Critical stable sectional area of downstream surge tank of hydropower plant with sloping ceiling tailrace tunnel

This paper investigates the critical stable sectional area (CSSA) of downstream surge tank (DST) of hydropower plant with sloping ceiling tailrace tunnel (SCTT). Firstly, two types of nonlinear mathematical model of hydropower plant with SCTT during transient process are established. Then, the criterion of operation stability of hydropower plant is obtained based on nonlinear stability theory. Formulae for CSSAs of DST under two types of mathematical model are derived. Finally, the similarities, differences, and relationships of different CSSAs are clarified. The effect mechanisms of the assumption of steady power output of hydro‐turbine and the SCTT on CSSA of DST are revealed. The recommended use conditions of different CSSAs are explained. The results indicate that, under the assumption of steady power output of hydro‐turbine, the formula for CSSA of DST of hydropower plant with SCTT is analytical and has a simple expression. Under the complete model, the formula for CSSA of DST of hydropower plant with SCTT is not analytical and has a complicated expression. The former formula is suggested to be given the priority of use because of good precision and concise expression. SCTT can improve the stability of hydropower plant and decrease the CSSA of DST.


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GUO and ZHU the variable load operation are required. 4,5 The operation stability of hydropower plant faces severe challenges. The analysis and control of the operation stability of hydropower plant are an important topic.
Hydropower plant is usually composed of hydro-turbine unit and pipeline.. 6 The tailrace system is an important part of pipeline system and usually determines the type of hydropower plant. Sloping ceiling tailrace tunnel (SCTT) is a novel tailrace system and is widely applied in hydropower plants. 7,8 During transient process, pressurized flow and free surface flow coexist in the SCTT. 9,10 The SCTT can decrease the pressure rise of water flow during transient process by using its special body type. However, there is a limitation of length of tailrace system for the application of SCTT. If the tailrace system length is greater than 500 m, the pressure rise of water flow during transient process cannot be effectively restricted. 11 For that situation, a downstream surge tank (DST) is necessary to be set on the SCTT. 12,13 The combined operation of SCTT and DST is an effective measure to regulate and decrease the pressure rise of water flow during transient process.
The stability of hydropower plant is affected by every part of the pipeline and hydro-turbine unit. 14,15 The variation of water head could change the stability of hydropower plant, and that change is caused by the turbine nonlinearity. 16 The effect of turbine nonlinearity on the stability of hydropower plant is closely related to control modes of governor. The controller of governor has an obvious effect on the stability of hydropower plant. A reasonable controller of governor can improve the stability of hydropower plant significantly. 17 The key for the design of controller of governor is the optimization of governor parameters. By using reasonable optimization algorithm, the optimal governor parameters can be determined to realize a stable hydropower plant. 18 For the grid-connected hydropower plant, the stability of hydropower plant is influenced by the power grid. The hydropower plant and power grid have coupling effect on the stability of the whole system. 19 When the DST is introduced into hydropower plant, the water level in DST oscillates periodically during transient process. 20 The water level oscillation (WLO) in DST exists the problem of stability. The stability of WLO in DST reflects the operation stability of hydropower plant. 21,22 If the WLO in DST is not stable, the hydropower plant would not be stable. The stability of WLO in DST is mainly affected by the sectional area of DST. 23 The sectional area of DST that makes the hydropower plant in the critical stable state is the critical stable sectional area (CSSA) of DST. For the design of hydropower plant with DST, the determination of CSSA of DST is the primary issue.
