Low-order thermal models of electrical machines are fundamental for the design and management of electric powertrains since they allow evaluation of multiple drive cycles in a very short simulation time and implementation of model-based control schemes. A common technique to obtain these models involves homogenization of the electrical winding geometry and thermal properties. However, incorrect estimation of homogenized parameters has a significant impact on the accuracy of the model. Since the experimental estimation of these parameters is both costly and time-consuming, authors usually prefer to rely either on simple analytical formulae or complex numerical calculations. In this paper, we derive a low-order homogenized model using the method of multiple-scales (MS) and show that this gives an accurate steady-state and transient prediction of hot-spot temperature within the windings. The accuracy of the proposed method is shown by comparing the results with both high-order numerical simulations and experimental measurements from the literature.

References

References
1.
Bennion
,
K.
,
Cousineau
,
E.
,
Feng
,
X.
,
King
,
C.
, and
Moreno
,
G.
,
2015
, “
Electric Motor Thermal Management R & D
,”
IEEE
Power and Energy Society General Meeting, Denver, CO, July 26-30, Paper No. PESGM2015P-002837.
2.
Huang
,
Z.
,
Márquez-Fernández
,
F. J.
,
Loayza
,
Y.
,
Reinap
,
A.
, and
Alaküla
,
M.
,
2014
, “
Dynamic Thermal Modeling and Application of Electrical Machine in Hybrid Drives
,”
International Conference on Electrical Machines (ICEM)
, pp.
2158
2164
.
3.
Popescu
,
M.
,
Staton
,
D.
,
Boglietti
,
A.
,
Cavagnino
,
A.
,
Hawkins
,
D.
, and
Goss
,
J.
,
2016
, “
Modern Heat Extraction Systems for Electrical Machines—A Review
,”
IEEE Trans. Ind. Appl.
,
52
(
3
), pp.
2167
2175
.
4.
Zhang
,
H.
,
2015
, “
Online Thermal Monitoring Models for Induction Machines
,”
IEEE Trans. Energy Convers.
,
30
(
4
), pp.
1279
1287
.
5.
Mellor
,
P.
,
Wrobel
,
R.
, and
Simpson
,
N.
,
2014
, “
AC Losses in High Frequency Electrical Machine Windings Formed From Large Section Conductors
,”
Energy Conversion Congress and Exposition (ECCE)
, pp.
5563
5570
.
6.
Boglietti
,
A.
,
Staton
,
D.
, and
Dipartimento
,
T.
,
2015
, “
Stator Winding Thermal Conductivity Evaluation: An Industrial Production Assessment
,”
Energy Conversion Congress and Exposition (ECCE)
, pp.
4865
4871
.
7.
Ayat
,
S.
,
Wrobel
,
R.
,
Goss
,
J.
, and
Drury
,
D.
,
2016
, “
Estimation of Equivalent Thermal Conductivity for Impregnated Electrical Windings Formed From Profiled Rectangular Conductors
,”
8th IET International Conference on Power Electronics, Machines and Drives (PEMD)
, pp.
1
6
.
8.
Nategh
,
S.
,
Wallmark
,
O.
,
Leksell
,
M.
, and
Zhao
,
S.
,
2012
, “
Thermal Analysis of a PMaSRM Using Partial FEA and Lumped Parameter Modeling
,”
IEEE Trans. Energy Convers.
,
27
(
2
), pp.
477
488
.
9.
Takatsu
,
Y.
,
Masuoka
,
T.
,
Nomura
,
T.
, and
Yamada
,
Y.
,
2016
, “
Modeling of Effective Stagnant Thermal Conductivity of Porous Media
,”
ASME J. Heat Transfer
,
138
(
1
), p.
012601
.
10.
Kundalwal
,
S. I.
,
Kumar
,
R. S.
, and
Ray
,
M. C.
,
2015
, “
Effective Thermal Conductivities of a Novel Fuzzy Fiber-Reinforced Composite Containing Wavy Carbon Nanotubes
,”
ASME J. Heat Transfer
,
137
(
1
), p.
012401
.
11.
Wrobel
,
R.
, and
Mellor
,
P. H.
,
2010
, “
A General Cuboidal Element for Three-Dimensional Thermal Modelling
,”
IEEE Trans. Magn.
,
46
(
8
), pp.
3197
3200
.
12.
Wrobel
,
R.
,
Mlot
,
A.
, and
Mellor
,
P. H.
,
2012
, “
Contribution of End-Winding Proximity Losses to Temperature Variation in Electromagnetic Devices
,”
IEEE Trans. Ind. Electron.
,
59
(
2
), pp.
848
857
.
13.
Baker
,
J. L.
,
Wrobel
,
R.
,
Drury
,
D.
, and
Mellor
,
P. H.
,
2014
, “
A Methodology for Predicting the Thermal Behaviour of Modular-Wound Electrical Machines
,”
Energy Conversion Congress and Exposition (ECCE)
, pp.
5176
5183
.
14.
Simpson
,
N.
,
Wrobel
,
R.
