An anisotropic magnetostriction model based on BP neural network combining Levenberg-Marquardt algorithm and particle swarm optimization

Abstract

In this paper, the dynamic magnetostriction of the electrical steel sheet is measured by a triaxial strain gauge based on a single-sheet tester under the alternating magnetic flux condition. The anisotropic magnetostriction property is analyzed, and an improved back propagation neural network assisted model is applied to model the anisotropic principal strain of magnetostriction along different excitation directions. To keep the evaluation cost at an acceptable level, the Levenberg-Marquardt algorithm combined with particle swarm optimization is developed. Finally, the proposed model is verified by comparing the measured magnetostriction and computed one.

Keywords

Anisotropic magnetostriction electrical steel BP neural network alternating magnetization

1. Introduction

The magnetostriction of the electrical steel sheet causes the vibration and undesirable noise in transformers and motors, which reduces the service life of the electrical devices and affects the human health. Therefore, it is essential to evaluate the magnetostriction in the electrical steel sheet [1, 2, 3, 4, 5]. For the dynamic magnetostriction in the electrical steel under purely sinusoidal induction, the back propagation neural network (BPNN) is widely used to predict the magnetostriction loops with different magnetic inductions along the rolling direction (RD) [6, 7]. However, when studying the anisotropic principal strain of magnetostriction along various excitation directions different from the RD, the BPNN based on the gradient descent (GD) algorithm is hard to reach the acceptable convergence condition due to the increase of the numbers of weights and thresholds of the neural network.

In this paper, with the help of the dynamic magnetostriction measurement device, the dynamic magnetostriction is measured by a triaxial strain gauge under some alternating inductions with different magnitudes and directions, and the anisotropic magnetostriction property is analyzed. A BPNN model assisted with the Levenberg-Marquardt (LM) algorithm and particle swarm optimization (PSO) algorithm is proposed to model the anisotropic magnetostriction property of the electrical steel under alternating magnetization. The validation and fast convergence ability of the model are verified.

2. Measurement of magnetostrictive property

The measurement device of magnetostriction is shown in Fig. 1. The exciting device adopts the single yoke structure, and a long strip of electrical steel sheet with 600 mm in length and 100 mm in width is chosen as specimen. The exciting voltage in the winding is controlled in a closed-loop way so that the sinusoidal magnetic flux density waveform at 50 Hz is generated in the sample. To investigate the anisotropic magnetostriction of the electrical steel sheet under different magnetizing directions, the specimen is cut with 15 ${}^{\circ}$ intervals from the RD to the transverse direction (TD) so that the magnetization direction $\theta_{B}$ can be regarded as varying from 0 ${}^{\circ}$ to 90 ${}^{\circ}$ with intervals of 15 ${}^{\circ}$ . To calculate the magnetostrictive principle strain, a triaxial strain gauge which can measure the strains at angles $\varphi$ of 0 ${}^{\circ}$ , 45 ${}^{\circ}$ and 90 ${}^{\circ}$ with respect to the RD is stuck on the surface of the specimen. The triaxial strain gauge is KFG-10-120-D17-11 from KYOWA, of which gauge factor, gauge grid length, gauge resistance at 24 ${}^{\circ}$ C, and adoptable thermal expansion are 2.10% $\pm$ 1.0%, 10 mm, 120.4 $\pm$ 0.4 $\Omega$ , and 11.7 ppm/ ${}^{\circ}$ C, respectively. During the measurement, the magnetic flux density amplitude $|B|$ is controlled from 0.5 T to 1.4 T with intervals of 0.1 T.

Figure 1.

Measurement device of magnetostriction. (a) Exciting device. (b) Triaxial strain gauge. (c) Configuration of exciting device.

Figure 2.

Magnetostrictive loops at three different detection directions when $\theta_{B}=$ 0 ${}^{\circ}$ , $|$ B $|=$ 0.8 T, 1.0 T, 1.2 T, 1.4 T. (a) $\varphi_{a}=$ 0 ${}^{\circ}$ . (b) $\varphi_{b}=$ 45 ${}^{\circ}$ . (c) $\varphi_{c}=$ 90 ${}^{\circ}$ .

Figures 2–4 show the measured magnetostrictive loops of three detection directions of the triaxial strain gauge when the magnetization direction $\theta_{B}$ is along the RD, TD and 45 ${}^{\circ}$ with respect to the RD, respectively. In these figures, the magnetostrictive loops, i.e. butterfly curves, are symmetric with the axis of B $=$ 0.0 T, which means the variation frequency of magnetostriction is twice than the excitation frequency, and their peak values increase with the increase of magnetization strength. In Fig. 2, when the magnetization direction is along RD, the magnetostriction along detection directions of 0 ${}^{\circ}$ and 45 ${}^{\circ}$ is positive and has the property of elongating the specimen, while that along detection directions of 90 ${}^{\circ}$ is contractive. Moreover, the largest elongation of nearly 8.0 $\mu$ m/m happens at the RD. In Fig. 3, when the magnetization direction is along TD, the magnetostriction along the TD is of elongation and attains to the nearly 7.0 $\mu$ m/m, while in Fig. 4 when the magnetization direction is at the angle of 45 ${}^{\circ}$ with respect to the RD, the largest elongation occurs at 45 ${}^{\circ}$ and is nearly just 2.3 $\mu$ m/m. Therefore, from the measurement results, the magnetostriction along different magnetization directions appears to be anisotropic.

Figure 3.

Magnetostrictive loops at three different detection directions when $\theta_{B}=$ 90 ${}^{\circ}$ , $|$ B $|=$ 0.8 T, 1.0 T, 1.2 T, 1.4 T. (a) $\varphi_{a}=$ 0 ${}^{\circ}$ . (b) $\varphi_{b}=$ 45 ${}^{\circ}$ . (c) $\varphi_{c}=$ 90 ${}^{\circ}$ .

Figure 4.

Magnetostrictive loops at three different detection directions when $\theta_{B}=$ 45 ${}^{\circ}$ , $|$ B $|=$ 0.8 T, 1.0 T, 1.2 T, 1.4 T. (a) $\varphi_{a}=$ 0 ${}^{\circ}$ . (b) $\varphi_{b}=$ 45 ${}^{\circ}$ . (c) $\varphi_{c}=$ 90 ${}^{\circ}$ .

3. Modeling of anisotropic magnetostriction based on the improved BPNN model

Figure 5 shows the general training scheme of the BPNN. The training sets are chosen from measured magnetic flux density $B_{x}(t)$ and $B_{y}(t)$ and measured magnetostriction along three detection directions $\lambda_{a}(t)$ , $\lambda_{b}(t)$ , $\lambda_{c}(t)$ . These values in the time domain are transformed into frequency components by the fast Fourier transform (FFT) and the first three dominant harmonics in Fourier series expansion are used to train the neural network in order to reconstruct magnetostriction waves more accurately and keep the estimation time within an acceptable level.

Figure 5.

General training scheme of the BPNN.

The key technique of establishing the BPNN model is to determine the number of hidden layers, the number of neurons in each layer, the type of transfer function and suitable training algorithm. Through various trainings from the GD, an optimal configuration of the BPNN is obtained, which includes 12 neurons in the input layer, 20 neurons in the hidden layer with a sigmoid transfer function, and 18 neurons in the output layer with a linear transfer function. The numbers of the weights and thresholds of the BPNN are 638.

In order to satisfy the training requirements of the BPNN with hundreds of weight thresholds, the conventional BPNN based on GD is improved by the LM algorithm combined with PSO. The LM algorithm uses the approximate second derivative of the information, and it has much faster convergence speed than the GD method. The general description of the LM algorithm is given as follows:

${\rm{\bf x}}(k+1)={\rm{\bf x}}(k)+\Delta{\rm{\bf x}}$ (1)

where x is the vector composed of the weights and thresholds, and $\Delta{\rm{\bf x}}$ is variation of the weights and thresholds. For the LM algorithm, $\Delta{\rm{\bf x}}$ is calculated as

$\Delta{\rm{\bf x}}=-\left[{J^{T}({\rm{\bf x}})J({\rm{\bf x}})+\mu I}\right]^{-% 1}J^{T}({\rm{\bf x}})e({\rm{\bf x}})$ (2)

where $e({\rm{\bf x}})$ is an error vector between calculated and expected values, $\mu$ is adaptive scaling factor, and $I$ is a unit matrix, and J $({\rm{\bf x}})$ is a Jacobian matrix.

