Research on prediction of nanocrystalline alloy hysteresis properties based on long short-term memory network

Abstract

In order to predict the hysteresis characteristics of nanocrystalline alloy materials at different frequencies, a data-driven hysteresis prediction model based on the encoder–decoder architecture, which combines long short-term memory network and feedforward neural network, is proposed in this paper. The data-driven based magnetic hysteresis prediction model can take advantage of the powerful nonlinear learning ability of artificial neural network to train and learn its magnetic hysteresis characteristics of nanocrystalline alloy materials at different frequencies. Firstly, based on the encoder–decoder architecture, a hysteresis prediction model is constructed by combining long short-term memory network and feedforward neural network. Subsequently, in order to obtain the training set and validation set used for the data-driven based hysteresis prediction model, the Jiles–Atherton (J–A) hysteresis model is identified based on the B–H measurement data of a small number of nanocrystalline alloy materials at different frequencies for expediency since it is quite cumbersome and time-consuming to get these B–H data by measurement. Finally, the validity and accuracy of the data-driven based hysteresis prediction model are proved by the validation set. The maximum error is about 10.29%. The results show that the hysteresis model of neural network is able to predict hysteresis characteristics with considering the effect of frequency, which provides a new way for the simulation of hysteresis characteristics.

Introduction

Soft magnetic materials are widely used in electrical equipment such as motors and transformers. Accurate prediction of its hysteresis characteristics is important for the design and optimization of power equipment¹. Compared with traditional soft magnetic materials such as ferrite and silicon steel sheet, nanocrystalline alloy materials have high saturation flux density, high permeability, low coercivity, and low loss. It has quite a potential in medium and high frequency power equipment². Especially with the power electronics of the power grid, there are a lot of high-order harmonic components in the power grid, and soft magnetic materials such as nanocrystalline alloys may have great potential application. However, in practical applications, nanocrystalline alloys exhibit different hysteresis characteristics at different frequencies. Therefore, accurate and efficient prediction of hysteresis characteristics of nanocrystalline alloys at different frequencies is very important for the design and optimization of power equipment.

At present, many scholars have carried out research on the simulation of hysteresis characteristics of soft magnetic materials and proposed various hysteresis models, such as the Preisach hysteresis model³, J–A hysteresis model⁴, neural network hysteresis model⁵, etc. Among them, the Preisach model describes the hysteresis characteristics of soft magnetic materials as the internal superposition of infinite hysteresis operators⁶. As a purely mathematical model, this method is simple and efficient for the simulation of magnetic hysteresis characteristics, but its disadvantage is that the determination of the distribution function is more difficult. Usually, the Lorentzian function, Gaussian function, Cosh function, and other methods can be used to approximate the Preisach distribution function, but more parameters are needed to be identified for such methods. Moreover, the measurement process of these parameters is complicated and the accuracy is limited⁷. In addition, the distribution function can also be calculated by using the method of first-order reversal curves, but the measurement method of first-order reversal curve is complicated and cumbersome, and errors in the actual measurement process can be introduced. Hence, it is difficult to be widely adopted⁸. J–A hysteresis model is a physical model used to describe hysteresis characteristics based on the pinning effect and the theory of domain wall motion⁹. This model divides the magnetization of soft magnetic materials into reversible magnetization and irreversible magnetization, and formulates the first order differential equations according to the energy balance equation to represent the model. Generally, intelligent optimization algorithms such as particle swarm optimization algorithm and genetic algorithm are used to identify the parameters of the J–A model. However, in the process of parameter identification, problems such as slow convergence speed and local optimal solution are prone to occur¹⁰. In addition, the hysteresis model based on artificial neural network has also been applied in the prediction of hysteresis characteristics. Liu and Chen¹¹ used a Back Propagation Neural Network (BPNN) to update the network’s parameters through error backpropagation to fit the nonlinear relationship between the input and output. The model only needs to use a small amount of hysteresis loop data for training to obtain a more accurate fitting effect.But the study did not explain the specific materials, nor did it consider the influence of frequency, temperature and other factors on hysteresis characteristics. Li and Zhu¹² proposed to combine BPNN with a genetic optimization algorithm to improve parameter identification of the J–A model, which can effectively improve the speed and accuracy of solving.The research method of hysteresis is still using the traditional hysteresis model, which does not play the advantage of neural network. Huang and Hu¹³ used BPNN to approximate the hysteresis loop, and the Preisach hysteresis model was used to establish the neural network expression of hysteresis loop under arbitrary magnetization conditions, which greatly simplified the numerical calculation process. Dong and Zhang¹⁴ used the convolutional neural network to simulate the magnetic characteristics of an electrical steel sheet. The residual module and batch normalization were introduced into the convolutional neural network model to reduce the number of model iterations and improve the model accuracy. These studies suggest that the hysteresis characteristics were initially modeled using only a simple BPNN. However, since the hysteresis characteristics of soft magnetic materials are affected by temperature, stress, frequency, and other factors^15,16. For this class of nonlinear multivariable problems, BPNN suffers from low efficiency.

