*4.2. Multi-Strategy Improved Grey Wolf Optimization Algorithm* 4.2.1. Sobol-Sequence Initialization Population Strategy

In the swarm intelligence algorithm, whether the initial population distribution is uniform will have a great impact on the optimization performance of the algorithm. GWO initializes the population randomly, resulting in the distribution of the initial population being extremely scattered, which will have a great impact on the algorithm's solving speed and optimization accuracy. Therefore, this paper initializes population individuals through the Sobol sequence. The Sobol sequence is a kind of low difference sequence [27], which is based on the smallest prime number, two. To produce a random sequence *X* ∈ [0, 1], an irreducible polynomial of the highest order *k* in base two is first required to produce a set of predetermined directional numbers *V* = [*V*1, *V*2, · · · , *V<sup>k</sup>* ], and then the index value of the binary sequence *i* = (· · · *i*3*i*2*i*1)<sup>2</sup> is required; then, the nth random number generated by the Sobol sequence is:

$$X\_i = i\_1V\_1 \oplus i\_2V\_2 \oplus \cdots \quad i = (\cdots i\_3i\_2i\_1)\_2 \tag{12}$$

The distribution of individuals with the same population size in the same dimensional space is shown in Figure 2. From Figure 2, it can be seen that the distribution of the population initialized using the Sobol sequence is more uniform than that generated randomly, which enables the population to traverse the entire search space better. *Sensors* **2023**, *23*, x FOR PEER REVIEW 7 of 25

**Figure 2.** The Sobol sequence and random method to generate individual distribution maps. **Figure 2.** The Sobol sequence and random method to generate individual distribution maps.

#### 4.2.2. Exponential Rule Convergence-Factor Adjustment Strategy 4.2.2. Exponential Rule Convergence-Factor Adjustment Strategy

The parameter *A* is an important parameter regulating global exploration and local development in GWO, which is mainly affected by convergence factor *f* . In GWO, when *A* 1 , the grey wolf population searches the entire search domain for potential prey, and The parameter *A* is an important parameter regulating global exploration and local development in GWO, which is mainly affected by convergence factor *f* . In GWO, when |*A*| > 1, the grey wolf population searches the entire search domain for potential prey, and when |*A*| ≤ 1, the grey wolf population will gradually surround and capture prey.

when *A* 1, the grey wolf population will gradually surround and capture prey. In GWO, the value of convergence factor *f* decreases linearly from 2 to 0 with the increase in the number of iterations, which cannot accurately reflect the complex random search process in the actual optimization process. In addition, in the process of algorithm iteration, the same method was used to calculate the enveloping step length for grey wolf In GWO, the value of convergence factor *f* decreases linearly from 2 to 0 with the increase in the number of iterations, which cannot accurately reflect the complex random search process in the actual optimization process. In addition, in the process of algorithm iteration, the same method was used to calculate the enveloping step length for grey wolf individuals with different fitness, which did not reflect the differences among individual

> / 2

> > in GWO decreases linearly with the in-

, which varies exponen-

'

*f*

(13)

*t T*

 *e*<sup>−</sup>

The curves of the linear convergence factor and exponential regular convergence factor proposed in this paper with the number of iterations are shown in Figure 3. As can be

*f*

crease in iterations, resulting in incomplete prey searches in the early stage and slow con-

tially, can thoroughly search for prey in the early stages of the algorithm, thereby enhanc-

individuals with different fitness, which did not reflect the differences among individual grey wolves. Therefore, this paper introduces an updated mode of convergence factor

'

=

*f*

vergence in the later hunting process. The convergence factor

seen from Figure 3, the convergence factor

ing its global optimization performance.

**Figure 3.** The convergence factor comparison curve.

grey wolves. Therefore, this paper introduces an updated mode of convergence factor based on exponential rule changes, whose equation is as follows: grey wolves. Therefore, this paper introduces an updated mode of convergence factor based on exponential rule changes, whose equation is as follows: '/2*tT*

 *e*<sup>−</sup>

=

, the grey wolf population searches the entire search domain for potential prey, and

*f*

, the grey wolf population will gradually surround and capture prey.

increase in the number of iterations, which cannot accurately reflect the complex random search process in the actual optimization process. In addition, in the process of algorithm iteration, the same method was used to calculate the enveloping step length for grey wolf individuals with different fitness, which did not reflect the differences among individual

is an important parameter regulating global exploration and local

**Figure 2.** The Sobol sequence and random method to generate individual distribution maps.

4.2.2. Exponential Rule Convergence-Factor Adjustment Strategy

development in GWO, which is mainly affected by convergence factor

The parameter

*A* 1

when

*A* 1 *A*

In GWO, the value of convergence factor

$$f' = 2e^{-t/T} \tag{13}$$

*f*

decreases linearly from 2 to 0 with the

. In GWO, when

The curves of the linear convergence factor and exponential regular convergence factor proposed in this paper with the number of iterations are shown in Figure 3. As can be seen from Figure 3, the convergence factor *f* in GWO decreases linearly with the increase in iterations, resulting in incomplete prey searches in the early stage and slow convergence in the later hunting process. The convergence factor *f* 0 , which varies exponentially, can thoroughly search for prey in the early stages of the algorithm, thereby enhancing its global optimization performance. tor proposed in this paper with the number of iterations are shown in Figure 3. As can be seen from Figure 3, the convergence factor *f* in GWO decreases linearly with the increase in iterations, resulting in incomplete prey searches in the early stage and slow convergence in the later hunting process. The convergence factor ' *f* , which varies exponentially, can thoroughly search for prey in the early stages of the algorithm, thereby enhancing its global optimization performance.

*Sensors* **2023**, *23*, x FOR PEER REVIEW 7 of 25

**Figure 3. Figure 3.** The convergence factor comparison The convergence factor comparison curve. curve.

#### 4.2.3. Adaptive Location-Update Strategy

In GWO, the initializing *α*, *β*, and *δ* solutions are recorded and retained until they are replaced by a better-fitting individual in the iterative process. In other words, if there is no better *α*, *β*, and *δ* solution in the population than that recorded in the t generation, the new population will still update its position toward wolves *α*, *β*, and *δ*. But when these three are in the local optimal area, then the whole population cannot obtain the optimal solution. Moreover, the average value of *X*1, *X*2, and *X*<sup>3</sup> in GWO cannot show the importance of *α*, *β*, and *δ*. Therefore, a new adaptive location-update strategy is proposed, which is expressed as follows: 

$$\begin{cases} W\_1 = \frac{|X\_1|}{|X\_1| + |X\_2| + |X\_3| + \varepsilon} \\ W\_2 = \frac{|X\_2|}{|X\_1| + |X\_2| + |X\_3| + \varepsilon} \\ W\_3 = \frac{|X\_3|}{|X\_1| + |X\_2| + |X\_3| + \varepsilon} \end{cases} \tag{14}$$

$$\mathbf{g} = \frac{T - t}{T} (\mathbf{g}\_{initial} - \mathbf{g}\_{final}) + \mathbf{g}\_{final} \tag{15}$$

where *g* is the inertia weight. The mathematical expression of grey wolf position update is shown in Equation (16).


