Vibration Characteristics of a Hydroelectric Generating System with Different Hydraulic-Mechanical-Electric Parameters in a Sudden Load Increasing Process

Abstract

In a sudden load increasing process (SLIP), the hydroelectric generating system (HGS) experiences a severe vibration response due to the sudden change of the hydraulic-mechanical-electric parameters (HMEPs). The instability of HGS limits the ability of sudden load increase, and its flexibility and reliability are reduced. Thus, in this study, a new transient nonlinear coupling model of HGS is proposed, which couples the hydro-turbine governing system (HTGS) and the hydro-turbine generator shafting system (HGSS) with the hydraulic-mechanical-electric coupling force, rotating speed, flow rate, hydro-turbine torque, electromagnetic torque, and guide vane opening. By using numerical simulation, the influences of different HMEPs on the vibration characteristics of HGS in SLIP are analyzed. The result shows that, compared with stable operating conditions, the vibration amplitude of HGS increases sharply in SLIP. The increase of the sudden load increasing amount, blade exit flow angle, mass eccentricity and excitation current, and the decrease in guide bearing stiffness and average air gap between the stator and rotor cause abnormal vibration of different degrees in the HGS. Hydraulic factors have the greatest influence on the nonlinear dynamic behavior of HGS. The maximum vibration amplitude of HGS in SLIP is increased by 70.46%, compared with that under stable operating conditions. This study provides reasonable reference for the analysis of the nonlinear dynamic behavior of HGS in SLIP under the multiple vibration sources.

1. Introduction

Hydropower is an environmentally friendly renewable energy, which has been widely used all over the world, and is of great significance to sustainable development [1,2,3]. HGS has the benefits of a quick startup and shutdown, flexible scheduling, and low operating costs [4,5,6]. When HGSs regulate the peak and frequency of the power system, it is impossible to avoid a series of transient processes, such as load increase and decrease [7]. Although the duration of the transient process is short, the parameters of the hydro-turbine, governor, and pipe are abnormal due to the sudden load change [8]. Compared with the large power system, the sudden load change has a greater impact on single-machine single-load power systems (SMSLPSs), which aggravates the impact of HMEPs on HGS, and causes severe vibration of the HGS [9,10]. For example, the water hammer superimposition occurred in Tianshengqiao-II HGS in Guangxi, China, due to the poor guide vane opening and closing strategies, which resulted in damage to the hydro-turbine runner and the surge tank [11].

The vibration of HGS is mainly caused by three factors: hydraulic, mechanical, and electric [12,13]. For hydraulic factors, Zhuang et al. proposed a hydraulic unbalance force model, and concluded that hydraulic factors determine the changing trend of HGSS dynamic behaviors [14]. Favrel et al. introduced a methodology to assess the vortex behavior, the pressure fluctuation, and the dynamic strain, and investigated the impact of precessing cavitation vortices on runner blades [15]. For mechanical factors, Al-Hussain et al. established a coupling model of lateral vibration and torsional vibration, and concluded that parallel misalignment is a source of both vibrations [16]. Yong-Wei T et al. established a coupling model of rub-induced force and temperature distribution, and found that the vibration and temperature increase when the rotor is running below its critical speed [17]. For electric factors, Calleecharan et al. derived an unbalanced magnetic pull (UMP) model, and found that the sensitivity of UMP to forcing frequency increases with the decrease in HGSS stiffness [18]. Xu et al. established a finite element rotor model and a UMP model, and concluded that UMP is limited by the saturation effects as the field current increases [19]. The above studies mainly focus on the vibration characteristics of HGS under stable operating conditions, and their application is limited. The transient process, as an important factor, effectively reflects the overall vibration characteristics of HGS.

In transient processes, the study of HGS vibration characteristics inevitably involves the time variant of parameters and the coupling of multi vibration sources due to the complex flow behavior [20,21]. Thus, the vibration characteristics of HGS in transient processes have not yet been further investigated. Zhang et al. established a hydraulic excitation model induced by a draft tube vortex, and analyzed the vibration characteristics of HGS in a sudden load increase of 10% [22]. Wu et al. proposed a coupling structure model of HGSS and the powerhouse, and researched the vibration characteristics of HGSS under mechanical and electromagnetic factors under start-up conditions [23]. The above studies only analyzed single or double vibration sources, but did not fully study the comprehensive influences of HMEPs on HGS vibration characteristics. Guo et al. established an HGSS model by using the dual-disc rotor system, and explored the vibration characteristics of the generator rotor and the hydro-turbine runner during the load rejection [24]. Although the simplified dual-disc rotor system reflects the vibration characteristics of the generator rotor and hydro-turbine runner, the vibration characteristics of other key parts of the HGSS, such as the guide bearing, are not obtained. In addition, the simplification of the model causes calculation errors.

Thus, this study proposes a new transient nonlinear coupling model of HGS, which couples HTGS and HGSS. The model is modularized, based on the real HGS parameters of China. The sub-models include: the water diversion system (WDS) model established by the method of characteristics (MOC), the generator and load model, the PID governor model, the hydro-turbine model and HGSS model established by the Riccati transfer matrix method (RTMM), and the Wilson-θ numerical integration method (WNIM). Through numerical simulation, the influences of different HMEPs on the vibration characteristics of HGS in SLIP under SMSLPS are investigated.

The innovations of this study lie in three main aspects: (1) a new coupling method is proposed, which adds HGSs, a hydro-turbine characteristic curve, and a PID governor to the HGS model; (2) the vibration characteristics of HGSS in SLIP are studied, which have not yet been further investigated; (3) considering the hydraulic-mechanical-electric coupling vibration source, the influences of different HMEPs on the vibration characteristics of HGS in SLIP are studied.

The structure of this paper is as follows: in Section 2, the transient nonlinear coupling model of HGS is established and verified. Section 3 analyzes the influences of different HMEPs on the vibration characteristics of HGSS in SLIP under SMSLPS. Finally, in Section 4, the main results are summarized.