For the hydropower plant with pressurized tailrace tunnel (PTT), the CSSA of DST is easily to be determined. The computational method and formula have been stated in many textbooks. 12,24 and national specifications. 25,26 But the body type of SCTT is totally different with that of PTT. The flow pattern in SCTT is also different with that in PTT. The combined operation of SCTT and DST introduces the coupling effect of free surface-pressurized flow in SCTT and WLO in DST. Under the coupling effect of the free surfacepressurized flow in SCTT, the phenomena and principle of WLO in DST are obviously different with those of hydropower plant with PTT. Therefore, the formula for CSSA of DST of hydropower plant with PTT cannot be applied to determine the CSSA of DST of hydropower plant with SCTT. The special formula for CSSA of DST of hydropower plant with SCTT is essential.
The research on the CSSA of DST of hydropower plant with SCTT is an important topic. However, there are no valuable researches and achievements on that topic. Until now, the computational method and formula for the CSSA of DST of hydropower plant with SCTT have not been proposed. Related researches mainly focus on the dynamic performance of hydropower plant with SCTT and the coupling effect between SCTT and surge tank. The representative literatures are stated as follows. In Ref.,27 a dynamic equation of penstock with SCTT is proposed and the stability of hydropower plant with SCTT is studied by Hopf bifurcation theory. The stability working principle of SCTT is revealed. In Ref.,28,29 the nonlinear control methods of dynamic performance of hydropower plant with SCTT are investigated. Both the Hopf bifurcation control and differential geometry theory are adopted, and the corresponding control strategies are designed. The designed control strategies prove to be better than the proportional-integral-derivative (PID) control strategy. In Ref.,30 the operation stability of hydropower plant with SCTT is analyzed by the method of numerical simulation. The effects of governor parameters and ceiling slope of SCTT are studied in detail, and the recommended values are given. In Ref.,31 the operation stability of hydropower plant with SCTT under small load disturbance is analyzed by using numerical simulation method. In that method, the pipeline, hydro-turbine unit, and governor are considered and calculated by a joint algorithm. In 32 the stability of hydropower plant with SCTT is investigated using the generalized Hamiltonian theory. The change law of energy of hydropower plant is clarified to explain the nonlinear dynamic behavior of hydropower plant with SCTT. In 33 a special operating condition, that is, primary frequency regulation, of hydropower plant with SCTT is studied by numerical simulation and field measurement. The governor parameters are optimized to improve the dynamic performance. In 34,35 the nonlinear model of hydropower plant with upstream surge tank and SCTT is established. The stability and dynamic performance of hydropower plant are analyzed. In 36 the coupling effect of DST and SCTT is studied. The optimization methods of the design parameters of DST and SCTT are proposed to improve the stability of hydropower plant during transient process.
This paper aims to study the CSSA of DST of hydropower plant with SCTT. The novelty of this paper is to: (1) establish the reasonable mathematical model of hydropower plant with SCTT during transient process for the derivation of CSSA of DST, (2) derive the formula for the CSSA of DST of hydropower plant with SCTT, (3) reveal the similarities, differences, and relationships of different CSSAs of DST, and (4) put forward the recommended use conditions of different CSSAs of DST.
In Section 2, two types of nonlinear mathematical model of hydropower plant with SCTT during transient process, that is, model considering the assumption of steady power output of hydro-turbine and complete model, are established for the derivation of CSSA of DST. State equations of hydropower plant with SCTT under two types of mathematical model are obtained. In Section 3, the criterion of operation stability of hydropower plant is obtained based on model of hydropower plant and stability analysis theory. The algebraic criterion of stability corresponding to the CSSA of DST is determined. Formulae for the CSSAs of DST under two types of mathematical model are derived according to the algebraic criterion of stability. In Section 4, the correctness of the formulae for CSSAs of DST is verified by numerical verification. The similarities, differences, and relationships of different CSSAs of DST are clarified by comparative analysis. The effect mechanisms of the assumption of steady power output of hydro-turbine and the SCTT on the CSSA of DST are revealed. The recommended use conditions of different CSSAs of DST are explained. In Section 5, the conclusions are given.

| MATHEMATICAL MODEL
The layout of pipeline system of hydropower plant with SCTT and the stability of WLO in DST during transient process are shown in Figure 1.

