# Signal Spectrum Analysis of Sediment Water Impact of Hydraulic Turbine Based on ICEEMDAN-Wavelet Threshold Denoising Strategy

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## Abstract

**:**

## 1. Introduction

## 2. Signal Denoising Fundamentals

#### 2.1. ICEEMDAN Algorithm

- There exists the original signal $x$ to be decomposed, and adding $i$ set of white noise ${w}^{\left(i\right)}$ to $x$ yields;$${x}^{\left(i\right)}=x+{\beta}_{0}{E}_{1}\left({w}^{\left(i\right)}\right)$$
- Decompose the signal ${x}^{\left(i\right)}$ using EMD and calculate the local mean of the signal to obtain the first-order residual ${r}_{1}$ and the first-order modal component ${IMF}_{1}$;$${r}_{1}=<M\left({x}^{\left(i\right)}\right)>$$$${IMF}_{1}=x-{r}_{1}$$
- Add white noise to the first-order residual and calculate the second-order residual ${r}_{2}$ and the second-order modal component ${IMF}_{2}$;$${r}_{2}={<M(r}_{1}+{\beta}_{1}{E}_{2}\left({w}^{\left(i\right)}\right))>$$$${IMF}_{2}={r}_{1}-{r}_{2}={r}_{1}-{<M(r}_{1}+{\beta}_{1}{E}_{2}\left({w}^{\left(i\right)}\right))>$$
- By analogy, the $k$th-order residual ${r}_{k}$ and the $k$th-order modal component ${IMF}_{k}$ are calculated;$${r}_{k}={<M(r}_{k-1}+{\beta}_{k-1}{E}_{k}\left({w}^{\left(i\right)}\right))>$$$${IMF}_{k}={r}_{k-1}-{r}_{k}$$
- Repeat step 4 until all modal components are obtained and the signal decomposition is completed.

#### 2.2. Wavelet Threshold Denoising

- Selecting appropriate wavelet basis functions and decomposition layers for discrete wavelet decomposition;
- Set the threshold $\lambda $ and select the threshold function to handle wavelet coefficients with different numbers of decomposition layers;
- Performing signal reconstruction to obtain the denoised signal.

#### 2.3. ICEEMDAN-Wavelet Threshold Combined Denoising

## 3. Signal Denoising Analysis

#### 3.1. Signal Generation

#### 3.2. Signal Pre-Processing

#### 3.3. Optimization of Wavelet Threshold Parameters

#### 3.4. Comparison of Denoising Methods

## 4. Experimental Signal Analysis

#### 4.1. Experimental Platform

#### 4.2. Experimental Materials and Equipment

#### 4.3. Signal Denoising

#### 4.4. Experimental Signal Characterization

#### 4.4.1. Spectral Characteristics

#### 4.4.2. Sample Entropy Characteristics

- Construct an m-dimensional vector using the data samples {$x\left(i\right)$,1 ≤ $i\text{}$≤ N};$${X}_{m}\left(i\right)=\left\{x\left(i\right),x\left(i+1\right),x\left(i+m-1\right)\right\},1\le i\le N-m+1$$
- Define the absolute value of the maximum difference between the two corresponding elements of the vector ${X}_{m}\left(i\right)$ and ${X}_{m}\text{}\left(j\right)$ as the distance $d$, i.e.,$$d={max}_{k=\mathrm{0,1},\dots ,m-1}\left(\left|x\left(i+k\right)-x\left(j+k\right)\right|\right)$$
- The number of statistical distances d should not be more significant than the similarity threshold $r$, ${B}_{i}$, for 1 ≤ $i$ ≤ $N-m$, which is defined as;$${B}_{i}^{m}\left(r\right)=\frac{{B}_{i}}{N-m-1}$$
- Calculate the mean ${B}_{avg}^{m}(r)$ of $N-m$ ${B}_{i}^{m}\left(r\right)$;$${B}_{avg}^{m}\left(r\right)=\frac{{\sum}_{i=1}^{N-m}{B}_{i}^{m}\left(r\right)}{N-m}$$
- Let the dimension be $m$ + 1, repeat steps 1 to 4, then calculate and obtain ${B}_{i}^{m+1}\left(r\right)$ and ${B}_{avg}^{m+1}\left(r\right)$;
- Calculate the sample entropy.$$SampEn\left(m,r,N\right)=-\mathrm{ln}\frac{{B}_{avg}^{m+1}\left(r\right)}{{B}_{avg}^{m}\left(r\right)}$$

