Centrifugal pump impeller defect identification by the improved adaptive variational mode decomposition through vibration signals

VMD decomposes the raw vibration signal into different modes ${u}_{k}$ having the sparsity with the original signal. The modes thus obtained must be centred on a central frequency ${w}_{k}.$ In the frequency domain, the sparsity of the mode ${u}_{k}$ is selected as the bandwidth [20, 34]. The following procedure is adopted to obtain the bandwidth of the mode: (1) Hilbert transform extracts the mode ${u}_{k}$ in order to draw frequency spectrum, (2) The obtained frequency spectrum is shifted to 'baseband' which is close to its respective central frequency, (3) In the end, ${L}^{2}$ norm computes the frequency bandwidth. The raw signal is decomposed in the following manner which is shown in equation (1) [20]:

$$\begin{eqnarray*}&&{\rm{\min }}\left\{\displaystyle \sum _{k}{\parallel \partial t\left[\left(\partial \left(t\right)+\displaystyle \frac{j}{\pi t}\right)\,* \,{u}_{k}\left(t\right)\right]{e}^{\left(i{w}_{k}t\right)}\parallel }_{2}^{2}\right\}\end{eqnarray*}$$

$$\begin{eqnarray}&&s.t.\,\displaystyle \sum _{k}{u}_{k}=f\left(t\right)\end{eqnarray} \tag{ 1 }$$

where, the input signal is represented by $f\left(t\right),$ the set of modes are indicated by $\left\{{u}_{k}\right\},$ central frequency is represented by $\left\{{w}_{k}\right\},$ whereas Dirac distribution $\partial t$ represents convolution. To make the optimization problem given in the equation (1) unconstraint, the Penalty factor $\alpha$ (data fidelity constraint) and Lagrangian multiplier λ is used. The obtained unconstraint problem is shown in equation (2).

$$\begin{eqnarray}&&\begin{array}{l} {\mathcal L} \left(\left\{{u}_{k}\right\},\left\{{w}_{k}\right\},\lambda \right)=\alpha {\displaystyle \sum _{{\boldsymbol{k}}}\parallel \partial t\left[\left(\partial \left(t\right)+\displaystyle \frac{j}{\pi t}\right)\,* \,{u}_{k}\left(t\right)\right]{e}^{-j{w}_{k}t}\parallel }_{2\,}^{2}\\ -{\parallel f\left(t\right)-\displaystyle \sum _{k}{u}_{i}\left(t\right)\parallel }_{2}^{2\,}+\left\langle \lambda \left(t\right),f\left(t\right)-\displaystyle \sum _{k}{u}_{i}\left(t\right)\,\right\rangle \end{array}\end{eqnarray} \tag{ 2 }$$

Equation (2) is solved by the alternate direction method of multipliers (ADMM) that gives the sequence of sub-optimization to find the saddle point. The obtained solutions of sub-optimization of ADMM directly optimizes the problem mentioned in equation (2) in the Fourier domain [20]. The detailed procedure of the algorithm is given in [20]. The modes ${u}_{k}$ and central frequency ${w}_{k}$ updates their values according to the ADMM optimization problem. The equation (2) updates the variational mode function (VMF) with respect to ${u}_{k}$ and after updating the VMF, the sub optimal problem is given in equation (3).

$$\begin{eqnarray}&&{u}_{k}^{n+1}=\,argmi{n}_{{u}_{k}\in X}\left\{\alpha {\parallel \partial t\left[\left(\partial \left(t\right)+\displaystyle \frac{j}{\pi t}\right)\,* \,{u}_{k}\left(t\right)\right]{e}^{-j{w}_{k}t}\parallel }_{2}^{2}\,\,+{\parallel f\left(t\right)-\displaystyle {\sum }_{i}{u}_{i}\left(t\right)+\displaystyle \frac{\lambda \left(t\right)}{2}\parallel }_{2}^{2}\right\}\end{eqnarray} \tag{ 3 }$$

The Fourier transform with tuned centre frequency helps in finding the optimum solution for equation (3). The modes ${\hat{u}}_{k}$ updates their values through equation ( 4).

$$\begin{eqnarray}&&{\hat{u}}_{k}^{n+1}\left(w\right)=\displaystyle \frac{\hat{f}\left(w\right)-\displaystyle {\sum }_{i\ne k\,}{\hat{u}}_{i}\left(w\right)+\tfrac{\hat{\lambda }\left(w\right)}{2}}{1+2\alpha {\left(w-{w}_{k}\right)}^{2}}\end{eqnarray} \tag{ 4 }$$

