Optimization Control on the Mixed Free-Surface-Pressurized Flow in a Hydropower Station

: The mixed free-surface-pressurized ﬂow in the tailrace tunnel of a hydropower station has a great impact on the pressure, velocity, and operation stability of the power station. In the present work, a characteristic implicit method based on the upwind differencing and implicit ﬁnite difference scheme is used to solve the mixed free-surface-pressurized ﬂow. The results of the characteristic implicit method agree well with the experimental results, which validates the accuracy of the method. Four factors that inﬂuence the amplitude of pressure ﬂuctuation are analyzed and optimized, and the results show that the relative roughness can inﬂuence the maximum pressure in the tailrace tunnel. Additionally, the maximum pressure decreases with the increase of the tunnel’s relative roughness. When the surface relative roughness increases from 0.010 to 0.018, the maximum pressure can decrease by 4.33%. The maximum pressure in the tailrace tunnel can be effectively restrained by setting vent holes in the ﬂat-topped tunnel section (tunnel (4)) and a vent hole at 81.25%L (L is the length of tunnel (4)), which can reduce the maximum pressure by 56.72%. Increasing the vent hole number can also reduce the maximum pressure of the mixed free-surface-pressurized ﬂow in the tailrace tunnel. An optimal set of two ventilation holes 10 m in diameter at 93.75%L and 56.25%L is proposed, which can reduce the maximum pressure by 15.30% in comparison with the single vent case.


Introduction
Energy is one of the most important basic elements of economic and social development, and the utilization of energy can greatly improve the living quality of humans. After hundreds of years of exploitation and utilization, traditional fossil energy is decreasing day by day. As a consequence, the development of renewable energy has become an important development direction for global energy. Every country takes the development of renewable energy, such as hydropower, wind energy, and solar energy, as an important means to meet the challenges of energy security and climate change [1][2][3][4]. Among them, hydropower, with its low power generation cost and high power generation efficiency, has become a great alternative in recent years [5][6][7][8][9][10].
With the development of the national economy and hydropower, many large-scale water conservancy and hydropower projects have been built in China, such as the Three Gorges power station, Baise Underground power station, Xiangjiaba power station, and Xiluodu power station. To ensure the safe operation of hydropower stations, sufficient hydraulic analysis and hydraulic calculation must be carried out for the hydraulic transition process. The purpose of the transition process calculation is to reveal the dynamic that the interface of the mixed free-surface-pressurized flow and positive surge wave is consistent, which is not suitable for some cases, such as a positive surge wave in a tailrace tunnel with a variable top height. Hamam and McCorquodale [25,26] proposed the rigid water body method to solve the mixed free-surface-pressurized flow, which assumes that water is incompressible and the flow velocity is uniform. Rigid water body theory is adopted for the liquid phase, and compressible flow theory is adopted for the gas phase. Li and Alex [16] developed the rigid water body method to calculate the bubble motion. However, the algorithm of the rigid water body method is complex, and there are many discrepancies with physical reality. Therefore, this method is rarely used to simulate the mixed free-surface-pressurized flow.
The existing work outlined above shows that the three calculation methods to solve the mixed free-surface-pressurized flow have their limits. Therefore, it is vital to propose a stable calculation format for the mixed free-surface-pressurized flow which can smoothly calculate and obtain results consistent with the actual situation, even when the water surface fluctuates greatly and the pressure fluctuates violently.
The basic structure of the present work is as follows. Firstly, the characteristic implicit scheme method [27] is used to solve the mixed free-surface-pressurized flow, and the experimental validation is carried out. Then, the influence of the relative roughness of the tailrace tunnel on the maximum pressure in the tunnel is analyzed. The restraining effect of setting single vent holes at different positions in the flat-topped tunnel section on the maximum pressure in the tunnel is discussed. The influence of ventilation holes with different diameters on the maximum pressure in the tunnel is calculated. The influence of setting multiple vent holes in the flat-topped tunnel section on suppressing the pressure fluctuation in the tunnel is also studied. Figure 1 shows the layout of the water conveyance system of a hydropower station. The downstream is a tailwater tunnel that starts at the altitude of 548.70 m and ends at the altitude of 577.00 m. As shown in Tables 1 and 2, the total length of tunnels (1), (2), (3), and (4) is 1541.10 m, where the mixed free-surface-pressurized flow may occur. The shape of tunnel (4) is rectangular with an arch crown; its width is 18 m, and its height is 20 m. The operating conditions are 825 m at the upstream and 597 m at the downstream, respectively, which means that the downstream water level is equal to the top of tunnel (4). method to solve the mixed free-surface-pressurized flow, which assumes that w incompressible and the flow velocity is uniform. Rigid water body theory is adop the liquid phase, and compressible flow theory is adopted for the gas phase. Li an [16] developed the rigid water body method to calculate the bubble motion. Howev algorithm of the rigid water body method is complex, and there are many discrep with physical reality. Therefore, this method is rarely used to simulate the mixe surface-pressurized flow.

