Refined Modeling and Dynamic Characteristics of Self-excited Vibration in the Main Inlet Valve of Pumped Storage Power Stations

Self-excited vibration of the main inlet valve poses a great threat to the safe operation of the pumped storage power system, thereby making an indirect hazardous effect on the frequency stability of the power grid. In order to avoid self-excited vibration, this paper focuses on the refined modeling and dynamic characteristics of self-excited vibration. Firstly, the flexible valve model is introduced and the refined pumped storage power system modeling considering self-excited vibration is established. The constructed model is then validated by comparing it with field data in the frequency domain. Secondly, the sensitivity of self-excited vibration to system parameters is investigated by using the modified Morris method. Once the sensitive parameters are found, the effect law of these parameters on the dynamic characteristics of self-excited vibration is studied in depth. Finally, engineering advice is proposed. The result indicates that the self-excited vibration is most sensitive to the length of the diversion pipe, valve clearance of the main inlet valve, and water hammer velocity. Moreover, it is verified by simulation that shorter upstream pipe length, lower water hammer velocity and smaller valve clearance help to restrain the amplitude and deterioration rate of self-excited vibration.


Introduction
A pumped storage power system (PSPS) plays an important role in keeping the safe and stable operation of the grid [1].The main inlet valve (MIV) is one of the main auxiliary facilities of a PSPS, and any unsuccessful operation of the MIV leads to the failure of the startup or shutdown processes of pumped storage units (PSUs) [2].However, water leakage gradually appears due to the wear and fatigue of MIV sealing after a long period of operation.Once the leakage increases to a critical amount, self-excited vibration (SeV) will occur due to the water hammer propagation and valve operation mechanics [3].Up to today, a considerable amount of accidents related to SeV have occurred in the PSPS in China.The first case happened in the PSPS in Guangdong Province in 2003 [4].In 2011, a SeV was reported in Beijing while the PSU was shutting down from pumping mode.Consequently, the operation alarm was triggered and the startup process of the generation model failed [5].In addition, two PSPSs in Zhejiang Province reported over four times SeV in 2014 [6].All reported cases resulted in the emergency shutdown of PSUs, which caused both huge safety hazards and tremendous economic loss to the PSPS.
Many researches have been done, trying to address the SeV problem in PSPS.Jeager [7] conducted a mechanical analysis of SeV by using the graphic method, which revealed that the maximum amplitude of SeV is likely to reach two times the static water pressure.Doerfler [8] investigated the effect of SeV on the axial oscillation of PSPS via theoretical analysis.Domestically, Zhou and Suo [9] studied the occurring criterion of SeV from the perspective of hydraulic resistance, and the unstable region was also revealed.In addition, the nonlinear vibration theory was applied to uncover the amplitude-frequency characteristics of SeV, which showed that the SeV response is composed of multiple frequency vibrations with positive damping.Ye [10] conducted case studies of real SeV failures in China, and the predisposition of SeV was verified as water leakage due to valve sealing fatigue.However, the main influencing factors related to the SeV phenomenon remained a myth.
The aforementioned research has addressed the SeV problem to some extent, whereas the most important parts, i.e., the effect mechanisms and engineering countermeasures are still missing.In order to address the SeV problem in PSPS, this paper focuses on the refined modeling and simulation of SeV, and the sensitivity of SeV response to system parameters is investigated in depth via the modified Morris method.Moreover, the effect law of sensitive parameters on SeV dynamics is revealed and the corresponding engineering advices are proposed.
The remainder of this paper is structured as follows.In section 2, the refined MIV model, PSU model, and water diversion system model are established, and the flexible valve model is adopted to describe the valve leakage characteristics.The established refined model is then validated in the frequency domain via empirical wavelet transform and Hilbert spectrum.In section 3, the SeV characteristics of MIV are investigated in depth.

Refined modeling of self-excited vibration in PSPS
A typical PSPS consists of MIV, PSU, and a water diversion system, which is presented in Fig. 1.To obtain a refined model of SeV in PSPS, the MIV is mathematically formulated according to the leakage mechanism of valves, the PSU is modeled via full-characteristics curves (FCC), and the water diversion system is mathematically described by the model of characteristics (MOC).

. Topological layout of PSPS (1) Main inlet valve
The MIV obeys the leakage characteristics of valves, a healthy MIV can be characterized by a plastic valve model, whereas a MIV after years of wear and fatigue follows flexible valve characteristics.In a plastic valve, the water leakage increases monotonously with the increase in water head.In a flexible valve, the water leakage increases at first and then decreases with the increase in water head, as is presented in Fig. 2. SeV often occurs in the MIV after a long period of operation.Therefore, the flexible valve model is adopted in this paper, which is formulated in Equation ( 1) [11].
(2) Pumped storage unit The FCC model is adopted in this paper to describe the dynamic behavior of PSU.From the perspective of FCC modeling, the dynamic characteristics of a PSU can be described by the moment function and discharge function, which are mathematically formulated in Equation (2).
where the moment and discharge of PSU are calculated according to Equation (3).
To avoid the S-shape characteristic in the FCC model, the improved Suter transformation method [12] is introduced in this paper.The transformed curves, i.e., WH and WM, are presented in Fig. 3. Once the transformed curves are obtained, the discharge and moment of PSU can be calculated via Equation (4).
(3) Water diversion system With the purpose of accurate simulation of hydraulic transients in penstocks and surges, MOC is adopted in this paper.The elasticity of the water column, water hammer, and penstock is considered in the MOC model, hence the momentum equation, as well as the continuity equation, can be formulated as follows.
(4) Simulation tool for PSPS A recently developed visual numerical simulation software [13] is adopted in this paper to realize the high-precision numerical simulation of SeV in PSPS.The aforementioned modeling methodologies of the main inlet valve, PSU, and water diversion system are embedded in the software.Based on the software, a typical "one pipeline-double units" PSPS model considering SeV is established as presented in Fig. 4. The PSPS model consists of two PSUs and two MIVs, however, only PSU1 and MIV1 are studied so as to reduce the complexity of the SeV problem.

Model validation based on empirical wavelet transform
To validate the rationality and accuracy of the proposed refined model, a comparative analysis between the simulation result and field data is conducted.However, the measured SeV signal is usually superposed with background noise due to the strong electromagnetic interference.Therefore, both the simulation result and measurement data are transformed and compared in the frequency domain. (

1) Simulation of SeV based on refined model
The initial working condition is as follows: the two PSUs and the corresponding MIVs operate at shut condition, and the water levels of upper and lower reservoirs are 733 m and 181 m, respectively.The upstream water head of MIV is 729 m, the valve leakage of PSU1 is 0.19 m 3 /s, the valve leakage of PSU2 is 0, and the water hammer velocity is 1000 m/s.The detailed parameters of penstocks are listed in Table 1.
Table 1.Detailed Parameters of Penstocks.Then, the initial parameters are input into the adopted visual numerical simulation software.A numerical simulation of the dynamic response of SeV in MIV under the initial condition is conducted, and the simulated result is presented in Fig. 5, where HSeV1 represents the upstream water head of MIV1.The EWT is adopted in this paper to transform the noise-embedded field data and simulation result from the time domain into the frequency domain, which is able to recognize major frequency components while neglecting minor spectral components in the spectrum.Step 1: Fast Fourier Transform is applied to the time-domain SeV response to obtain the original spectrum of SeV.
Step 2: EWT is introduced and the original spectrum in the frequency range [0, ʌ] is divided into N separated regions, i.e., n / , according to the shape of the original spectrum.The mathematical formulation of the aforementioned selected regions is presented in Equation ( 7), and the corresponding sketch map of segmentation is depicted in Fig. 6.
Apmlitude Fig. 6.Segmentation of original spectrum.The slash areas in Fig. 6 are the selected spectral regions.Then, the empirical scaling function and empirical wavelet function are defined as follows: where Ĳn and ȕ(x) are defined as below: Step 3: the EWT result of SeV can be obtained as follows: Step 4: Equation ( 8) is applied to the segmentation of the original spectrum, and we can obtain the EWT of the SeV response.
Based on the methodology presented above, the spectrum of simulated SeV response and measured SeV response can be obtained, which are depicted in Fig. 7.Moreover, the Hilbert spectrum corresponding to measured data and simulation result can also be obtained by using Hilbert-Huang Transform [14] in Fig. 8. From Fig. 7, we can get that the spectrum of simulated SeV presents a fine consistency with that of measured data, despite very slight deviations in spectrum amplitude.In addition, the Hilbert spectrums in Fig. 8 indicate that the time-frequency distribution of the measured data and the simulated result are almost identical, which further verifies the accuracy of the established refined model.

