1. Introduction
Hydropower, as an important clean energy source, is receiving increasing attention from countries around the world [
1,
2,
3]. The hydropower station, as an essential component of hydroelectric power generation, always has the mission of stability and safety of operation. The hydropower house is an important part of the hydropower station, which is not only the support structure of the hydro-generator unit, but also the channel for water to flow. With an increase in installed capacity and water head, the size of the hydropower station becomes more significant and the stiffness gradually decreases, which leads to serious stability problems in a large hydropower house. Against this background, the vibration of the hydropower house structure caused by various factors such as unit vibration and channel water flow is increasingly drawing attention [
4,
5,
6,
7]. Thus, using effective methods to analyze the structural vibration response of the hydropower house, identifying its vibration-transmission path and evaluating the stability of its vibration state are of great practical significance and engineering value.
The vibration system of hydropower units and the power house is a huge coupled hydraulic–mechanical–electromagnetic structural system, and the main sources of vibration include hydraulic, mechanical and electromagnetic factors [
8,
9]. The coupling effect of multiple sources creates huge difficulties and presents great challenges for research into vibration issues. The vibration prototype observation of the hydropower house structure based on the excitation of the working environment can obtain the required vibration parameters and response under the normal operation working conditions, but because the vibration signal of the hydropower house is a non-linear signal with varied noise, the information of the vibration characteristics is often drowned by the noise under the joint influence of multiple vibration sources, which affects the accuracy of the subsequent data analysis [
10].
In view of the various difficulties mentioned above, there are three thorny and step-by-step challenges that need to be addressed before proceeding with the evaluation of the vibration stability of hydropower house: (1) How to reduce the influence of noise and extract the characteristic information in the vibration signal with the help of effective methods; (2) How to effectively identify the transmission path of vibration energy during the operation of the hydropower house; (3) How to quantitatively evaluate the transmission strength of vibration energy and make recommendations for the optimization of the structure of the hydropower house.
In the field of signal processing, many studies have been conducted and many achievements have been made. The Fourier transform has been extensively applied in the field of signal processing, which not only can clarify the spectral features of signals, but also has the capability for precise resolution [
11,
12]. It is an effective tool for analyzing smooth signals but cannot correct and analyze local distortion of the signal. A digital filter is an improved Fourier-transform-based denoising method, which achieves denoising by mathematical operations on the difference equations of a discrete signal, but requires predefined technical specifications such as a passband cutoff frequency and stopband cutoff frequency [
13,
14]. The wavelet transform allows local transformation of the time and frequency of the signal, but there is no standard available for choosing the wavelet base function [
15]. Empirical mode decomposition (EMD) is an adaptive signal-analyzing method for processing nonlinear and non-stationary signals [
16,
17]. Barbosh [
18] introduces the application of the EMD method in complex vibration signal processing and modal identification of civil structures in detail. However, due to the defects of his computational theory, the decomposition process of EMD may cause boundary effects and mode mixing [
19,
20]. In order to reduce the interference of the mode mixing, the ensemble empirical mode decomposition (EEMD) method is proposed [
21]. Spinosa [
22] uses the EEMD method to effectively reduce the background noise of airframe-vibration data obtained from aircraft water-landing experiments. Gao [
23] uses the EEMD method to extract weak fault signals from bearing vibration signals to achieve early bearing fault prediction. However, the effect of EEMD decomposition depends on the integration time and the amplitude of the added white noise [
24]; the mode-mixing phenomenon will not be mitigated if the parameters are not chosen properly. With the purpose of overcoming the above problems, complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) is proposed. This method adds adaptive white noise at each stage of the decomposition and calculates a unique residual signal for each mode component, which results in negligible reconstruction error [
25,
26]. Mousavi [
27] applies CEEMDAN to bridge structural vibration signal processing to effectively identify the location and extent of bridge damage. However, although CEEMDAN can effectively decompose the signal and obtain the characteristic mode components, it cannot effectively filter out high-frequency white noise due to background noise [
28]. Singular value decomposition (SVD), as a typical orthogonal decomposition denoising method, can efficiently filter out high-frequency noise in the signal [
29].
