1
Sensitivity-Based Optimization of Interior Permanent Magnet
Synchronous Motor for Torque Characteristic Enhancement
S. Ahmadi1, A. Vahedi2, N. Taghavi2, T. Lubin1
1
2
Université de Lorraine, GREEN, F-54500, Nancy, France.
Center of Excellence for Power Systems Automation and Operation, Department of Electrical Engineering,
Iran University of Science & Technology, Tehran, Iran.
This paper presents a multi-objective optimal rotor design for an interior permanent magnet synchronous motor (IPMSM) based on
finite element analysis. Due to the importance of torque characteristic in electromagnetic design of IPMSMs, the main efforts of this
study are focused on finding a proper trade-off for its torque profile challenges. In this regard, in order to attain high average torque
and low torque ripple, this paper fully discusses the influence of several key factors such as the permanent magnet (PM) arrangements,
PM positions and PM sizes. Moreover, by using sensitivity analysis, effective optimization parameters, their initial value and their
appropriate variation intervals are determined which leads to accelerate the convergence process. The 2-dimensional finite element
model (FEM) of an IPMSM is used to perform a sensitivity analysis and establish a multi-objective FEM-based optimization.
Index Terms—Interior permanent magnet synchronous motor (IPMSM), multi-objective FEM-based optimization, torque profile,
sensitivity analysis.
I. INTRODUCTION
D
UE to the rapid development and implementation of NdFe-B magnets, IPMSMs play a significant role in many
industry applications [1]-[2]. IPMSMs provide a high torque
density, high reluctance torque, appropriate flux weakening
capability, high efficiency, and simple controllability [3]-[5].
In some applications such as traction application which
IPMSMs are promising electric machines, high torque density
and low torque ripple because of influence on the comfort and
the stability of vehicle, have a vital role, so a trade-off
between average torque and torque ripple should be taken into
consideration [6]-[9]. Produced torque in these machines is
intensively affected by parameters such as magnet size and
magnets position [10]-[11]. Hence, a multi-objective optimal
design on the design parameters of IPMSM is indispensable.
Owing to progress in computing methods in optimization
problems, FEM-based optimization has gotten more attention
for accurate design optimization of electric machines [12].
Some recently multi-objective optimization of IPMSMs have
been studied based on finite element analysis in [13]-[18].
However, [13] proposed a finite-element based multi-objective
optimization procedure using a new algorithm belonging to
the class of controlled random search algorithms for
minimizing weight and maximizing power output; however,
the torque ripple optimization has been ignored. Reference
[14] presents some fast optimization methods for IPMSMs
based on deep learning to maximize the average torque and
minimize the torque ripple; however, the side PMs effect is
not considered. In [15], torque ripples were optimized in
IPMSMs using a multi-island genetic algorithm, radial basis
function neural networks, and the orthogonal experimental
method; however, no effort is made to optimize the developed
torque. In [16], a multi-objective stratified optimization
strategy is proposed, where five optimization objectives are
divided into two levels consisting of priority level 1(output
torque, PM cost) and level 2(cogging torque, torque ripple,
efficiency); however, important design parameter, PMs
position relative to rotor center has been ignored. In[18], by
combining PSO algorithm with mesh adaptive direct search
and applying it to the design of a IPMSM, torque ripple is
minimized while the PMs angle and the torque ripple
minimization are not considered. In [19], an effective
multilevel optimization strategy using the Pearson correlation
coefficient analysis and cross-factor variance analysis for
high-dimensional multi-objective optimization design of an
IPMSM was proposed, however only V structure has been
considered and PMs position relative to rotor center has been
neglected. In [20]-[24] novel rotor types have been proposed
for the performance improvement of IPMSMs. However, the
manufacturing of the novel rotor made it more complicated
and more expensive than the regular rotors.
