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
iv
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.
v
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
vii
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
ix
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
x
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
ias<0, vgs=vgs,max
Rext
D
S
ias
Rds,on,min
(b) operation at negative haversine in DIM
(c) v-i characteristics
Figure 2.11: Equivalent circuit during dc injection mode.
The circuit needs to be operated in DIM for a minimum time interval (tDIM) that is
sufficient for obtaining an accurate estimate of Rs. Between two subsequent DIMs, the dc
27
injection circuit works in normal model. The length of the interval (tNM) is determined
based on how frequently an Rs estimate is required. When operating in NM between the
DIMs, there is no torque distortion in the motor and the power dissipation is significantly
lower than that under DIM. The suggested tDIM and tNM are 0.25s and 29.75s respectively
[37]. The overall circuit operates with very low power dissipation and short torque
pulsation intervals.
(a) operation in NM
(b) v-i characteristics
Figure 2.12: Equivalent circuit during normal mode.
2.3
Comparison of Different Temperature Estimation Schemes
Rules 430-32 (a) and 430-125 (b) of reference [38] stipulate that each motor shall be
protected against overload by either an integral protector or an external overload device.
Two major techniques in estimating the stator winding temperature, the thermal modelbased temperature estimation (TMTE) techniques and the parameter-based temperature
estimation (PTE) techniques, are presented in this chapter. Each of them has their own
advantages and disadvantages.
For the thermal model-based temperature estimation techniques, the stator winding
temperature is calculated from measurements of stator currents only. Therefore they are
28
non-intrusive and provide an estimated stator winding temperature within a certain
confidence level. In addition, they are robust and efficient. However, both the dualelement time-delay fuses and the thermal models with a single time constant cannot
provide an accurate temperature estimation at all times, while the complex thermal
networks need either full knowledge of the machine’s physical dimensions and
construction materials or high-precision temperature measurements from embedded
thermal sensors to perform the parameter estimation.
Parameter-based temperature estimation techniques, on the other hand, provide far
more accurate estimate of the temperature than the thermal model-based temperature
estimation techniques, since the stator resistance is a direct indicator of its temperature.
This type of method is capable of responding to all of the situations where the thermal
model parameters change due to the change in the machine operating condition or its
ambient temperature.
Since the stator temperature estimate is accurate, excessive
temperature rise due to abnormal cooling situations can be detected, which is normally
not possible using conventional TMTE method. However, as discussed in Section 2.2,
the induction machine model-based resistance estimation is highly susceptible to the
parametric errors from other measurements and parameters, while the resistance
estimation using dc injection may cause torque pulsations inside the motor and additional
power dissipation in the stator winding.
The detailed advantages and the disadvantages of the TMTE and PTE methods are
summarized below in Table 2.2.
2.4
Chapter Summary
Based on the discussion of the causes and consequences of the overload, three
methods to obtain stator winding temperature are discussed in this chapter. Due to the
scope of this research, two types of temperature estimation techniques developed so far,
29
the thermal model-based temperature estimation techniques and the parameter-based
temperature estimation techniques, remain the focus of this chapter.
Table 2.2: Comparison of different temperature estimation techniques
Parameter-based temperature
Thermal model-based temperature estimation
estimation
Single time
Complex thermal
constant
networks
Fuses
Advantages
z
z
Low cost
Maintenance free
Model-Based
z
z
State-of-the-art
z
Precise temperature
prediction
Fast and efficient
DC Injection
z
Fast and efficient
z
Fast and efficient
z
Accurate prediction of
z
Simple structure
z
Need no prior knowledge
temperature
z
Robust estimation
z
Need only stator
on machine parameters
currents
z
Rough
Disadvantages
approximation
z
Replacement
z
z
Rough approximation
z
May cause spurious
trips
Need machine
z
Susceptible to parametric
physical dimensions
errors or measurement
and construction
errors
z
Cause torque pulsation
z
Cause additional losses
materials
needed
z
Need high-precision
temperature
measurements from
the embedded
thermal sensors
As presented in Section 2.1, the thermal model-based temperature estimation
technique uses a thermal model to approximate the internal heating effects of the motor,
while the parameter-based temperature estimation technique derives the stator winding
temperature from the estimated stator resistance, discussed in Section 2.2.
The
underlying mechanisms of these two types of temperature estimation techniques are
briefly outlined. Their advantages and disadvantages are given in Section 2.3.
30
3
CHAPTER 3
INDUCTION MACHINE THERMAL ANALYSIS
As discussed in Chapter 2, microprocessor-based motor overload protection relays,
which represent 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 used in the industry. The thermal
capacitance and resistance are normally pre-determined by a set of parameters for a given
class of motors, such as full load current, service factor and trip class [22].
Once these parameters are set by plant operators or field engineers, the relay is
supposed to protect the motor utilizing the measured phase current magnitudes.
However, since these parameters are specified to protect the motor against the so-called
“transient overloads”, where inrush current ranges from 250% up to 1000% of the
motor’s rated current, the relay may not provide adequate overload protection during
“running overloads”, where the stator current is between 100% and 200% of the motor’s
rated current [6].
This inadequacy is the result of changes in the motor’s cooling
capacity. During a transient overload, such as a mechanical jam, an adiabatic internal
heating is assumed due to its large inrush current and short duration, and the temperature
inside the induction machine increases rapidly.
During a running overload, the
continuous rotation of the rotor provides a certain degree of cooling, which leads to a
gradual increase in the motor’s internal temperature. In addition, the cooling capability
can also change in service due to, for example, a broken cooling fan or a clogged motor
casing [15]. The conventional relay cannot adjust itself to yield an accurate estimate of
the stator winding temperature tailored to the specific motor’s cooling capability. The
inability of the conventional thermal model to respond to the changes in motor’s cooling
capability with updated thermal resistance may lead to an accelerated and often
31
irreversible deterioration of the stator winding insulation, and ultimately a reduced motor
life or even total motor failure [4].
In this chapter, the thermal behavior of small to medium size induction machines is
analyzed. Based on the analysis of the machine’s thermal behavior via the thermal
network with thermal resistors and thermal capacitors, a hybrid thermal model is
proposed and analyzed.
3.1
Analysis of Induction Machine Thermal Behaviors
It is assumed that the thermal behavior of the stator end winding or the rotor inside an
induction machine during running overloads can be modeled by a thermal network with
thermal resistors and thermal capacitors.
The thermal resistances and thermal
capacitances do not change over the time. The state-space representation of induction
machine thermal models is derived from the thermal network and the thermal behaviors
of the induction machine during various duty types are discussed.
3.1.1
State-space Representation of Induction Machine Thermal Models
For a typical induction machine, its thermal dynamics can be characterized by a
lumped-parameter thermal model with thermal conductors and thermal capacitors. The
state-space representation of this model is,
θ ( t ) = Aθ ( t ) + BPloss ( t )
(3.1)
y ( t ) = Cθ ( t )
(3.2)
where θ [ºC] is a vector containing nodal temperature rises above the ambient at different
locations inside a motor, including the stator winding, the rotor cage and the iron core;
Ploss [W] is a vector containing power losses, such as the stator I2R loss, the rotor I2R loss
and the core loss.