Research Object
The existing work outlined above shows that the three calculation methods t the mixed free-surface-pressurized flow have their limits. Therefore, it is vital to p a stable calculation format for the mixed free-surface-pressurized flow whi smoothly calculate and obtain results consistent with the actual situation, even wh water surface fluctuates greatly and the pressure fluctuates violently.
The basic structure of the present work is as follows. Firstly, the characteristic i scheme method [27] is used to solve the mixed free-surface-pressurized flow, and perimental validation is carried out. Then, the influence of the relative roughness tailrace tunnel on the maximum pressure in the tunnel is analyzed. The restrainin of setting single vent holes at different positions in the flat-topped tunnel section maximum pressure in the tunnel is discussed. The influence of ventilation holes w ferent diameters on the maximum pressure in the tunnel is calculated. The influ setting multiple vent holes in the flat-topped tunnel section on suppressing the p fluctuation in the tunnel is also studied. Figure 1 shows the layout of the water conveyance system of a hydropower s The downstream is a tailwater tunnel that starts at the altitude of 548.70 m and end altitude of 577.00 m. As shown in Tables 1 and 2, the total length of tunnels (1), and (4) is 1541.10 m, where the mixed free-surface-pressurized flow may occur. The of tunnel (4) is rectangular with an arch crown; its width is 18 m, and its height i The operating conditions are 825 m at the upstream and 597 m at the downstream, tively, which means that the downstream water level is equal to the top of tunnel

Mathematical Model
In recent work, the virtual slit method was mainly used to calculate the alternating full flow. According to the Preissmann model [17], a virtual slot on the top of the pipe or tunnel is assumed, and the slot will slightly influence the tunnel cross-section area A. The wave velocity of the free surface flow c is determined by c = gA/B, where B is the surface width and g is gravity. Then, the wave velocity of the pressurized flow a can be chosen as a = c. This is a way to simulate the pressurized flow by modifying the free surface wave velocity. Then, the free surface flow and the pressurized flow can be described by the same partial equations [28] as follows: where h is the flow depth, v is the flow velocity, i is the tunnel slope, and J f is the slope of the energy grade line.

Characteristic Implicit Method
With the rapid increase of the wave velocity c, the numerical calculation would be difficult to converge when the interface of the free surface flow and the pressurized flow passes the computational nodes. In the present work, the characteristic implicit method provides a differencing scheme with good stability and high accuracy to solve the mixed free-surface-pressurized flow. Since the system made up of the continuity and momentum equations is a hyperbolic system, upwind differencing can avoid nonphysical oscillations, and the implicit scheme is used for stability and accuracy.
To transform Equations (1) and (2) into the norm forms of the hyperbolic system and to replace v with Q/A, where Q is the volume flow, Equations (3) and (4) are derived as follows: Processes 2021, 9, 320 , and R is the hydraulic diameter. Equations (3) and (4) are differenced at the point of (m, n) by using the differencing scheme as follows. The forward differencing in time is found by For the upwind differencing in space, Equation (3) is differenced along the c + line: Equation (4) is differenced along the c − line: The substitution of Equations (5) and (6) into Equation (3), as well as the substitution of Equations (5) and (7) into Equation (4), lead to the system in Equation (8): where Equation (8) is constituted by a set of two nonlinear algebraic equations with six independent unknowns, two of which are the same in any two neighboring nodes, and a similar pair of equations are written for each of the M-2 internal points in the tunnel. Thus, there are two M-2 equations in 2 M unknowns. Equation (8) can provide one equation for each boundary point. The boundary condition at the end of the tunnel can provide two additional equations, so a unique solution can be obtained. In this paper, the new method is named the characteristic implicit method with first-order accuracy. By using the friction term and gravity term of the n + 1 time step, the computation precision and stability can be improved.

