The operation of electric railway causes several power quality problems to traction substations and outlet substations of power systems, including voltage fluctuation, voltage unbalance, voltage harmonics, and so on (Shen et al., 2021). In order to quantify these problems in practice, models of some common power quality problems including voltage rise, voltage drop, voltage flicker, and transient impact are defined (Singh et al., 2014; Wong et al., 2014; Shi et al., 2016), as expressed in (1–3).

1) Signal of voltage rise and drop f ( t ) = { sin ( ω t ) α ⁡ sin ( ω t ) t < t 1 , t > t 2 t 1 < t < t 2 (1) Where, t ₁ and t ₂ represent the start and end time of disturbance respectively; ω is the angle frequency of carrier voltage on power frequency; α is the normalized voltage amplitude, when α> 1, f(x) represents the voltage rise signal, when α< 1, f(x) represent the voltage drop signal.

2) Signal of voltage flicker f ( t ) = { sin ( ω t ) ( A + m c o s Ω t ) sin ( ω t ) t < t 1 , t > t 2 t 1 < t < t 2 (2) Where, m is the amplitude of amplitude-modulated (AM) wave, Ω is the angle frequency of AM wave.

3) Signal of transient voltage f ( t ) = { sin ( ω t ) sin ( ω t ) + ∑ i = 2 N A i ⁡ sin ( i ω t + φ i ) + A e − λ t t < t 1 , t > t 2 t 1 < t < t 2 (3)

It can be seen that the transient voltage consists of multi-frequency signals, such as mutations with damping attenuation A e − λ t and transient high-frequency signals ∑ i = 2 N A i ⁡ sin ( i ω t + φ i ) , A represents the amplitude and φ _i represents the phase deviation of the ith high-frequency signal. All these components lead to the power quality problems.

According to the analysis of formula (1-3), the definition of power quality problems mainly considers the harmonics in the frequency domain and amplitude characteristics of the time domain. Conventional Fourier transform mainly analyzes signal characteristics from the perspective of frequency domain (Boashash, 2015). However, it has limitation in rapid detection of the traction substation’s power quality problems in real-time. Wavelet transformation has the ability to express both time-domain and frequencydomain characteristics (Shen et al., 2020), but it is not easy to choose a suitable mother wavelet. After a comprehensive comparison, on the basis of discrete Fourier transform (DFT), this paper proposed the double discrete Fourier transform (DDFT) as the major tool to solve the electric power quality detection problems of traction substations.

To satisfy the real-time analysis of traction power quality, the DDFT algorithm is developed based on sliding-window iterative DFT (Zhan et al., 2016). Results of DFT usually reflect real-time information in the frequency domain, but they cannot distinguish all the power quality problems, which have an impact on the amplitude of fundamental frequency signals, e.g. voltage rise, voltage drop, and voltage flicker. Therefore, the proposed DDFT algorithm tries to extract fundamental sequences {X (k,t)} from the historical results of DFT transformation. The sequence is then used for Fourier transform for the second time. The purpose of this process is to obtain fundamental frequency information, which can be used to distinguish between temporary voltage rise, drop, and flicker problems.

Figure 2 shows the technological process of DDFT. It can be seen that the original signal is analyzed based on Fourier transformation by taking sliding windows as units. In this way, the computational cost is reduced, and the calculation is speeded up for real-time analysis. After that, the fundamental sequence in the historical windows is extracted at the same size of sliding window, and processed with DFT algorithm once again. Therefore, the DDFT algorithm can not only extract the variation of the fundamental signal and other frequency information, but also maintain the real-time signal analysis, both of which are essential for the power quality analysis in traction substations.

For a given voltage signal in the traction substation {x _n }, a series of transformed features are obtained through the DDFT algorithm. To determine the class that the power quality signal belongs to is a must for the decision-making of selecting appropriate control strategies and power compensation devices. This paper generated some typical power quality signals, and extracted their characteristics by DDFT. By regarding the mechanism of identifying power quality problems as the process of choosing a classifier, and collecting feature samples to be used to train the classifier, the classifier would complete the recognition of power quality problems. According to the above analysis, this paper applied ML-based algorithms to build the classifier.

1) synthetic power quality signals.

To train an optimal mechanism for power quality governance in a traction substation, the first task is to obtain the training dataset. In this paper, the training data is generated through the definitions of typical power quality problems. According to the definitions in (1-3), 10 sets of power quality signals with different parameters are constructed based on each mathematical definition.

2) Features extraction.

For different types of power quality problems, it is subjective and not applicable to determine the classes that each power quality signal belongs to through by observation directly. It absolutely does not work in the analysis of actual operation of electric railways. In order to discriminate different types of power quality problems quantitatively, the DDFT algorithm is proposed to extract features of voltage signal for fast analysis, instead of the original signal sequences. Due to the amplitude of high-frequency harmonics always being low, to reduce the dimension and computation cost, only the DC component and 5–50 Hz harmonic components are taken as the main feature variables. Table 1 gives out the feature values of four typical power quality signals.