In recent years, with the rapid development of artificial neural network technology, it has made great progress in dealing with complex nonlinear multivariable problems. Among them, the emergence of Long Short-Term Memory (LSTM) can not only deal with complex nonlinear multivariable problems but also learn the relationship between time series data. Because the hysteresis phenomenon of soft magnetic materials is due to the rotation and displacement of magnetic domains and domain walls inside the soft magnetic materials during the magnetization process, the change of magnetic flux density B lags behind the change of magnetic field strength H. The magnetic flux density B of a soft magnetic material at a certain time is not only related to the current magnetic field strength H but also closely related to the previous magnetization history. Therefore, the prediction of hysteresis characteristics of soft magnetic materials can be regarded as a time series problem, and the hysteresis characteristics of soft magnetic materials can be modeled using the LSTM network.

This paper first establishes a hysteresis characteristic prediction model based on the LSTM network, and introduces the structure and principles of the model. Then, the training set and validation set are obtained by the J–A hysteresis model. And different prediction models are obtained based on different training datasets. Finally, the feasibility of the prediction model is demonstrated by the validation set.

LSTM based construction of data-driven hysteresis prediction model

Encoder–decoder architecture

Encoder–decoder as a network architecture, the encoder converts the input into a fixed-dimensional vector C, which is then processed by the decoder to gradually generate the corresponding output sequence . The encoder–decoder architecture is shown in Fig. 1. With its flexibility, the architecture can select different network algorithms to solve the corresponding problems according to different task requirements, so it is widely used in sequence-sequence prediction problems. In this paper, the sequence of magnetic flux intensity B is predicted mainly by the sequence of magnetic field intensity H, so the encoder–decoder architecture is adopted.Where the input x is and the output y is . The process of encoding and decoding is realized by LSTM and Feedforward Neural Network (FNN).

Fig. 1.

Encoder–decoder architecture.

The hysteresis characteristics of magnetic materials are affected by frequency, temperature and stress. In this paper, in order to predict the hysteresis characteristic of the magnetic material at different frequencies, the frequency variable is added to the model training process. By training B–H data at different frequencies, and constantly updating the weight coefficients of the prediction model, the model can also realize the prediction of hysteresis characteristics considering the external influence factors. The overall hysteresis loop prediction model based on LSTM is shown in Fig. 2. Combining LSTM with FNN under the encoder–decoder architecture, the data sequence of magnetic field intensity H at different frequencies is taken as input. Firstly, the relationship between the data is learned through LSTM, and save the related sequence information such as H(t) trend into the hidden state and cell state. Then, the influence of the corresponding frequency F on hysteresis characteristics is characterized in the neural network through FNN, and the corresponding state information is updated. Finally, the updated state information is used to initialize the decoder and gradually generate the data sequence corresponding to the flux density B.

Fig. 2.

Hysteresis loop prediction model architecture based on LSTM.

Feedforward neural network

FNN is the most basic artificial neural network model, also known as Multilayer Perceptron (MLP)¹⁷. Through the connection between neurons, the input data is one-way transmitted through the input layer, hidden layer and output layer, and there is no cyclic structure. FNN is widely used because of its simple structure, and easy implementation. Therefore, the FNN is adopted to add frequency into network training, considering the influence of frequency on hysteresis loop. The network structure of FNN is shown in Fig. 3, where is the input value of the network corresponding to frequencies of different hysteresis loops; is the hidden state corresponding to different neurons in the hidden layer, which is obtained by the weighted sum of the input values; is the output value obtained after the forward propagation of the input data.