Experimental Validation
The purpose of this experiment was to verify the correctness of the characteristic implicit method for solving the mixed free-surface-pressurized flow. In this experimental model, the diversion system and tailwater system adopted the diversion mode that one tunnel distributed one machine. The tailrace adopted the four-in-one arrangement scheme, in which four branch tunnels converged to one main tunnel. Figure 2 shows the diagram and experimental rig of the tailrace tunnel.
The purpose of this experiment was to verify the correctness of the characteristic implicit method for solving the mixed free-surface-pressurized flow. In this experimental model, the diversion system and tailwater system adopted the diversion mode that one tunnel distributed one machine. The tailrace adopted the four-in-one arrangement scheme, in which four branch tunnels converged to one main tunnel. Figure 2 shows the diagram and experimental rig of the tailrace tunnel. The experimental tunnel was a model with a proportion scale λL = 30, according to the gravity similarity principle [29]. The prototype of the tailrace runner was the concrete surface, and its relative roughness was 0.014. According to the scale of relative roughness, the relative roughness of the model material was 0.0079, while the relative roughness of the plexiglass pipe was also 0.0079, so the tailrace channel model was made of plexiglass could meet the requirement.
The model structure and schematic diagram of the monitoring points on the main tailrace tunnel are shown in Figure 2a, in which the tailrace of the power station was connected with a large tailwater pond. The reason for this was that the tailrace of the prototype power station flowed into the natural river channel, and the tailrace water level of the power station was determined by the discharge flow of the discharge structure and the discharge flow of the power station. The discharge flow of the power station was relatively low, and the tailrace water level of the power station was relatively stable in the transient process. In the present experiment, the discharge flow was only from the power station, so it was necessary to manually adjust the actual tailrace level. In the unsteady experiment, a relatively stable boundary condition of downstream tailrace could be obtained by connecting a large pool with the model tailrace and adopting a wide weir at the pool end.
In the present experiment, the flow rate was measured by the rectangular, thinwalled weir, and the zero reading of the thin-walled weir was calibrated before the experiment. The water depth at each point of the main tunnel was measured by the pulsating pressure sensor. The pulsating pressure sensor was connected with the computer through a DJ800 multifunctional monitor to form a data acquisition and processing system. The system was used to measure the fluctuation of the water depth at various points in the tailrace system. To observe the obvious mixed free-surface-pressurized flow in the main tailrace tunnel, the downstream water level was set to 0.807 m. Under this condition, the first half of the tailrace was the pressurized flow, and the second half of the tailrace was the free surface flow. The discharge was calculated to be 0.066279 m 3 /s, according to the Rehbock weir formula [30]. When the butterfly valves at the inlets of four branch tunnels The experimental tunnel was a model with a proportion scale λ L = 30, according to the gravity similarity principle [29]. The prototype of the tailrace runner was the concrete surface, and its relative roughness was 0.014. According to the scale of relative roughness, the relative roughness of the model material was 0.0079, while the relative roughness of the plexiglass pipe was also 0.0079, so the tailrace channel model was made of plexiglass could meet the requirement.
The model structure and schematic diagram of the monitoring points on the main tailrace tunnel are shown in Figure 2a, in which the tailrace of the power station was connected with a large tailwater pond. The reason for this was that the tailrace of the prototype power station flowed into the natural river channel, and the tailrace water level of the power station was determined by the discharge flow of the discharge structure and the discharge flow of the power station. The discharge flow of the power station was relatively low, and the tailrace water level of the power station was relatively stable in the transient process. In the present experiment, the discharge flow was only from the power station, so it was necessary to manually adjust the actual tailrace level. In the unsteady experiment, a relatively stable boundary condition of downstream tailrace could be obtained by connecting a large pool with the model tailrace and adopting a wide weir at the pool end.