The system matrix A=−Ct-1Λt, where Ct [J/ ºC] is a diagonal matrix containing the
thermal capacitances of various components, and Λt [W/ ºC] is a symmetrical matrix of
32
thermal conductances between various parts of the motor. Since Ct is a diagonal matrix
and Λt is symmetrical, the eigenvalues of the system matrix A, denoted as λ1, λ2, …, λn,
are all real and negative [39]. The input matrix B is related to the thermal capacitance
matrix by B=Ct-1. The output matrix, C, represents the relationship between the nodal
temperature rises, θ, and the measured quantity, y.
An example of a full order thermal network is shown in Figure 3.1 [25]. Node 1, 2, 3,
4 and 5 represent stator core, stator slot winding, stator end winding, rotor cage and rotor
core, respectively. Ci [J/ ºC] is associated with the thermal capacitance at node i. Λij
[ºC/W] is the thermal conductance between nodes i and j. Pi [W] represents the power
loss at node i, and θi [ºC] is the temperature rise with respect to the motor ambient at
node i. The values of Ci and Λij are marked in Figure 3.1.
Figure 3.1: Thermal Network and Parameters [25].
The thermal network shown in Figure 3.1 is described by,
Ct θ + Λ t θ = Ploss
(3.3)
where θ=[θ1 θ2 θ3 θ4 θ5]T, Ploss=[P1 P2 P3 P4 P5]T with P5=0. The parameter matrices Ct
and Λt are,
Ct = diag [C1 C2 C3 C4 C5 ]
33
(3.4)
⎡ Λ11 + Λ12 + Λ15
⎢
−Λ12
⎢
Λt = ⎢
0
⎢
0
⎢
⎢⎣
−Λ15
−Λ12
0
0
Λ12 + Λ 23
−Λ 23
0
−Λ 23
Λ 23 + Λ 33 + Λ 34
−Λ 34
0
−Λ 34
Λ 34 + Λ 45
0
0
−Λ 45
−Λ15
⎤
⎥
0
⎥
⎥
0
⎥
−Λ 45 ⎥
Λ15 + Λ 45 ⎥⎦
(3.5)
According to Equations (3.1)-(3.2) , the state-space representation of the thermal
network is,
⎡ Λ11 + Λ12 + Λ15
⎢−
C1
⎢
⎢
Λ12
⎢
C2
⎢
⎢
0
A = −Ct−1Λ t = ⎢
⎢
⎢
0
⎢
⎢
⎢
Λ15
⎢
C5
⎣
Λ12
C1
−
Λ12 + Λ 23
C2
Λ 23
C3
−
0
0
Λ 23
C2
0
Λ 23 + Λ 33 + Λ 34
C3
0
Λ 34
C4
0
0
⎡1
B = Ct−1 = diag ⎢
⎣ C1
1
C2
1
C3
Λ 34
C3
−
Λ 34 + Λ 45
C4
Λ 45
C5
1
C4
Λ15
C1
⎤
⎥
⎥
⎥
0
⎥
⎥
⎥
0
⎥
⎥
⎥
Λ 45
⎥
C4
⎥
Λ15 + Λ 45 ⎥
−
⎥
C5
⎦
1⎤
⎥
C5 ⎦
(3.6)
(3.7)
To select temperature rise at the stator end winding (node 3) as the output, set
C=[0 0 1 0 0]T.
3.1.2 Induction Machine Thermal Behaviors under Different Duty Types
Many motor manufacturers supply motors with continuous running duty ratings,
designated as duty type S1 according to [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. According to Equations (3.1)-(3.7), the temperature rise in a
particular spot inside the motor with such duty type is described by,
θ ( t ) = ∑ θ ss ,i (1 − e −t τ ) + θ ambient
N
i
i =1
34
(3.8)
where τi=−1/λi is a time constant associated with the thermal characteristics of various
parts of the motor; θss,i is the steady-state temperature rise corresponding to τi if the
thermal equilibrium can be established inside the motor.
As indicted by Equation (3.8), all the eigenvalues of the system matrix manifest
themselves in terms of time constants in θ(t). For example, there are altogether 5 time
constants for the induction motor described by the sample thermal network of Figure 3.1:
τ1=23.91 sec, τ2=69.84 sec, τ3=85.06 sec, τ4=263.11 sec and τ5=2361.13 sec. The small
time constants correspond to fast thermal transients. They are usually associated with
localized heating inside the motor, such as a rapid temperature rise in the stator end
winding upon an increase in the motor’s load. The large time constants, on the other
hand, characterize the motor’s thermal behavior in the long run. They are generally
related to the overall heat dissipation capability of the motor.
When a motor is operated with duty type S1, both the fast and the slow thermal
transients can be observed. For a 7.5 hp TEFC test motor with parameters shown in
Table A.3, the temperature measurements captured at its stator end winding indicate that
there are two major thermal transients: a fast one with τ1=82 sec and a slow one with
τ2=1832 sec. As indicated in Figure 3.2(a), the fast thermal transient has a steady-state
temperature rise of θss,1=7.04 ºC, while the slow thermal transient has a steady-state
temperature rise of θss,2=47.79 ºC. Figure 3.2(a) also shows that the fast thermal transient
reached its steady-state value after approximately 500 sec from the instant when the
motor started.
From then on, the slow thermal transient becomes more and more
apparent as time increases.
In practice, however, 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.
equilibrium is usually not reached during the time with load.
35
Thermal
60
o
Temperature Rise ( C)
50
40
30
20
Temperature measured from the hottest thermocouple
Fast localized motor thermal transient τ1=82 sec
Slow overall motor thermal transient τ2=1832 sec
10
0
0
1000
2000
3000
4000
5000
6000
7000
Time (second)
(a) Motor operation with duty type S1
3500
100
Pin - Input power to the motor
θs - Hottest Thermocouple
Tc
90
∆tp
85
3000
2500
∆tv
2000
80
1500
75
1000
70
500
65
60
4000
Pin (W)
o
Stator Winding Temperature ( C)
95
0
4500
5000
5500
6000
6500
7000
Time (seconds)
(b) Motor operation with duty type S6
Figure 3.2: Stator winding temperature rise for the 7.5 hp TEFC motor.
36
In the case that the motor is operated with duty type S6, and the operation time at
constant load, ∆tp, is much shorter than the large time constants, the fast thermal
transients dominate the motor’s temperature response. As ∆tp increases, the fast thermal
transients reach their steady-state values, and the slow thermal transients begin to
dominate the motor’s temperature response. This trend continues until ∆tp is sufficiently
large that an overall thermal equilibrium is established inside the motor.
To illustrate the above discussion, the 7.5 hp test motor was subjected to periodic
duty cycles with a cyclic duration factor (∆tp/Tc) of 50%. Figure 3.2(b) shows the stator
end winding temperature, along with the input power, from one set of experiments with
∆tp=10 min. The result obtained by fitting the single thermal time constant to the
measured temperature is τ=239 sec.
For different values of ∆tp, the corresponding
thermal time constants are: τ=180 sec when ∆tp=5 min; τ=408 sec when ∆tp=20 min; and
τ=481 sec when ∆tp=30 min.
3.2
Hybrid Thermal Models of Induction Machines
As discussed in detailed the previous section as well as Section 2.1.3, thermal
networks, derived either from theoretical calculations based on a comprehensive
knowledge of the motor’s physical dimensions and construction materials [23]-[24], or
system identification schemes based on extensive temperature measurements at different
locations inside a motor [25]-[26], can be used to model induction machine temperature
rise from internal losses. However, for low-cost industrial applications with induction
machines up to 100 hp, the motor’s physical dimensions and construction materials are
often not readily available, and extensive temperature measurements at different locations
inside a motor require expensive installation of high-precision thermal sensors.