The Wiener filter is used to filter the signal prior to $1/{\left(w-{w}_{k}\right)}^{2}$ for current residual signal. The spectrum of each mode (${u}_{k}$) is obtained by Hermitian symmetric completion. The inverse Fourier transform of the real part of the filtered signal gives modes (u_k ) in the time domain. Equation (2) is optimized in such a way that the centre frequency does not originate in the reconstruction fidelity. The desired problem is defined in [20]:

$$\begin{eqnarray}&&{w}_{k}^{n+1}=argmi{n}_{{w}_{k}}\left\{\alpha {\parallel \partial t\left[\left(\partial \left(t\right)+\displaystyle \frac{j}{\pi t}\right)\,* \,{u}_{k}\left(t\right)\right]{e}^{\left(j{w}_{k}t\right)}\parallel }_{2}^{2}\right\}\end{eqnarray} \tag{ 5 }$$

Equation (5) is solved by the optimization to obtain the centre frequency which is shown in equation (6)

$$\begin{eqnarray}&&{w}_{k\,}^{n+1}=\displaystyle \frac{{\int }_{0}^{\infty }w{\left|{\hat{u}}_{k}\left(w\right)\right|}^{2}dw}{{\int }_{0}^{\infty }{\left|{\hat{u}}_{k}\right|}^{2}dw}\end{eqnarray} \tag{ 6 }$$

The centre frequency w_k in equation (6) is updated as per their corresponding mode at the centre gravity of the power spectrum.

The above equations suggested that four parameters viz, mode number (K), quadratic penalty factor (α), tolerance (τ), and convergence criterion (ε) are required in the procedure of VMD and need to be specified in advance. τ and ε usually adopt their default values, taken in original VMD as these parameters have little effect on decomposition results. But mode number K need not be specified in advance without having prior knowledge of the signal to be analyzed. As nothing can be said about the appropriateness of K and thus, it cannot ensure the accuracy and efficiency of decomposition results. Quadratic penalty factor α suppresses the noise interference in the signal and also controls the frequency bandwidth, thus should be chosen accordingly. The optimal selection of the combination of parameters is the key to VMD and is the motivation of the research.

ELM is a method used for both regression and classification purposes [35]. The single-layer feed-forward network (SLFN) is the foundation block of ELM as shown in figure 1. In ELM there are basically three layers viz, input, hidden, and output layers. For $M$ arbitrary samples $\left({X}_{j},{t}_{j}\right)\in {R}^{n}X\,{R}^{m},$ SLFN having $L$ hidden nodes and activation function $G({\alpha }_{i},{\beta }_{i},{X}_{i})$ is mathematically modelled as given in equation (7) [23, 35]:

$$\begin{eqnarray}&&{f}_{L}\left({X}_{j}\right)=\displaystyle \sum _{i=1}^{L}{\beta }_{i}G\left({\alpha }_{i}.{X}_{j}+{b}_{i}\right)={t}_{j},\,j=1,\ldots ,M.\end{eqnarray} \tag{ 7 }$$

Here, ${\alpha }_{i}$ and ${b}_{i}$ are learning parameters of hidden nodes, out of which ${\alpha }_{i}$ connects the input weight vector of input nodes to ith hidden node and ${b}_{i}$ denotes the threshold of the ith hidden node. ${\beta }_{i}$ and ${t}_{j}$ represents output weight and test point whereas activation function $G({\alpha }_{i},{\beta }_{i},{X}_{i})$ gives output for ith hidden node. The equation can be written in matrix form as shown in equation (8).

$$\begin{eqnarray}&&H\beta =T\end{eqnarray} \tag{ 8 }$$

where,

$$\begin{eqnarray}H=\left({\alpha }_{1},\ldots ,{\alpha }_{L},{b}_{1},\ldots ,{b}_{L},{X}_{1},\ldots ,{X}_{N}\right)={\left[\begin{array}{ccc}G\left({\alpha }_{1},{\beta }_{1},{X}_{1}\right) & \ldots & G\left({\alpha }_{L},{\beta }_{L},{X}_{1}\right)\\ \begin{array}{c}.\\ .\\ .\end{array} & \ldots & \begin{array}{c}.\\ .\\ .\end{array}\\ G\left({\alpha }_{1},{\beta }_{1},{X}_{N}\right) & \ldots & G\left({\alpha }_{L},{\beta }_{L},{X}_{N}\right)\end{array}\right]}_{N{\rm{X}}L}\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}&&\beta =\left[\begin{array}{c}{\beta }_{1}^{T}\\ \begin{array}{c}.\\ .\end{array}\\ {\beta }_{L}^{T}\end{array}\right],T=\left[\begin{array}{c}{T}_{1}^{T}\\ \begin{array}{c}.\\ .\end{array}\\ {T}_{N}^{T}\end{array}\right]\end{eqnarray} \tag{ 10 }$$