Figure 6.

Flowchart of the BP-LM algorithm.

Figure 6 shows the flowchart of an improved BP algorithm modified by LM algorithm. The convergence speed is greatly improved by adjusting the factor $\mu$ to control the search area. However, the above algorithm also leads to the local search minimum when the initial values of x is far away from the optimal values. To solve this problem, the PSO is applied to optimize the initial values of x.

Figure 7.

Flowchart of the optimization for initial weight and threshold.

Figure 7 shows the flowchart of the PSO for optimizing the initial weights and thresholds of the network. Before using the BP-LM algorithm to train the network, all the weights and thresholds of the network are used as a particle which are used to represent the individuals in the population, and the dimension $D$ for the population is determined by the number of neurons in each layer. The population of randomly generated particles is iterated according to the iterative formula of the PSO expressed as:

$\displaystyle V_{i}^{k+1}=\omega V_{i}^{k}+c_{1}r_{1}(P_{i}^{k}-X_{i}^{k})+c_{% 2}r_{2}(G^{k}-X_{i}^{k})$ (3) $\displaystyle X_{i}^{k+1}=X_{i}^{k}+X_{i}^{k+1}$ (4)

where $k$ is the number of the iterations; $\omega$ is inertial weight; $c_{1}$ and $c_{2}$ are nonnegative constants; $r_{1}$ and $r_{2}$ are random in a range [0, 1]; $X_{i}^{k}$ and $V_{i}^{k}$ are the position and velocity of the particle $i$ for iterated $k$ times respectively; $P_{i}^{k}$ and $G^{k}$ are the individual extremum and global extremum respectively. The new update optimal particle restores into the weights and thresholds of the network, then recalculates the mean square error (MSE) of the network. If the MSE is less than the specified error, the optimization process stops, otherwise the iteration will continue until the maximum number of iterations is reached.

Figure 8.

Compared convergence of the BPNN during the training period.

Figure 9.

Calculated and measured $\lambda$ and $\theta_{\lambda}$ when $|B|=$ 1.12 T and $\theta_{B}=$ 20 ${}^{\circ}$ . (a) $\lambda$ . (b) $\theta_{\lambda}$ .

Figure 8 compares the convergence during the training period of the conventional, LM trained and LM-PSO trained BPNN. It can be seen that the conventional GD trained BPNN is running into local minimum with the MSE of nearly 0.073 after 1000 times of iteration when the training algorithm of the BPNN is based on GD. The main reason is due to the increase of the numbers of the weights and thresholds. However, the BPNN based on the LM-PSO is able to meet the convergence conditions and greatly speed up the convergence compared with LM trained BPNN.

Figure 10.

Calculated and measured $\lambda$ and $\theta_{\lambda}$ when $|B|=$ 1.25 T and $\theta_{B}=$ 80 ${}^{\circ}$ . (a) $\lambda$ . (b) $\theta_{\lambda}$ .

Figure 11.

Experimental model of single phase transformer core.

In addition, based on the knowledge of plane strain in material mechanics, the principal strain $\lambda$ of magnetostriction and its direction $\theta_{\lambda}$ can be calculated by using the measurement data of a triaxial strain gauge. In order to the validation of the proposed BPNN model, we calculate the principal strain and its direction under two kinds of inductions with $|B|=$ 1.12 T, $\theta_{B}=$ 20 ${}^{\circ}$ and $|B|=$ 1.25 T, $\theta_{B}=$ 80 ${}^{\circ}$ , which are not the training sample. The computation values are compared with the measured ones as shown in Figs 9 and 10. The two figures show that there is good agreement between the measured and modeled curves.

