# Modified FOA Applied to Parameter Extraction of Flux-Gate Core

Authors: Wenjuan Jiang; Yunbo Shi; Wenjie Zhao
Journal: Journal of Sensors
 (2017)

---

## Abstract

The accuracy of the magnetic core model is important to the analysis and design of the flux-gate sensor. The Jiles-Atherton model (J-A model) is the mostly used model to describe the hysteresis characteristics of the flux-gate core. But the parameters of J-A model are difficult to identify. In this paper, Fruit Fly Optimization Algorithm (FOA) is proposed to identify the parameters of the J-A model. In order to enhance the performance of the identification, a Modified Fruit Fly Optimization Algorithm (MFOA) is applied to extract the parameters of the flux-gate core. The effectiveness of MFOA is verified through five typical test functions. The influence of variation factorh on the performance of MFOA is discussed. The impact of variation factor h on parameters extraction of hysteresis loop is studied. It is shown that MFOA with appropriate selection of variation factor h will get better performance in the accuracy, stability, and simulation time compared to those of PSO and FOA.


---

## Body

## 1. Introduction

Flux-gate sensor is the best vector magnetic sensor with integrated performance [1]. The magnetic property of iron core material is an important factor that influences the characteristics of flux-gate. The accuracy of the mathematical model of the iron core material affects the accuracy of the flux-gate analysis. Jiles-Atherton model (J-A model) [2] is based on the physical process of hysteresis and has the merits of less parameters and simplicity in expression, which makes it mostly used in practice. But the parameters are sensitive to the hysteresis loop, so they are difficult to be determined.Many methods have been proposed to determine the J-A model parameters, such as Genetic Algorithm (GA) [3], Particle Swarm Optimization (PSO) [4], Artificial Neural Network (ANN) [5], and Shuffled Frog Leaping Algorithm (SFLA) [6].Literature [7] has proposed using Fruit Fly Optimization Algorithm (FOA) and Modified Fruit Fly Optimization Algorithm (MFOA) to identify the J-A model parameters and shows good performance. But it has not verified the effectiveness of the algorithm and analyzed the influence of the variation factor h on the performance of the MFOA. So the main contributions of this paper are as follows: (1) proposing using dynamic variable step size to modify FOA; (2) discussing the effectiveness of MFOA by tests on five typical functions; (3) analyzing the influence of variation factor h on the performance of MFOA; (4) applying the MFOA to extract the parameters of hysteresis loop and analyze the simulation results compared with PSO and FOA.


## 2. J-A Hysteresis Model

J-A hysteresis model is a mathematical model for describing the nonlinear characteristics of magnetic cores, which was proposed by Jiles and Atherton in 1986 [2]. J-A model is generated from the physical process of the magnetization M which is divided into irreversible magnetization component Mirr caused by domain wall displacement and reversible magnetization component Mrev caused by domain wall bending. According to (1), the relationship between M and the magnetic field strength H can be described as (2):(1)M=Mrev+Mirr,Mrev=cMan-Mirr,dMirrdHe=1δkMan-M,He=H+αM,Man=Ms·coth⁡Hea-aHe,(2)dMdH=1-cMan-M+ckδdMan/dHδk-α1-cMan-M.Here,Man is the anhysteretic magnetization. He is effective magnetic field. δ is +1 for dH/dt>0 and -1 for dH/dt≤0. The five parameters that need to be identified are as follows: pinning factork, reversible magnetization coefficient c, coupling factor between magnetic domains α, form factor a, and saturation magnetization Ms.


## 3. MFOA and Its Performance

### 3.1. MFOA

Fruit Fly Optimization Algorithm (FOA) was put forward by Pan in 2012, which is described in detail in literature [8]. FOA uses fruit fly group collaboration mechanism and information sharing mechanism to search for the optimal solution, which causes it to have good ability of global optimization. Also the algorithm has the merits of simplicity, less parameters, small amount of calculation, and high precision, which makes it widely used in practical problems [9–11]. But when it encounters complex high dimension optimization problems, FOA may fall into local convergence with slow search speed, low efficiency, and poor stability.In early stage of search, the fruit fly uses smell to search for food, which can help them to search for the position with greater concentration of fruit. In the later stage of search, it has certain blindness and gambling to determine the position of fruit fly, which results in a waste of resources, low efficiency, and poor accuracy.In order to solve this problem, the dynamic variable step size is proposed. The direction of the movement of the fruit fly is determined by(3)Xk+1=x-axis+Random  Value∗Dk∗hk,Yk+1=y-axis+Random  Value∗Dk∗hk.Here,Dk is the distance between the optimal position of fruit fly for the kth iteration and the origin. Parameter h is variation factor which is in the range of [0.1,10] according to different optimization problems.


### 3.2. Simulation Experiments and Results Analysis

In order to verify the effectiveness and advantages of the MFOA, five functions are used to test the algorithm in this paper. Functionsf1 and f2 are the single-peak functions, and functions f3–f5 are multipeak functions. The five test functions are shown in Section 3.2.1. The simulation results are compared with those of the PSO algorithm and the original FOA.


