Application of improved multi-strategy MPA-VMD in pipeline leakage detection

Figures & data

Figure 1. Scatter plot of good point set initialization and random initialization.

Figure 2. Iterative plots of improved CF and original CF.

Figure 3. Curves of standard Cauchy distribution probability density function and Brownian motion probability density function distribution.

Figure 4. Scatter plot of standard Cauchy distribution and Brownian motion distribution.

Figure 5. Flow chart of denoising of optimized VMD based on improved MPA.

Table 1. Five standard test functions.

Function name	Dimension	Space	Function expression
Schwefel's Problem 1.2	50	$[-100,100]$	${\displaystyle f_1\left(x\right)=\sum _{i=1}^n{\left(\sum _{j=1}^i{x}_j\right)}^2}$
Generalized Rosenbrock's Function	50	$[-30,30]$	${\displaystyle f_2\left(x\right)=\sum _{i=1}^{n-1}\left[100\right(x_{i+1}-x_i^2{)}^2+(x_i-1{)}^2]}$
Easom2d	2	$[-100,100]$	$f_3\left(x\right)=-(-\cos (x\left(1\right))\cdot \cos (x\left(2\right))\cdot e^{(-(\left(x\right(1)-\pi {)}^2)-\left(x\right(2)-\pi {)}^2)})$
PeriodicEval	2	$[-10,10]$	$f_4\left(x\right)=-(1+(\sin \left(x\right(1\left)\right){)}^2+(\sin (x\left(2\right)){)}^2-0.1\cdot e^{(-x(1{)}^2-x\left(2{)}^2\right)})$
Shekel's Foxholes Function	2	$[-65,65]$	${\displaystyle f_5\left(x\right)={(\frac{1}{500}+\sum _{j=1}^{25}\frac{1}{j+\sum _{i=1}^2(x_i-a_{ij}{)}^6})}^{-1}}$

Table 2. Test results of standard test function.

	Average value					Standard deviation
Algorithm	$f_1\left(x\right)$	$f_2\left(x\right)$	$f_3\left(x\right)$	$f_4\left(x\right)$	$f_5\left(x\right)$	$f_1\left(x\right)$	$f_2\left(x\right)$	$f_3\left(x\right)$	$f_4\left(x\right)$	$f_5\left(x\right)$
MPA	−6.2346$\times {10}^{-4}$	0.0645	4.0599	$-$1.5708	$-31.9783$	0.0597	0.1881	1.2987	8.8858	6.5919$\times {10}^{-7}$
GWO	5.2328$\times {10}^{-4}$	0.0169	4.0691	3.1786	$-31.9764$	0.0339	0.0676	1.3055	6.7010	0.0027
PSO	$-0.1507$	0.5234	4.0535	0.0176	$-23.9659$	7.1464	0.8072	1.3077	11.0897	11.2967
IMPA	−1.8968$\times {10}^{-5}$	0.0034	2.2233	$-$6.2832	$-31.9783$	0.0012	0.0051	1.2987	2.2214	3.2696$\times {10}^{-8}$

Figure 6. Pipeline data waveform.

Figure 7. Pipeline signal decomposition spectrogram.

Figure 8. Line chart of dynamic entropy of pipeline signal.

Figure 9. Reconstructed signal waveform.

Figure 10. Original pipeline signal and reconstructed signal waveform.

Table 3. Comparison of denoising effects of actual data.

Method	K	α	SNR	MSE	MAE
MPA-VMD	7	3545	18.2449	0.0248	0.0173
GWO-VMD	10	4000	18.0231	0.0254	0.0180
PSO-VMD	9	1483	18.0835	0.0253	0.0178
IMPA-VMD	8	1393	18.2757	0.0231	0.0151