120 ⎤ ⎢ ⎥ ⎥⎢ ⎣ ⎦ 3 ⎣ 1 2 ⎦ ⎣ ib ( t ) + ∆i ⎦
(4.41)
Assuming that ωe=2πfe, and fe is the fundamental frequency, the period of each cycle at fundamental frequency is Te =
1 fe
. In addition, assuming that in each cycle, N samples
are measured, tn is the instant for the nth sample, therefore, tn = Nn ⋅ Te , then the corresponding positive sequence fundamental frequency phasor, Is′ , is obtained through the definition of the discrete Fourier transform, Is′ =
where WN = e − j 2π
N
1 N
N −1
∑ ⎡⎣ i ′ ( t ) ⋅W n=0
s qds
n
nk N
⎤ ⎦
(4.42)
and k=1.
Substituting Equation (4.41) into Equation (4.42), it becomes, 1 Is′ = N
N −1
⎧⎪ 2
∑ ⎨ 3 ⎡⎣1 n=0
⎩⎪
D ⎡ 2 1 ⎤ ⎡ia ( tn ) ⎤ nk ⎫⎪ 1 e j120 ⎤ ⎢ ⋅W + ⎦ ⎣ 1 2 ⎥⎦ ⎢ib ( tn ) ⎥ N ⎬⎪ N ⎣ ⎦ ⎭
∑ {(1 + j 3 ) ∆i ⋅W N −1 n=0
nk N
}
(4.43)
The first term on the right hand side of Equation (4.43) corresponds to the phasor, Is , from the true 3-phase currents. The second term on the right hand side of Equation (4.43) corresponds to a phasor, ∆Is , that is introduced due to the error of the measurement and data acquisition system. The worst case error bound of ∆Is is obtained by,
82
∆Is =
1 N
∑ {(1 + j 3 ) ∆i ⋅W N −1
nk N
n=0
(1 + j 3 ) ∆i ⋅ ≤ N
(4.44)
N −1
∑W n =0
}
nk N
= 2 ⋅ ∆i
Suppose that two experiments are performed and 2 sets of current measurements are acquired: Is ,1 and Is′,1 are the true and measured current phasors from the first experiment, respectively. Likewise, Is ,2 and Is′,2 are from the second experiment. When the distance between the two true load levels, Is ,1 − Is ,2 , is smaller than twice the worst-case error bound, 2 ∆Is , the current phasors constructed from the measurements are likely to overlap with each other, as indicated by Figure 4.16(a). Conversely, when the distance between the two measured current phasors, Is′,1 − Is′,2 , is smaller than 2 ∆Is , these two experiments may actually correspond to the same load level, where Is ,1 = Is ,2 = Is . Therefore, in order to have two load levels that are significantly different from each other, the distance between the measured current phasors must be greater than 2 ∆Is , or equivalently, 4∆i, as shown in Figure 4.16 (c). 4.3.3.2 Experimental Validation Experiments were performed on a 5 hp TEFC motor to validate the above analysis. By comparing the magnitudes of the current measured from a system with similar structure as the one shown in Figure 4.15 with the readings from the oscilloscope, the overall error of the current in the measurement and data acquisition system is quantified as ±5% of the full scale, where the full scale is 31.25 A.
83
Vs Is ,1
Is ,2 ∆Is
∆Is
< 2 ∆Is (a) The distance between Is ,1 and Is ,2 is less than 2 ∆Is
Vs
Is ,1
Is ,2 ∆Is
∆Is
= 2 ∆Is (b) The distance between Is ,1 and Is ,2 is equal to 2 ∆Is Figure 4.16: The phasor diagram to illustrate the load spread in inductance estimation algorithm.
84
b1
Vs a1
Is ,1
Is ,2
∆Is
∆Is > 2 ∆Is
(c) The distance between Is ,1 and Is ,2 is larger than 2 ∆Is Figure 4.16 (continued). The TEFC machine was supplied with rated voltages and run at different load levels. At each load level, two experiments were performed by loading the induction machine with almost identical loads. Two sets of experimental data were acquired at each load level: two experimental sets at no load with Is,1=4.62 A and Is,2=4.63 A, which are approximately 37% of the full load current (FLC); two sets at light load with Is,1=7.37 A and Is,2=7.69 A (62% FLC); two sets at heavy load with Is,1=9.14 A and Is,2=9.24 A (74% FLC); and two sets at almost full load with Is,1=12.15 A and Is,2=12.61 A (97% FLC). At a sampling frequency of 5 kHz, a “snapshot” with 5000 samples was taken at 10 seconds after the rated voltages were applied to the motor terminals; another two snapshots with the same length of data were taken at 15 seconds and 30 seconds after the motor had started, respectively. The proposed inductance estimation algorithm was applied to the experimental data, assuming only the stator resistance, Rs, was known. The first rows in both Equations (4.36) and (4.37) are computed from the first set of data (Is,1), and the 85
second rows in Equations (4.36) and (4.37) are computed from the second set of data (Is,2). The final results are shown in Table 4.5.
Table 4.5: Online inductance estimation results for the 5 hp TEFC test motor at same load level
Parameters
Standard test
ε (%)
Ls (mH)
σLs (mH)
74.3
7.24
—
Experimental sets
t = 10 sec
160.5
75.9
949.84
#1: Is,1=4.62 A;
t = 15 sec
110.4
73.9
921.96
Is,2=4.63 A
t = 30 sec
171.0
76.2
953.12
Experimental sets
t = 10 sec
80.2
8.49
17.43
#2: Is,1=7.37 A;
t = 15 sec
80.4
8.63
19.27
Is,2=7.69 A
t = 30 sec
80.3
8.53
17.90
Experimental sets
t = 10 sec
75.9
6.46
10.74
#3: Is,1=9.14 A;
t = 15 sec
78.5
7.26
5.61
Is,2=9.24 A
t = 30 sec
77.5
7.00
4.37
Experimental sets
t = 10 sec
73.5
6.12
15.36
#4: Is,1=12.15 A;
t = 15 sec
73.4
6.11
15.59
Is,2=12.61 A
t = 30 sec
73.8
6.17
14.74
In Table 4.5, the error, ε, is defined as the maximum error between the estimated inductance values and the inductance values obtained from the standard no load and locked rotor test in Appendix A. In all 4 cases shown in the table, the distance between the two current phasors are far smaller than twice the worst-case error bound (3.125 A). Therefore, the inductance estimation algorithm cannot guarantee reliable estimates of the motor inductances. Altogether 10 out of 12 conditions end up with a relative error larger than 10%. Compared with the results presented in reference [54], where none of the estimate yields a relative error larger than 10%, the performance of the inductance 86
estimation algorithm deteriorates significantly with identical load levels. In conclusion, a reasonable spread in load is recommended so that the inductance estimation algorithm can produce an accurate and reliable estimate of the motor inductances.
4.4
Online Rotor Resistance Estimation Algorithm
Both the stator and rotor temperatures fluctuate during a motor’s operation due to internal heating. Therefore, the rotor temperature estimation algorithm needs to exclude the influence of the stator resistance drift due to its temperature variation. By estimating the rotor resistance from only the voltage and current measurements plus the inductance values, obtained in the previous step, the effect caused by the stator temperature variation is removed. This leads to the novel solution that the estimated rotor temperature is independent of the stator temperature change. Since the voltage drop across the stator resistance, Is Rs , is parallel to the abscissa in Figure 4.8, Vsy is not related to Rs. Hence, by using the relationship between Vsy and Is, the rotor resistance can be estimated. Starting from the last two rows in Equation (4.21), and by solving for Iry and Irx, yields, ⎡ I ry ⎤ −Is ⎢I ⎥ = 2 2 2 2 ⎣ rx ⎦ r2 + ω e (1 − σ ) Ls
⎡ω e (1 − σ ) Ls r2 ⎤ ⎢ 2 2 2⎥ ⎢⎣ω e (1 − σ ) Ls ⎥⎦
(4.45)
Substituting Irx into the first row of Equation (4.21), gives, 2
⎛L ⎞ Rr ⎜ m ⎟ = sω e (1 − σ ) Ls ⎝ Lr ⎠
Vsy Is
− ω eσ Ls
ω e Ls −
Vsy Is
(4.46)
In Equation (4.46), s comes from the sensorless rotor speed detection algorithm, described later in Section 4.6; ωe=2π×60 rad/s for a mains-fed induction machine with 60 Hz power supply; Ls and σLs come from the online inductance estimation algorithm
87
described in Section 4.3; Vsy and Is are extracted from the real-time voltage and current measurements. Since only the temperature-independent inductance values, Ls and σLs, are used in Equation (4.46), the estimated rotor resistance is independent of the stator resistance, and hence it is independent of the stator temperature drift. In addition, although there are small parametric errors in Ls and σLs, Rr is relatively insensitive to such parametric errors [12]. Furthermore, although the rotor resistance is a function of both rotor temperature and slip frequency, the latter does not change significantly during normal motor operations. Therefore, the variation in the rotor resistance is mainly caused by a change in the rotor temperature.
4.5
Fast and Efficient Extraction of Positive and Negative Sequence Components
As discussed in Section 4.4, an accurate estimate of the rotor temperature is highly er er dependent on the accuracies of I qds , Vqds and the angle φ. When a 3-phase symmetrical
voltage supply is applied to a healthy motor with negligible parameter asymmetry, there are only positive sequence current and voltage components in the system, and the er er calculations of I qds , Vqds and φ are straightforward. However, motors are normally fed
by power supplies with some degree of unbalance. For example, the current and voltage spectra shown in Figure 4.4 indicate that the peak amplitudes of positive sequence fundamental frequency current and voltage components are 18.27 A and 191.58 V, respectively; and the peak amplitudes of their corresponding negative sequence counterparts are 1.01 A and 3.61 V. If the unbalance is defined as the ratio between the amplitude of the negative sequence component to that of the positive sequence component [8], then for the example above, the voltage unbalance is 1.88%, and the current unbalance is 5.53%. Such unbalance levels usually have adverse effects on the online rotor temperature estimation algorithm.
88
In this section, the effects of negative sequence fundamental frequency and other frequency components on the rotor temperature estimation algorithm are discussed. In case the rotor speed is measured by an independent tachometer, the Goertzel algorithm can be employed to achieve a fast and efficient extraction of the positive sequence fundamental frequency current and voltage components from their respective complex space vectors.
4.5.1 Estimation Error from Negative Sequence Fundamental Frequency and Other Frequency Components The online rotor temperature estimation algorithm is formulated from the induction machine steady-state positive sequence equivalent circuit.
Therefore, the positive
er er sequence components, I qds and Vqds should be replaced by i1 e and v1e when performing
the calculation. Correspondingly, φ is the angle between i1 e and v1e , as shown in Figure
4.17. In case the negative sequence current and voltage components are not properly er er e removed, i.e. I qds and Vqds are substituted by iqdse and vqds , estimation error is introduced
into the estimated rotor temperature. For an induction machine under steady-state operation, assuming the inherent parameter asymmetry among its 3 phases is negligible, the relationship between the complex stator current and phase-to-neutral voltage space vectors is represented by the positive and negative sequence components in the synchronous reference frame, e vqds ( t ) = v1e ( t ) + v2e ( t ) = i1 e ( t ) ⋅ Z1 + i2 e ( t ) ⋅ Z 2*
(4.47)
where Z1 and Z2 are the positive and negative sequence impedances of the motor, respectively [35], Z1 = Rs + jω e Ls +
sω e2 L2m Rr + j ⋅ sω e Lr 89
(4.48)
Z 2 = Rs + jωe Ls +
( 2 − s ) ωe2 L2m Rr + j ⋅ ( 2 − s ) ωe Lr
(4.49)
and Z2* is the complex conjugate of Z2. e Since iqdse and vqds move along their respective trajectories clockwise at a speed of
2ωe in Figure 4.17, their amplitudes oscillates at twice the fundamental frequency. In e addition, the angle between iqdse and vqds also oscillates at twice the fundamental
frequency. Such oscillations lead to an oscillatory error in the estimated rotor resistance and hence error in the estimated rotor temperature.
e vqds (t )
v2e ( t ) 2ω e
v1e ( t )
i1 e ( t ) i2 e ( t )
iqdse ( t ) Figure 4.17: Positive and negative sequence complex current and voltage space vectors in a synchronous reference frame.
90
Figure 4.18 illustrates the simulation results of the online rotor temperature estimation e algorithm for the 5 hp ODP motor when iqdse and vqds are used.
The fundamental
frequency of the power supply is 60 Hz, and the voltage unbalance is 1.88%. As e indicated by the top diagram in Figure 4.18, the angle between iqdse and vqds oscillates at
twice the fundamental frequency. This oscillation leads to a similar pattern of oscillatory error in the estimated rotor resistance, shown in the bottom diagram in Figure 4.18.
o
Angle ( )
40 38 36 34 32 30
Rotor Resistance (Ω)
28 0.38 0.36 0.34 0.32 0.30 0.28 0.26 0.24 0.95
0.96
0.97
0.98
0.99
1.00
Time (second)
Figure 4.18: Simulation results from the online rotor temperature estimation algorithm e are used. when iqdse and vqds e In practice, the use of iqdse and vqds may produce even larger errors in the estimated
rotor temperature due to the presence of current and voltage components at frequencies other than the fundamental frequency. Figure 4.19 shows the results from the online rotor temperature estimation algorithm when it is applied to the experimental data collected from the 5 hp ODP motor.
91
o
Angle ( )
40 38 36 34 32 30
Rotor Resistance (Ω)
28 0.38 0.36 0.34 0.32 0.30 0.28 0.26 0.24 0.95
0.96
0.97
0.98
0.99
1.00
Time (second)
Figure 4.19: Experimental results from the online rotor temperature estimation algorithm e are used. when iqdse and vqds Obviously, there is a significant amount of estimation error in the rotor temperature, and such a rotor temperature signal cannot be used as the reference signal to tune the thermal capacitance and resistance in a rotor temperature predictor [17],[55]. Figure 4.20 illustrates the rotor temperature estimated from i1 e and v1e for the 5 hp e ODP motor. As a comparison, the rotor temperature estimated directly from iqdse and vqds
is also plotted in the same figure. The use of the positive sequence components removes the estimation error.
Consequently, the estimated rotor temperature is of sufficient
accuracy and is suitable for tuning the thermal capacitance and resistance in the rotor temperature predictor.
92
0.38
Estimated rotor resistance (Ω)
0.36
0.34
0.32
0.30
0.28 e e Rr estimated directly from iqds and vqds
0.26
Rr estimated from i1e and v1e 0.24 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Time (second)
Figure 4.20: Experimental results from the online rotor temperature estimation algorithm when i1 e and v1e are used.
4.5.2 Goertzel Algorithm As discussed in the previous section, to obtain a reliable estimate of the rotor temperature, the positive sequence fundamental frequency current and voltage components need to be extracted from the online measurements in a fast and efficient manner.
Since the discrete Fourier transform has a similar form as the rotational
transformation discussed in Section 4.2.1.3, a family of algorithms that perform efficient computation of the discrete Fourier transform can be applied directly to the complex s space vectors, iqdss and vqds , to obtain the positive sequence fundamental frequency
components i1 e and v1e .
In case the rotor speed information is measured from an
independent tachometer, the Goertzel algorithm, originally developed to detect dual-tone multi-frequencies in touch-tone telephone service, is an ideal solution to achieve such a goal. 93
4.5.2.1 Extraction of Positive Sequence Fundamental Frequency Component The Goertzel algorithm convolves the complex space vector, in the form of a sequence of complex data constructed from discrete current or voltage samples according to Equation (4.4) or (4.8), with a system characterized by the following transfer function [56], − j 2π
fe
fs 1− e H ( z) = ⎛ f ⎞ 1 − 2 cos ⎜ 2π e ⎟ z −1 + z −2 fs ⎠ ⎝
(4.50)
where fs and fe are the sampling and fundamental frequencies, respectively; z−1 is the unit delay. Figure 4.21 illustrates the flowchart of the Goertzel algorithm. Loop ①, designated by the blocks inside the dashed rectangle on the left, is executed first.
Since the
coefficient 2·cos(2πfe/fs) is real and the factor −1 is a trivial operation in this loop, only 2 real multiplications and 2 complex additions are required to implement the poles of Equation (4.50) for each sample in a complex data sequence. Therefore, for a complex data sequence of length N, altogether 2N real multiplications and 2N complex additions are needed. Once loop ① is completed, its output is fed into loop ②, denoted by the blocks inside the dashed rectangle on the right. Since the complex multiplication by −e−j2πfe/fs, which is required to implement the zero of the transfer function in Equation (4.50), needs to be executed only once, 1 complex multiplication and 1 complex addition is needed in loop ② . Because each complex multiplication is composed of 4 real multiplications and 2 real additions, and each complex addition comprises 2 real additions, altogether 2N+4 real multiplications and 4N+4 real additions are needed to extract the positive sequence fundamental frequency component from the complex space vector by the Goertzel algorithm.
