Effect of turbine's torque and speed variation on hydraulic vibration analysis during transient processes

With the rapid development of large‐capacity hydraulic turbines and pumped‐storage power stations, hydraulic vibration analysis attracts increasing attention nowadays. To explore the influence of torque and speed variation on hydraulic vibration characteristics during transient processes and ensure the stability of hydropower stations, an improved analytical formula of turbine impedance involving the torque and speed variation was then derived. The free vibration analysis and forced vibration analysis using the transfer matrix method and the hydraulic impedance method were carried out. According to the numerical simulations, the obtained attenuation factors show differences in magnitude which illustrates that the torque variation and speed variation in transients do affect the damping rate of the vibration. In addition, the results indicated the natural angular frequencies of hydropower system were mainly dependent on the length of water conveyance system and pipe characteristics. It can be inferred form vibration mode shapes that for some specific frequencies, the magnitude of oscillatory pressure head and flow rate along the downstream tail tunnel are influenced by the hydraulic turbine. The correlation between the turbine impedance and guide vane openings tells that the hydraulic impedance is getting smaller as the opening increases. What's more, the corresponding free vibration analysis shows that the absolute value of the damping rate is closely related to the value of the flow rate.


| INTRODUCTION
With the rapid development of various kinds of intermittent renewable energies, hydropower stations including pumped-storage power stations are now attracting increasing attention in view of compensating for the power supply fluctuations. 1Hydropower plays great importance in renewable energy sources as it accounts for a relatively large proportion and its irreplaceable role in power grid regulation. 2 The variation of the operating conditions will inevitably lead to transients which may cause huge pressure fluctuation and operating stability problems of the system. 3Therefore, special attention should be concentrated on transient processes to avoid potential operation risks. 4n hydraulic transient processes, hydraulic vibration is a periodic hydraulic transient process that behaves like a small discharge oscillation accompanied by large pressure fluctuation, which differs from the water hammer and surge phenomenon in the pressurized water conveyance system. 5Generally, hydraulic vibration due to any disturbances will normally decay with time and finally disappear.However, sometimes the frequencies of external disturbances are similar or equal to the natural frequencies of the pressurized pipeline system, hydraulic resonance and a self-excited vibration could happen because of the system's inherent instabilities. 6][9] Numerous existing research have been carried out on the pressure fluctuations in hydropower systems, including experimental tests and numerical simulations.The experiment method is restricted because of its high costs and security risks. 10,11The numerical simulation, including one-dimensional method of characteristics (1D-MOC) and three dimensional (3D) numerical simulations based on computational fluid dynamics (CFD), is very popular in numerical investigations and extensively applied in pressure fluctuation simulations as well as the dynamic characteristics of the turbine during transient processes in hydropower stations.6][17] Furthermore, the 1D-3D coupled approach is gradually becoming a proper choice to study the transient processes in the whole hydropower system.Based on the 1D water conveyance system simulation and 3D turbine passage simulation, the pressure fluctuations in the pipeline and complex flow patterns through the turbine passage can be more specifically analyzed in the hydropower system. 18,19dditionally, for a water conveyance system with extremely long-distance pipelines, plenty of computation time is required when performing the time domain analysis since the process of convergence to another steady state is relatively slow. 20The hydraulic vibration theory of pressurized water conveyance systems in the frequency domain has attracted increasing attention since some earlier well-known incidents induced by hydraulic resonance have been identified in the research. 21,22The hydraulic vibration theory has been advanced nowadays based on the investigation progresses.Suo and Wylie modeled pipe wall viscoelasticity in oscillatory and nonperiodic flow and analyzed the frequency-dependent wave speed in a rock-bored tunnel due to the dynamic effect of the surrounding rock mass. 23,24Based on numerical calculation and experimental verification, the natural frequencies and pressure fluctuation characteristics are closely related to the wave speed variation, which is a function of the local air content and transient pressure. 25Except for the traditional hydropower stations, Zhou explored the possible self-excited oscillations in pumped storage power stations when the pump-turbine operates in the S-shaped regions of the hill chart. 26,27As a complement, the impedance of the guide vanes opening was proposed and further added to the impedance formula of the hydraulic turbine, the criteria that whether self-excited vibration might occur in the pump-turbine hydropower stations were improved. 28owever, the guide vane opening and the rotational speed are assumed to be unchanged in the numerical simulations of transient processes among hydraulic vibration analysis in hydropower systems.Thus, there are still some key issues that need to be further investigated, for instance, how to add the torque variation and speed variation into the impedance formula of the turbine and the consequent influence on the vibration characteristic of the system.This paper mainly focuses on the hydraulic vibration characteristics of pressurized water conveyance systems in hydropower stations considering the turbine's rotational speed variation in transients.In this paper, a new impedance formula of turbine including torque and speed variation was derived and the system's hydraulic vibration characteristics involving torque variation were distinguished.Then, the changing law of turbine's impedance at different frequencies were illustrated and the effect of guide vane opening on turbine's impedance and system's stability were revealed.
The structure of the paper is as follows.In Section 2, based on the power equation of the hydraulic turbine, the hydraulic impedance formula involving the torque and speed variation is derived.In Section 3, a 1D mathematical model of a pressurized water conveyance system is established to carry out the following free vibration analysis and forced vibration analysis.Besides, the correlations between the guide vane openings and impedance of the turbine and corresponding vibration characteristics strongly imply that the attenuation factor is relevant to the operating conditions of the turbine.In Section 4, the conclusions are given.

