# Sensor-Based Optimized Control of the Full Load Instability in Large Hydraulic Turbines

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Full Load Instability in the Francis Turbine and Experimental Set-Up

#### 2.1. General Description of the Unit and the Phenomena

_{t}in the same direction than the rotation. Finally, in position (3) or full load condition, the meridian velocity increases and c has a tangential component c

_{t}in the opposite direction than the rotation.

_{rot}), changes slightly with the operating conditions of the machine, such as wicket gate opening and outlet pressure. For certain conditions, a coincidence between the pulsations generated by the vortex rope and an acoustic frequency of the hydraulic circuit may occur which will greatly amplify them, making all the systems (hydraulic, mechanical and electric) unstable.

#### 2.2. Sensors Installed and Acquisition Strategy

_{rot}(vortex rope), these were not filtered. Nevertheless, for machines with a faster rotating speed, these fluctuations may be filtered if the same type of signal is used, making it useless. This adds a justification and a utility of the present study especially for high rotating speed units where fluctuations caused by the vortex rope may be filtered in the power signal.

## 3. Signal Analysis

#### 3.1. RMS Values

#### 3.2. FFT Analysis

_{rot}, where f

_{rot}is the rotating speed of the unit in Hz.

_{rope}, slightly varies depending the operating conditions of the machine as shown by Favrel et al. [13]. If a full load instability occurs, its oscillating frequency will be the vortex rope frequency, so that ${f}_{rope}={f}_{acoustic}$, where ${f}_{acoustic}$ is one of the natural frequencies of the hydraulic circuit. Therefore, to follow the variation of the oscillating characteristics of the signals, related to a possible onset of the full load instability it is proposed to follow the maximum value in a frequency band of the signal $X\left(f\right)$ that includes the vortex rope in all the situations, i.e., the interval $[{f}_{min}:{f}_{max}]$ has to contain ${f}_{rope}$ in all the operating conditions of interest. The indicator, for every analyzed signal, is obtained according to Equation (3):

#### 3.3. Envelope Characteristic Frequencies with Hilbert Transform

#### 3.4. Time-Frequency Analysis with Wavelet

## 4. Results and Analysis of an Optimized Acquisition Strategy

#### 4.1. Analysis of the Instability: Comparison with the Reduced Scale Model

#### 4.2. Time and Time-Frequency Characteristic of the Instability Onset

#### 4.3. Sensitivity to the Detect the Instability Onset

#### 4.3.1. RMS Indicators

#### 4.3.2. FFT Band

#### 4.3.3. Envelope through Hilbert Transform

#### 4.3.4. Linear Regression and Optimization

- Linearity with respect to ${\mathrm{x}}_{\mathrm{RMS}-\mathrm{power}}$: It is desirable that the indicators follow a linear correlation with the ${x}_{RMS-power}$ in order to simplify the control model of the instability and in order to predict the oscillating power (Figure 15).
- Slope of the correlation ${\mathrm{x}}_{\mathrm{RMS}-\mathrm{power}}$ vs. Indicator: Higher slopes will indicate a more sensitive indicator to detect a variation in the RMS of the oscillating power.
- Advancement with respect increasing power: If the two preceding indicators are achieved a third indicator is the advancement, which helps to advance the detection of the instability (reactivity). Graphically it can be seen as the value of the indicator, when ${x}_{RMS-power}$ starts to increase (approximately at t = 1.5 s in Figure 12, Figure 13 and Figure 14).

- ${x}_{RMS}$ of the pressure sensors,
- ${x}_{FFT-rope}$ for the pressure sensors,
- ${x}_{Envelope-rope}$ for ADT-highfreq.

## 5. Proposed Protection System to Increase the Operating Range of the Unit

_{controlled}) is send to the governor, which mechanically acts on the wicket gates and at the same time it is send to the central system in order to modify the order of going to a higher power. The block waits a certain amount of time (T

_{update}is about few seconds) to perform a new correction (Corr%). These two parameters can be precisely determined with a test measuring the exiting of the instability (closing WGO). Note that too large Corr (%) value or too short T

_{update}will perform an excessive and maybe unnecessary correction of the WGO signal.

_{update}, then no further corrections on the signals are performed and the machine will keep working in the actual WGO after the switches are commuted again. If the control signal keeps on ON-state after T

_{update}, new reductions of the WGO signal will be performed until the machine exits the instability.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Relative power fluctuations before reaching the part load instability and the full load instability.

**Figure 2.**General description of the analyzed unit (

**a**) and detail view of the runner velocities at the outlet of a generic Francis runner (

**b**).

**Figure 3.**Vortex rope characteristic at the Turbine outlet during the part load and the full load instability

**Figure 5.**Sensor ADT-high frequency. Mechanical phenomenon of the pressure wave (

**a**) and volume of cavitation oscillating (

**b**).

**Figure 11.**Evolution of the RMS indicators during the onset of the instability (${x}_{RMS-power}$ vs. ${x}_{RMS}$).

**Figure 12.**Evolution of the RMS indicators during the onset of the instability. Shifted trend according to initial value.