$$X(t+1) = \frac{W\_1 X\_1 + W\_2 X\_2 + W\_3 X\_3}{3} g + X\_1 \frac{t}{T} \tag{16}$$

#### 4.2.4. Cauchy–Gaussian Hybrid Mutation Strategy

In order to avoid the local optimization of the basic GWO algorithm, this paper introduces the Cauchy–Gaussian hybrid mutation strategy combining Cauchy and Gaussian distribution, and gives the best individuals the Cauchy–Gaussian perturbation. The

Cauchy–Gaussian operator can generate a large step length to avoid the algorithm falling into local optimality, and its expression is as follows:

$$X\_{new}^\*(t) = X^\*(t) \cdot \left(1 + \lambda\_1 \text{cuchy}(0, 1) + \lambda\_2 \text{Gauss}(0, 1)\right) \tag{17}$$

$$
\lambda\_1 = 1 - \frac{t^2}{T\_{\text{max}}^2} \tag{18}
$$

$$
\lambda\_2 = \frac{t^2}{T\_{\text{max}}^2} \tag{19}
$$

where *X* ∗ *new*(*t*) is the value obtained using Cauchy–Gaussian perturbation, *cauchy*(0, 1) is the Cauchy operator, and *Gauss*(0, 1) is the Gaussian operator. *Sensors* **2023**, *23*, x FOR PEER REVIEW 9 of 25

The pseudocode of MSGWO is shown in Figure 4.

**Figure 4.** The pseudocode of MSGWO. **Figure 4.** The pseudocode of MSGWO.

shown in Table 1.

2

1 1

2

( ) 100( ) ( 1)

+

*i i i*

= − + −

2 2 2

*j*

*n n i i i i F x x x* =<sup>=</sup>

= +

1 1

*F x x i n* <sup>4</sup> ( ) max ,1 = *i i* 

*F x x x x*

*i j F x x* = −

1

*n*

−

*i*

=

5 1 1

*n i*

1

*n i i F x x* =

1

2

3

( )

( )

( )

=

= **Table 1.** Benchmark functions.

#### *4.3. Improved Performance Test of Grey Wolf Optimization Algorithm*

*4.3. Improved Performance Test of Grey Wolf Optimization Algorithm* CEC23 sets of commonly used test functions are important examples of testing algorithm performance [28]. In an effort to test the performance of the MSGWO raised in this article, fifteen test functions in the CEC23 group of commonly used test functions were selected for verification, in which F<sup>1</sup> to F<sup>7</sup> were single-peak benchmark functions, F<sup>8</sup> to F<sup>13</sup> were multi-peak benchmark functions, and F<sup>14</sup> to F<sup>15</sup> were fixed-dimensional multi-peak test functions. The computing platform performance was based on IntelI CITM) i5-6500 CPU, 3.20 GHz main frequency, and 8 GB memory. The details of the test function are CEC23 sets of commonly used test functions are important examples of testing algorithm performance [28]. In an effort to test the performance of the MSGWO raised in this article, fifteen test functions in the CEC23 group of commonly used test functions were selected for verification, in which F<sup>1</sup> to F<sup>7</sup> were single-peak benchmark functions, F<sup>8</sup> to F<sup>13</sup> were multi-peak benchmark functions, and F<sup>14</sup> to F<sup>15</sup> were fixed-dimensional multi-peak test functions. The computing platform performance was based on IntelI CITM) i5-6500 CPU, 3.20 GHz main frequency, and 8 GB memory. The details of the test function are shown in Table 1.

**Function Dim Range Optima**

30 [−100, 100] 0

30 [−10, 10] 0

30 [−100, 100] 0

30 [−100, 100] 0

30 [−30, 30] 0


**Table 1.** Benchmark functions.

#### 4.3.1. Comparison Experiment between MSGWO and Standard Optimization Algorithm

In an effort to objectively verify the performance of MSGWO, the population size was set to 30 times, the maximum number of iterations was set to 500 times, and each algorithm was run independently 30 times. Algorithms to be compared in the experiment included the bat optimization algorithm (BOA) [29], whale optimization algorithm (WOA) [30], grey wolf optimization algorithm (GWO), gravity search algorithm (GSA) [31], particle swarm optimization algorithm (PSO) [32], and artificial bee colony algorithm (ABC) [33]. The parameters of all the comparison algorithms in the experiment were the same as those recommended in the original literature. The mean value and standard deviation of the optimal value of the simulation results were taken as the evaluation indexes of the algorithm performance, and the results are shown in Table 2. The test results shown in bold black in Table 2 are the best for comparison.

It can be seen from the data in Table 2 that MSGWO obtained the optimal mean and variance in functions F1–F4, F7, F9–F13, and F15. In the function F5, MSGWO obtained the best average value, but its stability was worse than BOA. In the function F6, MSGWO obtained the best average value, but its stability was worse than WOA and GWO. In the function F8, MSGWO achieved the best average, but its stability was the worst. In the function F14, MSGWO obtained the best average value, but its stability was worse than that of the ABC algorithm. It can be seen that MSGWO obtained the optimal average value in all the selected test functions. Although the stability of the algorithm was worse in some individual functions than that of some comparison algorithms, MSGWO still had better optimization performance on the whole.


**Table 2.** The compared results of MSGWO and standard optimization algorithms.

The simulation results show that MSGWO had better optimization performance under different benchmark test functions. This shows that compared with GWO, MSGWO enhances the local search ability, thus increasing the solution accuracy, and for multi-modal test functions, MSGWO has a strong local optimal avoidance ability, and can better find the optimal solution. When other algorithms have low optimization accuracy or even cannot converge, MSGWO still has high solving accuracy.

In order to explore the influence of improvement strategies on the algorithm convergence speed, the convergence curves of each algorithm under 15 benchmark test functions are shown in Figure 5. As can be seen from Figure 5, MSGWO has high precision and the fastest convergence rate of the optimal solution in the comparison algorithm, which effectively saves the optimization time.

4.3.2. Comparison Experiment between MSGWO and Improved Optimization Algorithm

In an effort to further test the performance of the MSGWO, the population size was set to 30 times, the maximum number of iterations was set to 500 times, and each algorithm was independently run 30 times. Comparative experimental analysis was conducted between MSGWO and GWO, MEGWO [34], mGWO [35], IGWO [36], and MPSO [37]. The mean

value and standard deviation of the optimal value of the simulation results were taken as the evaluation indexes of the algorithm performance, and the results are shown in Table 3. The test results shown in bold black in Table 3 are the best for comparison. are shown in Figure 5. As can be seen from Figure 5, MSGWO has high precision and the fastest convergence rate of the optimal solution in the comparison algorithm, which effectively saves the optimization time.