In the present experiment, the flow rate was measured by the rectangular, thin-walled weir, and the zero reading of the thin-walled weir was calibrated before the experiment. The water depth at each point of the main tunnel was measured by the pulsating pressure sensor. The pulsating pressure sensor was connected with the computer through a DJ800 multifunctional monitor to form a data acquisition and processing system. The system was used to measure the fluctuation of the water depth at various points in the tailrace system. To observe the obvious mixed free-surface-pressurized flow in the main tailrace tunnel, the downstream water level was set to 0.807 m. Under this condition, the first half of the tailrace was the pressurized flow, and the second half of the tailrace was the free surface flow. The discharge was calculated to be 0.066279 m 3 /s, according to the Rehbock weir formula [30]. When the butterfly valves at the inlets of four branch tunnels were suddenly closed, the phenomenon of mixed free-surface-pressurized flow in the main tailrace tunnel could be observed.
The flow rate condition was given at the upstream, and the flow rate of the four adits was assumed to be the same at 0.016570 m 3 /s. When the butterfly valve quickly closed, the flow rate became zero in a short time, and the variation of the flow rate was recorded. Although the downstream was a large pool, there were still small fluctuations in the water level, which were recorded and input into the calculation. Figure 3a-h shows the comparison curves of the water depth in the experiment and calculations. The results show that the water levels of the time domain agreed well between the experiment and calculations. The frequency results of the experiment and calculations also coincided well, and the dominant frequencies were both 0.1 Hz. It can be seen that the calculation results and the experimental results were consistent, and the change period of the water depth was nearly the same. The water level at point 1 exceeded the tunnel's top, so the flow in the transient process was pressurized. The water level at point 4 was below the tunnel top, so the flow in the transient process was a free surface flow. In the transient process, the water level at the second and third points would be below or exceed the tunnel's top, so the mixed free-surface-pressurized flow would occur. This is consistent with the experimental results that the separation interface between pressurized flow and free surface flow appeared near point 3 under the initial water level of 0.807 m. Figure 3e shows that when the water surface reached the tunnel's top at point 3, the free surface flow became the pressurized flow. The curve of the experimental results shows a large pressure fluctuation at this time, which agrees well with the calculated results. Due to the pressure fluctuation at measuring point 3, the pressure fluctuation also appeared at points 1 and 2. Therefore, the calculation model could accurately predict the interface of the free surface flow and the pressurized flow.
were suddenly closed, the phenomenon of mixed free-surface-pressurized flow in the main tailrace tunnel could be observed.
The flow rate condition was given at the upstream, and the flow rate of the four adits was assumed to be the same at 0.016570 m 3 /s. When the butterfly valve quickly closed, the flow rate became zero in a short time, and the variation of the flow rate was recorded. Although the downstream was a large pool, there were still small fluctuations in the water level, which were recorded and input into the calculation. Figure 3a-h shows the comparison curves of the water depth in the experiment and calculations. The results show that the water levels of the time domain agreed well between the experiment and calculations. The frequency results of the experiment and calculations also coincided well, and the dominant frequencies were both 0.1 Hz. It can be seen that the calculation results and the experimental results were consistent, and the change period of the water depth was nearly the same. The water level at point 1 exceeded the tunnel's top, so the flow in the transient process was pressurized. The water level at point 4 was below the tunnel top, so the flow in the transient process was a free surface flow. In the transient process, the water level at the second and third points would be below or exceed the tunnel's top, so the mixed freesurface-pressurized flow would occur. This is consistent with the experimental results that the separation interface between pressurized flow and free surface flow appeared near point 3 under the initial water level of 0.807 m.    Figure 3e shows that when the water surface reached the tunnel's top at point 3, the free surface flow became the pressurized flow. The curve of the experimental results shows a large pressure fluctuation at this time, which agrees well with the calculated results. Due to the pressure fluctuation at measuring point 3, the pressure fluctuation also appeared at points 1 and 2. Therefore, the calculation model could accurately predict the interface of the free surface flow and the pressurized flow.