| Basic equations and hydraulic impedance method
The numerical analysis method of hydraulic vibration is mainly based on the linear theory of oscillatory flow analysis, 5 the simplified motion equation and continuity equation of fluid transients in the pressurized pipeline systems are as follows.
Motion equation Continuity equation where Q is the instantaneous flow rate; H is the instantaneous pressure head; g is the gravity acceleration; A is the pipe cross-sectional area; D is the pipe inner diameter; a is the wave speed; t is the time; x is the coordinate along the axial direction of the pipe; f is the Darcy-Weisbach friction coefficient.
In hydraulic vibration analysis, both flow rate and pressure head are assumed to have a sinusoidal variation and can be decomposed into a steady average component H and Q , and an oscillatory component h′ and q′, Given the above specific hypothesis, the equations of motion and continuity in the time domain can be converted into the frequency domain, where complex values are generally used for calculation.The fluctuation value h′ and q′ can be written as, x st x st (5)   where H x and Q x are complex-valued oscillatory head and flow rate at position x; s is the complex frequency, the corresponding expression is s σ iω = + , σ is the attenuation factor, ω is the angular frequency; i is the imaginary unit number.
Substituting Equations ( 3), (4), and (5) into Equations ( 1) and ( 2), then eliminating the average terms, the linearized formula for oscillatory flow are derived as follows: where R, L, C, are defined as hydraulic resistance, inductance, and capacitance, respectively, R fQ gDA = / 2 ; L gA = 1/ ; C gA a = / 2 .Assuming the boundary conditions at the upstream reservoir are known and introducing the hyperbolic functions, then the complex-valued pressure head and flow rate are expressed as, where H U and Q U are the complex-valued pressure head and flow rate at the upstream inlet, respectively; In hydraulic vibration theory, the hydraulic impedance is defined as the ratio between the complex oscillatory head and flow rate.Thus, two oscillatory complex variables are merged into a single function as, where l is the pipe length; Z D and Z U are hydraulic impedance at the downstream end and upstream end, respectively.The impedance expressions of typical pipeline junctions, commonly used hydraulic elements and boundary conditions in the hydromechanical system have already been established. 5he transfer matrix method and hydraulic impedance method are always used together in hydraulic vibration analysis.Accordingly, the overall transfer matrix that relates the state vectors at the upstream and downstream end of the system can be built.Finally, free vibration analysis and forced vibration analysis can be carried out with appropriate boundary conditions.

