A Data-Driven Automatic Design of Induction
Motors Based on Tree Search and Reinforcement
Learning Considering Multiple Objectives
メタデータ

言語: English
出版者: IEEE Transactions on Magnetics
公開日: 2024-08-01
キーワード (Ja):
キーワード (En): Induction motors, Multi-fidelity
optimization, Multi-objective optimization,
Reinforcement learning
作成者: 佐藤, 孝洋, 渡邊, 浩太
メールアドレス:
所属: 室蘭工業大学, 室蘭工業大学

URL

http://hdl.handle.net/10258/0002000236

CMP-176

1

A Data-Driven Automatic Design of Induction Motors
Based on Tree Search and Reinforcement Learning
Considering Multiple Objectives
Takahiro Sato1, Member, IEEE, Kota Watanabe1, Member, IEEE
1

Muroran Institute of Technology, Muroran, Hokkaido, 050-8585, JAPANThe design problems of electric machines are mixed-integer ones, which are generally difficult to solve. To automatically obtain whole
designs of electric machines, in this paper, a multi-objective automatic design method for electric machines is presented based on thetree search-based reinforcement learning and multi-fidelity technique. In this method, various candidate designs are generated based on
the tree search using low-fidelity model. The value functions of the nodes are learned from the generated data. Moreover, high-fidelity
evaluations are applied to a limited number of better designs, and then, the multi-fidelity metamodel is constructed to correct the lowfidelity evaluation results. The present method is applied to the design problem an induction motor. It is shown that various designs are
obtained considering trade-off relation between two objectives based on high-accuracy evaluations.
Index Terms—Induction motors, Multi-fidelity optimization, Multi-objective optimization, Reinforcement learning.

I. INTRODUCTIONF

or electric machines, some optimization techniques, e.g.,
topology ones, have successfully been applied [e.g. 1-3].
However, in such optimizations, all the design processes of theelectric machines cannot be automated. This is because thedesign problems of the electric machines are mixed-integer
ones, which are generally difficult to solve. Namely, we haveto determine the integer variables like the number of slots andthe shape variables like the tooth width, simultaneously. In
recent years, to solve the mixed-integer problems of the electricmachines, some automatic design methods have been proposed[4,5]. Especially, the DEsign SEARrch Tree-based Automatic
design Method (DeSearTAM) proposed in [4] canautomatically generate all the integer and shape variables byutilizing self-generated design data based on the reinforcement
learning. However, the drawbacks of DeSearTAM are as
follows: 1) it is only applicable to single-objective problems, 2)
a large number of design data is necessary to adapt given design
problems, and thus, high-accuracy evaluation methods withheavy computational cost are difficult to be employed.
In this paper, we present a modified DeSearTAM to
overcome above mentioned drawbacks. First, to consider
multiple objectives in the reinforcement learning, ascalarization function is employed. Second, a multi-fidelity
optimization approach [6] is introduced. In short, a low-fidelity
model is used to generate various design data, and then, a highfidelity evaluation method is applied to the effective design datato correct the low-fidelity evaluation results.
In this work, the present method is applied to a design
problem of a squirrel-cage induction motor. It is shown thatwhole designs of the motor can automatically be obtained
considering the trade-off relation between two objectives.
Manuscript received June 8, 2023. Corresponding author: Takahiro Sato(e-mail: t-sato@mmm.muroran-it.ac.jp).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier (inserted by IEEE).

II. MULTI-OBJECTIVE AUTOMATIC DESIGN METHOD
A. Overview of conventional DeSearTAM [4]Let xfnd and xvar be fundamental integer and other shapevariables, respectively. In DeSearTAM, feasible combinationsof xfnd are listed as a tree structure. Then, xfnd are determinedthrough the tree search. Let sk,m be a k-th node of the depth m.
The transient probability from sk,m to sk’,m+1 are determined by
the node value, V(s), as follows: m ( k ', k ) =exp − V ( sk ', m +1 ) TjSm ( k )exp − V ( s j , m +1 ) T



,

(1)

where T is a temperature parameter, and Sm(k) is a set of thenodes reachable from the current node sk,m. In addition, V(s) isconstructed based on the design dataset. After determining xfnd,
xvar are estimated by a response surface, fVAR, as follows:(2)
xvar = f VAR ( xfnd ) ,and fVAR is also constructed from the design dataset. Finally, xvar
is optimized under the fixed xfnd. The resultant pair of (xfnd, xvar)
is stored in the database. By repeating this process, V(s) andfVAR are adapted to the given problem through self-generated
data [4].
B. Modified DeSearTAM for multi-objective problems
In this paper, the multi-objective DeSearTAM considering
multi-fidelity evaluations is proposed. This method is here
called MO-MF-DeSearTAM, whose basic algorithm is shownin Fig. 1. The key modified points are as follows.
1) Introducing scalarization function: To construct V(s) basedon multiple objectives, the scalarization function using
weighting coefficient vector wobj is employed. Let us denote
xs=[xfndT, xvarT]T as a best design attributed to a node s [4], and(xs) is the feature (objective) vector whose components are
the evaluation results of the machine performance. The node
value corresponding to s is determined as follows:(3)
V ( s ) = f scar ( ( xs ) , wobj ) ,0018-9464 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. (Inserted by IEEE.)CMP-176

