# Hydraulic Vibration and Possible Exciting Sources Analysis in a Hydropower System

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## Abstract

**:**

_{n}). The numerical results show all the attenuating factors are negative, which indicates the system itself is stable on the condition that all the hydraulic elements have steady operating performance. The free vibration analyses confirm that the frequency range of the vortex rope in the draft tube partly overlaps the natural frequencies of the hydropower system. Apart from the vortex rope, the runner rotational frequency is another common frequency that is approximately equal to the frequency of the 10th vibration mode. From the vibration mode shapes, it is inferred that a small disturbance in its frequency close or equal to a specific natural frequency of the vibration mode could induce large pressure oscillations in the tail tunnel. In light of the system’s response to different forcing frequencies, the vortex rope formed under off-design conditions and runner rotational frequency is verified to be the potential exciting source of a traditional hydropower system, and the frequency 0.2 f

_{n}is much more dangerous than other disturbances to the system.

## 1. Introduction

## 2. Research Object Description

_{r}= 1015 MW, the rated head H

_{r}= 202 m, the rated discharge Q

_{r}= 545.49 m

^{3}/s, and the rated rotational speed n

_{r}= 111.1 r/min. The maximum and minimum head of the prototype turbine are H

_{max}= 243.1 m and H

_{min}= 163.9 m, respectively. The test rig consists of a spiral casing, a scaled model turbine runner with 15 blades, and a draft tube.

_{11}and n

_{11}stand for the unit discharge and unit speed, respectively, which are defined as ${Q}_{11}=Q/({D}_{1}^{2}\sqrt{H})$ and ${n}_{11}=n{D}_{1}/\sqrt{H}$. From Figure 2, the optimal operating conditions for the turbine are guide vane opening a = 18.8°, unit speed n

_{11}= 53.07 r/min, unit flow rate Q

_{11}= 0.579 m

^{3}/s, and peak efficiency = 95.07%.

## 3. Materials and Mathematical Model

#### 3.1. Governing Equations

#### 3.2. Transfer Matrix and Hydraulic Impedance Methods

#### 3.2.1. Field Matrix

#### 3.2.2. Point Matrix

_{D}

_{1}= Q

_{U}

_{2}, H

_{D}

_{1}= H

_{U}

_{2}, the point matrix is,

_{s}, and l are the head loss coefficient, cross-sectional area, and water depth in the surge tank, respectively.

_{d}can be derived, ${\widehat{h}}_{d}(s)=\frac{\widehat{q}(s)}{sA}$.

#### 3.3. Hydraulic Vibration Analysis

#### 3.3.1. Free Vibration Analysis

_{U}= 0, H

_{D}= 0, which means the impedance values at the inlet and outlet are both zero. Hence, the characteristic equation of the system is written as below,

_{k}(k = 1,2,3,…).

#### 3.3.2. Forced Vibration Analysis

_{k}, the attenuate factor, while σ = 0 means the vibration amplitude was independent of time.

#### 3.4. Mathematical Model of Pressurized Flow in the System

## 4. Results and Discussions

_{i}is the instantaneous pressure, and $\overline{p}$ is the average pressure.

#### 4.1. Pressure Fluctuation in the Flow Passage

_{n}is the frequency with the greatest number of occurrences in the vaneless space and draft tube. Consequently, the pressure fluctuations in the vaneless space and draft tube are analyzed in the following section.

#### 4.1.1. Pressure Fluctuation in Vaneless Space

_{n}was manifested, caused by rotor–stator interaction, and with guide vane openings increasing, the runner rotational frequency reached 1.0f

_{n}at point CH4 under S02. A frequency of 0.2f

_{n}, affected by vortex rope in the draft tube, persists in all operating modes.

#### 4.1.2. Pressure Fluctuation in the Draft Tube

_{n}, and the frequency of 0.2~0.4f

_{n}captured in the experiment is in the range of the estimated frequency. Figure 9 shows the pressure fluctuations in the draft tube (CH6) under S01, S02, and S03. The time domain characteristics illustrate that the pressure fluctuation in the draft tube cone is the highest compared to other regions. From the frequency domain characteristics, we can infer that 0.2f

_{n}is the leading frequency that occurred in all conditions. Additionally, frequencies of 1.6f

_{n}and 1.5f

_{n}, measured at the inlet of the draft tube only, occurred in S01, and these may propagate from the upstream region affected by RSI.

#### 4.2. Numerical Computation

#### 4.2.1. Natural Frequencies

_{u}= 806.8 m and H

_{d}= 597.42 m, and both turbines worked at the rated head and discharge. Then, by solving the characteristic Equation (14), the complex frequencies were calculated and are listed in the Case 1 column in Table 5. Taking the effect of wave speed on numerical computation into consideration, the column Case 2 lists the results by adding a 10% increase to the wave speed.

_{n}should be reserved in Table 5. Further, all attenuating factors in Table 5 are negative, which means that all vibration modes would attenuate with time until a steady state is achieved, and the possibility of self-excited resonance can be excluded.

#### 4.2.2. Mode Shapes

_{U}= 1.0 m

^{3}/s at the upstream end. The rotational frequency of the prototype runner is f

_{n}= 1.852 Hz, and ω = 11.63 rad/s. According to the empirical formula, the estimated frequency range of vortex rope was f = (0.167~0.5) f

_{n}= 0.309~0.926 Hz, and ω = 1.94~5.82 rad/s. Based on the above analysis, only orders whose angular frequency is in the range of vortex rope frequency (4th, 5th, 6th) and close to the runner rotational frequency (10th) are selected for plotting in Figure 10, including oscillatory discharge and oscillatory pressure head, and the abscissa is the distance from the inlet, while the black line is the head and the red line is the flow rate. For frequencies approximately equal to those of the 4th, 5th, and 10th orders, a small disturbance could cause intense pressure oscillation in the tail tunnel. However, for the sixth order, there was no obvious pressure oscillation in the tail tunnel shown in Figure 10c. The mode shapes reveal the modulus of oscillatory value along the pipeline at different frequencies, and the amplitude of pressure fluctuation is at a minimum at the node and a maximum at the antinode. The locations of the node and antinode of the flow are opposite to those in the head. Since the upstream and downstream are reservoirs with a constant water level, the oscillatory head is zero at both ends and in all vibration modes.

#### 4.3. Comparative Analysis

_{n}measured for the vortex rope and the runner rotational frequency are emphasized here. Forced vibration was performed at three frequencies (0.2f

_{n}, 0.4f

_{n}, and f

_{n}) of oscillating discharge at the turbine point as the forcing function, with an expression ${q}^{\prime}=0.2\mathrm{sin}\omega t$, and the system’s responses are shown in Figure 11.

_{n}is the highest. On the contrary, the response to the runner rotational frequency is the lowest. It is clear that the oscillatory crest value of both pressure and discharge decreases as the disturbance’s frequency increases, which indicates that the lower frequency vibration is more severe and should be avoided, because severe pressure oscillation can burst or collapse the pipe due to pressure in excess of the designed pressure. The pressure fluctuation in the tail tunnel shown in Figure 11 is much higher than that in the upstream part of the system. For the blade frequency of 15f

_{n}with an extremely short exciting period, it is unnecessary to carry out targeted analyses, since the natural frequency of the higher order is not exact owing to the error in the estimated wave speed, and the corresponding higher-order vibration usually manifests as energy dissipation. It is confirmed that the leading frequencies of the vortex rope and the runner rotational frequency are closely related to natural frequencies, which may induce huge pressure fluctuations and even resonance along the water conveyance line. Under the actual operating conditions, real-time monitoring should concentrate on the frequency characteristics of vortex rope in the draft tube and pressure fluctuation in the vaneless space, in case these equal the natural frequency of the system.

## 5. Conclusions

_{n}in the vortex rope is the most dangerous disturbance, as this will cause huge pressure fluctuations in the water conveyance line. Besides this, the runner rotational frequency cannot be ignored either, as this may cause severe pressure fluctuations in the tail tunnel. According to the computation results, when the disturbance frequencies are similar to certain natural frequencies, vibration mitigations actions should be taken during the operating stage.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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No. | L(m) | D(m) | No. | L(m) | D(m) | No. | L(m) | D(m) |
---|---|---|---|---|---|---|---|---|