As the problem of vibration in a hydropower house has become a subject of increasing concern, researchers have started to analyze vibration from the perspective of the vibration mechanism and have tried to explore the energy-transfer path of vibration in a hydropower house. Xu [
30] studies the power transmission between the underground main powerhouse and auxiliary powerhouse based on the tidal current theory. They find that the vibration-transfer direction of the underground plant is mainly perpendicular to and along the flow direction of the bedrock. Lian [
31] concludes that the transmission of transverse vibration was larger than that of longitudinal and vertical vibration, while the transmission intensity of low-frequency vibration and rotational frequency vibration were basically equal. Wang [
32] carries out the calculation of structural sound intensity based on finite element theory for different parts of the hydropower plant, and realizes the vector visualization of the transmission path. Researchers often study the vibration of the hydropower house through numerical model calculations and have achieved good results, but the shortcomings and challenges cannot be ignored. Most previous studies have used computational analysis methods that assume the excitation of the house is usually composed of steady-state harmonic loads, which simplifies mathematical calculations but adds human interference.
In the field of vibration response analysis with the help of structural monitoring data, the transfer entropy (TE) has been widely used. Lindner [
33] uses Granger causality and transfer entropy to analyze the vibration of the site and finds that transfer entropy can determine the cause and effect of the vibration more accurately. Wang [
34] and Wang [
35] have also attempted to investigate the structural response by the transfer entropy based on data acquired from field tests rather than numerical simulations. Zhang [
36] applies the transfer entropy to the vibration-response analysis of the powerhouse structure to describe the vibration-transfer energy characteristics of different variables and the same variable in different directions. In this research, transfer entropy is introduced and verified as an innovative method that can analyze vibration-transmission paths based on structural vibration response data. However, using vibration monitoring data directly from field tests, the analysis process may be affected by strong background noise, resulting in inaccurate analysis results.
Considering the above problems and the advantage of CEEMDAN, SVD and TE in signal-analysis work, this paper proposes a new vibration-transmission path identification method for hydropower houses based on CEEMDAN-SVD-TE. Firstly, the accuracy and superiority of CEEMDAN-SVD-TE, which is higher than TE in information-transmission-direction identification, is verified by simulation-signal analysis. A large hydropower house is taken as the research object, and the CEEMDAN-SVD method is used to extract the tailwater-fluctuation signal as the characteristic signal for the transmission-path analysis; finally, the vibration-transmission path of tailwater fluctuation is analyzed using the transfer-entropy theory, and the information-transmission rate between different measurement points is calculated. This research can offer a dependable theoretical basis and technical support for the identification of the vibration mechanism and optimization of structural vibration reduction in hydropower houses.
This article is structured as follows.
Section 2 introduces the basic theory used in the following sections.
Section 3 presents the simulation analysis of the method proposed in the paper.
Section 4 shows the results for the analysis of the vibration-transmission path from the measured point signals in the hydropower house. Finally, this article is concluded in
Section 5.
2. Materials and Methods
2.1. CEEMDAN Algorithm
EMD [
16] can adaptively decompose the original signal into an intrinsic mode function (IMF) according to its own scale. However, due to the problem of EMD’s own computing theory, mode mixing often occurs in its decomposition. EEMD [
21] is an improvement on EMD; its calculation principle is to add the corresponding white noise to the original signal and eliminate the mode-mixing phenomenon in EMD decomposition at one time by suppressing and canceling the influence of decomposition noise through multiple integration. After multiple integration averaging, the influence of white noise on the decomposition results is offset, but a reconstruction error appears. The reconstruction error relies on the integration number, and increasing the integration number can diminish the reconstruction error, but it increases the computational volume to some extent and seriously affects the computational efficiency.
In order to solve the problems of mode mixing, calculation accuracy and computational efficiency, the CEEMDAN algorithm is proposed. Compared with the EEMD algorithm, the CEEMDAN algorithm adds a finite amount of adaptive white noise at each stage of EMD decomposition. When the number of integrations is small, its reconstruction error is almost zero, and the reconstructed signal is almost identical to the original signal. Therefore, the CEEMDAN algorithm can solve the mode-mixing phenomenon existing in the EMD algorithm and overcome the incompleteness as well as low computational efficiency of the EEMD algorithm.