## 5. Conclusions

- (1)
- In this paper, the optimized ICEEMDAN-wavelet threshold denoising method is employed to process the acoustic vibration signals of three operating conditions of the hydraulic turbine. The characteristics of the acoustic signals were analyzed using spectrograms and sample entropy. The results show that when clean water flows through the turbine, the sample entropy reach its smallest values and the dominant frequency component in the spectrogram is 59.39 Hz. When transitioning from clean water to the flood flow containing 2–4 mm sediment particles, the sample entropy is increasing and a high-frequency component higher than 59.39 Hz becomes the prominent frequency of the spectrogram. Meanwhile, the formation of high-frequency components increases with the sediment particle size. Therefore, based on the spectral characteristics and sample entropy characteristics of the sediment water flow, it can provide a reference for whether the hydraulic turbine needs a protective shutdown during flood seasons.
- (2)
- Starting from the acoustic vibration signals of the sediment water flow impacting the hydraulic turbine, the spectral characteristics and sample entropy characteristics of the signals under different operating conditions are analyzed. The results show apparent differences in the characteristics’ noise under different operating conditions, indicating that it is feasible to use the characteristics of acoustic vibration signals to analyze the operating conditions under which the hydraulic turbine is operating. Meanwhile, it can be concluded from the experimental results that the ICEEMDAN-wavelet threshold denoising method is effective in the background noise removal of acoustic vibration signals.
- (3)
- The analysis of acoustic vibration signals of the sediment water flow can provide a supplement for the existing hydraulic turbine condition monitoring system and a new avenue for subsequent research on early warning of hydraulic turbine failure.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Glossary

EMD | Empirical modal decomposition |

EEMD | Ensemble empirical modal decomposition |

CEEMD | Complete ensemble empirical modal decomposition |

CEEMDAN | Complete ensemble empirical modal decomposition with adaptive noise |

ICEEMDAN | Improved complete ensemble empirical modal decomposition with adaptive noise |

SampEn | Sample entropy |

$\beta $ | The ratio of the signal-to-noise ratio of the added noise relative to the original signal to the standard deviation of the added noise |

${w}^{\left(i\right)}$($i$ = 1, 2, …, $i$) | A series of Gaussian white noise with mean 0 and unit variance 1 |

${r}_{k}$ | The $k$th order residual |

${E}_{k}$ | The $k$th IMF component after EMD |

$M$ | The local mean of the decomposed signal |

<> | The mean value for all iterations |

${w}_{j,k}$ | Original wavelet coefficients |

${\widehat{w}}_{j,k}$ | Wavelet coefficients processed by the threshold function |

$sgn\left(\text{}\right)$ | The sign function |

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**Figure 3.**3D waveform and spectrum of IMF: (

**a**) IMF’s three-dimensional waveforms (Explanation: Blue is the first order IMF, red is the second order IMF, orange is the third order IMF, and so on up to the eighth order IMF.); (

**b**) Spectrum of IMF.

**Figure 5.**Calculation results of evaluation indexes: (

**a**) root mean square error; (

**b**) smoothness; and (

**c**) correlation coefficient.

**Figure 10.**Original signal of different operating conditions: (

**a**) clean water operating condition; (

**b**) 2–3 mm sediment operating condition; and (

**c**) 2–4 mm sediment operating condition.

**Figure 11.**IMF under different operating conditions (Explanation: Blue is the first order IMF, red is the second order IMF, orange is the third order IMF, and so on up to the eighth order IMF.): (

**a**) IMF under clean water operating condition; (

**b**) IMF under 2–3 mm sediment operating condition; and (

**c**) IMF under 2–4 mm sediment operating condition.

**Figure 13.**Filtered signal of different operating conditions: (

**a**) clean water operating condition; (

**b**) 2–3 mm sediment operating condition; and (

**c**) 2–4 mm sediment operating condition.

Denoising Method | Root Mean Square Error | Smoothness | Correlation Coefficient |
---|---|---|---|

ICEEMDAN | 1.6809 | 1.4121 | 0.9838 |

Wavelet threshold | 1.0817 | 1.1377 | 0.9939 |

ICEEMDAN-wavelet threshold | 0.8554 | 1.1114 | 0.9958 |

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## Share and Cite

**MDPI and ACS Style**

Bai, S.; Zeng, Y.; Dao, F.; Xiao, B.; Li, X.; Qian, J.
Signal Spectrum Analysis of Sediment Water Impact of Hydraulic Turbine Based on ICEEMDAN-Wavelet Threshold Denoising Strategy. *Water* **2023**, *15*, 4017.
https://doi.org/10.3390/w15224017

**AMA Style**

Bai S, Zeng Y, Dao F, Xiao B, Li X, Qian J.
Signal Spectrum Analysis of Sediment Water Impact of Hydraulic Turbine Based on ICEEMDAN-Wavelet Threshold Denoising Strategy. *Water*. 2023; 15(22):4017.
https://doi.org/10.3390/w15224017

**Chicago/Turabian Style**

Bai, Shufang, Yun Zeng, Fang Dao, Boyi Xiao, Xiang Li, and Jing Qian.
2023. "Signal Spectrum Analysis of Sediment Water Impact of Hydraulic Turbine Based on ICEEMDAN-Wavelet Threshold Denoising Strategy" *Water* 15, no. 22: 4017.
https://doi.org/10.3390/w15224017