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Effect of throttling orifice head loss on dynamic behavior of hydro-turbine governing system with air cushion surge chamber

This paper aims to study the effect of throttling orifice head loss on the dynamic behavior of the hydro-turbine governing system with air cushion surge chamber. Firstly, a nonlinear mathematical model of hydro-turbine governing system considering the nonlinear head loss of throttling orifice is established. Then, the nonlinear dynamic behavior of the hydro-turbine governing system is investigated by using the Hopf bifurcation theory and direct numerical integration. The stability and dynamic response of the system are detailed described. Finally, the effect mechanism of throttling orifice head loss on the dynamic behavior of system is revealed based on the comparison between the cases with nonlinear head loss and no head loss. The results indicate that the mathematical model considering the nonlinear head loss of throttling orifice for the hydro-turbine governing system is a fifth-order nonlinear state equation. The system goes through Hopf bifurcation at the bifurcation points and the type of Hopf bifurcation is supercritical. The domain at the bottom of the bifurcation line is the stable domain. The throttling orifice cannot affect the system stability through its head loss. The throttling orifice can affect the system dynamic response through its head loss. The throttling orifice head loss is favorable for the damping of the water level oscillation in the surge chamber.


Introduction
For the hydropower station, a chamber with its top closed and with compressed air between the water surface and the top of the chamber is called an air cushion surge chamber [1,2]. Air cushion surge chamber is an important pressure reduction facility in hydropower station. Compared with open surge tank, air cushion surge chamber is more efficient to decrease the water hammer pressure and improve the operating condition of hydro-turbine unit [3].
The throttling orifice is frequently used in air cushion surge chamber to reduce the surge amplitudes. The presence of the throttling orifice adds a nonlinear term of head loss to the hydro-turbine governing system, and its effect becomes predominant for dynamic behavior of the system. Because of the throttling orifice head loss, the modeling and characteristic analysis of the hydro-turbine governing system become more complicated. About the dynamic behavior of hydro-turbine governing system

Mathematical formulation
The pipeline and power generation system of hydropower station with air cushion surge chamber is shown in Figure 1. The basic equations for all subsystems of the hydro-turbine governing system shown in Figure 1, i.e. headrace tunnel, air cushion surge chamber, penstock, hydro-turbine, generator and governor, are presented as follows. Note that the definition and explanation for the notations and variables in this section are presented in Appendix A.
Dynamic equations of headrace tunnel and penstock: When the throttling orifice head loss of air cushion surge chamber is not considered, we have the dynamic equations of headrace tunnel and penstock as follows [2,8]: The throttling orifice head loss of air cushion surge chamber is denoted as T h . If T h is considered, the equation (1) and equation (2) can be rewritten as The head loss coefficient of throttling orifice is denoted as T  . Then we have . Then equation (5) can be transferred to the following form: Substitution of equation (6) into equation (3) and equation (4) gives the dynamic equations of headrace tunnel and penstock considering the throttling orifice head loss of air cushion surge chamber as follows: Continuity equation of air cushion surge chamber [2,8]: For the hydro-turbine governing system considering the throttling orifice head loss of air cushion surge chamber, the mathematical model is composed by equation (7) -equation (13). By integrating equation (7) -equation (13), we obtain the following fifth-order state equation for the hydro-turbine governing system.
Because of the existence of the nonlinear term 2 ( ) P H q q  , the hydro-turbine governing system equation (14) is a nonlinear dynamic system. The operating condition studied in this paper is that the unit goes through a load variation during normal output operation. g m is regarded as the load variation, i.e. the external disturbance. Therefore, the input signal for the nonlinear hydro-turbine governing system is g m . The state variables in equation (14), i.e. H q , P q , A h , x and y , are the output signals for the nonlinear hydro-turbine governing system. Except for the state variables and g m , the parameters in equation (14) are the characteristic parameters of the system.

Nonlinear dynamic behavior analysis
In this section, the nonlinear dynamic behavior of the hydro-turbine governing system with air cushion surge chamber is investigated by using the Hopf bifurcation theory and direct numerical integration. The stability and dynamic response of the system are detailed described and analyzed.
As one of the nonlinear mathematical theories, the Hopf bifurcation theory is adopted to study the nonlinear dynamic behavior of the hydro-turbine governing system equation (14). Hopf bifurcation theory is widely used in the area of the regulation and control of hydropower system. The principles and application procedures are detailed introduced in [11][12][13]. In this paper, the Hopf bifurcation theory is directly applied without multifarious presentations of the basic methodology.
For the hydro-turbine governing system, equation (14) can be transferred into the form of ( , ) is the state vector, and  is the bifurcation parameter. The equilibrium point E x of the system can be determined from ( , ) 0 (ii) The traversal coefficient A hydropower station with air cushion surge chamber is selected as example for the nonlinear dynamic behavior analysis. The basic data of the example are: 0 selected as the bifurcation parameter, and the corresponding bifurcation point is denoted as     Figure 3. Therefore, the system (14) goes through Hopf bifurcation at the bifurcation points and the type of Hopf bifurcation is supercritical. As a result, for a p K , the system is stable when The dynamic response of the system stabilizes at an equilibrium point. Then we can get that the state points at the bottom of the bifurcation line can make the system stable, which indicates that the domain at the bottom of the bifurcation line is the stable domain. Accordingly, the domain at the top of the bifurcation line is the unstable domain. For the state points in the unstable domain, the dynamic response of the system stabilizes at a limit cycle. The graphical representation of the stable domain and unstable domain is shown in Figure 2. Based on the stable domain, we can determine the dynamic states of the system. For different states points on the p i K K  plane, the dynamic behaviors are different. From the dynamic response processes of the system, we can evaluate the dynamic characteristics of the system. The dynamic response processes of the system can be obtained from equation (14) by using the direct numerical integration. In this section, three state points in Figure 2, i.e. 1 S , 2 S and 3 S , are selected for numerical simulation. 1 S locates in the stable domain and its coordinate value is (2.564, 3.5). 2 S locates on the bifurcation line and its coordinate value is (2.564, 4). 3 S locates in the unstable domain and its coordinate value is (2.564, 4.2). The function ODE45 in MATLAB [14] is adopted to solve equation (14). The results for the dynamic response processes of x and phase space trajectories of ( , , ) P q x y are shown in Figure 4.  Figure 4 shows that, for the state point 1 S , the phase space trajectory of ( , , ) P q x y stabilizes at an equilibrium point after several rounds of movement. Accordingly, the dynamic response process of x is damped and enters a new steady value after several periods of oscillation. For the state point 2 S , the phase space trajectory of ( , , ) P q x y stabilizes at a limit cycle. The dynamic response process of x is a persistent oscillation. For the state point 3 S , the phase space trajectory of ( , , ) P q x y firstly shows a emanative movement and then enters a stable limit cycle. Accordingly, the dynamic response process of x is emanative at the initial stage.