Fig. 3.

FNN basic structure.

In the process of forward propagation of data in different layers. The expression is as follows:

1

where is the input, is the output of the neuron, is the bias value corresponding to the neuron, is the weight corresponding to each pair of neurons in different layers, and is the activation function.

Long short-term memory network

LSTM was first proposed by Sepp Hochreiter¹⁸, aiming to solve the problem of disappearance gradient of Recurrent Neural Network (RNN) during the transmission of long data series and did not have long-term memory ability. Since the hysteresis characteristics of soft magnetic materials are not only related to the current magnetic field strength H, but also to the historical magnetization state, the prediction of hysteresis characteristics can also be regarded as a typical time series problem.The magnetization process of nanocrystalline alloy is shown in Fig. 4. In the initial state, as the magnetic field strength increases, the magnetization curve will gradually rise along a–b. After reaching point b, the magnetic field strength gradually decreases, and the hysteresis loop will gradually decline along the b–c. If the magnetic field strength begins to increase gradually after reaching point c, the hysteresis loop will rise along c-d, and the hysteresis information at point c needs to be recorded. If the magnetic field strength continues to decline after reaching point c, the hysteresis loop will decrease along c-e, and the hysteresis information of c has little influence on the subsequent changes of the hysteresis loop and can be selectively forgotten.

Fig. 4.

Magnetization process of nanocrystalline alloy.

Based on RNN, LSTM can avoid the problem of gradient disappearance during the transmission of long data series by adding input gate ,output gate , forget gate . By the control of the gate, and its combination with the cell state , the information that needs to be remembered or forgotten is determined so that it can better capture the long-term dependence relationship. The structure of LSTM after expansion according to time series is shown in Fig. 5.

Fig. 5.

LSTM basic structure.

The forget gate is used to determine the preservation and forgetting of historical information, which is mainly determined by the hidden state of the previous moment and the input of the current moment. The formula is as follows:

2

where and are divided into the weight matrix corresponding to the input data and the hidden state , and are the corresponding bias matrix respectively.

The function of the input gate is to determine how much information input at the current time is saved to the cell state . The formula is as follows:

3

4

5

where is the candidate memory unit, representing the multiplication of elements at corresponding positions in two matrices.

The output gate determines the information output at the current time, and updates the hidden state at the current time in combination with . The formula is as follows:

6

7

Data acquisition and model training

Magnetic material properties

For different magnetic materials, their various characteristics have a large difference, often used in power equipment among the magnetic materials are silicon steel sheet, ferrite, amorphous alloy and nanocrystalline alloy, their corresponding characteristics parameters are shown in the Table 1.

Table 1.

Material properties.

Parametersmaterial	Silicon steel	Ferrite	Amorphous alloy	Nanocrystalline alloy
Saturation flux density/T	2.03	0.5	1.56	1.25
Resistivity/( cm)	< 1	106	140	115
Maximum permeability (H m−1)	20,000	6000	4000–80,000	400,000
Curie temperature/	750	220	399	570
Operating frequency/kHz	< 0.4	10–400	5–100	20–200
Magnetostriction coefficient/	10	4	30	< 2
Laminated thickness/mm	0.23–0.33	–	0.02–0.04	0.025
Loss/(W kg−1)	1.2 (1.3 T, 50 Hz)	7.5 (0.2 T, 20 kHz)	0.18 (1.3 T, 50 Hz)	< 3.4 (0.2 T, 20 kHz)

It can be seen from the table that the saturation flux density and Curie temperature of the silicon steel sheet are the largest, but the resistivity and operating frequency are the smallest, resulting in excessive loss values at high frequency. The resistivity and operating frequency of ferrite are much higher than the silicon steel sheet, but its saturation flux density is smaller, resulting in a larger volume of power equipment at the same operating frequency. The overall characteristics of amorphous alloy are good, but the disadvantage is that the magnetostriction coefficient is the largest, and excessive vibration and noise will be generated in the application process. In addition to the high production cost, the nanocrystalline alloy has better characteristics than the other three materials in all aspects. Therefore the nanocrystalline alloy material was chosen to study its hysteresis characteristics.