| Basic equations
The pipeline system of hydropower plant with SCTT in Figure 1 contains penstock, DST, SCTT, hydro-turbine, generator, and governor. For the study of CSSA of surge tank, two types of mathematical model of hydropower plant during transient process are usually adopted. The first type of mathematical model is based on the assumption of steady output of hydro-turbine. In that assumption, the governor is extremely sensitive and the output of hydro-turbine unit can be kept unchanged. 12 The dynamic behavior of penstock, hydro-turbine, generator, and governor can be described by one equation, that is, equation of steady output of hydro-turbine. The second type of mathematical model is not based on the assumption of steady output of hydro-turbine. The power output of hydro-turbine unit changes during transient process. The dynamic behaviors of penstock, hydro-turbine, generator, and governor are described by their own equations. So, the second type of mathematical model is a kind of complete model of hydropower plant.
In this section, both the above two types of mathematical model of hydropower plant with SCTT during transient process are established for the derivation of CSSA of DST. Then, the characteristics and effects of mathematical models on the CSSA of DST can be analyzed and compared. The similarities, differences, and relationships between different CSSAs of DST can be revealed. The first type of mathematical model is denoted as Model A, and the second type of mathematical model is denoted as Model B. (1)  27 : For the SCTT, it is assumed that there is no interspace between the water and the ceiling of tailrace tunnel when the interface of free surface-pressurized flow moves. That assumption is valid when the ceiling slope gradient is less than 5%. The change of flow inertia of SCTT and the water level fluctuation in free surface flow section of SCTT are considered. The dynamic equation of SCTT is shown in Equation (3). Equation It should be noted that the definitions of variables and parameters are shown in Nomenclature. Moreover, q y = and m g = M g0 . The subscript of 0 refers to the initial value of variable.

| State equation of hydropower plant
For the hydropower plant, the whole pipeline system enters transient process under external disturbance. The water level in DST oscillates periodically. The stability of WLO in DST reflects the operation stability of hydropower plant. The stability of hydropower plant is mainly determined by the WLO in DST. When the WLO in DST is unstable, the hydropower plant is unstable. The stability of WLO in DST is mainly affected by the sectional area of DST. The sectional area of DST that makes the hydropower plant in the critical stable state is the CSSA of DST. For the convenience of the stability analysis of hydropower plant, the state equation is necessary to be obtained firstly. For Model A, the basic equations are Equations (1)- (3). According to the feature of basic equations and the research purpose of present study, the q y and z F are chosen as the state variables. Then, Equations (1)-(3) are converted to the following state equation.
Equation (9) is the state equation of Model A of hydropower plant. Model A is a second-order nonlinear system. When we let x = q y , z F T , Equation (9) can be rewritten as the standard form, that is, ̇x = M 1 x.
For Model B, the basic equations are Equations (2)-(8). According to the feature of basic equations and the research purpose of present study, the q y , z F , q t , x and y are chosen as the state variables. Then Equations (2)-(8) are converted to the following state equation.
Equation (10) is the state equation of Model B of hydropower plant. Model B is a fifth-order nonlinear system. When we let u = q y , z F , q t , x, y T , Equation (10) can be rewritten as the standard form, that is, u = M 2 u.
The stability of WLO in DST reflects the operation stability of hydropower plant. The sectional area of DST that makes the hydropower plant in the critical stable state is the CSSA of DST. So, the stability analysis of hydropower plant is the key for the derivation of CSSA of DST.
In this section, the operation stability of hydropower plant is firstly analyzed. Based on model of hydropower plant and stability analysis theory, the criterion of stability can be obtained. Then, aiming at the critical stable state of hydropower plant, the algebraic criterion of stability corresponding to the CSSA of DST can be obtained. Finally, according to the algebraic criterion of stability, the CSSA of DST can be derived. Moreover, the Lyapunov's first method [40][41][42][43][44][45] is used for the stability analysis of nonlinear dynamic systems in this paper.
In the following paragraphs, the CSSAs of DST under Model A and Model B are derived, respectively. At the origin of coordinates, the Jacobian matrix of

