Mathematical Problems in Engineering

Volume 2018, Article ID 3494785, 10 pages

https://doi.org/10.1155/2018/3494785

## A Novel Multidimensional Frequency Band Energy Ratio Analysis Method for the Pressure Fluctuation of Francis Turbine

^{1}Key Laboratory of Transients in Hydraulic Machinery, Wuhan University, Ministry of Education, Wuhan 430072, China^{2}Jiangxi Province Key Laboratory of Precision Drive & Control, Nanchang 330029, China^{3}Zhejiang Energy Beihai Hydropower Co., Ltd., Hangzhou 320002, China

Correspondence should be addressed to Qijuan Chen; nc.ude.uhw@nehcjq

Received 27 August 2017; Revised 27 November 2017; Accepted 26 December 2017; Published 21 January 2018

Academic Editor: Yuri Vladimirovich Mikhlin

Copyright © 2018 Weiyu Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The pressure fluctuation has multiple influence on the steady operation of Francis turbine, and the impact degree varies with the operation condition. In this paper, for the analysis of pressure fluctuation in the Francis turbine, a novel feature extraction method of multidimensional frequency bands energy ratio is proposed based on Hilbert Huang Transform (HHT). Firstly, the pressure fluctuation signal is decomposed into intrinsic mode functions (IMFs) by EEMD. Secondly, the Hilbert marginal spectrum is utilized to analyze the frequency characteristics of IMFs. Then, according to the inner frequency of IMFs, each of them is divided into high, medium, or low frequency band which are constructed based on the frequency characteristic of the pressure fluctuations in the Francis turbine. Afterward, the energy ratio of each frequency band to the original signal is calculated, which is to realize the feature extraction of multidimensional frequency band energy ratio. Actual applications verify that this method not only can extract the time-frequency characteristics but also can analyze the condition feature of the pressure fluctuation. It is a novel method for extracting the feature of pressure fluctuation in the Francis turbine.

#### 1. Introduction

Hydropower accounts for the largest proportion of renewable energy sources and plays an important role as a clean energy in the world's energy structure [1]. Due to the inherent structural characteristics of Francis Turbine, once it works at certain off-design conditions, the flow will be unsteady, so there will be a positive or negative circulation in the runner exit. Moreover, some hydraulic instability phenomenon such as runner inlet and outlet flow separation, channel vortex, and draft tube vortex will appear. Therefore, the Francis turbine can only operate stably in a certain load range, which is one of its inherent characteristics [2]. The draft tube vortex is one kind of hydraulic instability phenomenon, which occurs when the Francis turbine is running out of optimum conditions, especially under the partial load condition [3]. The pressure fluctuation caused by the draft tube vortex is one of the most important vibration sources of Francis turbine and has an important influence on the stability of operation [4, 5]. When the pressure fluctuation passes through the flow passage parts and structures, the vibration of the unit, the period swing of the shaft, and the vibration of the head cover of the Francis turbine may be caused [6, 7]. When the pressure fluctuation resonates with the water body in the pressure pipeline, it will cause the vibration, output oscillation of the hydropower generating unit, and even have an influence on the electric power system [8, 9]. In recent years, some large and giant Francis turbines have been put into operation, whose size and specific speed have increased, while relative stiffness has weakened; hence the hydraulic instability problems have become more serious [10]. Therefore, in order to improve the hydraulic stability of Francis turbine, it is necessary to research the inherent mechanism and performance characteristics of the pressure fluctuation and their influences on the operation stability of Francis turbine [11].

At present, there are many methods used to the pressure fluctuation signal processing, such as short-time Fourier transform and wavelet transform. However, their essential flaws limit their performance [12]. The actual measured pressure fluctuation signal has a distinct nonlinear and nonstationary characteristic. FFT assumes that the signal is linear and stationary, so it has an insurmountable limitation for the signal with strong shock mutations and nonstationary characteristics, which may lead to harmonic distortion problem. Wavelet analysis is an important method of time-frequency analysis; however, the selection problem of primary function still exists. HHT is a new method for nonlinear and nonstationary signal analysis. It can decompose signal into different scale fluctuations adaptively and produces a series of data sequences with different characteristic scales. Compared with FFT, HHT can deal with nonstationary and transient problems, and, compared with wavelet transform, HHT has the advantage of multiresolution, and there is no selection problem of primary function [13]. Currently, HHT analysis methods are used in many areas, like gear fault diagnosis [14, 15], bearing fault diagnosis [16, 17], wind power characteristics analysis [18], pneumatic conveying flow characteristics analysis [19], and so on. In the aspect of pressure fluctuation analysis and feature extraction, Feng and Chu used HHT method to analyze the pressure fluctuation signal in the draft tube during the start-up process and confirmed that the HHT method can effectively extract the low frequency and unsteady characteristics of the pressure fluctuation signal [20].