In order to explore the influence of improvement strategies on the algorithm convergence speed, the convergence curves of each algorithm under 15 benchmark test functions

*Sensors* **2023**, *23*, x FOR PEER REVIEW 12 of 25

**Figure 5.** The convergence curve of MSGWO is compared with that of standard algorithm. std 0.24 0.15 0.22 0.16 0.25 **0.13 Figure 5.** The convergence curve of MSGWO is compared with that of standard algorithm.

ducted between MSGWO and GWO, MEGWO [34], mGWO [35], IGWO [36], and MPSO [37]. The mean value and standard deviation of the optimal value of the simulation results were taken as the evaluation indexes of the algorithm performance, and the results are shown in Table 3. The test results shown in bold black in Table 3 are the best for compari-

**Table 3.** Comparison of experimental results between MSGWO and improved algorithms.

**F Index GWO MEGWO mGWO IGWO MPSO MSGWO**

mean 1.04 × 10−<sup>27</sup> 4.30 × 10−<sup>64</sup> 1.04 × 10−<sup>18</sup> 1.33 × 10−<sup>209</sup> 2.61 × 10−<sup>26</sup> **0** std 1.37 × 10−<sup>27</sup> 2.09 × 10−<sup>63</sup> 2.97 × 10−<sup>18</sup> 0 1.12 × 10−<sup>25</sup> **0**

mean 9.50 × 10−<sup>17</sup> 1.70 × 10−<sup>43</sup> 2.65 × 10−<sup>12</sup> 6.12 × 10−<sup>21</sup> 1.40 × 10−<sup>16</sup> **0** std 6.40 × 10−<sup>17</sup> 5.77 × 10−<sup>43</sup> 1.99 × 10−<sup>12</sup> 6.67 × 10−<sup>21</sup> 2.86 × 10−<sup>16</sup> **0**

mean 3.15 × 10−<sup>5</sup> 0.23 0.68 2.73 × 10−<sup>5</sup> 9.63 × 10<sup>2</sup> **0** std 9.68 × 10−<sup>5</sup> 0.48 0.81 9.57 × 10−<sup>5</sup> 4.81 × 10<sup>2</sup> **0**

mean 7.78 × 10−<sup>7</sup> 2.06 × 10−<sup>5</sup> 0.68 2.93 × 10−<sup>7</sup> 2.05 × 10−<sup>10</sup> **0** std 8.85 × 10−<sup>7</sup> 5.68 × 10−<sup>5</sup> 0.85 1.78 × 10−<sup>7</sup> 4.81 × 10−<sup>10</sup> **0**

mean 28.44 27.94 27.92 27.64 88.91 **27.08** std 0.82 9.97 0.58 **0.32** 1.89 × 10<sup>2</sup> 0.42

mean 0.90 0.49 0.41 0.43 0.41 **0.36** std 0.38 1.14 0.25 **0.19** 0.22 0.54

mean 2.07 × 10−<sup>3</sup> 1.01 × 10−<sup>3</sup> 4.68 × 10−<sup>3</sup> 2.80 × 10−<sup>3</sup> 1.68 × 10−<sup>3</sup> **6.58 ×10−<sup>5</sup>** std 7.10 × 10−<sup>4</sup> 9.10 × 10−<sup>4</sup> 1.90 × 10−<sup>3</sup> 1.10 × 10−<sup>3</sup> 8.87 × 10−<sup>4</sup> **6.62 ×10−<sup>5</sup>**

mean −5.70 × 10<sup>3</sup> −1.26 × 10<sup>4</sup> −5.33 × 10<sup>3</sup> −8.28 × 10<sup>3</sup> −8.12 × 10<sup>3</sup> **−5.47 ×10<sup>58</sup>** std 1.18 × 10<sup>3</sup> **2.15 ×10−<sup>12</sup>** 1.11 × 10<sup>3</sup> 1.69 × 10<sup>3</sup> 1.12 × 10<sup>3</sup> 1.81 × 10<sup>59</sup>

mean 1.03 × 10−<sup>13</sup> 5.27 × 10−<sup>15</sup> 1.26 × 10−<sup>10</sup> 6.25 × 10−<sup>14</sup> 6.22 × 10−<sup>15</sup> **8.88 ×10−<sup>16</sup>** std 2.23 × 10−<sup>14</sup> 1.50 × 10−<sup>15</sup> 9.69 × 10−<sup>11</sup> 8.96 × 10−<sup>15</sup> 7.38 × 10−<sup>15</sup> **0**

mean 3.63 0 37.94 27.09 23.92 **0** std 4.07 0 30.01 22.81 22.64 **0**

mean 3.02 × 10−<sup>3</sup> 0 3.83 × 10−<sup>3</sup> 3.37 × 10−<sup>3</sup> 0 **0** std 5.70 × 10−<sup>3</sup> 0 9.40 × 10−<sup>3</sup> 6.00 × 10−<sup>3</sup> 0 **0**

mean 0.07 0.05 0.05 6.58 × 10−<sup>2</sup> 0.42 **0.05** std 0.27 0.56 0.04 **2.00 ×10−<sup>3</sup>** 0.73 0.10

mean 0.71 0.46 0.63 0.66 0.45 **0.43**

son.

F1

F2

F3

F4

F5

F6

F7

F8

F9

F<sup>10</sup>

F<sup>11</sup>

F<sup>12</sup>

F<sup>13</sup>

4.3.2. Comparison Experiment between MSGWO and Improved Optimization Algorithm


**Table 3.** Comparison of experimental results between MSGWO and improved algorithms.

It can be seen from the data in Table 3 that for the optimization accuracy of the algorithm, MSGWO obtained the optimal average value in the function F1–F15. In terms of algorithm stability, the stability of the MSGWO was worse than that of the IGWO algorithm in F5; worse than those of the GWO, mGWO, IGWO, and MPSO algorithms in F6; the worst in F8; worse than those of the mGWO and IGWO algorithms in F12; worse than that of the MPSO algorithm in F14; and worse than that of MEGWO in F15. However, in the other nine test functions, its stability was better than the comparison algorithm, so the overall stability was still the best.

The convergence curves of the MSGWO algorithm and improved algorithms under 15 benchmark functions are shown in Figure 6. It can be seen from the convergence curves of each test function in Figure 6 that MSGWO has better local extreme value escape ability, overall optimization coordination, and convergence performance than the comparison algorithm.

ity was still the best.

gorithm.