2
Design Specification

Initialize MF-metamodel: κ=1
Generate weight vectors wobj
m 0
n 0
Extract current Pareto
design data

Change wobj
Construct V(s) and fVAR
using MF-metamodel

Apply high-fidelity
evaluations to
Pareto designs

Determine xfnd
based on tree search

Update MF-metamodel

Determine xvar
based on fVAR(xfnd)

m m+1
m > θM
yes
End

Optimize xvar using
MF-metamodel
n n+1
Otherwisen > θN

Otherwise

yesFig. 1. Basic procedure of MO-MF-DeSearTAM.

where fscar(, wobj) denotes the scalarization function, as which
the penalty-based boundary intersection (PBI) function [7] isemployed in this work.
2) Multi-fidelity (MF) metamodel: To adapt V(s) and fVAR,
various designs must be generated. Hence, a low-fidelity model
is employed in the data-generations. Then, high-fidelity
evaluations are applied to the better designs, i.e., Pareto
solutions. Based on two fidelity results, the low-fidelity results
are corrected using the MF-metamodel [6] as follows:(4) ( xs )  i   κ ( xs ,  )  i  ( xs )  i ,where [v]i denotes i-th component of vector v. Moreover,κ(x, ) is the correction vector, which is constructed to predictthe ratio of high- and low-fidelity results.
As a common issue when using metamodel, the output of themodels is sometimes incorrect if the input to the models is far
from the training dataset. To handle this problem, the Gaussian
process regression (GPR) is applied to construct κ(x, ), and weevaluate the uncertainly of [κ(x, )]i as follows:(5)
Ui ( xs ) = 100  2 i ,GP ( xs ,  ) fCDF −1 ( 0.975) i ,GP ( xs ,  ) ,where fCDF-1(0.975) ~ 1.9599 is the inverse cumulative
distribution function of normal distribution for 97.5%, andμi,GP(x, ) and σi,GP2(x, ) are average and variance of the i-th
output of GPR, respectively. When evaluating the candidate
designs, some penalty terms can be added based on Ui(x).

dataset. As a target spec, a 200kW motor is designed under thecondition that rated voltage, frequency, slip are 1100V, 100Hz,
0.02, respectively. This specification is based on the design for
railway traction [9]. However, we here only consider the rated
output at the rated slip for simplicity, because the purpose of
automatic design is to test whether we can obtain reasonable
designs as the given specifications by the proposed algorithm.
The objectives, f1 and f2, are defined as follows:(6)
f1 = − → min .,(7)
f 2 = M kg → min .,where η and Mkg are efficiency at the rated condition and total
mass. In addition, the following constraints are imposed:(8)
0.99  PO − PT PT  1.15,
smax  s p ,(9) max  O   ,

(10)
(11)
(12)
(13)

Bcore   B ,Bteeth   B ,Bteeth2   B ,Tc  T ,(14)
where PO=ωτO and τO denote the output and torque at rated
condition, respectively, τmax is the maximum torque and smax is
its slip, and sp is rated slip. Moreover, Bcore, Bteeth, Bteeth2 are theflux density at stator core back, stator teeth, rotor teeth,
respectively, and Tc is the temperature rise of the stator winding.
Moreover, PT=200kW is the target output, and θτ, θB, θT are thethresholds.
B. Low-fidelity evaluation
As a low-fidelity model, the classical T-equivalent circuit
method [10] (EqCir) is applied. The circuit constants areanalytically obtained from the motor’s geometry. Then, PO andτO are computed from the circuit analysis results. The otherfeatures are also evaluated from the geometrical shape [10]. In
addition, for simplicity, Tc is estimated from the following
simple modelling:(15)
Tc = 3I12 r1 (  S surf ) ,where γ=30 W/(m2degC) is assumed thermal conductivity of
the winding, Ssurf is surface of the winding, I1 and r1 are statorcurrent and stator resistance obtained from EqCir, respectively.
C. High-fidelity evaluation and MF-metamodel
As a high-fidelity model, we employ the finite element
method (FEM), using which the torque is computed. In the3) Conjunction of Multi-objective optimization: After
generating xfnd and xvar, multi-objective evolutionary
optimizations can be adopted to generate various candidate
designs. In this work, MOEA/DD-CMA [8] is employed.

