Forced and Self Oscillations of Hydraulic Systems Induced by Cavitation Vortex Rope of Francis Turbines

With economical energy market strategies based on instantaneous pricings of electricity as function of the demand or the predictions, operators harness more hydroelectric facilities to off-design operating points to cover the variations of the electricity production. Under these operating conditions, Francis turbines develop a cavitating swirling flow at the runner outlet which induces pressure fluctuations propagating in the whole hydraulic system. The core of this cavitating vortex is usually called vortex rope. At resonance conditions, the superimposition of the induced traveling waves gives rise to a standing wave leading to undesirable large pressure and output power fluctuations. The aim of this present work is to predict and to simulate this resonance phenomenon which may happen both in part load or full load operating conditions. The identification of the excitation sources induced by the cavitating vortex rope is performed with numerical simulations based on a three dimensional incompressible model, so called hydrodynamic (HD) model. The assumption of plane wave propagation in the water passages connected to the turbine is set since low surging frequencies are involved. Hence, propagation of these sources is simulated with a one dimensional compressible model, so called hydroacoustic (HA) model. The HA model covers the entire hydraulic system including the source region corresponding to the draft tube of the Francis turbine whereas the HD model covers only the source region. In this present work, a specific HA draft tube model has been developed. A momentum source modeling the forces induced by the flow acting on the draft tube wall is considered. Moreover, the fluctuating cavitation volume is considered as a mass source. Finally, a thermodynamic damping is introduced to model energy dissipation during a phase change between liquid and gas. Investigations at part load conditions aim to simulate the upper part load resonance phenomenon for which frequency of pressure fluctuations are experienced between 2 and 4 times the runner frequency. Measurements were carried out in the framework of the FLINDT project which is therefore the case study for validation. First of all, HA draft tube model parameters have been derived for the investigated operating point considering both single phase and two phase unsteady simulations with the HD model. An analysis of these parameters is performed and comparison between single phase and two phase simulation results is made. It is shown that the cavitation modeling in the HD model is necessary to find the vortex rope precession frequency which depends on the cavitation amount in the vortex core. However, the volume of vapor is underestimated and a correction factor on the Thoma number is necessary to get a good agreement between experiments and simulation results. Moreover it has been shown that the three dimensional flow in the elbow gives rise to HA sources able to excite the hydraulic system. Intensity of the sources are higher when two phase flow simulations are considered. Before simulating the upper part load resonance phenomenon, a preliminary validation of these HA parameters is performed by simulating a standard part load resonance where the vortex rope precession frequency, near 0.3 times the runner frequency, matches with the first eigenfrequency of the hydraulic system. In out of resonance conditions, maximum of pressure fluctuations amplitudes are experienced in the draft tube cone with an amplitude being 1% of the turbine head. However, when resonance occurs, maximum amplitude of pressure fluctuations reaches up to 7%. A good agreement is obtained with the order of magnitudes found in measurements available in the literature. After this preliminary validation, simulation of the upper part load resonance phenomenon has been tackled. It has been found that the mechanism inducing this phenomenon is related to an undesirable fluctuation of the cavitation volume which frequency can match with an eigenfrequency of the hydraulic system. However, this fluctuation is captured for a Thoma number much higher than the experimental one leading to a cavitation volume very small compared to the experiments. A prototype installation of four 478 MW Francis turbines located in the Canada's province British Columbia, has been chosen as the case study to analyze the full load instability phenomenon. Indeed, this instability occurred on prototype and reduced scale model as well. Hence, experimental measurements have been carried out on the reduced scale model aiming to use experimental data to validate the numerical simulations performed with the HA draft tube model. The mass source defined in this model, is described by a decisive parameter which is the mass flow gain factor. Extensively used in previous works for the analysis of this phenomenon, this parameter is defined to represent the effect of the HA fluctuations of the downstream flow rate to the cavitation volume on the mass source. In this present work, the same formulation is used and has been combined with the introduction of a new parameter: the thermodynamic damping. First of all, these HA parameters have been derived for the different investigated experimental operating points from single phase steady simulations. Then, using these computed parameters, a small perturbation stability analysis in the frequency domain has been carried out to identify the stability of the different operating points. The experimental unstable characteristic frequencies have been found out with this modal analysis. However, this analysis in the frequency domain does not give any information about the amplitude of the pressure fluctuations induced by the instability. Hence, time domain HA simulations have been performed. It has been shown that the using of constant HA draft tube model parameters leads to divergent time domain simulations, whereas nonlinear parameters depending on the pressure variable, lead to a limit cycle of finite amplitude fluctuations. Moreover, nonlinearity of the thermodynamic damping is decisive to reach this limit cycle. Finally, a methodology has been set up to predict the instability of the prototype from the investigations on the reduced scale model. A combination of measurements, numerical simulations and computation of the eigenmodes of the reduced scale model installed on test rig, allows the accurate calibration of the HA draft tube model parameters at the model scale. Finally, transposition of these parameters to the prototype according to similitude laws is applied for the stability analysis of the power plant

