# Hydraulic Transient Simulation of Pipeline-Open Channel Coupling Systems and Its Applications in Hydropower Stations

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model and Verifications

#### 2.1. Transient Modelling of Pipeline Flow

#### 2.2. Modelling of Open Channel Flow

**U**is the vector of conserved variables,

**F**is the vector of fluxes and each of its components is a function of the components of

**U**,

**S**is the source term, u is the flow velocity in the x-direction which is along with the channel, h is the water depth, q is the discharge per unit width, z is the bottom elevation of the channel, τ is a parameter on the friction of the channel wall and can be expressed as $\tau =g{n}^{2}u\left|u\right|/{h}^{1/3}$ with $n$ representing the Manning coefficient.

**A**is the Jacob matrix of the flux function

**F**(

**U**) and can be expressed as:

**U**

_{L}and

**U**

_{R}by

#### 2.3. Boundary Condition Based on Riemann Invariants

^{th}characteristic field with eigenvalue ${\lambda}_{i}$ is:

#### 2.4. The Simulation Process

## 3. Numerical Validations

^{2}. The wave speed of transient waves in the pipeline is 1000 m/s. The initial flow rate in the pipe is 0 m/s with the valve fully closed at the downstream end of the pipeline. Water is transmitted to the pipeline through the valve and the flow rate increases from 0 m

^{3}/s to 2m

^{3}/s in 5 s and keeps constant at 2 m

^{3}/s after 5 s. The friction losses in the tank and pipeline are neglected in this validation case.

## 4. Transient Simulation of a Hydropower Station with a Sand Basin

#### 4.1. System Configuration and Modelling

_{s}and the width is w. The initial water depth in the sand basin is h. the Manning coefficient in the sand basin is 0.014. The length of the tunnel between the upstream reservoir and the sand basin is L

_{1}and the length of the penstock between the sand basin and the bifurcation is L

_{2}. The sum of L

_{1}and L

_{2}is 3370 m and the diameter of the tunnel and penstock are both 8.7 m. The wave speeds of the transient wave in the pipelines are calculated based on the theoretical equation [5] and are around 1100 m/s. The Darcy-Weisbach coefficient is 0.02. Some other basic information about the hydropower station is shown in Table 1.

#### 4.2. Full Load-Rejection

#### 4.2.1. Simulation Results

_{1}= 1600 m, 2000 m and 2400 m, respectively, were simulated to illustrate the effects of the position of the sand basin on the transient performance of the hydropower station. The pressure heads at the inlet of the spiral case for these three scenarios were compared in Figure 8, which shows the maximum water hammer pressure decreases when the sand basin gets closer to the turbines. The wave fluctuations after 10 s are the water hammer waves and their periods are associated with the wave traveling time in the penstocks.

_{1}= 2400 m were conducted by changing the bottom elevation and the width of the sand basin, respectively. The comparison of the pressure heads at the inlet of the spiral case for these two scenarios with the original scenario is shown in Figure 9. The results show that the bottom elevation and width of the basin do not distinctively affect the water hammer pressure. This is because the pressure reaches its maximum shortly (within 5 s) after the load rejection, while the wave oscillation caused by the sand basin has a much large period (as shown in Figure 10).

_{1}= 2400 m are compared in Figure 10 for different sizes of the sand basin. The comparison shows the bottom elevation of the sand basin has a slight effect on the water level fluctuation. The width of the basin, however, affects the water level variation significantly with the period of the oscillation increasing and the magnitude decreasing for a wider basin.

_{1}= 2400 m, h = 10 m and w = 10 m. The comparison shows the overall trend of the water level oscillations at these three points are the same, but different wave oscillations with a short period are superimposed with the low-frequency oscillation. This is caused by the wave propagating in the horizontal direction along the sand basin during the transient process.

#### 4.2.2. Comparison with the Results by Modelling the Basin Tank as a Surge Tank

^{2}. The simulated results by treating the sand basin as a surge tank and those by modelling the sand basin as an open channel are compared in Figure 12 and Figure 13. When the sand basin is close to the turbines (L

_{1}= 2400 m), a slight difference can be found in the period and magnitude of the water level fluctuations in the sand basin. A more distinctive difference can be observed when the sand basin is far away from the turbines (L

_{1}= 400 m). Such differences can be ascribed to the fact that the flow velocity and friction in the sand basin along the horizontal direction are neglected in the surge tank modelling but incorporated in the FVM when modelling the sand basin.

#### 4.3. 5% Load-Rejection with Frequency Regulation

#### 4.3.1. Simulation Results

_{1}= 1600 m, 2000 m and 2400 m, respectively, were simulated to illustrate the effects of the position of the sand basin on the transient process. The rotational speed of the turbine and the water level in the sand basin for these three scenarios are compared in Figure 14 and Figure 15, respectively. Similar to having a surge tank in the system, the comparisons show that a shorter distance between the sand basin and the turbines can facilitate the stability of the transient process.