1 | 41.0 | 15.925 | 11 | 24.55 | 17.145 | 21 | 61.1 | 11.460 |

2 | 166.446 | 10.999 | 12 | 93.25 | 16.456 | 22 | 60.85 | 17.145 |

3 | 47.124 | 10.203 | 13 | 41.0 | 15.925 | 23 | 24.55 | 17.145 |

4 | 104.50 | 10.203 | 14 | 167.938 | 10.999 | 24 | 74.0 | 16.456 |

5 | 47.124 | 10.203 | 15 | 47.124 | 10.203 | 25 | 103.745 | 17.628 |

6 | 31.88 | 9.053 | 16 | 104.50 | 10.203 | 26 | 972.805 | 17.628 |

7 | 42.4 | 8.600 | 17 | 47.124 | 10.203 | 27 | 100.0 | 17.605 |

8 | 20.0 | 11.460 | 18 | 31.88 | 9.053 | 28 | 122.166 | 17.605 |

9 | 61.1 | 11.460 | 19 | 42.4 | 8.600 | 29 | 397.048 | 17.605 |

10 | 60.85 | 17.145 | 20 | 20.0 | 11.460 |

Mode | a (°) | n_{11}(r/min) | Q_{11}(L/s) | H_{m}(m) | P (MW) | P_{r}(%Pt) |
---|---|---|---|---|---|---|

S01 | 7.88 | 60.21 | 145.99 | 29.91 | 301.11 | 29.67 |

S02 | 12.63 | 60.15 | 249.77 | 30.32 | 607.72 | 59.87 |

S03 | 14.12 | 73.16 | 260.64 | 29.88 | 346.43 | 48.32 |

Mode | CH0 | CH1 | CH2 | CH3 | CH4 | CH5 | CH6 | CH7 | CH8 | CH9 | CH10 | CH11 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

S01 | 1.88 | 1.86 | 1.84 | 1.78 | 1.51 | 1.74 | 3.95 | 3.53 | 3.12 | 3.13 | 1.81 | 2.86 |

S02 | 1.91 | 1.88 | 2.67 | 3.02 | 2.14 | 2.54 | 2.68 | 2.86 | 2.63 | 3.36 | 1.54 | 2.28 |

S03 | 1.27 | 1.25 | 2.60 | 2.50 | 1.71 | 2.38 | 4.34 | 3.69 | 3.78 | 4.33 | 2.28 | 4.14 |

Mode | CH0 | CH1 | CH2 | CH3 | CH4 | CH5 | CH6 | CH7 | CH8 | CH9 | CH10 | CH11 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

S01 | 0.1 | 0.1 | 15 | 15 | 0.2 | 0.2 | 1.6 | 1.5 | 0.2 | 0.2 | 0.2 | 0.2 |

S02 | 0.0 | 0.1 | 0.2 | 0.2 | 1.0 | 0.2 | 0.2 | 0.2 | 0.4 | 0.4 | 0.2 | 0.2 |

S03 | 0.0 | 0.0 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 |

Order kth | Case 1 | Case 2 | ||
---|---|---|---|---|

σ | ω | σ | ω | |

1 | −0.0046 | 0.0284 | −0.0046 | 0.0284 |

2 | −0.0048 | 0.2597 | −0.0048 | 0.2599 |

3 | −0.0025 | 0.3640 | −0.0025 | 0.3640 |

4 | −0.0038 | 2.3428 | −0.0038 | 2.6017 |

5 | −0.0036 | 4.6627 | −0.0036 | 5.1799 |

6 | −0.3048 | 5.5526 | −0.3041 | 6.1735 |

7 | −0.0044 | 6.8170 | −0.0044 | 7.5732 |

8 | −0.0055 | 7.4851 | −0.0055 | 8.3165 |

9 | −0.0265 | 9.4424 | −0.0276 | 10.4924 |

10 | −0.0034 | 11.6914 | −0.0034 | 12.9902 |

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## Share and Cite

**MDPI and ACS Style**

Shen, A.; Chen, Y.; Zhou, J.; Yang, F.; Sun, H.; Cai, F.
Hydraulic Vibration and Possible Exciting Sources Analysis in a Hydropower System. *Appl. Sci.* **2021**, *11*, 5529.
https://doi.org/10.3390/app11125529

**AMA Style**

Shen A, Chen Y, Zhou J, Yang F, Sun H, Cai F.
Hydraulic Vibration and Possible Exciting Sources Analysis in a Hydropower System. *Applied Sciences*. 2021; 11(12):5529.
https://doi.org/10.3390/app11125529

**Chicago/Turabian Style**

Shen, Aili, Yimin Chen, Jianxu Zhou, Fei Yang, Hongliang Sun, and Fulin Cai.
2021. "Hydraulic Vibration and Possible Exciting Sources Analysis in a Hydropower System" *Applied Sciences* 11, no. 12: 5529.
https://doi.org/10.3390/app11125529