Predicting the eddy current loss of a large nuclear power turbo generator using a fuzzy c-means deep Gaussian process regression model

Highlights

• 10,000 samples of a nuclear generator eddy current loss are obtained by FEM.

• A novel fuzzy c-means deep Gaussian process regression model is established.

• The FCM-DGPR method has good performance on predicting the eddy current loss.

• Final result gives score evaluation with the highest R² of 0.9809.

• The FCM-DGPR method shows higher predictive ability than other methods.

Abstract

The distribution of the rotor slot wedge eddy current loss in large nuclear power turbo generators is complex and is influenced by many factors. Excessive eddy current loss leads to severe rotor heating, potentially leading to thermal accidents; therefore, the design precision of large generators must be improved. In this paper, a Fuzzy C-Means Deep Gaussian Process Regression (FCM-DGPR) method is proposed to predict the eddy current loss of a large generator in order to solve the problem of the insufficient accuracy of deep Gaussian process regression (DGPR) with increasing number of the data samples. First, the original dataset is obtained by the finite element method (FEM) and then normalized to construct the samples of the eddy current loss of a large nuclear power generator. Second, the training set is automatically clustered into different subsets by the fuzzy c-means algorithm, and each subset is used to train the DGPR model to obtain different sub models. The membership degree of each data point in the test set is calculated and used to evaluate the sub model of the data. Then, the sub model is used to predict the eddy current loss. Finally, the result is obtained by concatenating the results of each sub model. The results show that the goodness of fit (${\mathit{R}}^2$) is 0.9809, the root mean square error (RMSE) is 0.0271, the prediction error is small, and the model exhibits good prediction performance. Further experimental results show that the FCM-DGPR method is superior to the existing DGPR models and other models and is more suitable for predicting the eddy current loss of large generators.

Introduction

Large nuclear power turbo generators are key components of nuclear power plants, and their design and research play an important role in the development of the power industry [1], [2], [3]. In the operation process of a large turbo generator, the high-order harmonic generated by high-speed rotation will induce eddy currents in the rotor slot wedge and spacer strip of the generator, and the eddy current loss will directly affect the operation of the generator. Excessive eddy current loss will lead to strong rotor heating, resulting in a series of problems, such as local stress concentration and deformation that will not only affect the working efficiency of the generator but also lead to mechanical failure and serious accidents [4], [5], [6]. Therefore, accurate prediction of eddy current loss of a large nuclear power turbo generator is highly important for the design to ensure the smooth and stable generator operation.

In recent years, machine learning has been widely used in various fields of electrical engineering and has achieved many successes. Breker et al. classified low-voltage grids using machine learning techniques, in particular support vector machines (SVMs). Experiments were based on the data for 300 real rural and suburban low-voltage grids, and the results show that the approach can significantly better reflect domain expert assessments than a previously proposed technique [7]. Afrasiabi et al. designed a deep learning-based approach to predict the probability density function (PDF) of wind for several hours in the future. The convolutional neural network was utilized to enhance learning spatial features. The performance of the proposed approach in comparison with several state-of-the-art and previously investigated approaches was validated on two actual datasets [8]. Liu et al. proposed building the mapping from apparent resistivity data (input) to a resistivity model (output) directly by convolutional neural networks (CNNs) to study the inverse problem of electrical resistivity surveys (ERSs). The feasibility and efficiency of the proposed methods were demonstrated by six groups of experiments. Based on deep reinforcement learning [9]. Schenke et al. presented a first proof of concept based on controlling the phase currents of a permanent magnet synchronous motor in a field-oriented framework. Their results are promising and motivated further research in this field [10]. Stefenon et al. proposed an optimized ensemble extreme learning machine that outperforms the original EN-ELM method. Finally, their results show a significant increase in robustness and a faster training procedure compared to classical approaches [11].

These researchers have explored how to combine machine learning and soft computing methods to study specific scientific problems in the field of electrical engineering and solve critical technical problems. However, for the design of a large nuclear power turbo generator, due to its complex structure, the difficulty of measuring the internal magnetic distribution, the nonlinear electromagnetic model of the generator, and the difficult-to-decouple influencing factors, it is difficult to accurately calculate the eddy current loss. This further leads to the high cost of R&D and the difficulty of constructing experimental samples. On the other hand, the high value of a large nuclear power turbo generator requires high prediction accuracy. Therefore, in the study of the large nuclear power turbo generator body magnetic heat-loss problem, it is necessary to obtain a method that can accurately predict the eddy current loss even if the number of samples is small; the deep Gaussian process (DGP) can meet this requirement.

DGP is a novel deep network that was first proposed by Damianou et al. in 2013 [12]. It is a multilayer generalization of the Gaussian process and is equivalent to an infinite width hidden layer neural network in its form. The DGP is probabilistic and nonparametric. It can summarize the data more flexibly and estimate the uncertainty more accurately than other deep models. For smaller datasets, DGP still has good generalization ability, solving the problem of whether a deep abstract structure can be learned in smaller datasets.

With DGP gradually becoming a research hot spot, it has undergone many improvements. Bui et al. proposed an extended expectation propagation method called stochastic expectation propagation. This method can automatically find useful input information, and expand or compress the model and is a flexible form of Bayesian kernel design. Moreover, the method is highly efficient, scalable and easy to implement [13]. In 2016, Bui et al. proposed a differential deterministic approximation method based on an approximate expectation propagation algorithm and variational approximation that avoids the difficulty of calculation and analysis. For the edge problem in the approximate expectation propagation algorithm, a new extended probability backpropagation algorithm was proposed. The new method was evaluated on eleven datasets and compared with some of the latest algorithms of Bayesian neural networks. This new approximate Bayesian learning scheme enables the DGPR model to be applied to medium- and large-scale problems for the first time [14]. Dai et al. proposed an extensible deep nonparametric generation model by extending the DGP. The inference is carried out in a new scalable variational framework, and the posterior distribution of variation is reparameterized by a multilayer perceptron. The key aspect of this reformulation is to prevent the diffusion of variational parameters; otherwise, the number of parameters will increase proportionally with the increase in the number of samples. The new variational lower bound formula can allocate most of the calculation to the dataset of the mainstream deep learning task size [15]. Hugh et al. proposed a DGP based on expectation propagation and the Monte Carlo Markov chain (MCMC) method. This model uses the MCMC method to approximate the posterior distribution of the Gaussian process, reducing the parameters and better approximating the curve with lower computational complexity [16]. Zhao et al. applied the DGP to study the load eddy current loss prediction of large generator fasteners and achieved good results [17]. However, these studies still focus on the DGP in a small sample space, and the prediction accuracy of the DGP needs to be improved when the data space increases. Therefore, in this paper, a Fuzzy C-Means Deep Gaussian Process Regression (FCM-DGPR) method is proposed that can significantly improve the prediction accuracy of the DGP. To verify the effectiveness and reliability of the proposed method, experiments are carried out on the eddy current loss dataset of the large nuclear power turbo generator.

The remainder of this paper is organized as follows: Section 2 introduces the background of the FCM-DGPR method. Section 3 describes the proposed FCM-DGPR method in detail. Section 4 introduces the construction of the eddy current loss dataset of the large nuclear power generator and demonstrates the employment of the proposed method and experimental results. Finally, the conclusion and the next steps are presented in Section 5.