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

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 simplemethod for hydroturbinemodeling is proposed. By thismethod, 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.


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 steadystate 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 dischargepressure head model is represented by a valve equation [6][7][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 threedimensional 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.

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 0 Opening Torque environment.This model is a linear turbine model expressed as where the 6 transfer coefficients are updated according to the gate opening  and unit speed  11 by look-up table and interpolation method.
It can be seen from ( 1) that VTCHM is a type of SDHM with the transfer coefficients updating according to current  and  11 .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 A 0 ( 0 ,  ,0 ) to a new operating point A 1 ( 1 ,  ,1 ), if the transfer coefficients in A 1 and the initial operating point A 0 are used to calculate the torque, a result   ,1 corresponding to the point A  1 will be obtained.Apparently, there is error between the obtained torque   ,1 and the actual torque  ,1 .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.
From the above analysis, it can be known that the reason for the producing of the error is that the initial operating point keeps the same during the process of simulation.This makes the new operating point become so far away from the current operating point during large disturbance simulation that the linear approximation method may become no longer valid.In order to overcome this disadvantage of VTCHM, a variable initial operating point and transfer coefficients method is proposed in this paper to build Francis turbine model for HSTPS under large disturbance condition (for convenience, the model is expressed as VIOP-TCHM in the following part of this paper).By this method, both the initial operating point and transfer coefficients of Francis turbine model keep changing during the process of large disturbance simulation, so that the initial operating point can be close to the current operating point.Then, by contrast with VTCHM, more accurate result can be obtained.

Variable Initial Operating Point and Transfer Coefficients Method
As mentioned above, when the variable values of initial operating point are specified as constants in the process of simulation, the error may become larger as the current operating point gets further away from the initial operating point or the nonlinear characteristic of the turbine gets stronger.
In order to overcome this defect, a modeling method with both the initial operating point and the transfer coefficients updating in the process of simulation is proposed in this paper.
To give the derivation of this method, two assumptions are given first as follows.
Assumption 1.For general engineering, the variable values in transient process of hydroturbine are approximately equal to those in steady state.Assumption 2. Under small disturbance condition, the hydroturbine model can be linearized around the initial steady-state operating point.
Here, the first assumption is assumed to be valid for general engineering application [1,2,14], and the second one has been widely accepted and used in the area of turbine modeling [8,15,16].

Formula Derivation.
According to Assumption 1, the variable values of initial operating point at the end of each simulation step (the time step Δ in numerical integration methods [8]) can be calculated by those at the end of previous step plus an increment.It can be expressed as where Δ  ,−1 , Δ  −1 , Δ  −1 , Δℎ  −1 , and Δ  −1 are deviations of   , , , ℎ, and , respectively, between current and previous simulation step.When the system gets into steady state, all these deviations equal zero.
Note that the meanings of the deviations shown here are different from those in (1).In (1), the deviations are calculated from the initial values while, in (2) to ( 6), the deviations are calculated from the previous step results.To distinguish them clearly, superscript apostrophes are used in (2) to (6).
The water hammer head [17] is presented as where ℎ  can be calculated by (5).
And the transfer function of water hammer equation can be written for VIOP-TCHM as where the expression of  ℎ () differs depending on research purpose and water conduit length [1,7].The water hammer model of pressure conduit can be generally divided into inelastic water hammer model and elastic water hammer model, among which the inelastic water hammer model is much simpler and used more widely.According to [7,13,17], the inelastic water hammer model applies to short or medium length pressure conduit.Considering that this paper focuses on the hydroturbine modeling, the inelastic water hammer model is used for analysis.The transfer function of inelastic water hammer model is shown as where   [1,18] is water inertia time constant in s and can be calculated by where  is the length of pressure conduit in m;   is rated water flow in m 3 /s;   is local acceleration due to gravity in m/s 2 ;  is the cross section area of conduit in m 2 ; and   is rated hydroturbine operating head in m.