hc

1.
2.

nc

hst

wc
g
wr

III. AUTOMATIC DESIGN OF INDUCTION MOTOR
A. Problem definition
A squirrel-cage induction motor model shown in Fig. 2 isdesigned. This model has 5 integer variables and 12 shapevariables, which are defined to generate only the valid
geometries. These variables are determined starting from empty

Integer variables, xfnd:

ws

y

bh

x

3.

ts
tr
bw
rin

4.
5.

z

Num. of poles,
Num. of parallel
circuits in stator,
Num. of statorslots per pole perphase,
Num. of rotor slots,
Coil pitch.

Core material: 50A400r
Shape variables, xvar=[rin, g, wc, hc, nc, ws, ts, bw, bh, wr, tr, hst]

Fig. 2. Squirrel-cage induction motor model and its xfnd and xvar.

CMP-176
analysis, the voltage-source transient analysis is applied until
reaching steady-state condition. As a result, torque curves
against the slip are obtained.
Based on the low- and high- fidelity results, the MFmetamodel for PO and τO are constructed using GPR: κ=[κP, κτ]T.
The simplest approach to construct a metamodel is that all thevariables, xfnd and xvar, are input to GPR. However, to constructaccurate metamodel for any inputs, a huge number of teacher
data must be prepared. Hence, we define the inputs as follows:(16)κ ( x,  )  μGP ( xfnd , s, Bcore , Bteeth , Bteeth2 ) ,where μGP is the average of GPR. The teachers of GPR are theratio of FEM and EqCir results: PFEM/PO and τFEM/τO. The
reason why the magnetic flux density is used as the input is thatthe accuracy of EqCir results depends on the magnetic
saturation. The calculation flow for evaluating the machine
performance using the MF-metamodel is shown in Fig. 3.
D. Settings of Automatic Designs
The tree structure for the induction motor is designed asshown in Fig. 4. The feasible numbers of each integer variable
are set as follows:⚫ Number of poles, p: 2, 4, 8 (to receive a FE model that issymmetrical every 90 degrees)⚫ Number of parallel circuits in stator, nY: 1, 2
⚫ Number of stator slots per pole and phase, Nst: 1, 2, 3, 4
⚫ Number of rotor slots, Nrt: integer values with interval ofNrt/(3pNst) = [0.7, 1.3] and Nrt≠Nst.
⚫ Coil pitch, Nc: integer values with interval of βc=[0.7, 1.0],
where βc=(Nc-1)/(3Nst) is percent coil pitch.
In the tree search, V(s) is constructed considering theconstraints as follows:(17)
V ( s ) = f scar ( ( xs ) , wobj ) −  death N viol ,where Nviol denotes the number of violated constraint conditions,and αdeath=106 is a penalty coefficient. In addition, an additional
constraint condition is considered in order to prevent fromwrong search due to incorrect GPR output as follows:(18)
U k ( s )   UNC , for k = PO and  O .
As for the multi-objective optimization, the PBI function andthe same wobj used in the construction of V(s) are adopted. The
other hyper-parameters in MO-MF-DeSearTAM aresummarized in Table I. In summary, in the first data-generation(m=1), the tree search is repeated 1500 times. Then, FEM isapplied to the Pareto designs and the MF-metamodel is updated.
In the second data-generations (m=2), only the promising nodes
are aggressively searched by setting very small T: 10-6. Afterthe MF-metamodel is updated again, effective shape variables
are intensively searched (m=3).
E. Automatic design results
The resultant Pareto solutions obtained by first, second, andthird data-generations are shown in Fig. 5. The histogram of thetorque difference between corrected EqCir and FEM results forthe three data-generations are shown in Fig. 6, from which it
can be seen that the accuracy of the corrected EqCir results
increases with the number of metamodel updates. From theseresults, it can be seen that the reason why the Pareto front is

3
κPPO
κττ O
η

PO
τO

xfnd
xvar

loss

EqCir
one
design (low-fidelity)

Bcore
BteethBteeth2

Bcore
BteethBteeth2
Evaluation
results

MF-metamodelμGP(xfnd, s, Bcore, Bteeth, Bteeth2)

κP, κτ

Fig. 3. Calculation flow for evaluating machine performance of (xfnd, xvar).
Specification
Depth 1:
number of poles
Depth 2:
number of parallel circuits

2
1

4

8

2

Depth 3:
number of stator slots
Depth 4:
number of rotor slots
Depth 5:coil pitch