94
iqdss ( t )
s vqds (t )
∑
∑
∑
2cos(2π
i1 e ( t ) v1e ( t )
f −exp(− j2π e ) fs
fe ) fs
Figure 4.21: Flowchart of the Goertzel algorithm to extract positive sequence fundamental frequency components from the complex space vectors.
4.5.2.2 Performance of the Goertzel Algorithm Besides the Goertzel algorithm, there are 2 other algorithms commonly used to extract the positive sequence fundamental frequency components from the online current and voltage measurements: a) direct method based on the definition of the discrete Fourier transform; b) fast Fourier transform (FFT) based on the Cooley-Tukey algorithm [56]. The direct method needs 4N real multiplications and 4N−2 real additions to compute the positive sequence fundamental frequency component from a complex data sequence of length N. In addition, it needs to compute and store N−1 coefficients in advance. In contrast, the Goertzel algorithm cuts the multiplications to approximately one half, and it needs to compute and store only 2 coefficients in advance. Classical FFT algorithms, such as the decimation in time or decimation in frequency algorithms, usually need 2N·log2N real multiplications and 3N·log2N real additions when N is a power of 2. Similar to the direct method, it also needs to compute and store various coefficients in advance. 95
Table 4.6 compares the total computations needed by various algorithms to extract the positive sequence fundamental frequency component from the complex data sequence of length N.
Table 4.6: Total computations for each algorithm to extract positive sequence fundamental frequency component Method
Real Multiplications
Real Additions
Goertzel Algorithm
2N+4
4N+4
Direct Method
4N
4N−2
Classical FFT
2N·log2N
3N·log2N
For example, when N=2048, the Goertzel algorithm needs 4100 real multiplications and 8196 real additions to extract the positive sequence fundamental frequency component from the complex space vector. In comparison, classical FFT algorithm needs 45056 real multiplications and 67584 real additions to accomplish the same task. Figure 4.22 illustrates the performance of these 3 algorithms. The scale of the ordinate is in terms of 1000 operations. When N is not a number in the power of 2, such as a prime number, the performance of the classical FFT algorithms deteriorates significantly due to some special treatment needed to handle the uneven radices [57]. In contrast, the number of operations needed in the Goertzel algorithm is independent of such properties of N.
96
270k
Real additions - Goertzel algorithm Real multiplications - Goertzel algorithm Real additions - direct method Real multiplications - direct method Real additions - FFT algorithm Real multiplications - FFT algorithm
240k
Number of operations
210k 180k 150k 120k 90k 60k 30k 0k 256
512
1024
2048
4096
Length of the complex data sequence (N)
Figure 4.22: The performance of various algorithms in extracting positive sequence fundamental frequency components from complex space vectors. When extracting the positive sequence fundamental frequency components directly from the complex space vectors in real time, the Goertzel algorithm also outperforms the classical FFT algorithms. Since loop ① of the Goertzel algorithm is executed each time after a new set of current or voltage samples is available, and can be completed before the next set of data is available, most of the computations can be done along with the data acquisition process.
The FFT algorithms, on the other hand, cannot interleave the
computation process with the data acquisition process. They usually need to have the whole complex data sequence for batch processing, and this may cause time delay between the data acquisition process and the final outputs. Therefore, the Goertzel algorithm is an ideal choice if the motor’s thermal parameters need to be tuned online, plus the rotor speed is available from a tachometer measurement.
97
4.6
Sensorless Rotor Speed Detection from Current Harmonic Spectral Estimation
When a stiff voltage source with only fundamental frequency component is applied to the terminals of an induction machine, the stator current, as a response, is nevertheless not perfectly sinusoidal. Besides the fundamental frequency component inherent in the stator current spectrum, there are other spectral components produced by various sources, such as rotor slotting, air gap eccentricity and load-dependent oscillations. The rotor speed can be extracted from the rotor speed dependent slot harmonics and dynamic eccentricity harmonics in the stator current spectrum. 4.6.1 Rotor Slot Harmonics
There are steps in the waveform of the rotor magnetomotive forces (MMF) due to the finite number of slots on the rotor side. These steps in the rotor MMF affect the waveform of the air gap flux. In addition, the large reluctance of the rotor slot conductors and the flux saturation due to high field concentration at the rotor slot openings produce spatial variations in an induction machine’s air gap permeance. This air gap permeance interacts with the MMF from both the stator and the rotor to produce air gap flux. Variations in the air gap permeance manifest themselves in the air gap flux. Since the air gap flux is linked to the stator windings, the variations in the air gap flux are reflected in the stator current spectrum [58]-[62]. The rotor slot harmonic frequency, fsh, is related to the stator supply frequency, f1, via [61],
⎛ ⎞ 1− s f sh = f1 ⎜ kR ⋅ + nw ⎟ p 2 ⎝ ⎠
(4.51)
where R is the number of rotor slots; s is the per unit slip; p is the number of poles; nw=±1,±3,…, is the air gap MMF harmonic order; k=1,2,…, indicates rotor slot harmonics at different order. 98
Figure 4.23 shows the current harmonic spectrum of a 4-pole 5 hp motor with totally enclosed fan-cooled (TEFC) enclosure.
The dominant rotor slot harmonics, which
correspond to different values of nw, are marked by ‘○’ symbols in the figure. Stator current harmonic components at multiples of the 60 Hz fundamental frequency are marked by ‘□’ symbols in the figure. Other current harmonic components in the figure come either from interactions between rotor slot and eccentricity harmonics, or from other nonlinearities inherent in the induction machine. From Figure 4.23, the frequencies of the rotor slot harmonics, denoted as fsh, can be extracted directly from the current harmonic spectrum. Table 4.7 shows the results of rotor slot harmonic frequencies extracted directly from such a current harmonic spectrum. At the same time, the rotor speed is also recorded by an independent tachometer as 1766.13 rpm. Since the motor has 28 rotor slots and is connected to a 60 Hz power supply, the dominant rotor slot harmonic frequencies, fsh*, can also be calculated from Equation (4.51) for different nw with k=1 once the rotor speed is known from the tachometer. Table 4.7 also shows the results of this approach.
Table 4.7: Rotor slot harmonic frequencies from Figure 4.23 nw −5 −3 −1 1 3 5
fsh [Hz] 524.1 644.2 764.2 884.3 1004.3 1124.4
fsh* [Hz] 524.19 644.19 764.19 884.19 1004.19 1124.19
According to Table 4.7, fsh closely matches fsh*. This indicates that rotor speed can be detected from the stator current harmonic spectrum if fsh is properly extracted.
99
Figure 4.23: Rotor slot harmonics in the current harmonic spectrum.
4.6.2 Rotor Dynamic Eccentricity Harmonics
Rotor dynamic eccentricity harmonics are related to the finite tolerances during motor manufacturing process. Such harmonics are often caused by the rotor not being perfectly round, or by a round rotor not rotating on its geometric center, resulting in the nonuniform shape of the air gap. The varying distance between the rotor and the stator leads to variations in air gap permeance. Through interactions between the air gap permeance and the air gap MMFs, comparable to those in the rotor slot harmonics, the rotor dynamic eccentricity induces specific spectral contents in the stator current. The rotor dynamic eccentricity harmonic frequency, feh, is related to the stator supply frequency, f1, via [61],
⎛ 1− s ⎞ f eh = f1 ⎜ nd ⋅ + 1⎟ p 2 ⎠ ⎝
100
(4.52)
where nd=0,±1,…, is the order of rotor eccentricity. Figure 4.24 shows the current harmonic spectrum of the same 5 hp motor, but in the frequency range of 20~100 Hz. A direct reading from this current harmonic spectrum indicates that the rotor dynamic eccentricity harmonic components can be found at feh=30.53 Hz (nd=−1) and feh=89.47 Hz (nd=+1). Since the independent tachometer reading shows that this 4-pole motor is running at 1767.80 rpm, the frequencies for the rotor dynamic eccentricity harmonics are calculated as: feh*=30.54 Hz when nd=−1, and feh*=89.46 Hz when nd=+1 according to Equation (4.52). Therefore, the rotor dynamic eccentricity harmonics can also be used to yield speed information as the values of feh and feh* can be regarded as being equal to each other within the given measurement accuracy.
Figure 4.24: Rotor eccentricity harmonics in the current harmonic spectrum. In Figure 4.24, stator current harmonic components at 44.90 and 75.10 Hz are caused by the rotating elements in the motor and therefore also bear rotor speed information. 101
However, analysis of such harmonic components is beyond the scope of this work. A detailed discussion on such harmonic components can be found in [44]. 4.6.3 Sensorless Rotor Speed Detection
From the above discussion, the combined effects of the rotor slot and dynamic eccentricity harmonics are summarized as [61],
⎡ ⎤ 1− s f seh = f1 ⎢( kR + nd ) + nw ⎥ p 2 ⎣ ⎦
(4.53)
where fseh denotes the frequencies of rotor-related harmonic components. By estimating the frequencies of rotor slot and dynamic eccentricity harmonics from the current harmonic spectrum, the actual rotor speed can be determined accurately in a sensorless fashion. This algorithm is divided into two stages: an initialization stage and an online speed detection stage. During the initialization stage, the value of R and an optimal set of numbers for k, nd and nw are determined, so that fseh calculated from Equation (4.53) matches the frequencies of the dominant rotor slot harmonic components from the stator current harmonic spectrum. After that, the rotor speed is detected online to yield a slip estimate.
This slip estimate is independent of the motor electrical
parameters. A brief summary of the sensorless rotor speed detection algorithm is given below, further discussion can be found in [61]. During the initialization stage, one phase current is sampled and a fast Fourier transform (FFT) is performed on the sampled data to obtain the stator current harmonic spectrum. The frequencies of the rotor dynamic eccentricity harmonics are then extracted from the current harmonic spectrum in the 0~120 Hz range. A rough estimate of the slip is computed according to,
s = 1±
p ⎛ f eh ⎞ − 1⎟ ⎜ 2 ⎝ f1 ⎠
(4.54)
102
Once a rough estimate of slip is obtained, the same current harmonic spectrum is examined in a frequency range where slot harmonics are most likely to occur. The number of rotor slots is determined by trying various combinations of R, k, nw and nd so that the rotor slot harmonic frequencies predicted by Equation (4.53) matches the frequencies of the dominant harmonic components estimated from the current harmonic spectrum. Certain combinations of poles and rotor slots may not yield observable rotor slot harmonics [63]-[64]. However, the number of rotor slots is usually chosen by motor designers based on certain conventions to achieve desired motor performance, and there are only a finite number of choices for R in practice [65]. Hence, the rotor slot harmonics are apparent in the current harmonic spectra for a large class of motors. The number of rotor slots, R, can usually be identified after a few trials by setting k to 1, nd to 0, and selecting nw to correspond to the frequency where the slot harmonic is located. Upon the completion of the initialization algorithm, R, k, nd and nw are known quantities.
The subsequent online rotor speed detection algorithm estimates the
frequencies of the rotor slot harmonics from the current harmonic spectrum. The rotor speed is calculated by, s = 1−
p f seh f1 − nw ⋅ 2 kR + nd
(4.55)
Figure 4.25 shows the results of the sensorless rotor speed detection algorithm for the 5 hp TEFC motor. The ‘+’ symbols illustrate the rotor speeds detected from its slot harmonics versus the rotor speeds measured from an independent tachometer at different load levels.
103
1800
Estimated Speed (RPM)
1790
1780
1770
1760
1750
1740 1740
1750
1760
1770
1780
1790
1800
Measured Speed (RPM)
Figure 4.25: Relationship between the estimated and the measured speeds. As demonstrated in Figure 4.25, the sensorless rotor speed detection algorithm provides accurate tracking of the rotor speed. Therefore, it can be used as an acceptable substitute for the tachometer, in the calculation of the motor slip. 4.6.4 Experimental Validation
To validate the proposed scheme, experiments have been performed on three induction machines: the 5 hp TEFC motor, the 5 hp ODP motor and the 7.5 hp TEFC motor. Rotor speed is measured by a tachometer as a validation of the rotor speed detection algorithm. For the 5 hp TEFC motor, it is determined during the initialization stage that it has 28 rotor slots, and the most dominant rotor slot harmonic component corresponds to k=1, nd=0 and nw=−1 (Figure 4.23). At a sampling frequency of 5 kHz, a 10-second window is applied to the sampled current data, and the frequency of the most dominant rotor slot
104
harmonic component, fˆseh , is estimated from the stator current harmonic spectrum. By replacing fseh in Equation (4.55) with fˆseh and replacing R, nd, nw with their respective values obtained during the initialization stage, the slip is successfully estimated. The final result from the sensorless rotor speed detection algorithm is shown in Figure 4.26.
0.024
Per Unit Slip
0.023
0.022
0.021
slip measured from tachometer slip estimated from rotor slot harmonics 0.020 0
1000
2000
3000
4000
5000
6000
Time (second)
Figure 4.26: Result from the sensorless rotor speed detection algorithm for the test motor - 5 hp TEFC motor, Is=10.7 A (85% FLC). The accuracy of the estimated slip is determined largely by the accuracy of the estimated frequency of the dominant rotor slot harmonic component. When an error, designated by ∆fseh, occurs in the estimated rotor slot harmonic frequency, the corresponding error in the slip, ∆s, is, ∆s =
p ∆f seh f1 ⋅ 2 kR + nd
(4.56)
Frequency resolution in the current harmonic spectrum is an important factor in determining the accuracy of the dominant rotor slot harmonic frequencies. With a 1 105
second window, the frequency resolution is 1 Hz. For the 5 hp TEFC motor used in the experiment, this translates to an error of 0.0012 in per unit slip according to Equation (4.56). Since the rated per unit slip for a typical small to medium size mains-fed induction machine is 0.02~0.03, the relative error in slip due to the finite frequency resolution is 4~6% in this case.
0.033 0.032
Per Unit Slip
0.031 0.030 0.029 0.028 0.027
slip measured from tachometer slip estimated from rotor slot harmonics
0.026 0.025 0
1000
2000
3000
4000
5000
6000
Time (second)
Figure 4.27: Result from the sensorless rotor speed detection algorithm for the test motor - 5 hp ODP motor, Is=13.0 A (100% FLC). Similar conclusions can also be drawn from Figure 4.27 for the 5 hp ODP motor and from Figure 4.28 for the 7.5 hp TEFC motor, respectively. In all three cases, the sensorless rotor speed detection algorithm provides a reliable tracking of the motor speed.
106
0.021
0.020
Per Unit Slip
0.019
0.018
0.017
0.016
slip measured from tachometer slip estimated from rotor slot harmonics
0.015 0
1000
2000
3000
4000
5000
6000
Time (second)
Figure 4.28: Result from the sensorless rotor speed detection algorithm for the test motor - 7.5 hp TEFC motor, Is=19.7 A (101% FLC).
4.7
Chapter Summary
As discussed in Chapter 2 as well as references [11]-[12], direct stator winding temperature estimation based on the stator winding resistance is susceptible to parametric errors if signal injection method is not employed. Since the rotor temperature is highly correlated to the stator winding temperature due to the designs of small to medium size induction machines, as presented in Chapter 3, the rotor temperature can be used as an approximate indicator of the stator winding temperature.
This chapter proposed a
sensorless parameter estimation scheme for the small to medium size mains-fed induction machines as the first step to the online monitoring of the induction machine stator winding temperature. The online parameter estimation scheme includes an online inductance estimation algorithm, a rotor resistance estimation algorithm and a sensorless rotor speed detection algorithm based on the current harmonic spectral estimation. 107
Besides the stator
resistance, the overall scheme does not require any previous knowledge on motor parameters, and the rotor resistance is calculated without interrupting the normal motor operations. By using the complex space vectors to represent three-phase variables, the induction machine steady-state operation is modeled by a simplified equivalent circuit in the synchronous reference frame. Since the d-axis coincides with the rotor flux linkage, only two inductors, Ls and σLs, are present in the simplified equivalent circuit. The online inductance estimation algorithm derives the values of Ls and σLs from the above simplified equivalent circuit. It requires at least two distinct load levels to achieve an online estimate of the inductance values. The inductance seen from the motor stator terminals, Vsy/Is, was proposed as a convenient indicator of the load levels so that inductance values can be estimated successfully. Influences from numerical precision and A/D resolution in the data acquisition system on the inductance estimation algorithm were also analyzed, and recommendations were made based on this analysis. Once the inductance values are successfully estimated online, the rotor resistance can then be determined from the online current and voltage measurements. Since only the temperature independent inductance values are used in the rotor resistance estimation algorithm besides the current and voltage measurements, the estimated rotor resistance is then independent from the stator resistance temperature drift.
