In contrast to the popular signal-processing methods, such as empirical mode decomposition (EMD) (Chen et al., 2022) and wavelet transform (WT) (Wang et al., 2022), VMD can effectively alleviate the phenomenon of model aliasing (Qi et al., 2022). It decomposes the original signal to several mode components with a specific sparsity bandwidth. The constrained variational model is min u k , ω k ∑ k = 1 K ∂ t δ t + j / π t u k t e − j ω k t 2 2 s . t . ∑ k = 1 K u k = f t (1) where u k = u 1 , … , u K and ω k = ω 1 , … , ω K indicate the decomposed models and the corresponding center frequencies, respectively; K is the number of the components; ∂ t denotes the gradient calculation; d(t) indicates a pulse function; f(t) is the raw signal.

Equally, The Lagrange multiplier λ(t) and quadratic penalty parameter α has been applied to address reconstruction constraint presented in Eq. 1, and the augmented Lagrange function can be expressed as L u k , ω k , λ = α ∑ k ∂ t δ t + j / π t u k t e − j ω k t 2 2 + λ t , f t − ∑ k u k t + f t − ∑ k u k t 2 2 (2)

In this paper, the saddle point is obtained via the alternate direction method of multipliers (ADMM). The following VMD iterative operation is as follows:

(1) Initialize u ^ k 1 , ω k 1 , λ ^ 1 , n = 0

As an effective fault information quantification method, MPE can quantify the fault feature information of vibration signals (Liu et al., 2021). It aims to measure the random mutation behavior of time series at different scales, which is suitable for analyzing vibration signals produced by the random collision of free-conducting particles. The calculation process of MPE can be described as.

(1) For a given time series X = x i , i = 1,2 , ⋯ , N , the coarse-grained time series y j s can be obtained by using coarse graining processing as follows

Then the corresponding symbolic sequence S r = j 1 , j 2 , ⋯ , j m can be obtained. There are m! possible permutations, and the probability of each permutation P r r = 1,2 , ⋯ , R is counted.

(4) The permutation entropy Hp(m) of each coarse-grained sequence is normalized as follows

WOA is a new meta-heuristic optimization algorithm proposed by Mirjalili et al. (2016). It can provide a good global search with few adjustment parameters by imitating the bubble-net feeding maneuver of humpback whales. The calculation process of this method consists of two phases: the exploitation phase and the exploration phase. However, this algorithm cannot adjust the parameters according to position variation, causing great randomness of results. Also, it often gets stuck in a local optimum in the late iteration period (Jiang et al., 2020). Thus, a self-adapting strategy for weight coefficient identification and search scheme (Kong et al., 2020) is proposed as follows:

(1) Self-adapting weight adjustment strategy: in the exploitation phase, WOA assumes that the current best agent is the position of the target prey or nearest to the optimum. Then other search agents will gradually surround the best search agent to update their positions. The encircling prey behavior is represented as follows

To mimic the spiral movement of humpback whales, shrinking encircling and spiraling upward model are designed. The spiral equation used for updating position of the search agents can be expressed as M t + 1 = d t ⋅ e b h ⋅ cos 2 π h + M * t (13) where d t = M * t − M t , b is a constant as the shape parameter of the logarithmic spiral; h is a random number between [−1, 1].

When A < 1 , the algorithm assumed that the shrinking encircling mechanism and the spiral behavior are performed with the probability of 50%, respectively. In order to enhance the optimization ability and convergence speed of WOA, the weight coefficient ψ and the updated position of search agents can be described as ψ = φ 1 ⋅ X i u − X i l / t + φ 2 ⋅ M i w − M i b (14) M t + 1 = ψ ⋅ M * t − A ⋅ D   if   p < 0.5 d ⋅ e b h ⋅ cos 2 π h + ψ ⋅ M * t   if   p ≥ 0.5 (15) where X i u and X i l are the upper and lower limit of the variable, respectively; M i w is the current worst position vector; M i b is the current best position vector; φ₁ and φ₂ are constants; p is a random number in [0, 1].