Therefore, a simplified thermal model, which depends only on information from voltage
and current sensors, is preferred.
37
Hybrid thermal models are proposed in this section to correlate the voltage and
current measurements to the stator winding temperatures. The model parameters are
loosely associated with aspects of machine design and provide reasonable accuracy to the
estimation of the stator winding temperature during induction machine operation with
running overload.
3.2.1 Full Order Hybrid Thermal Model
The full order hybrid thermal model is first introduced in this section along with the
explanations of its parameters. Then the heat flow in this full order hybrid thermal model
is analyzed by correlating the machine thermal parameters with the machine design, and
conclusions are made with respect to the full order hybrid thermal model.
3.2.1.1 Definition of the Full Order Hybrid Thermal Model
A hybrid thermal model with lumped thermal capacitors and resistors is presented in
Figure 3.3 to safeguard the stator winding hot spot against excessive heating during
running overload conditions. It approximates the stator and rotor thermal characteristics.
The model parameters are loosely associated with aspects of the machine design.
Figure 3.3: Full order hybrid thermal model for an induction motor.
38
The quantities, θs and θr, are temperature rises [ºC] above ambient on the stator and
rotor side, respectively. The power input, Ps [W] is associated directly with the I2R loss
generated in the stator winding. In addition, under constant supply voltage, the core loss
generated in the stator teeth and the back iron contributes a fixed portion to the rise of θs.
The power input, Pr, is associated mainly with the I2R loss in the rotor bars and end rings.
These losses are calculated from the induction machine equivalent circuit.
The thermal resistance, R1 [ºC/W], represents the heat dissipation capability of the
stator to the ambient through the combined effects of heat conduction and convection; the
thermal resistance, R2 [ºC/W], is associated with the heat dissipation capability of the
rotor to its surroundings; the thermal resistance, R3 [ºC/W], is associated with the heat
transfer from the rotor to the stator across the air gap. Since the radiation only accounts
for a small amount of energy dissipated for most induction machines, its effect can be
safely ignored without introducing significant errors in the stator winding temperature
estimation [8].
The thermal capacitances, C1 and C2 [J/ ºC], are defined to be the energy needed to
elevate the temperature by one degree Celsius for the stator and rotor, respectively. The
capacitance, C1, represents the total thermal capacity of the stator windings, iron core and
frame, while C2 represents the combined thermal capacity of the rotor conductors, rotor
core and shaft.
The thermal characteristics of a motor manifest themselves in the rotor temperature,
which can be observed from the rotor resistance via a parameter-based temperature
estimator [13],[55]. Based on the estimated rotor temperature, the parameters of the
proposed hybrid thermal model can therefore be adapted to reflect the motor’s true
cooling capability.
Compared with the conventional thermal model using a single thermal capacitor and a
single thermal resistor, the hybrid thermal model incorporates the rotor losses and the
heat transfer between the rotor and the stator. When properly tuned to a specific motor’s
39
cooling capacity, it is capable of tracking the stator winding temperature during running
overloads, and therefore provides adequate protection against overheating. Furthermore,
the hybrid thermal model is of adequate complexity compared to the complex thermal
networks [23]-[26]. This makes it suitable for online parameter tuning.
3.2.1.2 Analysis of the Full Order Hybrid Thermal Model
According to the definition of C1 and C2, a motor’s stator and rotor thermal
capacitances are fixed once the motor is manufactured. The motor’s thermal resistances,
however, are largely determined by its working environment. When a motor experiences
an impaired cooling condition, R1, R2 and R3 vary accordingly to reflect the changes in the
stator and rotor thermal characteristics.
The knowledge of machine design can aid the parameter estimation and adaptation of
R1, R2, R3, C1 and C2 in the hybrid thermal model. NEMA standard MG-1 [5] specifies
the major physical dimensions of induction machines up to 100 hp, with squirrel-cage
rotors. Analysis of these design specifications gives a typical range for the thermal
parameters. Based on this range, the online parameter tuning algorithm is then optimized
to yield superior performance.
3.2.1.2.1 Analysis of the Full Order HTM Thermal Parameters
For most induction machines up to 100 hp, C1 is usually larger than C2 due to the
design of the motor.
R1 is typically smaller than R2 because of the superior heat
dissipation capability of the stator. R3 is usually of the same order of magnitude as R1,
and it correlates the rotor temperature to the stator temperature. These relationships are
justified by a more detailed look at the machine design.
1) Relationship between C1 and C2: For most small-size induction machines with
enclosures, the frame is an integral part of the machine. The frame acts as a heat sink
with a large thermal capacitance. Therefore, even though the bulk parts of both the stator
40
and rotor are laminated silicon steel of the same axial length, the thermal capacitance, C1
is often larger than C2 due to the inclusion of the frame. In addition, the stator winding
overhang provides some extra thermal capacity to C1. Normally, C1 is 2 to 3 times larger
than C2 [41].
2) Relationship between R1 and R2: The frame of a small to medium size TEFC
machine prevents the free exchange of air between the inside and outside of the case.
Consequently, the heat transfer by means of convection from the rotor to the external
ambient is restricted at running overload conditions. Only a limited amount of heat
created by the rotor is transferred by means of conduction from the rotor bars and the end
rings centripetally through the rotor iron to the shaft, and then flows axially along the
shaft, through the bearings and finally reaches the ambient [23],[42]. In contrast, the
stator dissipates heat effectively through the combined effects of conduction and
convection. The axial ventilation through the air gap carries part of the heat away from
the stator slot winding and the stator teeth, with the balance of the heat transferred
radially through the stator iron and frame to the external ambient.
As an evidence of the good thermal conductivities of the stator iron and frame, the
temperature of the stator slot winding is usually 5-10 °C lower than that of the stator end
winding [23]-[24]. Some TEFC motors are even equipped with pin-type cooling fins on
the surface of the frame, which makes it easier for the heat to be evacuated to the ambient
[8]. In conclusion, R2 is usually 4-30 times larger than R1 at rated condition [41],[43].
3) Relationship between R1 and R3: Most TEFC induction machines in the low power
range are characterized by a small air gap (typically around 0.25-0.75 mm) to increase
efficiency. Furthermore, only a limited amount of air is exchanged between the inside
and outside of the motor due to the enclosure. Therefore, among the rotor cage losses
that are dissipated through the air gap, a significant portion is transferred to the stator by
means of laminar heat flow, while the rest is passed to the endcap air inside the motor.
These rotor cage losses, combined with the stator losses, travel through the stator frame
41
and end shield and finally reach the ambient [8],[23].
Hence, the rotor and stator
temperatures are highly correlated due to this heat flow pattern.
Reference [26] estimates that 65% of the overall rotor losses are dissipated through
the air gap at rated condition. This indicates that R3 is much smaller than R2, and is
usually of the same order of magnitude as R1. Otherwise, the rotor losses would be
dissipated mainly along the shaft instead of through the air gap.