This estimated rotor
resistance provides the foundation for the subsequent rotor temperature estimation and the online tracking of the stator winding temperature. Due to the presence of the negative sequence fundamental frequency components and other frequency components in the complex space vectors, there are significant errors in the estimated rotor resistance if the acquired data are not pre-processed properly. Section 4.5 analyzed the possible causes of such estimation error and proposes a fast and efficient method based on the Goertzel algorithm to extract the positive sequence fundamental
108
frequency components from the complex space vectors if the rotor speed is measured from a tachometer. In case the rotor speed can not be obtained from an independent tachometer, a sensorless rotor speed detection algorithm was used to extract the rotor speed information directly from the online current measurement for a small to medium size mains-fed induction machine.
This rotor speed estimator eliminates the need for speed
measurements from tachometers. The online inductance estimation algorithm, the rotor resistance estimation algorithm and the sensorless rotor speed detection algorithm were all validated by extensive experiments.
The proposed estimator is fast, efficient and reliable, suitable for the
purpose of implementation on a low-cost hardware platform.
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5
CHAPTER 5
INDUCTION MACHINE SENSORLESS STATOR WINDING TEMPERATURE ESTIMATION
To track the stator winding temperature in an online fashion, the rotor temperature can be used as a reference signal to tune the thermal parameters in the reduced order hybrid thermal model. The tuned hybrid thermal model then reflects the actual motor’s cooling capability. Section 5.1 presents the online calculation of the rotor temperature from the estimated rotor resistance. Section 5.2 describes the procedure for the online adaptation of the hybrid thermal model parameters. Finally, a brief summary is given in Section 5.3 to conclude the sensorless stator winding temperature estimation algorithm for small to medium size induction machines. 5.1
Online Calculation of Rotor Temperature
The rotor resistance is a function of both rotor temperature and slip frequency. However, for a mains-fed induction machine, its slip frequency only varies within a small range during normal motor operation. Therefore, the change in the rotor resistance, as given by Equation (4.46), is mainly caused by the change in the rotor temperature. 5.1.1 Rotor Temperature Calculation
Assuming at time t1, that the rotor resistance is Rr(t1), and the corresponding rotor temperature is θr(t1) [ºC]; at time t2, the rotor resistance is Rr(t2), and the corresponding rotor temperature is θr(t2), the relationship between the rotor resistances and the rotor temperatures is, Rr ( t1 ) θ r (t1 ) + k = Rr ( t2 ) θ r ( t2 ) + k
(5.1)
110
where k is the temperature coefficient. For a typical rotor with aluminum bars of 62% volume conductivity at 25 °C, k is 225 [19]. Multiplying both the numerator and the denominator on the left hand side of Equation (5.1) by (Lm/Lr)2, Rr ( t1 ) ⋅ ( Lm Lr )
2
Rr ( t2 ) ⋅ ( Lm Lr )
2
=
θ r ( t1 ) + k θ r ( t2 ) + k
(5.2)
The rotor temperature at t2, θr(t2), is,
θ r ( t2 ) =
Rr ( t2 ) ⋅ ( Lm Lr ) Rr ( t1 ) ⋅ ( Lm Lr )
2
2
⋅ ⎡⎣θ r ( t1 ) + k ⎤⎦ − k
(5.3)
Since thermal processes are usually slow, the rotor temperature, θr(t1), is assumed to be the same as the motor’s ambient temperature during the first several seconds after the motor is energized. In Equation (5.3), the denominator, Rr(t1)·(Lm/Lr)2, is related to the rotor resistance at θr(t1), and its value is obtained from Equation (4.46). Given such information, each time the rotor resistance, Rr(t2)·(Lm/Lr)2, is updated from online current and voltage measurements according to Equation (4.46), the corresponding θr(t2) can be calculated from Equation (5.3). 5.1.2 Experimental Validation
Since most small to medium size mains-fed induction machines are characterized by small air gaps (typically around 0.25-0.75 mm) to increase their efficiency, the stator and rotor temperatures are highly correlated due to such designs [8]. Therefore, the stator winding temperatures can be used as approximate indicators of the real rotor temperatures during the normal and the so-called “running overload” conditions, where the stator current is between 100% and 200% of its FLC [6]. One heat-run is performed on each motor: for the 5 hp TEFC motor, Is=10.7 A (85% FLC); for the 5 hp ODP motor, Is=13.0 A (100% FLC); and for the 7.5 hp TEFC motor,
111
Is=19.7 A (101% FLC). The estimated rotor temperature rise, and the corresponding measured stator winding temperature rise, are plotted in Figure 5.1. Figure 5.1(a) shows the rotor temperature rise is not significantly different from the stator temperature rise throughout the duration of the motor’s operation. From the same figure, it is apparent that the rotor and the stator slot winding temperatures increase with almost the same thermal time constant. This is consistent with the conclusions made in [66]: the same thermal time constant can be seen from both the stator and rotor thermal transients under normal motor operation. Figure 5.1(a) also reveals that the estimated rotor temperature contains a certain amount of noise. Such noise comes partly from the output of the sensorless rotor speed detection algorithm. Due to their small amplitude, the rotor slot harmonics sometimes are dwarfed by other spectral components near them. In addition, the presence of other frequency components also introduces some spectral leakage effect, which may further obscure the slot harmonics. Therefore, the estimated slip from the rotor speed detection algorithm contains a certain amount of noise. Besides such noise from the sensorless rotor speed detection algorithm, the fundamental frequency positive sequence components are also distorted by various other frequency components due to the intrinsic nonlinearities of the induction machine. All these factors contribute to the presence of the noise in the estimated rotor temperature. Similar conclusions can be drawn from the experimental results presented in Figure 5.1(b) and Figure 5.1(c): the proposed rotor temperature estimator provides a reliable tracking of the motor internal temperature rise during normal motor operations, and this online tracking of the rotor temperature is achieved through the use of only voltage and current measurements.
112
60
o
Temperature rise ( C)
50
40
30
20
10
θs - Measured from the stator slot winding θr - Estimated from the rotor resistance
0 0
1000
2000
3000
4000
5000
6000
Time (second)
(a) 5 hp TEFC motor, Is=10.7 A (85% FLC) 60
o
Temperature rise ( C)
50
40
30
20
10
θs - Measured from the stator slot winding θr - Estimated from the rotor resistance
0 0
1000
2000
3000
4000
5000
6000
Time (second)
(b) 5 hp ODP motor, Is=13.0 A (100% FLC) Figure 5.1: Results from the rotor resistance estimation and the rotor temperature calculation algorithm for the test motors.
113
70
50
o
Temperature rise ( C)
60
40
30
20
θs - Measured from the stator end winding
10
θr - Estimated from the rotor resistance 0 0
1000
2000
3000
4000
5000
6000
Time (second)
(c) 7.5 hp TEFC motor, Is=19.7 A (101% FLC) Figure 5.1 (continued) Besides the rotor temperature, the slip frequency also affects the apparent rotor resistance seen from the stator terminal, for motors with deep bars or double cages on their rotors. Therefore, the calculation of the rotor temperature needs to take into account the influence from the slip frequency for those types of motors. If the relationship between the frequency and the apparent rotor resistance is known in advance, then the change in the rotor resistance, caused by the change in the slip frequency, must be subtracted from the total rotor resistance change before performing the calculation of the rotor temperature using Equation (5.3). 5.2
Online Adaptation of Reduced Order Hybrid Thermal Model
Online adaptation of reduced order hybrid thermal model is divided into two stages. First, the state-space equations are formulated for the reduced-order hybrid thermal
114
model.
Then, an online parameter tuning algorithm is developed for the online
adaptation of the reduced-order hybrid thermal model parameters in real time. 5.2.1 State-Space Representation of the Reduced Order Hybrid Thermal Model
From Figure 3.8, the hybrid thermal model is described by the following equations in the continuous-time domain, C1
dθ s ( t ) θ s ( t ) + = Ps ( t ) + 0.65Pr ( t ) + ∆Pr ( t ) dt R1
(5.4)
θ r ( t ) = θ s ( t ) + ⎡⎣0.65 Pr ( t ) + ∆Pr ( t ) ⎤⎦ ⋅ R3
(5.5)
When the motor is operated with rated current and runs at rated speed, the heat flowing from the rotor across the air gap to the stator is 65% of the rated rotor loss, Pr. Therefore, ∆Pr=0, and in this case, the temperature difference between the rotor and stator is usually around 10 °C [42]. In addition, at rated condition, the transfer function between the input, Ploss=Ps+0.65·Pr, and the output, θs, in the s-domain, is, G (s) =
θs ( s)
Ploss ( s )
=
θs ( s)
Ps ( s ) + 0.65 Pr ( s )
=
R1 1 + sτ th
(5.6)
where the thermal time constant, τth, is defined as the product of R1 and C1, i.e., τth=R1C1. For typical small to medium-size induction machines, their thermal time constants are related to their trip class and service factor [8]. Such thermal time constants typically range from 500 to 3000 seconds. The value of the thermal resistance, R1, is determined by the steady-state stator winding temperature rise and the corresponding power losses. Based on the efficiency levels given in [68] for squirrel-cage induction machines and typical rotor temperature rise in such machines at rated conditions, the value of R1 is usually between 0.1 and 10 ºC/W. Replacing s in Equation (5.6) with jω, and assuming typical values for τth and R1, the Bode diagram for the transfer function is plotted in Figure 5.2. 115
Bode Diagram 0
Magnitude (dB)
-20 -40 -60 -80
Phase (deg)
-100 0
-45
-90 -4
10
-3
10
-2
-1
10
10
0
10
1
10
Frequency (rad/sec)
Figure 5.2: Bode diagram for the frequency-response characteristics of the reduced order hybrid thermal model (τth=534 sec, R1=0.5 ºC/W). According to Equation (5.6), the reduced order hybrid thermal model can be regarded as a low-pass filter with a cutoff frequency ωc=1/τth. Signal components with frequencies below 1/τth are retained in the response θs, while those with frequencies above 1/τth are filtered out. The cutoff frequency is usually between 3.3×10-4 and 2×10-3 rad/sec. In conclusion, the high frequency components in Ploss, due to the load torque variations, are filtered out in θs because of the large thermal time constant and hence the slow thermal response.
An effective online parameter tuning algorithm should
concentrate on the identification of the rotor thermal parameters in a frequency range between 3.3×10-4 and 2×10-3 rad/sec. By assuming that samples are taken at an interval Ts, which is much smaller than the thermal time constant, and that the power losses are piecewise constant during Ts, the differential equations (5.4)-(5.5) are transformed into difference equations [51],
116
T − s ⎞ ⎛ ⋅ θ s ( n − 1) + R1 ⋅ ⎜ 1 − e τ th ⎟ ⎜ ⎟ ⎝ ⎠ ⋅ ⎡⎣ Ps ( n − 1) + 0.65 Pr ( n − 1) + ∆Pr ( n − 1) ⎤⎦
θs ( n ) = e
−
Ts
τ th
(5.7)
θ r ( n ) = θ s ( n ) + ⎡⎣0.65Pr ( n ) + ∆Pr ( n ) ⎤⎦ ⋅ R3
(5.8)
The state-space representation of the system described by Equations (5.7)-(5.8) is, x ( n ) = Ax ( n − 1) + B ⎡⎣u ( n − 1) + ∆u ( n −1) ⎤⎦
(5.9)
y ( n ) = Cx ( n ) + D ⎡⎣u ( n ) + ∆u ( n ) ⎤⎦
(5.10)
where x(n) is θs(n); A is e
−
Ts
τ th
T − s ⎛ τ th ; B is R1 ⎜1 − e ⎜ ⎝
⎞ T ⎟ ⋅ [1 1] ; u(n) is ⎡⎣ Ps ( n ) 0.65Pr ( n ) ⎤⎦ ; ⎟ ⎠
∆u(n) is ⎡⎣ 0 ∆Pr ( n ) ⎤⎦ ; y(n) is θr(n); C is 1; D is R3 ⋅ [ 0 1] . T
5.2.2 Online Parameter Tuning
The online parameter turning algorithm involves the identification of the thermal parameters, R1 and C1, from the relationship between the input, Ploss, and the output, θs. It is further divided into two stages (Figure 5.3). First, a frequency-selective digital filter is designed to suppress the intrinsic noises in the calculated Ploss and θs.
Then, a
simplified infinite impulse response least mean-square adaptive filter is used to identify the reduced order hybrid thermal model parameters in an online fashion.
Figure 5.3: Two-stage approach to the online parameter tuning algorithm. 117
5.2.2.1 Downsampling with prefiltering The system input, Ploss, and output θs, are calculated from the current and voltage measurements at an interval of one second. However, they are often corrupted by noises unrelated to the thermodynamics of the induction machine, such as those from measurement devices or mechanical vibration. If not properly filtered out, this noise may distort the true rotor thermal response and pose potential problems to the subsequent online parameter tuning. In addition, direct application of the online parameter tuning algorithm to the Ploss and θs signals may cause convergence problem when the time interval between successive Ploss or θs is small when compared to the true thermal time constant. The poles of the discrete transfer function often tends to 1 from inside the unit circle on the z-plane, and it may move outside the unit circle upon a small perturbation from the noise, resulting in an unstable system [67]. To solve such problems, a stage of downsampling with prefiltering is needed before the online parameter tuning algorithm is applied [56]. A finite impulse response (FIR) digital filter is designed in this stage to prefilter Ploss and θs so that the noise is effectively suppressed. For the filter design in the prefiltering stage, there are two major types of frequencyselective digital filters: finite impulse response filters and infinite impulse response filters. The former is preferred in this application due to its 1) stability at all frequencies regardless of the size of the filter, 2) property of generalized linear-phase, which is characterized by a constant group delay in the filtered signal, and 3) simplicity in implementation. Unlike IIR filters, FIR filters do not have the problem of spectrum factorization, and therefore, do not introduce additional poles into the filtered signal, which may complicate the subsequent parameter tuning algorithm and lead to an incorrect identification of the rotor thermal parameters. One disadvantage of FIR filters is their relatively larger delays than their IIR counterparts with equal performance. However, the online parameter tuning algorithm is 118
not a time-critical task and can therefore use such filters. In addition, FIR filters may require additional hardware in implementation, but recent developments in DSP chips have made this possible with only a marginal increase in production cost. Various FIR filter designs based on the window method are investigated, and the Kaiser window is selected for the FIR digital filter for its near-optimal performance in balancing the trade-off between the main-lobe width and the side-lobe area, as well as its design simplicity. The Kaiser window is defined as [56], 1/ 2 ⎧ ⎛ ⎡ ⎞ n −α 2 ⎪ I 0 ⎜ β ⎣1 − ( α ) ⎤⎦ ⎟ I 0 ( β ) , 0 ≤ n ≤ M w(n) = ⎨ ⎝ ⎠ ⎪ 0, otherwise ⎩
(5.11)
where α=M/2, and M is the order of the FIR filter; I0(·) represents the zeroth-order modified Bessel function of the first kind. The length and shape of a Kaiser window are controlled by (M+1) and β, respectively. By adjusting these two parameters, side-lobe amplitude is traded for main-lobe width. Assuming that the peak approximation error of a low-pass filter is δ, the passband cutoff frequency, ωp, is defined to be the highest frequency such that |H(ejω)|≥1-δ, and the stopband cutoff frequency, ωs, is defined to the lowest frequency such that |H(ejω)|≤δ. Therefore, the transition region has width ∆ω=ωs-ωp. Defining A=−20log10δ, the specifications of the performance for a desired Kaiser filter in the frequency domain are empirically related to M and β by, M=
A−8 2.285 ⋅ ∆ω
(5.12)
0.1102( A − 8.7), A > 50 ⎧ ⎪ 0.4 β = ⎨0.5842( A − 21) + 0.07886( A − 21), 21 ≤ A ≤ 50 ⎪ 0 A < 21 ⎩
(5.13)
Based on Equations (5.12) and (5.13), virtually no iteration or trial and error is needed when designing a Kaiser window.