Define an operator
that represents the process of EMD obtaining the
-th mode component
; let
be the white-noise-satisfying distribution of
and
is the amplitude coefficient of white noise added for the
-th time. The decomposition process for the CEEMDAN algorithm is shown below [
25]:
(1) The white noise
is added to the original signal, and
-th EMD decomposition is performed. An average operation is then performed on the result to obtain
.
(2) The first stage residual component can be calculated.
The white noise is added to the first-stage residual component, and the EMD is performed.
can then be calculated with the mean value of the first IMF.
For
, the
-th residual component can be calculated.
(3) White noise
is added to the
-th residual component and EMD decomposition performed.
can then be calculated with the mean value of the first IMF.
(4) Step (4) and Step (5) are repeated until the value of the residual component is less than two extremes, then the decomposition stops. Eventually the residual variable is obtained.
where
is the total number of modes in the decomposition process.
The reconstructed signal can be expressed as follows:
2.2. SVD Denoising Algorithm
Singular value decomposition (SVD) is a classical noise-reduction method, which is widely used in vibration signal processing [
29].
Assuming the signal
, the Hankel matrix can be constructed as follows [
37]:
where
is the matrix of
,
.
The singular value decomposition of
can be obtained. For any
orders real matrix
, there must be an orthogonal matrix
and an orthogonal matrix
to meet Equation (9).
where
;
;
or its transpose;
denotes the zero matrix;
represents the singular value obtained by decomposition, and satisfies
.
Assuming the optimal singular value order is , the denoised signal can be obtained by preserving the order singular value of the singular value matrix and reconstructing the matrix. The key problem is to determine the optimal order of singular value. If selected order is too low, the valid signal may be mistaken as noise, resulting in the loss of the valid signal. If selected order is too high, there will be a lot of residual noises, which affects the de-noising effect. The optimal order should preserve the valid signal in the maximal degree and filter out most of the noise.
A singular entropy increment is applied to identify the optimal order in this paper [
28]; the order when the singular entropy increment curve tends to be steady is chosen for the optimal singular value order
, which ensures that the signal features are preserved and the unfavorable noise is filtered effectively.
2.3. Transfer-Entropy Algorithm
Schreiber [
38] drew lessons from the basic theory of information entropy and extended it to form the transfer-entropy (TE) algorithm, which can characterize the relevance and information-transmission relationship among different time series with a unit of bit. This theory can quantify the information-transmission effect between related time series in the form of entropy and reflect the characteristics of information transmission. If the dynamic probabilities of a process
at time
is conditional only on previous
values, the process is called a
-th Markov process. The mathematic description of this transition probability is
Considering the influence of another process on these transition probabilities, the expression of coupling influence of and on is .
The transfer entropy
can be formulated as below.
When the transfer entropy of
to
is greater than the transfer entropy of
to
,
is referred to as the source of information transmission and
denotes the transfer entropy of
to
. The research of Nichols and Overbey [
39,
40] showed that defining the order
for both Markov processes
and
does not affect the directedness of process
to process
. Together with a time delay
added to
, Equation (11) can be simplified to Equation (12).
where
. The assumption of order
quantifies the information gain from
only.
If we use conditional probabilities
, Equation (12) can be rewritten in entropy form as Equation (13).
In order to quantitatively describe the transmission regularity of vibration energy, the information-transmission rate (
) index is introduced [
35,
36], which is calculated based on the transfer entropy and can effectively describe the information-transmission strength of vibration energy. For vibration signals
and
, the
is:
where
and
are the transfer entropy corresponding to the different transfer directions of signals
and
;
and
are the average value of
and
, respectively;
is the information-transmission rate; the direction of
> 0 is used as the positive direction of information transmission.
In the case of , when = 0, there is no information-transmission relationship between signal and , and the two signals are independent; When = 1, the information of signal is completely transmitted to the signal, is the source of information transmission. Therefore, can describe the information-transmission intensity of in the form of percentage, so as to quantitatively characterize the feature of information transmission between vibration signals.
2.4. COMBINED CEEMDAN-SVD-TE
The vibration signal of a hydropower house is a kind of nonlinear and non-stationary signal, including high-frequency white noise and electromagnetic noise. The CEEMDAN algorithm can effectively extract the required feature signal according to frequency, but because of the interference of strong background noise, a lot of high-frequency white noise cannot be separated accurately. To ensure the accuracy of feature-information extraction, the vibration signal needs to be further processed. As a classical denoising method, the SVD algorithm has a strong ability to filter the high-frequency random noise in the signal and extract the signal-feature information. The transfer-entropy algorithm can effectively analyze the correlation between signals, identify the source of information transmission and clarify the direction of vibration-signal transmission.
