## 1 Introduction
Unfortunately, two appropriate parameters (i.e., the penalty factor \(\hat{\alpha}\) and mode number \(K\)) need to be selected in advance when we apply VMD in fault detection. That is, the improper parameter setting may induce non-ideal and inaccuracy decomposition results, which further influence the subsequent feature extraction. To solve this problem, a lot of work has been carried out to optimize preferentially the important parameters of VMD. For instance, Isham et al. [29] proposed a signal difference average (SDA) to determine the mode number of VMD based on the similarity criterion between the sum of the sub-signals and the original input signal. Yang et al. [30] adopted a cross correlation coefficient to estimate adaptively the mode number of VMD. Liu et al. [31] used a simple criterion based on a detrended fluctuation analysis to select the mode number of VMD. Jiang et al. [32] proposed an improved VMD strategy with an empirical mode decomposition to determine the mode number \(K\) of VMD. Zhang et al. [33] adopted a mode-mixing judgment method for completing the optimization of the mode number of VMD. Li et al. [34] proposed an independence-oriented VMD method, which can effectively select the mode number based on peak searching and a similarity criterion. Zhao and Feng [35] designed the central frequency iteration rule to determine the mode number of VMD. Wang et al. [36] presented the permutation entropy optimization method to determine the mode number \(K\) of VMD. Lian et al. [37] proposed a neoteric method termed as adaptive variational mode decomposition, which can choose the mode number \(K\) of VMD based on the characteristic of the mode components. However, all parameter selection criteria presented in the above ref. [29; 30; 31; 32; 33; 34; 35; 36; 37] ignored the influence of the penalty factor \(\hat{\alpha}\) on the decomposition ability. Thus, on this basis, Shan et al. [38] applied the minimization criterion of the root mean square error (RMSE) to adaptively search for two VMD parameters. Isham et al. [39] adopted a differential evolution algorithm to select the optimized VMD parameter, where the statistical parameter ratio is utilized to construct the objective function. Zhu et al. [40] used an artificial fish swarm algorithm to choose adaptively the combination parameters of VMD, where the kurtosis is used as the optimization index. Yi et al. [41] applied particle swarm optimization to optimize the two parameters of VMD, where the ratio between the mean and variance of the cross-correlation coefficient is regarded as the objective function. Yan et al. [42] adopted a cuckoo search algorithm containing envelope entropy to optimize two key parameters (\(\hat{\alpha}\), \(K\)) of VMD. Wang et al. [43] adopted multi-objective particle swarm optimization with symbol dynamic entropy to select automatically the two parameters of VMD. Zhou et al. [44] applied the immunized fruit fly optimization algorithm containing the objective function named permutation entropy to determine the combined parameters of VMD. Wei et al. [45] proposed a whale optimization algorithm-based optimal VMD where envelope entropy is regarded as the objective function to find two optimal parameters of VMD. Miao et al. [46] presented an improved parameter-adaptive VMD, which can efficiently obtain two key parameters of VMD by using a grasshopper optimization algorithm containing the ensemble kurtosis. Although the objective function adopted in the above research separately investigates the impact properties of signals and the correlation between signals, it does so without considering simultaneously the impacting characteristics and correlation of signals, which indicates that the decomposition results of VMD may not be optimal. Grey wolf optimization (GWO), proposed recently by Mirjalili et al. [47], is a novel heuristic swarm intelligence optimization method, which integrates the predatory behavior of grey wolves with the characteristics of the leadership hierarchy.