J–A hysteresis model

The hysteresis prediction model of neural network needs a lot of data for the training of the model. Due to the large amount of training data obtained by the method of experimental measurement, it takes a long time, and errors in the actual measurement process can be introduced. In order to obtain a large amount of accurate and effective training data in a short time to verify the validity of neural network models, the J–A hysteresis model is adopted to obtain training data, Nelder-Mead simplex algorithm is used to identify parameters of hysteresis loops of nanocrystalline alloys at different frequencies. This method ensures data accuracy and reduces data acquisition time.

J–A hysteresis model divides the magnetization M into reversible magnetization and irreversible magnetization ¹⁹:

8

9

10

The non-hysteresis magnetization is expressed by Langevin function:

11

12

According to the above formula, the J–A hysteresis model can be expressed as:

13

where is the direction parameter having the value − 1 for 0 and +1 for 0, is the effective field; are the parameters to be identified, and c is the reversible coefficient; is saturation magnetization; a is the shape parameter of the magnetization curve without hysteresis; is the mean field parameter of the coupling inside the magnetic domain; k is the coupling parameter.

Nelder-Mead simplex algorithm is adopted to identify parameters in the J–A hysteresis model in this paper. The algorithm finds the optimal parameters through reflection, expansion, contraction, and does not need the objective function to be derivable, and can quickly converge to the local minimum value^20,21. By identifying parameters of hysteresis loops at different frequencies (1–20 kHz), the corresponding model parameters are obtained. The specific identification results are shown in Fig. 6, and the parameter values under different frequencies are shown in Table 2. The relative error of the hysteresis loop area is used to evaluate the identification results. The relative error is calculated as follows:

14

where is the area of the measured hysteresis loop, and is the area of the hysteresis loop fitted by the J–A model.

Fig. 6.

Parameter identification results at 1–20 kHz.

Table 2.

Identification parameter values at 1–20 kHz.

Frequency/kHz	Ms		k	a	c	Relative error/%
1			2.323	1.871		10.27
2			2.827	1.515		21.69
5			3.624	4.613		20.45
10			4.999	3.662		21.67
15			5.731	11.110		17.96
20			6.553	14.796		14.60

The average relative error of parameter identification is 17.78%. After the model parameters are determined, the relationship between M and H is obtained by solving equation (13), and B–H training data of different magnetic density peaks at different frequencies are obtained according to .

Training of prediction model

A 3-layer LSTM with 64,64,32 neurons in each layer was adopted to build the prediction model, and a 1-layer FNN with 32 neurons was adopted to add the frequency effect to the network model, and a lot of nanocrystalline alloy B–H data obtained by J–A hysteresis model at different frequencies were used to train the model. The specific training process is shown in Fig. 7.

Fig. 7.

Model training process.

The model training process is mainly divided into two parts, data processing and parameter updating.

1)Data processing

The B–H sequence data obtained in a stable period and the corresponding frequency F are used as the inputs of LSTM and FNN respectively:

Because the model prediction results is highly dependent on the quality of training data. Different dimensions of the original data B, H and F will lead to slower model convergence and inaccurate predictions. Therefore, it is necessary to standardize the input data of the model, which is processed in the following ways:

15

where x is the original data, is the mean value of the original data, is the standard deviation of the original data, and is the data after standardized processing.

The standardized data eliminated the dimensional differences, and then the data set was divided into a training set and a validation set in an 8:2 ratio for the training and validation of the model, respectively.

2)Parameter update

The processed data set is used for model input, and the corresponding predicted value is obtained after forward propagation. Mean-square error is adopted as the loss function to evaluate the prediction accuracy, and the loss function is taken as the objective function of the optimization algorithm. Its expression is as follows:

16

where is the predicted value and is the actual value.

In the neural network, the optimizer as an indispensable part is mainly used to update the weight parameters and biases of the model. Adam optimizer is widely used because of its simple implementation, high computational efficiency and low memory requirement. Therefore, the Adam optimizer was adopted to take the loss function as the optimization objective, and the weight parameters of the model were iteratively updated until the optimal number of iterations was reached, and the training was completed.