F<sup>14</sup>

F<sup>15</sup>

mean 4.53 1.78 2.00 1.70 1.99 **1.55** std 4.03 2.91 2.76 0.76 **0.36** 0.71

mean 2.47 × 10−<sup>3</sup> 3.07 × 10−<sup>4</sup> 1.04 × 10−<sup>3</sup> 8.62 × 10−<sup>4</sup> 5.68 × 10−<sup>4</sup> **3.46 ×10−<sup>4</sup>** std 6.00 × 10−<sup>3</sup> **3.42 ×10−<sup>15</sup>** 3.60 × 10−<sup>3</sup> 3.00 × 10−<sup>3</sup> 3.36 × 10−<sup>4</sup> 1.69 × 10−<sup>4</sup>

> It can be seen from the data in Table 3 that for the optimization accuracy of the algorithm, MSGWO obtained the optimal average value in the function F1–F15. In terms of algorithm stability, the stability of the MSGWO was worse than that of the IGWO algorithm in F5; worse than those of the GWO, mGWO, IGWO, and MPSO algorithms in F6; the worst in F8; worse than those of the mGWO and IGWO algorithms in F12; worse than that of the MPSO algorithm in F14; and worse than that of MEGWO in F15. However, in the other nine test functions, its stability was better than the comparison algorithm, so the overall stabil-

> The convergence curves of the MSGWO algorithm and improved algorithms under 15 benchmark functions are shown in Figure 6. It can be seen from the convergence curves

**Figure 6.** The convergence curves are compared between MSGWO and the improved algorithm. **Figure 6.** The convergence curves are compared between MSGWO and the improved algorithm.

tion algorithms (MEGWO, mGWO, IGWO, MPSO) using the Wilcoxon rank sum test at a significance level of 5%. The population size of all algorithms was set to 30, with 500 iterations. The *p* value of the test result was less than 0.05, indicating that there were significant differences between the comparison algorithms. The symbols "+", "−", and "=" of R indicate that the performance of MSGWO was better than, worse than, and equivalent to the comparison algorithm, respectively, and N/A indicates that a significance judgment

could not be made. The test results are shown in Tables 4 and 5, respectively.

**Table 4.** Wilcoxon rank sum test results for MSGWO and standard algorithms.

**F Index MSGWO–WOA MSGWO–GWO MSGWO–BOA MSGWO–GSA MSGWO–PSO MSGWO–ABC**

P 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> R + + + + + +

P 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> R + + + + + +

P 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> R + + + + + +

P 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> R + + + + + +

P 1.06 × 10−<sup>4</sup> 2.88 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.92 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> R + + + + + +

P 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 7.69 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 9.37 × 10−<sup>3</sup> R + + + + + +

P 2.13 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> 1.73 × 10−<sup>6</sup> R + + + + + +

In order to verify whether there were significant differences between MSGWO and other comparison algorithms, the Wilcoxon rank sum test was used for statistical analysis

4.3.3. Wilcoxon Rank Sum Test

F1

F2

F3

F4

F5

F6

F7

#### 4.3.3. Wilcoxon Rank Sum Test

In order to verify whether there were significant differences between MSGWO and other comparison algorithms, the Wilcoxon rank sum test was used for statistical analysis of the experimental data. For each test function, the results of 30 independent optimizations of MSGWO were compared with the 30 independent optimizations of the standard optimization algorithms (WOA, GWO, BOA, GSA, PSO, ABC) and improved optimization algorithms (MEGWO, mGWO, IGWO, MPSO) using the Wilcoxon rank sum test at a significance level of 5%. The population size of all algorithms was set to 30, with 500 iterations. The *p* value of the test result was less than 0.05, indicating that there were significant differences between the comparison algorithms. The symbols "+", "−", and "=" of R indicate that the performance of MSGWO was better than, worse than, and equivalent to the comparison algorithm, respectively, and N/A indicates that a significance judgment could not be made. The test results are shown in Tables 4 and 5, respectively.

**Table 4.** Wilcoxon rank sum test results for MSGWO and standard algorithms.


**Table 5.** Wilcoxon rank sum test results for MSGWO and improved algorithms.



**Table 5.** *Cont*.

As can be seen from Table 4, comparing the optimization results of MSGWO with those of WOA, GWO, BOA, GSA, PSO, and ABC on 15 test functions, the *p* values of the test results are all less than 0.05, and the R values are all +, indicating that the optimization results of MSGWO are significantly different from those of other six algorithms. Additionally, MSGWO is significantly better, which shows the superiority of the MSGWO algorithm statistically.

As can be seen from Table 5, compared with the optimization results of the five improved algorithms on 15 test functions, the *p* values of the test results of MSGWO are all less than 0.05, and R is +/=, which indicates that the optimization results of MSGWO are significantly different from the optimization results of the five improved algorithms, and MSGWO is significantly better. This result shows the superiority of the MSGWO algorithm statistically.

#### 4.3.4. Population Diversity Analysis of MSGWO

In an effort to further illustrate the effectiveness of the proposed algorithm, the diversity of population particles during evolution was analyzed. Population diversity measure-

ments can accurately evaluate whether a population is being explored or exploited [38], and the specific calculation formula is as follows: urements can accurately evaluate whether a population is being explored or exploited [38], and the specific calculation formula is as follows:

In an effort to further illustrate the effectiveness of the proposed algorithm, the diversity of population particles during evolution was analyzed. Population diversity meas-

As can be seen from Table 4, comparing the optimization results of MSGWO with those of WOA, GWO, BOA, GSA, PSO, and ABC on 15 test functions, the *p* values of the test results are all less than 0.05, and the R values are all +, indicating that the optimization results of MSGWO are significantly different from those of other six algorithms. Additionally, MSGWO is significantly better, which shows the superiority of the MSGWO algo-

As can be seen from Table 5, compared with the optimization results of the five improved algorithms on 15 test functions, the *p* values of the test results of MSGWO are all less than 0.05, and R is +/=, which indicates that the optimization results of MSGWO are significantly different from the optimization results of the five improved algorithms, and MSGWO is significantly better. This result shows the superiority of the MSGWO algo-

*Sensors* **2023**, *23*, x FOR PEER REVIEW 17 of 25

4.3.4. Population Diversity Analysis of MSGWO

rithm statistically.

rithm statistically.

$$I\_{\mathbb{C}}(t) = \sqrt{\sum\_{i=1}^{N} \sum\_{d=1}^{D} \left( \mathbf{x}\_{id}(t) - c\_d(t) \right)^2} \tag{20}$$

$$\mathbf{c}\_d(t) = \frac{1}{D} \sum\_{i=1}^{N} \mathbf{x}\_{id}(t) \tag{21}$$

*c*

where *I<sup>C</sup>* represents the dispersion between the population and the center of mass *c<sup>d</sup>* in each iteration, and *xid* represents the value of the *d* dimension of the ith individual at the time of iteration *t*. where *C I* represents the dispersion between the population and the center of mass *d* in each iteration, and *id x* represents the value of the *d* dimension of the ith individual at the time of iteration *t* .

