Dynamic evolution of a bulb hydroelectric generating unit considering effects of the blades
Introduction
With the rapid economy development of China in the last two decades, the electricity demand sharply increases [1], [2], [3]. In this process, to make up for the lack of the electricity supply, the water resource, as a clean energy, has been paid more and more attention to. The most typical one is the construction of a lot of hydropower stations for satisfying the electricity demand [4], [5], [6]. And, by 2017, The total installed capacity of the hydropower stations in China has reached to 340 GW [7], [8]. In the long-term operation of hydropower stations, some stability problems of the hydroelectric generating unit have been gradually exposed, which can not only influence the energy conversion efficiency, but also threat the safe operation of the whole hydropower stations [9], [10], [11].
For the hydroelectric generating unit, a mass of studies were put forward to research their stability. And these researches mainly include two aspects. The one hand is the stability analyses for the current state of the hydroelectric generating unit using some monitoring and test data. Based on a multidimensional frequency band energy ratio analysis method, Wang et al. [12] analyzed the pressure fluctuations characteristics in the Francis turbine at different operation conditions and discussed the relationships between the pressure fluctuation and the shaft vibration. Saeed et al. [13] applied a combined method of the numerical modeling and some typical artificial intelligence techniques to the condition monitoring and fault diagnosis of Francis turbine. Alberto et al. [14] analyzed the typical failure reasons of the runner blades based on some engineering cases. Using an integration of grey correlation analysis and entropy weights method, Li et al. [15] analyzed the safety performance of the unit under four heads and presented an optimal operational schedule for managing the electricity uncertainties. Trivedi et al. [16] investigated the changing laws of the pressure amplitudes in the Francis turbine with the increasing rotational speed and gotten the relationship between the pressure and the rotational speed. Lu et al. [17] presented an improved Hilbert-Huang transform method with an energy-correlation fluctuation criterion and then used it to analyze the vibration characteristics of the shaft for a Francis turbine. These analysis methods are advantageous to estimate the current state of the unit in the operation. However, if there are some serious faults in the unit, a large number of field experimentations could aggravate the faults and even cause unpredictable dangers. At the same time, limited by the operation time and monitoring data, the dynamic characteristics of the unit in the long time-scale or some extreme conditions cannot be gotten. Therefore, on the other hand, scholars attempt to establish some reliable and flexible mathematical models to study the system characteristics of the hydroelectric generating unit in the long time-scale or some extreme conditions instead of establishing field experimentations. Gustavsson et al. [18] analyzed the dynamic evolution of the shaft vibration considering the impact and contact between the runner and the discharge ring. Xu et al. [19] established a fractional-order model of the hydroelectric generating unit considering the effects of the unbalanced magnetic pull, and then presented the stability ranges of the unit with the changing of the exciting current. Combining a finite difference guide bearing model and a finite element rotor model, Bettig et al. [20] established an integrated hydro-generator rotordynamic model to calculate natural frequencies, and to analyze the stability of the unit. Due to that these studies only investigated the dynamic characteristics and evolutions of the hydroelectric generating unit under the single fault, many researchers have turned their eyes to the effects of the interaction of the multiple faults or instability factors. Huang et al. [21] analyzed the vibration characteristics of the hydroelectric generating unit considering the rotor rub-impact fault and the parallel misalignment fault. Considering the coupling effects of hydro-electric unbalance force, Kim et al. [22] studied the vibration characteristics of a Francis turbine using rotodynamic methods. In view of the bending and torsion coupled vibration of the generator rotor shaft system, Yan et al. [23] built a fractional model to analyze the dynamic characteristics of the system under multiple faults. For a large Francis hydroelectric generating unit, Xu et al. [24] modeled the oscillation modal interaction under the hydraulic-electrical-mechanical unbalance forces.
The previous studies about the stability of the hydroelectric generating unit mainly aim at the Francis hydroelectric generating unit. However, in the recent years, with the low-head water resources exploited largely, the installed capacity of the tubular hydroelectric generating unit is also rapidly increasing [25], [26], [27]. Thus, the stability analyses and studies of the tubular hydroelectric generating unit will be more essential for development of the hydroelectric engineering. On the one hand, for the stability estimations of the tubular hydroelectric generating unit in the current condition, the previous methods applied in the Francis hydroelectric generating unit are applicative, which has been also widely employed in the actual engineering [28], [29]. On the other hand, for the modeling of the tubular hydroelectric generating unit in the long time-scale or some extreme conditions, the mathematical models of the Francis hydroelectric generating unit in some respects are similar with that of the Francis hydroelectric generating unit and can be applied to the tubular hydroelectric generating unit, such as the mass eccentricity of the generator rotor model, the rub-impact fault model and generator electromagnetic imbalance model. However, because of the differences of the runner structure, the runner model of the Francis hydroelectric generating unit (assumed as a mass eccentricity disk) cannot be applied to the tubular hydroelectric generating unit. Specifically, the runner blades of the tubular turbine are anchored at only one end, which means that it is flexible under some unbalanced exciting forces. So, the flexural vibration of the blades needs to be considered. Namely, it is pivotal for the tubular hydroelectric generating unit to establish a reasonable mathematical model of the runner including the blades. At the same time, for covering the shortage of the stability studies for the tubular hydroelectric generating unit using the monitoring and test data in current conditions, the studies aiming at dynamic evolution characteristics of the tubular hydroelectric generating unit in the long time-scale and extreme conditions are necessary.