2. Transient Nonlinear Coupling Model of HGS

The HGS is composed of an upstream reservoir, a diversion tunnel, a surge tank, a pipe, an HGSS, a draft tube, and a downstream reservoir. The schematic diagram of the HGS is shown in Figure 1. The transient nonlinear coupling model of the HGS is established through the coupling of a hydro-turbine and the WDS, and the coupling of HTGS and HGSS. The schematic diagram of the HGS coupling model is shown in Figure 2.

2.1. Model of WDS

The WDS is composed of a reservoir, a diversion tunnel, a surge tank, and a pipe. MOC is used to transform the partial differential equation of the unsteady flow into an ordinary differential equation, and then the ordinary differential equation is approximately transformed into a differential equation for hydraulic transient analysis [25,26]. The hydraulic transient motion equation L₁ and the continuous equation L₂ are:

$$\{\begin{array}{l}{L}_1=\frac{\partial H}{\partial X}+\frac{V}{g}\frac{\partial V}{\partial X}+\frac{1}{g}\frac{\partial V}{\partial t}+\frac{f\left|V\right|V}{2\mathrm{g}D}=0\\ L_2=\frac{\partial H}{\partial t}+V\frac{\partial H}{\partial X}+\frac{a_w{}^2}{g}\frac{\partial V}{\partial X}+V\sin \sigma =0\end{array}$$

(1)

where H is the water head; X is the pipe length; V is the flow velocity; g is the gravitational acceleration; t is the time; f is the Darcy–Weisbach friction coefficient; D is the pipe diameter; a_w is the pressure wave velocity; and σ is the angle of the pipe inclined with the horizontal.

Equation (1) is solved by MOC combined with boundary conditions. The schematic diagram of MOC is shown in Figure 3. The flow rate and the water head at point p are calculated through the flow rate and the water head at point d and u, which are shown in Equations (2) and (3). More detailed descriptions of MOC are obtained in [27].

$$\{\begin{array}{l}{C}^{+}:Q_p=C_p-C_a{H}_p\\ C^{-}:Q_p=C_n+C_a{H}_p\end{array}$$

(2)

$$\{\begin{array}{l}{C}_a=\frac{gA}{a_w}\\ C_p=Q_d+\frac{gA}{a_w}{H}_d-\frac{f\Delta t}{2DA}{Q}_d\left|Q_d\right|+\frac{Q_d\Delta tg}{a_w{}^2}\sin \sigma \\ C_n=Q_u-\frac{gA}{a_w}{H}_u-\frac{f\Delta t}{2DA}{Q}_u\left|Q_u\right|-\frac{Q_u\Delta tg}{a_w{}^2}\sin \sigma \end{array}$$

(3)

where Q_p, Q_d, Q_u, H_p, H_d, and H_u are the flow rate and the water head at point p, d, and u, respectively; C_a, C_p, and C_n are intermediate variables; A is the cross-sectional area of the pipe; Δt is the time step.

2.2. The Boundary Conditions of WDS

When the WDS model established in 2.1 is used to calculate the flow rate and the water head at each point of the pipe, the corresponding boundary conditions are required. The schematic diagram of boundary conditions for each node of the WDS is shown in Figure 4.

(1)

Upstream and downstream reservoirs

Considering the large area of the reservoir, it is assumed that the water levels of the upstream and downstream reservoirs are constant during the transient flow, and the pipe inlet head loss is ignored. The boundary conditions of the upstream and downstream reservoirs are [28]:

$$\{\begin{array}{l}{H}_{p1}=H_{const1}\\ Q_{p1}=C_{n1}+C_{a1}{H}_{p1}\end{array}$$

(4)

$$\{\begin{array}{l}{H}_{p2}=H_{const2}\\ Q_{p2}=C_{p2}-C_{a2}{H}_{p2}\end{array}$$

(5)

where Q_p1 and Q_p2 are the flow rate at points p₁ and p₂, respectively; H_p1 and H_p2 are the water head at p₁ and p₂, respectively; H_const1 and H_const2 are the water head of upstream reservoir and downstream reservoir, respectively; and C_n1, C_p2, C_a1, and C_a2 are intermediate variables.

(2)

Surge tank

To reduce the water hammer pressure of the pipe in the transient process, a surge tank is built at the connection between the pressure diversion channel and the penstock for long WDS. The mathematical model is described in [29]. The boundary condition of surge tank is:

$$\{\begin{array}{l}{Q}_{d3}=C_{p3}-C_{a31}{H}_{d3}\\ Q_{u3}=C_{n3}+C_{a32}{H}_{u3}\\ Q_{d3}=Q_j+Q_{u3}\\ H_{d_3}=H_{u_3}=H_{p3}\\ H_{p3}=H_j+R_j\frac{Q_j\left|Q_j\right|}{A_{wj}^2}\\ H_j=H_{j,\Delta t}+\frac{(Q_j+Q_{j,\Delta t})\Delta t}{2A_j}\end{array}$$

(6)

where Q_d3 and Q_u3 are the flow rate at points d₃ and u₃, respectively; H_d3 and H_u3 are the water head at points d₃ and u₃, respectively; C_p3, C_n3, C_a31, and C_a32 are intermediate variables; Q_j and Q_j,Δt are the surge tank flow rate at this moment and the last moment, respectively; H_j and H_j,Δt are the surge tank water level at this moment and the last moment, respectively; H_p3 is the water head at point p₃; R_j is the hydraulic loss coefficient of surge tank inlet impedance; A_wj is the cross-sectional area of the impedance hole; and A_j is the cross-sectional area of surge tank.

(3)

Elbow pipe and bifurcated pipe

The front and rear sections of the elbow pipe satisfy the characteristic equation. The elbow pipe also satisfies the flow rate and water head balance equation, as shown in Equation (7). The boundary condition of elbow pipe is [29]:

$$\{\begin{array}{l}{Q}_{d4}=C_{p4}-C_{a41}{H}_{d4}\\ Q_{u41}=C_{n4}+C_{a42}{H}_{u41}\\ Q_{u41}=Q_{u42}\\ H_{u41}=H_{u42}+\xi \frac{V_{u42}^2}{2g}\end{array}$$

(7)

where Q_d4, Q_u41 and Q_u42 are the flow rate at points d₄, u₄₁, and u₄₂, respectively; H_d4, H_u41, and H_u42 are the water head at points d₄, u₄₁, and u₄₂, respectively; C_p4, C_n4, C_a41, and C_a42 are intermediate variables; ξ is the local head loss coefficient; and V_u42 is the flow velocity at point u₄₂.