1

*N*

A small population diversity measure indicates that particles converge near the population center, that is, develop in a small space. A large population diversity measure indicates that the particles are far from the center of the population, that is, they explore in a larger space. Unimodal function F<sup>1</sup> and multi-modal function F<sup>15</sup> of the commonly used test functions of CEC23 were selected as representatives to analyze the population diversity measurements of MSGWO and GWO, respectively. The experimental results are shown in Figure 7a,b. A small population diversity measure indicates that particles converge near the population center, that is, develop in a small space. A large population diversity measure indicates that the particles are far from the center of the population, that is, they explore in a larger space. Unimodal function F<sup>1</sup> and multi-modal function F<sup>15</sup> of the commonly used test functions of CEC23 were selected as representatives to analyze the population diversity measurements of MSGWO and GWO, respectively. The experimental results are shown in Figure 7a,b.

**Figure 7.** Population diversity measurement analysis. **Figure 7.** Population diversity measurement analysis.

As can be seen from Figure 7, the population diversity measure of the GWO algorithm decreased at the fastest speed in F<sup>1</sup> and F15, which is not conducive to sufficient space exploration in the early stage and is easy to fall into local optimization. In F1, the MSGWO As can be seen from Figure 7, the population diversity measure of the GWO algorithm decreased at the fastest speed in F<sup>1</sup> and F15, which is not conducive to sufficient space exploration in the early stage and is easy to fall into local optimization. In F1, the MSGWOalgorithm maintained a high level of population diversity in the early stage of evolution, fully satisfying the exploration of particles in the whole space, while the population diversity decreased rapidly in the middle and late stages of evolution, indicating that the algorithm has a good development ability. In F15, MSGWO population diversity fluctuated greatly and remained at a high level, indicating that the algorithm has a good global exploration ability.

#### **5. Bearing Fault Detection**

#### *5.1. Parameter Adaptive Multistable Stochastic Resonance Strategy*

In SR performance measurement indicators, signal-to-noise ratio (SNR) is commonly used and plays an important role. In this paper, the SNR is used as the target of optimization, that is, the fitness function. The formula for calculating the SNR is as follows [39]:

$$\text{SNR} = 10 \log\_{10} \frac{A\_l}{\sum\_{n=0}^{N/2} A\_n} \tag{22}$$

where *A<sup>t</sup>* is the amplitude of the target frequency, *A<sup>n</sup>* is the amplitude of frequencies other than the target frequency in the input signal, and *N* is the number of samples. *At An* other than the target frequency in the input signal, and *N* is the number of samples. Based on the above analysis, the flow chart of the bearing fault-detection method

=

SNR 10log *<sup>t</sup>*

=

*n*

0

*A*

*n*

(22)

is the amplitude of frequencies

*A*

 10 / 2

*N*

In SR performance measurement indicators, signal-to-noise ratio (SNR) is commonly used and plays an important role. In this paper, the SNR is used as the target of optimization, that is, the fitness function. The formula for calculating the SNR is as follows [39]:

algorithm maintained a high level of population diversity in the early stage of evolution, fully satisfying the exploration of particles in the whole space, while the population diversity decreased rapidly in the middle and late stages of evolution, indicating that the algorithm has a good development ability. In F15, MSGWO population diversity fluctuated greatly and remained at a high level, indicating that the algorithm has a good global ex-

Based on the above analysis, the flow chart of the bearing fault-detection method proposed in this paper is shown in Figure 8, and its specific steps are as follows: proposed in this paper is shown in Figure 8, and its specific steps are as follows:

*Sensors* **2023**, *23*, x FOR PEER REVIEW 18 of 25

*5.1. Parameter Adaptive Multistable Stochastic Resonance Strategy*

is the amplitude of the target frequency,

ploration ability.

where

**5. Bearing Fault Detection**

**Figure 8.** The flow diagram of the proposed algorithm.

**Figure 8.** The flow diagram of the proposed algorithm.

Step 1: Input noisy signals and initialize MSGWO parameters. The range of *a* is [0, 0.5]; the range of *b* , *c* , and *h* are [0, 10]. The maximum number of iterations is 200 and Step 1: Input noisy signals and initialize MSGWO parameters. The range of *a* is [0, 0.5]; the range of *b*, *c*, and *h* are [0, 10]. The maximum number of iterations is 200 and the number of grey wolf populations is 30.

the number of grey wolf populations is 30. Step 2: Run the MSGWO, calculate the SNR according to Equation (22), then update the individual position, iterate to the maximum number of iterations, and finally termi-Step 2: Run the MSGWO, calculate the SNR according to Equation (22), then update the individual position, iterate to the maximum number of iterations, and finally terminate the iteration.

nate the iteration. Step 3: Substitute the optimal solutions of *a*, *b*, *c*, and *h* into the SR system for operation, and subject the output of the SR system to fast Fourier transform to obtain the frequency domain. Then, analyze the output of the SR according to the frequency domain, and capture the fault frequency.

#### *5.2. CWRU Bearing Data Set*

In an effort to verify the applicability of the raised method in actual fault-signal detection, the open bearing-fault data set of CWRU was selected for the experiment [40], and the driving end bearing model 6205-2RS was used. Since the rotating speed of the bearing was 1750 rpm, the fault characteristic frequency of the inner ring was calculated to be 158 Hz. In the experiment, the sampling frequency was set to 12 kHz, and the data length of the signal was 12,000. The time domain and frequency domain waveforms of the input signal are shown in Figure 9, and the output signal-to-noise ratio was SNR = −37.77. As can be seen from Figure 9, the fault frequency of the original signal was difficult to capture in its frequency domain due to the influence of environmental noise. In order to ensure the accuracy of the experimental results, the average method of 30 experiments was adopted. The optimal parameters optimized by MSGWO were as follows: *a* = 0.033, *b* = 0.567, *c* = 0.082, and *h* = 0.086. We substituted the four parameters *a*, *b*, *c*, and *h* into the SR system to obtain the frequency domain waveform of its output, as shown in Figure 10. The output signal-to-noise ratio was SNR = −26.92, which was 10.85 dB higher

than that of the input. According to the frequency domain waveform diagram in Figure 10, it can be observed that there was a clear spike at the target frequency, and the amplitude of the peak frequency was much larger than the amplitude of other surrounding frequencies. It can be seen that the method in this paper can effectively detect the bearing fault signal. Figure 10, it can be observed that there was a clear spike at the target frequency, and the amplitude of the peak frequency was much larger than the amplitude of other surrounding frequencies. It can be seen that the method in this paper can effectively detect the bearing fault signal.

into the SR system to obtain the frequency domain waveform of its output, as shown

higher than that of the input. According to the frequency domain waveform diagram in

. We substituted the four parameters

SNR

 26.92

= −

*a* , *b* , *c*

operation, and subject the output of the SR system to fast Fourier transform to obtain the frequency domain. Then, analyze the output of the SR according to the frequency domain,