Effect of throttling orifice head loss on dynamic behavior
In this section, the effect mechanism of throttling orifice head loss on the dynamic behavior of system is revealed based on the comparison between the cases with nonlinear head loss and no head loss. Another hydropower station with air cushion surge chamber is selected as example for the analysis. The basic data are as follows: 0 Q =228.6m 3  , a T =9.46s. The other data and operating condition are the same with those of the example in Section 3.
The throttling orifice head loss can be evaluated by T  . Therefore, T  is selected as the control parameter for the effect analysis on dynamic behavior. Firstly, for the selected hydropower station with air cushion surge chamber under   (a) Phase space trajectory of (q P ,x,y) (b) Dynamic response process of h A Figure 6. Effect of throttling orifice head loss α T (s 2 /m 5 ) on dynamic response of the system. Figure 5 shows that the stable domain keeps the same when T  changes, which indicates that the throttling orifice head loss has no effect on the stability of the system. The mechanism is as follows: The stable domain is a static and steady concept. When the system enters a new steady state after transient process, the discharge through throttling orifice is zero. As a result, the throttling orifice head loss is also zero. Hence, the throttling orifice cannot affect the system stability through its head loss. The stable domain of the system is the same under different T  . Figure 6 shows that the phase space trajectory of ( , , ) P q x y and the dynamic response process of A h change obviously when T  changes, which indicates that the throttling orifice head loss has a significant effect on the dynamic response of the system. The result is different with that in Figure 5, and the reason is as follows. The dynamic response is a dynamic and unsteady concept. Under the unsteady state, the discharge through throttling orifice is not zero. As a result, the throttling orifice head loss is also not zero. Hence, the throttling orifice can affect the system dynamic response through its head loss. The dynamic response of the system is the different under different T  . From Figure 6(b) we can get that, with the increase of T  , the amplitude of A h decreases and the attenuation rate of A h increases. Based on the definition of A h we can know that the oscillation of A h is consistent with the water level oscillation in the surge chamber. A h can be adopted to evaluate the dynamic process characteristics of the water level oscillation in the surge chamber. Therefore, the following conclusion can be obtained: The throttling orifice head loss is favorable for the damping of the water level oscillation in the surge chamber.

Summary and conclusions
For the hydro-turbine governing system with air cushion surge chamber, the nonlinear mathematical model considering the nonlinear head loss of throttling orifice is established. The nonlinear dynamic behavior of the hydro-turbine governing system is investigated by using the Hopf bifurcation theory and direct numerical integration. The stability and dynamic response of the system are detailed described and analyzed. The effect mechanism of throttling orifice head loss on the dynamic behavior of system is revealed based on the comparison between the cases with nonlinear head loss and no head loss. Several conclusions can be drawn from this study: (1) The Hopf bifurcation theory provides a practical approach for analyzing the nonlinear dynamic characteristics of hydro-turbine governing system with air cushion surge chamber. Based on the Hopf bifurcation analysis, the stability and dynamic response of the system can be detailed described.
(2) The mathematical model considering the nonlinear head loss of throttling orifice for the hydro-turbine governing system is a fifth-order nonlinear state equation. The system goes through Hopf bifurcation at the bifurcation points and the type of Hopf bifurcation is supercritical. The domain at the bottom of the bifurcation line is the stable domain. For the state points in the stable domain and unstable domain, the dynamic response of the system stabilizes at an equilibrium point and a limit (3) The throttling orifice cannot affect the system stability through its head loss. The stable domain of the system is the same under different T  . The throttling orifice can affect the system dynamic response through its head loss. The dynamic response of the system is the different under different T  . The throttling orifice head loss is favorable for the damping of the water level oscillation in the surge chamber.

Acknowledgement
This work was supported by the J. Gust. Richert stiftelse in Sweden (Project No. 2017-00317).

Appendix
(2) Expressions of coefficients  x y x y x y q x y q y x x y q x q y x y x y x y x y x q q x y q y x x y q x q y y q x y .