For the state equation of hydropower plant under Model
Equation (12) is a quadratic equation. Based on the Lyapunov stability theory, the criterion of stability of the dynamic system expressed by Equation (12) is that b 1 > 0 and b 2 > 0 are met simultaneously. In the following paragraphs, b 1 > 0 and b 2 > 0 are analyzed to derive the CSSA of DST under Model A, respectively. gives.
Equation (13) is an inequality with respect to F. When Equation (13) Equation (14) is an inequality without F. For the actual hydropower plants, Equation (14) can usually be met. So, for b 2 > 0, F can be taken any positive value.
Equation (13) is met for partial value of F, while Equation (14) can usually be met for any positive value of F. Based on the above analysis about b 1 > 0 and b 2 > 0, Equation (13)  where F th−A represents the CSSA of DST under Model A. Equation (16) indicates that the equilibrium point of Model B is not at the origin of coordinates. The stability analysis of Lyapunov's first method is based on the coordinate values of state equation at the origin of coordinates. Therefore, the coordinate transformation is needed. Based on the expression of Equations (10) and (16), the following coordinate transformation is adopted.
Then Equation (10) Equation (18) is also a fifth-order nonlinear dynamic system. When we let w = (A, B, C, D, E) T , Equation (18)    , a 3 = a 33 a 45 a 54 − a 33 a 44 a 55 + a 34 a 43 a 55 − a 34 a 45 a 53 − a 35 a 43 a 54 + a 35 a 44 a 33 a 45 a 54 − a 33 a 44 a 55 + a 34 a 43 a 55 − a 34 a 45 a 53 − a 35 a 43 a 54 + a 35 a 44 a 53)

F(bc + T wy)
. Equation (20) is a fifth-order equation. Based on the Lyapunov stability theory, the criteria of stability of the dynamic system expressed by Equation (20)  The following expressions are defined for the convenience of derivation. It should be noted that both X i (i = 0, 1, 2, …, 14) and Y i (i = 1, 2, 3, …, 6) do not contain F. bc + T wy , We have a 0 = 1. So a 0 > 0 can always be met.
(b) a 1 > 0 and a 5 > 0 By substituting the expressions of a 1 and a 5 into a 1 > 0 and a 5 > 0 yields X 0 < 0 and X 13 + X 14 > 0, respectively. X 0 , X 13 , and X 14 do not contain F. So, the CSSA of DST under Model B does not depend on a 1 > 0 or a 5 > 0.
By substituting the expressions of a 0 , a 1 , a 2 , and a 3 into  a 0 , a 1 , a 2 , a 3 , a 4 , and a 5 into Δ 4 > 0 yields.
, When Equation (26) is taken as equality, we can obtain three roots, which are denoted as F 1 , F 2 , and F 3 , respectively. The above analysis indicates that the inequality intervals of F solved from the criteria of stability of hydropower plant must be left open intervals and are also continuous. Based on the feature of the solution interval of simple cubic inequality, the inequality interval of F solved from Equation (26) should be.

AND COMPARATIVE ANALYSIS
In Section 3, the formulae for the CSSAs of DST under Model A and Model B are derived according to the stability analysis of hydropower plant. The two CSSAs of DST are based on different assumptions of mathematical model. In order to verify the correctness of the obtained formulae for CSSAs of DST, the numerical verification is carried out firstly. Then, the comparative analysis is conducted to clarify the similarities, differences, and relationships of different CSSAs of DST. Moreover, the effect mechanisms of the assumption of steady power output of hydro-turbine and the SCTT on the CSSA of DST are also revealed from the comparative analysis. The recommended use conditions of different CSSAs of DST are explained. Equations (15) and (29) are the formulae for the CSSAs of DST under Model A and Model B, respectively. Based on the above basic data of hydropower plant, we can calculate the CSSAs of DST under Model A and Model B by using Equations (15) and (29). The results are shown in Table 1. In order to give a comprehensive analysis of the sectional area of DST on the stability of hydropower plant, more values of the sectional area of DST are also selected for the numerical simulation in the following paragraphs.