As per ELM theories, the values of ${\alpha }_{i}$ and ${b}_{i}$ are randomly assigned for all hidden nodes instead of being tuned. The solution of the above equation is estimated by using equation (11).

$$\begin{eqnarray}&&\beta ={H}^{* }T\end{eqnarray} \tag{ 11 }$$

where, ${H}^{* }$ is inverse for the output matrix $H.$ Moore-Penrose's generalized inverse is utilized for this purpose. The steps involved in the ELM algorithm is summarised below:

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Step 1: Assigning learning parameters i.e. ${\alpha }_{i}$ and ${b}_{i}$ of the hidden nodes randomly,

Step 2: Hidden layer output matrix $H$ is to be calculated.

Step 3: Inverse of the hidden layer output matrix is calculated using Moore-Penrose generalized inverse.

The measurement index is the key element for making VMD adaptive, which determines the efficacy of the decomposition results. Literature suggested that kurtosis and correlation coefficients are two important indices for fault diagnosis of rotating machines through vibration signals [36]. Kurtosis index is mainly dependent on the density distribution of impacts generated by fault. Hence, impacts having a large amplitude but with dispersing density distribution is neglected, if maximum kurtosis index is used as a candidate solution or fitness function to optimize parameters of VMD. The correlation coefficient quickly determines the similarity between two signals, but it is prone to the noises present in the signal of faulty components [36]. Their hybrid index i.e., weighted kurtosis index is built to assist as a fitness function to find optimal VMD parameters [37]. The weighted kurtosis index ($KCI$) is given in the following equation (12).

$$\begin{eqnarray}&&KCI=KI.\left|C\right|\end{eqnarray} \tag{ 12 }$$

where $KI$ represents the kurtosis index for input signal $x(n)$ and expressed as

$$\begin{eqnarray}&&KI=\,\displaystyle \frac{\tfrac{1}{N}\displaystyle {\sum }_{n=0}^{N-1}{x}^{4}\left(n\right)}{{\left(\tfrac{1}{N}\displaystyle {\sum }_{n=0}^{N-1}{x}^{2}\left(n\right)\right)}^{2}}\end{eqnarray} \tag{ 13 }$$

Here, the length of the signal is given by $N.$ Considering $E\left[.\right]$ a mathematical expectation, the correlation $C$ between $x$ and $y$ is expressed as

$$\begin{eqnarray}&&C=\,\displaystyle \frac{E\left[\left(x-\bar{x}\right)\left(y-\bar{y}\right)\right]}{E\left[{\left(x-\bar{x}\right)}^{2}\right]E\left[{\left(y-\bar{y}\right)}^{2}\right]}\end{eqnarray} \tag{ 14 }$$

Mirjalili et al [38] proposed the swarm intelligence (SI) based salp swarm algorithm (SSA) optimization technique. It imitates the swarm behaviour of salps or a chain of salps for food foraging in the deep ocean. In SSA, the initial random population is divided into two sections viz, leaders, and followers to construct the mathematical model of salp swarm [38]. The leader leads and guides the salp chain, whereas followers follow the leaders through the salp chain.

The basic idea of the salp swarm algorithm (SSA) and an ameliorated salp swarm algorithm (ASSA) proposed in this communication has been explained in the following subsections along with the inspiration behind the modification in SSA. Initialization of population, function evaluation and dividing the swarm are common processes for SSA and ASSA. Opposition-based learning is used to modify SSA and make it faster in terms of convergence. The process of updation of the position of leader and followers is modified through different equations given in respective subsections to design ASSA comprehensively.

3.2.1. Initialization of population

Initially, the population is generated in the search space ${x}_{ij}$ in a random manner through uniform distribution as shown in equation (15).

$$\begin{eqnarray}&&{x}_{ij}=\,{x}_{j}^{{\rm{\min }}}+{r}_{ij}\left({x}_{j}^{{\rm{\max }}}-{x}_{j}^{{\rm{\min }}}\right);\,\left(i=1,2,\ldots ,NP;j=1,2,\ldots ,D\right)\end{eqnarray} \tag{ 15 }$$

where ${x}_{ij}$ indicates the search space, $NP$ represents the populations, and $D$ is the dimension of search space. The upper bound and lower bound of the search space is denoted by ${x}^{\max }$ and ${x}^{\min }$ respectively. The population ${r}_{ij}$ is randomly generated in the range of (0,1).