#### 4.3.2. Comparison with the Results by Modelling the Basin Tank as a Surge Tank

_{1}= 2000 m, the simulated rotational speed of the turbine and the water level fluctuations in the sand basin are shown in Figure 16 and Figure 17, respectively. They are also compared with the results by modelling the sand basin as an open channel. Similar to the full load rejection simulation, the differences observed in the comparison can be ascribed to the neglected flow velocity and friction in the sand basin along the horizontal direction in the surge tank modelling. It can be concluded that treating the sand basin as a surge tank in the modelling may overestimate the operation stability of the system.

## 5. Conclusions

- The MOC-FVM coupling method can accurately simulate the pipeline-open channel coupling transient flow with the simulated parameters transmitted successfully at the coupling boundaries.
- The coupling method has been successfully applied to a hydropower station with a sand basin constructed between the upstream reservoir and turbines. The sand basin can be modelled as an open channel.
- The effects of the sand basin on the transient process are similar to a surge tank which can relieve water hammer pressures during load rejection scenarios and can benefit the frequency regulation process. By modelling the sand basin as an open channel, the flow velocity and the friction in the horizontal direction, which are neglected when modelling the sand basin as a surge tank, can be considered, and thus more reliable and accurate results can be obtained.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kim, S.-G.; Lee, K.-B.; Kim, K.-Y. Water hammer in the pump-rising pipeline system with an air chamber. J. Hydrodyn.
**2014**, 26, 960–964. [Google Scholar] [CrossRef] - Zhang, K.; Zeng, W.; Simpson, A.R.; Zhang, S.; Wang, C. Water Hammer Simulation Method in Pressurized Pipeline with a Moving Isolation Device. Water
**2021**, 13, 1794. [Google Scholar] [CrossRef] - Yang, W.; Yang, J.; Guo, W.; Zeng, W.; Wang, C.; Saarinen, L.; Norrlund, P. A Mathematical Model and Its Application for Hydro Power Units under Different Operating Conditions. Energies
**2015**, 8, 10260–10275. [Google Scholar] [CrossRef] - Karpenko, M.; Bogdevicius, M. Investigation into the hydrodynamic processes of fitting connections for determining pressure losses of transport hydraulic drive. Transport
**2020**, 35, 108–120. [Google Scholar] [CrossRef] - Wylie, E.B.; Streeter, V.L. Fluid Transients in Systems; Prentice Hall Inc.: Englewood Cliffs, NJ, USA, 1993. [Google Scholar]
- Urbanowicz, K. Modern Modeling of Water Hammer. Pol. Marit. Res.
**2017**, 24, 68–77. [Google Scholar] [CrossRef] - Chaudhry, M.H. Applied Hydraulic Transients, 3rd ed.; Springer: New York, NY, USA, 2014. [Google Scholar]
- Nicolet, C. Hydroacoustic Modelling and Numerical Simulation of Unsteady Operation of Hydroelectric Systems. Ph.D. Thesis, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 2007. [Google Scholar]
- Tijsseling, A.S. Fluid-structure interaction in liquid-filled pipe systems: A review. J. Fluids Struct.
**1996**, 10, 109–146. [Google Scholar] [CrossRef] - Covas, D.; Stoianov, I.; Mano, J.F.; Ramos, H.; Graham, N.; Maksimovic, C. The dynamic effect of pipe-wall viscoelasticity in hydraulic transients. Part I—Experimental analysis and creep characterization. J. Hydraul. Res.
**2004**, 42, 516–530. [Google Scholar] [CrossRef] - Vardy, A.E.; Brown, J.M.B. Transient, turbulent, smooth pipe friction. J. Hydraul. Res.
**1995**, 33, 435–456. [Google Scholar] [CrossRef] - Lai, W.; Khan, A.A. Numerical solution of the Saint-Venant equations by an efficient hybrid finite-volume/finite-difference method. J. Hydrodyn.
**2018**, 30, 189–202. [Google Scholar] [CrossRef] - Guo, Y.; Liu, R.-X.; Duan, Y.-L.; Li, Y. A Characteristic-Based Finite Volume Scheme for Shallow Water Equations. J. Hydrodyn.
**2009**, 21, 531–540. [Google Scholar] [CrossRef] - Yin, C.-C.; Zeng, W.; Yang, J.-D. Transient simulation and analysis of the simultaneous load rejection process in pumped storage power stations using a 1-D-3-D coupling method. J. Hydrodyn.
**2021**, 33, 979–991. [Google Scholar] [CrossRef] - Wang, C.; Nilsson, H.; Yang, J.; Petit, O. 1D–3D coupling for hydraulic system transient simulations. Comput. Phys. Commun.
**2017**, 210, 1–9. [Google Scholar] [CrossRef] - Zhang, X.-X.; Cheng, Y.-G.; Yang, J.-D.; Xia, L.-S.; Lai, X. Simulation of the load rejection transient process of a francis turbine by using a 1-D-3-D coupling approach. J. Hydrodyn.
**2014**, 26, 715–724. [Google Scholar] [CrossRef] - Zhang, X.-X.; Cheng, Y.-G. Simulation of Hydraulic Transients in Hydropower Systems Using the 1-D-3-D Coupling Approach. J. Hydrodyn.
**2012**, 24, 595–604. [Google Scholar] [CrossRef] - Yang, J.; Yang, J. 1-D MOC simulation software for hydraulic transients: TOPsys. Proc. IOP Conf. Ser. Earth Environ. Sci.
**2018**, 163, 12081. [Google Scholar] [CrossRef] - Zeng, W.; Yang, J.; Hu, J. Pumped storage system model and experimental investigations on S-induced issues during transients. Mech. Syst. Signal Pract.
**2017**, 90, 350–364. [Google Scholar] [CrossRef] - Hu, J.; Yang, J.; Zeng, W.; Yang, J. Transient Pressure Analysis of a Prototype Pump Turbine: Field Tests and Simulation. J. Fluids Eng.
**2018**, 140, 71102. [Google Scholar] [CrossRef] - Wang, C.; Yang, J.-D. Water Hammer Simulation Using Explicit-Implicit Coupling Methods. J. Hydraul. Eng.
**2015**, 141, 4014086. [Google Scholar] [CrossRef] - Toro, E.F. Shock-Capturing Methods for Free-Surface Shallow Flows; Wiley: Hoboken, NJ, USA, 2001. [Google Scholar]
- Toro, E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013. [Google Scholar]