Simulation Block Diagram.
According to Assumption 2, VIOP-TCHM can be linearized around the current initial operating point as According to (2)-( 6) and ( 11)-( 12), a block diagram of VIOP-TCHM can be obtained as shown in Figure 2, where  −1 represents a transport delay module.However, when this block diagram was used for large disturbance simulation, divergence phenomenon often occurred.The analysis reveals that the reason lies in the water hammer equation (9).When Δℎ  () is calculated by ( 9), there is a numerical differential link which will exemplify the numerical error [19] and lead to an unstable calculation result.
To avoid the numerical differential link, an improved block diagram of VIOP-TCHM is obtained by optimizing the calculation sequence.The details are as follows.
By the improved block diagram, the numerical differential link is changed into integration link.Therefore, the error exemplifying effect can be avoided.Simulations, even under quite serious disturbance conditions, have validated that stable results can be obtained by the block diagram shown in Figure 3.
With regard to the calculation of   ,   ,  ℎ ,   ,   , and  ℎ , the method in [13] was used in this paper.

Case Study
To verify the effectiveness of the proposed modeling method, a digital model of an actual hydroturbine unit in China was built, and comparisons between simulations and actual processes were conducted.

Introduction to Hydrounit for Case Study.
The unit simulated in this paper is a vertical shaft Francis turbine unit of an actual hydropower station at the dam toe.The hydraulic system of the unit consists of a pressure penstock with length of 175 m, a hydroturbine, and a tail water tunnel.There is no surge tank in upstream or downstream conduit.In addition, the pressure conduit has a quite short length of 277.4 m.Therefore, the inelastic water hammer model is employed for simulation.The basic parameters of the unit are provided in Table 1.
In Table 1,   is rated power output of the hydroturbine;  1 is runner diameter of the hydroturbine.

Simulation Model.
To highlight the focus of this paper, a single unit supplying a resistive isolated load is considered.In addition, since the main process concerned in this paper is the rotational speed-pressure head transient process, the electromagnetic process is ignored by using a first-order generator   model, and the electric network and load characteristic are simplified as a load torque represented by a step module.

System Model.
As can be seen in Figure 5, the system model consists of governor, servo system, turbine and pressure conduit system, and generator and isolated load.By this model, the rotational speed , feedback from the generator and isolated load model, is compared with the reference value firstly.Then, the comparison result is used to calculate a gate opening control signal   by the governor model.After that,   is processed by the servo system model to be the servomotor opening output, which equals turbine gate opening  when the nonlinearity between servomotor opening and gate opening is ignored.Receiving the new gate opening value , the turbine and pressure conduit system model will work to update the flow , water head ℎ, and   mechanical torque   to new states.Then, with the new   , speed  will be updated after some response time by the generator and isolated load model.Note that when written in per unit, the rotational speed is equal to the frequency.6 is a typical PID controller.In this diagram,  is the same as that in Figure 5;  ref is speed reference;   ,   , and   are proportional, integral, and differential gains, respectively;  1 is differential time constant; and   is the control output of the governor model.

Servo System Model.
Figure 7 shows the block diagram of the servo system model.Here,  0 is a forward gain representing the electrohydraulic converter;  1 and  2 are the forward and feedback gains of the ancillary servomotor, respectively; and  1 and   are the response time constants of ancillary servomotor and main servomotor, respectively.

Turbine and Pressure Conduit System
Model.The proposed VIOP-TCHM shown in Figure 3 is used as the turbine and pressure conduit system model.Its 6 transfer coefficient tables are calculated by the method proposed in [13].

Generator and Isolated Load Model.
The block diagram of generator and isolated load model is shown in Figure 8, where   is damping coefficient;  0 is initial steady-state rotational speed; and   is generator load torque which is set by a step function module.

Results and Analysis.
To validate the effectiveness of the proposed VIOP-TCHM, simulations were carried out.Considering the concern of this paper is the modeling method of hydroturbine for HSTPS under large disturbance condition, and load rejections may be those of the most serious large disturbances the hydrounit can encounter.Hence some load rejection simulations were conducted.The values of parameters for simulation are listed in Table 2, and the simulation step Δ was set to 0.015 s.The results of 25%, 50%, and 75% rated load rejection simulations are shown in Figures 9, 10, and 11, respectively.Their corresponding initial parameter values are listed in Table 3.In addition, the results of no-load frequency disturbance simulation with the frequency reference stepping from 1.0 p.u. to 1.02 p.u. then back to 1.0 p.u. are shown in Figure 12.The initial variable values of no-load frequency disturbance simulation are listed in Table 4.

