1 Introduction

Pumped storage is a green, low-carbon, clean and flexible regulating power supply with the most mature technology, the best economy and the most large-scale development conditions [1]. It has a good coordination effect with wind power, solar power, nuclear power and thermal power. Accelerating the development of pumped storage is an urgent requirement for building a new type of power system with new energy as the main body, is an important support for ensuring the safe and stable operation of the power system, and is an important guarantee for the large-scale development of renewable energy. Due to the long length of the diversion channel, the high pressure on the pressure side, and the design of the unit biased towards the runoff type, the vaneless area space between the runner inlet and the active guide vane is greatly limited. The vaneless area of the pump turbine forms a high-frequency and strong-amplitude dynamic and static interference pressure pulsation. The propagation, interference and superposition of the first and higher harmonics of the pulsation in the flow channel of the water guide mechanism, the volute flow channel and the pressure steel pipe may cause vibration, noise and various instability problems in various parts. For the plant vibration of hydropower station, there are three kinds of vibration sources, namely hydraulic vibration source, mechanical vibration source and electromagnetic vibration source [2]. Related studies show that the main vibration source of pumped storage power station unit and plant is hydraulic vibration source [3, 4], which is pressure pulsation in the flow channel of the unit. The common hydraulic vibration sources include hydraulic excitation and resonance caused by the dynamic and static interference in the vaneless area of the pump turbine, as well as pressure pulsation in the draft tube. Some of the pumped storage power stations that have been put into operation in China have relatively strong plant vibration. In the later period, the plant vibration is reduced to a certain extent by taking some measures such as replacing the runner [5,6,7,8]. The measured statistics show that the main vibration frequency of the pumped storage power station with strong plant vibration is basically the dynamic and static interference frequency of the pump turbine and its multiple frequency.

In this paper, the vibration characteristics of the measured plant structure are calculated and counted, and the specific influence of the pressure pulsation generated by the operation of typical different cascade combination units on the vibration characteristics of the plant structure is analyzed.

2 Investigation on Vibration Characteristics of Unit and Plant of Pumped Storage Power Station

When the unit operates under different loads, the main frequency of plant vibration response is multiple frequency of the overcurrent frequency of the unit. The main excitation source of the plant vibration is pulsating pressure in the flow channel of the unit.

The matching combination of the number of runner blades and the number of active guide vanes of the unit is called cascade combination. The cascade combination of early pumped storage power station units is mainly 9 + 20 or 7 + 20. With the improvement of research and development capacity of domestic units, the types of pump turbines with new cascade combinations continue to emerge, such as (5 + 5)/16, 9/22, 13/22, 11/20 and so on, among which (5 + 5)/16, 9/22 type have been the main type of medium and high head section. The statistical analysis of the plant vibration data obtained from the test of the operating power station shows that the main frequency of the plant vibration is mainly 2 or 3 times the overcurrent frequency, which is closely related to the cascade combination type of the unit.

The on-site vibration test analysis results of multiple research institutions are summarized, and the main frequency characteristics of the plant vibration of multiple pumped storage power stations are obtained, as shown in Table 1.

Table 1. Statistics on vibration frequency of some pumped storage power plants in China
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The power stations with cascade combination of 9 runner blades and 20 guide vanes of the unit include Pushihe pumped storage power station, Heimifeng pumped storage power station (Unit 1#~3#), and Xianyou pumped storage power station. At present, there are many units of this type of cascade combination. When the unit operates at full load, the main frequency of the plant vibration is mainly reflected as 2 times the overcurrent frequency of the runner blade.

The power stations with cascade combination of 7 runner blades and 16 guide blades of the unit mainly include Shisanling pumped storage power station. When the unit of this type of cascade combination operates at full load, the main frequency of the plant vibration is mainly reflected as 2 times the overcurrent frequency of the runner blade.

The power stations with cascade combination of 7 runner blades and 20 guide blades of the unit mainly include Baishan pumped storage power station. When the unit of this type of cascade combination operates at full load, the main frequency of the plant vibration is mainly reflected as 3 times the overcurrent frequency of the runner blade.

The power stations with cascade combination of 9 runner blades and 26 guide blades of the unit include Tianhuangping pumped storage power station and Yixing pumped storage power station. When the unit of this type of cascade combination operates at full load, the main frequency of the plant vibration is mainly reflected as 3 times the overcurrent frequency of the runner blade.

3 Hydraulic Excitation Law of Different Cascade Combination Units

The vibration source of pumped storage power station units and plants is mainly hydraulic excitation. The hydraulic excitation characteristics of units with different cascade combinations are closely related to the number of guide vanes and the number of runner blades [9].

The main reason for the hydraulic excitation produced by the unit operation is that the outlet edge of the movable guide vane of the pump turbine is relatively thick. When the unit is running, the water flow passes through the movable guide vane, and the wake effect leads to a uneven flow field at the outlet of the guide vane. Under the action of water pressure, the water flow entering the rotating runner also produces regular potential flow disturbance at the inlet of the runner. The vaneless area between the runner and the movable guide vane will produce dynamic and static interference, and the pressure pulsation in the volute will show a periodic law as a whole [10]. The pressure pulsation in the volute causes vibration and noise of the plant, and the plant vibration and structural noise also show a similar periodic law.

The pressure field formed by the interaction between the runner blade and the guide vane can be described by the following equation [11]:

$$ f(x) = \frac{B}{2}\cos [mZ_{{\text{r}}} \omega_{{\text{n}}} t - (mZ_{{\text{r}}} - nZ_{{\text{g}}} )\theta_{{\text{s}}} + \varphi_{{\text{n}}} - \varphi_{{\text{m}}} ] + \frac{B}{2}\cos [mZ_{{\text{r}}} \omega_{{\text{n}}} t - (mZ_{{\text{r}}} + nZ_{{\text{g}}} )\theta_{{\text{s}}} - \varphi_{{\text{n}}} - \varphi_{{\text{m}}} ] $$
(1)

where, B is the average amplitude of pressure pulsation. m and n are integers. θs is an angular coordinate, which is closely related to the angular coordinates of the static system (volute, guide vane and top cover) and the rotating system (runner). Zg is the number of active guide vanes. Zr is the number of runner blades. ωn is the rotating speed of the unit.

Equation (1) shows that pressure pulsation is a function of time t and spatial angle θs. It has different low-order and high-order hydraulic excitation modes, which is determined by the cascade combination composed of the number of runner blades and the number of guide vanes.

The pitch diameter number k is described as

$$ k = mZ_{{\text{r}}} \pm nZ_{{\text{g}}} $$
(2)

The minimum pitch diameter number k1 and the maximum pitch diameter number k2 are described as

$$ k_{1} = mZ_{{\text{r}}} - nZ_{{\text{g}}} $$
(3)
$$ k_{2} = mZ_{{\text{r}}} + nZ_{{\text{g}}} $$
(4)
$$ k_{{{\text{min}}}} = {\text{Min}}(\left| k \right|) = {\text{Min}}(\left| {k_{1} } \right|) $$
(5)

It can be seen From Eq. (1) that there are many hydraulic excitation modes in the flow channel of the unit. The higher the harmonic order of the hydraulic excitation mode is, the larger the corresponding pitch diameter number |k| is, and the smaller the vibration amplitude of the hydraulic excitation force is. On the contrary, the lower the harmonic order of the hydraulic excitation mode is, the smaller the corresponding pitch diameter number |k| is [12], and the larger the vibration amplitude of the hydraulic excitation force is. Therefore, the harmonic corresponding to the minimum pitch diameter number k1 is the main vibration excitation force of pressure pulsation, and the hydraulic excitation force corresponding to kmin is the max predominant vibration excitation force of pressure pulsation.

During the operation of the pumped storage unit, the flow components such as volute and guide vane are forced to vibrate in the hydraulic excitation mode generated by the dynamic and static interference. The frequency and vibration amplitude of the hydraulic excitation force are determined by the cascade combination mode and the unit speed.

The frequency of the hydraulic excitation force on the runner is the overcurrent frequency of the active guide vane \(f_{{\text{r}}}\) and its multiple frequency:

$$ f_{{\text{r}}} = nZ_{{\text{g}}} f_{{\text{n}}} $$
(6)

The frequency of the hydraulic excitation force on the guide vane, head cover and volute is the overcurrent frequency of the runner blade \(f_{{\text{s}}}\) and its multiple frequency:

$$ f_{{\text{s}}} = mZ_{{\text{r}}} f_{{\text{n}}} $$
(7)

where, \(f_{{\text{n}}}\) is the unit rotation frequency.

For the pumped storage power station that has been put into operation, when the unit speed and cascade combination are known, the amplitude and frequency characteristics of the hydraulic excitation force can be predicted and evaluated according to Eqs. (1)~(7).

The main excitation force of the plant vibration of pumped storage power station is hydraulic excitation. The pressure pulsation is transmitted to the peripheral concrete support through the head cover and volute, and then is transmitted to the plant structure, causing plant structure vibration and structural noise. Equations (1)~(7) can reflect the plant vibration characteristics of pumped storage power station to a certain extent.

4 Vibration Characteristics of Measured Pumped Storage Power Plant

The authors have obtained a lot of vibration data of the plant structure of pumped storage power station with various cascade combinations through measured tests. The typical hydropower stations tested include Baishan hydropower station, Heimifeng hydropower station, Shisanling hydropower station, Pushihe hydropower station and Jinzhai hydropower station. The minimum pitch diameter numbers of the unit in the above typical pumped storage power stations are also counted, as shown in Table 2. At the same time, the vibration data of these plant structures are analyzed and compared. It should be noted that, due to the obvious vibration of 4# unit in Heimifeng pumped storage power station, the runner has been replaced. The cascade combination of the replaced 4# unit has been transformed from the original 9/20 to the current (6 + 6)/20[8].

Table 2. Statistics for minimum pitch diameter of typical pumped storage power plant units
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4.1 Analysis of Vibration Spectrum Characteristics of Measured Pumped Storage Power Station Plant

The frequency spectrum analysis of the measured data of the plant vibration of typical pumped storage power stations was carried out. From the calculated frequencies of the hydraulic excitation mode corresponding to the minimum pitch diameter number ( see Table 2), it can be seen that the main frequencies of the actual plant vibration are very consistent with the frequencies of hydraulic excitation mode corresponding to k1, which shows that the hydraulic excitation force of different cascade combination units is mainly the hydraulic excitation mode with n = 1 or n = 2 and k = kmin.

For the power stations with the minimum pitch diameter kmin = 1 of the unit, such as Baishan pumped storage power station, and the power stations with the minimum pitch diameter kmin = 2, such as Pushihe pumped storage power station and Shisanling pumped storage power station. The vibration spectrum of the plant structure of the three pumped storage power stations shows obvious dominant frequency, and the dominant frequency is m times the overcurrent frequency of the runner blade. Vibration fourier spectrograms of Baishan power station are shown in Fig. 1.

Fig. 1.
figure 1

Three orthogonal direction’s vibration frequency spectrograms of a measuring point in Baishan power station

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For the power stations with the minimum pitch diameter kmin = 4 of the unit, such as Jinzhai pumped storage power station and Heimifeng pumped storage power station (4# unit), although the main vibration frequency of the plant structure includes m times frequency of the overcurrent frequency of the runner blade, the vibration amplitude corresponding to the frequency is not significant. Vibration fourier spectrograms of Jinzhai power station are shown in Fig. 2. The plant vibration energy is distributed in a wide frequency band. The vibration amplitude of the plant structure of the two pumped storage power stations are significantly smaller than that of other power stations.

Fig. 2.
figure 2

Three orthogonal direction’s vibration frequency spectrograms of a mseasuring point in Jinzhai power station

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4# unit of Heimifeng power station adopts the cascade combination type of (6 + 6)/20. 6 + 6 is a runner type with six long blades and six short blades. Its hydraulic excitation mode is more complex and diverse than that presented by a single length runner blade. The main frequency characteristics of the hydraulic excitation force contain the main frequency characteristics of the hydraulic excitation mode of the cascade combination of 12/20 and 6/20. There are many characteristic frequencies in the spectrum, but the vibration amplitude is small.

4.2 Statistical Comparison of Measured Vibration Response of Pumped Storage Power Station Plant

There are obvious differences in the vibration amplitude of the plant structure in several power stations with different cascade combinations that are investigated. It can be seen from the measured statistical results (Table 3) that the plant structural vibration of Heimifeng pumped storage power station (the cascade combination is (6 + 6)/20) and Jinzhai pumped storage power station (the cascade combination is 13/22) is significantly small. The common feature of these two combinations is kmin = 4.

Table 3. Statisticals for plant vibration of different pumped storage power stations during full load generating operation of units
Full size table

From the perspective of spectrum characteristics, the plant vibration energy distribution characteristics of Heimifeng pumped storage power station and Jinzhai pumped storage power station are that the dominant frequency is not obvious, the vibration energy is distributed in a wide frequency band, and the vibration amplitude of the plant structure is significantly reduced. It shows that the plant vibration amplitude is obviously reduced by using the cascade combination mode with the number of pitch diameter kmin = 4, which is beneficial for the vibration reduction of the plant.

5 Conclusions

The excitation force amplitudes of pressure pulsation of the dynamic and static interference of the units with different cascade combinations are significantly different. The frequency spectrum analysis results of the measured vibration data in the typical pumped storage power plant structures also reflect the frequency characteristics of the dynamic and static interference, indicating that the main vibration source of pumped storage power plant vibration is hydraulic excitation force.

  1. (1)

    When the minimum pitch diameter number of the unit is 1 or 2, the vibration spectrum of the plant structure shows a significant single dominant frequency, which is m times the overcurrent frequency of the runner blade.

  2. (2)

    When the minimum pitch diameter number of the unit is 4, although the main vibration frequency of the plant structure contains m times frequency of the runner blade, the amplitude of the main vibration frequency is not large. The plant vibration energy is distributed in a wide frequency band, and the vibration amplitude of the plant structure has a significant decrease.