# The Benefits of Variable Speed Operation in Hydropower Plants Driven by Francis Turbines

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Constant to Variable Speed Operation System

#### 2.1. Back-to-Back Frequency Converter

#### 2.2. Double-Fed Asynchronous Machine

#### 2.3. Variable Frequency Transformer

#### 2.4. Direct Current Transmission

## 3. Francis Turbine Operating at Variable Head and Speed

^{3}/s), and H is the net head (m).

^{2}), and g is the gravitational constant (m/s

^{2}).

_{m}is the meridional mean speed (m/s), c

_{u}is the projection of the mean speed to the tangential direction (m/s), c is the mean absolute speed in the rotor entrance (m/s), β is the angle between the tangential velocity and the mean relative speed (degrees), and α is the angle between the tangential velocity and the mean absolute speed of the rotor entrance (degrees).

^{3}/s, head 60 m, speed 360 rpm, efficiency 0.948, and an input diameter of 1.280 m, operates at a 10% increase in the head. Therefore, on applying the aforementioned equations, the turbine has to operate at a speed of 377.6 rpm. The resulting velocity triangle is depicted in Figure 8. The increase in power was 15.4% in the flow and speed was 4.9%, thus maintaining the rated efficiency.

## 4. Experimental Method

#### 4.1. Experimental Testing Setup

_{I}and NM

_{II}. As long as the downstream level is constant (NJ), the efficiency of the hydraulic turbine with the Francis runner (THF) can be obtained and compared with the two different gross heads at constant and variable operational speeds.

^{3}/s), water density ρ (kg/m

^{3}), and local gravity acceleration g (m/s

^{2}), as described in (10). The turbine power is the power available in the mechanical shaft. It is a function of the torque M (Nm) and speed n (rpm), calculated using (11):

^{2}), $\mathrm{z}$ are heights in points 1, 2, and ${\mathrm{S}}_{\mathrm{m}}$ (m).

#### 4.2. Efficiency Testing Measurements

## 5. Application to Furnas Hydropower Plant

^{2}at the maximum upstream level and a perimeter of about 3500 km, which is equal to almost half of the Brazilian coastline. The total volume of water is 22,590 km

^{3}with a storage volume of 17,217 km

^{3}. This power plant has an installed power of 1280 MVA and a rated head of 98.83 m [67].

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Low specific speed (

**a**), normal specific speed (

**b**), and high specific speed (

**c**) Francis runners suitable for high, average, and low heads, respectively.

**Figure 14.**Location map of the Furnas Hydropower plant [66] (

**a**), and Grande River cascade and partial Brazilian hydropower system (

**b**).

**Figure 15.**Operation characteristics of the Furnas hydro power plant. Gross head variation over the years (

**a**), gross head duration curve (

**b**), and generated power duration curve (

**c**).

Q (m^{3}/s) | H (m) | n (rpm) | n_{qA} |
---|---|---|---|

0.156 | 4.16 | 450 | 183 |

Quantity | Value | Unit |
---|---|---|

Weir width | 0.600 | (m) |

Width of the channel | 1.000 | (m) |

Gravity acceleration | 9.785 | (m/s^{2}) |

Distance from bottom to floor | 0.750 | (m) |

Sensor | Symbol | Range | Accuracy Class |
---|---|---|---|

Static pressure transmitter | ${\mathrm{p}}_{\mathrm{e}}$ | 0–5 m | 0.02% |

Differential pressure transmitter | $\Delta {\mathrm{h}}_{\mathrm{WK}}$ | 0–1 m | 0.1% |

Static pressure transmitter | ${\mathrm{p}}_{\mathrm{s}}$ | 0–1 m | 0.1% |

Static pressure transmitter | ${\mathrm{p}}_{\mathrm{v}}$ | 0–1 m | 0.1% |

Water temperature | ${\mathrm{t}}_{\mathrm{w}}$ | 0–100 °C | 0.1% |

Rotating speed meter | - | 0–1000 rpm | 1 rpm |

Torque transducer | - | 0–50 Nm | 0.1% |

Generator power meter | ${\mathrm{P}}_{\mathrm{e}}$ | 0–20 kW | 0.5% |

System power meter | ${\mathrm{P}}_{\mathrm{g}}$ | 0–20 kW | 0.5% |

Head (m) | Flow (m^{3}/s) | Speed (rpm) | Turbine Efficiency (%) |
---|---|---|---|

2.948 | 0.158 | 450 | 68.33 |

2.954 | 0.157 | 450 | 80.88 |

2.952 | 0.152 | 450 | 89.24 |

2.951 | 0.149 | 450 | 95.48 |

2.952 | 0.139 | 450 | 100.0 |

2.950 | 0.136 | 450 | 94.74 |

Head (m) | Flow (m^{3}/s) | Speed (rpm) | Turbine Efficiency (%) |
---|---|---|---|

2.178 | 0.130 | 450 | 34.06 |

2.217 | 0.127 | 450 | 45.82 |

2.174 | 0.127 | 450 | 56.20 |

2.141 | 0.122 | 450 | 67.40 |

2.163 | 0.120 | 450 | 65.71 |

2.194 | 0.103 | 450 | 70.89 |

Head (m) | Flow (m^{3}/s) | Speed (rpm) | Turbine Efficiency (%) |
---|---|---|---|

2.185 | 0.134 | 387 | 61.42 |

2.174 | 0.133 | 389 | 71.43 |

2.132 | 0.130 | 386 | 88.78 |

2.232 | 0.131 | 383 | 80.59 |

2.199 | 0.120 | 384 | 94.95 |

2.199 | 0.112 | 387 | 96.55 |

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**MDPI and ACS Style**

Bortoni, E.; Souza, Z.d.; Viana, A.; Villa-Nova, H.; Rezek, Â.; Pinto, L.; Siniscalchi, R.; Bragança, R.; Bernardes, J., Jr.
The Benefits of Variable Speed Operation in Hydropower Plants Driven by Francis Turbines. *Energies* **2019**, *12*, 3719.
https://doi.org/10.3390/en12193719

**AMA Style**

Bortoni E, Souza Zd, Viana A, Villa-Nova H, Rezek Â, Pinto L, Siniscalchi R, Bragança R, Bernardes J Jr.
The Benefits of Variable Speed Operation in Hydropower Plants Driven by Francis Turbines. *Energies*. 2019; 12(19):3719.
https://doi.org/10.3390/en12193719

**Chicago/Turabian Style**

Bortoni, Edson, Zulcy de Souza, Augusto Viana, Helcio Villa-Nova, Ângelo Rezek, Luciano Pinto, Roberto Siniscalchi, Rafael Bragança, and José Bernardes, Jr.
2019. "The Benefits of Variable Speed Operation in Hydropower Plants Driven by Francis Turbines" *Energies* 12, no. 19: 3719.
https://doi.org/10.3390/en12193719