The proposed strategy adjusts the weight coefficient from two aspects. In early iteration period, φ 1 ⋅ X i u − X i l / t is not affected by the population distribution, which could obtain a bigger weight to avoid the algorithm falling into small-scale search. It could enhance the local searching ability of the algorithm. With the increasing of iteration times, the effect of φ 1 ⋅ X i u − X i l / t on the weight becomes smaller. If the optimal solution is not obtained in late iteration period, φ 2 ⋅ M i w − M i b could improve convergence speed of the algorithm by getting larger step size.

(2) Self-adapting search adjustment strategy: when A ≥ 1 , humpback whales will search for prey in a random manner depending on the position of each other. For avoiding getting stuck in a local optimum, the probability threshold F is used to optimize the randomly selected search scheme. The probability threshold is expressed as

Then, a random number f in [0,1] is compared with F. If f < F , the position of randomly selected whale individual is updated according to the Eq. 17 and the positions of other whale individuals remain unchanged. M t + 1 = M r a n d − A ⋅ D ′ (17) where M r a n d is a randomly selected search agent; D ′ = C ⋅ M r a n d − M t .

Otherwise, the positions of other whale individuals are updated according to the Eq. 18 M r a n d = M r min + γ ⋅ M r max − M r min (18) where γ is a random number in [0,1]; M _rmax and M _rmin are the maximum and the minimum value of M _rand, respectively.

The scheme could generate a set of random solutions in global range with a high probability in the early iteration period and enhance the global search capability of WOA. The flowchart of the proposed SAWOA-MPE method is shown in Figure 1.

Deep forest (DF), as a kind of deep-learning model, has an excellent performance in processing small-scale datasets by adjusting fewer hyper-parameters (Boualleg et al., 2019; Zhou et al., 2019). Considering the paucity of particle fault data in practical GIL engineering, this article introduces the DF algorithm to realize the identification of particle faults of GIL. The signal processing of DF consists of two parts: multi-grained scanning and cascade forest. Inspired by the excellent performance of convolutional and recurrent neural networks in handling feature relationships, multi-grained scanning is applied to enhance the classification performance of deep forest. Then, cascade forest, a deep procedure, is used for signal classification.

(1) Color/Grayscale figures: multi-grained scanning is a crucial part of deep forest, as shown in Figure 2. It uses sliding windows to scan raw features, generating the corresponding feature vectors. A sliding window of length q performs sampling on the Q-dimensional raw vector. It is supposed that the sliding step is g, and in total G (G=(Q-q)/g+1) feature vectors of q-dimension are produced. Then, both random forest and completely random forest are used to train these subsample vectors. New trained features from the two forests are concatenated and imported to the cascade forest for further processing.

For improving computational efficiency by decreasing the invalid cascade layer, the coefficient of determination Z² is used as the index of cascade layer expansion, which is given as: Z 2 = ∑ i = 1 n z ^ i − z ¯ i 2 ∑ i = 1 n z i − z ¯ i 2 , i = 1,2 , … , k (19) where z _i is the true value; z ¯ i is the average value of the sum of z _i ; z ^ i is the diagnosis value.

When Z i 2 > Z i − 1 2 , the cascade layer continues to expand, otherwise the expansion of cascade layer would be stopped. The structure of cascade forest is shown in Figure 3.

By incorporating VMD, SAWOA-MPE and DF, a fault diagnosis approach for free-conducting particles within GIL is proposed. The flowchart is shown in Figure 4. The specific steps are summarized as follows:

(1) The raw vibration signal is decomposed by VMD to obtain multiple intrinsic mode functions (IMFs). The correlation coefficient between each IMF and the raw vibration signal is calculated, and the IMFs corresponding to high correlation coefficients (correlation coefficients are larger than 0.2) are selected to restructure the vibration signal.

In the VMD algorithm, the penalty parameter α and the decomposition scale K are two vital parameters, which may affect the performance of VMD decomposition. With taking the signal-to-noise ratio (SNR) into account, the penalty parameter α is chosen to be 2000 to ensure the detail retention ability and denoising ability of VMD (Ren et al., 2019). The optimal K can retain the significant feature information of raw signals to avoid over-decomposition in the VMD process. Thus, in this article, the decomposition scale K is determined by the center frequency method. The variation of center frequency of different K values under Type-F working condition, for instance, is shown in Table 3.