The complexity of optimization grows with addition of more
objective functions, design parameters and inappropriate
design parameters variation interval; therefore, in this study an
effective multi-objective optimal design method for IPMSMs
has been presented. First, by using finite element method, we
investigate the influence of various design parameters such as
magnet width and thickness, position of central magnet
relative to rotor center, side magnet width and thickness and
side magnet angle on average torque and torque ripple.
Afterwards, an initial state based on the results obtained from
the sensitivity analysis is determined for optimization. Finally,
using optimization toolbox of ANSYS-Maxwell and the
inbuilt BFGS method which does not need large sample
points, optimization problem is run.
II. METHODOLOGY FOR OPTIMAL DESIGN OF IPMSM
The purpose is to effective multi-objective optimal design
of IPMSM with rated speed of 2369 rpm and rated output
power of 250 kW. Design structure of the studied IPMSM is
illustrated in Fig. 1 and the stator specifications are tabulated
in Table I.
2
III. SENSITIVITY ANALYSIS
So as to investigate the effect of rotor parameters on
average torque and torque ripple, the specified optimization
parameters are varied according to tabulated intervals
indicated in Table II.
Sensitivity analysis of average torque and torque ripple
relative to rotor parameters, are illustrated in Fig. 2 – Fig. 13.
TABLE II
VARIATION INTERVAL OF ROTOR PARAMETERS FOR SENSITIVITY ANALYSIS AND
OPTIMIZATION ALGORITHM
Var.
x1
x2
x3
x4
x5
x6
Sensitivity analysis
From
To
25
40
19.5
25.5
2
5.5
55
65
5
8.5
30
50
Optimization algorithm
From
To
28
31
20
25
4
5
57
61
6
7
37.5
42.5
Unit
deg
mm
mm
mm
mm
mm
Fig. 1. Proposed structure of IPMSM and optimization
variables.
Based on the structure presented in Fig. 1, selected
parameters for multi-objetive optimal design of rotor include
side magnet angle, side magnet width, side magnet thickness,
central magnet position relative to rotor center, central magnet
thickness and central magnet width.
TABLE I
STATOR SPECIFICATIONS OF THE STUDIED IPMSM
Quantity
Unit
Value
Stator stack length
Stator bore diameter
Stator yoke diameter
Air gap length
Number of stator slots
Number of poles
Number of turns per phase
Rated phase current (rms)
Rated phase voltage (rms)
Phase connection
Frequency
Type of permanent magnet
Rated speed
Conductor cross section
Permanent magnet residual flux
density
Relative permeability of
magnet
mm
mm
mm
mm
A
V
Hz
rpm
mm2
549
192
283.5
2
36
6
48
169.7
653
Y
118.45
Nd-Fe-B
2369
26.67
T
1.23
-
1.09
Mentioned Parameters are denoted by x1, x2, x3, x4, x5 and
x6, respectively and are illustrated in Fig.1. Due to symmetric
properties, one pole pitch of proposed structure for IPMSM is
depicted in Fig. 1.
By accurate investigating the sensitivity of average torque
and torque ripple relative to rotor parameters, besides initial
state of optimization algorithm, the appropriate variation
intervals for rotor parameters are chosen.
Fig. 2. Effect of side magnet angle variation on average
torque.
Fig. 3. Effect of side magnet angle variation on torque ripple.
3
Fig. 4. Effect of side magnet width variation on average
torque.
.
Fig. 5. Effect of side magnet width variation on torque ripple.
Fig. 6. Effect of side magnet thickness variation on average
torque.
Fig. 7. Effect of side magnet thickness variation on torque
ripple.
Fig. 8. Effect of side magnet position relative to rotor center
on average torque.
Fig. 9. Effect of central magnet position relative to rotor center
on torque ripple.
4
Fig. 10. Effect of central magnet thickness variation on
average torque.
Fig. 11. Effect of central magnet thickness variation on torque
ripple.
Fig. 13. Effect of based magnet width variation on torque
ripple.
Since the purpose of optimal design is to maximize average
torque and to minimize torque ripple, determining the
appropriate variation interval for all of parameters is required.