In light of the above analyses, comparing with previous works, there are three main advantages making our research attractive. First, in previous works, the mass concentration method is mainly applied in the modeling of the shaft system, in which the effects of the shaft are ignored. Here, a continuum method is employed to build the model of the shaft, in which the bending and torsional vibration of the shaft are both considered. Secondly, the blade model of the bulb hydroelectric generating unit (BHGU) is originally established by simplifying the blade as a cantilever beam. Finally, the dynamic evolutions of the shaft system for the BHGU in the long time-scale and some extreme conditions are analyzed and the effects of the blades on them are also discussed.
The rest of the paper is organized as follows: The model of the whole shaft system is established in Section 2. Based on this model, the dynamic evolution characteristics of the shaft system with some system parameters are discussed in Sections 3 and 4 closes the paper.
Section snippets
Mathematical modeling
For a BHGU, the schematic diagram of the shaft system is shown in Fig. 1. And it is mainly composed of the generator rotor, the shaft, the bearings and the runner. For the convenience of the modeling, the total shaft system of the BHGU can be resolved into two parts. The first one is the blade part which includes all blades. And the second is the disk-shaft part including the disk-1, the shaft and the disk-2, which would be regarded as a shaft with variable cross-sections.
The coordinate systems
Analyses and discussions
The BHGU is a complicated nonlinear system. Therefore, the nonlinear dynamic analysis methods, such as the bifurcation diagram, the largest Lyapunov index, and the phase diagram, will be used to analyze the dynamic characteristics of them. The numerical simulation will be carried out using Gear's method (ode15s solver in MATLAB R2016b). For the simulation variables, the initial values of them are all selected as 1 × 10-6 [23]. The step size is 0.001 [23] and the simulation parameters comes from
Conclusions
In this research, considering the effects of the blades, a novel continuum shaft system model of the BHGU has been built, which is verified by comparing with the test data. At the same time, the superiority of the new model over the previous model without blades is presented. Then, the dynamic evolution processes of the BHGU with some system parameters are presented and the effects of the blade on these processes are also obtained. Here, there are four main conclusions need to be concerned
Conflict of interest
None declared.
Acknowledgment
This work was supported by the Natural Science Foundation of China (grant number 51679171).
References (33)
- et al.
A life-cycle comparison of the energy, environmental and economic impacts of coal versus wood pellets for generating heat in China
Energy
(2017) Feasibility analysis of river off-grid hydrokinetic systems with pumped hydro storage in rural applications
Energy Convers Manage
(2015)- et al.
Comprehensive review of renewable energy curtailment and avoidance: a specific example in China
Renew Sustain Energy Rev
(2015) - et al.
Present situation and future prospect of hydropower in China
Renew Sustain Energy Rev
(2009) - et al.
Parallel chance-constrained dynamic programming for cascade hydropower system operation
Energy
(2018) - et al.
China׳s small hydropower and its dispatching management
Renew Sustain Energy Rev
(2015) - et al.
Hydropower development situation and prospects in China
Renew Sustain Energy Rev
(2018) - et al.
Research on a layered coupling optimal operation model of the Three Gorges and Gezhouba cascade hydropower stations
Energy Conv Manage
(2014) - et al.
Hydro turbine failure mechanisms: an overview
Eng Fail Anal
(2014) - et al.
3D fluid–structure modelling and vibration analysis for fault diagnosis of Francis turbine using multiple ANN and multiple ANFIS
Mech Syst Signal Proc
(2013)
Safety assessment of hydro-generating units using experiments and grey-entropy correlation analysis
Energy
Investigations of unsteady pressure loading in a Francis turbine during variable-speed operation
Renew Energy
Improving Hilbert-Huang transform for energy-correlation fluctuation in hydraulic engineering
Energy
Evaluation of impact dynamics and contact forces in a hydropower rotor due to variations in damping and lateral fluid forces
Int J Mech Sci
Fractional-order modeling and dynamic analyses of a bending-torsional coupling generator rotor shaft system with multiple faults
Chaos, Solitons Fractals
Modeling oscillation modal interaction in a hydroelectric generating system
Energy Convers Manage
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