In an effort to verify the applicability of the raised method in actual fault-signal detection, the open bearing-fault data set of CWRU was selected for the experiment [40], and the driving end bearing model 6205-2RS was used. Since the rotating speed of the bearing was 1750 rpm, the fault characteristic frequency of the inner ring was calculated to be 158 Hz. In the experiment, the sampling frequency was set to 12 kHz, and the data length of the signal was 12,000. The time domain and frequency domain waveforms of the input

can be seen from Figure 9, the fault frequency of the original signal was difficult to capture in its frequency domain due to the influence of environmental noise. In order to ensure the accuracy of the experimental results, the average method of 30 experiments was

, and

*h*

into the SR system for

SNR

 37.77

*a* = 0.033,

. As

= −

*a* , *b* , *c*, and

, which was 10.85 dB

and

*h* = 0.086

in Figure 10. The output signal-to-noise ratio was

*Sensors* **2023**, *23*, x FOR PEER REVIEW 19 of 25

signal are shown in Figure 9, and the output signal-to-noise ratio was

adopted. The optimal parameters optimized by MSGWO were as follows:

Step 3: Substitute the optimal solutions of

and capture the fault frequency.

*5.2. CWRU Bearing Data Set*

*b* = 0.567 *c* = 0.082

*h*

**Figure 9.** Time domain waveform and FFT spectrum of CWRU input signal. **Figure 9.** Time domain waveform and FFT spectrum of CWRU input signal.

**Figure 10.** The FFT spectrum of the output signal processed by the raised algorithm. **Figure 10.** The FFT spectrum of the output signal processed by the raised algorithm.

In the case of the same parameters, the raised method was compared with five bearing fault-detection methods based on the improved algorithms to optimize the SR parameters. In an effort to ensure the accuracy of the experimental results, the method of averaging 30 experiments was adopted. The comparison experiment results are shown in Table 6. The test results shown in black bold in Table 6 are the best results for comparison. In the case of the same parameters, the raised method was compared with five bearing fault-detection methods based on the improved algorithms to optimize the SR parameters. In an effort to ensure the accuracy of the experimental results, the method of averaging 30 experiments was adopted. The comparison experiment results are shown in Table 6. The test results shown in black bold in Table 6 are the best results for comparison.

**Table 6.** Comparison of experimental parameter results based on CWRU dataset. **Table 6.** Comparison of experimental parameter results based on CWRU dataset.


According to the data in Table 6, compared with five bearing fault-detection methods based on improved algorithms to optimize SR parameters, the raised method had the highest SNR, but the convergence speed was slower than that of bearing fault-detection According to the data in Table 6, compared with five bearing fault-detection methods based on improved algorithms to optimize SR parameters, the raised method had the highest SNR, but the convergence speed was slower than that of bearing fault-detection methods based on IGWO and MPSO. Since the SNR was taken as the evaluation index in

methods based on IGWO and MPSO. Since the SNR was taken as the evaluation index in bearing fault detection, the proposed method had some advantages over the five bearing

In an effort to further verify the applicability of the raised method in actual faultsignal detection, the bearing data set of the MFPT in the United States was selected as the research object [41] to detect the outer-ring signal of the faulty bearing. The input shaft speed of the selected outer ring fault signal was 25 Hz, the load was 25, and the fault characteristic frequency was calculated to be 162 Hz. The time domain and frequency domain waveform of the input signal are shown in Figure 11. According to Figure 11, due to the influence of ambient noise, the fault frequency of the original signal was submerged in the noise and was difficult to be captured in its frequency domain. In an effort to ensure the accuracy of the experimental results, the average method of 30 experiments was

into the SR system to obtain the frequency domain waveform of its output, as shown in Figure 12. According to the frequency domain waveform diagram in Figure 12, it can be observed that the amplitude of the target frequency was the largest in its frequency

. We substituted the four parameters

*a* = 0.500,

*a* , *b* , *c*, and

adopted. The optimal parameters optimized by MSGWO were as follows:

SNR −28.35 −28.51 −28.27 −28.37 −28.32 **−26.92**

, and

*h* = 0.409

*5.3. MFPT Bearing Data Set*

*b* = 9.571 , *c* = 0.019

*h*

bearing fault detection, the proposed method had some advantages over the five bearing fault-detection methods based on the improved algorithm to optimize the SR parameters.

#### *5.3. MFPT Bearing Data Set*

In an effort to further verify the applicability of the raised method in actual fault-signal detection, the bearing data set of the MFPT in the United States was selected as the research object [41] to detect the outer-ring signal of the faulty bearing. The input shaft speed of the selected outer ring fault signal was 25 Hz, the load was 25, and the fault characteristic frequency was calculated to be 162 Hz. The time domain and frequency domain waveform of the input signal are shown in Figure 11. According to Figure 11, due to the influence of ambient noise, the fault frequency of the original signal was submerged in the noise and was difficult to be captured in its frequency domain. In an effort to ensure the accuracy of the experimental results, the average method of 30 experiments was adopted. The optimal parameters optimized by MSGWO were as follows: *a* = 0.500, *b* = 9.571, *c* = 0.019, and *h* = 0.409. We substituted the four parameters *a*, *b*, *c*, and *h* into the SR system to obtain the frequency domain waveform of its output, as shown in Figure 12. According to the frequency domain waveform diagram in Figure 12, it can be observed that the amplitude of the target frequency was the largest in its frequency domain and was much larger than the amplitude of other surrounding frequencies. This further proves that the raised method can detect the bearing fault signal effectively. *Sensors* **2023**, *23*, x FOR PEER REVIEW 21 of 25 domain and was much larger than the amplitude of other surrounding frequencies. This further proves that the raised method can detect the bearing fault signal effectively. *Sensors* **2023**, *23*, x FOR PEER REVIEW 21 of 25 domain and was much larger than the amplitude of other surrounding frequencies. This further proves that the raised method can detect the bearing fault signal effectively.

**Figure 11.** Time domain waveform and FFT spectrum of MFPT input signal. **Figure 11.** Time domain waveform and FFT spectrum of MFPT input signal.

**Figure 12.** The FFT spectrum of the output signal processed by the proposed algorithm. **Figure 12.** The FFT spectrum of the output signal processed by the proposed algorithm.

**Table 7.** Comparison of experimental parameter results based on MFPT dataset.

**Table 7.** Comparison of experimental parameter results based on MFPT dataset.

this article has certain advantages over the comparative method.

this article has certain advantages over the comparative method.