| Hydraulic impedance formula of hydraulic turbine
In previous research, the hydraulic impedance of the turbine is defined as the slope M at the operating point in the head-discharge curve of the turbine. 25The value of M is usually assumed to be a real constant value if the guide vane opening remains unchanged during the transient process.That is, the turbine impedance, Z M = T1 , the point matrix of a turbine can be expressed as, To reflect the operating characteristics of a hydraulic turbine, a more accurate hydraulic impedance formula was proposed.Based on the turbine's model characteristics curve, the head balance condition, flow continuity condition, and similarity law, the unit flow rate, and unit speed are involved in the impedance expression and written as, 29   ( ) where n is the runner rotational speed; Q 11 is the unit flow rate; n 11 is the unit speed; D 1 is the diameter of the runner.However, the previous two formulas of turbine impedance neglect the torque and speed variation during the transient processes.In the actual transients, the torque and speed of the turbine will change with the flow rate oscillation, and to understand the vibration characteristics of the system more accurately, it is important to take the torque and speed variation into consideration.
Assuming the working efficiency of a turbine to be a constant over the oscillating range of flow rate and head variations.The power equation of hydraulic turbine is expressed as, where P is the power of the turbine; η is the efficiency at corresponding operating point; γ 0 is the specific gravity of water; T is the torque, T T T T T e + T st , T and ω s are the average torque and rotational speed respectively; T T , Q T , and H T are the oscillatory amplitude of torque, flow rate, and pressure head corresponding to complex frequency s.
The dynamic equation describing the rotation of the turbine during the transient is, where I is the inertia of the turbine.
Substituting T T e ′ = T st into Equation ( 13) and integrating over time, the expression of a complexvalued oscillating rotational speed can be obtained, s T st (14)   Expanding the instantaneous terms in Equation ( 12) by using average items and oscillatory items, then replacing the oscillating torque and oscillating rotational speed item with Equations ( 13) and (14).Based on the definition of hydraulic impedance, the improved formula of the turbine impedance can be finally obtained and written as, According to the similarity law,

and T T D H
= 11 1 3 , then eliminating the term H, the formula of torque can be derived as, where T 11 is unit torque of the turbine and Q 11 is unit flow rate of the turbine.
The instantaneous torque T and flow rate Q are then replaced by average term and fluctuating term, respectively, As a first approximation, only the following terms are used here, Eliminating the average terms, (19) Equation ( 15) then can be written as below, 3 | RESULTS AND DISCUSSIONS

| Case description
As shown in Figure 1, the hydropower system consists of an upstream reservoir, water diversion tunnel, hydraulic turbine, surge tank, tail tunnel, and downstream reservoir in sequence.The detailed information of tunnels including length and the inner diameter are listed in Table 1, additionally, the corresponding roughness of the penstock and tunnel are 0.012 and 0.014, respectively.Besides, the diameter of the surge tank is 45.01 m and the initial water depth inside is 50.91 m.The designed parameters of the turbine used in the hydropower system are listed in Table 2 and the characteristic curves of the turbine are shown in Figure 2, here, the Q 11 , n 11, and T 11 stand for the unit flow rate, unit speed, and unit torque, respectively.The wave speed in penstock is approximately 1100 and 900-1000 m/s in the tunnel.

| Free vibration analysis
Considering the unit parameters of the turbine, the latter two impedance formulas are used in the following calculation.The operating condition in this section is selected as the designed parameters in  2, the other unit parameters at the operating point are Q 11 = 0.7086 m 3 /s, dQ 11 /dn 11 = −0.00472,T 11 = 1144.55N•m, the corresponding flow rate is Q = 542.586m 3 /s which approaching the designed operating condition.Combining with the parameters and boundary conditions, the free vibration analyses are carried out for the hydropower system and the obtained natural frequencies are listed in Table 3. Column Case 1 lists the results based on Equation (11), column Case 2 lists the results using Equation ( 21) which involves the torque and rotational speed variation characteristics.
It is clear that the angular frequencies corresponding to the same order are almost the same if omit the differences after the decimal point which means that the turbine's torque and speed variation have few influences on the natural angular frequencies of the hydropower system.On the other hand, the results meet well with the previous results which verify the correctness of the newly derived equation.The sign of the attenuation factors are all negative which indicates the pressure or discharge fluctuation will attenuate with time and finally disappear, namely, the system itself is stable.The theoretical period of the surge tank located at the downstream tail tunnel could be calculated according to the theoretical equation T π lA gA = 2 /( ) s l , the obtained period T = 182.24s, the period corresponding to the first order is 182.12 and is in accordance with the theoretical period of surge tank which can also verify the correctness of numerical calculations.