The front and rear sections of the bifurcated pipe satisfy the characteristic equation. The head loss at the joint is ignored. According to the energy equation and continuity equation, the boundary condition of the bifurcated pipe is [29]:

$$\{\begin{array}{l}{Q}_{d5}=C_{p5}-C_{a5}{H}_{d5}\\ Q_{u51}=C_{n51}+C_{a51}{H}_{u51}\\ Q_{u52}=C_{n52}+C_{a52}{H}_{u52}\\ Q_{u53}=C_{n53}+C_{a53}{H}_{u53}\\ H_{d5}=H_{u51}=H_{u52}=H_{u53}=H_p\\ Q_{d5}=Q_{u41}+Q_{u52}+Q_{u53}\end{array}$$

(8)

where Q_d5, Q_u51, Q_u52, and Q_u53 are the flow rate at points d₅, u₅₁, u₅₂, and u₅₃, respectively; H_d5, H_u51, H_u52, and H_u53 are the water head at points d₅, u₅₁, u₅₂, and u₅₃, respectively; C_p5, C_n51, C_n52, C_n53, C_a5, C_a51, C_a52, and C_a53 are intermediate variables.

(4)

Hydro-turbine

The water head of the hydro-turbine is the difference between the inlet of the volute and the outlet of the draft tube. The flow loss of the hydro-turbine is ignored. The boundary condition of the hydro-turbine is [24]:

$$\{\begin{array}{l}{Q}_{d6}=Q_{u6}=Q_t\\ Q_{d6}=C_{p6}-C_{a61}{H}_{d6}\\ Q_{u6}=C_{n6}+C_{a62}{H}_{u6}\\ H_t=H_{d6}-H_{u6}\end{array}$$

(9)

where Q_d6 and Q_u6 are the flow rate at points d₆ and u₆, respectively; H_d6 and H_u6 are the water head at points d₆ and u₂, respectively; C_p6, C_n6, C_a61, and C_a62 are intermediate variables; and Q_t is the flow rate of the hydro-turbine, which is obtained from the hydro-turbine model.

2.3. Model of Hydro-Turbine

The modeling process of the hydro-turbine depends on the characteristic curve of hydro-turbine, as the boundary of the pipe, its characteristics need to be. When the guide vane opening, rotating speed, and water head are known, the characteristic line equation of the pipe is solved simultaneously. The hydro-turbine characteristic data are transformed into characteristic curves through the polynomial interpolation method, as shown in Figure 5. The calculation formulas of the flow rate, torque, and speed are [30]:

$$\{\begin{array}{l}{Q}_{11}=f\left(n_{11},a\right),Q_t=Q_{11}{D}_1^2\sqrt{H_t}\\ M_{11}=f\left(n_{11},a\right),M_t=M_{11}{D}_1^3{H}_t\\ n_t=\frac{n_{11}\sqrt{H_t}}{D_1}\end{array}$$

(10)

where Q₁₁ is the unit flow rate; M₁₁ is the unit torque; n₁₁ is the unit rotating speed; and a is the guide vane opening.

2.4. Model of Generator and Load

In this study, the parallel operation of the HGSS is regarded as a rotating rigid body, and electromagnetic power, torque angle, and the change of the field winding flux are not considered. Thus, the first-order generator model is selected. In addition, HGSS still bears the grid load, and the generator electromagnetic torque is not zero in SLIP. The model of generator and load [31] is:

$$\{\begin{array}{l}{M}_t-M_{\mathrm{g}}=J\frac{d\omega }{dt}\\ m_{\mathrm{g}}=C_g+A_g+e_g(n-n_r)\\ n_t=n_{(t-\Delta t)}+\frac{\Delta t}{T_a}\left[1.5\left(m_{t(t-\Delta t)}-m_{g(t-\Delta t)}\right)-0.5\left(m_{t(t-2\Delta t)}-m_g{}_{(t-2\Delta t)}\right)\right]\end{array}$$

(11)

where M_t is the driving torque; M_g is the electromagnetic torque; ω is the angular velocity; J is the moment of inertia; m_g is the relative electromagnetic torque; C_g is the relative load torque; A_g is the step value of load torque; n_r is the relative rated speed; e_g is the load self-regulation coefficient; n is the relative rotating speed; and T_a is the inertia time constant.

2.5. Model of PID Governor

In SLIP, the control mode of HGS is changed to frequency control under SMSLPS in order to ensure that the HGS tracks the load change automatically, and maintains the frequency near the rated frequency [32]. Where K_P, K_I, and K_D are the proportional, integral, and differential coefficients, respectively; x₁ is the intermediate variable; x₂ is the output variable; T_y is the servomotor response time; and b_p is the permanent difference coefficient. The equations are shown in Equations (12) and (13) [33]. The schematic diagram of the PID governor is shown in Figure 6.

$$\{\begin{array}{l}{\dot{x}}_1=-K_I{n}_t-b_{p}e\\ {\dot{x}}_2=-\frac{1}{T_y}{x}_2+\frac{1}{T_y}e\end{array}$$

(12)

$$e=x_1-K_P{n}_t-K_D{\dot{n}}_t$$

(13)

When transformed in Equations (12) and (13), the state equation of the PID governor model is:

$$\left[\begin{array}{l}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}-b_p& 0\\ 1/T_y& -1/T_y\end{array}\right]\left[\begin{array}{l}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{cc}{b}_p{K}_P-K_I& b_p{K}_D\\ -K_P/T_y& -K_D/T_y\end{array}\right]\left[\begin{array}{l}{n}_t\\ {\dot{n}}_t\end{array}\right]$$

(14)

2.6. Model of HGSS

The HGSS is composed of runners, turbine shafts, generator shafts, generator rotors, and guide bearings. This study discretizes the vertical HGSS with a continuous mass distribution, and transforms it into a multi degree of freedom nodes with the centralized mass and the centralized moment of inertia [34]. These nodes are generally selected in the generator rotors, bearings, runners, and shaft diameter changes. In this section, the structural model of HGSS is established through the combination of RTMM and WNIM. The HGSS is divided into 18 nodes and 17 shaft segments for modeling. The schematic diagram of the HGSS is shown in Figure 7.

(1)

WNIM

WNIM is a stepwise integration method. The displacements of each node at time t + θΔt and the displacements, velocities, and accelerations of each node at time t are known to obtain the displacements, velocities, and accelerations of each node at time t + Δt [35]. The expression of WNIM is

$$\{\begin{array}{l}{\ddot{x}}_{t+\Delta t}=\frac{6}{{\theta }^3\Delta t^2}(x_{t+\theta \Delta t}-x_t)-\frac{6}{{\theta }^2\Delta t}{\dot{x}}_t+(1-\frac{3}{\theta }){\ddot{x}}_t\\ {\dot{x}}_{t+\Delta t}={\dot{x}}_t+\frac{\Delta t}{2}({\ddot{x}}_{t+\Delta t}+2{\ddot{x}}_t)\\ x_{t+\Delta t}=x_t+{\dot{x}}_t\Delta t+\frac{\Delta t^2}{6}({\ddot{x}}_{t+\Delta t}+2{\ddot{x}}_t)\end{array}$$

(15)

where x is the displacement. When θ ≥ 1.37, WNIM is unconditionally stable.

(2)

Differential equations of motion:

$$\{\begin{array}{l}{m}_i{\left\{\begin{array}{l}\ddot{x}\\ \ddot{y}\end{array}\right\}}_i=-{\left[\begin{array}{cc}{K}_{xx}& K_{xy}\\ K_{yx}& K_{yy}\end{array}\right]}_i{\left\{\begin{array}{l}x\\ y\end{array}\right\}}_i-{\left[\begin{array}{cc}{C}_{xx}& C_{xy}\\ C_{yx}& C_{yy}\end{array}\right]}_i{\left\{\begin{array}{l}\dot{x}\\ \dot{y}\end{array}\right\}}_i+{\left\{\begin{array}{l}{P}_x^L-P_x^R\\ P_y^L-P_y^R\end{array}\right\}}_i+{\left\{\begin{array}{l}{F}_x(t)\\ F_y(t)\end{array}\right\}}_i\\ J_{di}{\left\{\begin{array}{l}\ddot{\phi }\\ \ddot{\psi }\end{array}\right\}}_i=-\omega {\left[\begin{array}{cc}0& J_p\\ -J_p& 0\end{array}\right]}_i{\left\{\begin{array}{l}\dot{\phi }\\ \dot{\psi }\end{array}\right\}}_i+{\left\{\begin{array}{l}{M}_x^R-M_x^L\\ M_y^R-M_y^L\end{array}\right\}}_i+{\left\{\begin{array}{l}{L}_{ax}(t)\\ L_{ay}(t)\end{array}\right\}}_i\end{array}$$

(16)

where subscripts i, x, y, d, and p are the ith node, x-direction, y-direction, radial direction and axial direction, respectively; superscripts L and R are the left and right sections of each node; m is the mass; x, y, ψ, and φ are the displacements; K is the stiffness coefficient; C is the damping coefficient; P and M are the shear force and the shear moment, respectively; and F_a and L_a are additional forces and additional moments, respectively.

(3)

RTMM

The state vectors of each node are divided into f = {M_x, Q_x, M_y, Q_y}^T and e = {x, y, ψ, φ}^T. Combined with the differential equation of motion, the transfer matrix at time t + θΔt is:

$${\left\{\begin{array}{l}f\\ e\end{array}\right\}}_{i+1}=\left[\begin{array}{cc}{B}_{11}& B_{12}\\ B_{21}& B_{22}\end{array}\right]\left[\begin{array}{cc}{T}_{11}& T_{12}\\ T_{21}& T_{22}\end{array}\right]{\left\{\begin{array}{l}f\\ e\end{array}\right\}}_i+{\left\{\begin{array}{l}{T}_{13}\\ T_{23}\end{array}\right\}}_i=\left[\begin{array}{cc}{u}_{11}& u_{12}\\ u_{21}& u_{22}\end{array}\right]{\left\{\begin{array}{l}f\\ e\end{array}\right\}}_i+{\left\{\begin{array}{l}{F}_f\\ F_e\end{array}\right\}}_i$$

(17)