Large Disturbance Simulation.
As can be seen from Figure 9, at  = 4.5th second, 25% rated load rejection was imposed through changing the load torque from initial value to 0. Because the turbine torque   was controlled by governor, when the load torque   stepped to zero, the turbine torque cannot change to zero immediately but kept nearly the same until governor started to regulate.Consequently, a net accelerating torque   =   emerged, and then the unit began to speed up.When the speed was high enough, as shown in Figure 9(c), at  = 4.86th second, the governor worked to decrease the gate opening to reduce the turbine torque.However, the speed still kept increasing until the turbine torque became smaller than the damping torque which is composed of water friction, air friction, and   From the curves shown in Figure 9, it can be observed that the contours of the VIOP-TCHM result and the field test result are consistent.In addition, the rising time period, descending process time period, and peak value on the frequency curves of VIOP-TCHM are quite close to those of the field test.This indicates that, through using the VIOP-TCHM as the turbine model for HSTPS, a quite accurate result can be obtained.Note that the turbine head curve was not recorded in the field test; hence the turbine head comparison between simulation result and test result was not conducted.However, the simulation process of turbine head is consistent with the typical load rejection process.From Figures 10 and 11, it can be seen that the simulation curves under 50% and 75% rated load rejection conditions look similar to those under 25% rated load rejection condition, except that the peak value of frequency becomes larger and a valley emerged on the frequency curve as the initial load increasing, which may have something to do with the parameter settings of the governor.
Figures 10 and 11 also show that the simulation result is quite close to the field test result which indicates that the VIOP-TCHM is quite accurate.

Small Disturbance Simulation.
To verify the performance of VIOP-TCHM further, small disturbance simulations were also conducted.The result is shown in Figure 12.In the beginning, the unit was operating under steadystate no-load operating condition.Then, a step disturbance was imposed on the unit suddenly at  = 1.81st second by making the value of governor reference frequency change from 1.0 p.u. to 1.02 p.u.At  = 2.02nd second, the gate opening started to respond and increase sharply, and, almost at the same time, turbine head started to decrease.At  = 3.08th second, turbine head reached the minimum value 0.9196 p.u.Likewise, at  = 63.17thsecond, there was a step change of reference frequency from 1.02 p.u. to 1.0 p.u.This led to the decrease of the gate opening at  = 63.25thsecond and the increase of turbine head.In this process, it can be seen that the curves obtained by taking VIOP-TCHM as the hydroturbine model and those by field test have similar contours, which validates the accuracy of the VIOP-TCHM under small disturbance condition.

Conclusion
Considering that CGIHM, the most widely used large disturbance hydroturbine model in HSTPS, may fail to give results for Francis turbine simulation in some cases, a simple variable initial operating point and transfer coefficients method is proposed in this paper for Francis turbine modeling.By this method, VIOP-TCHM is constructed.To validate the accuracy of VIOP-TCHM, both of large disturbance and small disturbance numerical simulations were carried out.It was shown that the result obtained by VIOP-TCHM was consistent with that by field test.Therefore, with characteristics of simple structure, definite physical meaning, and accurate calculation result, VIOP-TCHM can be effectively used in the HSTPS involving Francis turbine modeling under large disturbance condition.

Figure 1 :
Figure 1: A schematic diagram to show the problem when VTCHM is used for large disturbance simulation.
Obtain x and g from output of Output m t and h Compute m t and h by the block

Figure 6 :
Figure 6: Governor model for case study.

Figure 7 :
Figure 7: Servo system model for case study.

Figure 8 :
Figure 8: Generator and isolated load model for case study.

Table 1 :
Basic parameters of the unit.

Table 2 :
Parameters for simulation.

Table 3 :
Initial values for load rejection simulations.

Table 4 :
Initial values for no-load frequency disturbance simulation.Thereafter, because the frequency was still much larger than reference value, the governor kept decreasing the gate opening to reduce the speed.At  = 12.07th second, the governor started to increase the gate opening to avoid overregulation of the frequency, and turbine head reached the minimum value 0.9832 p.u. after about 0.88 seconds since the gate opening started to increase.When, at  = 25th second, the frequency came back to the frequency deadband of governor and gate opening stabilized around the noload opening, turbine head restored to the value before the disturbance and the unit got into a new steady state.
mechanical friction, and so forth.When gate opening started to decrease, turbine head increased sharply.As shown in Figure9(b), the maximum turbine head 1.135 p.u. emerged at  = 7.26th second, that is, about 2.4 seconds after the moment when the gate opening started to decrease.At  = 9.61st second, the frequency increased to the maximum value; meanwhile the gate opening reached around the no-load opening.