#### 3.2.1. Test Functions

(1)
f
1
–
S
p
h
e
r
e
M
o
d
e
l
(4)
f
1
x
=
∑
i
=
1
n
x
i
2
,
-
100
≤
x
i
≤
100
,
min
⁡
f
1
x
=
f
1
0
,
…
,
0
=
0
(2)
f
2
–
E
x
p
o
n
e
n
t
i
a
l
F
u
n
c
t
i
o
n
(5)
f
2
x
=
exp
⁡
0.5
∑
i
=
1
n
x
i
-
1
,
-
1.28
≤
x
i
≤
1.28
,
min
⁡
f
2
x
=
f
2
0
,
…
,
0
=
0
(3)
f
3
–
A
c
k
l
e
y
F
u
n
c
t
i
o
n
(6)
f
3
x
=
-
20
exp
⁡
-
0.2
1
n
∑
i
=
1
n
x
i
2
-
exp
⁡
1
n
∑
i
=
1
n
cos
⁡
2
π
x
i
+
20
+
e
,
–
32
≤
x
i
≤
32
,
min
⁡
f
3
x
=
f
3
0
,
…
,
0
=
0
(4)
f
4
–
G
r
i
e
w
a
n
k
F
u
n
c
t
i
o
n
(7)
f
4
x
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
x
i
i
+
1
,
–
600
≤
x
i
≤
600
,
min
⁡
f
4
x
=
f
4
0
,
…
,
0
=
0
(5)
f
5
–
R
a
s
t
r
i
g
i
n
F
u
n
c
t
i
o
n
(8)
f
5
x
=
∑
i
=
1
n
x
i
2
-
10
cos
⁡
2
π
x
i
+
10
,
–
5.12
≤
x
i
≤
5.12
,
min
⁡
f
5
x
=
f
5
0
,
…
,
0
=
0
.


#### 3.2.2. Simulation Experiments and Results Analysis

In simulations, the dimension of the five test functions is 30, and maximum number of iterations is 3000, variation factorh of MFOA is 2, and the inertia factor and the acceleration constants of PSO are η=0.75, c1=2, and c2=2.5, respectively.The optimal value, the average value of 20 times, variance, and running time are shown in Table1.Table 1
Simulation results for three methods.
Function
Method
Optimal value
Mean value
Variance
Time/s
f
1
PSO
1.535
e
-
5
8.729
e
-
5
7.584
e
-
9
1.549
FOA
2.177
e
-
6
2.418
e
-
1
1.196
e
-
1
4.164
MFOA
2.083
e
-
307
3.922
e
-
307
0
14.48
f
2
PSO
2.836
e
-
4
0.182
0.136
1.864
FOA
4.045
e
-
3
4.034
e
-
1
3.190
4.285
MFOA
0
0
0
12.08
f
3
PSO
3.191
e
-
4
7.484
e
-
4
5.902
e
-
8
2.611
FOA
1.081
e
-
3
6.061
e
-
1
4.777
e
-
1
4.617
MFOA
0
0
0
12.37
f
4
PSO
1.920
e
-
4
2.182
e
-
2
5.253
e
-
4
2.964
FOA
1.446
e
-
7
4.819
e
-
2
5.311
e
-
3
5.057
MFOA
0
0
0
12.86
f
5
PSO
1.059
2.463
e
+
1
1.128
e
+
2
2.104
FOA
4.340
e
-
4
8.374
e
-
1
5.193
e
-
1
4.575
MFOA
0
0
0
12.35From Table1, the following can be seen: (1) running time: the running time of PSO algorithm is the shortest, and the running time of MFOA is the longest which is about 6 times that of PSO and about 3 times that of FOA; (2) the precision of the algorithm: MFOA has the highest accuracy which is far higher than the FOA and PSO algorithm; (3) variance: the variance values of MFOA for the five test functions are all 0, which states that MFOA has the best stability, FOA for single-peak function also shows instability, and PSO for multipeak function f5 is unstable. During operations, PSO algorithm and FOA are easy to fall into local convergence, especially for multipeak function f5.The optimization process of the five test functions for three algorithms is shown in Figures1–5. Figure 1 shows function value with iterations for function f1 with partially enlarged detail. In order to see the process clearly, Figures 2–4 are the partially enlarged detail figures. From Figures 1–5, it can be seen that MFOA has best convergence precision and stability. The optimization precision and stability of PSO algorithm are higher than those of FOA expect f5 but much lower than MFOA; PSO and FOA methods show instability and low precision for complex multipeak function f5.Figure 1
Optimization process off1.Figure 2
Optimization process off2.Figure 3
Optimization process off3.Figure 4
Optimization process off4.Figure 5
Optimization process off5.In the iterative process, the location search of fruit flies using dynamic variable step size makes fruit flies search in a wider range and improves the diversity and development of fruit flies’ positions, which can help the algorithm escape from local optimal and improve the precision of the algorithm.So MFOA has high convergence precision and strong ability to jump out of local extreme value. To ensure the capability of algorithm to jump out of the extreme, fruit flies are needed to search the best position in a wider range, so the simulation time for 3000 iterations of MFOA is longer than that of FOA.The factors that influence the performance of MFOA are population size, the maximum number of iterations, and the variation factorh. In this paper, simulations have been performed to study the influence of the parameter h. For each test function, the MFOA method is used to find the optimal solutions at h = [0.1, 0.3, 0.5, 0.7, 0.9, 1, 1.1, 1.3, 1.5, 2] and the algorithm runs for 20 times for each h. The curves of average optimal value with h for five test functions are shown in Figure 6, the curves of variance with h are shown in Figure 7, and the curves of simulation time with h are depicted in Figure 8.Figure 6
Optimal value withh for five functions.Figure 7
Variance withh for five functions.
(a)
Full picture
(b)
Partially enlarged pictureFigure 8
Simulation time withh for five functions.From Figure6, the following is found: (1) the optimal values decrease with the increase ofh; (2) when h∈[0.1,0.9], the algorithm has poor precision, and whenh> 1, the algorithm has better precision; (3) the bigger h value is, the better the precision is; (4) when h achieves some value, the optimal value changes little.From Figure7, the following is shown: (1) the variance decreases with the increase of h; (2) when h>1, the algorithm has better stability; (3) the bigger h value is, the better the stability is; (4) when h achieves some value, the variance tends to 0.From Figure8, it can be seen that when h∈[0.1,1.1], the simulation time is almost the same, but when h>1.3, the simulation time is increasing quickly with h.From the analysis above, the following is stated: (1) whenh=1.1, the algorithm has better precision, better stability, and better convergence rate; (2) bigger h has better precision and stability, but the simulation time increases at the same time when h>1.3.