3.2.2. Opposition-based learning

Nature-inspired optimization algorithms start by selecting global optima randomly and then initialize every individual randomly in a given search space. To discover the appropriate solution, each individual updates their position based on its intellect and behaviour. The computation time required by these methods is determined by the initial guesses. However, the computation time can be reduced if these techniques start checking their own opposite solution [39]. One with a better solution, either from randomly generated or its opposite guess, is selected. Starting with a closed guess and verifying it with its fitness function not only lowers computing time but also enhances the convergence speed. This approach can be continuously applied to each solution for the current population, along with initialization. The same concept has been introduced to modify the basic salp swarm algorithm. The initialization of the population in opposition-based learning is done in the following manner as shown in equation (16).

$$\begin{eqnarray}&&{x}_{{o}_{ij}}=\,{x}_{j}^{{\rm{\max }}}+{x}_{j}^{{\rm{\min }}}-{x}_{ij}\end{eqnarray} \tag{ 16 }$$

where, ${x}_{{o}_{ij}}\,$ represents the salp population from opposition-based learning.

3.2.3. Computation of fitness function

The fitness of the swarm is evaluated by using equation (17). The best function value, ${F}_{b}(i)$ is obtained mathematically and is written as

$$\begin{eqnarray}&&{F}_{b}\left(i\right)=min\left\{F\left(x\left(i\right)\right)\right\}\end{eqnarray} \tag{ 17 }$$

where, i = 1, 2, ..., NP

The best salp position is saved corresponding to the best function value, ${F}_{b}\left(i\right).$

3.2.4. Division of the swarm

3.2.5. Position updation of the leader

Like any other swarm-based optimization technique, the salp position is the candidate solution stored in a two-dimensional matrix called $X$ for $m$-dimensional search problem. $m$ is the number of design variables. In ${x}_{ij},$ the optimal position represents the best food source called $F.$ The position of leaders is updated as per equation (18).

$$\begin{eqnarray}&&{X}_{j}^{1}=\left\{\begin{array}{c}{F}_{j}+\,{c}_{1}\left(\left(u{b}_{j}-l{b}_{j}\right){c}_{2}+\,l{b}_{j}\right),\,\,{c}_{3}\geqslant P\\ {F}_{j}-\,{c}_{1}\left(\left(u{b}_{j}-l{b}_{j}\right){c}_{2}+\,l{b}_{j}\right),\,\,{c}_{3}\lt P\end{array}\right.\end{eqnarray} \tag{ 18 }$$

where ${X}_{j}^{1}$ is the position of the leader (first salp), $P$ is the probability, on the basis of which leader's position is decided and ${F}_{j}$ is a food source in $jth$ dimension. $u{b}_{j}\,and\,\,l{b}_{j}$ are upper bound and lower bound respectively of $jth$ dimension. ${c}_{1},$ ${c}_{2}$ and ${c}_{3}$ are random variables, out of which ${c}_{1}$ plays a vital role in balancing exploitation and exploration. ${c}_{1}$ is defined in equation (19) as

$$\begin{eqnarray}&&{c}_{1}=2{e}^{-{\left(\displaystyle \frac{4k}{L}\right)}^{2}}\end{eqnarray} \tag{ 19 }$$

Here, $k$ indicates the current iteration, the maximum iteration is represented by $T.\,$ Random variables ${c}_{2}$ and ${c}_{3}$ are randomly generated between 0 and 1. The probability $P$ is defined in equation (20) as

$$\begin{eqnarray}&&P=tanh\left|S\left(i\right)-DF\right|\end{eqnarray} \tag{ 20 }$$

where, $S(i)$ represents the fitness of ${x}_{ij},$ $DF\,$ indicates the best fitness obtained in all iterations. Equation (20) has been introduced as a modification that helps in updating the position of the leaders.

3.2.6. Position updation of the followers

The follower position is updated according to equation (21), which is based on Newton's Law of Motion.

$$\begin{eqnarray}&&{X}_{j}^{i}=\displaystyle \frac{1}{2}\,\left(\left(1-\alpha \right){X}_{j}^{i}+\alpha {X}_{j}^{i-1}\right)\end{eqnarray} \tag{ 21 }$$

where $2\leqslant {\rm{i}}\leqslant {\rm{L}},$ ${X}_{j}^{i}$ indicates the position of ith follower in salp chain of jth dimension. And α is weighting factor defined in following equation (22)

$$\begin{eqnarray}&&\alpha =rand\left(-var1,var1\right)\end{eqnarray} \tag{ 22 }$$

where,

$$\begin{eqnarray}&&var1=atanh\left(-\displaystyle \frac{1}{\max \,T}+1\right)\end{eqnarray} \tag{ 23 }$$

Equations (22), and (23) are incorporated in equation (21) as modifications in the basic salp swarm which updates the follower's position.