For example, in case of side magnet angle, it can be observed
that the points corresponding to the maximum average torque
and minimum torque ripple are approximately coincident with
each other. Hence it is obvious that the optimal value for side
magnet angle is located in immediate vicinity of points
corresponding to maximum average torque and minimum
torque ripple. According to Fig. 10 and Fig. 11 it can be drawn
that as the central magnet thickness is stepped up, average
torque is stepped up and consequently it goes to saturation.
Torque ripple also consists of a minimum value as side
magnet thickness is stepped up. In this case, since points
corresponding to average torque knee value and minimum of
torque ripple are coincident with each other, the proper
variation interval for central magnet thickness for optimization
process would be determined around knee point of average
torque. In case of other parameters, regarding to increment
rate or decrement rate of average torque and torque ripple,
appropriate
corresponding
parameter
variation
for
optimization algorithm could be determinded.
IV. MULTI-OBJECTIVE OPTIMAL DESIGN OF ROTOR
A. Objective Function and Optimization Variables
Since the purpose of optimal design is to maximize average
torque and to minimize torque ripple, objective function is
proposed as (1).
(1)
Fig. 12. Effect of central magnet width variation on average
torque.
Where w1 and w2 stand for weighting value of average
torque and torque ripple which are set at 1 and 2 by priority to
torque ripple minimization, respectively. G1 and G2 imply on
desired value of normalized average torque and normalized
torque ripple, respectively. Tav denotes average torque and
Tripple denotes torque ripple that is defined as (2)
(2)
5
Normalized value of average torque and torque ripple are
expressed by using (3) and (4), respectively.
(3)
analysis and are tabulated in Table III. Moreover outcomes of
optimization are demonstrated in Table III. As it is discernible,
the torque ripple is reduced by 64% and the torque average is
increased from 1008 N.m to 1093 N.m.
Fig. 14 illustrates electromagnetic torque of optimal and
basic design.
(4)
Normalized value of average torque and torque ripple
should be chosen in such a way value that they fall constantly
between 1 up to 10. As such, G1 and G2 are set at 10 and 1,
respectively.
TABLE III
DEMONSTRATION OF OPTIMIZATION RESULTS.
Basic
Optimal
Var.
unit
Design
Design
x1
30
30
deg
x2
24
24.6
mm
x3
4
5
mm
x4
59
59.9
mm
x5
7
6.9
mm
x6
40
42
mm
Tav
1008
1093
N.m
Tripple
9.08
5.82
percent
Fig. 14. Demonstration of electromagnetic
corresponding to basic design and optimal design
V. CONCLUSION
B. Multi-objective Optimization by BFGS Method
BFGS optimization method carries out the optimization
based on the second order Taylor expansion of multivariable
function f at vicinity of vector Xk which is presented as (5) [7].
(5)
Where βk is known as Hessian Matrix of f in Kth iteration.
By applying the gradient operator to both sides of (5),
minimum point can be stated as (6):
(6)
Where Pk stands for search direction in kth iteration in order
to calculate P0, assumption of H0=I could be acceptable. (7)
represents the updating relation that is exerted on optimizer
variables where αk is determined by line search.
Xk+1=Xk+αk Pk
torque
(7)
By defining Sk and Yk as (8) and (9), respectively, inverse
of Hessian matrix in iteration of k+1 is updated by (10) [7].
(8)
(9)
(10)
C. Optimization Results
Initial state for optimization algorithm are the values
assigned to basic design which are obtained by sensitivity
In this paper, a multi-objective optimal rotor design of an
IPMSM by BFGS method that is one of Quasi Newton
methods has been presented. In order to determine range of
optimization variables and to assign initial state of
optimization algorithm, a sensitivity analysis has been
performed and finite element method is used as an analysis
tool. Furthermore, average torque and torque ripple are
compared. The proposed method can be generalized for all the
IPMSMs topologies.
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