## 3.1. MFOA

Fruit Fly Optimization Algorithm (FOA) was put forward by Pan in 2012, which is described in detail in literature [8]. FOA uses fruit fly group collaboration mechanism and information sharing mechanism to search for the optimal solution, which causes it to have good ability of global optimization. Also the algorithm has the merits of simplicity, less parameters, small amount of calculation, and high precision, which makes it widely used in practical problems [9–11]. But when it encounters complex high dimension optimization problems, FOA may fall into local convergence with slow search speed, low efficiency, and poor stability.In early stage of search, the fruit fly uses smell to search for food, which can help them to search for the position with greater concentration of fruit. In the later stage of search, it has certain blindness and gambling to determine the position of fruit fly, which results in a waste of resources, low efficiency, and poor accuracy.In order to solve this problem, the dynamic variable step size is proposed. The direction of the movement of the fruit fly is determined by(3)Xk+1=x-axis+Random  Value∗Dk∗hk,Yk+1=y-axis+Random  Value∗Dk∗hk.Here,Dk is the distance between the optimal position of fruit fly for the kth iteration and the origin. Parameter h is variation factor which is in the range of [0.1,10] according to different optimization problems.


## 3.2. Simulation Experiments and Results Analysis

In order to verify the effectiveness and advantages of the MFOA, five functions are used to test the algorithm in this paper. Functionsf1 and f2 are the single-peak functions, and functions f3–f5 are multipeak functions. The five test functions are shown in Section 3.2.1. The simulation results are compared with those of the PSO algorithm and the original FOA.


### 3.2.1. Test Functions

(1)
f
1
–
S
p
h
e
r
e
M
o
d
e
l
(4)
f
1
x
=
∑
i
=
1
n
x
i
2
,
-
100
≤
x
i
≤
100
,
min
⁡
f
1
x
=
f
1
0
,
…
,
0
=
0
(2)
f
2
–
E
x
p
o
n
e
n
t
i
a
l
F
u
n
c
t
i
o
n
(5)
f
2
x
=
exp
⁡
0.5
∑
i
=
1
n
x
i
-
1
,
-
1.28
≤
x
i
≤
1.28
,
min
⁡
f
2
x
=
f
2
0
,
…
,
0
=
0
(3)
f
3
–
A
c
k
l
e
y
F
u
n
c
t
i
o
n
(6)
f
3
x
=
-
20
exp
⁡
-
0.2
1
n
∑
i
=
1
n
x
i
2
-
exp
⁡
1
n
∑
i
=
1
n
cos
⁡
2
π
x
i
+
20
+
e
,
–
32
≤
x
i
≤
32
,
min
⁡
f
3
x
=
f
3
0
,
…
,
0
=
0
(4)
f
4
–
G
r
i
e
w
a
n
k
F
u
n
c
t
i
o
n
(7)
f
4
x
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
x
i
i
+
1
,
–
600
≤
x
i
≤
600
,
min
⁡
f
4
x
=
f
4
0
,
…
,
0
=
0
(5)
f
5
–
R
a
s
t
r
i
g
i
n
F
u
n
c
t
i
o
n
(8)
f
5
x
=
∑
i
=
1
n
x
i
2
-
10
cos
⁡
2
π
x
i
+
10
,
–
5.12
≤
x
i
≤
5.12
,
min
⁡
f
5
x
=
f
5
0
,
…
,
0
=
0
.