Analysis of prediction results

Prediction results without considering frequency effect

Firstly, B–H data of different magnetic flux density peaks at the same frequency of nanocrystalline alloy materials are selected as the training set, and the hysteresis prediction model without considering the frequency effect is obtained. Then, the data of the validation set at this frequency is selected as the input of the model. The comparison results between the predicted value obtained from the prediction model and the fitted value obtained from the J–A model are shown in Fig. 8, and the corresponding B–H loop area is shown in Table 3. Relative error is used to evaluate the prediction results.

Fig. 8.

Different Bm predicted values at 10 kHz correspond to B–H loops.

Table 3.

Different magnetic flux density Bm at 10 kHz corresponds to the area of hysteresis loop.

Bm/T	Fitted value	Predicted value	Relative error/%
0.5	9.030	8.899	1.45
1.0	19.285	18.491	4.12
1.2	22.209	21.309	4.05

As shown in Table 3, the error between the predicted value and the fitted value under different maximum magnetic flux density Bm is within 5%, which proves prediction model accuracy.

Consider the frequency effect prediction results

B–H data of nanocrystalline alloy materials at different frequencies (1–20 kHz) were selected as the training set, and the final frequency-dependent hysteresis model was obtained through continuous iterative training. Then, the data of the validation set at different frequencies are selected as the input of the model. The comparison results between the predicted value and the fitted value are shown in Fig. 9, which shows that they are very close to each other. The B–H loop area of the predicted and fitted values at each frequency is shown in Table 4, and the relative error is also used to evaluate the prediction results.

Fig. 9.

The predicted value and the fitted value at 1–20 kHz correspond to the B–H loop.

Table 4.

Predicted and fitted values at different frequencies.

Frequency/kHz	Bm/T	Fitted value	Predicted value	Relative error/%
1	0.5	1.937	1.738	10.29
	1.0	8.962	8.135	9.23
	1.2	10.612	9.906	6.65
2	0.5	3.154	2.982	5.46
	1.0	10.018	9.012	10.04
	1.2	12.709	11.413	10.19
5	0.5	7.009	6.916	1.33
	1.0	14.066	13.946	0.85
	1.2	16.907	16.402	2.98
10	0.5	7.567	7.309	3.40
	1.0	19.146	18.976	0.89
	1.2	22.128	21.209	4.15
15	0.5	10.449	10.206	2.33
	1.0	22.384	21.860	2.34
	1.2	25.175	24.108	4.24
20	0.5	12.464	12.225	1.92
	1.0	24.843	24.219	2.51
	1.2	26.938	26.122	3.03

It can be seen from Table 4 that the area of hysteresis loop increases with the increase of frequency. The maximum error between the predicted results and the fitted results is 10.29%, the minimum error is 0.85%, and the average relative error is 4.55%. These results suggest that LSTM-based data-driven hysteresis prediction model can accurately predict the hysteresis characteristic of nanocrystalline alloy materials at different frequencies.

Conclusion

This paper adopts mainly the LSTM network to model the hysteresis characteristics of nanocrystalline alloy materials at different frequencies. And the J–A hysteresis model was used to obtain a large amount of training data. Then, the hysteresis models of neural network with and without frequency effect were obtained through different training sets, and the validation set was used to verify the effectiveness of the models. The maximum error of the hysteresis model without considering the frequency effect is 4.12%. The maximum error of the hysteresis model considering the frequency effect is 10.29%. It is proved that the hysteresis model of neural network can simulate hysteresis loops accurately after training with different training sets. However, a limitation of this study is that predictive models of neural networks are data-driven models, which train the model mainly through a large amount of data and learn its complex relationships without considering actual physical properties. Future studies will use a large amount of actual measurement data and introduce relevant physical constraints to further improve the model.

Author contributions

Hailin Li, Bo Zhang, Yongpeng Shen, Lei Zhang, Kun Liu: discussion of the idea, formal analysis, data curation, methodology. Hailin Li, Bo Zhang: writing—original draft, writing—review & editing, project administration.

Data availability

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.