Mathematical Problems in Engineering

Volume 2016, Article ID 1504659, 13 pages

http://dx.doi.org/10.1155/2016/1504659

## Transient Simulations in Hydropower Stations Based on a Novel Turbine Boundary

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China

Received 11 March 2016; Accepted 31 May 2016

Academic Editor: Jian Guo Zhou

Copyright © 2016 Yanna Liu 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

Most accidents in hydropower stations happened during transient processes; thus, simulation of these processes is important for station design and safety operation. This study establishes a mathematical model of the transient process in hydropower stations and presents a new method to calculate the hydraulic turbine boundary based on an error function of the rotational speed. The mathematical derivation shows that the error function along the equal-opening characteristic curve is monotonic and has opposite signs at the two sides, which means that a unique solution exists to make the error function null. Thus, iteration of the transient simulation is unique and monotonous, which avoids iterative convergence or false solution and improves the solution efficiency compared with traditional methods. Simulation of an engineering case illustrates that the results obtained by the error function are reasonable. Then, the accuracy and feasibility of the mathematical model using the proposed solution are verified by comparison with model and field tests.

#### 1. Introduction

Because hydropower stations play an important role in the peak regulation and valley filling of a power grid, the hydraulic turbine needs to frequently change its operating conditions and experiences many transient processes. The transient process in hydropower stations, including the interactions among hydraulics, mechanism, and electricity, is complicated. The closure of guide vanes and spherical valve induces a change in the flow inertia, which causes changes in the turbine rotational speed and hydraulic pressure in the piping system. When the working condition dramatically changes during transients, drastic changes in the water-hammer pressure and high rotational speed may lead to serious accidents that will endanger the safety of the hydraulic structure and turbine unit [1–3] and affect the power grid stability [4]. Therefore, simulating the transient process of hydropower stations is necessary. The calculation accuracy is directly related to the design of the water diversion system, safe operation of the hydropower plant, and power quality.

The calculation methods of the water-hammer pressure are analytical [5, 6], graphical [7, 8], and computer-numerical [9]. The analytical method is based on the chain equation of Allievi and the simplifications of the hydraulic turbine as a valve. This method is suitable for simple tube Pelton turbines. On the basis of the analytical method, the graphical method combines the water-hammer wave reflection and superposition with the turbine characteristic curves and guide-vane closing scheme. The traditional numerical simulation can be classified into two types: time- and frequency-domain methods. The time-domain method includes the method of characteristics (MOC) [8–13], finite difference (FD) [14, 15], and finite volume (FV) [16, 17]. In these methods, the water delivery system is divided into a limited number of calculated cells. MOC is mostly applied owing to its advantages: high efficiency of computation, easy implementation for boundary conditions, easy parallelization with FD and FV for pipeline processing [18, 19], and so on. By using MOC, various transition processes can be simulated. In the frequency-domain method [9], the basic equations are linearized to obtain the transfer function of the pipeline. Combined with the transfer functions of the governor and turbine, obtaining the dynamic responses of the hydropower system becomes easy. Irrespective of the time or frequency domain, the transient simulation model includes the pipe model and the turbine boundary.

The turbine boundary in the transient simulations can be divided into two categories [20]. One is based on the geometric size of the turbine [21–23]. The other is the characteristic curve based on the model tests, which is generally adopted in the transient simulations. The characteristic curves are usually transformed into curves with the unit speed as an independent variable and the unit discharge and unit torque as dependent variables. To enable transient simulation, the turbine characteristic curves are formulated into piecewise polygonal functions using an auxiliary grid [24–26]. Then, the operating point consisting of unit parameters is expressed as By combining the two equations and other boundary equations, a one-element cubic equation of the square root of the head can be obtained, and the equation can be solved using the Newton–Simpson iterative method [27]. The roots obtained by this method are correlated to the initial values and it can only obtain the root near the initial values. Meanwhile the first-order derivative of the function which is taken as the denominator should be neither too small nor null; otherwise, the iteration cannot be carried out. Generally, we take the root of last instant as the initial value. If the unit parameters obtained from the root of the iteration just lie in the line segment assumed previously, the root is considered as the correct solution and it will be taken as the initial values of the iteration in next instant. If the unit parameters are not in the assumed line segment, we should extend the search range and solve the cubic equation again until we get the right line segment or search all over the equal-opening curve. If all of the line segments of the equal-opening curve cannot get a right root, we will reduce the accuracy of the iteration properly and search again. If all the methods proposed above cannot get a root, then we will select the root whose operating point is the closest to that of last instant as the right solution. Obviously, this calculation method suffers from two shortcomings. First, the root search direction is ambiguous; when the desired operating point is beyond the line segment, it is not clear whether the next search should target the last or next segment of the current line segment. Second, the roots obtained by the Newton–Simpson iterative method are correlated to the initial values; if the initial values are incorrect, the iteration results may be beyond the assumed line segment. Therefore, the equation may either become unsolvable or be incorrectly solved.

This study establishes a mathematical model of a transient process by MOC and a novel turbine boundary, as shown in Figure 1. In terms of the novel turbine boundary, an error function on the rotational speed is constructed, and a new solution method based on the monotonicity of this error function is presented. The novel solution method is theoretically analyzed, and the superiority of the new solution is validated by transient simulations, model test, and field test.