$$\begin{array}{c}{\left[u_{22}\right]}_i={\left[B_{21}\right]}_i{\left[T_{12}\right]}_i+{\left[B_{22}\right]}_i,{\left\{F_f\right\}}_i={\left[B_{11}\right]}_i{\left\{T_{13}\right\}}_i,{\left\{F_e\right\}}_i={\left[B_{21}\right]}_i{\left\{T_{13}\right\}}_i,\left[T_{23}\right]={\left\{\begin{array}{cccc}0& 0& 0& 0\end{array}\right\}}^T,\\ {\left[T_{11}\right]}_i={\left[T_{22}\right]}_i=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],{\left[T_{12}\right]}_i=\left[\begin{array}{cccc}0& K_2& 0& K_3\\ W_1& 0& W_2& 0\\ 0& -K_3& 0& K_2\\ W_3& 0& W_4& 0\end{array}\right],{\left[B_{11}\right]}_i={\left[B_{22}\right]}_i={\left[u_{11}\right]}_i=\left[\begin{array}{cccc}1& l& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& l\\ 0& 0& 0& 1\end{array}\right],\\ {\left[u_{12}\right]}_i=\left[\begin{array}{cccc}l{W}_1& K_2& l{W}_2& K_3\\ W_1& 0& W_2& 0\\ l{W}_3& -K_3& l{W}_4& K_2\\ W_3& 0& W_4& 0\end{array}\right],{\left[B_{21}\right]}_i={\left[u_{21}\right]}_i=\left[\begin{array}{cccc}{a}_2& a_3& 0& 0\\ a_1& a_2& 0& 0\\ 0& 0& a_2& a_3\\ 0& 0& a_1& a_2\end{array}\right],\\ \left[T_{13}\right]=\left\{\begin{array}{c}-K_2{\phi }_t-\frac{6J_{di}}{\theta \Delta t}{\dot{\phi }}_t-2J_{di}{\ddot{\phi }}_t-K_3{\psi }_t-2J_{pi}\omega {\dot{\psi }}_t-\frac{1}{2}{J}_{pi}\omega \theta \Delta t{\ddot{\psi }}_t-L_x(t+\theta \Delta t)\\ (2m+\frac{1}{2}\theta \Delta t{C}_{xx}){\ddot{x}}_t+\frac{1}{2}\theta \Delta t{C}_{xy}{\ddot{y}}_t+(\frac{6m}{\theta \Delta t}+2C_{xx}){\dot{x}}_t+2C_{xy}{\dot{y}}_t+(\frac{6m}{{(\theta \Delta t)}^2}+\frac{3C_{xx}}{\theta \Delta t})x_t+\frac{3}{\theta \Delta t}{C}_{xy}{y}_t+F_x(t+\theta \Delta t)\\ -K_2{\psi }_t-\frac{6J_{di}}{\theta \Delta t}{\dot{\psi }}_t-2J_{di}{\ddot{\psi }}_t+K_3{\phi }_t+2J_{pi}\omega {\dot{\phi }}_t+\frac{1}{2}{J}_{pi}\omega \theta \Delta t{\ddot{\phi }}_t-L_y(t+\theta \Delta t)\\ (2m+\frac{1}{2}\theta \Delta t{C}_{yy}){\ddot{y}}_t+\frac{1}{2}\theta \Delta t{C}_{yx}{\ddot{x}}_t+(\frac{6m}{\theta \Delta t}+2C_{yy}){\dot{y}}_t+2C_{yx}{\dot{x}}_t+(\frac{6m}{{(\theta \Delta t)}^2}+\frac{3C_{yy}}{\theta \Delta t})y_t+\frac{3}{\theta \Delta t}{C}_{yx}{x}_t+F_y(t+\theta \Delta t)\end{array}\right\},\end{array}$$

(18)

$$\begin{array}{c}{K}_1=\frac{6m_i}{{(\theta \Delta t)}^2},K_2=\frac{6J_{di}}{{(\theta \Delta t)}^2},K_3=\frac{3J_{pi}\omega }{\theta \Delta t},W_1=-K_{xx}-\frac{3}{\theta \Delta t}{C}_{xx}-\frac{6m}{{(\theta \Delta t)}^2},W_2=-K_{xy}-\frac{3}{\theta \Delta t}{C}_{xy},\\ W_3=-K_{yx}-\frac{3}{\theta \Delta t}{C}_{yx},W_4=-K_{yy}-\frac{3}{\theta \Delta t}{C}_{yy}-\frac{6m}{{(\theta \Delta t)}^2},a_1=\frac{l}{EJ},a_2=\frac{l^2}{2EJ},a_3=\frac{l^3}{6EJ}.\end{array}$$

(19)

where l is the length of the shaft segment; and EJ is the bending stiffness.

The Riccati transformation is:

$${\left\{f\right\}}_i={\left[S\right]}_i{\left\{e\right\}}_i+{\left\{P\right\}}_i$$

(20)

where [S]_i and {p}_i are the undetermined matrices.

It is obtained from Equations (17) and (20):

$$\{\begin{array}{l}{\left[S\right]}_{i+1}={\left[u_{11}S+u_{12}\right]}_i{\left[u_{21}S+u_{22}\right]}_i^{-1}\\ {\left\{P\right\}}_{i+1}={\left\{u_{11}P+F_f\right\}}_i-{\left[S\right]}_{i+1}{\left[u_{21}P+F_e\right]}_i\end{array}$$

(21)

From the boundary conditions {f}₁ = 0 and {e}₁ ≠ 0, [S]₁ = 0 and {p}₁ = 0 is obtained. In addition, from Equation (21), [S]_i and {p}_i are obtained. The boundary condition {f}_{N + 1} = 0 and Equation (21) are combined to obtain:

$${\left\{e\right\}}_{N+1}=-{\left[S\right]}_{N+1}^{-1}{\left\{P\right\}}_{N+1}$$

(22)

Thus, the displacements of each node at time t + θΔt are obtained:

$${\left\{e\right\}}_i={\left[u_{21}S+u_{22}\right]}_i^{-1}\left({\left\{e\right\}}_{i+1}-{\left[u_{21}P+F_e\right]}_i\right)$$

(23)

2.7. Additional Forces on HGSS

(1)

Nonlinear oil film force

Most of the guide bearings on HGSS are oil-lubricated sliding bearings with tilting pads. The oil film force of the bearing has a non-linear relationship with the speed and displacement of the journal. The expression of nonlinear oil film force [36] is:

$$\left\{\begin{array}{l}{F}_x\\ F_y\end{array}\right\}=\sigma \cdot \left\{\begin{array}{l}{f}_x\\ f_y\end{array}\right\}=-\frac{{[{(x-2y^{\prime })}^2+{(y+2x^{\prime })}^2]}^{1/2}}{1-x^2-y^2}\cdot \left\{\begin{array}{l}3x\cdot V(x,y,\alpha )-\sin \alpha \cdot G(x,y,\alpha )-2\cos \alpha \cdot S(x,y,\alpha )\\ 3y\cdot V(x,y,\alpha )+\cos \alpha \cdot G(x,y,\alpha )-2\sin \alpha \cdot S(x,y,\alpha )\end{array}\right\}$$

(24)

$$V(x,y,\alpha )=\frac{2+(y\cos \alpha -x\sin \alpha )\cdot G(x,y,\alpha )}{1-x^2-y^2}$$

(25)

$$G(x,y,\alpha )=\frac{2}{{(1-x^2-y^2)}^{1/2}}\left[\frac{\pi }{2}+\arctan \frac{y\cos \alpha -x\sin \alpha }{{(1-x^2-y^2)}^{1/2}}\right]$$

(26)

$$S(x,y,\alpha )=\frac{x\cos \alpha +y\sin \alpha }{1-{(x\cos \alpha +y\sin \alpha )}^2}$$

(27)

$$\alpha =\arctan \frac{y+2x^{\prime }}{x-2y^{\prime }}-\frac{\pi }{2}\mathrm{sgn}(\frac{y+2x^{\prime }}{x-2y^{\prime }})-\frac{\pi }{2}\mathrm{sgn}(y+2x^{\prime })$$

(28)

$$\sigma =\mu \omega R_b{L}_b{\left(\frac{R_b}{C_b}\right)}^2{\left(\frac{L_b}{D_b}\right)}^2$$

(29)

where μ is the lubricating oil viscosity; R_b is the guide bearing radius; L_b is the guide bearing length; C_b is the guide bearing clearance; and D_b is the guide bearing diameter.

