Mathematical Problems in Engineering

Volume 2015, Article ID 971678, 10 pages

http://dx.doi.org/10.1155/2015/971678

## Research on Francis Turbine Modeling for Large Disturbance Hydropower Station Transient Process Simulation

^{1}School of Power and Mechanical Engineering, Wuhan University, Wuhan 430072, China^{2}College of Water Resources and Architectural Engineering, Northwest A&F University, Yangling 712100, China

Received 8 July 2015; Accepted 3 November 2015

Academic Editor: Renata Archetti

Copyright © 2015 Guangtao Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the field of hydropower station transient process simulation (HSTPS), characteristic graph-based iterative hydroturbine model (CGIHM) has been widely used when large disturbance hydroturbine modeling is involved. However, by this model, iteration should be used to calculate speed and pressure, and slow convergence or no convergence problems may be encountered for some reasons like special characteristic graph profile, inappropriate iterative algorithm, or inappropriate interpolation algorithm, and so forth. Also, other conventional large disturbance hydroturbine models are of some disadvantages and difficult to be used widely in HSTPS. Therefore, to obtain an accurate simulation result, a simple method for hydroturbine modeling is proposed. By this method, both the initial operating point and the transfer coefficients of linear hydroturbine model keep changing during simulation. Hence, it can reflect the nonlinearity of the hydroturbine and be used for Francis turbine simulation under large disturbance condition. To validate the proposed method, both large disturbance and small disturbance simulations of a single hydrounit supplying a resistive, isolated load were conducted. It was shown that the simulation result is consistent with that of field test. Consequently, the proposed method is an attractive option for HSTPS involving Francis turbine modeling under large disturbance condition.

#### 1. Introduction

As energy demand grows and technology advances, more and more hydropower stations are being built or planned to be built. In addition, many existing hydropower stations need to be overhauled or improved to meet new requirements after operating for some time. In this case, to ensure the operating safety of hydropower station with newly designed hydraulic system and wicket gate closing and opening law, detailed hydropower station transient process simulation (HSTPS), which can offer predictions of maximum speed, maximum pressure, and minimum pressure [1], should be conducted before the construction or improvement of hydropower station. In the past, hydropower station transients have caused many catastrophes like equipment damage and casualties [1, 2]. Therefore, the research on HSTPS is of great importance.

Generally, the key to obtain an accurate result of HSTPS lies in the model. Hydropower station transient process model mainly consists of four submodels, that is, (1) pressure conduit system model, (2) large disturbance hydroturbine model (LDHM), (3) governor and servomotor model, and (4) generator and load model. Among these submodels, pressure conduit system model [1, 2], governor and servomotor model [1], and generator and load model [1, 3] have been well studied. However, because of limited knowledge of the complicated dynamic characteristic of hydroturbine and the flowing process of water through hydroturbine, the LDHM is still being developed. In general, there are mainly two types of LDHM widely used for HSTPS. One is characteristic graph-based iterative hydroturbine model (CGIHM), for which the transient turbine pressure head and rotational speed are calculated by iterations on the steady-state hydroturbine characteristic graph [1, 2]. This type of model can provide relatively accurate results in most cases and is convenient to use. Hence it has been used widely for HSTPS, especially when reaction hydroturbine like Francis or Kaplan turbine is applied [2] (for simplicity, only Francis turbine is discussed in this paper hereon). However, because of the using of iterative algorithm, slow convergence or no convergence problems may be encountered due to some reasons like special characteristic graph profile, inappropriate iterative algorithm, or inappropriate interpolation algorithm [4, 5], and so forth. Another popular type of LDHM is valve-assumption hydroturbine model (VHM), for which the hydroturbine is assumed as a valve and the turbine discharge-pressure head model is represented by a valve equation [6–8]. This model is simple, analytical, and appropriate to simulate the impulse turbine; hence it has been used widely in power system simulation [7, 8]. However, the valve-assumption of VHM may not be valid for Francis turbine [2], because the turbine discharge is not only the function of turbine head and gate opening, but also the function of the rotational speed which changes dramatically for most of the HSTPSs.

Besides the above two popular types of LDHM, some other types of LDHM have also been investigated by researchers. The one described in [9, 10] was realized by an internal characteristic analysis method. This model has been used in some engineering cases, and good consistency between simulation results and field test results was claimed. But this method needs detailed turbine parameters such as flow areas and outflow angle of turbine runner [9], which are difficult to acquire. As another method of representing the turbine characteristic graph, neural network hydroturbine model also needs iterations in its simulation process and suffers from nonconvergence problem [11]. In [12], a three-dimensional flow model was used to represent the hydroturbine characteristics and it was indicated that this model could provide rational results without the need for hydroturbine characteristic graph. However, to build the three-dimensional model of hydroturbine, very detailed turbine parameters such as the geometric parameters of turbine water passage are needed, whereas these parameters are quite difficult to acquire. As a result, to the knowledge of the authors, there seems to be no appropriate LDHM for HSTPS when CGIHM cannot offer convergence results. So this paper focuses on this problem and proposed a simple and accurate type of LDHM for HSTPS.

The paper is arranged in the following sequence, after introduction of the problems of existing LDHMs for HSTPS in Section 1, the advantage and disadvantage of an existing turbine modeling method are discussed in Section 2. Based on the discussion in Section 2, a new Francis modeling method under large disturbance condition for HSTPS is proposed in Section 3. In Section 4, a model of a single Francis hydrounit supplying a resistive, isolated load is built, and then the effectiveness of the proposed method is verified by comparing the simulation result with the field test result under several typical disturbance conditions. Finally, conclusions are given in the last section.

#### 2. Variable Transfer Coefficients Hydroturbine Model

It is worth noting that a variable transfer coefficients hydroturbine model (VTCHM) was adopted in [13] to build a hydroturbine regulating system under MATLAB/Simulink environment. This model is a linear turbine model expressed aswhere the 6 transfer coefficients are updated according to the gate opening and unit speed by look-up table and interpolation method.

In (1), , , , , and are deviations of turbine torque per unit (hereinafter referred to as p.u.) based on rated turbine torque, discharge p.u. based on rated discharge, gate opening p.u. based on maximum opening, operating head p.u. based on rated head, and speed p.u. based on rated speed, respectively. They are expressed as , , , , and , where denotes the current simulation step, , and 0 denotes the initial value before the first simulation step. The 6 transfer coefficients can be written as , , , , , and . These coefficients change with and in the simulation process. The unit speed [1] can be expressed as .

It can be seen from (1) that VTCHM is a type of SDHM with the transfer coefficients updating according to current and . It is because of the changing of the transfer coefficients in the simulation process that the result is relatively more accurate than the one obtained when the transfer coefficients are fixed. However, this model should not be used for large disturbance hydroturbine simulation, because it is still linearized around the initial operating point. This means that, no matter where the real operating point is, and are calculated from that fixed operating point in the simulation process. Consequently, error would be increased when the simulation operating point gets far away from the initial operating point. To demonstrate this reason clearly, the schematic diagram of a torque-opening characteristic is shown in Figure 1. As can be seen from Figure 1, when the operating point changes from initial point to a new operating point , if the transfer coefficients in and the initial operating point are used to calculate the torque, a result corresponding to the point will be obtained. Apparently, there is error between the obtained torque and the actual torque . And the error may become larger when the current operating point gets further away from the initial operating point or the nonlinear characteristic of the turbine becomes stronger.