Self-excited vibration characteristics of the main inlet valve
The SeV of MIV is the consequence of internal disturbance, whereas the key influencing factors and their effect mechanisms have not been fully addressed.Hence, the sensitivity of SeV of MIV to several structural and hydraulic parameters is studied in depth by using the modified Morris method in this section.Subsequently, once the key influencing factors are revealed, the specific effect law of key parameters on SeV is further investigated.

Sensitivity analysis of SeV to system parameters (1) Modified Morris method
To investigate the sensitivity of SeV to system parameters, the modified Morris method is adopted in this paper.From the perspective of the modified Morris method, the global sensitivity of a system can be quantitatively evaluated via the sensitivity discriminant factor, denoted as SN [15].To do so, the candidate parameters are set as a sequence with fixed step value, each member of the sequence is input into the proposed refined model, the model output, i.e., SeV response, is captured and the sensitivity of SeV to those parameters is evaluated according to Equation (12).
Based on the value of SN, the sensitivity level of SeV to system parameters is divided into four grades: a) Highly sensitive, if (2) Sensitivity analysis based on the modified Morris method The sensitivity of SeV to the length of the diversion pipe, altitude of MIV, the diameter of MIV, sealing clearance of MIV, valve clearance of MIV, area of MIV sealing cover, upstream water level, downstream water level, and Water hammer velocity is investigated in this section by using modified Morris method.Each of the above parameters varies from 70% to 130% of their rated value with a step value of 10%.The corresponding SN are listed in Table 2. From Table 2, we can get that the |SN| corresponding to the downstream water level is far smaller than 0.05, which indicates that SeV is not sensitive to downstream water level.The |SN|s corresponding to the altitude of MIV, the diameter of MIV, sealing clearance of MIV, area of MIV sealing cover, area of MIV sealing cover, and upstream water level are larger than 0.05 but smaller than 0.2, which indicates that SeV is medium sensitive to those parameters.In addition, the |SN|s corresponding to the length of the diversion pipe, valve clearance of MIV, and water hammer velocity are larger than 0.5, which means that SeV is sensitive to these parameters.

Effect of system parameters on self-excited vibration
Now that the sensitivity of SeV to system parameters is revealed, the detailed effect of system parameters on SeV is investigated in this section.According to sensitivity analysis, the SeV is sensitive to the length of the diversion pipe, valve clearance of MIV, and water hammer velocity, hence these system parameters are selected and their effect on SeV response is further studied.Three cases are constructed as follows: (i) The length of pipe L2 is set as 756 m while other parameters are set as their default values; (ii) The valve clearance of MIV is set as 1.1 times the default value while other parameters are set as their default values; (iii) The water hammer velocity is set as 1030 m/s while other parameters are set as their default values.
For each case, the system parameters are input into the refined SeV model, the SeV responses are then calculated, which are depicted in Figs. 9.In addition, the measured SeV response corresponding to default parameters is also presented in the figures.
response to system parameters is then investigated by adopting the modified Morris method.Finally, the specific effect law of system parameters on SeV response is studied in depth and several engineering advices are proposed.The conclusions are summarized as follows: (1) The SeV phenomenon can be characterized by incorporating a flexible valve model, and the established refined model precisely describes the SeV response according to model validation.In addition, the model simulation result indicates that unlimited SeV will occur if no countermeasure is applied.
(2) The length of the diversion pipe, water hammer velocity, and valve clearance of MIV are the top three global sensitive factors of SeV with respect to the modified Morris method.The decrease of diversion pipe length leads to a 20% of deterioration rate in amplitude, and the decrease in water hammer velocity helps restrain the oscillation amplitude by over 17%.
(3) A shorter upstream pipe length, lower water hammer velocity, and smaller valve clearance lead to a smaller amplitude and deterioration rate of SeV.Therefore, a shorter upstream pipe length and suitable pipe material are advised in engineering practice.

Fig. 5 .
Fig. 5. Simulation result of the SeV in MIV.(2) Transformation of field data and simulation result based on Empirical wavelet transform (EWT)The EWT is adopted in this paper to transform the noise-embedded field data and simulation result from the time domain into the frequency domain, which is able to recognize major frequency components while neglecting minor spectral components in the spectrum.

Table 2 .
Sensitivity of SeV to System Parameters.