(2)

Mechanical unbalance force

Due to uneven mass distribution and installation accuracy, mass eccentricity occurs at the generator rotor and the hydro-turbine runner, which causes the rotation center to shift. The expression of mechanical unbalance force [37] is:

$$F_{muf}=me{\omega }^2$$

(30)

where F_muf is the mechanical unbalance force; and e is the eccentricity.

(3)

UMP

When the air gap between the stator and the rotor is uneven due to manufacturing or installation errors, the rotor produces a lateral unbalanced magnetic tension. The expression of UMP [38] is:

$$\{\begin{array}{l}{F}_{x-ump}=\frac{R_r{L}_r\pi }{4{\mu }_0}{F}_j{}^2(2{\Lambda }_0{\Lambda }_1+{\Lambda }_1{\Lambda }_2+{\Lambda }_2{\Lambda }_3)\cos \eta \\ F_{y-ump}=\frac{R_r{L}_r\pi }{4{\mu }_0}{F}_j{}^2(2{\Lambda }_0{\Lambda }_1+{\Lambda }_1{\Lambda }_2+{\Lambda }_2{\Lambda }_3)\sin \eta \end{array}$$

(31)

$${\Lambda }_n=\{\begin{array}{cc}\frac{{\mu }_0}{{\delta }_0}\frac{1}{\sqrt{1-{\epsilon }^2}}\hfill & \hfill (n=0)\\ \frac{2{\mu }_0}{{\delta }_0}\frac{1}{\sqrt{1-{\epsilon }^2}}{(\frac{1-\sqrt{1-{\epsilon }^2}}{\epsilon })}^n\hfill & \hfill (n>0)\end{array}$$

(32)

where F_x-ump and F_y-ump are the unbalanced magnetic pulls in x-direction and y-direction, respectively; R_r is the generator rotor radius; L_r is the generator rotor length; μ₀ is the air permeability; F_j is the fundamental magnetomotive force of the excitation current; ε is the relative eccentricity; δ₀ is the average air gap between stator and rotor; and η is the rotating angle.

(4)

Hydraulic excitation force

When the hydro-turbine performs energy conversion, it also brings complicated hydraulic vibration problems to the HGSS. The reasons for this include many aspects, such as operation under non-design conditions, draft tube pressure pulsation, unequal sealing gaps, and excessive axial water thrust, etc. Zhuang [14] proposed an excitation force model on a single blade of the hydro-turbine runner. Superposition of the forces on all the blades of the hydro-turbine runner is the hydraulic excitation force, and its expression is

$$\{\begin{array}{l}{F}_{x-hef}=\frac{n_b\gamma C_y{Q}^2{F}_b\cos \left({\beta }_m-\lambda \right)}{8g\cos \lambda }\left[\frac{1}{{\left(s_1\pi D_1{b}_0\sin {\beta }_1\right)}^2}+\frac{1}{{\left(s_2\pi D_2^2\sin {\beta }_2\right)}^2}+\frac{2\cos \left({\beta }_2-{\beta }_1\right)}{s_1{s}_2{\pi }^2{D}_1{D}_2^2{b}_0\sin {\beta }_1\sin {\beta }_2}\right]\cos \eta \\ F_{y-hef}=\frac{n_b\gamma C_y{Q}^2{F}_b\cos \left({\beta }_m-\lambda \right)}{8g\cos \lambda }\left[\frac{1}{{\left(s_1\pi D_1{b}_0\sin {\beta }_1\right)}^2}+\frac{1}{{\left(s_2\pi D_2^2\sin {\beta }_2\right)}^2}+\frac{2\cos \left({\beta }_2-{\beta }_1\right)}{s_1{s}_2{\pi }^2{D}_1{D}_2^2{b}_0\sin {\beta }_1\sin {\beta }_2}\right]\sin \eta \end{array}$$

(33)

$${\beta }_m=\arcsin \left(\frac{2\left(s_1{D}_1{b}_0+s_2{D}_2^2\right)}{\sqrt{{\left(\frac{s_2{D}_2^2}{\sin {\beta }_1}\right)}^2+{\left(\frac{s_1{D}_1{b}_0}{D_2^2\sin {\beta }_2}\right)}^2+\frac{2s_1{s}_2{D}_1{D}_2^2{b}_0\cos \left({\beta }_2-{\beta }_1\right)}{\sin {\beta }_1\sin {\beta }_2}}}\right)$$

(34)

$${\beta }_1=\arctan \left(\frac{2Q\tan {\alpha }_1}{\pi D_1^2\omega s_1{b}_0\tan {\alpha }_1-2Q}\right)$$

(35)

where F_x-hef and F_y-hef are the hydraulic excitation forces in x-direction and y-direction, respectively; n_b is the number of runner blades; γ is the unit weight; λ = arctan(C_x/C_y); C_y is the lift coefficient; C_x is the drag coefficient; F_b is the maximum area of the blade; β_m is the direction angle of the relative average speed; s₁ and s₂ are the entrance excretion coefficient and exit excretion coefficient, respectively; D₁ and D₂ are the entrance diameter and exit diameter of hydro-turbine runner, respectively; b₀ is the height of guide vane; β₁ and β₂ are the blade entrance flow angle and the blade exit flow angle, respectively; and α₁ is the guide vane opening angle.

2.8. Model Validation

This study selects the real HGS parameters of Qinghai Province, China, to establish the model, as well as the comprehensive verification based on the operation characteristics and vibration characteristics of HGS. The HGS parameters of Qinghai Province in China are shown in

Table 1. The HGS parameters of Qinghai province in China.