Fig. 4. Design search tree for squirrel-cage induction motor model.
TABLE I
HYPERPARAMETER SETTINGS
MF-MO-DESEATAM
Num. of data-generation: θM
31500 (m=1), 100 (m=2),
Num. of iteration in data-generation: θN
10 (m=3)(1,0), (0.25,0.75),
Num. of wobj
(0.5,0.5), (0.75,0.25), (0,1)
10 (m=1 and n<1000)
Temperature parameter: T
0.1 (m=1 and n>1000)
10-6 (m=2,3)
MOEA/DD-CMA
Num. of candidate solutions
50
Num. of iterations:250
Thresholds in constraints
Thresholds for maximum torque ratio: θτ
1.5
Thresholds for flux density: θB
1.4 T
Thresholds for temperature-rise: θT
120 degC
Thresholds for uncertainty: θUNC
10%

shifted to upper left is that the EqCir evaluations become moreaccurate considering the magnetic saturation of the iron cores.
In this automatic design, the computational time of each datageneration is less than 1 day, and the dominant computational
cost is the FEM evaluations. In this case, the total number of
FEM analysis is about 700. Because these analyses canperfectly be parallelized, the computational burden of MO-MFDeSearTAM depends on the number of parallel computations.
The resultant shapes marked in Fig. 5 and theircorresponding torque curves are shown in Fig. 7. We can see
from Fig. 7(a) that the torque curve evaluated by EqCir andFEM are not coincided with each other in the first datageneration. In the second data-generation, as shown in Fig. 7(b),p and diameter of the resultant design are changed. This isbecause, other designs that are not mainly evaluated in the firstdata-generation are searched depending on the corrected EqCir
results. Through these searches and metamodel updates, in thethird data-generation, two types of resultant shapes withdifferent p and diameter are obtained as shown in Figs. 7(c)-CMP-176

4450

Second (corrected)
Second400

First (Eq.Cir)
First

1500

p: 4, nY: 1,
Core length: 167mm,
Diameter: 476.8mm,
Num. turns: 4

Third (corrected)
Third

500

FEM

Torque (Nm)total weights (kg)

550

(B)

350
300

(II)

250
88.5

(I)

(III)
89.5

90.5

EqCir

500

0

(A)

91.5 92.5 93.5
efficiency (%)

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0slip (-)

94.5

95.5

(a)

Fig. 5. Resultant Pareto designs obtained by 1st, 2nd, 3rd data-generations.
100
Third
Second
First
80

6000

Torque (Nm)

5000

60
40

0

0

5 10 15 20 25 30 35 40 45 50
difference ratio (%)
Fig. 6. Histograms of torque difference between FEM and corrected EqCir
evaluations, which denotes κττO, in 1st, 2nd, 3rd data-generations.

EqCir

3000

2000

0
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0slip (-)(b)
6000

Torque (Nm)

5000
4000

FEM

EqCir

3000
2000
1000

p: 8, nY: 2,
Core length: 120mm,
Diameter: 621.2mm,
Num. turns: 70

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0slip (-)(c)
6000

p: 4, nY: 2,
Core length: 221mm,
Diameter: 448.0mm,
Num. turns: 7

IV. CONCLUSIONS
To solve the multi-objective mixed-integer problem ofelectric machines, MO-MF-DeSearTAM has been proposed.
The proposed method has been applied to the design problemof the induction motor. It has been shown that the accuracy ofthe low-fidelity model is improved with the MF-metamodel
updates, and the Pareto solutions can be obtained considering
all the design variables based on high-accuracy evaluations.
For the future work, MO-MF-DeSearTAM will be applied toother problems, such as permanent magnet motor designs.

5000

Torque (Nm)7(e). In addition, the torque curves evaluated by corrected
EqCir and FEM are in good agreement. From these results, it isconcluded that accuracy of the low-fidelity model can be
improved by the MF-metamodel, and various designs can beobtained considering the trade-off relation between two
objectives based on high-accuracy evaluation results.
Finally, it is noted that, once the dataset for a certain problem
is obtained, similar problems can easily be solved, because V(s),fVAR, κ can be constructed from the stored design data.

4000

FEM

1000

p: 8, nY: 2,
Core length: 119mm,
Diameter: 638.0mm,
Num. turns: 7

20

4000

FEM
EqCir

3000
2000
1000
0
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0slip (-)(d)
6000

p: 4, nY: 2,
Core length: 234mm,
Diameter: 426.2mm,
Num. turns: 7

5000

Torque (Nm)frequency (%)

1000

4000

FEM

EqCir

3000
2000
1000

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(e)
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