There are many kinds of pressure fluctuations in the Francis turbine, which have different effects on the operation reliability of the unit [21, 22]. At present, most of the analysis methods of pressure fluctuation signal focus on the extraction of amplitude and frequency characteristics, without considering its condition features. Moreover, the frequency characteristics of different pressure fluctuation have not been analysed specifically. In this paper, a multidimensional frequency band energy feature extraction method for the pressure fluctuation of Francis turbine is proposed. The HHT was performed on the pressure fluctuation signal for the time-frequency analysis, and a multidimensional frequency band concept for pressure fluctuation of Francis turbine is proposed, which helps to find out the underlying mechanism affecting the pressure fluctuation amplitude and proved the feasibility of this method on the pressure fluctuation analysis of Francis turbine.

#### 2. Multidimensional Frequency Band Energy Ratio Feature Extraction Method

Pressure fluctuations in the Francis turbine may have different frequencies and operating condition features. In terms of frequency characteristics, there are several kinds of frequencies related to vortex, rotation, and the blades [23]. In recent years, it has been continually found some kinds of pressure fluctuations with other frequencies existing in some large Francis turbines, for example, pressure fluctuation with a frequency of 1 to 4 times the rotation frequency [24–26] and frequency related to the number of the leaf and the rotation frequency [21]. In terms of operating condition feature, under low partial load, pressure fluctuation with rotation frequency and other higher frequencies may appear [26, 27]. Under the partial load condition, the draft tube vortex pressure fluctuation with a low frequency is taken as the main factor of the pressure fluctuation which may occur in the flow channel [10]. Under the high load area, Karman Vortex, channel vortex, higher part load pressure fluctuation, and so on may appear [28, 29]. Thus, it can be seen that there will be different types of pressure fluctuation under diverse operating conditions, and their frequency characteristics are also different. Therefore, the analysis of pressure fluctuation is not only for the extraction of the amplitude and frequency characteristics, but also for the analysis of the condition feature.

The most common method of the pressure fluctuation partition for the Francis turbine is to divide it into three parts of low, medium, and high according to their frequency characteristics [21, 27]: low frequency pressure fluctuation (it is generally about 1/6~1/2 times of the rotational frequency); medium frequency pressure fluctuation (it is about 1~4 times of the rotational frequency, most of which are caused by the rotation of the runner); high frequency pressure fluctuation (the frequency may be related to the number of guide vanes, the number of blades, and the rotation speed of the unit). Therefore, this paper constructs the high-medium-low three frequency bands according to the frequency characteristics of the pressure fluctuation which may occur in the Francis turbine. By means of mode analysis combined with Principal Component Analysis (PCA), the energy ratios of pressure fluctuation belonging to different frequency bands are computed. Thus, the amplitude and frequency characteristics of the pressure fluctuation components are extracted, and the condition feature of the pressure fluctuation can be analyzed. Based on this method, the dominant frequency components that increase the amplitude of the pressure fluctuation can be analyzed.

According to the above-mentioned frequency characteristics of pressure fluctuation, three bands are constructed: the low frequency band of 0~2.5 Hz (the rotation frequency of the test Francis turbine) representing the low frequency vortex pressure fluctuation of the draft tube; the medium frequency band with a range of 2.5 Hz~10 Hz representing the frequency of higher part load pressure fluctuation and analogous rotational pressure fluctuation; and the high frequency band whose frequency is more than 10 Hz representing leaf number of frequency of pressure fluctuation.