### 3.2.2. Simulation Experiments and Results Analysis

In simulations, the dimension of the five test functions is 30, and maximum number of iterations is 3000, variation factorh of MFOA is 2, and the inertia factor and the acceleration constants of PSO are η=0.75, c1=2, and c2=2.5, respectively.The optimal value, the average value of 20 times, variance, and running time are shown in Table1.Table 1
Simulation results for three methods.
Function
Method
Optimal value
Mean value
Variance
Time/s
f
1
PSO
1.535
e
-
5
8.729
e
-
5
7.584
e
-
9
1.549
FOA
2.177
e
-
6
2.418
e
-
1
1.196
e
-
1
4.164
MFOA
2.083
e
-
307
3.922
e
-
307
0
14.48
f
2
PSO
2.836
e
-
4
0.182
0.136
1.864
FOA
4.045
e
-
3
4.034
e
-
1
3.190
4.285
MFOA
0
0
0
12.08
f
3
PSO
3.191
e
-
4
7.484
e
-
4
5.902
e
-
8
2.611
FOA
1.081
e
-
3
6.061
e
-
1
4.777
e
-
1
4.617
MFOA
0
0
0
12.37
f
4
PSO
1.920
e
-
4
2.182
e
-
2
5.253
e
-
4
2.964
FOA
1.446
e
-
7
4.819
e
-
2
5.311
e
-
3
5.057
MFOA
0
0
0
12.86
f
5
PSO
1.059
2.463
e
+
1
1.128
e
+
2
2.104
FOA
4.340
e
-
4
8.374
e
-
1
5.193
e
-
1
4.575
MFOA
0
0
0
12.35From Table1, the following can be seen: (1) running time: the running time of PSO algorithm is the shortest, and the running time of MFOA is the longest which is about 6 times that of PSO and about 3 times that of FOA; (2) the precision of the algorithm: MFOA has the highest accuracy which is far higher than the FOA and PSO algorithm; (3) variance: the variance values of MFOA for the five test functions are all 0, which states that MFOA has the best stability, FOA for single-peak function also shows instability, and PSO for multipeak function f5 is unstable. During operations, PSO algorithm and FOA are easy to fall into local convergence, especially for multipeak function f5.The optimization process of the five test functions for three algorithms is shown in Figures1–5. Figure 1 shows function value with iterations for function f1 with partially enlarged detail. In order to see the process clearly, Figures 2–4 are the partially enlarged detail figures. From Figures 1–5, it can be seen that MFOA has best convergence precision and stability. The optimization precision and stability of PSO algorithm are higher than those of FOA expect f5 but much lower than MFOA; PSO and FOA methods show instability and low precision for complex multipeak function f5.Figure 1
Optimization process off1.Figure 2
Optimization process off2.Figure 3
Optimization process off3.Figure 4
Optimization process off4.Figure 5
Optimization process off5.In the iterative process, the location search of fruit flies using dynamic variable step size makes fruit flies search in a wider range and improves the diversity and development of fruit flies’ positions, which can help the algorithm escape from local optimal and improve the precision of the algorithm.So MFOA has high convergence precision and strong ability to jump out of local extreme value. To ensure the capability of algorithm to jump out of the extreme, fruit flies are needed to search the best position in a wider range, so the simulation time for 3000 iterations of MFOA is longer than that of FOA.The factors that influence the performance of MFOA are population size, the maximum number of iterations, and the variation factorh. In this paper, simulations have been performed to study the influence of the parameter h. For each test function, the MFOA method is used to find the optimal solutions at h = [0.1, 0.3, 0.5, 0.7, 0.9, 1, 1.1, 1.3, 1.5, 2] and the algorithm runs for 20 times for each h. The curves of average optimal value with h for five test functions are shown in Figure 6, the curves of variance with h are shown in Figure 7, and the curves of simulation time with h are depicted in Figure 8.Figure 6
Optimal value withh for five functions.Figure 7
Variance withh for five functions.
(a)
Full picture
(b)
Partially enlarged pictureFigure 8
Simulation time withh for five functions.From Figure6, the following is found: (1) the optimal values decrease with the increase ofh; (2) when h∈[0.1,0.9], the algorithm has poor precision, and whenh> 1, the algorithm has better precision; (3) the bigger h value is, the better the precision is; (4) when h achieves some value, the optimal value changes little.From Figure7, the following is shown: (1) the variance decreases with the increase of h; (2) when h>1, the algorithm has better stability; (3) the bigger h value is, the better the stability is; (4) when h achieves some value, the variance tends to 0.From Figure8, it can be seen that when h∈[0.1,1.1], the simulation time is almost the same, but when h>1.3, the simulation time is increasing quickly with h.From the analysis above, the following is stated: (1) whenh=1.1, the algorithm has better precision, better stability, and better convergence rate; (2) bigger h has better precision and stability, but the simulation time increases at the same time when h>1.3.