119
Assuming Ps, Pr and θr are calculated once each second, and selecting ωp=0.01 rad/sec, ωs=0.05 rad/sec and δ=0.01. From Equations (5.12) and (5.13), M=351 and β=3.40. The shape of the designed Kaiser window in the time domain and its response in the frequency domain are plotted in Figure 5.4.
Figure 5.4: Kaiser window in the time- and frequency- domain. After the Kaiser window, the Ploss and θr signals are downsampled by a factor of 10 to provide stability margins to the subsequent online parameter tuning block. 5.2.2.2 Online parameter tuning Based on the previous discussion, at rated condition, the stator winding temperature can either be derived from the estimated rotor resistance, designated as θs', or calculated from the reduced order hybrid thermal model, designated as θs. The online parameter tuning algorithm utilizes the temperature difference between θs' and θs to tune the thermal parameters R1 and C1. The stator winding temperature θs' is obtained from the estimated rotor temperature according to the procedures outlined in Chapter 4 as well as in Section 5.1. It is assumed that this stator winding temperature is 10 ºC less than the estimated rotor temperature at 120
rated condition. Since the localized heating effects of the stator winding (fast thermal transients) usually occur at the first several hundred seconds from the motor start, this relationship between the stator and rotor temperatures is assumed 500 seconds from the motor start and after the rotor temperature rise is at least 10 ºC above the motor ambient. Rewriting Equations (5.9)-(5.10) in a simplified scalar form,
θ s ( n ) = a1 ⋅θ s ( n − 1) + b1 ⋅ Ploss ( n − 1)
(5.14)
where a1=exp(−Ts/τth), b1=(1−a1)R1, Ploss=Ps+0.65·Pr. The error between the stator winding temperature derived from the estimated rotor resistance, θs', and the stator winding temperature calculated from the reduced order hybrid thermal model, θs, is,
e ( n ) = θ s′ ( n ) − θ s ( n )
(5.15)
and the mean-square error is,
ξ (n) =
{
}
{
2 2 1 1 E ⎡⎣ e ( n ) ⎤⎦ = E ⎡⎣θ s′ ( n ) − θ s ( n ) ⎤⎦ 2 2
}
(5.16)
where E{·} denotes the expectation. The online parameter tuning algorithm adjusts the rotor thermal parameters R1 and C1, or equivalently, a1 and b1, according to certain update rules so that ξ(n) is minimized. As a necessary condition to minimize ξ(n), the following equation must be satisfied [69],
∇ξ ( n ) = E {e ( n ) ⋅∇e ( n )} = 0
(5.17)
Since θs'(n) is obtained from the estimated rotor resistance, it is independent of a1 and b1. Only θs(n) is dependent on a1 and b1. Substituting Equation (5.15) into Equation (5.17) to obtain,
∇ξ ( n ) = − E {e ( n ) ⋅∇θ s ( n )} = 0
(5.18)
The expected values are replaced by their instantaneous counterparts based on the least mean-square approach,
121
ˆ ξ ( n ) = −e ( n ) ⋅∇θ ( n ) = 0 ∇ s
(5.19)
Expanding the gradient of θs(n) in Equation (5.19), yields, ⎧ ∂θ s ( n ) ∂θ ( n − 1) = θ s ( n − 1) + a1 ⋅ s ⎪ ∂a1 ⎪ ∂a1 ⎨ ⎪ ∂θ s ( n ) = P ( n − 1) + a ⋅ ∂θ s ( n − 1) 1 loss ⎪ ∂b ∂b1 ⎩ 1
(5.20)
Assuming that a1 and b1 are updated concurrently with the measurements of θs'(n) and Ploss(n), a1 and b1 become time-dependent series, a1(n) and b1(n), respectively. Consequently, the rotor temperature calculated from the simplified rotor thermal model in Equation (5.14) becomes,
θ s ( n ) = a1 ( n ) ⋅θ s ( n − 1) + b1 ( n ) ⋅ Ploss ( n − 1)
(5.21)
The update rules for a1(n) and b1(n) are given by the following equations, ∂ξ ( n ) ∂θ ( n ) ⎧ = a1 ( n ) + µ ⋅ e ( n ) s ⎪a1 ( n + 1) = a1 ( n ) − µ ∂a1 ( n ) ∂a1 ( n ) ⎪ ⎨ ⎪ b ( n + 1) = b ( n ) − µ ∂ξ ( n ) = b ( n ) + µ ⋅ e ( n ) ∂θ s ( n ) 1 1 ⎪ 1 ∂b1 ( n ) ∂b1 ( n ) ⎩
(5.22)
Substituting a1(n) and b1(n) for a1 and b1 in Equation (5.20), yields, ∂θ ( n − 1) ⎧ ∂θ s ( n ) = θ s ( n − 1) + a1 ( n ) s ⎪ ∂a1 ( n ) ⎪ ∂a1 ( n ) ⎨ ⎪ ∂θ s ( n ) = P ( n − 1) + a ( n ) ∂θ s ( n − 1) 1 loss ⎪ ∂b ( n ) ∂b1 ( n ) ⎩ 1
(5.23)
To achieve an online tuning of the rotor thermal parameters in a recursive manner, the step size µ in Equation (5.22) is usually chosen to be small enough so that a1(n)≈a1(n−1) and b1(n)≈b1(n−1), therefore,
122
⎧ ∂θ s ( n − 1) ∂θ s ( n − 1) ≈ ⎪ ∂a1 ( n − 1) ⎪ ∂a1 ( n ) ⎨ ⎪ ∂θ s ( n − 1) ≈ ∂θ s ( n − 1) ⎪ ∂b ( n ) ∂b1 ( n − 1) 1 ⎩
(5.24)
Based on Equation (5.24), approximation of Equation (5.23) is given as follows, ∂θ ( n − 1) ⎧ ∂θ s ( n ) ≈ θ s ( n − 1) + a1 ( n ) s ⎪ ∂a1 ( n − 1) ⎪ ∂a1 ( n ) ⎨ ⎪ ∂θ s ( n ) ≈ P ( n − 1) + a ( n ) ∂θ s ( n − 1) 1 loss ⎪ ∂b ( n ) ∂b1 ( n − 1) ⎩ 1
(5.25)
Denoting ψa1(n)=∂θs(n)/∂a1(n), ψb1(n)=∂θs(n)/∂b1(n), Equation (5.25) is further simplified to, ⎧⎪ ψ a1 ( n ) ≈ θ s ( n − 1) + a1 ( n )ψ a1 ( n − 1) ⎨ ⎪⎩ψ b1 ( n ) ≈ Ploss ( n − 1) + b1 ( n )ψ b1 ( n − 1)
(5.26)
and Equation (5.22) is simplified to, ⎧⎪ a1 ( n + 1) = a1 ( n ) + µ ⋅ e ( n )ψ a1 ( n ) ⎨ ⎪⎩ b1 ( n + 1) = b1 ( n ) + µ ⋅ e ( n )ψ b1 ( n )
(5.27)
By computing θs(n), e(n), ψa1(n), ψb1(n), a1(n) and b1(n) at each step from Equations (5.15)-(5.27), the parameters of the rotor thermal model can be determined recursively. A flowchart of the online parameter tuning algorithm is illustrated in Figure 5.5. Once the tuning process is completed, the reduced order hybrid thermal model is therefore able to predict the stator temperature in an online fashion.
123
Figure 5.5: Flowchart of the online parameter tuning algorithm.
5.2.3 Experimental Validation
To validate the proposed algorithm, experiments were performed on the 7.5 hp TEFC motor with parameters shown in the Appendix. The motor is operated at rated load for sufficient time until it reaches its thermal equilibrium. The thermal parameters are acquired from the experiment when the motor is loaded with Is=19.7 A, which is 124
approximately 101% of the FLC.
The parameters are identified as: R1=0.18 ºC/W,
R3=0.12 ºC/W, C1=8800 J/ ºC for the reduced order hybrid thermal model. Then the hybrid thermal model is used to predict the stator winding temperature when the motor is loaded with Is=22.5 A (115% FLC) until it reaches its thermal equilibrium. Both the stator temperature measured from the stator thermocouples, and the stator temperature predicted by the hybrid thermal model are plotted in Figure 5.6. It is still assumed that 65% of the rotor I2R loss is transferred across the air gap to the stator side
120
300
100
250
80
200
60
150
40
100
2
Stator I R loss 2 Rotor I R loss
θs - thermocouple
20
Power Losses (W)
o
Stator winding temperature rise ( C)
inside the motor in this case.
50
θs - HTM prediction 0 0
1000
2000
3000
4000
5000
6000
7000
0 8000
Time (second)
Figure 5.6: Stator winding temperature predicted by the the reduced order hybrid thermal model at Is=22.5 A, with the thermal parameters identified from heat run at Is=19.7 A. As seen in Figure 5.6, the reduced order hybrid thermal model, identified from the proposed online parameter adaptation algorithm, provides the tracking of the stator winding temperature change at motor running overload with reasonable accuracy.
125
Although the sensorless hybrid thermal model identification algorithm identifies the reduced order hybrid thermal model parameters and provides tracking of the dominant stator winding thermal dynamics with reasonable accuracy, as demonstrated by the experimental results shown in Figure 5.6, the proposed reduced order hybrid thermal model does not take into account the change of a motor’s cooling capability due to the variation of the speed of circulating air in a motor’s air gap. This is often caused by the change of the rotor speed in a drive controlled motor. In case there is a prolonged change of rotor speed, the thermal resistances, R1 and R3, needs to be tuned again to reflect such a change in the motor’s thermal behavior. In addition, the power loss caused by the PWM switching frequency may also influence the thermal behavior of the motor when it is connected to a drive and it might need to be incorporated into the loss calculation to yield an accurate estimate of the stator winding temperature for such motors. 5.3
Chapter Summary
This chapter focuses on the development of an adaptive sensorless hybrid thermal model parameter identification algorithm. The overall algorithm consists of a sensorless rotor temperature estimator and an online thermal model parameter identification scheme. The sensorless rotor temperature estimator provides an estimate of the rotor temperature from the rotor resistance.
After that, the online hybrid thermal model
parameter identification scheme performs a downsampling of the Ploss and θr signals after passing them through a digital lowpass antialiasing filter, and then identifies the thermal parameters through a recursive online parameter tuning algorithm. Once the thermal parameters of the reduced order hybrid thermal model are successfully determined, the dominant stator winding thermal behavior can be captured with reasonable accuracy, in an online fashion.
The stator winding temperature,
predicted by such a reduced order hybrid thermal model from the motor losses, serves as a piece of critical information for the motor overload protection relays. 126
The overall algorithm avoids possible temperature spikes from the rotor resistance estimation during an electromagnetic transient through the use of a Kaiser lowpass filter. Besides voltage and current sensors, no additional sensors are needed by the proposed algorithm. Therefore, the proposed online hybrid thermal model identification algorithm is suitable for real-time implementation on low-cost hardware platforms.
127
6
CHAPTER 6
EXPERIMENTAL SETUP AND IMPLEMENTATION OF VARIOUS TESTS
Experiments have been performed to validate the proposed stator winding temperature estimation scheme. This chapter discusses the experimental setup as well as implementation of various tests. The experimental setup is divided into the motor-load subsystem and various measurement subsystems. The measurement subsystems include the current and voltage measurement subsystem, the speed measurement subsystem and the temperature measurement subsystem.
Each measurement subsystem is further divided into the
hardware platform and the software for data acquisition purposes. Experiments have been implemented to emulate motor operations under various conditions, such as motor operations with unbalanced voltage supply and motor operations with impaired cooling caused by clogged motor casing. In addition, to study the thermal behavior of motors driving conveyor belts or operating wood-cutting saws, motor operations with continuous-operation periodic duty cycles are also implemented. 6.1
Experimental Setup
The overall experimental setup to validate the proposed stator winding temperature estimation algorithm is shown in Figure 6.1. The whole experimental setup is divided into 4 subsystems: 1) Motor and load 2) Current and voltage measurement subsystem 3) Rotor speed measurement subsystem 4) Temperature measurement subsystem 128
K-Type Thermocouple K-Type Thermocouple
Voltage Signal Voltage Signal Voltage Signal Current Signal Current Signal Current Signal
129
Figure 6.1: Overall experimental setup to validate the proposed stator winding temperature estimation algorithm.
K-Type Thermocouple
Each measurement subsystem includes a hardware platform and software written for the data acquisition purposes.
The detailed description and specifications of each
subsystem is given Sections 6.1.1-6.1.4. 6.1.1 Motor and Load
A 10 hp Westinghouse Life-Line dc dynamometer is connected to resistor boxes to provide load to the test motor. Both the test motor and the dc dynamometer are mounted on the same work bench, as shown in Figure 6.2. 6.1.1.1 Test Motors Extensive experiments have been performed on 3 test motors: a 5 hp TEFC motor, a 5 hp ODP motor and a 7.5 hp TEFC motor. The nameplate data of these motors are shown in Table 6.1.
In addition, Tables A.1-A.3 in Appendix A give detailed electrical
parameters of the test motors.
Table 6.1: Nameplate data of motors used in the experiments Motor brand
Leeson WattSaver
Marathon Electric
US Electrical Motors
Model Frame Enclosure Insulation class Design Horsepower Volt (V) Full load current (A) Rated RPM Service factor Efficiency (%) Power factor (%)
C184TTFS6334 184T TEFC F B 5 230/460 12.5/6.2 1755 1.15 90.2 83.5
184TTDB4026 184T ODP F B 5 208-230/460 14.4-13/6.5 1745 1.15 87.5 81.5
H7E2D 213T TEFC F B 7.5 208-230/460 20.8-19.6/9.8 1765 1.15 89.5 79.9
130
(a) Motor and dc machine
(b) Resistor boxes Figure 6.2: The motor-load configuration. The test motor is directly coupled to the dc machine. Therefore, the base plate on the work bench is adjusted for each motor so that the motor’s shaft is aligned to the dc machine’s shaft despite the change of the motor’s frame size. In addition, since the test motors have been disassembled to install thermocouples inside, comprehensive tests have been performed on the test motors after the instrumentation and before they are installed on the work bench. A Baker Instrument D12R motor tester, shown in Figure 6.3, is used 131
to verify the motor’s integrity. The tests performed on each motor using the D12R motor tester include a dc HiPot test, a phase to phase surge test and a dc resistance test.
Figure 6.3: Baker D12R digital motor tester [70]. The dc HiPot test detects insulation faults between the stator winding and the motor frame. Such insulation faults may occur anywhere in winding insulation, slot liner insulation, wedges, varnish, and sometimes phase paper. Polarization index can also be obtained in this test. The surge test detects either stator winding inter-turn fault or phase-to-phase insulation faults. Each test motor is verified to be free from any stator winding faults before the installation and heat runs. Since the dc resistance test provides value of the stator phase-to-phase resistance by injecting dc current into the stator windings and measuring the dc voltage across the stator windings at motor terminals, this test is used to obtain accurate values of the stator resistance values, Rs. Rs values shown in Tables A.1-A.3 in Appendix A are obtained through this test.
132
6.1.1.2 DC Generator The dc machine is connected to resistor boxes and used as the load to the test motor. The specifications of the dc machine are given in Table 6.2.
Table 6.2: Nameplate data of the dc machine Specifications
Output as Generator
Output as Motor
Input as Generator
Power Volt (V) Current (A) RPM
7.5 kW 125 60 1750-3600
7.5 kW 115 60 1450-3600
10 hp 125 60 1750-3600
Since the field circuit of the dc machine is separately excited, an independent dc voltage supply is used to provide the excitation voltage and function as a regulator of the load. This independent dc voltage supply is connected to the field circuit of the dc dynamometer via a rheostat. Assuming that the armature inductance of the dc machine is negligible, the simplified equivalent circuit of such a dc machine is illustrated in Figure 6.4.
Figure 6.4: DC machine equivalent circuit. From the equivalent circuit of this separately excited dc machine,
133
ea − Ra ia = RL ia
(6.1)
ea = K Φω m
(6.2)
Tm = K Φia
(6.3)
where ea and ia are the armature voltage and current, respectively; K is the armature constant; Φ is the flux per pole; ωm is the shaft speed of the dc machine; Tm is the torque developed by the armature. Substituting ea and ia in Equation (6.1) with Equations (6.2) and (6.3), respectively, yields,
ωm =
Ra + RL
(KΦ)
2
Tm
(6.4)
From Equation (6.4), the torque, Tm, is increasing linearly with respect to the shaft speed during the acceleration of the ac motor-dc generator coaxial shaft, and therefore the transient change of the load level in the experiment is better approximated by a ramp signal rather than a step change signal. 6.1.2 Current and Voltage Measurements
The current and voltage measurement subsystem has central importance in all measurement systems. Fast response to electromagnetic transient, high accuracy, good linearity, small offset and thermal drift in current and voltage measurements are desired for the purpose of accurate estimation of the stator winding temperature. In addition, proper signal conditioning techniques must be applied to suppress the noise and high frequency interference inherent in the raw measurements.