3.2.1.2.2 Analysis of the Full Order HTM Transfer Function
Assuming Ps and Pr are the inputs and θr is the output, the state space equations that
describe the hybrid thermal model in Figure 3.3 are,
⎡ 1 ⎛1 1 ⎞
⎤
1
⎡1
− ⎜ + ⎟
⎢
⎥
⎢
R3C1
⎡θ s ⎤ ⎢ C1 ⎝ R1 R3 ⎠
⎥ ⎡θ s ⎤ + ⎢ C1
=
⎢⎥ ⎢
⎥⎢ ⎥
1
1 ⎛ 1
1 ⎞ ⎥ ⎣θ r ⎦ ⎢ 0
⎣θ r ⎦ ⎢
− ⎜ + ⎟
⎢
⎢⎣
R3C2
C2 ⎝ R2 R3 ⎠ ⎥⎦
⎣
⎡θ ⎤
y = [ 0 1] ⎢ s ⎥
⎣θ r ⎦
⎤
0⎥
P
⎥ ⎡⎢ s ⎤⎥
1 ⎥ ⎣ Pr ⎦
C2 ⎥⎦
(3.9)
(3.10)
Since the magnitude of the core loss is independent of the motor load level under
constant supply voltage, and the core loss is far less than the stator I2R loss for most
modern induction machines, it is therefore neglected in Ps for the sake of convenience in
the analysis. By taking into account only the I2R losses in the stator and rotor, Ps and Pr
are,
Ps = 3I s2 Rs
Pr = 3I s2
(3.11)
s 2ω e2 L2m
Rr
Rr2 + s 2ω e2 L2r
(3.12)
Assuming K is the ratio of Pr to Ps,
K=
s 2ω 2 L2
Pr
R
= 2 e2 m2 2 ⋅ r
Ps Rr + s ω e Lr Rs
42
(3.13)
Since the temperature increases gradually inside the motor at running overload
conditions, K remains constant if the rotor speed does not change.
By substituting Pr with KPs in Equation (3.9) and taking the Laplace transform, the
transfer function between the input, Ps, and the output, θr, in the s-domain, is,
Y (s) =
where
b1s + b0
Ps ( s )
a2 s + a1s + a0
a0=R1+R2+R3;
(3.14)
2
a1=C1R1R3+C1R1R2+C2R2R3+C2R1R2;
a2=C1C2R1R2R3;
b0=R1R2+R1R2K+R2R3K; b1=C1R1R2R3K.
When the input to the hybrid thermal model is a step signal with a magnitude c, the
output, in the time domain, is,
y ( t ) = α ⋅ u ( t ) + β1eλ1t + β 2eλ2t
where α =
b0 c
a0
; u(t) is a unit step; λ1 =
− a1 + a12 − 4 a2 a0
2 a2
(3.15)
; λ2 =
− a1 − a12 − 4 a2 a0
2 a2
; β1 =
( b1λ1 + b0 )c
a2 ( λ1 − λ2 ) λ1
;
β 2 = a( ( λ −λ ))λ .
b1λ2 + b0 c
2
2
1
2
In Equation (3.15), α represents the magnitude of the steady-state rotor temperature;
β1eλ t and β 2 eλ t correspond to rotor thermal transient.
1
2
Assuming that R1 and C1 are the bases for the thermal resistances and capacitances,
respectively, i.e. R1=1.0 p.u. and C1=1.0 p.u., then according to the previous discussion,
R2=5.0~31 p.u., R3=1.0~10 p.u., C2=0.3~0.5 p.u.. By varying the values of R2, R3 and C2
within their respective ranges, the ratios of β1 to β2 and λ1 to λ2 are obtained.
The change in |β1/β2| with respect to the changes in R2 and C2 is plotted in Figure
3.4(a), while the change in |λ1/λ2| with respect to the changes in R2 and C2 is plotted in
Figure 3.4(b). Varying R3 in conjunction with R2 or C2 yields similar results.
43
25
|β1/ β2|
20
15
10
5
30
25
0.5
20
0.45
15
0.4
10
0.35
5
R2 (p.u.)
0.3
C2 (p.u.)
(a) Change in |β1/β2| with respect to the changes in R2 and C2
0.45
|λ1/ λ2|
0.4
0.35
0.3
0.25
30
25
0.5
20
0.45
15
0.4
10
R2 (p.u.)
0.35
5
0.3
C2 (p.u.)
(b) Change in |λ1/λ2| with respect to the changes in R2 and C2
Figure 3.4: Identification of the dominant component in the rotor thermal transient.
44
As shown in Figure 3.4, |β1| is at least 6 times larger than |β2|, and |λ1| is 30-45% of
|λ2|. These results indicate that β1eλ1t corresponds to a slow exponential change with
large magnitude, while β 2 e λ2t corresponds to a fast but rather small thermal transient.
Since the latter term vanishes quickly, β1eλ1t is identified as the dominant component in
the rotor thermal transient. Similar conclusions can also be drawn for the stator thermal
transient.
80
60
o
Temperature ( C)
40
20
0
α·u(t)
-20
β1eλ t
1
-40
β2eλ t
2
-60
θr(t)
-80
0
500
1000
1500
2000
2500
3000
Time (second)
Figure 3.5: Different components in a typical rotor thermal transient.
With typical full order HTM parameters from [41], Figure 3.5 illustrates the
simulation results of the rotor temperature rise at rated condition according to Equation
(3.15). As shown in this figure, β 2 e λ2t decays to zero quickly, and the rotor thermal
transient is captured chiefly by β1eλ1t . Therefore, Equation (3.15) is simplified as,
45
y ( t ) ≈ α ⋅ u ( t ) + β1e = α ⋅ u ( t ) + β1e
λ1t
−
t
τ th
(3.16)
where τth is the thermal time constant obtained from the rotor temperature, estimated by a
parameter-based temperature estimator.
As a conclusion, the analysis of the HTM transfer function identifies β1eλ1t as the
dominant component in the rotor thermal transient. It also associates the thermal time
constant with λ1. Simulations reveal that the stator winding thermal transient has the
same thermal time constant as the rotor, which is the consequence of the strong
correlation between the rotor and stator temperatures at running overload conditions.
3.2.1.2.3 The Role of R1 in the HTM
Since a small to medium size induction motor’s internal losses are dissipated chiefly
through the stator to the ambient, the thermal resistance, R1, is therefore the most
significant factor in determining the steady-state rotor temperature and its thermal time
constant in a rotor thermal transient. Due to the strong correlation between the rotor and
stator temperatures, R1 is also the most important factor in determining the steady-state
stator temperature and its thermal time constant.
The steady-state stator winding temperature is formulated from Equation (3.9) as,
⎛
θ s ( t ) t =∞ = ⎜1 +
⎝
KR2′ − 1 ⎞
⎟ R1c
1 + R2′ + R3′ ⎠
(3.17)
where K, c and R1 have the same definition as in the previous section; R'2 is the ratio of
R2 to R1, or the per unit value of R2; similarly, R'3 is the ratio of R3 to R1, or the per unit
value of R3.
By assigning K and c typical values at the rated condition [41] and varying R'2 and R'3
within their respective ranges, the simulation results of the steady-state stator winding
temperature, θs(t)|t=∞, are plotted in Figure 3.6.
temperature at rated conditions.
46
The results are normalized to the
As shown in Figure 3.6, the per unit change in θs(t)|t=∞ due to the changes in R'2 and
R'3 is between 0.8 and 1.0. Therefore, this fairly flat surface implies the lack of variations
in θs(t)|t=∞ with respect to the changes in R2 and R3. Compared with its insensitivity to R2
and R3, θs(t)|t=∞ is in proportion to R1 according to Equation (3.17). Such a phenomenon
is explained by the fact that R1 is directly connected to the stator side in the HTM, and
that the motor’s internal losses are dissipated chiefly through R1.
1.1
θs (t)|t=∞ (p.u.)
1
0.9
0.8
0.7
40
30
10
8
20
6
10
R2 (p.u.)