The following is the pseudocode for the proposed ASSA Algorithm:

Initialize the salp population NP1 ${x}_{ij}$ using equation (15).

Apply opposition-based learning on this initial population to get NP2 members.

Calculate objective function on NP1 and NP2 using equation (25).

Select best NP members out of (NP1 + NP2).

Select members, $xbest(j)$ with best function value and designate as $gbest.$

while $(t\leqslant T)$

Calculate the objective function of each individual.

Update ${c}_{1}$ by equation (19)

for i = 1: size of salp population

if i <= half of the salp population

generate ${c}_{2}$ and ${c}_{3}$ randomly within [0,1]

calculate the value of P using equation (20)

update the position of the leaders using the equation (18)

else if i > half of the salp population

generate the variable var1 using equation (23)

create α using equation (22)

update the position of the follower using equation (21)

end

end

updated salp position

end

amend the salps based on upper and lower bounds of variables.

evaluate the objective function on the updated salp position

update the value xbest and gbest

end

In the case of large data, generally, filter type algorithms are used to rank the features. Feature filter is simply the function of correlation or information which returns the relevant index $J\left(S| D,C\right)$ [40]. This index computes how relevant is the given feature subset $S$ for task $C$ which can be a classification or approximation of data $D.$ Relevant index is calculated for each individual feature ${X}_{i},\,i=1\ldots N$ in order to know the ranking order $J\left({X}_{{i}_{1}}\right)\leqslant J\left({X}_{{i}_{2}}\right)\ldots .\leqslant J\left({X}_{{i}_{N}}\right).$ Features with lowest rank are generally filtered out. Pearson correlation coefficient (PCC) is one of the methods which determines the rank of features which is based on correlation measurement [40]. PCC is computed by using equation (24):

$$\begin{eqnarray}&&e\left(X,C\right)=\displaystyle \frac{E\left(XC\right)-E\left(X\right)E\left(C\right)}{\sqrt{{\sigma }^{2}\left(X\right){\sigma }^{2}\left(C\right)}}=\,\displaystyle \frac{\displaystyle {\sum }_{i}\left({x}_{i}-{\bar{x}}_{i}\right)\left({c}_{i}-{\bar{c}}_{i}\right)}{\sqrt{\displaystyle {\sum }_{i}{\left({x}_{i}-{\bar{x}}_{i}\right)}^{2}{\displaystyle {\sum }_{j}\left({c}_{j}-{\bar{c}}_{j}\right)}^{2}}}\end{eqnarray} \tag{ 24 }$$

where $X$ is a feature with value $x\,,$ and class $C$ is a class having values $c.$ If ${\rm{e}}\left(X,C\right)$ is ± 1, then $X$ and $C$ are dependent on each other but if $e\left(X,C\right)$ is zero, then $X$ and $C$ are uncorrelated. Error function $e\left(X\unicode{x00303}C\right)={\rm{erf}}\left(\left|e\left(X,C\right)\right|\sqrt{N/2}\right)$ is used to compute the probability of how two variables are correlated to each other. Decreasing values of error function $e\left(X\unicode{x00303}C\right)$ serve as feature ranking and order the feature list accordingly.

A method to make VMD adaptive is proposed based on ameliorated salp swarm algorithm (ASSA) by using maximum weighted kurtosis as the fitness function as given in equation (25). Here, the original salp swarm algorithm is improved by introducing opposition-based learning in it, thus named ASSA. The proposed optimization algorithm is made to find the minimum value; thus, the required maximization problem is converted into a minimization problem by introducing a negative sign in the fitness function, as shown in equation 25. Ranking of features is also proposed to see how features are relevant to fault diagnosis using PCC. It is further passed through ELM to compute training and testing accuracy.

$$\begin{eqnarray}&&\left\{\begin{array}{c}objective\,function=mi{n}_{\upsilon (k,\alpha )}\left(-KC{I}_{i}\right)\\ s.t,\,\,\,\,\,\,\,\,\,\,\,k=2,3,\ldots ,7\\ \alpha {\epsilon }\left[1000,\ldots ,10,000\right]\end{array}\right.\end{eqnarray} \tag{ 25 }$$

where, $KC{I}_{i}$ $(i=1,2,\ldots ,\,K)$ indicates the weighted kurtosis for decomposition modes of VMD. $\upsilon (k,\alpha )$ are parameters of VMD which are to be optimized. Mode number $k$ varies in the interval of [2, 7], and the quadratic penalty factor ${\rm{\alpha }}$ takes value in an interval of [1000,10000]. The range for parameters has been decided based on an extensive literature survey [23, 34, 41].

The detailed steps of the proposed work are given below:

• The raw vibration signal is decomposed by VMD with set ranges of parameters that are to be optimized. Store the value of the objective function for each iteration.
• Initialize parameters of ASSA with $N$ population and $L$ number of maximum iterations.
• Obtain the modes after decomposition of vibration signal using VMD. Then, compute the objective function for each mode.
• If $t\geqslant T,$ then the required condition is achieved, the iteration will end. Otherwise, put $t=t+1$ and continue the iteration.
• Save the combination of optimal parameters based on the minimum value of an objective function. And, store the mode corresponding to a maximum weighted kurtosis index, thus known as a sensitive mode.
• Characteristics features are obtained from sensitive mode and arranged in ranked order using PCC. Store the obtained data.
• Further obtained data is feed into ELM to determine the training and testing accuracy of the model.

The whole methodology is shown in the form of flow chart as given in figure 2.

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The proposed method was applied to the data set acquired from the test rig of the centrifugal pump. The schematic of the pump test rig and its pictorial view is shown in figures 3(a) and (b), respectively. The pump is operated at 2800 rpm (i.e., 46.67 Hz). The pump's specification is given in table 1. Two bearings are used to support the shaft of the pump. These bearings are named Bearing 1 (closer to impeller; bearing number 6203-ZZ) and Bearing 2 (away from the impeller, bearing number 6202-ZZ). The impeller is attached to the rotor shaft and is encased in a casing. The impeller is made up of revolving vanes that pull water axially through the impeller's eye and impart kinetic energy to the water. This kinetic energy forces the water to flow in a radially outward direction through the casing. This energy gets transferred between water and vanes through impacts which generate the chaotic phenomenon in the water.

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The vibration signals are acquired by a uniaxial accelerometer having a sensitivity of 100 mV g⁻¹. The accelerometer is mounted near the impeller casing as shown in figure 3(b). The vibration signal is acquired using a 24-bit, four-channel DAQ of National Instruments make, with a sampling frequency of 70 kHz. To investigate the signal, 0.1 seconds of data with 7000 data points are taken. The study is carried out under various impeller conditions, as illustrated in figure 4. The adaptive VMD technique is used to process the raw signal acquired from the centrifugal test rig. The two parameters of VMD, i.e., mode number (K) and quadratic penalty factor (α), are optimized using ameliorated salp swarm algorithm. The other parameters of VMD are taken, as suggested in [20]. The suggested algorithm ASSA obtains the optimal combination of VMD parameters. With these optimal combinations of parameters, the most prominent mode is selected based on maximum weighted kurtosis. This prominent mode is processed for further analysis.

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Initially, the data is acquired for the pump operating at 2800 rpm (i.e., 46.67 Hz operating frequency) with normal (no defect) impeller in place. The pump has a constant operating speed for all types of impeller conditions under study. The raw signal in the time domain for defect-free impeller condition is shown in figure 5(a). This raw signal is converted into a frequency domain and represented in figure 5(b). The characteristic frequency of 47 Hz is also shown in this figure which is equivalent to the pump operating frequency. Since the pump is operating at constant rpm in all health conditions that's why FFT corresponding to operating frequency will remain the same for every condition. The raw signal is decomposed into different modes by adaptive VMD based on ASSA. From ASSA, the mode number, K and quadratic penalty factor, $\alpha$ comes out to 3 and 1000 respectively. The different modes are shown in figure 5(c). For each mode, the weighted kurtosis is calculated. The third mode has the highest weighted kurtosis value, 12.73, and has been chosen for feature extraction. Twenty signals are collected for examination, with each signal subjected to no-defect, clogged, blade cut, and wheel cut impeller conditions.

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Similarly, under clogged impeller conditions, the data is collected and divided into distinct modes using adaptive VMD with 3 mode number (K) and 4000 quadratic penalty factors, which is optimised by ASSA. In this case, the three modes have weighted kurtosis values of 1.99, 1.95, and 3.08. The third mode has the highest weighted kurtosis value and is thus chosen for further processing. The time-domain signal, frequency domain signal and decomposed modes are shown in figures 6(a)–(c), respectively.

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The raw signal for blade cut impeller condition is shown in figure 7(a). The corresponding frequency domain signal is shown in figure 7(b). The computed optimized value of parameters K and $\alpha$ are 3 and 1000, respectively, for this signal. The raw signal is split into three modes at these parameters, as illustrated in figure 7(c). Weighted kurtosis values for the first, second, and third modes are 1.62, 1.60, and 8.62, respectively. The third mode was chosen for further examination because it has the highest value of weighted kurtosis.