Fast analog to digital
conversion is also a necessary step in the data acquisition process. This A/D conversion enables the storage of measurement data on computer hard drive or other data storage devices for the subsequent data analysis and visualization. In this section, the hardware platform for the current and voltage measurements are first described.
Then the software program, which is written to facilitate the 134
configuration and control of the data acquisition devices, is briefly discussed and its major functions are outlined. 6.1.2.1 Hardware Platform The hardware platform used in the experimental validation of the proposed sensorless stator winding temperature estimation algorithm includes a set of closed loop Hall effect transducers, a set of signal conditioning devices and an analog to digital conversion card. 6.1.2.1.1 Closed loop Hall Effect Transducers Closed loop Hall effect transducers are used in the current and voltage measurement subsystem, as shown in Figure 6.5(b). Compared to its open loop counterpart (Figure 6.5(a)), which amplifies the Hall generator voltage to provide an output voltage, the closed loop Hall effect transducers use the Hall generator voltage to create a compensation current in a secondary coil to create a total flux, as measured by the Hall generator, equal to zero. Operating the Hall generator in a zero flux condition eliminates the drift of gain with temperature. An additional advantage to this configuration is that the secondary winding will act as a current transformer at higher frequencies, significantly extending the bandwidth and reducing the response time of the transducer [71]. LEM LA 55-P closed loop Hall effect current transducers are selected to measure the 3-phase line currents of the test motor. LEM LV 25-P closed loop voltage transducers are used to measure the line-to-line voltages of the test motor. The voltage transducers also provide galvanic isolation between the high voltage primary circuit and the low voltage secondary measurement circuit.
135
(a) Open loop Hall effect transducer
(b) Closed loop Hall effect transducer Figure 6.5: The Hall effect transducers [71]. Figure 6.6 shows the schematic of the current and voltage transducers connections on a printed circuit board. By selecting proper burden resistors on the secondary sides of the current and voltage transducers, the outputs of the transducers are normalized to voltage signals between −5V and +5V. The normalized voltage signals are then fed into the signal conditioning devices to suppress the noise and high frequency interference inherent in the raw analog measurements.
136
Figure 6.6: Schematic of the current and voltage transducers on PCB.
6.1.2.1.2 Signal Conditioning Devices The output signals from the closed loop Hall effect current and voltage sensors may contain various components uncorrelated to the true electromagnetic dynamics of the measured motor-load system, such as the signals caused by the thermal drift of the Hall effect sensors, or signals caused by the carrier frequency components in a nearby motor drive employing pulse width modulation (PWM) scheme. It is necessary to suppress such components using analog circuitry before the signal is digitized. A set of devices with signal conditioning extensions for instrumentation (SCXI) from National Instrument, including a SCXI-1000 chassis, a SCXI-1305 AC/DC coupling BNC terminal block, a SXCI-1141 lowpass filter module, is used in the experiment to provide proper signal conditioning capability. 137
The SCXI-1000 chassis, as shown in Figure 6.7, is a compact AC-powered chassis that houses any SCXI modules. The chassis is equipped with a timing circuitry for highspeed multiplexing, and it also provides a low-noise signal conditioning environment to the SCXI analog filter modules.
Figure 6.7: SCXI-1000 4-slot chassis. The SCXI-1305 AC/DC coupling BNC terminal block, as shown in Figure 6.8, provides an interface between the outputs of the Hall effect sensors and the SCXI-1141 analog filter module. The floating and ground-referenced signal configuration switches inside SCXI-1305 are switched to position ‘G’ since the output voltage signals from the Hall effect sensors are already grounded. In addition, the AC/DC coupling configuration switches are switched to position ‘DC’ to capture the signals with components from dc up to the cutoff frequency determined by the SCXI-1141 analog filter.
(a) SCXI-1305 BNC terminal block Figure 6.8: SCXI-1305 AC/DC coupling BNC terminal block [72].
138
(b) Closed loop Hall effect transducer Figure 6.8 (continued). The SCXI-1141 filter module is an eighth-order elliptic lowpass filter. It is a hybrid of a switched-capacitor and a continuous-time architecture, thus providing good cutoff frequency control while avoiding the sampling errors found in conventional switchedcapacitor designs.
(a) SCXI-1141 lowpass filter module Figure 6.9: SCXI-1141 8-channel lowpass elliptical filter module [73].
139
(b) Typical passband response of the SCXI-1141 filter module
(c) Phase response characteristics of the SCXI-1141 filter module Figure 6.9 (continued). The SCXI-1141 filter module prevents aliasing by removing all signal components with frequencies greater than the Nyquist frequency. In addition, since the SCXI-1141 module stopband begins at 1.5 times the cutoff frequency, the Nyquist frequency should be at least 1.5 times the cutoff frequency. Thus, the rate at which the DAQ device samples a channel should be at least 3 times the filter cutoff frequency to acquire
140
meaningful data [73]. In the experiments, the sampling rate is chosen at 5 kHz for each channel, and thus the cutoff frequency determined by such a sampling rate is 1.667 kHz. Such a cutoff frequency is high enough to observe most of the machine dynamics in the acquired current and voltage measurements, such as the rotor slot harmonics. 6.1.2.1.3 Analog to Digital Conversion A National Instruments PCI-6036E card with a hardware sampling rate of 200 kS/sec is used to accomplish the task of sampling the analog output signals from the SCXI-1141. Three phase currents and three line-to-line voltages are acquired to achieve a certain level of redundancy. The sampling frequency is selected to be 5 kHz. This means that the 3 current and 3 voltage channels are sampled sequentially, as shown in Figure 6.10. The time between adjacent samples from different channels within one batch is 5 µs, and this is related to the PCI-6036E hardware sampling rate of 200 kS/sec.
The time of
consecutive samples from one channel is determined by the 5 kHz sampling frequency, in this case, it is 200 µs.
Figure 6.10: PCI-6036E data acquisition scheme. The PCI-6036E card provides 16-bit A/D conversion resolution. Such a resolution is sufficiently high to distinguish electromagnetic dynamics of small magnitudes, such as the rotor slot harmonics, as shown in Figure 4.23-Figure 4.24.
141
6.1.2.2 Data Acquisition Software Software program has been written to acquire and store the sampled and digitized current and voltage data from the PCI-6036E card. The program is written with National Instruments LabView, and it is divided into two parts: 1) a front panel for control and configuration of the peripheral devices, including the SCXI-1305, SCXI-1141 and PCI6036E (Figure 6.11(a)); 2) a block diagram composed of graphical objects representing terminals, constants, structures, functions and wires to represent the logical relationships and operations among each components (Figure 6.11(b)). Since the current and voltage measurement data are stored in one computer while the speed and temperature measurement data are stored in another computer, the acquired data need to be time-stamped so that data from different computers can be synchronized for the subsequent data analysis and visualization. Dedicated LabView functions have been written and incorporated into the data acquisition program to realize such objectives.
(a) Front panel of the LabView data acquisition program Figure 6.11: The LabView data acquisition program for current and voltage measurements.
142
(b) Block diagram of the LabView data acquisition program Figure 6.11 (continued).
6.1.3 Speed Measurement
The rotor speed is measured by an Extech non-contact photo tachometer with an error range of ±0.1% of the full range. A piece of reflective tape is attached to the coupling between the induction motor and the dc machine. The digital tachometer is installed in a position perpendicular to the motor-load shaft, as shown in Figure 6.12(a). The rotor speed information is acquired by a dedicated software program, shown in Figure 6.12(b), through the serial communication via a RS-232 cable between the computer and the tachometer.
143
(a) Extech non-contact photo tachometer
(b) Software program for data acquisition from the tachometer Figure 6.12: Measuring the rotor speed with a non-contact photo tachometer.
144
Since changes in rotor speed are relatively slow compared to the electromagnetic transient, the speed is measured at an interval of 1 second. This one second interval is sufficiently fast for the purpose of sensorless induction machine parameter estimation and the subsequent stator winding temperature estimation. 6.1.4 Temperature Measurement
To validate the proposed sensorless stator winding temperature estimation algorithm, accurate and reliable temperature measurements must be obtained from the motor stator windings as the reference signals for the subsequent analysis and comparisons. Since a motor’s overall thermal process is slow compared to the current and voltage signals, the sampling rate of the temperature measurement subsystem is selected as 0.1 Hz. However, since the outputs from thermocouples are dc voltages in the mV range, they are highly susceptible to the electromagnetic interference and other noise, therefore, shields with aluminum screens are used to cover the thermocouple wires to minimize the influences from the motor ambient and to increase the reliability of temperature readings. Similar to the current and voltage measurement subsystem, the temperature measurement subsystem is divided into two parts: the hardware platform and the data acquisition software. 6.1.4.1 Hardware Platform The hardware platform for the temperature measurement consists of 9 K-type thermocouples for each motor, a 16-channel thermocouple monitor, a general purpose interface bus (GPIB) cable and a GPIB interface card. 6.1.4.1.1 K-type Thermocouples K-type thermocouples are instrumented at different locations of the induction machine stator winding, including the stator slot winding and the stator end winding, as
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shown in Figure 6.13, to capture the stator winding temperature change during various modes of motor operation.
(a) Thermocouple in the stator slot
(b) Thermocouple in the stator end winding Figure 6.13: The locations of the thermocouples. Due to the non-homogenous ventilation inside the motor, the temperature variation between different locations of stator windings may be up to 5~10 °C. Stator winding hot 146
spots usually occur at the stator end winding located on the opposite end of the ventilation fan instead of locations close to the stator slot windings. To illustrate this, the temperature measurements recorded at different locations of the stator windings for the 5 hp TEFC motor are plotted in Figure 6.14. The incongruous temperature distribution inside stator windings due to the location and the corresponding ventilation condition can be observed in this figure: there are around 5~10 °C temperature differences between the end windings and the slot windings, at both the drive end (DE) and the non-drive end (NDE).
Temperature Measurements (° C) Temperature Measurements (° C)
Drive-End Temperature Measurements (° C)
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140
120 100 80 60 End Winding Temperature Slot Winding Temperature
40 20
0
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Non Drive-End Temperature Measurements (° C)
120 100 80 60 End Winding Temperature Slot Winding Temperature
40 20
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4000 Time (sec)
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Figure 6.14: Temperature measurements for slot and end windings (Is=150% FLC).
6.1.4.1.2 Thermocouple Monitor
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A Stanford Research Systems SR630 16-channel thermocouple monitor, as shown in Figure 6.15(a) is used to acquire the outputs from 9 K-type thermocouples instrumented in the motor and transform them into temperature readings.
(a) SR630 thermocouple monitor
(b) Thermocouple junction voltage compensation Figure 6.15: SR630 thermocouple monitor. The SR630 thermocouple monitor is connected to the thermocouples at junctions C and D, as indicated by Figure 6.15(b). The temperature of thermocouple materials between A-C and B-D are measured with a low cost, high resolution semiconductor detector, and the ‘expected voltage drops’ across A-C and B-D, induced by the contact of different materials, are then tabulated. By measuring the terminal voltages between C and D, and deducting from it the expected voltage drops across contact points A-C and B-
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D, the voltage drop across the thermocouple junction inside the induction machine is calculated. This voltage drop is then translated to the temperature units and acquired for the subsequent data analysis. Normally, the accuracy of K-type thermocouples is ±0.5 °C. To assure that the temperatures are measured correctly, calibration is performed after each motor has been instrumented with the thermocouples. By using one calibrated Fluke 54II hand held thermometer, the actual measurement of the SR630 thermocouple monitor is found to be within ±0.6 °C over the temperature range of 0 °C to 160 °C for the SR630 thermocouple monitor. 6.1.4.2 Data Acquisition Software A software program is written with National Instruments LabView to configure and acquire temperature measurements from the SR630 thermocouple monitor, through the use of GPIB between the SR630 thermocouple monitor and the computer. Figure 6.16 shows the front panel and the block diagram of the LabView program written for such purposes. The overall temperature data acquisition program is divided into 3 stages. First, the computer sends a sequence of initialization commands to the SR630 thermocouple monitor. This sequence of commands resets the internal counter of the SR630 and synchronizes its internal clock with the computer clock.
In addition, the types of
thermocouples and the units of the temperature data are also configured. Once the initialization stage is completed, a ‘start’ command is sent to the SR630 to begin the temperature measurement and data acquisition process. Altogether 12 channels are used in the experiment, as shown in Figure 6.16(b). The first 9 channels measure the dc voltage outputs from thermocouples embedded in the motor and translate them into temperature data, and the last 3 channels measure the motor ambient temperatures and
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thus provide reference signals when calculating the motor stator winding temperature rise above its ambient.
(a) Front panel of the LabView data acquisition program
(b) Block diagram of the LabView data acquisition program Figure 6.16: LabView program for data acquisition of temperature measurements.
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Since the SR630 has a limited internal memory to store measurement data, the data acquisition program interrupts the temperature measurement process every 100 second by default. The temperature data are then fetched from the SR630 internal memory through the GPIB cable and are stored in the computer. Once this process is completed, the temperature measurement process resumes and new temperature measurements are collected from the thermocouples. At the end of each experiment, the stop button on the front panel of the LabView program is pressed, and the computer terminates the temperature measurement and data acquisition process by sending a sequence of commands to stop the operation of SR630. The commands stop the scanning of thermocouples for temperature measurements, read all remaining temperature measurements from the SR630 internal memory to the computer and close the communication link between the SR630 and the computer. When the temperature measurements are made at a sampling rate of 0.1 Hz, the interchannel delay is usually around 1 second.
Interpolation techniques, such as the
polynomial interpolation or spline interpolation, are needed when data points between two adjacent temperature measurements are desired. 6.2
Implementation of Various Tests
Induction machines are operated under various conditions, such as operations with unbalanced voltage supply, with impaired cooling caused by clogged motor casing, or operations with continuous-operation periodic duty cycles when driving conveyor belts or operating wood-cutting saws. These types of operating conditions are implemented in the experiments to simulate the real motor operations and the experimental data are used to validate the proposed stator winding temperature estimation algorithm.
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6.2.1 Motor Operation with Unbalanced Voltage Supply
The unbalanced voltage supply is created by 3 STACO 3PN1010 single-phase variable autotransformers, as shown in Figure 6.17(a). The wiring diagram of these autotransformers is shown in Figure 6.17(b). By adjusting the output voltage of the individual autotransformer, a certain level of voltage unbalance is created at the motor terminals. For a 3-phase system with floating neutral, there are two different definitions of the voltage unbalance: IEC standard definition and NEMA standard definition [8]. The IEC standard definition defines the voltage unbalance as the ratio between the amplitude of the negative sequence component, V2, to that of the positive sequence component, V1,
(a) 3 single-phase variable autotransformers Figure 6.17: Experimental setup to create unbalanced voltage supply.
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Power Supply
A
a
B
b
C
c
Motor Terminals
N
(b) Wiring diagram of the transformers Figure 6.17 (continued).
Voltage Unbalance =
V2 × 100% V1
(6.5)
While the NEMA standard defines the voltage unbalance as the ratio between the maximum deviation from the mean value of 3 line voltage magnitudes, Vab, Vbc and Vca, to the mean value of these 3 line voltages, Voltage Unbalance =
max deviation from mean value of Vab , Vbc and Vca × 100% mean value of Vab , Vbc and Vca
(6.6)
Since the negative sequence equivalent circuit of an induction machine is different from its positive sequence counterpart, the voltage unbalance definition given in Equation (6.5) is a better representation of motor operation under unbalanced supply. Therefore the IEC standard definition of voltage unbalance is used in the subsequent quantification of the induction machine operation. 6.2.2 Motor Operation with Impaired Cooling
For a three-phase induction machine, most of the heat generated by the motor losses is dissipated to its ambient through the combined effects of heat conduction and convection [8]. The motor’s heat dissipation capability can change in service due to, for example, a broken cooling fan or a clogged motor casing [15]. If such changes are not
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properly monitored, it may lead to the accumulation of excessive heat on the rotor of the induction machine, and this may cause severe thermal stress to the rotor structure, resulting in rotor conductor burnout or even total motor failure [4].
Therefore, an
accurate and reliable real-time tracking of the rotor temperature is needed to provide an adequate warning of imminent rotor over-heating due to a change of the motor’s cooling capability. The impaired cooling conditions caused by broken cooling fans have already been studied by other researchers [15]-[16], therefore, this work focuses on the impaired cooling conditions caused by the clogged motor casings. To simulate such operating conditions, the motors are covered with a thermally insulated blanket, as shown in Figure 6.18.