Based on the working characteristics of the hydropower house and the advantages of each method, the CEEMDAN-SVD-TE method is introduced in this paper. The flow chart of the CEEMDAN-SVD-TE is presented in
Figure 1, and the principal procedures can be summarized as follows:
- (1)
The collected vibration signal is decomposed to IMFs from high frequency to low frequency by CEEMDAN processing, and the IMF component can represent the vibration characteristics of the structure.
- (2)
The optimal singular value order is selected according to the singular entropy increment theory, and the high-frequency noise in the IMF signal is filtered out utilizing the singular value decomposition algorithm.
- (3)
The IMF component is reconstructed by CEEMDAN-SVD joint filtering, and the target feature signal is obtained.
- (4)
The reconstructed feature signals are analyzed for transfer effects with the help of the transfer-entropy algorithm to identify the direction of information transmission between signals.
3. Simulation Analysis
3.1. CEEMDAN-SVD Simulation
To test the validity of the joint CEEMDAN-SVD filtering, a simulation signal
with superimposed low-frequency noise and high-frequency noise is constructed with sampling frequency
, sampling time
. The equation is as follows:
High-frequency white noise:
where
is the time;
is the number of samples;
is white noise.
A comparison of the time history for signals
and
is shown in
Figure 2, and a comparison of the power spectral density plot is shown in
Figure 3.
Because part of the true frequency of the pure signal will be masked by noise, the presence of noise will influence the accuracy of feature-information extraction. Filtering and denoising the acquired signal is the key step of signal analysis. In this paper, the SVD, EEMD, CEEMDAN and CEEMDAN-SVD methods are used to reduce the noise of , and the noise-reduced results of each method are compared to verify the good applicability of the CEEMDAN-SVD algorithm for noisy signal filtering.
The comparison of the power spectrum after noise reduction by the filtering methods are shown in
Figure 4.
From
Figure 4a, we can see that although the SVD algorithm can filter out the high-frequency noise in the original signal well, there is still a large amount of residual low-frequency noise.
Figure 4b presents the filtering effect of EEMD, and its filtering accuracy is low because EEMD is greatly affected by background noise. In
Figure 4c, the filtering accuracy of CEEMDAN for high-frequency noise is low, but the effect of filtering low-frequency noise is excellent. From
Figure 4d, CEEMDAN-SVD has a good filtering effect for both low-frequency and high-frequency noise.
Through the analysis of the filtering effect comparison, among the four filtering methods, CEEMDAN-SVD filter can effectively remove high-frequency noise while filtering low-frequency noise, and the accuracy of feature-signal extraction is higher than the other three methods, which can effectively extract the target signal.
To quantitatively evaluate the filtering capabilities of SVD, EEMD, CEEMDAN and CEEMDAN-SVD, the signal-to-noise ratio (
SNR) and root mean square error (
RMSE) are applied to assess the filtering capabilities of the above four methods.
where
represents the original signal;
represents the signal after noise reduction.
If the
SNR is larger and the
RMSE is smaller, this shows that the denoising effect of this method is better. The denoising performance of SVD, EEMD, CEEMDAN and CEEMDAN-SVD are compared, as shown in
Table 1.
The two indexes calculated in
Table 1 show that among the four methods of SVD, EEMD, CEEMDAN and CEEMDAN-SVD, the maximum
SNR of CEEMDAN-SVD is 4.35 and the minimum
RMSE is 1.07. According to the evaluation principles of
SNR and
RMSE, the CEEMDAN-SVD algorithm has the best denoising performance, indicating that the method can effectively filter out low-frequency noise and high-frequency noise and is more suitable for the analysis of low-
SNR signals.
3.2. Transfer-Entropy Simulation
In order to apply the transfer-entropy algorithm to the analysis of the vibration-transmission path of the hydropower house, it is necessary to determine the efficiency of the transfer-entropy algorithm in the identification on the direction of information transmission.
Construct two correlated simulation signals
and
:
where
;
;
is the correlation coefficient.