Cavitation influence on hydroacoustic resonance in pipe

In pipe systems, pressure and flow fluctuations below cutoff frequency propagate as plane waves along pipes. Depending on the pipe length and propagation velocity, resonance leading to high amplitude pressure fluctuation may occur. At low pressure, cavitation is an important source of fluctuation. Beside its active role in the mechanism of noise generation, the cavitation reflects partially the incoming plane waves. This may modify the values of the eigenfrequencies of the system consisting of the pipe, the contained fluid and the vapor cavity. The influence of cavitation is experimentally investigated in a hydroacoustic resonator: a straight pipe connecting two tanks. At three quarters of the pipe length, a bluff body is placed cross flow to generate periodic vortex wake cavitation in a limited section of the pipe. The analysis of the wall pressure measurements along the hydroacoustic resonator results is performed with the help of a one-dimensional transient model of the pipe including the compliance of the cavities created in the wake of the bluff body. The results of the numerical simulations enable the determination of both the eigenvalues within the resulting system of equations and the mode shape of the pressure fluctuations corresponding to the experimental results

Simulation of water column separation in Francis pump-turbine draft tube

The paper presents the modelling, simulation and analysis of the transient behaviour of a 340 MW pump-turbine in case of emergency shutdown in turbine mode with focus on possible draft tube water column separation. The model of a pumped storage power plant with simplified layout is presented. This model includes a penstock feeding one 340MW pump-turbine with the related rotating inertia and a tailrace tunnel. The model of the tailrace tunnel allowing for water column separation simulation is introduced. The simulation results of the transient behaviour of the pump-turbine in case of emergency shutdown in generating mode, with and without downstream water column separation model are presented for different degree of severity triggered by the submergence and the tailrace tunnel length. The amplitudes of the pressure peaks induced by the cavity collapse are analysed with respect to the pressure drop magnitude and tailrace dimensions. The maximum and minimum pressure amplitudes obtained along the tailrace tunnel are analysed for different test case conditions

Determination of Hydroacoustic Draft Tube Parameters by High Speed Visualization during Model Testing of a Francis Turbine

Francis turbines can experience critical instabilities at high load operating points, limiting their maximum power output. The swirling flow developed in the draft tube produces a cavitating axisymmetric volume, acting as an internal energy source leading to a self-excited surge phenomenon. The pulsation of the vortex rope corresponds to one of the eigenfrequencies of the hydraulic system. Efforts to accurately characterize, simulate and predict this phenomenon have been undertaken by several researchers, using a 1-D hydroacoustic model of the full load vortex rope. The key physical parameters are the mass flow gain factor, standing for the excitation mass source of the hydraulic system, the cavitation compliance factor, representing the wave speed and the thermodynamic damping, modeling the energy dissipation between the liquid and the gas. These parameters need to be determined either numerically or experimentally. The aim of the present investigation is to determine the mass flow gain factor and the cavitation compliance using experimental data obtained during a measurement campaign on a reduced scale Francis turbine model and to compare the results to existing CFD data

Hydroacoustic Simulation of Rotor-Stator Interaction in Resonance Conditions in Francis Pump-Turbine

Combined effect of rotating pressure field related to runner blade and wakes of wicket gates leads to rotor stator interactions, RSI, in Francis pump-turbines. These interactions induce pressures waves propagating in the entire hydraulic machine. Superposition of those pressure waves may result in standing wave in the spiral casing and rotating diametrical mode in the guide vanes and can cause strong pressure fluctuations and vibrations. This paper presents the modeling, simulation and analysis of Rotor-Stator Interaction of a scale model of a Francis pump-turbine and related test rig using a one-dimensional approach. The hydroacoustic modeling of the Francis pump-turbine takes into account the spiral casing, the 20 guide vanes, the 9 rotating runner vanes. The connection between stationary and rotating parts is ensured by a valve network driven according to the unsteady flow distribution between guide vanes and runner vanes. Time domain simulations are performed for 2 different runner rotational speeds in turbine mode. The simulation results are analyzed in frequency domain and highlights hydroacoustic resonance between RSI excitations and the spiral case. Rotating diametrical mode in the vaneless gap and standing wave in the spiral case are identified. The influence of the resonance on phase and amplitude of pressure fluctuations obtained for both the spiral case and the vaneless gap is analyzed. The mode shape and frequencies are confirmed using eigenvalues analysis