4
0
2
R3 (p.u.)
Figure 3.6: Normalized steady-state stator winding temperature.
Besides the significant influence that R1 has on the steady-state stator winding
temperature, it is also one of the most important factors in determining the thermal time
constant in the stator thermal transient.
47
350
300
∆ τth (%)
250
200
150
100
50
0
400
300
400
300
200
200
100
100
0
∆ R1 (%)
0
∆ R2 (%)
(a) Variations in R1, R2 and their effects on the thermal time constant
300
250
∆ τth (%)
200
150
100
50
0
400
300
400
300
200
200
100
∆ R1 (%)
100
0
0
∆ R3 (%)
(b) Variations in R1, R3 and their effects on the thermal time constant
Figure 3.7: Effects of variations in R1, R2 and R3 on τth.
48
Figure 3.7 illustrates the variations in R1, R2 and R3 and their effects on the thermal
time constant. Both R1 and R2 are increased to 500% of their original values [41] to
emulate the change of motor’s cooling capability under impaired cooling conditions. The
relative change in the thermal time constant, which is approximated by the negative
reciprocal of λ1, is plotted in Figure 3.7(a). Similarly, Figure 3.7(b) illustrates the relative
change in the thermal time constant when R1 and R3 are increased.
Corresponding to an increase in ∆R1, marked increases in ∆τth are observed in both
Figure 3.7(a) and (b). In contrast to this, increments in R2 and R3 do not produce as
significant changes in ∆τth as R1. In conclusion, R1 is the dominant thermal parameter in
determining the steady-state stator temperature, it is also critical to the thermal time
constant in both the rotor and stator thermal transients.
3.2.2 Reduced Order Hybrid Thermal Model
As discussed in Section 3.2.1, induction machines with enclosures in the low power
range are normally shielded by frames to provide basic protection against mild hazards,
such as dust and moderate rain. Consequently, only a limited amount of air is exchanged
between the inside and outside of the motor due to the enclosure, and hence the heat
transfer by means of convection from the rotor directly to the external ambient, is
restricted. In addition, this type of machine is also characterized by a small air gap
(typically around 0.25-0.75 mm) to increase the efficiency [8]. Therefore, a significant
portion of the heat caused by the rotor cage losses, is transferred across the air gap to the
stator by means of laminar heat flow; while the remainder of the heat is passed to the
endcap air inside the motor. The heat from the rotor cage losses, combined with the heat
from the stator losses, travels through the stator frame and end shield and finally reaches
the ambient [23]. In conclusion, the stator and rotor temperatures are highly correlated
due to this heat flow pattern.
49
Since the internal heat of a small to medium size induction machine is dissipated
chiefly through the stator to the ambient due to its design, the stator thermal properties in
such a machine are the most significant factors in determining both the steady-state motor
internal temperatures and the transient state motor thermal responses, as indicated in
Section 3.2.1.2.3. The rotor thermal properties in such a machine, on the other hand, are
usually insignificant compared to their stator counterparts [46]. Consequently, the rotor
thermal properties can be replaced by a percentage of the heat flowing from the rotor to
the stator. It is commonly assumed that 65% of the heat created by the rotor cage losses
is transferred across the air gap at rated condition [26],[42].
From the above discussion, as well as the analysis presented in Section 3.2.1 for the
full order hybrid thermal model, a reduced order hybrid thermal model with lumped
thermal resistors and capacitor is proposed here for low-cost, small to medium size
induction machines (Figure 3.8). This thermal model approximates the stator and rotor
thermal characteristics, and the model parameters are loosely associated with various
aspects of the machine design. By estimating the stator winding temperature rise above
its ambient from only voltage and current measurements, the proposed thermal model
safeguards the stator winding hot spot against excessive heating during the running
overload conditions, where the stator current is between 100% and 200% of the motor’s
rated current.
50
Figure 3.8: Reduced order hybrid thermal model for an induction motor.
As outlined before, approximately 65% of the heat created by the rotor cage losses is
transferred across the air gap to the stator and ambient at rated speed. This figure may be
different at other operating points. Therefore, ∆Pr is incorporated in the reduced order
HTM to indicate the variation of the heat transferred from the rotor across the air gap to
the stator at different load levels. The voltage, V, in Figure 3.8, represents the rotor
conductor temperature rise observed from an induction machine parameter-based
temperature estimator.
The hybrid thermal model correlates the stator winding temperature with the rotor
conductor temperature by modeling their relationship with a thermal resistance, R3,
across the air gap. In addition, with the stator winding temperature estimated from the
thermal model and the rotor conductor temperature estimated from the induction machine
parameter-based temperature estimator, the hybrid thermal model also unifies the thermal
model-based and the induction machine parameter-based temperature estimators.
Furthermore, it has only three parameters, R1, R3 and C1, that need to be determined.
Simple offline tests similar to those described in reference [41] yield fairly accurate
values for R1, R3 and C1. After the successful identification of R1, R3 and C1, an optimal
estimate of the stator winding temperature can be obtained from an adaptive Kalman
filter, which is constructed atop the hybrid thermal model.
3.3
Chapter Summary
To track the stator winding temperature accurately and account for the disparities in
thermal operating conditions for different motors of the same rating, this chapter presents
a hybrid thermal model for small to medium size induction machines with enclosures
based on the analysis of induction machine thermal behaviors.
The induction machine thermal behavior is analyzed in Section 3.1. The analysis
involves representing the induction machine thermal network with state-space equations
and decomposing stator winding temperature rise collected from experimental data into
51
different components. Induction machine thermal behaviors under different duty types
are also discussed in this section.
Hybrid thermal models are proposed in Section 3.2 to correlate the voltage and
current measurements to the stator winding temperatures. The model parameters are
loosely associated with aspects of machine design and provide reasonable accuracy to the
estimation of the stator winding temperature during induction machine operation with
running overload. The hybrid thermal model utilizes the rotor resistance as an indicator
of rotor temperature, and consequently the motor’s thermal operating conditions. This
model is therefore capable of reflecting the motor’s true thermal characteristics, in an online fashion.
52
4
CHAPTER 4
INDUCTION MACHINE ONLINE PARAMETER ESTIMATION
The objective of this work is to develop and implement a fast, efficient and reliable
algorithm to estimate the stator winding temperature in an online fashion with only
voltage and current measurements from the terminals of a small to medium size mainsfed induction machine. In addition, motor cooling system condition monitoring is also
explored for complete stator winding protection. The ultimate goal of this work is to
provide a comprehensive set of algorithms on motor overload protection to the next
generation microprocessor-based protective relays.
As the first step to provide a real time tracking of the stator winding temperature, this
chapter proposes an online parameter estimation scheme for small to medium size mainsfed induction machines.
An overall architecture is proposed in Section 4.1 as a
framework to obtain online estimates of the motor parameters, such as inductance values
and rotor resistance. Each element in the framework is elaborated in the subsequent
sections in this chapter. Section 4.2 presents an induction machine electrical equivalent
circuit using complex space vectors.
Section 4.3 formulates the online inductance
estimation algorithm from the complex space vector model of the induction machine.
Analysis of the online inductance estimation algorithm is also presented in Section 4.3.
Section 4.4 derives the online rotor resistance estimation algorithm.
Section 4.5
discusses the possible causes of the errors in the estimated rotor resistance and proposes a
method to remove such estimation errors. Section 4.6 presents a method to obtain the
rotor speed information directly from the online current measurement.