, and
Mellor
,
P. H.
,
2013
, “
Estimation of Equivalent Thermal Parameters of Impregnated Electrical Windings
,”
IEEE Trans. Ind. Appl.
,
49
(
6
), pp.
2505
2515
.
15.
Idoughi
,
L.
,
Mininger
,
X.
,
Bouillault
,
F.
,
Bernard
,
L.
, and
Hoang
,
E.
,
2011
, “
Thermal Model With Winding Homogenization and FIT Discretization for Stator Slot
,”
IEEE Trans. Magn.
,
47
(
12
), pp.
4822
4826
.
16.
Galea
,
M.
,
Gerada
,
C.
,
Raminosoa
,
T.
, and
Wheeler
,
P.
,
2012
, “
A Thermal Improvement Technique for the Phase Windings of Electrical Machines
,”
IEEE Trans. Ind. Appl.
,
48
(
1
), pp.
79
87
.
17.
Wiener
,
O. H.
,
1912
,
Die theorie des mischkörpers für das feld der stationären strömung. 1. abhandlung: Die mittelwertsätze für kraft, polarisation und energie
,
BG Teubner
,
Leipzig, Germany
.
18.
Hashin
,
Z.
, and
Shtrikman
,
S.
,
1963
, “
A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials
,”
J. Mech. Phys. Solids
,
11
(
2
), pp.
127
140
.
19.
Milton
,
G. W.
,
1981
, “
Bounds on the Transport and Optical Properties of a Two-Component Composite Material
,”
J. Appl. Phys.
,
52
(
8
), pp.
5294
5304
.
20.
Kanzaki
,
H.
,
Sato
,
K.
, and
Kumagai
,
M.
,
1992
, “
A Study of an Estimation Method for Predicting the Equivalent Thermal Conductivity of an Electric Coil
,”
Heat Transfer Jpn. Res.
,
21
(
2
), pp.
123
138
.
21.
Siesing
,
L.
,
Reinap
,
A.
, and
Andersson
,
M.
,
2014
, “
Thermal Properties on High Fill Factor Electrical Windings: Infiltrated vs Non Infiltrated
,”
International Conference on Electrical Machine (ICEM)
, pp.
2218
2223
.
22.
Bensoussan
,
A.
,
Lions
,
J.-L.
, and
Papanicolaou
,
G.
,
1978
,
Asymptotic Analysis for Periodic Structures
,
North-Holland
,
Amsterdam, The Netherlands
.
23.
Sanchez-Palencia
,
E.
,
1980
,
Non-Homogeneous Media and Vibration Theory
,
Springer-Verlag
,
Berlin, Germany
.
24.
Yu
,
Q.
, and
Fish
,
J.
,
2002
, “
Multiscale Asymptotic Homogenization for Multiphysics Problems With Multiple Spatial and Temporal Scales: A Coupled Thermo-Viscoelastic Example Problem
,”
Int. J. Solids Struct.
,
39
(
26
), pp.
6429
6452
.
25.
Asakuma
,
Y.
, and
Yamamoto
,
T.
,
2013
, “
Effective Thermal Conductivity of Porous Materials and Composites as a Function of Fundamental Structural Parameters
,”
Comput. Assisted Methods Eng. Sci.
,
20
(
2
), pp.
89
98
.
26.
Hales
,
J.
,
Tonks
,
M.
,
Chockalingam
,
K.
,
Perez
,
D.
,
Novascone
,
S.
,
Spencer
,
B.
, and
Williamson
,
R.
,
2015
, “
Asymptotic Expansion Homogenization for Multiscale Nuclear Fuel Analysis
,”
Comput. Mater. Sci.
,
99
, pp.
290
297
.
27.
White
,
J.
,
2015
, “
Analysis of Heat Conduction in a Heterogeneous Material by a Multiple-Scale Averaging Method
,”
ASME J. Heat Transfer
,
137
(
7
), p.
071301
.
28.
Bruna
,
M.
, and
Chapman
,
S. J.
,
2015
, “
Diffusion in Spatially Varying Porous Media
,”
SIAM J. Appl. Math.
,
37
(
2
), pp.
215
238
.
29.
Cioranescu
,
D.
, and
Paulin
,
J. S. J.
,
2012
,
Homogenization of Reticulated Structures
,
Springer Science & Business Media
,
Berlin, Germany
.
30.
Mei
,
C. C.
, and
Vernescu
,
B.
,
2010
,
Homogenization Methods for Multiscale Mechanics
,
World Scientific
,
Singapore
.
31.
Jóesson
,
H.
, and
Halle
,
B.
,
1996
, “
Solvent Diffusion in Ordered Macrofluids: A Stochastic Simulation Study of the Obstruction Effect
,”
J. Chem. Phys.
,
104
(
17
), pp.
6807
6817
.
32.
Trefethen
,
L. N.
,
2000
,
Spectral Methods in MATLAB
,
SIAM
,
Philadelphia, PA
.
33.
Hastings
,
W. K.
,
1970
, “
Monte Carlo Sampling Methods Using Markov Chains and Their Applications
,”
Biometrika
,
57
(
1
), pp.
97
109
.
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