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 equalopening 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.
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 waterhammer pressure and high rotational speed may lead to serious accidents that will endanger the safety of the hydraulic structure and turbine unit [
The calculation methods of the waterhammer pressure are analytical [
The turbine boundary in the transient simulations can be divided into two categories [
This study establishes a mathematical model of a transient process by MOC and a novel turbine boundary, as shown in Figure
Characteristic grid.
The mathematical model of the transient process comprises the pipe, turbine boundary, generator, and speed governor models.
Two sets of characteristic curves are generally used to represent the turbine characteristics. Here, the traditional characteristic curves are grouped into a nonuniform
Nonuniform
Spacecurved surface of the characteristic curves
Spacecurved surface of
Parameterization plane
The measured data points
According to the onedimensional continuity and momentum equations of the pipe flow, the equations along the characteristic lines at the front and back of the turbine joint using MOC are as follows [
The equations for the unit parameters are expressed as
The first derivative of the differential equation of the generator is expressed as
Under an offgrid operation,
The guidevane opening is defined in advance; thus
Parameter
If the turbine is operating under a frequency or power control, (
The unknowns in these equations include the heads
During the calculation of the load rejection in the transient process, the closure law of the guide vane is given, and the guidevane opening is known at every instant. Thus, solving the abovementioned boundary conditions is simplified to seeking the operating points that satisfy the equations. With a known opening value, the equations can be solved as follows [
(1) We assume a value for parameter
Then, the two roots are obtained as follows:
(2) We insert
(3) We define
(4) We insert
The flowchart is shown in Figure
Flowchart of the solution process of the turbine boundary equations based on the space surface.
Ω is a monotonic function in the search segment of the equalopening curve with parameter
Ω has opposite signs at the two boundary points of this segment.
From Assumptions
According to the directional derivative theorem, the directional derivative of Ω in line segment
We define
Therefore, the sign of
The following parts will consider
Evidently,
The signs of
Key points of the characteristic curves.
Table
Signs of the different terms at different segments.
Segment 











Pump region  AB  +  +  +  +  +  +  +  −  − 
BC  +  +  +  +  +  +  +  −  −  


Pump braking region  CD  +  +  +  +  +  +  +  +  + 


Turbine region  DE  +  +  +  +  +  +  +  +  + 
EF  +  +  +  −  +  −  +  +  +  
FG  +  +  +  −  +  −  +  −  −  
GH  +  +  +  −  −  +  +  −  −  
HI  +  −  −  −  −  +  +  −  −  


Braking region  IJ  +  −  −  −  −  +  +  −  − 


Reversepump region  JK  +  −  −  −  −  +  +  −  − 
KL  +  +  +  −  −  +  +  −  − 
Assuming
Variation in
Consequently, except for the BC segment and the neighborhoods of points H and K, the sign of
Assuming
Figure
Variation in
When
When
In summary,
If the current operating point
Similarly, when
If
In summary, when
Based on the systematic discussions of
To validate the proposed solution, simulations of the load rejection set in a pumpedstorage power station were carried out and presented in this section. The details of the pumpedstorage power station, along with simulation results of both the traditional method and proposed solution, are presented as follows.
The simulated pumpedstorage power station has four units (300 MW each and 1200 MW in total) supplied by a single tunnel, and two surge chambers were installed on both upstream and downstream. Figure
Basic pumpturbine information of the engineering cases.
Rated 
Rated 
Rated 
Rated 
Rotational inertia (kg⋅m^{2})  Calculated 
Calculated 
Calculated 

306.1  428.6  419  83.64  6000000  457.00  73.18  306.1 
Schematic diagram of the piping system of a hydropower station.
During the simulation, all turbines simultaneously rejected their loads, except for one that experienced a guidevane mechanical failure and its opening remained unchanged. For the other three turbines with no malfunction, their guide vanes held their position for 10 s and then completely shut in 15 s. Figures
Comparison of the results of the traditional and new solutions.
Variation in trajectories
Variation in flows
Variation in pressure in spiral case
Variation in pressure in draft tube
Until 20 s, the operating trajectories and variations in the flow and pressure at the inlet and outlet of the rotating wheel obtained by traditional method are more logical and credible with no obvious fluctuation. After 20 s, the operational trajectories become thicker at the reverseSshaped region of the turbine characteristic curve and even deviate from the scheduled equalopening curve. Obvious vibrations occur in both the flow and pressure values, indicating that the traditional method is unable to determine the correct operating points. In contrast, no obvious vibration is observed in the calculation results obtained by the proposed method in this paper, and the calculation results until 20 s coincide with those obtained by the traditional solution. Therefore, this new method is clearly superior to the traditional solution and can improve the calculation accuracy of the transient process.
The traditional methods is to solve the head first and then seek the operating point. There may be more than one solutions to the oneelement cubic equation and the roots obtained by the Newton–Simpson iterative method are correlated to the initial values. Because lack of sufficient accordance, we take the root of last instant as the initial value. But when the right root is the further one, we will not obtain the right root based on the current initial value. Reducing the iterative accuracy or specifying one root according to last instant may result in bigger errors, and the unit parameters may deviate from the equalopening curve and the transient parameters will jump up and down nonphysically.
The new method determines the unit parameters and the head at the same time. In the solving process, the equations of transient process and the operating point of the equalopening curve are mutually restrained. All roots obtained are consistent with the physical conditions in the calculation of the transition process: the operating points are in the equalopening curve; the rotational speeds calculated from speed equation and head equation are equal to each other. Therefore the present new solution method avoids the appearance of wrong roots. The results obtained are in line with the physical phenomenon without obvious vibrations.
A pumpedstorage station model consisting of model units, piping systems, measuring system, and generator system was established in the laboratory. The schematic of the model is shown in Figure
Basic unit information of the pipe system.
Pipe section  Inlet  Upstream main pipe  Upstream branch pipe  Upstream branch pipe  Downstream main pipe  Outlet 