**Figure 13.**Evolution of the band ${f}_{rope}$ on the FFT and RMS of the power signal (${x}_{FFT-rope}$ vs. ${x}_{RMS-power}$).

**Figure 14.**Evolution of the band ${f}_{rope}$ on the Envelope and RMS of the power signal (${x}_{Envelope-rope}$ vs. ${x}_{RMS}$).

**Table 1.**Main characteristics of the sensors installed: ADT 10, AT 9, AGA 12, PSC 10, PDT 10, B07_01.

Sensor Name | Physical Unit | Location, Direction | Sensitivity |
---|---|---|---|

ADT 10 | $\frac{\mathrm{m}}{{\mathrm{s}}^{2}}$ | Draft tube wall Radial direction | $10\mathrm{mV}/\frac{\mathrm{m}}{{\mathrm{s}}^{2}}$ |

AT 9 | $\frac{\mathrm{m}}{{\mathrm{s}}^{2}}$ | Turbine guide bearing Radial direction | $10\mathrm{mV}/\frac{\mathrm{m}}{{\mathrm{s}}^{2}}$ |

AGA 12 | $\frac{\mathrm{m}}{{\mathrm{s}}^{2}}$ | Generator bearing Axial direction | $10\mathrm{mV}/\frac{\mathrm{m}}{{\mathrm{s}}^{2}}$ |

PSC 10 | Pa | Draft tube pick-up pressure hole radial | $400\mathrm{mV}/\mathrm{bar}$ |

PDT 10 | Pa | Spiral casing pick-up pressure hole radial | $400\mathrm{mV}/\mathrm{bar}$ |

B07_01 | $\mathsf{\mu}\mathrm{m}/\mathrm{m}$ | Runner blade trailing edge | $3265\mathrm{mV}/\mathsf{\mu}\mathrm{m}/\mathrm{m}$ |

**Table 2.**Summary of the correlation between relative indicators values and relative RMS of the oscillating power values.

Sensor | RMS | FFT | Envelope | ||||||
---|---|---|---|---|---|---|---|---|---|

Slope | Intercept | R^{2} | Slope | Intercept | R^{2} | Slope | Intercept | R^{2} | |

ADT-low freq. | 0.39 | 0.007 | 0.96 | 1.26 | −0.26 | 0.65 | 0.61 | 0.13 | 0.83 |

ADT-high freq. | 0.38 | −0.008 | 0.94 | 1.26 | −0.26 | 0.65 | 0.64 | 0.13 | 0.93 |

PDT | 0.83 | 0.01 | 0.96 | 0.63 | 0.1 | 0.94 | 1.18 | −0.13 | 0.85 |

PSC | 0.80 | 0.003 | 0.96 | 0.62 | 0.13 | 0.93 | 1.14 | 0.01 | 0.82 |

Parameter | Present Case | Comments |
---|---|---|

Window length (FFT) | 4 s | Less is not recommendable (resolution in frequency) |

Refresh rate (FFT) | 0.5 s | Enough small to have a reactive system |

Sampling rate | 10 samples/s | >5·${f}_{rope}$ |

${f}_{rope}\pm {f}_{tol}$ | $0.8125\pm 0.3$ Hz | To be determined experimentally |

${x}_{th-PSC},{x}_{th-PDT}$ | 0.15 and 0.1 | To be determined experimentally according to IEC 41 |

Corr (%) | 1% | To be determined experimentally testing the exit of the instability |

${T}_{update}$ | 2 | To be determined experimentally testing the exit of the instability |

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**MDPI and ACS Style**

Presas, A.; Valentin, D.; Egusquiza, M.; Valero, C.; Egusquiza, E. Sensor-Based Optimized Control of the Full Load Instability in Large Hydraulic Turbines. *Sensors* **2018**, *18*, 1038.
https://doi.org/10.3390/s18041038

**AMA Style**

Presas A, Valentin D, Egusquiza M, Valero C, Egusquiza E. Sensor-Based Optimized Control of the Full Load Instability in Large Hydraulic Turbines. *Sensors*. 2018; 18(4):1038.
https://doi.org/10.3390/s18041038

**Chicago/Turabian Style**

Presas, Alexandre, David Valentin, Mònica Egusquiza, Carme Valero, and Eduard Egusquiza. 2018. "Sensor-Based Optimized Control of the Full Load Instability in Large Hydraulic Turbines" *Sensors* 18, no. 4: 1038.
https://doi.org/10.3390/s18041038