53
4.1
The Overall Architecture of the Sensorless Parameter Estimation Algorithm
The overall architecture of the proposed approach to the stator winding temperature
estimation is illustrated in Figure 4.1.
Figure 4.1: Overall architecture for the induction machine sensorless parameter
estimation algorithm.
The entire algorithm takes as its input only the motor terminal voltage and current
measurements, plus the stator resistance at room temperature. The results it produces are
estimates of motor inductances and rotor resistance values. The whole algorithm is fast,
efficient and reliable due to its simple and robust structure.
4.2
Complex Space Vector Modeling of Induction Machines
The use of complex space vectors can significantly simplify the representation of an
induction machine with a uniform air gap.
The complex space vectors also carry
information on the positive and negative sequence fundamental frequency components,
as well as other frequency components. In addition, since a complex space vector is a
54
sequence of complex numbers from current or voltage samples, it is best suited for
frequency domain analysis via the discrete Fourier transform (DFT).
4.2.1 Complex Space Vector Representation of Three Phase Variables
Complex current and phase-neutral voltage space vectors can be constructed directly
from the online current and voltage measurements. Such complex space vectors contain
not only the positive sequence fundamental frequency components, but also the negative
sequence fundamental frequency components, as well as components at other
frequencies. Since the induction machine is usually represented by electrical quantities in
a synchronous reference frame, a rotational transformation is introduced in this section to
achieve such shift from one reference frame to another reference frame. Finally, the
relationship between complex space vectors and phasors is discussed, so that the complex
space vector modeling of induction machines is correlated to the phasor representation of
induction machines.
4.2.1.1 Construction of Complex Space Vectors
In a stationary reference frame, the complex current space vector of a three-phase
induction machine is defined as,
iqdss ( t ) =
D
D
2⎡
ia ( t ) + ib ( t ) ⋅ e j120 + ic ( t ) ⋅ e j 240 ⎤
⎦
3⎣
(4.1)
where ia, ib and ic are currents in phase a, b and c, respectively. The superscript s in iqdss
denotes that the complex current space vector is defined in the stationary reference frame.
The subscript, qds, indicates that this complex space vector can be further decomposed to
a q-axis component and a d-axis component, with the q-axis coincides with the direction
of phase a stator winding [35].
For an induction machine with a floating neutral point, the three-phase currents
satisfy the following relationship,
55
ia ( t ) + ib ( t ) + ic ( t ) = 0
(4.2)
In addition,
D
D
1 + e j120 + e j 240 = 0
(4.3)
Substituting Equations (4.2) and (4.3) into (4.1), the final result, expressed in a matrix
form, is,
iqdss ( t ) =
D
⎡ 2 1 ⎤ ⎡ia ( t ) ⎤
2⎡
1 e j120 ⎤ ⎢
⎦ ⎣ 1 2 ⎥⎦ ⎢ ib ( t ) ⎥
3⎣
⎣
⎦
(4.4)
Similarly, the complex phase-to-neutral voltage space vector is,
s
vqds
(t ) =
D
D
2⎡
van ( t ) + vbn ( t ) ⋅ e j120 + vcn ( t ) ⋅ e j 240 ⎤
⎣
⎦
3
(4.5)
where n designates the neutral point of the motor; van is the voltage drop between the
input terminal of the phase a winding and the neutral point, and so forth for vbn and vcn.
Normally, the neutral point of an induction machine is inaccessible, and only the line
voltages, vab, vbc and vca, can be measured. These line voltages are related to the phaseto-neutral voltages by,
⎧vab ( t ) = van ( t ) − vbn ( t )
⎪
⎨ vbc ( t ) = vbn ( t ) − vcn ( t )
⎪v (t ) = v (t ) − v (t )
cn
an
⎩ ca
(4.6)
When the inherent parameter asymmetry among the three phases of the machine is
negligible, there is no zero sequence voltage. Therefore,
van ( t ) + vbn ( t ) + vcn ( t ) = 0
(4.7)
Selecting vab and vbc as two independent variables, and substituting Equations (4.6)
and (4.7) into (4.5), yields,
s
vqds
(t ) =
D
⎡1 1⎤ ⎡ vab ( t ) ⎤
2⎡
1 e j120 ⎤ ⎢
⎥
⎥⎢
⎣
⎦
3
⎣ 0 1⎦ ⎣ vbc ( t ) ⎦
56
(4.8)
If the 3-phase currents are balanced and of a single frequency, the trajectory of the
complex current space vector, constructed from (4.4), is an ideal circle. However, in
practice, due to the presence of the negative sequence fundamental frequency component
as well as other frequency components in the 3-phase currents [44], this trajectory is no
longer an ideal circle. Figure 4.2 shows the trajectory of a complex current space vector
in the stationary reference frame. This complex current space vector is obtained from the
current measurements made at the terminals of a 5 hp open drip proof test motor
(parameters shown in the Appendix).
The trajectory of the corresponding complex
phase-to-neutral voltage space vector is also shown in the same figure.
q-axis voltage amplitude (V)
-200 -150 -100
-50
0
50
100
150
200
200
trajectory of complex
voltage space vector
20
150
100
10
50
trajectory of complex
current space vector
0
0
-50
-10
-100
-20
-150
-30
d-axis voltage amplitude (V)
d-axis current amplitude (A)
30
-200
-30
-20
-10
0
10
20
30
q-axis current amplitude (A)
Figure 4.2: Trajectories of the complex current and phase-to-neutral voltage space vectors
in a stationary reference frame.
57
4.2.1.2 Frequency Components in a Complex Space Vector
The complex current and voltage space vectors contain not only positive and negative
sequence fundamental frequency components, but also other frequency components.
When only positive and negative sequence fundamental frequency components are
present in the 3-phase stator currents, they are described by,
ia ( t ) = I1 cos (ωet + θ1 ) + I 2 cos ( −ωet + θ 2 )
ib ( t ) = I1 cos (ω e t + θ1 − 120D ) + I 2 cos ( −ω et + θ 2 − 120D )
0≤t ≤T
(4.9)
ic ( t ) = I1 cos (ω et + θ1 − 240D ) + I 2 cos ( −ω et + θ 2 − 240D )
where I1 and I2 are the amplitudes of the positive and negative sequence components,
respectively; ωe [rad/s] is the angular speed corresponding to the supply frequency.
The corresponding complex current space vector is formed by substituting (4.9) into
(4.4) and simplifying the result according to the Euler formula and the trigonometric
identities [45],
iqdss ( t ) = i1 s ( t ) + i2 s ( t ) = I1e
where i1 s ( t ) = I1e
j (ω e t +θ1 )
j (ω e t +θ1 )
and i2 s ( t ) = I 2 e
+ I 2e
j ( −ω e t +θ 2 )
j ( −ω e t +θ 2 )
(4.10)
.
Figure 4.3 illustrates this complex current space vector in a stationary reference
frame. Note that the positive direction is defined as counterclockwise.
As indicated in Figure 4.3, iqdss is the complex current space vector obtained in (4.10).
The projections of this vector onto the a, b and c axes correspond to ia, ib and ic,
respectively. In addition, iqdss is the vector sum of two components: a positive sequence
component, i1 s , that rotates counterclockwise at a frequency ωe, and a negative sequence
component, i2 s , that rotates clockwise at the same frequency.
Besides the positive and negative sequence fundamental frequency components, other
frequency components also manifest themselves in the complex space vectors. These
58
frequency components can be represented by vectors of constant amplitudes and rotate at
other frequencies, either counterclockwise or clockwise in this stationary reference frame.