| Vibration mode shapes
With an assumption that an external disturbance exists at the upstream reservoir, the fluctuation amplitude of the | 1141 flow rate is |Q U | = 1.0 m 3 /s, then combined with the natural frequencies listed in Table 3, the oscillatory head and flow rate along the tunnel for some selected orders are calculated and plotted in Figure 3.The abscissa is the axis distance to the upstream reservoir, the left ordinate is the modulus of the oscillatory pressure head corresponding to the black line and the right side represents the oscillatory flow rate corresponding to the red line.
The solid lines are the mode shapes based on frequencies in Case 1, the dashed lines are the results calculated from data in Case 2. The sign "T" and "S" labeled in Figure 3 represent the location of the turbine and surge tank, respectively.The mode shapes reveal the modulus of oscillatory value along the water conveyance system at different frequencies, and the fluctuating pressure head is the minimum at a node point and maximum at a loop point, and the flow rate changes exactly opposite to the head.Severe pressure fluctuations at the loop point may burst the pipe due to excessive pressure or collapse the pipe due to subatmospheric pressure.Consequently, the mode shapes provide a reference for the pipe construction and location of the surge tank, the surge tank is invalid in preventing the transmission of pressure waves from one side to the other side if a pressure node is formed at its base.It is clear in Figure 3 that the pressure node and loop according to both turbine impedance formulas are approximately the same, and the amplitude at lower frequency is higher than that at a higher frequency, which verifies that the lower frequency vibration is more dangerous and should be well avoided in proper vibration reduction measures.In addition, it is obvious in Figure 3C,D,F that the surge tank is located at the pressure node, which means that, if pressure fluctuates at these frequencies, there will be no discharge enter into or out and the surge tank is invalid in the pressure regulating.In Figure 3C,F, it indicates that the speed and torque variation in transients do affect the vibration characteristics of the system since the oscillatory pressure and flow rate amplitude along the downstream side of the surge tank are both lower than before.T A B L E 3 complex frequencies calculated from two different methods.

| Time-domain analysis
According to Equation ( 5), the time history of oscillatory head and flow rate can be written as below From Equations ( 22) and ( 23), it can be found that the time history of oscillatory pressure head and flow rate have the same changing law, the difference between h and q is the initial amplitude and phase angle, therefore, only the time history of the oscillatory pressure head corresponding to the first order is presented here.Assuming a small arbitrary initial oscillatory amplitude value of 1 m, the time history of oscillatory pressure head for the first order is plotted in Figure 4.
It is clear in Figure 4 that the damping rate of vibration has differences.For the same frequency with an extremely small attenuation factor σ, the damping of vibration is slow.Both periods corresponding to the first order in Figure 4 approach the theoretical period of the surge tank but there is a difference after taking the speed | 1143 variation and torque variation into consideration in the mathematical model.The first period in Case 2 is 184.81 s and it is slightly higher than the first period in Case 1 which is 182.02 s.As shown in Figure 4, the damping trend is almost the same as before, however, when taking the torque variation and speed variation into the numerical simulation, the time history of oscillatory pressure head lag behind the former results about 30 s, and as a consequence, the extreme value is slightly lower at the first wave loop.