Model Parameter	Symbol	Value	Unit
Length of the tunnel	TL	5693	m
Length of the surge tank	STL	2.3	m
Length of the penstock behind the surge tank	PBSTL	84	m
Length of the sloping penstock	SPL	58.97	m
Length of the penstock behind the sloping penstock	PBSPL	562.44	m
Length of pipe 1/2/3	PL1/PL2/PL3	360.6	m
Diameter of the tunnel	TD	3.2	m
Diameter of the surge tank	STD	6	m
Diameter of the surge tank impedance	STID	2.3	m
Diameter of the penstock behind the surge tank	PBSTD	3.2	m
Diameter of the sloping penstock	SPD	3	m
Diameter of the penstock behind the sloping penstock	PBSPD	3	m
Diameter of pipe 1/2/3	PD1/PD2/PD3	1.73	m
Upstream water level	Hstart	3456	m
Downstream water level	Hend	3252.05	m
Rated output	Pr	11	MW
Rated flow rate	nr	600	rpm
Rated water head	Qr	6.8	m3/s
Runner diameter	D1	1.4	m
Generator rotor diameter	D2	3.144	m
Guide bearing stiffness	K	1 × 108	N/m
Generator rotor mass eccentricity	e1	1 × 10−4	m
Runner mass eccentricity	e2	1 × 10−4	m
Uniform air gap thickness	δ0	2 × 10−2	m
Air permeability	μ0	4π × 10−7	H/m

.

First, the operation characteristics of HGSS #1, #2 and #3 in the simultaneous load rejection condition are simulated, and the simulation results are compared with the experimental results from [39].

Table 2. The comparison of the simulation results and the experiment results.

Result	Upper and Lower Water Level (m)	HGSS	The Maximum Pressure Head at the Volute (m)	The Minimum Pressure Head at the Volute (m)	The Maximum Speed Rise Rate (%)	The Maximum and Minimum Surge Level in Surge Tank (m)
The simulation	3456.00 3252.05	#1	252.75	180.01	58.98	3471.7 3453.0
		#2	252.76	180.03	58.96
		#3	252.74	180.01	58.95
The experiment	3456.00 3252.05	#1	243.45	185.18	59.07	3476.29 3442.43
		#2	243.12	184.92	58.97
		#3	243.36	184.94	59.03

shows the comparison of the simulation results and the experiment results.

From Table 2, it can be seen that the maximum error of the maximum and minimum pressure at the volute are 3.9% and 2.79%, respectively; the maximum error of the maximum speed rise rate is 0.15%; and the error of the maximum and minimum surge in surge tank are 0.13% and 0.31%, respectively. The above comparisons show that the maximum error between the simulation and the experiment do not exceed 4%, and indicate that the established model is correct and reasonable in operation characteristics.

Second, 50%-, 75%-, and 100%-rated load conditions in an operable load area are selected to simulate the vibration characteristics of the HGS, and the simulation results are compared with the National Standard of China “Specification installation of hydraulic turbine generator units. GB/T 5864-2003” [40]. For the HGSS in this study, the allowable absolute vibration amplitudes of the upper guide bearing, the lower guide bearing, the flange, and the water guide bearing are 0.022 mm, 0.076 mm, 0.106 mm, and 0.2 mm, respectively. The vibration characteristics of different guide bearings and the flange are shown in Figure 8.

The vibration characteristics of HGSS #1, #2, and #3 are similar, and HGSS #1 is used as an example. As shown in Figure 6, the upper guide bearing and the lower guide bearing are mainly affected by the mechanical unbalance force and UMP at the generator rotor, and the vibration amplitudes are the largest under the 100%-rated load condition, which are 0.0096 mm and 0.008 mm, respectively. The flange and the water guide bearing are mainly affected by the hydraulic excitation force, and the vibration amplitudes are the largest under the 50%-rated load condition, which are 0.017 mm and 0.098 mm, respectively. The above results show that the vibration amplitudes of HGS do not exceed the allowable vibration amplitude in National Standard of China, and indicate that the established model is correct and reasonable in vibration characteristics.

3. Numerical Simulation and Result Analysis

3.1. Hydraulic Factors

(1)

Sudden load increasing amount

For HGS in this study, 50–100%-rated load area is the operable area. Thus, in this subsection, the vibration characteristics of HGSS #1, #2, and #3 which collectively suddenly increase 30%, 40%, and 50%-rated load from the 50%-rated load (ΔP = 30%, ΔP = 40%, and ΔP = 50%) are analyzed. The operation and vibration characteristics of the hydro-turbine runner under different sudden load increasing amounts are shown in Figure 9, Figure 10 and Figure 11, respectively.

From Figure 9, the PID governor tracks the load change automatically, and opens the guide vane quickly. The rotating speed decreases by 11.16%, 12.87%, and 15.21%, and the water head decreases by 6.67%, 7.52%, and 8.46%, with the increase of sudden load increasing amount, respectively.

From Figure 10, it can be seen that Area I and Area III are the stable operating areas; Area II is the operating area of SLIP. When t = 16 s, the maximum vibration response occurs at the hydro-turbine runner. When ΔP = 50%, the vibration amplitudes of the hydro-turbine runner in x-direction and y-direction in Area II are the largest, which increase by 69.89% and 69.95%, compared with that of Area I, respectively. When ΔP = 30%, the vibration amplitudes of the hydro-turbine runner in x-direction and y-direction in Area III are the largest, which increase by 33.79% and 34.1%, compared with that of Area I, respectively. Thus, the hydraulic excitation force causes the vibration amplitude under a partial load to be greater than that under a rated load. In addition, as demonstrated in Figure 11, the greater the sudden load increasing amount, the greater the axis trajectory and vibration amplitude in Area II, and the smaller axis trajectory and vibration amplitude in Area III.

(2)

Blade exit flow angle

When the Francis HGSS was operating in the non-optimal load condition area, the direction of the water flow from the blade exit had a circumferential component, which caused the vortex. Thus, in this subsection, the vibration characteristics of HGS under ΔP = 50% and the blade exit flow angles, taken as 0.05 rad, 0.1 rad, and 0.15 rad, are analyzed, respectively. The vibration characteristics of the hydro-turbine runner under different blade exit flow angles are shown in Figure 12 and Figure 13.