(a) Motor before applying the thermally insulated blanket Figure 6.18: Experimental setup to create impaired cooling conditions.
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(b) Motor after applying the thermally insulated blanket Figure 6.18 (continued). Since the motor heat dissipation capability is obstructed by the thermally insulated blanket, the thermal equilibrium established during normal motor operations is no longer valid. Experimental results in Chapter 7 will demonstrate that there are steady increases in both the rotor conductor and stator winding temperatures when the motor’s cooling capability is impaired. 6.2.3 Motor Operation with Continuous-operation Periodic Duty Cycles
Many motor manufacturers supply motors with continuous running duty ratings, designated as duty type S1 according to reference [40], as default options to their clients. These motors are supposed to be operated at constant loads for sufficient time until they reach their thermal equilibriums. In practice, however, such motors are often subjected to continuous-operation periodic duties, denoted as duty type S6 [40]. Each cycle of this duty type consists of a time of operation at constant load, ∆tp, and a time of operation at no-load, ∆tv (Figure 6.19). Thermal equilibrium is usually not reached during the time on 155
load. For small to medium size mains-fed induction machines, this periodic duty type can be found in many industrial applications, such as motors driving conveyor belts and motors operating wood-cutting saws.
Figure 6.19: Continuous operation periodic duty – duty type S6 [40]. To provide proper motor overload protection and at the same time ensure plant productivity, protection relays need to be configured for those motors [22], [74]. These protection relays usually use thermal models with a single thermal time constant to 156
simulate the motor’s internal heating. Such thermal models are derived from the heat transfer of a thermally homogenous object [22].
The thermal time constant is an
important parameter in these thermal models. In practice, the value of the thermal time constant is predetermined for a motor protection relay by electrical installation engineers or plant operators, usually under the assumption that the motor is operated with duty type S1. Despite the fact that thermal models with a single thermal time constant are widely used to approximate the motors’ thermal dynamics, the true thermal behavior of an induction machine is influenced by various components with dissimilar thermal characteristics inside the machine, as discussed in Chapters 2 and 3 . Therefore, for a motor operated with duty type S6, its real thermal time constant is highly dependent on its operation time at constant load within one load cycle.
The difference between
different thermal time constants, obtained at various portions of the temperature-rise curve, can be as large as 17 minutes for a typical motor [75]. The mismatch between the real thermal time constant in the motor and its predetermined counterpart in the relay, and the fact that the motor’s thermal time constant varies with respect to the duty cycle within one period, often results in disparities between the motor’s real and estimated stator winding temperature. Since the latter one is used as an indicator for the purpose of motor overload protection, such disparities may lead to nuisance trips of motors and unnecessary downtime of assembly lines or even the entire plant. To avoid such trips for motors operated with periodic duty cycles, its stator winding temperature must be tracked accurately. This requires the overload protection relay to adopt either a value that matches the motor’s true thermal time constant corresponding to ∆tp within that load cycle, or to use temperature feedbacks from certain sensors to correct the temperature estimates for the stator winding.
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(a) Schneider Electric/Square D manual starter, smart relay and contactor
(b) Interface of the logic configuration software Zelio-Soft Figure 6.20: Experimental setup to create continuous-operation periodic duty cycles. To provide an accurate estimate of the stator winding temperature for a small to medium size mains-fed induction machine with periodic duty cycles, experiments have been performed on the 7.5 hp TEFC motor. The periodic duty cycles is created by 158
switching the resistor boxes in and out using a Schneider Electric/Square D SR1-B101FU 6 inputs 4 outputs smart relay, in conjunction with an LC1-D65 3-pole contactor. Both devices are connected to a 120V/60 Hz ac power supply through a GV2ME06 manual starter, as shown in Figure 6.20(a). Once the manual starter is switched on, the smart relay is connected to the power supply. The Zelio-Soft logic configuration software, as shown in Figure 6.20(b), is used to configure the smart relay from a PC via serial communication. The contactor is switch on and off by the smart relay periodically, so that the resistor boxes are switched in and out, and thus simulating the intermittent change between no load and load for a motor. 6.3
Chapter Summary
Conventional split-core current transformers usually provide accurate measurement of currents up to 400 Hz. Since the proposed stator winding temperature estimation scheme utilizes the rotor slot harmonics to estimate the rotor speed, current and voltage must be measured accurately.
To achieve such high accuracies in current and voltage
measurements, Hall effect sensors are used. In addition, since the current and voltage measurements are converted into digital signals for the data analysis and visualization, they must be properly conditioned before the A/D conversion. National Instruments SCXI devices are used to achieve such an objective. Temperatures at various locations of the stator winding, including the stator slot winding and the stator end winding, are measured through the use of K-type thermocouples and a 16-channel thermocouple monitor. Although there are 5~10 °C temperature differences between the stator end windings and the stator slot windings, the temperature rises at both locations exhibit the same pattern and can thus be approximated by the hybrid thermal model described in Chapter 3.
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The rotor speed is measured by a non-contact photo tachometer. For the convenience of the subsequent data analysis and visualization, all measurements are read and stored in computers via data acquisition software in text file format. Besides normal motor operations, other typical motor operations, such as the motor operation with unbalanced voltage supply, motor operations with impaired cooling condition and motor operations with continuous-operation periodic duty cycles, are also simulated and studied in the lab environment.
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7
CHAPTER 7
INDUCTION MACHINE ONLINE THERMAL CONDITION MONITORING
Online thermal condition monitoring is a very important aspect of a comprehensive induction machine condition monitoring scheme [76]. Two typical motor operations, the motor operation under impaired cooling condition and the motor operation under continuous-operation periodic duty cycles, have been singled out and studied. Experimental results are presented to explain the online thermal monitoring of induction machines under such operations. 7.1
Induction Machine Thermal Monitoring under Impaired Cooling Condition
Comprehensive studies have been performed by other researchers to analyze the effects of impaired cooling on the motor internal temperature rise [15]-[16]. However, such studies mainly focus on the stator resistance change over the time when the motor’s cooling vents are blocked. As discussed in Chapter 3, since most small to medium size induction machines are characterized by small air gaps (typically around 0.25-0.75 mm) to increase their efficiency, the stator and rotor temperatures are highly correlated due to the heat flow patterns associated with such designs. Therefore, by monitoring the rotor temperature, the impaired cooling condition can also be detected. To validate the above conclusion, the impaired cooling conditions are created by obstructing the motor heat dissipation with a thermally insulated blanket, as described in Section 6.2.2. One heat-run is performed on each motor: for the 5 hp TEFC motor, the stator rms current is Is=10.25 A which is approximately 82% of its full load current (FLC); for the 5 hp ODP motor, Is=12.6 A (97% FLC); and for the 7.5 hp TEFC motor, Is=19.1 A (97% FLC). 161
In addition, since the 3-phase supply voltages are inherently unbalanced, the 3-phase current at motor terminals are also unbalanced. The voltage and current unbalances are 0.74% and 1.48%, respectively, for the 5 hp TEFC motor; 1.88% and 5.53% for the 5 hp ODP motor; 0.95% and 5.43% for the 7.5 hp TEFC motor. The estimated rotor temperature rises, derived from the online rotor temperature estimation algorithm based on the complex space vectors and the Goertzel algorithm given in Chapters 4 and 5, are plotted in Figure 7.1. The measured stator winding temperature rises are also plotted in the same figure.
100 90
70
o
Temperature rise ( C)
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60 50 40 30 20
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0 0
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Time (second) (a) 5 hp TEFC motor, Is=10.25 A (82% FLC) Figure 7.1: Rotor temperatures estimated for motors with impaired cooling conditions and unbalanced supply.
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o
Temperature rise ( C)
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60 50 40 30 20
θs - Measured from the stator slot winding
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Time (second)
(b) 5 hp ODP motor, Is=12.6 A (97% FLC) 110 100
o
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θs - Measured from the stator end winding
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θr - Estimated from the rotor resistance
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Time (second)
(c) 7.5 hp TEFC motor, Is=19.1 A (97% FLC) Figure 7.1 (continued).
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7000
Compared with healthy motor operations at approximately the same load levels (Figure 5.1), in which the steady-state stator winding temperature rise is 41 °C for the 5 hp TEFC motor, 55 °C for the 5 hp ODP motor and 57 °C for the 7.5 hp TEFC motor, there are steady increases in both the rotor and stator temperatures when the same motors are subjected to impaired cooling, as indicated by Figure 7.1. This phenomenon is caused by the change of the motor’s heat dissipation capability, and consequently the thermal equilibrium cannot be established within the motor. In addition, Figure 7.1 also demonstrates that the rotor temperature estimates are consistent with the corresponding stator winding temperature measurements. Despite a certain amount of noise in the estimated rotor temperature, caused primarily by the rotor speed measurement noise, the rotor temperature is not significantly different from the corresponding stator winding temperature. Therefore, the estimated rotor temperature can be used as an indicator to provide real time monitoring of the motor’s cooling capability. 7.2
Induction Machine Thermal Monitoring under Continuous-operation Periodic Duty Cycles
As discussed in Sections 3.1.2 and 6.2.3, in order to track stator winding temperature tracked accurately for motors operated with periodic duty cycles, an overload protection relay either needs to adopt a thermal time constant that matches the motor’s true thermal time constant corresponding to ∆tp within a specific load cycle, or to use temperature feedbacks from certain sensors to correct the temperature estimates for the stator winding. This section discusses the use of the estimated rotor temperature in conjunction with the conventional thermal model to track the stator winding temperature during each duty cycle. A proportional integral observer is proposed to unify the thermal model-based and the induction machine parameter-based temperature estimators (Figure 7.2). 164
This
observer differs from the conventional Luenberger observer by introducing an integration path to provide an additional degree of freedom. This freedom is used to make the final estimated stator winding temperature less sensitive to the mismatches between the real thermal time constant in the motor and the one adopted by the thermal model [78].
Figure 7.2: Block diagram of the sensorless stator winding temperature estimator based on a proportional integral observer.
7.2.1 Proportional Integral Observer
In the continuous-time domain, the hybrid thermal model is described by the following equations, θˆ ( t ) = A mθˆ ( t ) + B m Ploss ( t )
(7.1)
y ( t ) = Cθˆ ( t )
(7.2)
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where θˆ ( t ) designates the stator winding temperature, θs(t), calculated by this thermal model; Am=−1/(R1C1); Bm=1/C1; Ploss(t)=Ps(t)+0.65Pr(t)+∆Pr(t); y(t)=θr(t)−[0.65Pr(t)+ ∆Pr(t)]·R3; and C=1.
The thermal behavior of a motor, denoted as a ‘plant’ when viewed from the control system perspective, is assumed to be characterized by,
θ ( t ) = A p θ ( t ) + B p Ploss ( t )
(7.3)
y ( t ) = Cθ ( t )
(7.4)
where Ap and Bp are system and input matrices of the motor thermal process. By comparing Equations (7.3)-(7.4) with Equations (7.1)-(7.2), the differences between the plant and the model are: ∆A=Ap−Am and ∆B=Bp−Bm. The proposed PI observer is described by,
θˆ ( t ) = A mθˆ ( t ) + B m Ploss ( t ) + K p ⎡⎣ y ( t ) − Cθˆ ( t ) ⎤⎦ + K i v ( t )
(7.5)
v ( t ) = y ( t ) − Cθˆ ( t )
(7.6)
where Kp and Ki are the proportional and integral gains, respectively; v(t) is the integral of the difference between y(t) and Cθˆ ( t ) . Defining the difference between θˆ and θ as the error, e, e ( t ) = θˆ ( t ) − θ ( t )
(7.7)
When the plant dynamics are described by a first-order differential equation in (7.3), all matrices degenerate to scalars, and vectors to variables. In addition, C=1. By subtracting Equation (7.3) from Equation (7.5) and simplifying the result in the s-domain, yields, e(s) =
e ( 0) ∆A ⋅θ s ( s ) + ∆B ⋅ Ploss ( s ) − s + K p − Am + K i s s + K p − Am + K i s
where e(0) is the initial value for the error.
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(7.8)
There are two terms on the right hand side of Equation (7.8): e1 and e2. By selecting appropriate values for Ki and Kp so that the characteristic polynomial, s2+(Kp−Am)s+Ki, satisfies the Routh-Hurwitz stability criterion, e1 will decay to zero. Assuming a zero initial condition, θs can be solved from Equation (7.3) in the sdomain as,
θs ( s) =
B p Ploss ( s )
(7.9)
s − Ap
When a motor is operated with periodic duty cycles, Ploss can be described by a step change each time the motor experiences a transition in load, i.e., Ploss(s)=K/s. In this case, by substituting Equation (7.9) into the second term in Equation (7.8), e2 becomes, e2 ( s ) =
−∆BKs + ∆BAp K − ∆AB p K
s + ( K p − Am − Ap ) s 2 + ( K i + Am Ap − K p Ap ) s − Ki Ap 3
(7.10)
It can be shown from the final value theorem that the steady-state error e2 approaches zero given a pair of properly selected Ki and Kp values. 7.2.2 Operation of the Proportional Integral Observer
When the motor is operated at constant load during ∆tp, the thermal model-based stator winding temperature estimator is compensated by a correction term, derived from the sensorless rotor temperature estimator. Hence the PI observer achieves a closed-loop tracking of the stator winding temperature. When the motor is operated at no load during ∆tv, the estimated rotor temperature is no longer available due to the absence of the rotor slot harmonics. In this case, the thermal model-based temperature estimator operates independently to produce an estimate of the stator winding temperature. With properly tuned Kp and Ki values, the PI observer not only compensates for an incorrect initial estimate of the stator winding temperature in the thermal model, but also provides good tracking of the stator winding temperature even when the thermal time
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constant of the thermal model does not reflect the actual motor thermal dynamics within a load cycle. 7.2.3 Experimental Results
Experiments were performed on the 7.5 hp test motor to validate the proposed sensorless adaptive stator winding temperature estimation scheme. A dc generator with resistor banks was connected to the induction machine as an adjustable load. Hall-effect voltage and current transducers were used to collect motor terminal voltage and current information.
The voltages and currents were sampled at 5 kHz.
The motor was
instrumented with thermocouples on its stator winding, and a thermocouple reader was used to acquire and store the temperature readings when the motor was operated with periodic duty cycles. The thermal model time constant was set according to the following guidelines: with a class F insulation, the motor is assumed to have a 115 ºC temperature rise above the ambient in its stator winding when operated at 115% of its rated load; and the motor is assumed to be able to withstand 6 times its FLC for a maximum of 20 seconds before it reaches its thermal operating limit from a cold state [8]. In the first set of experiments, a cyclic load with a period of Tc=60 min and a cyclic duration factor of 50% was applied to the motor. In each load cycle, the motor was operated at a constant overload with Is=24.4 A, which was approximately 125% of the motor’s full load current (FLC), for ∆tp=30 min. After that, the resistor banks were disconnected so that the motor was operated at the no load condition for ∆tv=30 min. The initial stator winding temperature at the beginning of the experiment was 49.9 ºC while the ambient temperature was 25.9 ºC. The estimated stator winding temperature from the thermal model alone is plotted in Figure 7.3(a). The power loss and the measured stator winding hot spot temperature are also shown in the same figure. As a comparison, Figure 7.3(b) shows the results of the 168
proposed sensorless stator winding temperature estimation algorithm. Ploss is calculated by assuming that ∆Pr=0 in the hybrid thermal model, i.e.: Ploss(t)=Ps(t)+0.65Pr(t), where Ps and Pr are calculated by Equations (3.11)-(3.12). The initial stator winding temperature rise above the motor ambient is 24.0 ºC. However, since the thermal model itself did not have any knowledge of this information, it started the calculation by assuming a zero initial condition, i.e., θs(0)=0. Consequently, the estimated stator winding temperature was below the measured temperature for approximately 250 seconds, as shown in Figure 7.3(a). Such a discrepancy, between the estimated and the actual stator winding temperatures, indicates that there is a possibility of insufficient protection against motor overheating by the overload relay. Depending on the initial stator winding temperature and the load level, the duration of this insufficient protection may be even longer, and may sometimes lead to an accelerated and irreversible deterioration of the stator winding insulation, and ultimately a reduced motor life or even total motor failure. The proposed adaptive stator winding temperature estimator, on the other hand, includes the output from the sensorless rotor temperature estimator as a feedback signal. Therefore, it can track the stator winding temperature closely even though the thermal model started the calculation from an incorrect initial condition. As shown in Figure 7.3(b), the estimated stator winding temperature lies mostly within 10 ºC from its measured counterpart. Since the hot spot temperature is usually 10 ºC higher than the average stator winding temperature, the test motor is considered to be properly protected.