In Table 1, feature values of four typical power quality signals are presented, features of the corresponding original signals are also extracted and displayed. Here, X0 represents the DC component, and X1-X5 represent the 5–50 Hz harmonic components, respectively. It is seen that voltage rise and drop signals mainly reflect difference with other signals on the DC component and 5 Hz harmonics component. The voltage flicker signals mainly have some influence on the 10 Hz harmonics component. For more thorough feature analysis, a nonlinear model is required.

3) Establishing classifiers and strategies library.

To build a suitable classification model for analyzing power quality problems, the DDFT algorithm is applied to extract feature vectors x_i at first. Taking feature vectors x_i = [X (0,t), X (1,t), ⋯X (N-1,t)] as inputs, the class labels of power quality problems as outputs, different ML classification models are established, such as SVM, ELM, NN (Ouyang, 2021; Ouyang et al., 2021; Tang et al., 2021). Finally, the optimal ML model is selected. The essence of the proposed method is to construct classifiers based on ML algorithms, and to the constructed classifier to distinguish power quality problems automatically.

According to the above description, some typical power quality signals are generated as training data, and used to build classification models. Meanwhile, the governance strategies are given out according to expert systems and form the strategies library. Then, the intelligent governance system for traction substations’ power quality problems is completed by combining classifiers and strategies libraries. In order to validate performance of the proposed system, including how effective it is to use the DDFT algorithm to extract features of power quality problems, the performance of using ML to classify different power quality problems, and the effectiveness and feasibility of the intelligent governance system in traction substation are discussed.

For analyzing the effectiveness of the proposed approach quantitatively, this paper introduces the confusion matrix (Xiong et al., 2017) in performance evaluation. Here, four commonly used indicators are defined in (4), such as Recall (R), Precision (P), Accuracy (Acc), and Error Rate (ER) (Ohsaki et al., 2017). { A c c = T P + T N T P + F P + T N + F N R = T P / ( T P + F N ) P = T P / ( T P + F P ) E R = 1 − A c c (4)

In this paper, taking an actual traction substation in Wuhan China as an example, there are in total 1,000 voltage signals collected as the testing dataset. The frequency of these signals is 50Hz, and each signal has a period of 50 ms. First, necessary data preprocessing is needed, and the dataset is normalized. Then the power quality problems of the collected signals are tested according to the given intelligent governance system. According to the records of operators in traction substations, the distribution of these signals is displayed as 43 voltage variation, 16 voltage flicker, 19 voltage transient singles and 922 other signals. In other signals, most of the signals are normal signals, and a few of the signals are other undefined signals in this paper. Two typical signals collected from actual traction substations are shown in Figure 3.

In Figure 3, the variation of voltage signals for two typical power quality problems are shown. Figure 3A shows the signal of voltage flicker, and Figure 3B shows the impact of transient voltage signal. It is known that it is difficult for operators to determine what power quality problems a collected signal belongs to, and the period of a set of signals is too fast to capture. This is also the reason why an intelligent system is required to be developed for governing the power quality problems in traction substations.

To evaluate the performance of the proposed approach in the classification of power quality problems, 120 sets of actual signals are used for validation. Since normal signals take up more than 90% of the testing datasets, for classifying different types of power quality problems effectively, only 42 sets of normal signals are selected, and combined with other typical signals as the validation dataset. According to the description in Figure 1, the validation data is processed by DDFT to extracted feature vectors, then feature vectors are input into the obtained classifier. Based on the classification results, the suitable governance strategies could be determined according to the strategies library to control the harmful effects of power quality problems. In other words, the performance of classifying power quality problems of voltage signals of traction substation is directly related to the performance of the proposed intelligent governance system. Therefore, statistical analysis is completed to discuss the performance of the proposed approach. According to the definitions in (Eq. 4), values of four indicators are shown in Table 2.

In Table2, the performance of four indicators are presented. For comparison, three ML models using feature vectors extracted from DFT are also analyzed. From the results of Table 2, it is seen that all these methods have values of almost 85% on Recall, Precision, and Accuracy indicators, and the classifier based on ELM in power quality problems classification performs the best. Therefore, ELM could be selected for the construction of final intelligent system. On the other hand, comparing the feature vectors from DFT and DDFT, it is seen that the performance has been improved. Therefore, it is validated that the proposed approach has better performance on classification of power quality problems. While, for the governance of traction substations in the real world, the proposed system combined the advanced approach with the control strategies library, therefore it not only achieves a fast and accurate identification of power quality problems, but also makes a fast decision to govern traction substations’ power quality by taking the corresponding measures from the strategies library. In this way, the proposed system can realize the intelligent governance of power quality in traction substations and guarantee the safety operation of electrified railways.