Unstable Operation of Francis Pump-Turbine at Runaway: Rigid and Elastic Water Column Oscillation Modes

This paper presents a numerical simulation study of the transient behavior of a 2x340MW pump-turbine power plant, where the results show an unstable behavior at runaway. First, the modeling of hydraulic components based on equivalent schemes is presented. Then, the 2 pump-turbine test case is presented. The transient behavior of the power plant is simulated for a case of emergency shutdown with servomotor failure on Unit 1. Unstable operation at runaway with a period of 15 seconds is properly simulated using a 1-dimensional approach. The simulation results points out a switch after 200 seconds of the unstable behavior between a period of oscillations initially of 15 seconds to a period of oscillation of 2.16 seconds corresponding to the hydraulic circuit first natural period. The pressure fluctuations related to both the rigid and elastic water column mode are presented for oscillation mode characterization. This phenomenon is described as a switch between a rigid and an elastic water column oscillation mode. The influence of the rotating inertia on the switch phenomenon is investigated through a parametric study

Non-Linear Stability Analysis of a Reduced Scale Model Pump-Turbine at Off-Design Operation

Nowadays, the pump-storage power plants are a proven solution for storing electricity at large scale and offering flexibility to the power management. Therefore, the hydraulic machines are increasingly subject to off-design operation, start-up and shutdown sequences. However, the fast and frequent switching between pumping mode and generating mode presents technical challenges. In the present study, the reduced scale model of a low specific speed pump-turbine is investigated in generating mode at off-design conditions. The operation in the typical “S-shaped” curve of pump-turbine may become unstable and the machine may switch back and forth from generating mode to reverse pumping mode preventing the correct experimental survey of this part during the model testing. The instability has been solved by a testing procedure imposing a restriction of section and a control valve for being able to increase the energy losses. This procedure, commonly used in model testing of pump-turbines, significantly improves the stability of the machine and allows for the survey of the entire “S-curve”. The aim of the present investigation is to understand and explain the origin of the switch to reverse pumping mode. Thus, a hydro-acoustic test rig model was developed with the In-house EPFL SIMSEN software and a comparison between the systems with and without a restriction of section was studied. A numerical analysis indicates that the operating points of a pump-turbine system are defined by the solution of the equation relating the test rig characteristic and the energy-discharge characteristic of the hydraulic machine for a given rotational speed and a constant guide vanes opening. Furthermore, the addition of a restriction alters the curvature of the test rig characteristic and creates a new degree of freedom to achieve stable operating points in the “S-curve”. Finally, to ensure the stability of each operating points described by the numerical model, an eigenvalue study of the non-linear hydraulic system is necessary

Numerical Simulation of Nonlinear Self Oscillations of a Full Load Vortex Rope

Self excited instabilities or oscillations of a cavitating full load vortex rope occur due to an interaction between the gas volume and the acoustic waves. From the onset of the oscillations, the amplitudes grow until they reach a maximum, called the “limit cycle”. The aim of this paper is to predict and to simulate this full load instability with its corresponding “limit cycle”. The test case is a reduced scale model installed on test rig in the Laboratory for Hydraulic Machines at the EPFL. An advanced hydro acoustic vortex rope model is developed to take into account the energy dissipation due to thermodynamic exchange between the gas and the surrounding liquid. Three key hydro acoustic parameters are set up using both steady CFD simulations and analytical models. First of all, parameters are assumed to be constant and time domain simulation is divergent without reaching the limit cycle. However frequency of instability is well predicted. Then inclusion of nonlinear parameters is found to lead to a limit cycle of finite amplitude. Prediction is compared with results from experiments and is in good agreement. It is shown that nonlinearity of the viscoelastic damping parameter, modelling the energy dissipation, is decisive to reach the limit cycle. Moreover, an energy approach is developed to understand the interaction process between the mass source and the system dissipation to reach the equilibrium at the limit cycle. It brings out that over one period the dissipation can provide energy to the system whereas the mass source dissipates to ensure the equilibrium