*5.4. Bearing-Fault Diagnosis of Crystal Growing Furnace*

*5.4. Bearing-Fault Diagnosis of Crystal Growing Furnace*

**Figure 12.** The FFT spectrum of the output signal processed by the proposed algorithm. In the case of the same parameters, the raised method was compared with five bear-In the case of the same parameters, the raised method was compared with five bearing fault-detection methods based on the improved algorithms to optimize the SR param-In the case of the same parameters, the raised method was compared with five bearing fault-detection methods based on the improved algorithms to optimize the SR parameters. In an effort to ensure the accuracy of the experimental results, the method of averaging

ing fault-detection methods based on the improved algorithms to optimize the SR parameters. In an effort to ensure the accuracy of the experimental results, the method of aver-

eters. In an effort to ensure the accuracy of the experimental results, the method of averaging 30 experiments was adopted. The comparison experiment results are shown in Ta-

a 0.500 0.495 0.500 0.472 0.052 0.500 b 10.00 2.173 8.554 8.247 8.968 9.571 c 0.025 0.488 0.054 3.728 1.287 0.019 h 0.328 0.185 0.257 0.069 0.122 0.409 Time 21.51 25.35 35.52 34.79 22.91 **19.95** SNR −26.56 −27.75 −26.82 −27.21 −27.62 **−26.42**

a 0.500 0.495 0.500 0.472 0.052 0.500 b 10.00 2.173 8.554 8.247 8.968 9.571 c 0.025 0.488 0.054 3.728 1.287 0.019 h 0.328 0.185 0.257 0.069 0.122 0.409 Time 21.51 25.35 35.52 34.79 22.91 **19.95** SNR −26.56 −27.75 −26.82 −27.21 −27.62 **−26.42**

According to the data in Table 7, compared with five bearing fault-detection methods

based on the improved algorithms to optimize SR parameters, the method raised in this article had a larger SNR and better time performance. Therefore, the method proposed in

According to the data in Table 7, compared with five bearing fault-detection methods

In this paper, the crystal lifting and rotating mechanism of a crystal growing furnace

was taken as the actual test object, as shown in Figure 13. The crystal growing furnace is the major equipment for producing wafers. The mechanism is composed of two

In this paper, the crystal lifting and rotating mechanism of a crystal growing furnace

was taken as the actual test object, as shown in Figure 13. The crystal growing furnace is the major equipment for producing wafers. The mechanism is composed of two

based on the improved algorithms to optimize SR parameters, the method raised in this article had a larger SNR and better time performance. Therefore, the method proposed in

**GWO IGWO MEGWO mGWO MPSO MSGWO**

**GWO IGWO MEGWO mGWO MPSO MSGWO**

30 experiments was adopted. The comparison experiment results are shown in Table 7. The test results shown in black bold in Table 7 are the best results for comparison.


**Table 7.** Comparison of experimental parameter results based on MFPT dataset.

According to the data in Table 7, compared with five bearing fault-detection methods based on the improved algorithms to optimize SR parameters, the method raised in this article had a larger SNR and better time performance. Therefore, the method proposed in this article has certain advantages over the comparative method. *Sensors* **2023**, *23*, x FOR PEER REVIEW 22 of 25

#### *5.4. Bearing-Fault Diagnosis of Crystal Growing Furnace*

In this paper, the crystal lifting and rotating mechanism of a crystal growing furnace was taken as the actual test object, as shown in Figure 13. The crystal growing furnace is the major equipment for producing wafers. The mechanism is composed of two Mitsubishi HG-KR73 servo motors, the crystal lift motor is used to lift the crystal upward, and the crystal rotating motor is used to drive the crystal to spin during the growth process. Because the stability of crystal rotating is an important factor to determine the crystal formation and crystal quality, it is necessary to accurately monitor the fault of the crystal rotating motor. The experiment object was the motor of a certain type of electronic-grade silicon single-crystal growing furnace. The purpose was to detect the failure frequency of the crystal rotating motor. A certain type of three-dimensional vibration sensor was used in the experiment, and its connection with the motor is shown in Figure 14. As shown in Figure 14, the vibration sensor was adsorbed on the motor, and information such as vibration displacement, vibration speed, and vibration frequency can be collected. The deceleration ratio of the crystal rotating system was 100:1, that is, when the crystal rotating speed was 10 rad/min, the speed of the crystal rotating motor was 1000 rad/min. Mitsubishi HG-KR73 servo motors, the crystal lift motor is used to lift the crystal upward, and the crystal rotating motor is used to drive the crystal to spin during the growth process. Because the stability of crystal rotating is an important factor to determine the crystal formation and crystal quality, it is necessary to accurately monitor the fault of the crystal rotating motor. The experiment object was the motor of a certain type of electronic-grade silicon single-crystal growing furnace. The purpose was to detect the failure frequency of the crystal rotating motor. A certain type of three-dimensional vibration sensor was used in the experiment, and its connection with the motor is shown in Figure 14. As shown in Figure 14, the vibration sensor was adsorbed on the motor, and information such as vibration displacement, vibration speed, and vibration frequency can be collected. The deceleration ratio of the crystal rotating system was 100:1, that is, when the crystal rotating speed was 10 rad/min, the speed of the crystal rotating motor was 1000 rad/min.

**Figure 13.** Crystal growing furnace and crystal lifting and rotating mechanism. **Figure 13.** Crystal growing furnace and crystal lifting and rotating mechanism.

The vibration signal of the motor collected by the vibration sensor is shown in Figure

15. As can be seen from Figure 15, the time domain signal of the actual motor fault collected by the vibration sensor is very weak, completely submerged in the noise, and the frequency domain signal cannot distinguish the fault frequency. The method proposed in this paper was used to detect the fault frequency of the crystal rotating motor, and the test results are shown in Figure 16. It can be seen from Figure 16 that the algorithm increased the frequency domain amplitude of the fault signal and effectively detected that the fault

**Figure 14.** Vibration sensor installation position.

frequency of the crystal motor was 35 Hz.

**Figure 14.** Vibration sensor installation position. **Figure 14.** Vibration sensor installation position.

The vibration signal of the motor collected by the vibration sensor is shown in Figure 15. As can be seen from Figure 15, the time domain signal of the actual motor fault collected by the vibration sensor is very weak, completely submerged in the noise, and the frequency domain signal cannot distinguish the fault frequency. The method proposed in this paper was used to detect the fault frequency of the crystal rotating motor, and the test results are shown in Figure 16. It can be seen from Figure 16 that the algorithm increased the frequency domain amplitude of the fault signal and effectively detected that the fault frequency of the crystal motor was 35 Hz. The vibration signal of the motor collected by the vibration sensor is shown in Figure 15. As can be seen from Figure 15, the time domain signal of the actual motor fault collected by the vibration sensor is very weak, completely submerged in the noise, and the frequency domain signal cannot distinguish the fault frequency. The method proposed in this paper was used to detect the fault frequency of the crystal rotating motor, and the test results are shown in Figure 16. It can be seen from Figure 16 that the algorithm increased the frequency domain amplitude of the fault signal and effectively detected that the fault frequency of the crystal motor was 35 Hz. *Sensors* **2023**, *23*, x FOR PEER REVIEW 23 of 25 *Sensors* **2023**, *23*, x FOR PEER REVIEW 23 of 25