Analysis of Influencing Factors for the Mixed Free-Surface-Pressurized Flow
Many factors can affect the occurrence and development of mixed free-surface-pressurized flow, which are important for the construction design of hydraulic engineering and the operation of hydraulic machinery.

Influence of the Tunnel Relative Roughness
When the mixed free-surface-pressurized flow occurs in the tailrace tunnel, the wave velocity at the interface between the free surface flow and the pressurized flow will change suddenly and cause great pressure fluctuation. According to Equations (3) and (4), the surface relative roughness of the tunnel wall will affect the amplitude of the pressure fluc-

Analysis of Influencing Factors for the Mixed Free-Surface-Pressurized Flow
Many factors can affect the occurrence and development of mixed free-surfacepressurized flow, which are important for the construction design of hydraulic engineering and the operation of hydraulic machinery.

Influence of the Tunnel Relative Roughness
When the mixed free-surface-pressurized flow occurs in the tailrace tunnel, the wave velocity at the interface between the free surface flow and the pressurized flow will change suddenly and cause great pressure fluctuation. According to Equations (3) and (4), the surface relative roughness of the tunnel wall will affect the amplitude of the pressure fluctuation.
Tunnels with the relative roughness of 0.010/0.012/0.014/0.016/0.018 were selected to calculate the influence of the tunnel surface relative roughness on the mixed free-surfacepressurized flow. The maximum pressure in the tunnel with different relative roughnesses was calculated and shown in Table 3, and the results show that the maximum pressure in the tunnel decreased with the increase of the tunnel's relative roughness. This means that increasing the relative roughness of the tunnel surface could suppress the maximum pressure in the tunnel. The reason for this is that the increasing relative roughness will increase the hydraulic loss, which makes the maximum pressure lower. When the surface relative roughness increased from 0.010 to 0.018, the maximum pressure of the tunnel could decrease by 4.33%. However, increasing the relative roughness of the tunnel surface would reduce the power generation efficiency of the hydropower station. Therefore, a relative roughness of 0.014 is recommended upon consideration of the balance of economy and construction.

Influence of Vent Position
The vent can greatly discharge pressure when the mixed free-surface-pressurized flow appears. To study the effect of vent holes at different positions on the maximum pressure in the tunnel, the water level of the tailrace tunnel under the mixed free-surface-pressurized flow was calculated, as is shown in Figure 4. The diameter of the vent hole was set as D = 10 m in the calculation. In Figure 4, the red line represents the water level in the tunnel when the mixed free-surface-pressurized flow occurred, the blue line represents the tunnel's top, and the black line represents the tunnel's bottom.
Tunnels with the relative roughness of 0.010/0.012/0.014/0.016/0.018 were selected to calculate the influence of the tunnel surface relative roughness on the mixed free-surfacepressurized flow. The maximum pressure in the tunnel with different relative roughnesses was calculated and shown in Table 3, and the results show that the maximum pressure in the tunnel decreased with the increase of the tunnel's relative roughness. This means that increasing the relative roughness of the tunnel surface could suppress the maximum pressure in the tunnel. The reason for this is that the increasing relative roughness will increase the hydraulic loss, which makes the maximum pressure lower. When the surface relative roughness increased from 0.010 to 0.018, the maximum pressure of the tunnel could decrease by 4.33%. However, increasing the relative roughness of the tunnel surface would reduce the power generation efficiency of the hydropower station. Therefore, a relative roughness of 0.014 is recommended upon consideration of the balance of economy and construction.

Influence of Vent Position
The vent can greatly discharge pressure when the mixed free-surface-pressurized flow appears. To study the effect of vent holes at different positions on the maximum pressure in the tunnel, the water level of the tailrace tunnel under the mixed free-surfacepressurized flow was calculated, as is shown in Figure 4. The diameter of the vent hole was set as D = 10 m in the calculation. In Figure 4, the red line represents the water level in the tunnel when the mixed free-surface-pressurized flow occurred, the blue line represents the tunnel's top, and the black line represents the tunnel's bottom.    On the other hand, the vents in the tunnel could be though nels, and the wave velocity in the vent could be calculated to be a velocity formula. Therefore, the maximum values of the pressu greatly reduced by setting proper vent positions. This method ca the surge shafts to guarantee the stable operation of the hydropo