## 3.2.1. Test Functions

(1)
f
1
–
S
p
h
e
r
e
M
o
d
e
l
(4)
f
1
x
=
∑
i
=
1
n
x
i
2
,
-
100
≤
x
i
≤
100
,
min
⁡
f
1
x
=
f
1
0
,
…
,
0
=
0
(2)
f
2
–
E
x
p
o
n
e
n
t
i
a
l
F
u
n
c
t
i
o
n
(5)
f
2
x
=
exp
⁡
0.5
∑
i
=
1
n
x
i
-
1
,
-
1.28
≤
x
i
≤
1.28
,
min
⁡
f
2
x
=
f
2
0
,
…
,
0
=
0
(3)
f
3
–
A
c
k
l
e
y
F
u
n
c
t
i
o
n
(6)
f
3
x
=
-
20
exp
⁡
-
0.2
1
n
∑
i
=
1
n
x
i
2
-
exp
⁡
1
n
∑
i
=
1
n
cos
⁡
2
π
x
i
+
20
+
e
,
–
32
≤
x
i
≤
32
,
min
⁡
f
3
x
=
f
3
0
,
…
,
0
=
0
(4)
f
4
–
G
r
i
e
w
a
n
k
F
u
n
c
t
i
o
n
(7)
f
4
x
=
1
4000
∑
i
=
1
n
x
i
2
-
∏
i
=
1
n
cos
⁡
x
i
i
+
1
,
–
600
≤
x
i
≤
600
,
min
⁡
f
4
x
=
f
4
0
,
…
,
0
=
0
(5)
f
5
–
R
a
s
t
r
i
g
i
n
F
u
n
c
t
i
o
n
(8)
f
5
x
=
∑
i
=
1
n
x
i
2
-
10
cos
⁡
2
π
x
i
+
10
,
–
5.12
≤
x
i
≤
5.12
,
min
⁡
f
5
x
=
f
5
0
,
…
,
0
=
0
.


## 3.2.2. Simulation Experiments and Results Analysis

In simulations, the dimension of the five test functions is 30, and maximum number of iterations is 3000, variation factorh of MFOA is 2, and the inertia factor and the acceleration constants of PSO are η=0.75, c1=2, and c2=2.5, respectively.The optimal value, the average value of 20 times, variance, and running time are shown in Table1.Table 1
Simulation results for three methods.
Function
Method
Optimal value
Mean value
Variance
Time/s
f
1
PSO
1.535
e
-
5
8.729
e
-
5
7.584
e
-
9
1.549
FOA
2.177
e
-
6
2.418
e
-
1
1.196
e
-
1
4.164
MFOA
2.083
e
-
307
3.922
e
-
307
0
14.48
f
2
PSO
2.836
e
-
4
0.182
0.136
1.864
FOA
4.045
e
-
3
4.034
e
-
1
3.190
4.285
MFOA
0
0
0
12.08
f
3
PSO
3.191
e
-
4
7.484
e
-
4
5.902
e
-
8
2.611
FOA
1.081
e
-
3
6.061
e
-
1
4.777
e
-
1
4.617
MFOA
0
0
0
12.37
f
4
PSO
1.920
e
-
4
2.182
e
-
2
5.253
e
-
4
2.964
FOA
1.446
e
-
7
4.819
e
-
2
5.311
e
-
3
5.057
MFOA
0
0
0
12.86
f
5
PSO
1.059
2.463
e
+
1
1.128
e
+
2
2.104
FOA
4.340
e
-
4
8.374
e
-
1
5.193
e
-
1
4.575
MFOA
0
0
0
12.35From Table1, the following can be seen: (1) running time: the running time of PSO algorithm is the shortest, and the running time of MFOA is the longest which is about 6 times that of PSO and about 3 times that of FOA; (2) the precision of the algorithm: MFOA has the highest accuracy which is far higher than the FOA and PSO algorithm; (3) variance: the variance values of MFOA for the five test functions are all 0, which states that MFOA has the best stability, FOA for single-peak function also shows instability, and PSO for multipeak function f5 is unstable. During operations, PSO algorithm and FOA are easy to fall into local convergence, especially for multipeak function f5.The optimization process of the five test functions for three algorithms is shown in Figures1–5. Figure 1 shows function value with iterations for function f1 with partially enlarged detail. In order to see the process clearly, Figures 2–4 are the partially enlarged detail figures. From Figures 1–5, it can be seen that MFOA has best convergence precision and stability. The optimization precision and stability of PSO algorithm are higher than those of FOA expect f5 but much lower than MFOA; PSO and FOA methods show instability and low precision for complex multipeak function f5.Figure 1
Optimization process off1.Figure 2
Optimization process off2.Figure 3
Optimization process off3.Figure 4
Optimization process off4.Figure 5
Optimization process off5.In the iterative process, the location search of fruit flies using dynamic variable step size makes fruit flies search in a wider range and improves the diversity and development of fruit flies’ positions, which can help the algorithm escape from local optimal and improve the precision of the algorithm.So MFOA has high convergence precision and strong ability to jump out of local extreme value. To ensure the capability of algorithm to jump out of the extreme, fruit flies are needed to search the best position in a wider range, so the simulation time for 3000 iterations of MFOA is longer than that of FOA.The factors that influence the performance of MFOA are population size, the maximum number of iterations, and the variation factorh. In this paper, simulations have been performed to study the influence of the parameter h. For each test function, the MFOA method is used to find the optimal solutions at h = [0.1, 0.3, 0.5, 0.7, 0.9, 1, 1.1, 1.3, 1.5, 2] and the algorithm runs for 20 times for each h. The curves of average optimal value with h for five test functions are shown in Figure 6, the curves of variance with h are shown in Figure 7, and the curves of simulation time with h are depicted in Figure 8.Figure 6
Optimal value withh for five functions.Figure 7
Variance withh for five functions.
(a)
Full picture
(b)
Partially enlarged pictureFigure 8
Simulation time withh for five functions.From Figure6, the following is found: (1) the optimal values decrease with the increase ofh; (2) when h∈[0.1,0.9], the algorithm has poor precision, and whenh> 1, the algorithm has better precision; (3) the bigger h value is, the better the precision is; (4) when h achieves some value, the optimal value changes little.From Figure7, the following is shown: (1) the variance decreases with the increase of h; (2) when h>1, the algorithm has better stability; (3) the bigger h value is, the better the stability is; (4) when h achieves some value, the variance tends to 0.From Figure8, it can be seen that when h∈[0.1,1.1], the simulation time is almost the same, but when h>1.3, the simulation time is increasing quickly with h.From the analysis above, the following is stated: (1) whenh=1.1, the algorithm has better precision, better stability, and better convergence rate; (2) bigger h has better precision and stability, but the simulation time increases at the same time when h>1.3.