Among all the frequency components, the fundamental frequency positive and negative
sequence components usually have the largest amplitudes. Figure 4.4 shows the current
and voltage harmonic spectra that are derived from the complex current and voltage
space vectors shown in Figure 4.2.
i1 s ( t )
iqdss ( t )
i2 s ( t )
Figure 4.3: Positive and negative sequence fundamental frequency components in a
complex current space vector in the stationary reference frame.
In Figure 4.4, the positive part of the spectrum corresponds to the positive sequence
components at various frequencies, while the negative part corresponds to the negative
sequence components. Therefore, the stems at 60 Hz correspond to the positive sequence
fundamental frequency current and voltage components, while the stems at −60 Hz
correspond to the negative sequence fundamental frequency components.
59
4.2.1.3 Rotational Transformation
To simplify the calculation, the induction machine equations are often expressed in a
synchronously rotating reference frame, and the complex space vectors constructed in
this reference frame are denoted by the superscript e.
The relationship between a
complex current space vector in the stationary reference frame and its counterpart in the
synchronous reference frame is [35],
Voltage Amplitude (V)
Current Amplitude (A)
iqdse ( t ) = iqdss ( t ) e− jωet
(4.11)
10
1
0.1
100
10
1
0.1
-300
-240
-180
-120
-60
0
60
120
180
240
300
Frequency (Hz)
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.
If the complex current space vector in the stationary reference frame contains only
positive and negative sequence fundamental frequency components, as described by
(4.10), then its counterpart in the synchronous reference frame is,
iqdse ( t ) = i1 e ( t ) + i2 e ( t ) = I1e jθ1 + I 2 e
j −2ω t +θ
where i1 e ( t ) = I1e jθ1 and i2 e ( t ) = I 2 e ( e 2 ) .
60
j ( −2ωe t +θ 2 )
(4.12)
Figure 4.5 illustrates such a complex current space vector in the synchronous
reference frame. In this synchronous reference frame, i1 e corresponds to the positive
sequence current component. This complex space vector has fixed position with respect
to the real and imaginary axes.
i2 e corresponds to the negative sequence current
component. This complex space vector rotates clockwise at a speed of 2ωe. The circle is
the trajectory of iqdse . In each cycle, e.g.: 16.67 ms for a 60 Hz power supply, iqdse
completes 2 revolutions along this trajectory. Similar relationship can also be found for
the complex phase-to-neutral voltage space vector.
imaginary axis
2ωe
i1 e ( t )
θ1
i2 e ( t )
O
e
qds
i
real axis
(t )
Figure 4.5: Positive and negative sequence complex current and voltage space vectors in
a synchronous reference frame.
4.2.1.4 Relationship between Complex Space Vectors and Phasors
The positive sequence components in complex space vectors are associated with the
phasor representation of an induction machine. Assuming a phasor, Is , is represented by,
Is = I1e jθ1
(4.13)
61
Then the positive sequence component in the complex current space vector, i1 s ( t ) , is
related to this phasor by,
i1 s ( t ) = Is e jωet
(4.14)
Correspondingly, the complex space vector in the synchronous reference frame is
related to the same phasor by,
i1 e ( t ) = Is
(4.15)
A similar relationship also exists between the positive sequence component in a
complex phase-neutral voltage space vector and its phasor counterpart [45].
4.2.2 Complex Space Vector Representation of Induction Machines
When complex space vectors are used to model the steady-state operation of an
induction machine, there are usually 3 types of equivalent circuit that can be used to
describe such a motor’s operation. Among them, the rotor flux based equivalent circuit is
especially useful in the subsequent derivation and analysis of the induction machine
sensorless parameter estimation scheme.
In a positive sequence synchronous reference frame, where d-axis coincides with the
rotor flux linkage, the complex space vector representation of a 3-phase symmetrical
induction machine under steady-state operation, is [35],
er
er
Vqds
= ( Rs + jω eσ Ls ) I qds
+ jω e (1 − σ ) Ls ( − jI dser )
(4.16)
2
R ⎛L ⎞
0 = r ⎜ m ⎟ ( − I qser ) + jω e (1 − σ ) Ls ( − jI dser )
s ⎝ Lr ⎠
(4.17)
where Rs is the stator resistance; Rr is the rotor resistance referred to the stator side; Lls,
Llr and Lm are the stator leakage, rotor leakage and mutual inductance, respectively; the
stator self inductance, Ls, is the sum of Lls and Lm; the rotor self inductance, Lr, is the sum
of Llr and Lm; the leakage factor, σ, is defined as, σ=1−Lm2/LsLr; ωe [rad/s] is the angular
speed corresponding to the synchronous speed; s is the per unit slip.
62
er
The complex current space vector I qds
is related to its q-axis component, I qser , and the
d-axis component, I dser , by,
er
I qds
= I qser − jI dser
(4.18)
Here, the superscript ‘r’ designates that the quantities are expressed in the rotor flux
linkage oriented reference frame.
The steady-state positive sequence equivalent circuit of an induction machine,
described by Equations (4.16)-(4.17), is shown in [35],
jω eσ Ls
er
I qds
Rs
− I qser
− jI dser
jω e (1 − σ ) Ls
er
Vqds
Rr ⎛ Lm ⎞
⎜ ⎟
s ⎝ Lr ⎠
2
Figure 4.6: Steady-state positive sequence equivalent circuit using complex vectors.
Figure 4.6 places all of the ‘leakage’ on the stator side. Therefore, the magnetizing
branch depicts the total magnetizing current producing the rotor flux.
For a mains-fed induction machine, Sections 4.2.1.3 and 4.2.1.4 have already shown
that the complex space vectors in the synchronous reference frame are closely related to
er
er
the phasors. Therefore, by replacing I qds
with Is , Vqds
with Vs , − I qser with Ir , − jI dser with
Im , Equations (4.16)-(4.18) are transformed to,
Vs = ( Rs + jωe Ls ) Is + jωe (1 − σ ) Ls Ir
(4.19)
⎡ R ⎛ L ⎞2
⎤
0 = jωe (1 − σ ) Ls I s + ⎢ r ⎜ m ⎟ + jω e (1 − σ ) Ls ⎥ Ir
⎢⎣ s ⎝ Lr ⎠
⎥⎦
(4.20)
63
The steady-state positive sequence equivalent circuit of an induction machine with
phasor representation is shown in Figure 4.7 [35].
jω eσ Ls
Is
Ir
Im
Rs
Vs
jω e (1 − σ ) Ls
Rr
s
⎛ Lm ⎞
⎜ ⎟
⎝ Lr ⎠
2
Figure 4.7: Steady-state positive sequence motor equivalent circuit using phasors.
In Figure 4.7, Vs and Is are stator per-phase input voltage and current phasors,
respectively; Ir and Im are the rotor and magnetizing current, respectively.
4.3
Online Inductance Estimation Algorithm
The inductance estimation algorithm plays a critical role in the overall rotor
temperature estimation scheme.
By calculating the temperature-independent motor
inductances, Ls and σLs, from the motor terminal voltage and current measurements,
without interrupting normal motor operations, this algorithm provides the necessary
information to the subsequent rotor resistance estimation and rotor temperature
calculation algorithm. The stator winding resistance, Rs, at ambient temperature, is the
only motor parameter assumed to be known at this stage.