Influence of the Vent Diameter
According to the formula of the wave velocity for the open ch decreased with the increase of the vent diameter. It can be infer the ventilation holes affected the maximum pressure inside the different diameters were set at 81.25%L, and the maximum pressu surface-pressurized flow occurred in the tunnel was calculated, a Three vent diameters of 5 m, 10 m, and 15 m were set to inv the vent diameter on the suppression effect on the maximum pr results show that the maximum pressure in the tunnel decreased vent diameter. When the diameter of the vent hole increased fro the maximum values of the pressure in the tunnel were 68.541 m 66.026 mH2O, with a decrease of 3.53% and 3.67%, respectively. T ing the diameter had a slight effect on the maximum pressure su It can be seen that the vent at 81.25%L had the best suppression effect on the maximum pressure in the tunnel, and the corresponding maximum pressure was 66.124 mH 2 O with a decrease of 56.72%. When the free surface flow turned into the pressurized flow, the wave velocity increased instantaneously, which caused a rapid increase of the pressure fluctuation in the tunnel. On the one hand, according to the design manual of the hydropower station, the wave velocity of the tailwater tunnel of the prototype power station was 1160 m/s under the pressurized flow. On the other hand, the vents in the tunnel could be thought of as small open channels, and the wave velocity in the vent could be calculated to be about 19 m/s by the wave velocity formula. Therefore, the maximum values of the pressure in tunnel 4 could be greatly reduced by setting proper vent positions. This method can be an alternative with the surge shafts to guarantee the stable operation of the hydropower station.

Influence of the Vent Diameter
According to the formula of the wave velocity for the open channel, the wave velocity decreased with the increase of the vent diameter. It can be inferred that the diameter of the ventilation holes affected the maximum pressure inside the tunnel. Vent holes with different diameters were set at 81.25%L, and the maximum pressure when the mixed free-surface-pressurized flow occurred in the tunnel was calculated, as is shown in Figure 5.

Influence of the Vent Number
A vent can effectively reduce the maximum pressure in the tunnel, so the influence of the vent number should be further investigated. Because the vent position at 93.75%L could effectively suppress the maximum pressure at the upstream of the tunnel, the first vent position was set at 93.75%L. Then, the second or third vent was set at different positions, and five cases with different vent numbers and corresponding positions were determined, as is shown in Table 4.

Influence of the Vent Number
A vent can effectively reduce the maximum pressure in the tunnel, so the influence of the vent number should be further investigated. Because the vent position at 93.75%L could effectively suppress the maximum pressure at the upstream of the tunnel, the first vent position was set at 93.75%L. Then, the second or third vent was set at different positions, and five cases with different vent numbers and corresponding positions were determined, as is shown in Table 4.  Figure 6 shows the maximum pressure of the mixed free-surface-pressurized flow in the tunnel with vents. The results show that the maximum pressure in the tunnel was Processes 2021, 9, 320 13 of 15 56.006 mH 2 O for vent positions at 93.75%L and 56.25%L, which was reduced by 15.30% in comparison with the single-vent case. For the three vent positions at 93.75%L, 68.75%L, and 43.75%L, the maximum pressure in the tunnel was 55.398 mH 2 O, which was reduced by 16.22% in comparison with the single-vent case. It can be concluded that increasing the vent number can strengthen the suppression effect on the maximum pressure; however, more vents may influence the safety of the tunnel structure.
Processes 2021, 9, x FOR PEER REVIEW 13 of 15 Figure 6 shows the maximum pressure of the mixed free-surface-pressurized flow in the tunnel with vents. The results show that the maximum pressure in the tunnel was 56.006 mH2O for vent positions at 93.75%L and 56.25%L, which was reduced by 15.30% in comparison with the single-vent case. For the three vent positions at 93.75%L, 68.75%L, and 43.75%L, the maximum pressure in the tunnel was 55.398 mH2O, which was reduced by 16.22% in comparison with the single-vent case. It can be concluded that increasing the vent number can strengthen the suppression effect on the maximum pressure; however, more vents may influence the safety of the tunnel structure.