ESTIMATING TEMPERATURES

AT INACCESSIBLE LOCATIONS

© RUBBERBALL PRODUCTIONS
RUCHIR SAHEBA, MARIO ROTEA, OLEG WASYNCZUK,
STEVEN PEKAREK, and BRETT JORDAN

A
virtual thermal sensor (VTS) is developed to tures can further be used with diagnostic and prognostic
estimate temperatures at various locations in an algorithms to detect failures and estimate the future capa-
electric machine. The VTS is a model-based dy- bility of the machine.
namical observer that requires only a single tem- The simplest approach to thermal monitoring is to install
perature sensor installed at an easily accessible temperature sensors, such as thermocouples, resistance tem-
location in the machine. Experimental results demonstrate perature detectors (RTDs), at the desired locations. However,
that the VTS can provide useful estimates of temperatures there are typically limitations on where these sensors can
at various locations in the machine in real time despite be placed. To address this problem, model-based approaches
modeling uncertainty and unknown initial temperatures. can be used to enable the estimation of critical temperatures
Temperature histories strongly influence an electric with a minimal number of hardware sensors, preferably in-
machine’s condition and future capability. High tempera- stalled at easily accessible locations in the machine.
tures inside an electric machine, especially the winding Methods for estimating winding temperatures include
temperatures, affect the functioning of the machine. For the parameter-based method and temperature estimation
instance, winding insulation materials suffer irreversible using a thermal model. The parameter-based method [1]
damage when thermal stresses exceed their permissible uses estimates of temperature-dependent parameters, such
limits. These temperature-dependent changes affect the as stator resistance, to estimate the winding temperature.
machine’s performance, can shorten its useful life, and Alternatively, thermal models for temperature estimation
may lead to failure. Continuous monitoring of critical [2], [3] consist of a lumped-parameter thermal network of
component temperatures inside an electric machine can thermal resistances and capacitances. A thermal network
be used to prevent thermal overloading and ensure safe model along with measurements of electrical and mechani-
operation under nominal conditions. Machine tempera- cal variables can be used to estimate the winding tempera-
tures. The thermal model may involve only a single state
Digital Object Identifier 10.1109/MCS.2009.934991 corresponding to the stator winding temperature [2] or

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several states corresponding to the temperature of various
machine components, such as frame, iron, winding, and
rotor [3]. Lumped-parameter thermal models are the natu-
ral choice when temperatures of various machine compo-
nents are desired.
Thermal monitoring schemes combining the thermal
model and the parameter-based method are described in
[2] and [4]. The basic idea is to blend these two approaches
using a Kalman filter structure, where the thermal model
forms the basis for the Kalman filter and where the input
data for the Kalman filter consist of measurements and the
estimates obtained from parameter-based methods. A
Kalman filter is developed in [4] to yield optimal winding-
temperature estimates for a permanent-magnet machine.
The states of the thermal model for the Kalman filter are
the temperatures of the winding and the aggregated stator FIGURE 1 Permanent-magnet machine with temperature sensors.
and rotor core. The input data for the Kalman filter are The machine is a wye-connected, three-phase permanent-mag-
obtained from winding-temperature estimates based on net synchronous machine whose rated power is 3.8 kW. Resis-
tance temperature detectors, with wiring shown as white wires,
the stator resistance and temperature measurements from are installed on the machine to monitor the temperatures of vari-
the motor surface. An adaptive Kalman filter is developed ous components. The temperature sensors have an accuracy of
in [2] to provide optimal stator-winding-temperature esti- 60.35 0 oC over 0–100 oC.
mates for an induction machine. The thermal model for the
Kalman filter has a single state corresponding to the stator A lumped-parameter thermal model is used to design a
winding temperature. The input data for the Kalman filter Kalman filter that estimates the temperatures of the ma-
consist of the rotor-winding-temperature estimates based chine’s components, such as rotor, stator, and windings.
on the rotor-winding resistance. The Kalman filter combines the thermal model with tem-
This article presents a VTS that estimates the temperatures perature measurements from a single temperature sensor
of various components of an electric machine using conven- installed on the machine to estimate the unmeasured com-
tional sensors, such as those for current, speed, and ambient ponent temperatures. The error covariance matrix of the
temperature, as well as a single temperature sensor installed Kalman filter [5] is used to select the best temperature sensor
on the machine. A schematic of the VTS architecture and its location. Experimental results with the Kalman filter dem-
details are discussed in “Virtual Thermal Sensor Architec- onstrate that the filter estimates converge to the measured
ture.” The VTS is based on a lumped-parameter thermal temperatures and track their variations in real time.
model of the machine and a Kalman filter that blends the esti- The methodology in this article is demonstrated using a
mates provided by the thermal model with temperature mea- 3.8-kW permanent-magnet machine. Figure 1 shows the
surements to estimate the temperatures of the unmeasured machine with the sensors, which are RTDs, installed to vali-
machine components. Although the focus is on air-cooled date the VTS. The temperature sensors have an accuracy
machines, the method of analysis and algorithm design of 60.35 oC over 0–100 oC. The machine is a wye-connect-
applies to machines with other cooling media as well. ed, three-phase permanent-magnet synchronous machine.
The lumped-parameter thermal model is based on [3]. The parameters of the machine are listed in Table 1.
This model consists of a thermal network and a loss model. This work on thermal sensing is part of an ongoing effort
The thermal network, which consists of interconnected directed at developing diagnostic and prognostic methods
thermal resistances and capacitances, estimates the tem- to assess the condition and remaining life of electrical
peratures of various machine components given the major
heat sources in the machine. The loss model calculates
values of the major heat sources in the machine based on TABLE 1 Permanent-magnet machine parameters.
the measured electrical and mechanical variables. Baseline
values for all model parameters are obtained from dimen- Rated power 3.8 kW
sional information and material properties of the machine Max current 100 A
under consideration. While the baseline model estimates Rated torque 13 N-m
the trends seen in the measurements with qualitative fidel- Rated speed 3600 rpm
ity, this model lacks quantitative accuracy. A parameter- Stator resistance 0.022 V
estimation method is used to improve the accuracy of the Stator self-inductance 0.4 mH
EMF constant 0.0112 V-s/rad
model by tuning critical model parameters, such as frame
Number of pole pairs 6
capacitance and resistance, from experimental tests.