**Figure 8.**Comparison of the pressure heads at the inlet of the spiral case with different positions of the sand basin.

**Figure 9.**Comparison of the pressure heads at the inlet of the spiral case with different sizes of the sand basin.

**Figure 10.**Comparison of the water levels in the sand basin with different sizes of the sand basin (L

_{1}= 2400 m).

**Figure 11.**Comparison of the water levels in the sand basin at different positions (L

_{1}= 2400 m, h = 10 m, w = 10 m).

**Figure 12.**Comparison of the water levels in the sand basin with different modelling methods (L

_{1}= 2400 m, h = 10 m, w = 10 m).

**Figure 13.**Comparison of the water levels in the sand basin with different modelling methods (L

_{1}= 400 m, h = 10 m, w = 10 m).

**Figure 14.**Comparison of the turbine rotational speed with different locations of the sand basin (h = 10 m, w = 10 m).

**Figure 15.**Comparison of the water level fluctuations in the sand basin with different locations of the sand basin (h = 10 m, w = 10 m).

**Figure 16.**Comparison of the turbine rotational speed with different modelling methods (L

_{1}= 2000 m, h = 10 m, w = 10 m).

**Figure 17.**Comparison of the water level fluctuations in the sand basin with different modelling methods (L

_{1}= 2000 m, h = 10 m, w = 10 m).

Runner Inlet Diameter (m) | Guide Vane Height (m) | Upstream Water Level (m) | Downstream Water Level (m) | Rated Rotational Speed (rpm) | Rated Output (Mw) | Rated Flow Rate (m^{3}/s) | Rotational Inertia (t.m ^{2}) |
---|---|---|---|---|---|---|---|

2.3 | 0.7 | 1073 | 829 | 429.6 | 10.54 | 29 | 726 |

Temporary Droop | Differential Time Constant | Time Lag in Servomotor | Dashpot Time Constant |
---|---|---|---|

0.3 | 0.3 s | 0.05 s | 5.0 s |

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**MDPI and ACS Style**

Zeng, W.; Wang, C.; Yang, J.
Hydraulic Transient Simulation of Pipeline-Open Channel Coupling Systems and Its Applications in Hydropower Stations. *Water* **2022**, *14*, 2897.
https://doi.org/10.3390/w14182897

**AMA Style**

Zeng W, Wang C, Yang J.
Hydraulic Transient Simulation of Pipeline-Open Channel Coupling Systems and Its Applications in Hydropower Stations. *Water*. 2022; 14(18):2897.
https://doi.org/10.3390/w14182897

**Chicago/Turabian Style**

Zeng, Wei, Chao Wang, and Jiandong Yang.
2022. "Hydraulic Transient Simulation of Pipeline-Open Channel Coupling Systems and Its Applications in Hydropower Stations" *Water* 14, no. 18: 2897.
https://doi.org/10.3390/w14182897