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The time-domain signal for the wheel-cut impeller condition is shown in figure 8(a) and the corresponding frequency signal is shown in figure 8(b). Following the process described in the previous scenarios, the optimal settings for mode number and penalty factor are 3 and 2000, respectively. The three obtained modes are depicted in figure 8(c). The weighted kurtosis for the first, second, and third modes, respectively, is 1.54, 3.87, and 11.63. As a dominant mode, the mode with 11.63 as weighted kurtosis is chosen.

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The adaptive VMD technique based on ASSA yields a total of 80 significant modes of vibration signals (20 each for healthy (no-defective), clogged, blade cut, and wheel cut impeller conditions) employing weighted kurtosis as a measurement index. Following that, eleven features are retrieved from prominent modes decomposed through adaptive VMD. Table 2 provides a list of features and their definitions. The features extracted are normalized in the range of [0 1] using the following mathematical expression:

$$\begin{eqnarray}&&Normalized\,feature=\displaystyle \frac{Feature-\,{\rm{\min }}\left(Feature\right)}{\max \left(Feature\right)-\,{\rm{\min }}\left(Feature\right)}\end{eqnarray} \tag{ 26 }$$

Table 2. Extracted features with their definition.

S. no.	Features	Definitions
1.	Standard Deviation (${x}_{std}$)	${x}_{std}=\,\sqrt{\displaystyle {\sum }_{i=1}^{N}{\left(x\left(i\right)-{x}_{m}\right)}^{2}/N}$
2.	Peak (${x}_{p}$)	${x}_{p}=max\left|x\left(i\right)\right|$
3.	Skewness (${x}_{ske}$)	${x}_{ske}=\displaystyle {\sum }_{N}^{i=1}{\left(x\left(i\right)-{x}_{m}\right)}^{3}/N$
4.	Kurtosis (${x}_{kur}$)	${x}_{kur}=\,\displaystyle {\sum }_{N}^{i=1}{(x\left(i\right)-{x}_{m})}^{4}/N{x}_{std}^{4}$
5.	Root Mean Square (${x}_{rms}$)	${x}_{rms}=\,\sqrt{\displaystyle {\sum }_{i=1}^{N}x{\left(i\right)}^{2}/N}$
6.	Peak Factor (PF)	$PF=\,{x}_{p}/{x}_{rms}$
7.	Square Root Amplitude (${x}_{sra}$)	${x}_{sra}={\left(\displaystyle {\sum }_{i=1}^{N}\sqrt{\left|x\left(i\right)\right|}/N\right)}^{2}$
8.	Shape Factor (SF)	$SF={x}_{rms}/\left(\displaystyle \sum _{N}^{i=1}\left|x\left(i\right)\right|/N\right)$
9.	Impulse Factor (IF)	$IF={x}_{p}/\left(\displaystyle {\sum }_{i=1}^{N}\left|x\left(i\right)\right|/N\right)$
10.	Wavelet Packet Decomposition (WPD) Energy	$WP{D}_{i}=\displaystyle {\sum }_{n=1}^{N}{\left|{x}_{i}\left(n\right)\right|}^{2}/\displaystyle {\sum }_{k=0}^{{2}^{j}-1}\displaystyle {\sum }_{n=1}^{N}{\left|{x}_{k}\left(n\right)\right|}^{2}$
11.	Permutation entropy	$\tfrac{-1}{n-1}\displaystyle {\sum }_{j=1}^{n!}{P}_{j}^{,}{\mathrm{log}}_{2}\left({P}_{j}^{,}\right)$

Here, $x$ represents data; $N$ represents the number of data points; ${x}_{m}$ is the average value of $x;$ ${x}_{k}(n)$ represents the decomposition coefficient for $kth$ sequence and $j$ is the level of $WPD$ decomposition.

The efficacy of the proposed optimization algorithm (ASSA) is tested on twenty-three classification functions. These functions are widely used in literature and defined in table A1 at annexure. The proposed optimization algorithm ASSA is compared with other art of optimizations i.e., SSA, GWO, GOA, SCA and HGPSO in terms of mean, standard deviation, best, worst and median. The comparison is given in table A2. Based on the minimum value of standard deviation, ASSA proved its superiority in nineteen benchmark functions that are Sphere, Schwefel 2.22, Schwefel 1.2, Schwefel 2.21, Rosenbrock, Quartic, Rastrigin, Griewank, Penalized, Penalized 2, Foxholes, Kowalik, Six-hump camel back, Branin, Goldstein-price, Hartman 3, Shekel 5, shekel 7 and shekel 10. For Schwefel and Ackley, the techniques SCA and GWO show better results. HGPSO shows the best results for Step and Hartman 6 benchmark functions. Results obtained showed the overall effectiveness of ASSA in comparison to other optimization techniques.