| Frequency response analysis
The oscillation driven by a known external disturbance is defined as forced vibration.In a steady forced vibration, the attenuation factor is null and the complex frequency only contains an imaginary part, that is, s iω = .Herein, based on the hydropower system in Figure 1 and given a typical external boundary disturbance with a fluctuating pressure head at the surface of the upstream reservoir, which can be described as sinusoidal variation, The state vectors at the upstream and downstream end are related by the overall transfer matrix, the expression is where the matrix c c { , } T 1 2 is the external forcing function acting on the upstream boundary, then expanding Equation (25) yields, (26) Since at the upstream end h U = 0 and downstream end, h D = 0, Equation ( 26) can be written as, Substituting Equation (28) into Equation (27), it becomes, Finally, the modulus of the flow rate oscillation at the downstream end is, Based on Equation ( 30), the flow rate oscillation at the downstream end response to the disturbance can be obtained.The amplitude of the disturbance is assumed to be K = 0.01, then the impedance modulus calculated by the two formulas (Equations 11 and 21) are recorded in Figure 5.Moreover, the frequency response diagram at the downstream end with a disturbance frequency ranging from 0.001 to 20 rad/s is plotted in Figure 6.
As clearly shown in Figure 5, the impedance modulus according to Equation ( 11) remains constant and is independent of disturbance frequency, while Equation (21) indicates that the hydraulic impedance of the turbine is relevant to the disturbance frequency.It tells that once the system is driven by a forcing function, the turbine impedance decreases when the forcing frequency continues increasing, upon the frequency is higher than 0.3 rad/s, it can be approximately regarded as an unchanged value that is equal to the real part and it is still much smaller than that of the former results.It is shown in Figure 6 that once the disturbance frequency is equal to the natural frequency of the system, the pressure peak appears, and severe resonance phenomena will inevitably occur under these frequencies.Additionally, the runner-related frequency and vortex rope in the draft tube is thought to be the main disturbance in hydropower systems.For the turbine in this paper, the runner rotational speed is 11.21 rad/s, and the frequency range of vortex rope in the draft tube under off-design operating condition is about 0.2-0.4times that of runner rotational frequency in early papers, 30 which is about 2.24-4.48rad/s.This obtained frequency range of the possible vortex rope includes the second frequency of the system, and the runner rotational frequency approaches the fifth order, both of them will affect the security of the system, and vibration reduction measures should be taken to avoid possible resonance if dangerous frequencies are monitored during the normal operation process.The frequency response of flow rate at the downstream outlet shows that the peak value is slightly higher than before after considering speed and torque variations.

| Influence of guide vane openings
A series of operating conditions at different guide vane openings are selected for further analysis since the turbine impedance change with the operating conditions.With the premise that the turbine continues working at the designed head and rotational speed and the guide vane openings ranging from 18 to 30 mm with detailed parameters listed in Table 4, the free vibration analysis is then performed.The modulus of turbine impedance corresponding to different operating conditions is recorded in Figure 7, for the reason that when the frequency is higher than 0.3 rad/s, the impedance modulus is almost a constant, only the turbine impedance corresponding to second-order are recorded.
Figure 7 indicates that the modulus of turbine impedance is relevant to the operating conditions, it is clear that the impedance modulus decrease when the guide vane openings increase, also, the value is always smaller than before for different guide vane openings.Based on the obtained turbine impedance, further free vibration analysis is carried out and the results are recorded in Table 5.
Since the obtained angular frequencies are almost the same as the results in Table 3 and it indicates that the natural frequencies of the hydropower system are independent of the operating condition.However, the difference is shown in attenuation factors, the absolute value of attenuation factors increases with the increase of guide | 1145 vane openings, which illustrates that the damping speed of the fluctuating pressure is considerably dependent on the operating conditions.

| CONCLUSION
The hydraulic vibration characteristics in the hydropower system are investigated in this paper, including free vibration analysis and frequency response analysis using the transfer matrix method and hydraulic impedance method.Based on the power equation of the hydraulic turbine, an improved impedance formula involving the torque and speed variation is derived and then applied to hydraulic vibration analysis.Finally, the turbine impedance is calculated and analyzed under different operating conditions.The torque and speed variation has a nonnegligible influence on the value of turbine impedance.The free vibration analysis reveals that the turbine impedance does affect the complex frequencies.Furthermore, the vibration mode shapes are almost the same along the system except that the pressure and flow rate modulus is slightly lower along the tail tunnel for some frequencies when using the improved impedance formula.The frequency response analysis tells that the turbine impedance decreases with the increase of forced frequency until approaching a constant, besides, the resonance peak appears to be higher when involving the torque and speed variation.At length, the impedance decreases with the increase of guide vane opening and the attenuation factors of the system are dependent on the operating conditions.
Accordingly, the torque and speed variation should be taken into consideration in hydraulic vibration analysis and more off-design conditions for traditional turbines including pumped storage power stations should be further investigated.

2
Comprehensive characteristic curves of the turbine.(A) Q 11 versus n 11 and (B) T 11 versus n 11 .

F I G U R E 4
Time history of oscillatory pressure head.I G U R E 5 The impedance of turbine.

Table 2
, and the opening of the guide vane is 26°.According to the similarity law, the unit speed is n Sketch of the pressurized system layout.
SHEN ET AL.
T A B L E 5 attenuation factors at different GVOs with Equation(15).