The measured pressure fluctuation signal contains several vibration modes. In this paper, the HHT algorithm is used to analyze the mode components of the pressure fluctuation and do the time-frequency analysis. The HHT algorithm includes two parts [30]: Empirical Mode Decomposition (EMD) and the Hilbert transform. A signal can be decomposed into a finite set of IMFs by using the EMD without a priori basis assumption. The first extracted IMF corresponds to the highest frequency component of the signal, whereas the lower frequency components are represented by higher-order IMFs. The result of EMD decomposition may have the problem of mode mixing [31]; that is, an IMF contains extremely different characteristic time scales, or similar feature time scales are distributed in different IMFs. EEMD is an improved method of EMD, which can well solve the modal aliasing problem of EMD [32]. Therefore, EEMD is utilized to decompose the pressure fluctuation signal in this paper.

After mode decomposition, the original signal can be expressed as a combination of a certain number of IMF components and a residual component, as shown in

Then, the frequency characteristic of each IMF component is analyzed by Hilbert marginal spectrum. As the IMF component is stable, it has the physical meaning to calculate the instantaneous frequency of the IMF component by the Hilbert transform [30]. The equation can be expressed as

Constructing the interpretation signal with the intrinsic mode function components,

The instantaneous amplitude and phase can be expressed as

Thus, the instantaneous frequency is expressed as follows:

The original signal without can be expressed as the sum of all the explanatory signals’ real part:

The Hilbert spectrum can accurately describe the change of the signal amplitude with the changing of time and frequency under the entire frequency domain, as it is shown in the following:

Hilbert marginal spectrum can show that the signal amplitude changes with frequency in the whole frequency domain, and it can reflect the actual frequency component of the signal more accurately [33]. The magnitude of the marginal spectrum indicates that the sum of the amplitudes of the signal at each time of the frequency can not only improve the frequency resolution but also remove other uncorrelated frequency components [34]. Therefore, in this paper, the Hilbert marginal spectrum is used to extract the frequency characteristics of IMF components. The calculation formula is shown in the following:

Finally, according to the frequency characteristics of the IMF component shown in Hilbert marginal spectrum, the IMFs are divided into the corresponding frequency band to calculate the total energy ratios of each frequency band. The pressure fluctuation multidimensional frequency band energy ratio characteristic is represented by the energy distribution coefficients . The energy ratios of high, medium, and low frequency band are expressed by , , . In this way, the extraction of multidimensional frequency band energy proportion characteristic of pressure fluctuation is realized. The calculation formula is shown in the following:where , , and represent the energy sum of the high frequency band, medium frequency band, and low frequency band, respectively; is the total energy of the pressure fluctuation signal, and is the th IMF component.

#### 3. Case Study and Discussion

Most of the pressure fluctuations in the Francis turbine are reflected in the draft tube, in which the measured value at the taper pipe part is the most important representative of the hydraulic stability [6]. Therefore, this paper takes the measured signal from the taper pipe part of a 200 MW Francis turbine whose rotation frequency is 2.5 Hz as an example to discuss the method and analyze the inner mechanism of pressure fluctuation of large Francis turbine.

##### 3.1. Time Domain Analysis and Multidimensional Frequency Band Energy Distribution Analysis by Using FFT

The amplitude and frequency characteristics of pressure fluctuation and their variation law can be visualized in the waves of different load signals. After analysis and comparison, the analysis results of the signals which are measured from a same working conditions are not obvious, so, in the analysis below, only one signal analysis result of all conditions is selected for display. Figure 1 shows the original waveform of the pressure fluctuation signal under load of 60 MW, 100 MW, 140 MW, and 180 MW. It can be seen from the figure that pressure fluctuation signals are obvious nonstationary, and the frequency components of the signal are changing with time. The fluctuation is much intense at 100 MW and 140 MW and tends to be stable at 180 MW. Figure 2 shows the trend curve of pressure fluctuation amplitude with load. It can be seen that the pressure fluctuation does not change significantly in the range of 0~40 MW, and the amplitude is smaller. The amplitude increases rapidly starting from 40 MW. The maximum point of 230 kPa occurs under 100 MW, and the relative value is 19%. The secondary amplitude of 130 kPa appears in the 80 MW with a relative value of 11%. With the load increases to full, the amplitude decreases gradually. As shown in the right -axis of the figure, the relative value of the pressure fluctuation is less than 2% within the 0–40 MW, more than 3% from 60 MW to 140 MW, which is greater than 8% in the range of 80 MW to 120 MW and greater than 15% around the 100 MW.