## 4. Parameters Extraction of J-A Model

The fitness function in this paper is as follows:(9)fitness=1N∑i=1NBcali-Bi2.Here,Bcali is the calculated value of B(Hi) and Bi is experimental data at Hi.


### 4.1. Diagram

Figure9 is the diagram of the parameters extraction of J-A model applying MFOA.Figure 9
The implementation procedure of MFOA.


### 4.2. Parameters Extraction of J-A Model

#### 4.2.1. The Influence of Factorh on the Extraction Result

From Section3.2.2, it can be seen that the factor h influences the performance of the MFOA. Therefore, the influence of the factor h on the extraction performance has been discussed in this section. Here,h= [0.1, 0.3, 0.5, 0.7, 0.9, 1.1, 1.3, 1.5, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Figures 10–12 show the curves of fitness value, variance, and simulation time with h, respectively.Figure 10
Fitness value withh for parameters extraction of J-A model.Figure 11
Variance withh for parameters extraction of J-A model.Figure 12
Simulation time withh for parameters extraction of J-A model.From Figure10, the following can be found: (1) when h∈[0.1,0.7], the fitness value decreases with the increase of h; (2) when h>1.1, the fitness value increases; (3) when h=0.9, the fitness value (0.005 T) is the best. From Figure 11, the following can be seen: (1) when h=0.7, h=0.9, and h=1.1, the variances are the smallest, which are 3.44×10-6, 2.59×10-6, and 2.55×10-6, respectively; (2) when h>1.3, the variance is increased with the increase of h. Figure 12 shows the following: (1) when h∈[0.1,0.9], the simulation time increases with the increase of h; (2) when h>2, the simulation time changes in a small way, which is about 130 s. So it is stated that when h=0.9, the MFOA has good extraction performance in accuracy, stability, and simulation time.


#### 4.2.2. Parameters Setting

MFOA, FOA, and PSO are used to identify the parameters of J-A model according to the experimental hysteresis loop of the nonoriented steel V3250-50A taken from literature [3]. The parameters of PSO algorithm are set according to literature [12]: swarm size 100, inertial factor 0.75, and acceleration constants c1=2 and c2=2.5. The swarm size of FOA and MFOA is also set to 100. The variation factor h of MFOA is 0.9 according to Section 4.2.1. The maximum iterations for three methods are all 50.