Equivalent diameter (m)  0.597  0.357  0.202  0.300  0.426  0.675 
Length (m)  2.251  24.154  2.195  4.891  16.285  1.062 
Basic unit information of the pumpturbines.

Inlet diameter (mm)  Outlet diameter (mm)  Guidevane height (mm)  Number of blades  Number of guide vanes  Rated 
Rated 
Rated 
Rotational inertia (N⋅m^{2}) 

37.91  280  146.34  24.44  9  20  1000  10.54  49.1  66.4 
Pumpedstorage station model.
The basic information on the different sensors used in the experiments is presented in Zeng et al. [
A load rejection experiment was conducted on the model station. The guidevane opening of the pumpturbine was kept constant, and the experimental data are shown as the solid line in Figure
Comparison of the simulated parameters and experimental data.
Pressure in the spiral case
Pressure in the draft tube
Rotating speed
A load rejection test of a pumpedstorage power station in South China was conducted by ALSTOM. This section presents the comparison of the test results with the simulation results obtained through the mathematical model proposed in Sections
Basic unit information of the pumpturbines.
Rated 
Rated 
Rated 
Rated 
Rotational inertia (kg⋅m^{2})  Tested 
Tested 
Tested 

306.1  500  517.4  66.72  6000000  535.03  62.78  298 
Two pressure sensors were installed on the unit. One was located downstream of the ball valve 3 m away from its center line. The other was located at the ell of the draft tube. The altitude of this sensor was 133.12 m. By comparing the results of field test and simulation, we can clearly see and thus could arrive at the conclusion that the simulation has good accuracy and can reflect the hydraulic transients of the load rejection with acceptable errors. These errors may be caused by the inaccurate characteristic curve, imprecise parameters of the piping system, and errors in the installation location of the sensors and crosssectional area of the tubes. In addition, the onedimensional characteristic method cannot simulate the pressure fluctuation, as clearly shown in Figures
Comparison of the simulation and field test.
Closing law of guide vanes and spherical valve
Variation in pressure in spiral case
Variation in pressure in draft tube
Variation in rotational speed
This study has presented a mathematical model of the hydraulic transient process and proposed a new method to solve the turbine boundary. The new solution was based on error function Ω of the rotational speed. According to the change in the values and signs of the error function along the equalopening characteristic curve, we can discriminate and search the root of the transient equations; when
This study has shown that
The simulation of the engineering case illustrates that the operating trajectories obtained by the method proposed in this paper are more reasonable and credible, and the pressure at the end of the spiral case and at the entrance of the draft tube obtained by this method does not sharply jump. In contrast with the model and the field test results, the simulation results can accurately reflect the change process of the transient parameters in the load rejection. Therefore, this method is clearly superior to the traditional solution and can improve the simulation accuracy of the transient process of a hydropower station.
Interpolated coefficients of
Interpolated coefficients of
Diameter of runner inlet [m]
Selfregulation coefficient of power network
Flywheel moment [kg·m^{2}]
Pressure head [m]
Rotating inertia (=
Resistance moment of generator [kg·m]
Driving force moment of hydraulic turbine [kg·m]
Unit torque [kg·m]
Specific speed (m·kW)
Rotational speed [r/min]
Unit speed [r/min]
Rated power [MW]
Crosssection discharge [m^{3}/s]
Unit discharge [m^{3}/s]
Time [s]
Position of an operating point in opening curve
Relative opening degree
Square root of head [
Absolute guidevane opening [mm] or [°]
Angular velocity (=
Value of last time step
Generator parameter
Maximum value
Rated value
Head of the runner outlet
Hydraulic turbine parameter
Head of the runner inlet.
The authors declare that there are no competing interests regarding the publication of this paper.
Yanna Liu developed the theory, produced the results, and wrote the paper; Jiandong Yang conceived the project, provided measurement data of hydropower plants, and supplied guidance as a supervisor; Jiebin Yang performed part of programming work and conducted part of case studies and discussions; Chao Wang performed part of programming work and gave technical support; Wei Zeng performed the model test and polished the paper.
This work was supported under the Key Program of the National Natural Science Foundation of China (Grant no. 51039005).