# Finite Volume Method for Modeling the Load-Rejection Process of a Hydropower Plant with an Air Cushion Surge Chamber

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Models of Hydropower Plant Hydraulic Transients

#### 2.1. Water Hammer Control Equations

_{0}is the slope of the pipe.

#### 2.2. Control Equations of Air Cushion Surge Chamber

_{a}and V

_{a}are the absolute pressure head and volume of the air within the air chamber, and their initial values are H

_{a}

_{0}and V

_{a}

_{0}, respectively; and k is the polytropic exponent. The adiabatic process with k = 1.4 is often used for the fast transients whereas the isothermal with k = 1.0 is often presented for the slower compressions; an intermediate polytropic case with k = 1.2 is often suggested as a reasonable compromise.

_{a}is the absolute head equal to the gage plus barometric pressure heads

_{P}is the piezometric head at the bottom of air chamber; Z

_{s}is the elevation of the air–water interface in air chamber; H

_{atm}is the absolute barometric pressure head; Q

_{s}is the inflow rate to the air chamber; R

_{s}is the head loss coefficient of the impedance hole of the air chamber; A

_{s}is the cross-section area of the air chamber.

_{t}are the flow rates at the inlet and outlet pipe of the bottom of the air chamber.

_{P}, is associated with the inflow rate to the air chamber Q

_{s}, which can be expressed as

#### 2.3. Control Equations of Hydraulic Turbine

## 3. Numerical Solution by Using the Second-Order Finite Volume Method

**U**

_{i}is the average value of

**u**within [i − 1/2, i + 1/2]; the superscripts n and n + 1 indicate the t and t + Δt time levels, respectively; and

**f**

_{i}

_{+1/2}and

**f**

_{i}

_{−1/2}are the mass and momentum fluxes at the control interfaces i − 1/2 and i + 1/2, which are determined by solving a local Riemann problem at each cell interface [2,3].

#### 3.1. Computation of Flux Term

^{n}, t

^{n}

^{+1}] can be calculated by

**A**; ${\mathbf{U}}_{L}^{n}$= average value of

**u**to the left of interface i + 1/2 at time n; and ${\mathbf{U}}_{R}^{n}$ = average value of

**u**to the right of interface i + 1/2 at time n.

**f**

_{i}

_{+1/2}is computed according to the following steps [6]:

_{i}

_{−1/2}, x

_{i}

_{+1/2}], and ${\mathbf{U}}_{i}^{n}$ at the extreme points are,

_{i}is a suitably chosen slope vector. The MINMOD limiter was used here to increase the order of accuracy of a scheme while avoiding spurious oscillations. Namely,

_{i}

_{−1/2}, x

_{i}

_{+1/2}], the boundary extrapolated values ${\mathbf{U}}_{i}^{L}$, ${\mathbf{U}}_{i}^{R}$ in Equation (16) are evolved by a time 0.5Δt according to

**f**

_{i}

_{+1/2}, the conventional Riemann problem with data can be calculated by

^{n}, t

^{n}

^{+1}] is obtained.

#### 3.2. Incorporation of Source Term

#### 3.3. Virtual-Boundary Strategy

**f**

_{1/2},

**f**

_{N}

_{+1/2}, which are required in order to update the extreme cells I

_{1}and I

_{N}to the next time level n + 1.

_{−1}and I

_{0}adjacent to I

_{1}and virtual control volumes I

_{N}

_{+1}and I

_{N}

_{+2}adjacent to I

_{N}were proposed to realize second-order Godunov scheme at the upstream and downstream control volumes of the computational domain, respectively. The corresponding fluxes

**f**

_{1/2}and

**f**

_{N}

_{+1/2}were computed in the same method as the interior control volumes.

## 4. Numerical Solution by Using Method of Characteristics

^{+}along characteristic lines from interior (fixed grid) point A to point P, and integration of C

^{−}along characteristic lines from interior (fixed grid) point B to point P, can be written as

_{P}, B

_{P}, C

_{M}, and B

_{M}are known constants when the equations are applied.

_{P}, B

_{P}, C

_{M}, and B

_{M}along the C

^{+}and C

^{−}characteristic lines as follows:

^{2}); as shown in Figure 4, Q

_{PR}and H

_{PR}are the flow rate and pressure head at R node; Q

_{PS}and H

_{PS}are the flow rate and pressure head at S node; their values can be calculated by interpolation,

## 5. Results and Discussion

#### 5.1. Water Hammer Problem in a Simple Reservoir–Pipe–Valve System

_{S}= 32). At Cr = 0.3, in order to reach the same numerical accuracy, MOC needed N

_{S}= 256, while only N

_{S}= 32 was used in the second-order FVM scheme. Table 1 displays that the for the same computation accuracy, the computation time in the MOC scheme (0.19 s with N

_{S}= 256) was about 5 times that in the second-order FVM scheme (0.037 s, N

_{S}= 32). Therefore, when Cr < 1.0, the second-order FVM scheme is more efficient than the MOC scheme for the same computation accuracy.