4. Application

To check the effectiveness of the proposed method to model the anisotropic magnetostriction property, a single phase transformer core model is made as shown in Fig. 11, which is laminated by 10 pieces of oriented electrical steel sheet (30ZH105), and the number of turns in the winding is 160. Three triaxial strain gauges are stuck on the surface of the core to measure the magnetostriction waveform at three different points marked with point ‘ $a$ ’, point ‘ $b$ ’, and point ‘ $c$ ’.

Figure 12.

Measured and calculated values for measuring position ‘ $a$ ’. (a) Flux density locus. (b) $\lambda$ . (c) $\theta_{\lambda}$ .

Figure 13.

Measured and calculated values for measuring position ‘ $b$ ’. (a) Flux density locus. (b) $\lambda$ . (c) $\theta_{\lambda}$ .

Figure 14.

Measured and calculated values for measuring position ‘ $c$ ’. (a) Flux density locus. (b) $\lambda$ . (c) $\theta_{\lambda}$ .

Figures 12–14 show the measured flux density locus and the principal strain at three different test points, respectively, in which the flux density locus are measured by a $B-H$ vector sensor. From these figures, it can be seen that the flux density locus in one exciting period of both position ‘ $a$ ’ and ‘ $c$ ’ are alternating, while the flux density of position ‘ $b$ ’ is rotating. Under the alternating magnetic flux in Figs 12 and 14, the measured and calculated principal strain $\lambda$ have good agreement, and the error from $\theta_{\lambda}$ is still within an acceptable range. The error in $\theta_{\lambda}$ is brought about the difference between the single sheet and the laminated sheet in magnetostrictive property. However, in Fig. 13, under the rotating magnetic flux, the error of measured and calculated values are large than that expected. This is due to lack of training data of the rotational magnetostriction, the model is only trained by magnetostriction under the alternating magnetic field.

If utilizing the proposed BPNN to model the rotational magnetostriction, it will be difficult since the rotating magnetic field covers different magnetization values, inclination angles and axial ratios, which needs a large amount of training sets and results in the low computation efficiency. In order to ensure the calculation speed and the accuracy of the model, it is necessary to further improve the neural network or use another model to replace it, which is our further research work.

5. Conclusion

The magnetostrictive property in the electrical steel sheet is measured under the alternating magnetizations, and the experimental results illustrate the magnetostriction property has highly strong anisotropy along different magnetization directions. An improved BPNN assisted model is proposed to model the anisotropic magnetostriction, in which a BPNN combining the LM algorithm and PSO is applied to keep the evaluation cost at an acceptable level. The validation of proposed model is verified by comparing modeled and experimentally measured magnetostriction of a single phase transformer core, and it is also referred that the proposed model has good modelling accuracy for alternating magnetization and is not suitable for rotational one.

Footnotes

Acknowledgments

This work was supported by National Natural Science Foundation of China under Grant 51777128.

References

Mohammed

O.A.

et al., Acoustic noise signal generation due to magnetostrictive effects in electrical equipment, in: Twenty Second National Radio Science Conference2005, pp. 45–52.

Mose

A.J.

et al., Contribution of magnetostriction to transformer noise, in: Universities Power Engineering Conference2010, pp. 1–5.

Vandevelde

et al., Magnetic forces and magnetostriction in electrical machines and transformer core, IEEE Trans. Magn.39(3) (2003), 1618–1621.

Belahcen

, Vibrations of rotating electrical machines due to magneto-mechanical coupling and magnetostriction, IEEE Trans. Magn.42(4) (2006), 971–974.

Somkun

et al., Measurement and modeling of 2-D magnetostriction of nonoriented electrical steel, IEEE Trans. Magn.48(2) (2012), 2711–714.

Hilgert

et al., Neural-network-based model for dynamic hysteresis in the magnetostriction of electrical steel under sinusoidal induction, IEEE Trans. Magn.44 (2008), 874–877.

Hilgert

et al., Comparison of magetostriction models for use in calculations of vibrations in magnetic cores, IEEE Trans. Magn.44(6) (2008), 874–877.