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 Virtual Thermal Sensor Architecture
T
he main building blocks of the VTS are
the thermal model and a corrective al-
gorithm. The thermal model consists of Thermal Model
the thermal network and the loss model.
The thermal network is a physics-based
Heat Generation
lumped-parameter model that calculates Measured + + Rates
the thermal states, which are the spatially Currents Loss Thermal Thermal
Measured + Model + Measured Network States
averaged temperatures of various compo- Speed Ambient
nents of the machine. The calculation re- Temperature
quires the ambient temperature and heat Temperature
generation rates at various locations due Residual
to machine power losses such as copper, Corrective Residual Motor
Algorithm Generator Frame
iron, and friction losses. The loss model Temperature
calculates the heat generation rates based
on the measured electrical and mechanical Kalman Filter
variables of the machine, such as phase
currents and rotor speed.
Estimation of thermal states of the Virtual Thermal Sensor
machine using the thermal model re-
quires precise knowledge of initial tem- FIGURE S1 Virtual thermal sensor (VTS) architecture. The main building blocks of
peratures inside the machine. Errors in the VTS are the thermal model and the Kalman filter. The thermal model, consist-
the initial temperatures inside the ma- ing of the thermal network and loss model, calculates the machine thermal states
chine cause the thermal model estimates based on machine currents, speed, and ambient temperature. These estimates
are prone to errors when the initial temperatures inside the machine are unknown.
to be inaccurate. The VTS compensates
Hence, the thermal model estimates are combined with temperature measure-
for inaccurate initial conditions by using ments using the Kalman filter to yield estimates of thermal states despite uncer-
feedback from measurements to reduce tainty in initial temperatures.
estimation errors. Temperatures calculat-
ed from the model are compared against measured temper- network in such a manner that the residuals are minimized.
atures to generate temperature residuals. These residuals This combination of the thermal model estimates with the
are processed by a corrective algorithm, the Kalman Filter, corrective algorithm and temperature measurements con-
which adjusts the inputs to the loss model or the thermal stitutes the VTS.

systems in advanced aircraft. The criticality of electric the machine. The thermal model has two subsystems, the
machines in advanced aircraft [6] dictates that these thermal network and the loss model. The thermal network,
machines be replaced based on the number of flight hours or which is an interconnection of thermal resistances, capaci-
a similar usage metric, which is not cost effective. A solution tances, and current sources, describes the heat transfer
to this problem is to replace those machines that have lim- occurring in the machine. The thermal network estimates
ited life remaining. The remaining life of an electric motor or the component temperatures based on the power losses
generator can be estimated only if key temperatures of the occurring in the machine. The loss model calculates these
machine are known, such as the temperature of the wind- power losses based on the measured electrical currents and
ings. Hence, a VTS that delivers estimates of the machine rotor speed.
temperatures may enable the replacement of only those
machines that have limited life remaining, thereby ensuring Thermal Network
reliable and cost-effective operation. The thermal components of interest are shown in Figure 2;
the spatially averaged temperatures of these components,
THERMAL MODEL excluding the air gap and the end cap air, are referred to as
The thermal model provides estimates of the temperatures the thermal states of the machine.
of the various components of an electric machine. The inputs Each thermal component of the machine is modeled as a
to the thermal model are measurements of the ambient network node whose potential represents the spatially
temperature, electrical currents, and rotor speed, while the averaged temperature of the component. The nodes are
outputs are the temperatures of the various components of interconnected by means of thermal resistances, which

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 Thermal monitoring is fundamental to diagnostics
and prognostics of electric machines.

represent the resistance to heat transfer between the nodes. where R0 is the value of the winding resistance at ambient
The formulas for thermal resistances are obtained from temperature u 0, a is the temperature coefficient of resis-
conduction and convection heat transfer modeling of each tance, whose value is 0.00393 8C21 for copper, and u 4 and u 6
component [3]. These formulas specify the thermal resis-
tances in terms of the machine dimensions and material
properties, such as thermal conductivity, specific heat, and 2 1 3
heat transfer coefficients. Thermal capacitances are intro-
5
duced at the nodes to represent heat storage of the compo- 6
nent. Current sources are introduced at the nodes to 7
represent heat generation due to power losses occurring in 8
the component. The resulting thermal network of the
machine is shown in Figure 3. 4
9
To specify the thermal network, values of all of the ther-
mal resistances, capacitances, and current sources are
required. The values of thermal resistances and capacitances
1) Frame 2) Stator Yoke 3) Stator Teeth
can be determined from the machine dimensions, material 4) Slot Winding 5) Air Gap 6) End Winding
properties, and constant heat transfer coefficients [3]. 7) End Cap Air 8) Rotor Iron 9) Rotor Shaft
The power losses denoted by P2, P3, P4, and P6 occur in
the components corresponding to the stator yoke, stator
FIGURE 2 Division of the machine into thermal components. The
teeth, slot winding, and end winding, respectively. The
machine is divided into nine thermal components, which form the
losses are calculated using measured electrical currents basis for the thermal network. The spatially averaged tempera-
and speeds. The losses are represented with current sources tures of these components, excluding the air gap and the end cap
in the thermal network as shown in Figure 3. air, are the thermal states of the machine.

Power Loss Model

The major heat losses occurring in the permanent-magnet
machine are the copper losses and iron losses. The copper R1a
1
losses depend on the phase currents and the winding resis- R17 R12d
R27 2 R22d
tance, which changes with the winding temperature. The Ct,1 2d
iron losses, which are modeled using approximate expres- 7
P2 Ct,2 R2d4d
R37
sions for eddy-current losses and hysteresis losses, depend 4d
on the rotor speed and maximum flux density in the R44d R3d4d
R19
R67 6 R46 4 R34 3 R33d
machine. The total copper losses are distributed between 3d
the slot winding and the end winding as P4 and P6, respec- R45 P3 R53d
P6 Ct,6 Ct,3
tively. The total iron losses are divided between the stator P4 Ct,4 5
yoke and teeth as P2 and P3, respectively.
R78 R88d R58d
The total copper loss PCu is calculated using the instan- 8
8d
taneous values of the phase currents ia, ib, and ic as well as Ct,9
Ct,8 R98d
the per-phase winding resistance R as 9

PCu 5 ( i2a 1 i2b 1 i2c ) R W. (1) FIGURE 3 Thermal network of the permanent-magnet machine.
The potential at each colored circled node represents the spatially
The per-phase winding resistance R is assumed to be a averaged temperature of the corresponding machine component.
function of the average temperature of the winding The nodes are interconnected by means of thermal resistances,
which represent the resistance to heat transfer between the
expressed by
nodes. Thermal capacitances are introduced at the nodes to rep-
resent heat storage of the component. Current sources are intro-
u4 1 u6
R 5 R0 c 1 1 a d, (2) duced at the nodes to model heat generation due to losses
2 occurring in the component.