#### 4.2.3. Simulation Results and Analysis

The parameters of J-A model obtained by three methods are as follows:(1)
MFOA:Ms = 1.2426 × 106, a=63.86, α=11.79×10-5, c=0.77, k=60.97.(2)
FOA:Ms = 1.2446 × 106, a=51.26, α=9.076×10-5, c=0.34, k=66.70.(3)
PSO:Ms = 1.2312 × 106, a=71.55, α=11.89×10-5, c=0.53, k=69.22.Figure13 depicts the hysteresis loops according to the parameters obtained. The errors between calculated data and measured data for three methods are shown in Figure 14, which gives the part when dH/dt>0. The blue dot line (obtained by PSO) shows the error is small when the hysteresis loop is becoming saturated, and the biggest error −0.205 T is obtained at H=100 A/m. The green line (obtained by FOA) shows the biggest error is −0.1507 T at H=6.75 A/m and the error is bigger than that of MFOA and PSO when H is becoming saturated. The red line (obtained by MFOA) shows the biggest error is 0.09 T at H=-100 A/m and the error is bigger than that of PSO when H is becoming saturated.Figure 13
Calculated and measuredB-H curves.Figure 14
Error curves of three methods whendH/dt>0.Table2 shows convergence values of fitness function (the best, worst, and mean), variance, and simulation time. From Table 2, it can be observed that the fitness value achieved by MFOA is better than the minimum fitness values achieved by PSO and FOA. The precision of MFOA is the best of the three methods. The variance of the convergence values for 20 times obtained by MFOA is 2.590×10-6 which is the smallest of the three methods. The smaller the variance is, the better the stability of the algorithm is. So the stability of MFOA is much better than that of FOA and PSO. The simulation time for PSO is much longer than that of FOA and MFOA.Table 2
The results of convergence values.
Methods
Best
Worst
Mean
Variance
Time/s
PSO
0.0125
0.0589
0.0279
2.976
e
-
4
4932
FOA
0.0060
0.0286
0.0086
4.973
e
-
5
150
MFOA
0.0039
0.0079
0.0050
2.590
e
-
6
137


## 4.1. Diagram

Figure9 is the diagram of the parameters extraction of J-A model applying MFOA.Figure 9
The implementation procedure of MFOA.


## 4.2. Parameters Extraction of J-A Model

### 4.2.1. The Influence of Factorh on the Extraction Result

From Section3.2.2, it can be seen that the factor h influences the performance of the MFOA. Therefore, the influence of the factor h on the extraction performance has been discussed in this section. Here,h= [0.1, 0.3, 0.5, 0.7, 0.9, 1.1, 1.3, 1.5, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Figures 10–12 show the curves of fitness value, variance, and simulation time with h, respectively.Figure 10
Fitness value withh for parameters extraction of J-A model.Figure 11
Variance withh for parameters extraction of J-A model.Figure 12
Simulation time withh for parameters extraction of J-A model.From Figure10, the following can be found: (1) when h∈[0.1,0.7], the fitness value decreases with the increase of h; (2) when h>1.1, the fitness value increases; (3) when h=0.9, the fitness value (0.005 T) is the best. From Figure 11, the following can be seen: (1) when h=0.7, h=0.9, and h=1.1, the variances are the smallest, which are 3.44×10-6, 2.59×10-6, and 2.55×10-6, respectively; (2) when h>1.3, the variance is increased with the increase of h. Figure 12 shows the following: (1) when h∈[0.1,0.9], the simulation time increases with the increase of h; (2) when h>2, the simulation time changes in a small way, which is about 130 s. So it is stated that when h=0.9, the MFOA has good extraction performance in accuracy, stability, and simulation time.


### 4.2.2. Parameters Setting

MFOA, FOA, and PSO are used to identify the parameters of J-A model according to the experimental hysteresis loop of the nonoriented steel V3250-50A taken from literature [3]. The parameters of PSO algorithm are set according to literature [12]: swarm size 100, inertial factor 0.75, and acceleration constants c1=2 and c2=2.5. The swarm size of FOA and MFOA is also set to 100. The variation factor h of MFOA is 0.9 according to Section 4.2.1. The maximum iterations for three methods are all 50.


### 4.2.3. Simulation Results and Analysis

The parameters of J-A model obtained by three methods are as follows:(1)
MFOA:Ms = 1.2426 × 106, a=63.86, α=11.79×10-5, c=0.77, k=60.97.(2)
FOA:Ms = 1.2446 × 106, a=51.26, α=9.076×10-5, c=0.34, k=66.70.(3)
PSO:Ms = 1.2312 × 106, a=71.55, α=11.89×10-5, c=0.53, k=69.22.Figure13 depicts the hysteresis loops according to the parameters obtained. The errors between calculated data and measured data for three methods are shown in Figure 14, which gives the part when dH/dt>0. The blue dot line (obtained by PSO) shows the error is small when the hysteresis loop is becoming saturated, and the biggest error −0.205 T is obtained at H=100 A/m. The green line (obtained by FOA) shows the biggest error is −0.1507 T at H=6.75 A/m and the error is bigger than that of MFOA and PSO when H is becoming saturated. The red line (obtained by MFOA) shows the biggest error is 0.09 T at H=-100 A/m and the error is bigger than that of PSO when H is becoming saturated.Figure 13
Calculated and measuredB-H curves.Figure 14
Error curves of three methods whendH/dt>0.Table2 shows convergence values of fitness function (the best, worst, and mean), variance, and simulation time. From Table 2, it can be observed that the fitness value achieved by MFOA is better than the minimum fitness values achieved by PSO and FOA. The precision of MFOA is the best of the three methods. The variance of the convergence values for 20 times obtained by MFOA is 2.590×10-6 which is the smallest of the three methods. The smaller the variance is, the better the stability of the algorithm is. So the stability of MFOA is much better than that of FOA and PSO. The simulation time for PSO is much longer than that of FOA and MFOA.Table 2
The results of convergence values.
Methods
Best
Worst
Mean
Variance
Time/s
PSO
0.0125
0.0589
0.0279
2.976
e
-
4
4932
FOA
0.0060
0.0286
0.0086
4.973
e
-
5
150
MFOA
0.0039
0.0079
0.0050
2.590
e
-
6
137