#### 5.2. Hydraulic Transients in Actual Hydropower Plant

#### 5.2.1. Project Overview

#### 5.2.2. Comparison to Field Experimental Data

_{i}is the grid length of the ith pipe section, m; a

_{i}is the wave speed of the ith pipe section, m·s

^{−1}; Cr

_{i}is the Courant number of the ith pipe section.

_{i}is calculated by the following equation.

_{i}is the grid number of the ith pipe; L

_{i}is the length of the ith pipe, m.

_{i}is an integer, it is necessary to adjust the Courant number Cr of the ith pipe, and the principle of adjustment is as follows: the range of the Courant number is less than or equal to 1, and preferably equal to 1 or close to 1. Using the abovementioned method, the wave speed a, grid number N, and Cr of each segment in the second-order FVM calculation can be obtained, as shown in Table 5.

#### 5.2.3. Effect of Air Chamber Parameters on the Error of MOC Scheme

- A.
- The effect of static water depth in design condition

_{0}is used, keeping P

_{0}·V

_{0}constant. When the cross-section area of air chamber remains unchanged, the P

_{0}·l

_{0}value can be treated constant, in which l

_{0}is the air length of air chamber. So, the static water depth L

_{s}

_{0}under design condition can be derived from the total height of the air chamber and air length. In this paper, five static water depths L

_{s}

_{0}were selected to study their effects on 100% load rejection hydraulic transients of the power station. The MOC and FVM simulation results are shown in Table 7 and Figure 11.

_{s}

_{0}, the maximum rotational speed gradually increased. The error of MOC calculation had a trend of decreasing and then increasing, with a slight change between 0.35% and 0.36%.

- B.
- The effect of air cushion height

_{0}increased. Five air cushion height (air length) l

_{0}were selected to study their effects on 100% load rejection hydraulic transients of the power station. The MOC and FVM simulation results are shown in Table 8 and Figure 12.

_{0}, the maximum rotational speed gradually decreased. The error of MOC calculation had a trend of decreasing, with change between 0.355% and 0.375%. It indicates that the simulation effect of MOC became worse with the increase of air cushion height. For high head hydropower plants, when the air cushion height is large, it is advisable to use FVM for simulation in order to ensure the calculation accuracy.

- C.
- Effect of polytropic exponent k

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Grid diagram introducing virtual cells at the boundaries of reservoir and hydropower unit.

**Figure 4.**Definition of the sketch in the x–t plane with a fixed MOC grid for the water hammer problem.

**Figure 7.**Comparison of pressure heads calculated by second-order FVM and MOC with different grid numbers at Cr = 0.3.

**Figure 9.**Comparison of the calculation results (FVM and MOC) and the measured data in Field Experiment Case A of Load Rejection.

**Figure 10.**Comparison of the calculation results (FVM and MOC) and the measured data in Field Experiment Case B of Load Rejection.

**Figure 11.**The maximum rotational speeds calculated by FVM and MOC with different design air-chamber water depths and the MOC computation error.

**Figure 12.**The maximum rotational speeds calculated by FVM and MOC with different air cushion heights and the MOC computation error.

**Figure 13.**The maximum rotational speeds calculated by FVM and MOC with different polytropic exponents and the MOC computation error.

Number of Grids | MOC Calculation Time/s | FVM Calculation Time/s |
---|---|---|

32 | 0.012 | 0.037 |

128 | 0.069 | 0.555 |

256 | 0.19 | 1.849 |

Pipe Number | Pipe Length/m | Wave Speed/(m·s^{−1}) | Roughness |
---|---|---|---|

L1 | 15.39 | 976.4 | 0.014 |

L2 | 169.26 | 976.4 | 0.014 |

L3 | 20.77 | 976.4 | 0.014 |

L4 | 56.4 | 976.4 | 0.015 |

L5 | 26.6 | 976.4 | 0.014 |

L6 | 100.33 | 1202.3 | 0.013 |

L7 | 5.4 | 1210.8 | 0.013 |

L8 | 14 | 1045.1 | 0.014 |

L9 | 70.94 | 1045.1 | 0.014 |

L10 | 25.52 | 1152.75 | 0.014 |

L11 | 13.6 | 1152.75 | 0.014 |

Unit Parameters | Numerical Value |
---|---|

Single machine capacity (MW) | 150 |

Rated head (m) | 105.8 |

Rated flow rate (m^{3}·s^{−1}) | 148.8 |

Rated speed (r·min^{−1}) | 200 |

Power Rating (kW) | 139,000 |

Rotational inertia (t·m^{2}) | 10,920 |

Pipe Number | MOC (Scheme 1) | MOC (Scheme 2) | ||||
---|---|---|---|---|---|---|