4.3.1 Derivation of the Inductance Estimation Algorithm
This section first formulates the online inductance estimation algorithm from the
induction machine steady-state positive sequence fundamental frequency equivalent
64
circuit.
Then the experimental results are given to validate the proposed online
inductance estimation algorithm.
4.3.1.1 Theoretical Formulation
Starting from the equivalent circuit shown in Figure 4.7, it is useful to orient the plane
Cartesian coordinate system so that the x-axis is aligned with the current phasor Is , as
shown in Figure 4.8.
Vsy
jω eσ Ls Is
Vs
Is Rs
jω e (1 − σ ) Ls Im
I rx
Is
Ir
I ry
Vsx
Im
Figure 4.8: Phasor diagram of the equivalent circuit.
In Figure 4.8, denoting Vsx and Vsy as the projection of Vs onto the x-axis and y-axis,
respectively; and similarly Irx and Iry for Ir , Equations (4.19) and (4.20) are then
expanded to a matrix form,
⎡Vsy ⎤ ⎡ ω e Ls
⎢V ⎥ ⎢
Rs
⎢ sx ⎥ = ⎢
⎢ 0 ⎥ ⎢ω e (1 − σ ) Ls
⎢ ⎥ ⎢
0
⎣ 0 ⎦ ⎣⎢
0
ω e (1 − σ ) Ls ⎤
⎥ ⎡ Is ⎤
−ω e (1 − σ ) Ls
0
⎥ ⎢I ⎥
ω e (1 − σ ) Ls ⎥ ⎢ ry ⎥
r2
⎥ ⎢⎣ I rx ⎥⎦
r2
−ω e (1 − σ ) Ls
⎦⎥
(4.21)
First, Iry and Irx are solved from the first two rows of Equation (4.21), respectively,
65
I ry =
I rx =
Rs I s − Vsx
ωe (1 − σ ) Ls
(4.22)
Vsy − ω e Ls I s
(4.23)
ω e (1 − σ ) Ls
Then, the expression of r2 is obtained from the last row of Equation (4.21),
r2 = ω e (1 − σ ) Ls
I ry
(4.24)
I rx
Substituting Equations (4.22)-(4.23) into (4.24), yields,
r2 = ω e (1 − σ ) Ls
Rs I s − Vsx
Vsy − ω e Ls I s
(4.25)
Substituting Equations (4.22)-(4.23) and (4.25) back into the 3rd row of Equation
(4.21), after simplification, gives,
ωe I sVsy (1 + σ ) Ls − ωe2 I s2σ L2s = Vs2 + I s2 Rs2 − 2 Rs I sVsx
(4.26)
Rewriting Equation (4.26) in a compact matrix form,
⎡⎣ωe I sVsy
Defining
⎡(1 + σ ) Ls ⎤
2
2 2
−ωe2 I s2 ⎤⎦ ⎢
⎥ = Vs + I s Rs − 2 Rs I sVsx
2
σ
L
s
⎣
⎦
ξ1 = (1 + σ ) Ls
,
ξ 2 = σ L2s
,
(4.27)
u = ⎡⎣ω e I sVsy
−ω e2 I s2 ⎤⎦
T
,
ξ = [ξ1 ξ 2 ] = ⎡⎣(1 + σ ) Ls σ L2s ⎤⎦ , y = Vs2 + I s2 Rs2 − 2 Rs I sVsx , and rewriting Equation
T
T
(4.27) as,
uT ξ = y
(4.28)
The values of y and u depend on the operating point of the motor. For a given motor,
different load levels lead to different values of y and u. To estimate the inductances, ξ,
for such a motor, the voltage and current measurements at different load levels are
required.
Suppose the voltage and current measurements are taken at n load levels (n≥2), the
following vectors can be formed,
66
⎡u1T ⎤
⎢ T⎥
u
U=⎢ 2⎥,
⎢# ⎥
⎢ T⎥
⎣⎢u n ⎦⎥
⎡ y1 ⎤
⎢y ⎥
y = ⎢ 2⎥
⎢#⎥
⎢ ⎥
⎣ yn ⎦
(4.29)
where ui and yi are calculated from the voltage and current measurements obtained at the
ith load level. Therefore, Equation (4.28) now can be written as,
Uξ = y
(4.30)
With voltage and current measurements taken at more than 2 load levels, Equation
(4.30) turns out to be a set of overdetermined linear equations. This linear system can be
solved by standard techniques, such as the Moore-Penrose inverse.
By taking the Moore-Penrose inverse of U, designated as U†, the minimum least
square solution of the inductances, ξ, is obtained as [47],
ξ = U† y
(4.31)
Once the parameter vector, ξ=[ξ1 ξ2]T, is determined from Equation (4.31), the
inductances Ls and σLs are calculated by solving the following quadratic equation [48],
x 2 − ξ1 x + ξ 2 = 0
(4.32)
Denoting the roots of Equation (4.32) as x1 and x2, with x1>x2, 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; L̂ 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.
109
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
145
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)
140
140
120
100
80
60
End Winding Temperature
Slot Winding Temperature
40
20
0
1000
2000
3000
4000
Time (sec)
5000
6000
7000
Non Drive-End Temperature Measurements (° C)
120
100
80
60
End Winding Temperature
Slot Winding Temperature
40
20
0
1000
2000
3000
4000
Time (sec)
5000
6000
7000
Figure 6.14: Temperature measurements for slot and end windings (Is=150% FLC).
6.1.4.1.2 Thermocouple Monitor
147
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-
148
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
149
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.
150
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.
151
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.
152
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
153
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.
154
(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.
157
(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.
159
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.
160
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).
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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)
80
60
50
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
7000
8000
9000
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.
162
100
90
70
o
Temperature rise ( C)
80
60
50
40
30
20
θs - Measured from the stator slot winding
10
θr - Estimated from the rotor resistance
0
0
1000
2000
3000
4000
Time (second)
(b) 5 hp ODP motor, Is=12.6 A (97% FLC)
110
100
o
Temperature rise ( C)
90
80
70
60
50
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=19.1 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)
165
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 θ̂ 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.
166
(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
167
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.
169
140
o
Stator Winding Temperature ( C)
500
120
300
80
60
Ploss (W)
400
100
200
Ploss - Calculated from v, i Measurements
40
θs - Measured from the Hottest Thermocouple
θs - Estimated from the Thermal Model Alone
20
0
1000
2000
3000
4000
5000
6000
100
7000
Time (seconds)
(a) Temperature estimation by the thermal model alone
140
o
Stator Winding Temperature ( C)
500
120
80
300
60
Ploss (W)
400
100
200
Ploss - Calculated from v, i Measurements
40
θs - Measured from the Hottest Thermocouple
θs - Estimated from the PI Observer
20
0
1000
2000
3000
4000
5000
6000
100
7000
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%).
170
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].
171
140
o
Stator Winding Temperature ( C)
500
120
300
80
60
Ploss (W)
400
100
200
Ploss - Calculated from v, i Measurements
40
θs - Measured from the Hottest Thermocouple
θs - Estimated from the Thermal Model Alone
20
0
500
1000
1500
2000
2500
3000
100
3500
Time (seconds)
(a) Temperature estimation by the thermal model alone
140
o
Stator Winding Temperature ( C)
500
120
80
300
60
Ploss (W)
400
100
200
Ploss - Calculated from v, i Measurements
40
θs - Measured from the Hottest Thermocouple
θs - Estimated from the PI Observer
20
0
500
1000
1500
2000
2500
3000
100
3500
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%).
172
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.
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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.
204