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 This article presents a virtual thermal sensor that estimates the temperatures
of various components of an electric machine using conventional sensors as
well as a single temperature sensor installed on the machine.

are the temperatures of the slot winding and the end wind- State-Space Representation” for details regarding the deri-
ing, respectively, relative to the ambient temperature. vation of these equations. These equations are reduced to
The total copper loss is divided between the slot wind- seven ordinary differential equations, which are repre-
ing and end winding according to the volume of copper in sented in state space as
each component. Letting h represent the ratio of the volume
dx
of copper in the slot winding to the sum of the volume of C 5 Ax 1 Bu, (9)
dt
copper in the slot winding and end winding, the current
sources P4 and P6 introduced at nodes 4 and 6 of Figure 3 where x 5 3 u 1 u 2 u 3 u 4 u 6 u 8 u 9 4 T represents the vector of
can be obtained as thermal states and u i denotes the relative temperature of
node i with respect to the ambient temperature u 0. The
P4 5 hPCu W, (3) vector u is of the form
P6 5 ( 1 2 h ) PCu W. (4)
u5 c d.
PFe
(10)
The total iron loss PFe is calculated as the sum of the PCu
eddy current loss Pe and the hysteresis loss Ph [7], which
are given by The coefficient matrices A, B, and C, which depend
on the thermal resistances and capacitances shown in
Pe 5 KeB2v 2s W/m3, (5) Figure 3, are specified in “Thermal Model State-Space Rep-
b 3 resentation.” The vector u is a function of the thermal
Ph 5 KhB v s W/m , (6)
states since the copper losses depend on the winding
where B is the peak flux density in the machine, v s is the temperature. The output equation for the measured tem-
stator angular frequency, and b, Ke, and Kh are constants peratures is given by
that depend on the iron material used in the machine [7].
The total iron loss is divided between the stator yoke y 5 Cx, (11)
and teeth according to the volume of iron in each compo-
nent. Letting g represent the ratio of the volume of the where C is a zero-one matrix that represents the compo-
stator yoke to the sum of the volumes of the stator yoke and nents being measured.
teeth, the current sources P2 and P3 introduced at nodes 2 The vector u given by (10) is a function of the state
and 3 of Figure 3 can be obtained as vector x. To derive the Kalman filter, it is necessary to
make this dependence explicit. From (1) and (2) u can be
P2 5 gPFe W, (7) rewritten as
P3 5 ( 1 2 g ) PFe W. (8)
u 5 u0 1 Dx, (12)
With values of thermal resistances and capacitances
obtained from formulas based on machine dimensions where
and material properties, along with values of current
u0 5 c d
PFe
sources obtained from the loss model based on the mea- (13)
( i2a 1 i2b 1 i2c ) R0
sured electrical currents and rotor speed, the thermal net-
work can now be fully determined as explained in the and
next section.
0 0 0 0 0 0 0
STATE-SPACE REPRESENTATION D5 £ ( i2a 1 i2b 1 i2c ) R0a ( i2a 1 i2b 1 i2c ) R0a §
.
OF THE THERMAL MODEL 0 0 0
2 2
0 0
The equations describing the thermal states of the model
can be obtained by applying Kirchoff’s current law to each (14)
node of the network in Figure 3. See “Thermal Model Substituting u in (9), we obtain

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 dx
5 Ax 1 Bu0, (15) TABLE 2 Numerical values of thermal resistances.
dt

Baseline value Improved value

where the coefficient matrices are given by
Parameter (°C/W) (°C/W)
R1a 0.2000 0.1884
A 5 C 21 ( A 1 B D ) , (16)
R17 0.9629 0.9629
B 5 C 21B. (17) R27 14.7422 14.7289
R19 0.8040 0.8040
Equation (15) represents the thermal model of the
R12d 0.0115 0.0096
machine, which is time varying due to the presence of the
R22d –0.0039 –0.0033
instantaneous phase currents in the matrix A. The thermal
model can be solved to obtain estimates of the thermal states R37 80.6865 80.6796
of the machine. These estimates represent the temperature R2d4d 0.0120 0.0100
rise of the components above the ambient temperature u 0. R78 2.5841 2.5810
Adding the ambient temperature to the solution of (15) R67 0.6032 0.6380
yields the temperature of the components. Hence, a tem- R44d 0.1403 0.2148
perature sensor for measuring the ambient temperature is R34 0.0864 0.1345
required to implement the thermal model. R33d –0.0264 –0.0219
R3d4d 0.0734 0.0611
BASELINE MODEL R53d 1.1017 1.0872
Baseline values of all model parameters can be obtained R46 0.0017 0.0244
from machine dimensions, material properties, and con- R88d –0.0898 –0.0747
stant heat transfer coefficients. The thermal model based R58d 0.8258 0.7930
on the baseline values of model parameters is referred to R45 1.7653 1.839
as the baseline model. The numerical values of the ther- R98d 1.3497 1.2655
mal resistances and capacitances of the baseline model
are shown in tables 2 and 3, respectively.
Experiments are conducted by operating the motor at
the constant rotor speed of 800 rev/min with sinusoidal
TABLE 3 Numerical values of thermal capacitances.
phase currents whose peak values are between 40 A and
65 A. The measured data consist of temperature measure-
Baseline value Improved value
ments of five components, namely, frame, stator yoke, Parameter (°C/W) (°C/W)
stator teeth, slot winding, and end winding. The tempera- 1797.8 4478
Ct,1
tures of the rotating components, namely, shaft and rotor,
Ct,2 199.5570 199.5570
are not measured. To compare the thermal model esti-
Ct,3 383.9875 383.9875
mates with the measured data, the matrix C in (11) is
Ct,4 805.8445 805.8445
chosen as
Ct,6 445.8064 445.8064
Ct,8 849.5137 849.5137
1 0 0 0 0 0 0 Ct,9 128.9220 128.9220
0 1 0 0 0 0 0
C 5 E0 0 1 0 0 0 0U.
0 0 0 1 0 0 0
component. Some mismatch between the time constants of
0 0 0 0 1 0 0
the model and the physical system exists. The maximum
discrepancy between the model estimates and the mea-
Figure 4 compares the model estimates (dotted) and sured data is approximately 5 oC.
measured data (solid) for sinusoidal phase currents with Comparison between the model estimates (dotted) and
peak value 50 A for 2700 s. The measured data show that the measured data (solid) are shown in Figure 5 for sinusoidal
highest temperatures occur in the end winding with slightly phase current with amplitudes of 45 A for the first 870 s,
lower temperatures in the slot winding. The lowest temper- 55 A for the next 850 s, and 45 A for the last 480 s. The
ature occurs at the frame. The temperatures of the compo- response of the thermal model to a change in heat input,
nents rise slowly with time constants of at least 1500 s. which is due to the change in phase current amplitudes, is
The model estimates follow the same trend in compo- faster than the physical system. The maximum discrepancy
nent temperatures as seen in the data, where the end wind- between the model estimates and the measured data is
ing is the hottest component and the frame is the coolest approximately 5 oC.