Over 20 separate runs, the algorithms on classical functions were compared using mean, standard deviation, best, worst, and median. This, however, does not compare the particular run. This suggests that excellence may be acquired through accident. As a result, it is critical to compare the outcomes of each run and determine the significance of the results. To determine the significance level of each run, the Wilcoxon rank-sum statistical test was performed at a 5% significance level and the corresponding P-values for each benchmark are computed and tabulated in table 3. If the P-values are less than 0.05, it is strong evidence against the null hypothesis. This also implies that the best method produced better final objective function values that did not happen by coincidence. For the statistical analysis, the best algorithm in each test function is chosen and separately compared with other algorithms. The best algorithm in the given study is chosen based on the lowest standard deviation value. If there are many algorithms with the same standard deviation, the one with the lowest mean value is the best. Because the best algorithm cannot be compared to itself, N/A, which stands for Not Applicable, has been written for the best algorithm in each function.

As it can be seen from the table, ASSA obtained the best results for 18 functions i.e., F1, F3-F14, F16-F18, and F21-F23. For other functions such as F8 and F19, SCA and SSA respectively show better results. HGPSO is selected as the best algorithm for F2, F15 and F20. As per results in tables 4 and 5, ASSA performs better than other compared algorithms, which demonstrates that the superiority of this algorithms is statistically significant. According to the No-free lunch theorem (NFL) theorem [42], it can solve the problem that cannot be solved efficiently by other algorithms.

Table 4. Weight of each feature obtained after applying PCC.

S. No.	Features	Weight of features
1.	Standard Deviation (${x}_{std}$)	0.2743
2.	Peak (${x}_{p}$)	0.0552
3.	Skewness (${x}_{ske}$)	0.0789
4.	Kurtosis (${x}_{kur}$)	0.1435
5.	Root Mean Square (${x}_{rms}$)	0.5223
6.	Peak Factor (PF)	0.0552
7.	Square Root Amplitude (${x}_{sra}$)	0.0711
8.	Shape Factor (SF)	0.1619
9.	Impulse Factor (IF)	0.0259
10.	Wavelet Packet Decomposition (WPD) Energy	0.0915
11.	Permutation entropy	0.0850

The selection of optimal parameters viz, mode number, and quadratic penalty factor (controls frequency bandwidth) plays a vital role in the selection of VMD parameters, which can be obtained by Ameliorated salp swarm algorithms (ASSA) as proposed in this work. Opposition-based learning along with the position updation concept is introduced in a basic salp swarm algorithm (SSA), which makes it faster. The comparison is made between the basic salp swarm and ameliorated salp swarm algorithm. The convergence curve for both algorithms is shown in figure 9. It has been observed from figure 9 that ASSA converges at a faster rate than that of SSA.

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The proposed ASSA algorithm has also been compared with other algorithms in terms of accuracy considering VMD parameters optimization problem. The results were obtained in bar plot as shown in figure 10. It is clear from the figure that the ASSA outperforms other optimizations techniques.

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Pearson correlation coefficient (PCC) is applied to show the relevance of the extracted features. Based on this coefficient, features are ranked in the order of their decreasing value of the coefficients (weights). The weights of the features listed in table 2 are computed by utilizing PCC and tabulated in table 4. The weights are also plotted in the form of bars for comparative analysis as shown in figure 11. From both table 4 and figure 11, it is clear that the feature 'root mean square' at Sl. No. 5 is the most prominent feature out of all 11 features as it is having the highest weight. Standard deviation holds the second position.

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Based on prominent features, the data set is prepared. The obtained data set is made input to the ELM model, which classifies the different fault conditions. The parameters of ELM are taken as suggested in [35]. Accordingly, the regularization coefficient is taken as 1, kernel type assigned as RBF kernel, and kernel parameter is taken as 0.01. The training and testing accuracy found by the proposed algorithm is 100% and 97.5%, respectively, with a training time of 0.0012 seconds. The proposed work with the ELM classifier is compared with other classification techniques whose results are given in table 5. It clearly shows that the proposed work with ELM outperforms the other classification techniques and is found to be reliable in identifying defect conditions from the vibration signal. The proposed method has also been analysed on different sizes of the data. The two sizes of the data have been considered for this purpose viz., 40 × 11 and 80 × 11. The results of the analysis are tabulated in table 5. It has been observed from table 5, that the training accuracy is higher for larger data sizes. This may be due to the fact that the classifiers are based on supervised learning that means a larger data size trains the classifier more efficiently.