a/(m·s^{−1}) | N | Cr | a/(m·s^{−1}) | N | Cr | |

L1 | 961.875 | 4 | 1.000 | 976.400 | 31 | 0.983 |

L2 | 984.070 | 43 | 1.000 | 976.400 | 346 | 0.998 |

L3 | 1038.500 | 5 | 1.000 | 976.400 | 42 | 0.987 |

L4 | 1007.143 | 14 | 1.000 | 976.400 | 115 | 0.995 |

L5 | 950.000 | 7 | 1.000 | 976.400 | 54 | 0.991 |

L6 | 1194.405 | 21 | 1.000 | 1202.300 | 166 | 0.995 |

L7 | 1350.000 | 1 | 1.000 | 1210.800 | 8 | 0.897 |

L8 | 1166.667 | 3 | 1.000 | 1045.100 | 26 | 0.970 |

L9 | 1043.235 | 17 | 1.000 | 1045.100 | 133 | 0.980 |

L10 | 1063.333 | 6 | 1.000 | 1152.750 | 44 | 0.994 |

L11 | 1133.333 | 3 | 1.000 | 1152.750 | 23 | 0.975 |

Pipe Number | FVM | ||
---|---|---|---|

a/(m·s^{−1}) | N | Cr | |

L1 | 976.400 | 3 | 0.761 |

L2 | 976.400 | 43 | 0.992 |

L3 | 976.400 | 5 | 0.940 |

L4 | 976.400 | 14 | 0.969 |

L5 | 976.400 | 6 | 0.881 |

L6 | 1202.300 | 20 | 0.959 |

L7 | 1210.800 | 1 | 0.897 |

L8 | 1045.100 | 3 | 0.896 |

L9 | 1045.100 | 16 | 0.943 |

L10 | 1152.750 | 5 | 0.903 |

L11 | 1152.750 | 2 | 0.678 |

**Table 6.**Maximum rotational speed during 100% load rejection in two field experiment cases and the calculation results of MOC and FVM.

Experiment Case | Experimental Rotational Speed (r·min ^{−1}) | FVM (r·min ^{−1}) | MOC (Scheme 1) (r·min^{−1}) | MOC (Scheme 2) (r·min^{−1}) |
---|---|---|---|---|

A | 283.84 | 282.416 | 281.455 | 280.946 |

B | 279.2 | 279.076 | 278.081 | 277.672 |

**Table 7.**Comparison of the maximum rotational speeds calculated by FVM and MOC with differently designed air chamber water depths.

Design Water Depth(m) | Maximum Rotational Speed (FVM) | Maximum Rotational Speed (MOC) |
---|---|---|

(r·min^{−1}) | (r·min^{−1}) | |

4.8 | 278.068 | 277.068 |

5.4 | 278.55 | 277.555 |

6 | 279.076 | 278.081 |

6.6 | 279.626 | 278.629 |

7.2 | 280.154 | 279.148 |

**Table 8.**Comparison of the maximum rotational speeds calculated by FVM and MOC with different air cushion heights.

Air Cushion Height (m) | Maximum Rotational Speed (FVM) | Maximum Rotational Speed (MOC) |
---|---|---|

(r·min^{−1}) | (r·min^{−1}) | |

4 | 279.076 | 278.081 |

5 | 278.202 | 277.203 |

6 | 277.459 | 276.451 |

7 | 279.626 | 275.836 |

8 | 276.363 | 275.336 |

**Table 9.**Comparison of the maximum rotational speeds calculated by FVM and MOC with different polytropic exponents.

Polytropic Exponent | Maximum Rotational Speed (FVM) | Maximum Rotational Speed (MOC) |
---|---|---|

(r·min^{−1}) | (r·min^{−1}) | |

1 | 279.077 | 278.082 |

1.1 | 279.077 | 278.081 |

1.2 | 279.076 | 278.084 |

1.3 | 279.076 | 278.082 |

1.4 | 276.076 | 278.081 |

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## Share and Cite

**MDPI and ACS Style**

Lu, J.; Wu, G.; Zhou, L.; Wu, J.
Finite Volume Method for Modeling the Load-Rejection Process of a Hydropower Plant with an Air Cushion Surge Chamber. *Water* **2023**, *15*, 682.
https://doi.org/10.3390/w15040682

**AMA Style**

Lu J, Wu G, Zhou L, Wu J.
Finite Volume Method for Modeling the Load-Rejection Process of a Hydropower Plant with an Air Cushion Surge Chamber. *Water*. 2023; 15(4):682.
https://doi.org/10.3390/w15040682

**Chicago/Turabian Style**

Lu, Jianwei, Guoying Wu, Ling Zhou, and Jinyuan Wu.
2023. "Finite Volume Method for Modeling the Load-Rejection Process of a Hydropower Plant with an Air Cushion Surge Chamber" *Water* 15, no. 4: 682.
https://doi.org/10.3390/w15040682