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 Thermal Model State-Space Representation
T u5 u9 u8
he equations describing the thermal states of the machine
0 5 u 8d c d2
1 1 1
1 1 2 2 . (S15)
can be obtained by applying Kirchoff’s current law to each R58d R98d R88d R58d R98d R88d
node of the network in Figure 3. Applying Kirchoff’s current law
The current sources P2, P3, P4, and P6 are from (7), (8), (3),
to node 1, we have
and (4), respectively. Using the definition of u from (10), (S3)–
d u 1 u 1 2 u 0 u 1 2 u 7 u 1 2 u 9 u 1 2 u 2d (S9) can be represented by
Ct,1 1 1 1 1 5 0, (S1)
dt R1a R17 R19 R12d
dx
where u i is the absolute temperature of node i. The absolute C 5 Y11x 1 Y12z 1 Bu, (S16)
dt
temperature of a node u i can be expressed as
while (S10)–(S15) can be represented by
u i 5 u i 1 u 0, (S2)
0 5 Y21x 1 Y22z, (S17)
where u i is the relative temperature of the node with respect
to the ambient temperature u 0. Using the relative temperatures where x 5 3 u 1 u 2 u 3 u 4 u 6 u 8 u 9 4 T, z 5 3 u 2d u 3d u 4d u 5 u 7 u 8d 4 T,
for all nodes and rearranging, (S1) becomes the matrix C is given by
d u1 u7 u9 u 2d
52 u 1 c d1
1 1 1 1
Ct,1 1 1 1 1 1 .
dt R12d R17 R19 R1a R17 R19 R12d
(S3) Ct,1 0 0 0 0 0 0
0 Ct,2 0 0 0 0 0
Similarly, applying the Kirchoff’s current law to all the nodes of
0 0 Ct,3 0 0 0 0
the thermal network and rearranging, we obtain
C5G 0 0 0 Ct,4 0 0 0 W, (S18)
d u2 u7 u 2d 0 0 0 0 Ct,6 0 0
5 2u 2 c d1
1 1
Ct,2 1 1 1 P2, (S4)
dt R27 R22d R27 R22d 0 0 0 0 0 Ct,8 0
0 0 0 0 0 0 Ct,9
d u3 u7 u4 u 3d
5 2u 3 c d1
1 1 1
Ct,3 1 1 1 1 1 P3,
dt R37 R34 R33d R37 R34 R33d
(S5)
d u4 and the matrix B is given by
5 2u 4 c d
1 1 1 1
Ct,4 1 1 1
dt R46 R44d R34 R45
0 0
u6 u 4d u3 u5
1 1 1 1 1 P4, (S6) g 0
R46 R44d R34 R45
g21 0
d u6 u7 u4 B5G 0 h W. (S19)
5 2u 6 c d1
1 1
Ct,6 1 1 1 P6, (S7) 0 h21
dt R67 R46 R67 R46
0 0
d u8 u7 u 8d
5 2u 8 c d1
1 1 0 0
Ct,8 1 1 , (S8)
dt R78 R88d R78 R88d

d u9 u1 u 8d The matrices Y12, Y11, Y22, and Y21 are given by

5 2u 9 c d1
1 1
Ct,9 1 1 , (S9)
dt R19 R98d R19 R98d

u1 u2 u 4d
05 u 2d c d2
1 1 1
1 1 2 2 , (S10) 1 1
R12d R22d R2d4d R12d R22d R2d4d 0 0 0 0
R12d R17
u 4d u3 u5
05u 3d c d2
1 1 1 1 1
1 1 2 2 , (S11) 0 0 0 0
R3d4d R33d R53d R3d4d R33d R53d R22d R27
1 1
u 3d u4 u 2d
05u 4d c d2
1 1 1 0 0 0 0
1 1 2 2 , (S12) R33d R37
R3d4d R44d R2d4d R3d4d R44d R2d4d
Y12 5 I 0 0
1 1
R44d R45
0 0 Y, (S20)
u 3d u4 u 8d
05u 5 c d2
1 1 1
1 1 2 2 , (S13) 1
R53d R45 R58d R53d R45 R58d 0 0 0 0 0
R67
05u 7 c d
1 1 1 1 1 1 1
1 1 1 1 0 0 0 0
R17 R27 R37 R67 R78 R78 R88d
u1 u2 u3 u6 u8 1
2 2 2 2 2 , (S14) 0 0 0 0 0
R17 R27 R37 R67 R78 R98d

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 1 1 1 1 1
2c 1 1 1 d 0 0 0 0 0
R1a R17 R19 R12d R19
1 1
0 2c 1 d 0 0 0 0 0
R27 R22d
1 1 1 1
0 0 2c 1 1 d 0 0 0
R33d R34 R37 R34
1 1 1 1 1
Y11 5 I 0 0 2c 1 1 1 d 0 0 0 , (S21)
R34 R46 R34 R45 R44d Y
1
0 0 0 0 0 0
R46
1 1
0 0 0 0 0 2c 1 d 0
R88d R78
1 1 1
0 0 0 0 0 2c 1 d
R19 R19 R98d

1 1 1 1
c 1 1 d 0 2 0 0 0
R2d4d R22d R12d R2d4d
1 1 1 1 1
0 c 1 1 d 2 2 0 0
R3d4d R33d R53d R3d4d R53d
1 1 1 1 1
2 2 c 1 1 d 0 0 0
R2d4d R3d4d R2d4d R3d4d R44d
Y22 5 I Y, (S22)
1 1 1 1 1
0 2 0 c 1 1 d 0 2

