Mathematical model and parametric uncertainty analysis of a hydraulic generating system
Introduction
In the next 30 years, global hydropower capacity will be doubled roughly from the current 1 billion kW to 2 billion kW [1,2]. The ongoing challenge with increasing number of hydropower stations is stability evaluation of HGSs. Conversion efficiency and unit vibration are two important indexes in evaluating the stability of a HGS. Historically, stability modeling has been split in two directions (see Fig. 1), focusing on the hydro-turbine governing systems (HTGSs) [3] and the shaft systems of hydro-turbine generator units (SSHTGs) [4].
The HTGS consists of penstocks, hydro-turbines, governors, generators, and surge tanks. The configurations of subsystems are various for each hydropower station. Since most of the differences come from penstocks and generators, we sum up the impacts of these two subsystems on dynamic characteristics of HTGS in Table 1 and Table 2. The HTGS models that have been recently developed provide new theories to design controller using high dimensional equations. Sarasua et al. proposed two governor tuning criteria for a long penstock pumped-storage plant [5]. Li et al. introduced Hamiltonian theory to investigate transient stability of a HGS [6]. Riasi et al. made sensitivity analysis of transient flow and numerical analysis of the hydraulic transient response [7,8].
The SSHTG is usually simplified as the generator rotor, generator shaft, turbine runner, and turbine shaft. The SSHTG models are established based on the forces, which usually include oil film forces [4], asymmetric magnetic pull forces [23], and damping forces [24]. The main target of SSHTG models was studied to improve the modeling accuracy. Xu et al. proposed a fractional order model that broadens ranges of amplitude responses by tuning the value of the fractional order [23]. Zeng et al. integrated the SSHTG into the framework of the generalized Hamiltonian system to investigate its vibration characteristics [19].
For the HTGS models, they concern with structures of hydropower stations and ignore dynamic forces acting on the SSHTG. By the way, such SSHTG models involved with the forces and neglect structures of penstocks. They also independently studied the stability for each subsystem. Also, they assumed that parameters were deterministic. In real power stations, some parameters of HTGS and SSHTG are not precisely known or cannot be measured, of which the uncertainties make a difference in efficiency and vibration.
Motivated by the above discussions, first, we propose expressions of unbalanced hydraulic forces on the unit shafting to model hydraulic generating system. We consider both the structures and the forces to make the model. Second, we investigate whole sensitivity and uncertainty of parameters regarding for conversion efficiency and unit vibration. Third, we verify the model with two proposed conventional models.
This paper is structured as follows. We present an HGS model in Section 2. In Section 3, we analyze the parameters in the model. In Section 4, we investigate the sensitivity and uncertainty analysis of the parameters for conversion efficiency and unit vibration and explain the HGS model in the uncertainty theory framework. Finally, conclusions are given in Section 5.
Section snippets
Turbine runner modeling
The lift force of flowing water acting on a runner blade is [25]where is the average value of the relative velocity around the blade; is the average circulation. Hydraulic forces acting on the blade of the turbine runner see Fig. 2. Under Kutta-Joukowski condition, the joint force of the blade is [25]where is the liquid weight around the runner blade; is the lift coefficient of the runner blade; is the resistance coefficient of runner blade,
Parametric uncertainty
The model of the turbine runner is a link between the penstock model and SSHTG model. Since uncertainty in runner model parameters is not taken into account, it is expected that there would exist some inaccuracy when the runner blade and the flowing water interrelate in operation. Based on the proposed model of Eq. (25), we choose six critical parameters from the turbine runner model: the relative height of the guide vane (xx = b0/D1), the diameter ratio (pp = D2/D1), the angle (b2 = ), the
Uncertainty outputs
Combining the models of the turbine runner, the SSHTG, and the penstock, the mathematical model of HGS is established considering parametric uncertainty. As we all know, there is a close correspondence between the conversion efficiency and the unit vibration. Hence, it is very important to understand their relationship by the probability distributions under the impact of the uncertain parameters. Here, calculate ten thousand times to the model to obtain the probability density function of the
Conclusions
A rigorous study on parametric uncertainties in modeling HGS is presented. First, the hydro-turbine runner-a key component of the mathematical model is first proposed considering the inlet and outlet velocity vectors as well as the unbalanced hydraulic forces based on the Kutta-Zhoukowski condition. Second, uncertain outputs of conversion efficiency and unit vibration are investigated, and a normal distribution (i.e., average value = 0.786, and the standard deviation is 0.106) for conversion
Acknowledgements
This work was supported by the Scientific Research Foundation of the National Natural Science Foundation Outstanding Youth Foundation (51622906), National Science Foundation (51479173), Fundamental Research Funds for the Central Universities (201304030577), scientific research funds of Northwest A&F University (2013BSJJ095), the Scientific Research Foundation on Water Engineering of Shaanxi Province (2013slkj-12), the Science Fund for Excellent Young Scholars from Northwest A&F University, and
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