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o
Stator Winding Temperature ( C)
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Ploss (W)
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θs - Measured from the Hottest Thermocouple θs - Estimated from the Thermal Model Alone
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(a) Temperature estimation by the thermal model alone 140
o
Stator Winding Temperature ( C)
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θs - Measured from the Hottest Thermocouple θs - Estimated from the PI Observer
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Time (seconds)
(b) Temperature estimation by the proportional integral observer Figure 7.3: Performance of the sensorless adaptive stator winding temperature estimator (Tc=60 min, cyclic duration factor 50%).
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Figure 7.3(a) also indicates that motor overload protection schemes based on the motor thermal model alone often provide excessive overprotection to the motor. For example, at t=5388 sec in Figure 7.3(a), the stator winding temperature predicted by the motor thermal model is 138.1 ºC, but the hot spot stator winding temperature measured by the thermocouple only reaches 94.2 ºC.
Such large discrepancies often lead to
unnecessary trips of motors before they reach their thermal limits. As a comparison, Figure 7.3(b) shows that the proposed algorithm can avoid such trips by closely tracking the stator winding temperature. Experiments were also performed on the test motor with the same cyclic duration factor but at a shorter period (Tc=30 min) and a slightly different overload level. In each load cycle, the motor was operated at a constant overload with Is=23.5 A (120% FLC) for ∆tp=15 min. After that, the load was removed and the motor was operated at no load for ∆tv=15 min. The experimental results are shown in Figure 7.4. Similar conclusions can be drawn from these figures. In addition, both Figure 7.3(b) and Figure 7.4(b) indicate that the estimated stator winding temperature by the proposed PI observer contains certain amount of noise. This noise comes mainly from the output of the sensorless rotor temperature estimator due to the intrinsic nonlinearities of the induction machine and the distortion of the fundamental frequency positive sequence components by various other frequency components in the power supply [79].
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o
Stator Winding Temperature ( C)
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Ploss (W)
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Ploss - Calculated from v, i Measurements
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θs - Measured from the Hottest Thermocouple θs - Estimated from the Thermal Model Alone
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Time (seconds)
(a) Temperature estimation by the thermal model alone 140
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Stator Winding Temperature ( C)
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Time (seconds)
(b) Temperature estimation by the proportional integral observer Figure 7.4: Performance of the sensorless stator winding temperature estimator (Tc=30 min, cyclic duration factor 50%).
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7.3
Chapter Summary
For an induction machine, its heat dissipation capability often changes in service due to various factors, such as a broken cooling fan or a clogged motor casing. Therefore, its thermal operating condition needs to be monitored continuously to ensure safe and reliable motor operation. The rotor temperature provides a good indicator of the motor’s overall thermal operating condition. Therefore, it can be used as a reference signal to monitor the motor’s actual cooling capability. Experimental results validate the proposed scheme, which provides consistent estimates of rotor temperatures under impaired cooling conditions, even with certain level of unbalance in the power supply. The estimation error in the rotor temperature is reduced significantly due to the use of only positive sequence current and voltage components.
Consequently, this rotor temperature signal is sufficiently accurate to
reflect the true motor cooling capability. The overall scheme is fast, efficient and robust, and is suitable for implementation on a low-cost hardware platform with a modest requirement of processor speed and memory. When a mains-fed induction machine is operated with continuous-operation periodic duty cycles, the thermal time constant of a simplified motor thermal model needs to be adjusted accordingly to reflect the motor’s dominant thermal transients during one load cycle. Based on the analysis of the thermal behavior for small to medium size mains-fed induction machines, a sensorless adaptive stator winding temperature estimator is proposed to provide closed-loop tracking of the stator winding temperature for motors operated with periodic duty cycles.
This estimator utilizes the rotor temperature,
estimated from voltage and current measurements, as a feedback signal to adjust the thermal model output. A proportional integral observer is constructed to eliminate the steady-state error when there are mismatches between the real thermal time constant in the motor and its predetermined counterpart in the relay. The proposed algorithm tracks the stator winding temperature for the test motor under periodic duty cycles with 173
satisfactory performance, even when the thermal model is started with an incorrect initial estimate of the stator winding temperature rise. Experimental results also show that the performance of the proposed PI observer deteriorates due to the presence of noise in the estimated rotor temperature. The noise is related mainly to the intrinsic nonlinearities of the induction machine and the distortion of the fundamental frequency positive sequence components by various other frequency components in the power supply. To achieve a smooth estimate of the stator winding temperature with the proposed PI observer, future research will focus on the tuning of the PI observer and the development of effective filtering techniques to suppress the noise that are uncorrelated with the thermal dynamics of the induction machine.
174
8
CHAPTER 8
CONCLUSIONS, CONTRIBUTIONS AND RECOMMENDATIONS
8.1
Conclusions
The objective of this work was to develop a reliable, consistent and practical stator winding temperature estimation scheme for small to medium size mains-fed induction machines. The motivation for sensorless stator winding temperature estimation scheme is to provide an accurate indicator to the motor overload protection relays. By utilizing this estimated temperature, the relays can provide comprehensive motor overload protection against stator winding overheating, which may lead to stator winding insulation degradation or even total motor failure. Direct measurements of rotor temperature by means of thermal sensors, such as thermocouples, temperature-sensitive stick-ons, temperature-sensitive paint or infrared cameras, usually involves expensive instruments or dedicated wiring back to a motor control center. Therefore, they are not suitable for many low cost induction machine applications.
Dual-element time-delay fuses provide economical means of motor
overload protection. However, this type of devices trips the motor based only on a crude estimate of the stator winding temperature and is subject to either spurious trips or underprotection. Most microprocessor-based motor overload protective relays, representing the state-of-the-art in motor protection, rely on the motor heat transfer models to predict the stator winding temperature. Thermal models with a single thermal capacitor and a single thermal resistor are widely adopted in the industry. But since the values of the thermal capacitor and thermal resistor are usually predetermined by plant operators or
175
electrical installation engineers, such thermal models cannot respond to changes in the motor’s cooling capability. The induction machine parameter-based temperature estimator, on the other hand, derives average stator winding and rotor conductor temperatures from the stator and rotor resistances, respectively. This type of method can be further divided into two approaches: 1) induction machine model-based stator winding temperature estimator and 2) signalinjection based stator winding temperature estimator.
The former is based on the
induction machine equivalent circuit and was developed to improve field orientation performance or speed estimation accuracy in the low speed range. The latter creates a dc bias in the stator supply voltage and uses the dc components of the voltage and current measurements to calculate the stator resistance, and subsequently the stator winding temperature.
The induction machine model-based temperature estimator does not
produce any motor torque oscillation while estimating the stator winding temperature and is therefore noninvasive, but it suffers from inaccuracies in the estimates of the stator resistance at high speed motor operation, rendering it unsuitable to most applications with mains-fed induction machines.
The signal-injection based temperature estimator is
invasive in nature because it requires an extra injection circuit connected in series with one phase of the mains-fed machine. It also introduces small motor torque pulsations during the injection mode.
In addition, the series resistance of the cable from the
injection point to the motor terminal needs to be properly compensated for. To improve the accuracy in the estimate of the stator winding temperature in a noninvasive manner, a hybrid thermal model was proposed in this work to unify the thermal model-based and the induction machine parameter-based temperature estimators. The structure of the proposed HTM is based on the analysis of the thermal behavior of a small to medium size induction machine during various modes of operations. The hybrid thermal model is also able to capture the dominant thermal dynamics of the induction machine without sacrificing the estimation accuracy significantly. 176
In addition, the
proposed hybrid thermal model is of reasonable complexity and correlates the stator winding temperature with the rotor conductor temperature without requiring explicit knowledge of the machine’s physical dimensions and construction materials. Since the hybrid thermal model utilizes the rotor temperature as an indicator of the motor’s overall cooling capability, the rotor temperature must be estimated accurately. As the first step toward the development of an accurate stator winding temperature estimation scheme, the sensorless online parameter estimation scheme was proposed for small to medium size induction machines.
The overall architecture of the online
parameter estimation scheme is outlined in Chapter 4. The online parameter estimation scheme includes an online inductance estimation algorithm, a rotor resistance estimation algorithm and a sensorless rotor speed detection algorithm based on the current harmonic spectral estimation. Besides the stator resistance, the overall scheme does not require any previous knowledge on motor parameters, and the rotor resistance is calculated without interrupting the normal motor operations. In the proposed online parameter estimation scheme, the complex current and voltage space vectors were used to simplify the representation of the induction machine electrical equivalent circuit under steady-state operation. Since the complex space vectors are related to the phasor representation via rotational transformation, phasors were also used to describe the induction machine steady-state equivalent circuit.
The stator self
inductance, Ls, and the stator transient inductance, σLs, were derived from such simplified representations. The rotor resistance was determined from the online current and voltage measurements once the inductance values were successfully estimated. Since only the temperature independent inductance values were used in the rotor resistance estimation algorithm besides the current and voltage measurements, the estimated rotor resistance was shown to be independent from the stator resistance temperature drift. This estimated
177
rotor resistance serves as the foundation for the subsequent rotor temperature estimation and the online tracking of the stator winding temperature. The rotor temperature was estimated from the rotor resistance assuming that the rotor temperature variation was the sole cause of the rotor resistance change.
The rotor
temperature then serves as a reference signal to the subsequent adaptive identification of the hybrid thermal model parameters. In the adaptive identification of the hybrid thermal model parameters, a frequency-selective digital antialiasing filter using a Kaiser window was designed to remove the dynamics, which are unrelated to the rotor thermal characteristics, from the estimated rotor temperature after a downsampling stage. After that, the hybrid thermal model parameters are identified online based on the minimization of the mean squared error between the hybrid thermal model output and the estimated rotor temperature. Once the model parameters are obtained, the relationship between the motor losses and the motor temperatures is established, and the stator winding temperature is predicted from the motor losses, in an online fashion. Aside from the normal motor operations, another two typical motor operations: the motor operation under impaired cooling condition and the motor operation under continuous-operation periodic duty cycles were also studied. Previous research on the detection of impaired cooling condition focused on the use of stator resistance as an indicator of the change of the motor’s cooling capability [15]-[16], however, since most small to medium size induction machines dissipate the internal heat to the ambient through the stator enclosures, plus the rotor and the stator are separated only by small air gaps (typically around 0.25-0.75 mm), the rotor temperature can also be used as an indicator of the motor’s cooling capability. The experimental results validated the above claim and the rotor temperature was shown to be a good indicator of the motor’s cooling capability. The motor operation with continuous-operation periodic duty cycles is a common operation mode for motors driving conveyor belts or motors operating wood-cutting 178
saws. Each cycle of this duty type consists of a time of operation at constant load, ∆tp, and a time of operation at no-load, ∆tv. Since the rotor resistance cannot be estimated during the no load operation due to the almost zero current induced in the rotor conductors, the hybrid thermal model was used in conjunction with a proportional integral observer to produce an accurate tracking of the stator winding temperature during each load cycle. When the motor is operated at constant load during ∆tp, the hybrid thermal model-based stator winding temperature estimator is compensated by a correction term, derived from the sensorless rotor temperature estimator, and achieves a closed-loop tracking of the stator winding temperature. When the motor is operated at no load during ∆tv, the estimated rotor temperature is no longer available and the hybrid thermal model-based temperature estimator operates independently to produce an estimate of the stator winding temperature. 8.2
Contributions
The research performed in this work on sensorless stator winding temperature estimation for induction machine includes the following: 1) analysis of induction machine thermal behavior under various modes of motor operations (Chapter 3); 2) online machine parameter estimation (Chapter 4); 3) online adaptive thermal model parameter identification (Chapter 5); 4) online monitoring of the motor cooling capability (Chapter 7); 5) tracking of the induction machine stator winding temperature during continuous-operation periodic duty cycles (Chapter 7). Many original contributions have been made in all five of the above-mentioned areas, which are summarized in the following: 1) Induction machine thermal behaviors under various modes of motor operations were analyzed in this work. Since an induction machine is constituted of many different components, such as the stator winding, the rotor cage and the iron core, it is not a thermally homogeneous object and different components inside the induction machine 179
demonstrate different thermal behaviors.
A detailed thermal network is needed to
characterize the thermal behaviors for all components in the motor, and this high order thermal model often complicates the subsequent online identification of the thermal model parameters. It was shown in this work that the thermal behaviors of different components inside the induction machine can be divided into two groups in terms of their peak magnitudes and duration: the fast thermal transients and the slow thermal transients. The fast thermal transients are associated with the localized heating inside the motor, and they are characterized by small thermal time constants, usually within 100 seconds, and modest temperature rises. The slow thermal transients, on the other hand, are associated with the motor’s overall heat dissipation capability, and they are characterized by large thermal time constants, ranging from 1×102 to 1×104 seconds, and significant temperature rises. Consequently, the slow thermal transients often dominate the overall thermal response of the induction machine when compared with their fast counterparts. Hybrid thermal models were proposed in this work to capture such dominant induction machine thermal behaviors and provide reasonable accuracies in estimating the stator winding temperature from the power losses. 2) A sensorless online parameter estimation scheme was proposed for small to medium size induction machines in this work.
The sensorless online parameter
estimation scheme requires current and voltage measurements as its inputs. When the rotor speed is measured from an independent tachometer, it was shown that the positive sequence fundamental frequency current and voltage components could be extracted efficiently from the current and voltage space vectors via the use of the Goertzel algorithm.
It was demonstrated through experimental results that such a scheme
significantly reduces the computation efforts and internal memory needed. For example, with 2048 data samples as the complex space vector, classical FFT algorithm needs 45056 real multiplications and 67584 real additions to extract the positive sequence fundamental frequency component, while the proposed method using the Goertzel 180
algorithm only needs 4100 real multiplications and 8196 real additions to accomplish the same task. Therefore, the proposed method is more suitable for implementation on low cost hardware platforms. When the rotor speed is not provided by the tachometer, a sensorless rotor speed detection algorithm was introduced.
It was shown that the
algorithm estimates the rotor speed by making use of the rotor slot and dynamic eccentricity harmonics in the line current. It was also shown in this work that the proposed speed detection algorithm usually produces accurate estimates of the rotor speeds given enough length of current measurements in the time domain. With a 1 second window, a typical error introduced to slip in the rotor speed estimation algorithm due to the finite frequency resolution is 4~6%. 3) An online adaptive thermal model parameter identification algorithm was proposed and validated in this work. It was shown that the motor cooling capability could be captured by the online adaptation of the hybrid thermal model parameters. Once the thermal model parameters are properly tuned, the hybrid thermal model then predicts the stator winding temperature in an online fashion from the motor losses. As indicated in Section 5.2.3, a typical thermal time constant, obtained from the proposed algorithm, is τth=R1×C1=1584 seconds, instead of the commonly assumed 534 seconds. 4) An online scheme to monitor the motor cooling capability was proposed and validated by the experimental results in this work.
It was shown that the rotor
temperature can be used as an indicator of the motor’s overall cooling capability besides the direct measurement of stator temperature-related quantities, such as the stator winding resistance or stator winding temperature. 5) A scheme to provide online tracking of the induction machine stator winding temperature during continuous-operation periodic duty cycles was proposed in this work. Since the conventional Luenberger observer can only compensate for the error introduced at the beginning of the tracking, and it cannot provide proper correction to the error between the hybrid thermal model and the real induction machine, a proportional integral 181
observer is proposed to achieve an online compensation to both thermal model error and initial condition temperature error. It was shown in this work that such an observer provide an additional degree of freedom in the observer design and the stator winding temperature is captured by such a closed-loop tracking system. Compared with the output from the conventional overload relay, which has a maximum error of 48.5 °C in the estimates of the stator winding temperature, the proposed method based on the proportional integral observer has a maximum error of 14.7 °C in the estimates of the stator winding temperature.
This 14.7 °C error comes mainly from the dynamics
uncorrelated with the motor’s thermal behavior.
In this case, it comes from the
mechanical disturbance mainly in the form of the estimated rotor speed. 8.3
Recommendations for Future Work
Although this work has presented contributions to various areas of stator winding temperature estimation, there are several directions in which further research could build on the results presented in this work. An important aspect of the future research for the sensorless stator winding temperature estimation scheme is to adapt the algorithms to the medium-voltage large size machines. These large machines differ from their small to medium size low-voltage counterparts mainly by their 2300 V and 4000 V designs using vacuum-pressureimpregnated (VPI) form windings, and the copper or its alloy as the material for the rotor cage instead of aluminum. Since medium-voltage large size induction machines are important assets of industrial plants and usually involved in critical industry processes, such as the petrochemical process, they require more sophisticated overload protection. Due to their different designs, the per unit resistance values of medium-voltage large size induction machines are typically smaller than those of small to medium size induction machines, as shown in Table 4.4. Therefore, a smaller rotor resistance means that if the baseline rotor resistance comes with the same amount of error as in the small to 182
medium size induction machines, then the estimated rotor temperature will have a larger relative error due to the division operation in the rotor resistance estimation algorithm, as given in Equation (5.3). In addition, unlike their small to medium size counterparts, the large size induction machines are very often soft started. During this prolonged starting stage, the localized heating effects of the machine are no longer negligible.
This may require the
modification of the hybrid thermal model, which was initially developed for small to medium size induction machines. To illustrate the above discussion, Figure 8.1(a) plots the stator winding temperature rise, collected from the 7.5 hp TEFC motor during a heat run with full load current. The single thermal time constant approximation to the stator winding temperature rise is also plotted on the same figure. It can be seen from Figure 8.1(a) that a certain amount of error exists between the approximated and the measured stator winding temperature rise, especially during the first several hundred seconds.
Figure 8.1(b) shows the
approximated and the measured stator winding temperature rise during the first 1000 seconds.
As indicated in Figure 8.1(b), the stator winding temperature rise is not
completely captured by the thermal model with a single thermal time constant. From Figure 8.1(b), the error between the approximated and the measured stator winding temperature is usually within 5~10 °C for the small to medium size induction machine. Therefore, such an error can be ignored without introducing significant errors into the final estimated stator winding temperature.
However, for large size machines, this
temperature error becomes more significant because the machines are energized by soft starters and undergo prolonged start-up stage. Future research on the modeling of the localized heating effects of the large size induction machine is expected to improve the estimation accuracy in the stator winding temperature.
183
Normal Full Load Operation 60
40
o
Temperature rise ( C)
50
30
20
θs: thermocouple
10
θs=A(1-e-t/τ) 0 0
1000
2000
3000
4000
5000
6000
7000
Time (second)
(a) 0≤t≤7000 seconds Normal Full Load Operation 30
20
o
Temperature ( C)
25
15
10
θs: thermocouple
5
θs=A(1-e-t/τ) 0 0
200
400
600
800
1000
Time (second)
(b) 0≤t≤1000 seconds Figure 8.1: Stator winding temperature rise for the 7.5 hp TEFC motor.
184
Besides the adaptation of the existing algorithms to account for the electrical and thermal properties of large size induction machines, another important aspect of the future research to provide comprehensive overload protection scheme for these machines is to find ways to correlate the temperatures from RTDs or thermistors to the temperature estimated from the thermal model, and to make good use of the knowledge of the motor design in such a correlation process.
Since large induction machines are usually
protected with various types of thermal sensors, such as the RTDs or thermistors, proper signal processing of measurements from these temperature sensors may afford useful information to the stator winding temperature and the motor’s cooling capability.
185
APPENDIX A MOTOR PARAMETERS
Table A.1: Parameters of the 5 hp TEFC Motor Hp Poles Vrated Irated Nm,rated Rs, 25 ºC
5 4 230 [V] 12.5 [A] 1755 [RPM]
Rr, 25 ºC
0.237 [Ω] 0.071 [H] 3.0 [mH] 4.5 [mH]
Lm Lls Llr
0.332 [Ω]
Table A.2: Parameters of the 5 hp ODP Motor Hp Poles Vrated Irated Nm,rated Rs, 25 ºC
5 4 230 [V] 13 [A] 1745 [RPM]
Rr, 25 ºC
0.290 [Ω] 0.059 [H] 2.6 [mH] 3.9 [mH]
Lm Lls Llr
0.354 [Ω]
186
Table A.3: Parameters of the 7.5 hp TEFC Motor Hp Poles Vrated Irated Nm,rated Rs, 25 ºC
7.5 4 230 [V] 19.6 [A] 1765 [RPM]
Rr, 25 ºC
0.134 [Ω] 0.043 [H] 1.1 [mH] 1.6 [mH]
Lm Lls Llr
0.148 [Ω]
187
APPENDIX B RELATIONSHIP BETWEEN TEMPERATURE AND RESISTIVITY
The resistivity, ρ [Ω·m], of copper or aluminum changes with respect to the ambient temperature. Table B.1 illustrates such relationship between the temperature and the resistivity [77].
Table B.1: Relationship between temperature and resistivity Temperature (°C)
ρ (×10−8 Ω·m)
Copper
Aluminum
−173
0.348
0.442
−123
0.699
1.006
−73
1.046
1.587
0
1.543
2.417
20
1.678
2.650
25
1.712
2.709
27
1.725
2.733
127
2.402
3.870
227
3.090
4.990
327
3.792
6.130
Figure B.1 plots the curves of resistivity versus temperature for the copper and aluminum, respectively. Through the curve fitting, the slope is 6.86×10−8 Ω·m/°C for copper and 1.14×10−10 Ω·m/°C for aluminum. If the resistivity at 25 °C is chosen as the reference value, then the temperature coefficient is 0.0040 for copper and 0.0042 for aluminum.
188
7
-8
Resistivity (×10 Ω m)
6
5
4
3
2
Copper Aluminum
1
0 -200
-100
0
100
200
300
400
o
Temperature ( C)
Figure B.1: Relationship between resistivity and temperature.
189
APPENDIX C SINGULAR VALUE DECOMPOSITION AND MOORE-PENROSE INVERSE
The Moore-Penrose inverse method is used in Section 4.3.1 to obtain intermediate results for motor inductance values. This inverse method is based on the singular value decomposition (SVD) theorem and can yield accurate inductance values with minimal least squares norm. Compared to the conventional matrix inverse, the Moore-Penrose inverse method is robust and fault tolerant. In this appendix, the singular value decomposition theorem is described. Then, the flowchart showing how to calculate the singular value decomposition and the MoorePenrose inverse is given. Two examples are given to illustrate the calculation methods outlined in the flowchart. Finally, advantages of the Moore-Penrose inverse over the conventional matrix inverse are discussed. Theorem: Every matrix A ∈ \ m×n can be factored as, A = UΣV T
where U ∈ \ m×m is orthogonal, V ∈ \ n×n is orthogonal, and Σ ∈ \ m×n has the form, Σ = diag (σ 1 , σ 2 ," , σ p )
with p = min ( m, n ) . The Moore-Penrose inverse of A is, A † = VΣ † UT
The singular value decomposition (SVD) and the Moore-Penrose inverse techniques are outlined in Figure C.1. Two examples are then given to further illustrate the techniques.
190
191
λ2 "
λp ⎤⎦
Figure C.1: The calculation of the singular value decomposition and the Moore-Penrose inverse.
∑ = diag ⎡⎣ λ1
⎡ 2 2⎤ Example C.1: Find the Moore-Penrose inverse of matrix A = ⎢ ⎥. ⎣ −1 1 ⎦ First, compute AAT and find its eigenvalues, ⎡8 0 ⎤ ⎡8 − λ ⇒ det ( AAT − λΙ ) = det ⎢ AAT = ⎢ ⎥ ⎣0 2⎦ ⎣ 0
0 ⎤ =0 2 − λ ⎥⎦
(8 − λ )( 2 − λ ) = 0 ⇒ λ1 = 8, λ2 = 2 Second, the corresponding unit eigenvectors are found, ⎧ ⎡8 0 ⎤ ⎡ u11 ⎤ ⎡u ⎤ T = 8 ⎢ 11 ⎥ ⎡ u11 ⎤ ⎡1 ⎤ ⎪⎧ AA u1 = λ1u1 ⎪⎢ ⎢ ⎥ ⎥ ⇒ ⎨ ⎣0 2 ⎦ ⎣u12 ⎦ ⎨ ⎣u12 ⎦ ⇒ ⎢ ⎥ = ⎢ ⎥ u1 = 1 ⎣u12 ⎦ ⎣ 0 ⎦ ⎩⎪ ⎪ u112 + u122 = 1 ⎩ ⎧ ⎡8 0 ⎤ ⎡ u21 ⎤ ⎡ u21 ⎤ T = 8 ⎡ u21 ⎤ ⎡0 ⎤ ⎪⎧ AA u2 = λ2u2 ⎪⎢ ⎢ ⎥ ⎢ ⎥ ⇒ ⎨ ⎣0 2 ⎥⎦ ⎣u22 ⎦ ⎨ ⎣u22 ⎦ ⇒ ⎢ ⎥ = ⎢ ⎥ u2 = 1 ⎣u22 ⎦ ⎣1 ⎦ ⎩⎪ ⎪ 2 2 + u22 =1 u21 ⎩
Then the left matrix, U, is,
u21 ⎤ ⎡1 0 ⎤ ⎡u U = ⎢ 11 ⎥=⎢ ⎥ ⎣u12 u22 ⎦ ⎣0 1 ⎦ Similarly, by computing AT A , its eigenvalues, and the corresponding unit eigenvectors, the right matrix, V, is determined, ⎡v V = ⎢ 11 ⎣ v12
v21 ⎤ ⎡ =⎢ v22 ⎥⎦ ⎣⎢
1 2 1 2
⎤ ⎥ 1 ⎥ 2⎦
−1 2
The matrix Σ is,
⎡ λ1 Σ=⎢ ⎢⎣ 0
0 ⎤ ⎡2 2 ⎥=⎢ λ2 ⎥⎦ ⎢⎣ 0
Therefore, the SVD of A is,
192
0 ⎤ ⎥ 2 ⎥⎦
0 ⎤⎡ ⎥⎢ 2 ⎥⎦ ⎢⎣
⎡1 0 ⎤ ⎡ 2 2 A = UΣV = ⎢ ⎥⎢ ⎣ 0 1 ⎦ ⎢⎣ 0 T
1 2
⎤ ⎥ 1 2⎥ ⎦
−1 2
1 2
T
Finally, the Moore-Penrose inverse of A is, ⎡ A † = VΣ† UT = ⎢ ⎣⎢
1 2 1 2
⎤ ⎡ 42 ⎥⎢ 1 ⎥ ⎢⎣ 0 2⎦
0 ⎤ ⎡1 0 ⎤ ⎡ 14 ⎥ ⎥ = ⎢1 2 ⎢0 1 ⎣ ⎦ ⎣4 ⎥ 2 ⎦
−1 2
⎤ 1 ⎥ 2 ⎦
−1 2
Since A is full rank, A† is the same as A−1, which is,
⎡1 A −1 = ⎢ 14 ⎣4
−1 2 1 2
⎤ ⎥ ⎦
⎡2 2⎤ Example C.2: Find the Moore-Penrose inverse of matrix A = ⎢ ⎥ . Note that in this ⎣1 1 ⎦ case rank(A)=1. First, compute AAT and find its eigenvalues, ⎡8 4⎤ ⎡8 − λ ⇒ det ( AAT − λΙ ) = det ⎢ AAT = ⎢ ⎥ ⎣4 2⎦ ⎣ 4
4 ⎤ =0 2 − λ ⎥⎦
(8 − λ )( 2 − λ ) − 16 = 0 ⇒ λ1 = 10, λ2 = 0 Second, the corresponding unit eigenvectors are found, ⎧ ⎡8 4 ⎤ ⎡ u11 ⎤ ⎡u ⎤ T = 10 ⎢ 11 ⎥ ⎡ u11 ⎤ ⎡ ⎪⎧ AA u1 = λ1u1 ⎪⎢ ⎢ ⎥ ⎥ u u ⇒ ⇒ 4 2 ⎨ ⎨⎣ ⎦ ⎣ 12 ⎦ ⎣ 12 ⎦ ⎢u ⎥ = ⎢ u1 = 1 ⎣ 12 ⎦ ⎢⎣ ⎩⎪ ⎪ 2 2 u11 + u12 = 1 ⎩
⎤ ⎥ 1 5⎥ ⎦ 2 5
⎧ ⎡ 8 4 ⎤ ⎡ u21 ⎤ ⎡u ⎤ 1 T = 0 ⎢ 21 ⎥ ⎡ u21 ⎤ ⎡ 5 ⎤ ⎪⎧ AA u2 = λ2u2 ⎪⎢ ⎢ ⎥ ⎥ ⇒ ⎨ ⎣ 4 2 ⎦ ⎣u22 ⎦ ⎨ ⎣u22 ⎦ ⇒ ⎢ ⎥ = ⎢ −2 ⎥ u2 = 1 ⎣u22 ⎦ ⎣⎢ 5 ⎦⎥ ⎩⎪ ⎪ 2 2 1 u u + = 21 22 ⎩
Then the left matrix, U, is, u21 ⎤ ⎡ ⎡u U = ⎢ 11 ⎥=⎢ ⎣u12 u22 ⎦ ⎢⎣
193
2 5 1 5
⎤ ⎥ −2 5⎥ ⎦ 1 5
Similarly, by computing AT A , its eigenvalues, and the corresponding unit eigenvectors, the right matrix, V, is determined, v21 ⎤ ⎡ =⎢ v22 ⎥⎦ ⎢⎣
⎡v V = ⎢ 11 ⎣ v12
⎤ ⎥ 1 2⎥ ⎦
−1 2
1 2 1 2
The matrix Σ is,
⎡ λ1 Σ=⎢ ⎢⎣ 0
0 ⎤ ⎡ 10 0 ⎤ ⎥=⎢ ⎥ λ2 ⎥⎦ ⎣ 0 0⎦
Therefore, the SVD of A is, ⎡ A = UΣV = ⎢ ⎢⎣ T
⎤ ⎡ 10 0 ⎤ ⎡ ⎥ ⎥⎢ −2 ⎢ 5⎥ ⎦ ⎣ 0 0 ⎦ ⎣⎢
2 5
1 5
1 5
⎤ ⎥ 1 ⎥ 2⎦
−1 2
1 2 1 2
T
In SVD, since λ2=0 has no contribution in A (null mode), A is obtained from only the first column of U and the first row of V, ⎡ 25 ⎤ A = ⎢ 1 ⎥ 10 ⎡⎣ ⎣⎢ 5 ⎥⎦
1 2
1 2
⎤ ⎦
Finally, the Moore-Penrose inverse of A is obtained from the above equation by, ⎡ A =⎢ ⎢⎣ †
⎤ ⎥ 1 2⎥ ⎦ 1 2
1 10
⎡ ⎣
2 5
1 5
⎡1 ⎤ = ⎢5 1 ⎦ ⎣5
1 10 1 10
⎤ ⎥ ⎦
Since A is rank deficient, only A† is available for subsequent calculation, while A−1 is not defined. The above examples only illustrate the basic calculation techniques for a matrix Moore-Penrose inverse. A more thorough description of the SVD and Moore-Penrose inverse techniques, together with the implementation of its computation, is given in reference [47]. As illustrated in Example C.2, the Moore-Penrose inverse has certain advantages over the conventional matrix inverse. For example, if two sets of data are taken at identical load levels, then the matrix U in Equation (4.30) is rank deficient. The conventional 194
matrix inverse technique cannot indicate this singularity problem until the last step, where the matrix inverse is calculated. Compared to the conventional matrix inverse, the Moore-Penrose inverse based on the SVD can indicate the singularity problem midway by examining either the eigenvalues of U or the matrix condition number, which are produced as the interim results. If any of its eigenvalues approaches zero, or the condition number is too large, a singularity problem may occur. In this case, the online inductance estimation algorithm pauses and waits until more data are available. In addition, the Moore-Penrose inverse can be applied to a matrix U constructed from more than 2 sets of data. As a conclusion, the Moore-Penrose inverse based on the singular value decomposition is robust, efficient and can handle data collected at more than 2 operating points from the motor without compromising the accuracy in the final estimated motor inductances.
195
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VITA
Zhi Gao was born in Nanjing, China on June 24, 1977. He received his Bachelor of Engineering degree and Master of Science in Electrical Engineering degree, both from Zhejiang University, China in 1999 and 2002, respectively. In August 2002, he began his study at Georgia Institute of Technology, Atlanta, GA, where he is pursuing his doctoral study in Power Electronics, Diagnostics and Control of Electrical Machines. He worked as a Graduate Teaching Assistant in the School of Electrical and Computer Engineering in 2002, and later as a Graduate Research Assistant from 2003 to 2006. He is also a student member of the IEEE and the IEEE Power Engineering Society’s Georgia Tech Student Chapter.
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