Table 3 exhibits that, when K = 9, some IMF components with similar center frequency begin to appear, such as IMF1 to IMF6, i.e., over-decomposition occurs in the process. Thus, K is determined as eight for vibration signals of Type-F working condition. The time-frequency diagram of IMF components is shown in Figure 8. Further, the correlation coefficients are calculated to determine which IMF contains rich fault information. Once the correlation coefficient exceeds 0.2, the corresponding IMFs are chosen to reconstruct vibration signals.

To improve the identification performance of free-conducting particle faults, MPE is extracted to reflect the randomness and mutability of vibration signals. According to the previous researches, there are four parameters should be selected with special care when calculating MPE: the scale factor s, embedding dimension m, delay time τ, and the length of time series N (Jin et al., 2021). It should be noted that setting the initial value of the four parameters is the key to achieving optimal MPE. In the SAWOA algorithm, the iterative ranges of the four parameters are set as, s ∈ 2,18 , m ∈ 3,7 , τ ∈ 1,5 , N ∈ 2000,5000 . Additionally, we set the number of search agents to 50, the maximum number of iterations to 30, φ ₁ = 1e⁻⁴, φ ₂ = 1e⁻⁴. Meanwhile, the original WOA algorithm is also applied to this article. The parameters of MPE optimized by WOA and SAWOA are illustrated in Table 4.

It can be seen from Table 4, for the same signals, the MPE parameters optimized by the two algorithms are different. Additionally, the optimal parameters of MPE under different working conditions are also different. To intuitively show the superiority of the SAWOA algorithm, the results of unoptimized MPE (m = 6, s = 12, τ = 1, N = 5,000) (Zhao et al., 2020) and optimized MPE (WOA-MPE and SAWOA-MPE) are shown in Figure 9. The scale factors obtained by the two optimization algorithms are chosen according to the minimum value in Table 4.

It can be seen from Figure 9A, the entropy values of unoptimized MPE under six working conditions are intertwined, which can hardly distinguish the different states. Since WOA comes with parameter optimization capability, the MPE curves under different operating conditions in Figure 9B become more independent, with only a few intersections. More particles can cause more complicated motion behavior, which is likely to produce similar fault characteristics. For this reason, the vibration signals of working conditions E and F may have a higher overlapped fault feature space. As shown in Figure 9B, the MPE curves remain slightly intersected under the working conditions E and F even after WOA-MPE processing. Therefore, the entropy calculated by WOA-MPE still not be highly effective for characterizing the feature information of different free-conducting particle faults. Meanwhile, Figure 9C shows that the optimization effect of SAWOA-MPE is significantly better than that of the first two methods. Almost all entropy variety curves are entirely independent, especially the curves under working conditions E and F are no longer interleaved. In addition, the distances between entropy curves of different working conditions increase obviously, which can verify the superiority of SAWOA-MPE.

(1) Parameters Selection of Deep Forest: in the experiment of this article, as shown in Table 1, free-conducting particle faults within GIL are divided into six categories. 180 groups of sample data are collected for each fault category, and each group contains 5,000 sampling points. Then the feature vectors of each group signal are extracted by the VMD-SAWOA-MEP method. The total 1,080 groups of feature datasets are divided into the training sample sets and testing sample sets according to the ratio of 4:1, then fed into the DF model for fault identification.

Deep forest is composed of multi-grained scanning and cascade forest. According to Table 4; Figure 9, eight-dimension feature vectors are used as the raw input vectors for deep forest. Thus, in the multi-grained scanning phase, the dimensions of sliding windows are set as 2, 4, and 6 (i.e., Q/4, Q/2, 3Q/4, Q is the dimension of feature vector), respectively. The feature vectors generated by sliding the window are input into a random forest and a completely random forest, each of which contains 500 trees.