As shown in Equations (21) and (22), it can be seen that the signal consists of the signal with correlation coefficient and part of the interference signal. From the perspective of the signal composition, as the correlation coefficient increases, the proportion of signal in signal increases and the proportion of interference signal decreases. The signal should be more easily considered as the source of the signal. On the basis of this, this section identifies the applicability of transfer entropy in the direction identification and quantitative analysis of transfer effects between signals by varying the correlation coefficient in the signal and analyzing the changes in and with different correlation coefficients .
The correlation coefficients
of signals
and
are set to 0.2, 0.4, 0.6 and 0.8, respectively, and the transfer entropy
and
between the simulated signals
and
is calculated. The curves of the transfer-entropy values with the change in time are shown in
Figure 5 and the information-transmission rate with different correlation coefficients are shown in
Table 2.
(1) When the correlation coefficient is 0.2, 0.4, 0.6, 0.8, are all greater than , indicating that a large amount of information is transmitted from to , and the pure signal is the source of information transmission. This is consistent with the actual simulation signal-information-flow direction; the signal is constructed according to signal and the signal has no interference signals and has the main characteristics of the signal. From the perspective of calculating transfer entropy, the amount of information flowing from to is much larger than that in the opposite direction, and more significant transfer effect can be obtained.
(2) When the correlation coefficient is 0.2, the values of and are close to each other, which is not conducive to the identification of the transmission direction between two signals, and is 12.30%. As the correlation coefficient becomes larger, the difference between and becomes larger, and the transmission effect becomes more and more obvious. When the correlation coefficient is 0.8, is 93.57%. This result indicates that the transfer entropy is highly sensitized to the transmission direction of the information flows between two signals, and the transfer-entropy curve can accurately reflect the direction of information transmission between two signals.
(3) In this section, simulation signals and are constructed, and the reasonableness of the transfer entropy for judging the direction of information transmission between two signals is verified by changing the magnitude of the correlation coefficient . increases with the increasing correlation coefficient , which is consistent with the correlation of the constructed signals, indicating that the transfer entropy can not only reflect the directionality of information transmission between two signals, but also accurately quantify the correlation degree between the two signals from the perspective of entropy.
3.3. CEEMDAN-SVD-TE Simulation
To further study the possible influence of noise on the transfer-entropy calculation results and to clarify the usefulness of the joint CEEMDAN-SVD-TE method, the same set of Gaussian white noise was added to the signals
and
with correlation coefficient
of 0.8; the signals
and
were noise-reduced with the help of CEEMDAN-SVD method, and the transfer-entropy curves of
and
signals before and after noise reduction were compared. Here, take the
signal as an example to show the effect of CEEMDAN-SVD noise reduction, the power spectrum of the
signal before and after noise reduction can be seen in
Figure 6.
Figure 6 shows that the 25 Hz characteristic frequency of the signal after noise addition is drowned by the high-frequency white noise, and the high-frequency white noise in the original signal can be effectively filtered out with the help of the CEEMDAN-SVD method. After CEEMDAN-SVD noise reduction, the characteristic main frequencies of the pure signal (25 Hz and 200 Hz) are highlighted again.
The transfer-entropy curves of
and
before and after noise reduction are presented in
Figure 7. As shown in the figure, without noise reduction of
and
, the values of
and
are interleaved and converge, indicating that the characteristic information of the signals may have been submerged by the high-frequency noise and the information-transmission effect between the signals is weakened. According to
Table 2 and
Figure 7, the
obtained from the signals of
and
without noise is 93.57%,the
obtained from the signals of
and
after noise reduction is 90.22%, and the
of
and
without noise reduction is 12.08%. These results indicate that the
can reflect the information-transmission strength between two signals, but the noise in the signals will weaken the transfer effect between the signals and have a negative effect on the computation results of transfer entropy. Using the CEEMDAN-SVD method can prevent the noise from the interference of the signal transmission direction analysis and effectively improve the accuracy of information-transmission-direction identification.
In summary, the transfer-entropy algorithm can effectively determine the transmission source of the signal, but its results are easily affected by noise signals. Integrating the CEEMDAN-SVD noise-reduction method with the transfer-entropy algorithm can effectively reduce the negative impact of noise on the transfer-entropy calculation results and obtain real and reliable transmission-path results.