## 4.2.1. The Influence of Factorh on the Extraction Result

From Section3.2.2, it can be seen that the factor h influences the performance of the MFOA. Therefore, the influence of the factor h on the extraction performance has been discussed in this section. Here,h= [0.1, 0.3, 0.5, 0.7, 0.9, 1.1, 1.3, 1.5, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Figures 10–12 show the curves of fitness value, variance, and simulation time with h, respectively.Figure 10
Fitness value withh for parameters extraction of J-A model.Figure 11
Variance withh for parameters extraction of J-A model.Figure 12
Simulation time withh for parameters extraction of J-A model.From Figure10, the following can be found: (1) when h∈[0.1,0.7], the fitness value decreases with the increase of h; (2) when h>1.1, the fitness value increases; (3) when h=0.9, the fitness value (0.005 T) is the best. From Figure 11, the following can be seen: (1) when h=0.7, h=0.9, and h=1.1, the variances are the smallest, which are 3.44×10-6, 2.59×10-6, and 2.55×10-6, respectively; (2) when h>1.3, the variance is increased with the increase of h. Figure 12 shows the following: (1) when h∈[0.1,0.9], the simulation time increases with the increase of h; (2) when h>2, the simulation time changes in a small way, which is about 130 s. So it is stated that when h=0.9, the MFOA has good extraction performance in accuracy, stability, and simulation time.


## 4.2.2. Parameters Setting

MFOA, FOA, and PSO are used to identify the parameters of J-A model according to the experimental hysteresis loop of the nonoriented steel V3250-50A taken from literature [3]. The parameters of PSO algorithm are set according to literature [12]: swarm size 100, inertial factor 0.75, and acceleration constants c1=2 and c2=2.5. The swarm size of FOA and MFOA is also set to 100. The variation factor h of MFOA is 0.9 according to Section 4.2.1. The maximum iterations for three methods are all 50.


## 4.2.3. Simulation Results and Analysis

The parameters of J-A model obtained by three methods are as follows:(1)
MFOA:Ms = 1.2426 × 106, a=63.86, α=11.79×10-5, c=0.77, k=60.97.(2)
FOA:Ms = 1.2446 × 106, a=51.26, α=9.076×10-5, c=0.34, k=66.70.(3)
PSO:Ms = 1.2312 × 106, a=71.55, α=11.89×10-5, c=0.53, k=69.22.Figure13 depicts the hysteresis loops according to the parameters obtained. The errors between calculated data and measured data for three methods are shown in Figure 14, which gives the part when dH/dt>0. The blue dot line (obtained by PSO) shows the error is small when the hysteresis loop is becoming saturated, and the biggest error −0.205 T is obtained at H=100 A/m. The green line (obtained by FOA) shows the biggest error is −0.1507 T at H=6.75 A/m and the error is bigger than that of MFOA and PSO when H is becoming saturated. The red line (obtained by MFOA) shows the biggest error is 0.09 T at H=-100 A/m and the error is bigger than that of PSO when H is becoming saturated.Figure 13
Calculated and measuredB-H curves.Figure 14
Error curves of three methods whendH/dt>0.Table2 shows convergence values of fitness function (the best, worst, and mean), variance, and simulation time. From Table 2, it can be observed that the fitness value achieved by MFOA is better than the minimum fitness values achieved by PSO and FOA. The precision of MFOA is the best of the three methods. The variance of the convergence values for 20 times obtained by MFOA is 2.590×10-6 which is the smallest of the three methods. The smaller the variance is, the better the stability of the algorithm is. So the stability of MFOA is much better than that of FOA and PSO. The simulation time for PSO is much longer than that of FOA and MFOA.Table 2
The results of convergence values.
Methods
Best
Worst
Mean
Variance
Time/s
PSO
0.0125
0.0589
0.0279
2.976
e
-
4
4932
FOA
0.0060
0.0286
0.0086
4.973
e
-
5
150
MFOA
0.0039
0.0079
0.0050
2.590
e
-
6
137


## 5. Conclusions

In this paper, a modified FOA is proposed and the effects of the variation factorh of MFOA are studied through five test functions. The results show that the variation factor h of MFOA has important influence on the accuracy, stability, and simulation time of the algorithm. The reasonable selection of h can improve the comprehensive performance of the algorithm. FOA and MFOA are applied to optimize the parameters of J-A model. The simulation results show that MFOA with reasonable selection of h is better than PSO and FOA in precision and stability.In this paper, the parameters of J-A model are extracted from static hysteresis loop. But the hysteresis loop varies with the changes of temperature and the frequency of the excitation magnetic field. So more work should be done on the following: (1) the improvement of J-A model with temperature and frequency; (2) parameters extraction of the improved J-A model.



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*Source: 9671378-2017-03-14.xml*