Similar to the related study (Boualleg et al., 2019), in the cascade forest phase, random forests and completely random forests are used to increase the balance between bias and variance in this study. Notably, the number of forests and trees can significantly affect the identification accuracy and processing time. However, in most studies, the number of each type forest and trees are fixed at 4 and 500, respectively (Zhou et al., 2019; Jia et al., 2021). In order to achieve a balance between identification accuracy and processing time, we vary the number of each type forest (No. F = 2 and No. F = 4) and the number of trees contained per forest (100–1,000 with step size 100). By using five-fold cross validation, the identification accuracy and the processing time of diagnosis model are shown in Figure 10.

It can be seen from Figure 10, the highest identification accuracy of the diagnosis models can reach 99.5%, when the number of each type forest is set to 2 or 4. In addition, the training process becomes slower as the number of trees increases. Consequently, considering the identification accuracy and the processing time, we set the number of each type forest to 2 and the number of trees contained per forest to 400. The number of cascade levels is determined by the coefficient of determination Z². As shown in Figure 11, the coefficient of determination Z² presents an upward trend, i.e., the diagnosis accuracy of the model is constantly improving. When the levels of cascade forest increase to 5, the diagnosis accuracy becomes stable, and the training process of the whole model is stopped.

(2) Comparative Study: in this comparative study, we compare the proposed method with some typical learning-based classification models and fault features. Five-fold cross validation is used to reduce the risk of overfitting and handle the potential randomness in the diagnostic validation stage. Several parameters of the DF model are shown in Table 5.

Four metrics, including accuracy, recall, precision, and F1 score, are used to evaluate the robustness and generalization of different fault diagnosis methods. The diagnostic results of free-conducting particle faults using MPE feature vectors are shown in Figure 12; Table 6. It is observed that, due to the random motion behavior of free-conducting particles in AC electric field, when the unoptimized MPE is used as a feature vector, it is difficult to distinguish the different free-conducting particle faults. The classification accuracy drops as the number of particles increases. It is because, in AC electric field, the interaction force between the particles and the distortion degree of electric field is intensified by an increasing number of particles, which can cause more complicated particle motion behavior, richer fault characteristic information, and higher overlapped feature space. In contrast, the diagnosis model achieves higher performance using optimized MPE, especially for multi-particle faults. The detection accuracy using MPE, WOA-MPE and SAWOA-MPE are respectively 76.8%, 98.6%, and 99.5%. It should emphasize that, for the five free-conducting particles fault (Type-F), the detection accuracy is 100% by using SAWOA-MPE. Thus, the proposed SAWOA-MPE-DF model can effectively tackle overlapped feature problems between the particle faults.

In this comparative experiment, a comprehensive time-frequency domain feature vector is used as a contrastive feature vector, which contains nine time-domain features (root mean square, absolute mean value, mean square, kurtosis, skewness, waveform factor, peak factor, impulsion index, and margin index) and five frequency-domain features (mean value, standard deviation, root mean square, center frequency, and frequency kurtosis). Referring to the related researches (Wiener et al., 2021; Wu et al., 2022), the combination of particle collision frequency f_c and maximum amplitude A _max in the time-domain signal is used as another contrastive feature vector. Then SAWOA-MPE is compared with the two categories of feature vectors by using the DF diagnosis model. Meanwhile, the proposed method is compared with the four other diagnosis models, namely convolutional neural network (CNN), k-nearest neighbors (KNN), support vector machine (SVM), and random forest (RF), to verify its superiority. The comparison results of different diagnosis methods are illustrated in Table 7.

It is observed that there are significant performance differences among the feature vectors. Based on deep forest, SAWOA-MEP, achieves the highest performance in classifying free-conducting particle faults compared to other feature vectors: SAWOA-MEP (99.5%), WOA-MPE (98.6%), MPE (76.8%), time-frequency domain feature vector (70.1%), and the combination of particle collision frequency and maximum amplitude (48.6%). The maximum difference of classification accuracy between these feature vectors is 50.9%. For using SAWOA-MPE, as the feature vector, deep forest still gets the best performance compared to CNN, and other typical machine learning-based methods, such as KNN, SVM, and RF., By contrast, CNN, may lead to overfitting for small datasets in the learning process, which can hardly process the task with small-scale training data. In summary, the proposed diagnosis method has a good application in the field of free-conducting particle fault detection with small datasets.