Mitsubishi HG-KR73 servo motors, the crystal lift motor is used to lift the crystal upward, and the crystal rotating motor is used to drive the crystal to spin during the growth process. Because the stability of crystal rotating is an important factor to determine the crystal formation and crystal quality, it is necessary to accurately monitor the fault of the crystal rotating motor. The experiment object was the motor of a certain type of electronic-grade silicon single-crystal growing furnace. The purpose was to detect the failure frequency of the crystal rotating motor. A certain type of three-dimensional vibration sensor was used in the experiment, and its connection with the motor is shown in Figure 14. As shown in Figure 14, the vibration sensor was adsorbed on the motor, and information such as vibration displacement, vibration speed, and vibration frequency can be collected. The deceleration ratio of the crystal rotating system was 100:1, that is, when the crystal rotating

speed was 10 rad/min, the speed of the crystal rotating motor was 1000 rad/min.

Brush ring Crystal rotating motor

Crystal lifting and rotating mechanism

Crystal lifting motor

Hoisting wheel room

**Figure 13.** Crystal growing furnace and crystal lifting and rotating mechanism.

**Figure 15.** Original vibration signal of crystal rotating motor. **Figure 15.** Original vibration signal of crystal rotating motor. **Figure 15.** Original vibration signal of crystal rotating motor.

**Figure 16.** Spectrum amplitude of motor fault. **Figure 16.** Spectrum amplitude of motor fault.

**Figure 16.** Spectrum amplitude of motor fault.

**6. Conclusions**

**6. Conclusions**

Taking bearing fault-signal detection as the research object, this paper proposes a bearing fault-detection method based on an improved grey wolf algorithm to optimize

Taking bearing fault-signal detection as the research object, this paper proposes a bearing fault-detection method based on an improved grey wolf algorithm to optimize multistable stochastic resonance parameters, aiming at the problems that multistable sto-

tion algorithm is prone to local optimization and low convergence accuracy. This method improved the grey wolf optimization algorithm. Firstly, the Sobol sequence was used to initialize the grey wolf population to improve the diversity of the population. Secondly, the exponential rule convergence factor was used to balance the global search and local development stages of the algorithm. At the same time, the adaptive position-update strategy was introduced to improve the accuracy of the algorithm. Additionally, we used Cauchy–Gaussian hybrid variation to improve the ability of the algorithm to escape from the local optimal area. The performance of the proposed algorithm was verified using experiments with 15 benchmark test functions in the CEC23 group of common test functions. The results show that the multi-strategy improved grey wolf optimization algorithm has better optimization performance. Then, the improved grey wolf optimization algorithm was used to optimize the parameters of the multistable stochastic resonance algorithm, so as to realize the detection of bearing fault signals. Finally, the bearing data sets of Case Western Reserve University and the Association for Mechanical Fault Prevention Technology were analyzed and diagnosed with the proposed bearing fault-detection method, and the optimization results were compared with other improved algorithms. At the same time, the method proposed in this paper was used to diagnose the fault of the bearing of

tion algorithm is prone to local optimization and low convergence accuracy. This method improved the grey wolf optimization algorithm. Firstly, the Sobol sequence was used to initialize the grey wolf population to improve the diversity of the population. Secondly, the exponential rule convergence factor was used to balance the global search and local development stages of the algorithm. At the same time, the adaptive position-update strategy was introduced to improve the accuracy of the algorithm. Additionally, we used Cauchy–Gaussian hybrid variation to improve the ability of the algorithm to escape from the local optimal area. The performance of the proposed algorithm was verified using experiments with 15 benchmark test functions in the CEC23 group of common test functions. The results show that the multi-strategy improved grey wolf optimization algorithm has better optimization performance. Then, the improved grey wolf optimization algorithm was used to optimize the parameters of the multistable stochastic resonance algorithm, so as to realize the detection of bearing fault signals. Finally, the bearing data sets of Case Western Reserve University and the Association for Mechanical Fault Prevention Technology were analyzed and diagnosed with the proposed bearing fault-detection method, and the optimization results were compared with other improved algorithms. At the same time, the method proposed in this paper was used to diagnose the fault of the bearing of

#### **6. Conclusions**

Taking bearing fault-signal detection as the research object, this paper proposes a bearing fault-detection method based on an improved grey wolf algorithm to optimize multistable stochastic resonance parameters, aiming at the problems that multistable stochastic resonance system parameters are difficult to select and basic grey wolf optimization algorithm is prone to local optimization and low convergence accuracy. This method improved the grey wolf optimization algorithm. Firstly, the Sobol sequence was used to initialize the grey wolf population to improve the diversity of the population. Secondly, the exponential rule convergence factor was used to balance the global search and local development stages of the algorithm. At the same time, the adaptive position-update strategy was introduced to improve the accuracy of the algorithm. Additionally, we used Cauchy–Gaussian hybrid variation to improve the ability of the algorithm to escape from the local optimal area. The performance of the proposed algorithm was verified using experiments with 15 benchmark test functions in the CEC23 group of common test functions. The results show that the multi-strategy improved grey wolf optimization algorithm has better optimization performance. Then, the improved grey wolf optimization algorithm was used to optimize the parameters of the multistable stochastic resonance algorithm, so as to realize the detection of bearing fault signals. Finally, the bearing data sets of Case Western Reserve University and the Association for Mechanical Fault Prevention Technology were analyzed and diagnosed with the proposed bearing fault-detection method, and the optimization results were compared with other improved algorithms. At the same time, the method proposed in this paper was used to diagnose the fault of the bearing of the lifting device of a single-crystal furnace. The experimental results show that this method can be used to detect the bearing fault signal and can effectively enhance the fault signal in the noise. Compared with other optimized bearing fault-detection methods based on improved intelligent algorithms, the proposed method has the advantages of fast convergence, high parameter optimization accuracy, and strong robustness.

In the future, this paper will study the following two aspects: Firstly, the MSGWO needs to be further improved to improve its stability due to its poor stability in individual test functions. Secondly, the bearing fault-detection method proposed in this paper will be applied to the bearing fault detection of rotating machinery in different industries, and the corresponding improvement will be made according to the actual detection results, so as to improve the applicability of the bearing fault-detection method proposed in this paper to different industries.

**Author Contributions:** Conceptualization, W.H.; methodology, W.H.; software, G.Z.; writing original draft preparation, G.Z.; writing—review and editing, W.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the National Natural Science Foundation (NNSF) of China (NOs. 62073258, 62127809, 62003261).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.

**Data Availability Statement:** Our source code is available on https://github.com/Zfutur1/Codeinformation.git.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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