From Figure 12 and Figure 13, when t = 15 s, the maximum vibration response occurs at the hydro-turbine runner. When β₂ = 0.15 rad, the vibration amplitudes of the hydro-turbine runner in x-direction and y-direction are the largest, which increase by 67.36% and 66.54%, compared with that of Area I, respectively. The greater the blade exit flow angle, the greater the axis trajectory and vibration amplitude of the hydro-turbine runner.

3.2. Mechanical Factors

(1)

Mass eccentricity

Uneven mass distribution of the rotating parts caused by poor materials and insufficient installation accuracy is common, which causes the vibration of the HGS. Thus, in this subsection, the vibration characteristics of the HGS under ΔP = 50% and the mass eccentricities, taken as 0.4 mm, 0.6 mm, and 0.8 mm, are analyzed, respectively. The vibration characteristics of the generator rotor and the hydro-turbine runner under different mass eccentricities are shown in Figure 14 and Figure 15.

From Figure 14 and Figure 15, when t = 18 s and t = 15 s, the maximum vibration response occurs at the generator rotor and the hydro-turbine runner, respectively. When e = 0.8 mm, the vibration amplitudes of the generator rotor and the hydro-turbine runner in x-direction and y-direction are the largest, which increase by 40.71%, 37.25%, 65.3%, and 64.95%, compared with that of Area I, respectively. The axis trajectory and vibration amplitude of the HGS increase with mass eccentricity increase. In addition, the mass eccentricity has a great impact on the generator rotor, but a lesser impact on the hydro-turbine runner, as the hydro-turbine runner is greatly affected by hydraulic factors.

(2)

Guide bearing stiffness

The guide bearing is used to bear the radial force during the operation, and limit the vibration of HGSS. Thus, the unreasonable guide bearing stiffness leads to the severe vibration of HGS. In this subsection, the vibration characteristics of the HGS under ΔP = 50% and different stiffnesses of the upper guide bearing, lower guide bearing, and water guide bearing are analyzed, respectively. The guide bearing stiffness is taken as 1.0 × 10⁸ N × m⁻¹, 3.0 × 10⁸ N × m⁻¹, and 5.0 × 10⁸ N × m⁻¹, respectively. The vibration characteristics of the guide bearing under different stiffnesses are shown in Figure 16 and Figure 17.

As demonstrated in Figure 16 and Figure 17, the maximum vibration response of upper guide bearing, lower guide bearing, and water guide bearing occurs at t = 22 s, 17 s, and 15 s, respectively. When K = 1.0 × 10⁸ N × m⁻¹, the vibration amplitudes of the upper guide bearing, lower guide bearing, and water guide bearing in x-direction and y-direction are the largest, which increase by 41.24%, 36.35%, 63.54%, 60.01%, 70.46%, and 70.17%, compared with that of Area I, respectively. In conclusion, the stiffness of the water guide bearing has the greatest influence on vibration response, and the stiffness of the lower guide bearing has less influence on the vibration response than that of the upper guide bearing. The axis trajectory and vibration amplitude at guide bearing increase as guide bearing stiffness decreases.

3.3. Electric Factors

(1)

Excitation current

The change of excitation current causes the change of UMP, which leads to the vibration of the HGS. Thus, in this subsection, the vibration characteristics of HGS under ΔP = 50% and the excitation currents, taken as 50% I_j, 75% I_j, and 100% I_j, are analyzed, respectively. The vibration characteristics of the generator rotor under different excitation currents are shown in Figure 18 and Figure 19.

As can be observed in Figure 18 and Figure 19, when t = 21 s, the maximum vibration response occurs at the generator rotor. When 75% excitation current is conducted, the vibration amplitudes of the generator rotor in x-direction and y-direction are the largest, which increase by 40% and 34.67%, compared with that of Area I, respectively. The axis trajectory and vibration amplitude at the generator rotor increase as the excitation current increases.

(2)

Average air gap between stator and rotor

An uneven air gap between the stator and rotor produces greater UMP, and aggravates the vibration of the HGSS. Thus, in this subsection, the vibration characteristics of HGS under ΔP = 50% and the average air gaps between the stator and rotor, taken as δ₀ = 20 mm, δ₀ = 30 mm, and δ₀ = 40 mm, are analyzed, respectively. The vibration characteristics of the generator rotor under different average air gaps between the stator and rotor are shown in Figure 20 and Figure 21.

As demonstrated in Figure 20 and Figure 21, when t = 21 s, the maximum vibration response occurs at the generator rotor. When δ₀ = 20 mm, the vibration amplitudes of the generator rotor in x-direction and y-direction are the largest, which increase by 40% and 34.54%, compared with that of Area I, respectively. The axis trajectory and vibration amplitude at the generator rotor increase as the average air gap between the stator and rotor decreases.

4. Conclusions

This study proposes a new transient nonlinear coupling model of HGS to analyze the influences of different HMEPs on the vibration characteristics of HGS in SLIP under SMSLPS. The main conclusions obtained are as follows:

(1)

For hydraulic factors, the greater the sudden load increasing amount and blade exit flow angle, the greater the vibration of the hydro-turbine runner. The maximum vibration amplitude is increased by 69.95%, compared with that under stable operating conditions. Moreover, the vibration under a partial load is greater than that under a rated load. Hydraulic factors determine the tendency of vibration at hydro-turbine runner.

(2)

For mechanical factors, the vibration of HGSS increases as mass eccentricity increases and guide bearing stiffness decreases. The maximum vibration amplitude is increased by 70.46%, compared with that under stable operating conditions. The mass eccentricity has a great impact on the generator rotor, but a lesser impact on the hydro-turbine runner. The influence of the water guide bearing stiffness on the vibration response is greater than that of the upper guide bearing and the lower guide bearing.

(3)

For electric factors, the vibration of the generator rotor increases as the excitation current increases and the average air gap between the stator and rotor decreases. The maximum vibration amplitude is increased by 40%, compared with that under stable operating conditions.

More importantly, this study provides reasonable reference for the analysis of the nonlinear dynamic behavior of HGS in SLIP, and the stable operation of the hydropower station.