FEBRUARY 2010
R53d R45 R58d R53d R58d

«
1 1 1 1 1
0 0 0 0 c 1 1 1 1 d 0
R17 R27 R37 R67 R78
1 1 1 1
0 0 0 2 0 c 1 1 d
R58d R58d R88d R98d

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IEEE CONTROL SYSTEMS MAGAZINE 49
 1 1
2 2 0 0 0 0 0
R12d R22d
1
0 0 2 0 0 0 0
R33d
1
0 0 0 2 0 0 0
R44d
Y21 5 I Y. (S23)
1
0 0 0 2 0 0 0
R45
1 1 1 1 1
2 2 2 0 2 2 0
R17 R27 R37 R67 R78
1 1
0 0 0 0 0 2 2
R88d R98d

Expressing z in terms of x from (S17) and substituting into (S24) becomes

(S16), we obtain

5 Y11x 1 Y12 5 2Y22 Y21x 6 1 Bu.

dx dx
C 21
(S24) C 5 Ax 1 Bu, (S26)
dt dt
Defining the matrix A by
which is the same as (9). Equation (S26) is the desired state-
21
A 5 Y11 2 Y12Y 22 Y21, (S25) space representation of the thermal model of the machine.

MODEL TUNING
50 Figures 4 and 5 show that the baseline model is consistent
with the physical system in estimating the temperatures of
45 the machine components. However, quantitative discrep-
ancies exist between the model estimates and the mea-
Temperature (°C)

40 sured data. The time constants of the model and the

physical system have some disparity as well. We now use
35 a parameter-estimation technique to refine some key
model parameters to improve the estimates provided by
30
the thermal model.
25
To reduce the gap between model estimates and test
data, key model parameters are tuned using system identi-
20 fication tools. The Matlab System Identification Toolbox [8]
0 500 1000 1500 2000 2500 is used to tune these parameters.
Time (s) The tuned model parameters are the frame-to-ambient
thermal resistance R1a, the frame thermal capacitance Ct,1,
Measured End Winding Estimated End Winding
the equivalent winding thermal conductivity kc, the lamina-
Measured Slot Winding Estimated Slot Winding tion thermal conductivity kl, and the varnish thermal con-
Measured Teeth Estimated Teeth ductivity kv. The equivalent winding thermal conductivity is
Measured Yoke Estimated Yoke the thermal conductivity of windings taking into account the
Measured Frame Estimated Frame aggregate effects of the winding insulation, copper conduc-
tors, and end-winding curvature. The thermal model is most
FIGURE 4 Thermal states of the baseline model for 50-A phase
sensitive to the frame-to-ambient thermal resistance R1a and
currents. The dotted curves denote baseline model estimates, hence this parameter is tuned. Comparisons of the baseline
whereas the solid curves denote test data. The baseline model model with measured data in the previous section reveal that
estimates follow the same trend in component temperatures as the measured thermal states rise slowly compared to the
seen in the data, where the end winding is the hottest component model estimates, especially the frame temperatures. This
and the frame is the coolest component. The maximum discrep-
ancy between the model estimates and the measured data is
comparison indicates that the frame thermal capacitance Ct,1
approximately 5 oC. This discrepancy is due to mismatch between is underestimated. One reason for the underestimation of
the time constants of the model and the physical system. the frame thermal capacitance is that the motor mounting

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 The main building blocks of the virtual thermal sensor are the thermal model
and a corrective algorithm based on a Kalman filter.

thermal capacitance, which must be added to the frame ther-

mal capacitance, is neglected in the baseline model. 50
45 A 55 A 45 A
Comparison of the baseline model with the data
shows that the differences between the end winding 45
and slot winding temperatures for the physical system

Temperature (°C)
40
are larger than that estimated by the baseline model.
This mismatch can be improved by tuning the equiva-
35
lent winding thermal conductivity because the thermal
resistance R46 between the nodes of the thermal network 30
corresponding to the slot winding and the end winding
is inversely proportional to the equivalent winding ther- 25
mal conductivity kc. The varnish thermal conductivity
kv appears in the thermal resistances connected to the 20

0
0

0
00

00

00

00

00

00

00
nodes corresponding to the slot winding and the end

20

40

60

80
10

12

14

16

18

20

22
winding. By tuning the equivalent winding thermal Time (s)
conductivity and the varnish thermal conductivity, the
Measured End Winding Estimated End Winding
goal is to obtain improved estimates of the winding Measured Slot Winding Estimated Slot Winding
temperatures. Model performance can be further im- Measured Teeth Estimated Teeth
proved by tuning the lamination thermal conductivity Measured Yoke Estimated Yoke
kl since it appears in several thermal resistances of the Measured Frame Estimated Frame
thermal network.
The parameter-estimation algorithm used for tuning FIGURE 5 Thermal states of the baseline model for 45-55-45-A
the key model parameters is the prediction-error minimi- phase currents. The dotted curves denote baseline model esti-
zation (pem) method of the Matlab System Identification mates, whereas the solid curves denote test data. The response
of the thermal model to a change in heat input, which is due to the
Toolbox. This method minimizes a quadratic prediction-
change in phase-current amplitudes, is faster than the physical
error criterion, which is a function of the difference between system. The maximum discrepancy between the model estimates
the measured output and the output estimated by the and the measured data is approximately 5 oC.
model, using an iterative search algorithm to obtain opti-
mal values of the parameters.
To obtain an improved thermal model, the parameter-
TABLE 4 Comparison of the baseline and improved parameter
estimation algorithm is used with the input-output data values. The parameters that have a large change in their
corresponding to 50-A phase currents shown in Figure 4. values after tuning are the frame thermal capacitance Ct,1
The improved thermal model is then validated using input- and the equivalent winding thermal conductivity kc. The
output data corresponding to Figure 5. The numerical large value of the frame capacitance is due to the slow rise
of measured frame temperature in comparison with the
values of the thermal resistances and capacitances of the
frame temperature estimated by the baseline model. The
improved model are shown in tables 2 and 3, respectively. thermal resistance between the slot winding and the end
The numerical values of the tuned model parameters are winding is inversely proportional to kc. The baseline value
shown in Table 4. The parameters that have large changes of kc makes the simulated thermal states of the end winding
after system identification are the frame thermal capaci- and the slot winding within 0.2 8C of each other. However,
test data indicate differences of 1–2 8C between the two
tance Ct,1 and the equivalent winding thermal conductivity
thermal states, suggesting a smaller value of the equivalent
kc. The large value of the frame capacitance is due to the winding thermal conductivity kc.
slow rise of the measured frame temperature in compari-
son with the frame temperature estimated by the baseline Parameter Baseline values Improved values
model. The thermal resistance between the slot winding R1a ( °C/W ) 0.2 0.1884
and the end winding is inversely proportional to the equiv- Ct,1 ( J/°C ) 1797.8 4478
alent winding thermal conductivity kc. Hence, a large ini- k1 (W/m-°C ) 15 18.0185
kc (W/m-°C ) 385 27.0148
tial estimate of the equivalent winding thermal conductivity
kv (W/m-°C ) 0.44 0.2874
makes the simulated thermal states of the two components

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 High temperatures inside an electric machine, especially the winding
temperatures, affect the functioning of the machine.

within 0.2 8C of each other. However, the test data indicate installed on the machine, while the sensor location is opti-
differences of 1–2 8C between the two thermal states, sug- mally chosen to minimize the estimation errors.
gesting a smaller value of the equivalent winding thermal
conductivity kc. Kalman Filter
Comparison of the improved model estimates (dotted) A Kalman filter [9] is an optimal state estimator that uses
with the measured data from Figure 5 (solid) is shown in knowledge of the model, measurements of the output, and
Figure 6. The estimation errors for all states are less than statistical properties of the process disturbances and sensor
3 8C. Figures 5 and 6 show that the time constants of the noise to estimate the states.
improved thermal model are in better agreement with the A discrete-time model for the Kalman filter is
data compared to the time constants of the baseline model. obtained by discretizing (11) and (15) and adding sto-
The improved model forms the basis for the VTS described chastic inputs. The result is the linear stochastic differ-
in the next section. ence equation

VIRTUAL THERMAL SENSOR x ( n 1 1 ) 5 AD x ( n ) 1 BDu ( n ) 1 w ( n ) , (18)

The VTS combines the thermal model estimates with tem-
perature measurements from a sensor using a Kalman filter with measured temperatures
structure and provides estimates of all the thermal states
despite modeling uncertainty and unknown initial ther- y ( n ) 5 Cx ( n ) 1 v ( n ) , (19)
mal states. The VTS uses only one temperature sensor
where the coefficient matrices are

AD 5 exp ( At ) , (20)
50 t
45 A 55 A 45 A
BD 5 3 exp ( As ) dsB, (21)
45
0
Temperature (°C)

40
A and B are given by (16), (17), and t is the sampling time.
35
Process noise w ( n ) is added to take into account modeling
uncertainty, and measurement noise v ( n ) is added to take
30 into account measurement error. The process and measure-
ment noises are assumed to be independent white noise
25 characterized by

20 E 3 w ( n ) 4 5 0, (22)
0
0

0
00

00

00

00

00

00

00

E 3 w ( n ) w ( m ) 4 5 Spdnm,
20

40

60

80
10

12

14

16

18

20

22

T
(23)
E 3 v ( n ) 4 5 0,
Time (s)
(24)
E 3 v ( n ) v ( m ) T 4 5 Smdnm.
Measured End Winding Estimated End Winding
Measured Slot Winding Estimated Slot Winding (25)
Measured Teeth Estimated Teeth
Measured Yoke Estimated Yoke The nonnegative-definite matrices Sp and Sm are the
Measured Frame Estimated Frame process noise covariance and measurement noise covari-
ance matrices, respectively, and dnm is the Kronecker
FIGURE 6 Thermal states of the improved model for 45-55-45-A delta function.
phase currents. The dotted curves denote improved model esti- The Kalman filter generates two estimates of the thermal
mates, whereas the solid curves denote test data. The errors for
state x ( n ) in (18), namely, x ( n ) , the a priori state estimate at
all states estimated by the improved model are less than 3 oC.
Figures 5 and 6 show that the time constants of the improved ther- step n given the measurements up to time n 2 1, and x^ ( n ) ,
mal model are in better agreement with the data compared to the the a posteriori state estimate at time n after the measure-
time constants of the baseline model. ment at time n arrives. The a priori and a posteriori

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 Methods for estimating winding temperatures include the parameter-based
method and temperature estimation using a thermal model.

estimation errors associated with the Kalman filter esti- K ( n ) 5 S ( n ) CT 3 CS ( n ) CT 1 Sm 4 21. (34)
mates are defined by

e(n) 5 x(n) 2 x(n), (26) Equations (30)–(34) are solved recursively at each time step
to obtain the optimal estimates of the thermal states.
and
Sensor Location
e ( n ) 5 x ( n ) 2 x^ ( n ) , (27) We consider a single temperature sensor installed on the
machine to provide the measurement y ( n ) in (19). An
respectively. From (26) and (27), the uncertainty associated additional sensor is used to measure the ambient tempera-
with the Kalman filter estimates can be quantified using ture. To guide the selection of the sensor location, we ana-
the covariance matrices lyze the Kalman filter estimation errors.
The mean square error (MSE) of estimates of the states
S ( n ) 5 E 3e ( n ) e ( n ) T 4 , (28) of interest can be used to evaluate the loss of accuracy
S ( n ) 5 E 3e ( n ) e ( n ) 4 ,
T
(29) associated with a particular sensor location. The MSE is
defined as
where S ( n ) is the a priori estimation error covariance

a E 3 e ( n ) W ( n ) e ( n )4 ,
matrix and S ( n ) is the a posteriori estimation error covari- 1 N T
MSE 5 (35)
ance matrix. N n51
The Kalman filter equations can be categorized into two
groups of equations, time update and measurement update. where e ( n ) is the a posteriori estimation error defined in
The time update equations provide the a priori estimates (27), N is the estimation window length, and W ( n ) is a
of the state and estimation error covariance matrix at time weighting matrix that can be used to select the states of
n given measurements up to time n 2 1. The a priori esti- interest. Sensor locations that correspond to small MSE are
mates are obtained by projecting forward the estimates at preferred.
time n 2 1 based on the model of the system. These esti- The MSE can be computed as [9]
mates are given by

a trace 3 W ( n ) S ( n )4 ,
1 N
MSE 5 (36)
x ( n ) 5 AD x^ ( n 2 1 ) 1 BDu ( n 2 1 ) (30) N n51

and where S ( n ) is the a posteriori error covariance defined in

(29). Note that S ( n ) can be obtained from the Riccati dif-
S ( n ) 5 ADS ( n 2 1 ) ATD 1 Sp. (31) ference equation [9]

The measurement update equations provide the a poste- S ( n 1 1 ) 5 ADS ( n ) ATD 1 Sp

riori estimates of the state and estimation error covariance 2 ADS ( n ) CT 3 CS ( n ) CT 1 Sm 4 21CS ( n ) ATD. (37)
matrix at time n after the measurement at time n arrives.
The a posteriori estimates are obtained by correcting the a Thus, (36) and (37) can be used to calculate the MSE. It is
priori estimates with the new measurement that arrives at evident from (20), (16), and (14) that AD in (37) depends
time n. These estimates are given by on the phase currents. Hence, (37) is solved with three
values of AD, where each value is in effect over one third
x^ ( n ) 5 x ( n ) 1 K ( n ) 3 y ( n ) 2 Cx ( n )4 (32) of the data length N. The three values of AD are obtained
by using the minimum, the maximum, and an interme-
and diate value of the phase currents. To compare the MSE
values for various sensor locations, the thermal model
S ( n ) 5 3 I 2 K ( n ) C 4S ( n ) , (33) needs to be normalized. The final values of the thermal
states obtained from Figure 4 are used to normalize
where K ( n ) is the Kalman filter gain matrix given by the model.

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 A lumped-parameter thermal model is used to design a Kalman filter
that estimates the temperatures of the machine’s components,
such as rotor, stator, and windings.

The values of MSE considering all states are given in W ( n ) is selected to be the 7 3 7 diagonal matrix whose
Table 5. To obtain the MSE for all states, W ( n ) is chosen fourth and fifth diagonal entries are set to one and whose
to be the 7 3 7 identity matrix. The results in Table 5 sug- remaining entries are zero. The results in Table 6 suggest
gest that when estimates of all the thermal states are that when estimates of only the winding temperatures are
desired, placing the sensor in either the frame, yoke, or desired, the slot winding or the end winding is the best
teeth is the best option, while placing the sensor in the location for the sensor. This result is expected because the
slot winding or end winding yields less accurate temper- best option for obtaining accurate estimates of a component
ature estimates. is to measure that component. Also, the estimates of the
The values of MSE considering only the slot-winding winding temperatures are more accurate for sensor loca-
and end-winding states are given in Table 6. In this case, tions closer to the winding.
The most feasible sensor location is the frame due to its
accessibility. The frame location is also optimal when the
TABLE 5 Mean square error (MSE) considering all of the
thermal states. The MSE values represent the mean square entire thermal state is desired. When only the winding
estimation error of all of the thermal states given that the temperatures are desired, the slot winding or the end wind-
sensor is placed at a particular location. For example, the ing is the best location.
MSE for all states is 0.0014 when the sensor is placed on
the frame. The sensor location corresponding to the smallest
MSE is preferable since it results in more accurate estimates
EXPERIMENTAL RESULTS
of the thermal states. Thus, when estimates of all of the In this section, a Kalman filter is designed using the
thermal states are desired, placing the sensor in either the improved thermal model and temperature measurements
frame, yoke, or teeth is the best option. from the frame. The Kalman filter is executed in real time,
and its performance is demonstrated by comparing the
Nondimensional estimated temperatures with the temperature measure-
Sensor Location Mean Square Error ments. Real-time ambient temperature measurements are
Frame 0.0014
needed in addition to the frame measurements to execute
Yoke 0.0014
Teeth 0.0014 the Kalman filter.
Slot winding 0.0018 The measurement used in the Kalman filter is the frame
End winding 0.0019 temperature. Measuring frame temperature corresponds to
the matrix C in (19) given by

TABLE 6 Mean square error (MSE) considering slot-winding C 5 3 1 0 0 0 0 0 0 4.

and end-winding thermal states. The MSE values represent The measurement noise covariance matrix Sm is estimated
the mean square estimation error of the end winding and the
slot winding given that the sensor is placed at a particular from the measurements. We select the process noise covari-
location. For example, the MSE for the slot winding and end ance matrix Sp that achieves the desired tracking speed for
winding is 0.00003 when the sensor is placed on the slot the Kalman filter, by trial and error. The values of the cova-
winding. The sensor location corresponding to the smallest riance matrices are given by
MSE is preferable since it results in more accurate estimates
of the thermal states. Thus, when estimates of only the end-
winding and slot-winding states are desired, placing the m 5 0.0557 °C,
S1/2 (38)
sensor in the slot winding is the best option.
S1/2
P 5 0.0894I7 °C, (39)
Nondimensional
Sensor Location Mean Square Error where I 7 is the 7 3 7 identity matrix. The initial thermal states
Frame 0.00017 for the Kalman filter are assumed to be random values that
Yoke 0.00015 result in large (,10 8C) initial estimation errors for all states.
Teeth 0.00011
Figure 7 shows the performance of the Kalman filter
Slot winding 0.00003
End winding 0.00004 when the machine is operated at a constant rotor speed of
800 rev/min, and the phase currents are sinusoidal with

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 55 60
40 A 40 A 55 A 45 A 65 A 55 A 0A
60 A
50 55

45 50
Temperature (°C)

Temperature (°C)
40 45
35 40
30 End Winding 35
End Winding
Slot Winding Slot Winding
25 Teeth 30
Teeth
Yoke Yoke
20 Frame 25
Frame
15 20
0

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00
50

50
10

15

20

25

30

35

40

45

10

15

20

25

30

35

40
Time (s) Time (s)
(a) (a)
4
40 A 60 A 40 A 55 A 45 A 65 A 55 A 0A
4
2
2
0
Temperature (°C)

–2 Temperature (°C) 0

–4 –2
End Winding
–6 End Winding –4 Slot Winding
Slot Winding Teeth
–8 Teeth Yoke
–6 Frame
Yoke
–10 Frame
–8
–12
0

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00
50

50
10

15

20

25

30

35

40

45

10

15

20

25

30

35

40
Time (s) Time (s)
(b) (b)

FIGURE 7 Kalman filter performance for 40-60-40-A phase cur- FIGURE 8 Kalman filter performance for 55-45-65-55-0-A phase cur-
rents. (a) shows the Kalman filter state estimates, while (b) shows rents. (a) shows the Kalman filter state estimates, while (b) shows the
the estimation error, which is calculated as the difference estimation error, which is calculated as the difference between the
between the measured temperatures and the Kalman filter esti- measured temperatures and the Kalman filter estimates. The initial
mates. The initial thermal states of the Kalman filter are assumed thermal states of the Kalman filter are assumed to be random values
to be random values that result in large (~10 °C) initial estimation that result in large (~10 oC) initial estimation errors for all states. The
errors for all states. The Kalman filter is provided with frame- temperature measurements used with the Kalman filter are the frame-
temperature measurements. Despite the large initial errors in temperature measurements. Despite the large initial errors in estima-
estimation, all temperatures are tracked within 3 °C of the mea- tion, all temperatures are tracked within 3 oC of the measured data in
sured data in less than 5 min. The filter tracking errors remain less than 5 min. The filter tracking errors remain within 3 oC thereafter,
within 3 °C thereafter. despite the frequent changes in the current amplitude.

amplitude 40 A for the first 1000 s, 60 A for the next 1050 s, the rest of the experiment. Despite the large initial error in
and 40 A for the last 2450 s. The initial condition for the estimation and the frequent changes in machine currents,
Kalman filter is chosen such that large initial estimation all temperatures are tracked within 3 oC of the measured
errors (,10 8C) result, as can be seen from the figure. In less data in less than 5 min. The filter tracking errors remain
than 5 min, all thermal states are tracked within 3 oC of the within 3 oC thereafter, despite the frequent changes in the
measured data. The estimation errors in all states are within current amplitude.
3 oC thereafter.
Figure 8 shows the performance of the Kalman filter for CONCLUSIONS
the case where the rotor speed is 800 r/min. However, the A VTS that tracks the thermal states of a permanent-magnet
current amplitudes are now changed more frequently and machine in the presence of modeling uncertainties and
have values of 55 A for the first 820 s, 45 A for the next 790 s, unknown initial thermal states is presented. The VTS is a
65 A for the next 770 s, 55 A for the next 840 s, and 0 A for Kalman filter that combines a lumped-parameter thermal

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model of the machine and temperature measurements from Machinery Committee, including secretary (2001–2002),
a single temperature sensor installed on the machine frame. vice chair (2003–2004), and chair of the Generator Subcom-
The VTS performance is demonstrated using a 3.8-kW per- mittee (2005–2006). He has authored or coauthored over
manent-magnet machine. The results show that the VTS can 100 technical papers, the undergraduate textbook Electro-
estimate the unmeasured thermal states of the machine with mechanical Motion Devices, and the graduate-level textbook
less than 3 oC error in less than 5 min. The results give confi- Analysis of Electric Machinery. He is the chief technical offi-
dence in the use of model-based Kalman filtering as an effec- cer of PC Krause and Associates, a business specializing in
tive tool for virtual thermal sensing in electric machines. the analysis and simulation of power systems.
Steven D. Pekarek received the B.S.E.E. and Ph.D. in
ACKNOWLEDGMENT electrical engineering from Purdue University in 1991 and
The authors are grateful to Prof. Douglas Adams from 1996, respectively. From 1997 to 2004, he was an assistant
the School of Mechanical Engineering at Purdue Univer- and then associate professor of electrical and computer
sity for his helpful comments on temperature sensing engineering at the University of Missouri-Rolla. He is
and instrumentation. presently a professor of electrical and computer engineer-
ing at Purdue University and chair of the energy sources
AUTHOR INFORMATION and systems area. He has been an active member of the
Ruchir Saheba received the B.E degree in instrumentation IEEE Power Electronics Society, served the Future Energy
and control engineering from Nirma Institute of Technology Challenge as secretary (2003) and chair (2005), was the
in 2003 and the M.S. degree in engineering from Purdue Uni- technical chair of the 2007 IEEE Applied Power Electron-
versity in 2006. He is currently pursuing his Ph.D. degree in ics Conference, and was chair of the 2008 IEEE Applied
mechanical engineering at the University of Massachusetts. Power Electronics Conference. He is an associate editor
His research interests include temperature estimation in for IEEE Transactions on Power Electronics and IEEE Trans-
electric machines and structural control of wind turbines. actions on Energy Conversion.
Mario A. Rotea (rotea@utdallas.edu) graduated with Brett Jordan received the B.S. degree from Ohio Universi-
a degree in electronic engineering from the University of ty in 1994 in electrical engineering and the M.S.E.E. degree in
Rosario in 1983. He received the master’s degree in elec- electrical engineering from Wright State University in 1997.
trical engineering in 1988 and the Ph.D. in control sci- He joined Stanley Electric, where he worked in halogen bulb
ence and dynamical systems in 1990 from the University manufacturing. In 2001, he joined the Energy, Power, and
of Minnesota. He is currently a professor and head of the Thermal Division as a project engineer, where he is respon-
Mechanical Engineering Department at the University of sible for the research and development of fault-tolerant elec-
Texas, Dallas, where he is also a professor of electrical en- trical distribution systems, power semiconductor devices,
gineering. He began his academic career at Purdue Uni- power electronics, and prognostics and health management
versity, where he was a professor of aeronautics and astro- systems. He is a registered professional engineer in Ohio.
nautics. He was a professor and head of the Mechanical
and Industrial Engineering Department from 2007 to 2009
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[2] Z. Gao, T. G. Habetler, R. G. Harley, and R. S. Colby, “An adaptive Kalman
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research interests include health management and model Industry Applications Society, Hong Kong, China, Oct. 2005, vol. 1, pp. 2–9.
[3] P. H. Mellor, D. Roberts, and D. R. Turner, “Lumped parameter thermal
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University of Texas at Dallas, 800 W. Campbell Rd., EC-38, [4] P. Milanfar and J. H. Lang, “Monitoring the thermal condition of perma-
nent-magnet synchronous motors,” IEEE Trans. Aerosp. Electron. Syst., vol.
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Oleg Wasynczuk received the B.S.E.E. degree from [5] T. J. Harris, J. F. MacGregor, and J. D. Wright, “Optimal sensor location
Bradley University in 1976 and the M.S.E.E. and Ph.D. with application to a packed bed tubular reactor,” AIChE. J., vol. 26, no. 6,
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degrees from Purdue University in 1977 and 1979, respec- [6] J. A. Rosero, J. A. Ortega, E. Aldabas, and L. Romeral, “Moving to-
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of the IEEE. He has held several positions on the Electric Springer-Verlag, 1999.

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