| Numerical verification
By substituting the CSSAs and other values of sectional area of DST under Model A into Equation (9), we can determine the dynamic processes of WLO in DST. The external disturbance is selected as q y0 = −0.10 and the solution of Equation (9) is realized by ode45 in MATLAB. 46 The results of dynamic processes of WLO in DST under Model A are shown in Figure 2(A). By using the same method, the dynamic processes of WLO in DST under Model B can be determined from Equation (18) and the results are shown in Figure 2(B). Figure 2 shows that: (

| Comparative analysis
Equations (15) and (29) are the formulae for the CSSAs of DST under Model A and Model B of hydropower plant with SCTT, respectively. Equation (15) is obtained based on the assumption of steady power output of hydro-turbine, and Equation (29) is obtained from the complete mathematical model of hydropower plant with SCTT. For hydropower plant with PTT, the Thoma CSSA of surge tank is the most widely used formula. For the hydropower plant with PTT, the formula for the Thoma CSSA of DST is 12 Equation (30) is also obtained based on the assumption of steady power output of hydro-turbine. In this section, the similarities, differences, and relationships of Equations (15), (29), and (30) are studied. The effect mechanisms of the assumption of steady power output of hydro-turbine and the SCTT on the CSSA of DST are revealed from the comparative analysis. The recommended use conditions of different CSSAs of DST are explained.
The hydropower plant with SCTT in Section 4.1 is taken as an example for the comparative analysis. By substituting the basic data of the hydropower plant into Equation ( into Equation (18), we can determine the dynamic processes of WLO in DST using ode45 in MATLAB. The results for the dynamic processes of WLO in DST are shown in Figure 3. From Equations (15), (29), and (30), Table 2, and Figure 3, we can get that: (1) The formulae of F th−A and F th−T are analytical, and the values of F th−A and F th−T can be easily solved from Equations (15) and (30). In order to make the above results more convincing, the two models under several different operating conditions are compared. Specifically, the operating conditions with different H 0 are selected. In Table 2 and Figure 3, H 0 is taken as 70.70m. In the following analysis, H 0 is taken as 65.00m and 75.00m, respectively. When H 0 is taken as 65.00m or 75.00m, F th−A , F th−B , and F th−T can be determined by using Equations (15), (29), and (30), respectively. The results of F th−A , F th−B , and F th−T when H 0 is taken as 65.00m and 75.00m are shown in Table 3 and Table 4, respectively. By substituting the values of F th−A , F th−B , and F th−T into Equation (18), we can determine the dynamic processes of WLO in DST using ode45 in MATLAB when H 0 is taken as 65.00 m and 75.00 m. The results are shown in Figures 4 and 5.
The results and rules in Table 3, Table 4, Figures 4 and 5 are the same with those in Table 2 and Figure 3. Therefore, the formulae for the CSSAs of DST under Model A and Model B of hydropower plant with SCTT, that is, Equations (15) and (29), are always applicable under different operating conditions. Under different operating conditions, the rules for the similarities, differences, and relationships of different CSSAs of DST are the same.
To sum up, the stability of hydropower plant with SCTT is obviously different with that of hydropower plant with PTT. SCTT can improve the stability of hydropower plant significantly and decrease the CSSA of DST obviously. For the determination of CSSA of DST of hydropower plant with SCTT, F th−T is not applicable. F th−A and F th−B are the correct and reasonable formulae for the determination of CSSA of DST of hydropower plant with SCTT. F th−B has the highest precision and a complicated expression. F th−A has a good precision and a concise expression. Because the expression of F th−B is much more complex than that of F th−A , the calculation time corresponding to F th−B is much longer than that corresponding to F th−A (Table 2). In actual engineering applications, F th−A is suggested to be given the priority of use if both the calculation precision and calculation difficulty are considered. If the calculation difficulty is not unconstrained and the requirement for calculation precision is high, F th−B is suggested to be given the priority of use. The main conclusions are as follows: