# Dynamics Analysis for Hydroturbine Regulating System Based on Matrix Model

Authors: Jiafu Wei; Hao Wang; Yuanchu Cheng; Xinran Guo
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The hydraulic turbine model is the key factor which affects the analysis precision of the hydraulic turbine governing system. This paper discusses the basic principle of the hydraulic turbine matrix model and gives two methods to realize. Using the characteristic matrix to describe unit flow and torque and their relationship with the opening and unit speed, it can accurately represent the nonlinear characteristics of the turbine, effectively improve the convergence of simulation process, and meet the needs of high precision real-time simulation of power system. Through the simulation of a number of power stations, it indicates that, by analyzing the dynamic process of the hydraulic turbine regulating with 5-order matrix model, the calculation results and field test data will have good consistency, and it can better meet the needs of power system dynamic simulation.


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## Body

## 1. Introduction

As an important part of power system operation, the accuracy of the model is of great significance to the stability analysis of power system [1, 2]. In the simulation of power system, the mathematical model of hydraulic turbine and governing system is often simplified to improve the calculation speed [3, 4], such as the use of linear model. The calculation error caused by the simplified model of the hydraulic system can be accepted for studying some problems, but not for the research for the long-term dynamic stability simulation of hydropower system, the stable simulation of fluctuation of surge shaft, the startup of turbine generator or sudden load rejection, and hydraulic-machine-electric coupling process simulation, and control and regulation problem into saturation and other nonlinear state; because of the incomplete data and the influence of the nonlinear conditions, the accuracy and reliability of the calculation are seriously affected, and it will even produce the unreasonable conclusions.Therefore, it is necessary to analyze and study the mathematical model of hydraulic turbine governing system. In order to describe the nonlinear characteristics of hydraulic turbine, a simplified nonlinear model is proposed by IEEE. In this model, the turbine is simplified as a valve. Considering the influence of the rigid water hammer and the no-load flow on the turbine model, a simplified analytical model of nonlinear hydroturbine was obtained [5, 6]. This model has good accuracy in the case of small fluctuations, and the model has been widely used in the United States’ western Power Consortium [7, 8]. In article [9], an improved nonlinear model of turbine is derived based on the generalized basic equation of hydraulic turbine; the model is in good agreement with the actual data when analyzing the transient stability and damping of power system. Chang presented a hydraulic turbine model based on internal characteristics of turbine [10]. According to the specific parameters of the hydraulic turbine generator unit and the geometric structure and the movement rule of the control mechanism, the mathematical formula is deduced, and it has definite physical meaning. The simulation using the internal characteristic model has high accuracy with the actual data in the large opening, but it is difficult to measure the parameters of the hydraulic turbine structure [11]. The other methods are to deal with the comprehensive characteristic curve. Article [12] used the two-order differential equation model to describe the turbine system and used the least squares method to identify the parameters and orthogonal surface to fit the characteristics of hydraulic turbine. Good results are obtained in the case of small fluctuations. The BP neural network is used to model the characteristics of the turbine, which improves the convergence of water hammer calculation [13, 14]. However, the commonly used method in the project is still the tabular interpolation of the comprehensive characteristic curve of the turbine [15, 16], while the common interpolation of two-dimensional table will cause derivative’s discontinuity in the whole domain, which leads to increased number of iterations or even no convergence.In order to make up for the deficiency of the numerical table interpolation method in the calculation of turbine characteristic parameters, in this paper, the characteristics matrix is used to model the turbine, and the practical results of several projects are given.


## 2. Modeling Principle of Characteristic Matrix Method

For the convenience of calculation and discussion, it is assumed that the unit flow rate and the unit torque table of a certain type of turbine have been obtained, as shown in Table1. The data in the table represents the condition of n equal opening lines and n unit speeds, where α is the opening, m11 is unit torque, q11 is unit discharge, and x11 is the unit speed of the point. By observing the data in Table 1 and combining with the characteristic curve of turbine, under a given opening α, this paper can use n-1-order polynomial to represent the unit discharge and the unit torque. For example, unit flow can be expressed as follows:(1)q11=A1+A2x+⋯+Anxn-1=A1A2⋯An1x⋮xn-1.Table 1
Unit flow and unit torque table.
α
x
11
x
11
1
x
11
2
⋯
x
11
n
α
1
α
2
⋮
α
n
m
11
11
m
11
12
⋯
m
11
1
n
m
11
21
m
11
21
⋯
m
11
2
n
⋮
⋮
⋯
⋮
m
11
n
1
m
11
n
2
⋯
m
11
n
n
α
1
α
2
⋮
α
n
q
11
11
q
11
12
⋯
q
11
1
n
q
11
21
q
11
21
⋯
q
11
2
n
⋮
⋮
⋯
⋮
q
11
n
1
q
11
n
2
⋯
q
11
n
nIt is obvious that the polynomial coefficients about the unit speed are the function of the openingα. The specific form is shown as follows:(2)A1=a11+a21α+a31α2+⋯+an1αn-1A2=a12+a22α+a32α2+⋯+an2αn-1⋮An=a1n+a2nα+a3nα2+⋯+annαn-1.For an opening, the coefficients can be expressed in matrix form:(3)A1A2⋮Ani=a11a21⋯an1a12a22⋯an2⋮⋮⋮⋮a1na2n⋯ann1αi⋮αin-1,i=1⋯nA1A2⋯Ani=1αi⋮αin-1Ta11a21⋯an1a12a22⋯an2⋮⋮⋮⋮a1na2n⋯annT,i=1⋯n.Combining formulas (1) to (3), one has(4)q11=1αi⋮αin-1Ta11a21⋯an1a12a22⋯an2⋮⋮⋮⋮a1na2n⋯annT1x⋮xn-1.For the convenience of presentation, use the following expressions:(5)xijn×n=11⋯1x111x112⋯x11n⋮⋮⋯⋮x111n-1x112n-1⋯x11nn-1αijn×n=1α1⋯α1n-11α2⋯α2n-1⋮⋮⋯⋮1αn⋯αnn-1aqijn×n=a11a21⋯an1a12a22⋯an2⋮⋮⋮⋮a1na2n⋯annT.According to (4), one can get(6)q11ijn×n=αijn×naqijn×nx11ijn×n.For unit torque, the same can be obtained:(7)m11ijn×n=αijn×namijn×nx11ijn×n.The matrixaqijn×n and amijn×n is defined as the characteristic matrix of the unit flow rate and unit torque of turbine. The characteristic matrix terms in (6) and (7) can be calculated from the unit flow table and the unit torque table of the turbine curves. So the turbine characteristic matrix can be concluded by formulas (8) and (9).(8)aqij=aij-1q11ijx11ij-1(9)amij=aij-1m11ijx11ij-1(10)x11=nDH.n is speed, D represents the diameter of the runner, and H is current head. Through calculating the characteristic matrix, a matrix model of the hydroturbine is got.At a certain time, according to turbine headH and speed n, the unit speed x11t can be calculated with (10); then combining the current guide vane opening, the unit flow rate and unit torque at any time can be quickly obtained by calculating the turbine characteristic parameters with characteristic matrix.(11)q11t=1,αt,αt2,…,αtn-1aqij1x11t⋮x11tn-1m11t=1,αt,αt2,…,αtn-1amij1x11t⋮x11tn-1.The above method can be used to solve the flow and torque at any time, but, in the calculation of the characteristic matrix, the method of selecting the points from the unit flow and torque table is limited. It can only obtainn opening and n unit speed and n×n corresponding points of unit flow rate and unit torque as the input of the arithmetic. In order to improve the flexibility of access points, the method is improved by taking formula (2) directly into formula (1). A matrix equation of (n×n)×(n×n) is needed to obtain all the coefficients. Then the solution of each element of the characteristic matrix is seen in the following formula: (12)a11a21⋮an×n=1α1α12⋯α1n-1x1α1x1α12x1⋯α1n-1x1⋯x1n-1α1x1n-1α12x1n-1⋯α1n-1x1n-11α2α22⋯α2n-1x2α2x2α22x2⋯α2n-1x2⋯x2n-1α2x2n-1α22x2n-1⋯α2n-1x2n-1⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋯⋮⋮⋮⋮⋮1αn×nαn×n2⋯αn×nn-1xn×nαn×nxn×nαn×n2xn×n⋯αn×nn-1xn×n⋯xn×nn-1αn×nxn×nn-1αn×n2xn×nn-1⋯αn×nn-1xn×nn-1-1q1q2⋮qn×n.Accordingly, this paper can use the same method to solve the characteristic matrix of unit torque. Using the improved method to solve the feature matrix can remove the restriction of the point selection. The required characteristic matrix can be calculated as long as the points are linearly independent.(13)a11a21⋮an×n=1α1α12⋯α1n-1x1α1x1α12x1⋯α1n-1x1⋯x1n-1α1x1n-1α12x1n-1⋯α1n-1x1n-11α2α22⋯α2n-1x2α2x2α22x2⋯α2n-1x2⋯x2n-1α2x2n-1α22x2n-1⋯α2n-1x2n-1⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋯⋮⋮⋮⋮⋮1αn×nαn×n2⋯αn×nn-1xn×nαn×nxn×nαn×n2xn×n⋯αn×nn-1xn×n⋯xn×nn-1αn×nxn×nn-1αn×n2xn×nn-1⋯αn×nn-1xn×nn-1-1m1m2⋮mn×n.


## 3. Case Study with 5-Order Matrix Model of Hydraulic Turbine

### 3.1. Case 1

#### 3.1.1. The Model and Parameters

The main parameters of a hydropower station are as follows: rated headHr=216 m, rated speed nr=150 r/min, rated flow Qr=360.3 m3/s, rated power Pr=714 MW, the maximum of the opening amax=28.5 mm (model), head loss coefficients are 0.0098 and 8.402×10-4, and their water flow inertia time constants are 1.3 s and 0.65 s, respectively. The time constant of the surge tank Tj=482 s; impedance hole loss coefficient f0=0.0049. The governor adopts the parallel PID microcomputer governor which is consistent with the actual governor control law and structure; the simulation model is shown in Figure 1. The parameters are set according to the actual governor parameters, permanent speed droop bp = 0.04, proportional gain Kp = 4.5, integral gain Ki = 0.1, differential gain Kd = 2, differential time constant Tn = 0.2, and servomotor time constant Ty = 6.39.Figure 1
The simulation model of governor.The mathematical model of the generator uses 1-order model; the transfer function is(14)GSS=1TaS+eg.In the formula,Ta is the unit inertia time constant; eg is the generator load self-regulation coefficient.


#### 3.1.2. Simulation Results and Analysis

Simulation step is 0.01 s andH=164 m. The simulation was carried out in small and large disturbance with 5-order matrix model of hydraulic turbine and IEEE proposed turbine model. When the disturbance is small, the frequency change process is more consistent, but the absolute position of the servomotor is not the same. Then the simulations for 25% and 50% load rejection results are shown in Figures 2 and 3. The solid lines indicate simulation results of 5-order turbine matrix model and the dashed line represents simulation results using IEEE recommendation model. The blue lines are the prototype test results. The simulation results in Tables 2 and 3 show that when the disturbance is large, the simulation results of the two models are quite different. First, the maximum frequency of 5-order matrix and prototype is obviously different with IEEE proposed turbine model. Besides, the simulation results using 5-order turbine matrix model coincide better with the prototype test results in the law of gate opening movement.Table 2
The result of 25% load rejection.
Variable
5-Order matrix
Prototype
IEEE
Frequency maximum (Hz)
52.50
52.51
51.96
Time of maximum frequency (s)
9.21
9.25
7.62
Frequency minimum (Hz)
49.99
49.98
49.52
Time of minimum frequency (s)
20.82
21.03
15.73
Time of stabilization (s)
42.13
41.35
48.23
Time of servomotor reversing (s)
13.22
13.24
10.09Table 3
The result of 50% load rejection.
Variable
5-Order matrix
Prototype
IEEE
Frequency maximum (Hz)
58.20
58.18
57.12
Time of maximum frequency (s)
9.98
10.01
10.02
Frequency minimum (Hz)
49.98
49.96
48.85
Time of minimum frequency (s)
37.83
37.80
23.72
Time of stabilization (s)
46.85
47.32
51.24
Time of servomotor reversing (s)
24.78
24.76
18.54Figure 2
Simulation comparison of 25% load rejection.Figure 3
Simulation comparison of 50% load rejection.


### 3.2. Case 2

The main parameters of a hydropower station are as follows: rated headHr=73 m, rated speed nr=150 r/min, rated flow Qr=180.2 m3/s, rated power Pr=115 MW, actual operating head H=71.5 m, the maximum of the opening amax=34 mm (model), and water flow inertia time constant Tw=1.47 s. The governor model adopts the parallel PID microcomputer governor which is consistent with the actual governor control law and structure.Initial condition isH=73 m and P=90 MW, with large net mode; the frequency disturbance is ±0.2 HZ.Using the characteristic matrix to establish the 5-order hydroturbine matrix model, the simulation result of primary frequency regulation is shown in Figure4. The black line indicates the power of the prototype experiment result, and the blue line represents the simulation power according to the law of gate opening of the prototype.Figure 4
The power output using prototype and 5-order matrix models in primary frequency regulation mode after frequency disturbance.Then load regulation test is also simulated, the initial condition isH=71.5 m and P=115 MW, with large net mode, and the range of load regulation is 0~100%; the result of simulation is shown in Figure 5.Figure 5
The power regulation using prototype and 5-order matrix models in large net.From the cases, the 5-order matrix model has similar transition process and steady-state value to the prototype. So the 5-order matrix model can also coincide well with the prototype in the condition of the primary frequency modulation.


## 3.1. Case 1

### 3.1.1. The Model and Parameters

The main parameters of a hydropower station are as follows: rated headHr=216 m, rated speed nr=150 r/min, rated flow Qr=360.3 m3/s, rated power Pr=714 MW, the maximum of the opening amax=28.5 mm (model), head loss coefficients are 0.0098 and 8.402×10-4, and their water flow inertia time constants are 1.3 s and 0.65 s, respectively. The time constant of the surge tank Tj=482 s; impedance hole loss coefficient f0=0.0049. The governor adopts the parallel PID microcomputer governor which is consistent with the actual governor control law and structure; the simulation model is shown in Figure 1. The parameters are set according to the actual governor parameters, permanent speed droop bp = 0.04, proportional gain Kp = 4.5, integral gain Ki = 0.1, differential gain Kd = 2, differential time constant Tn = 0.2, and servomotor time constant Ty = 6.39.Figure 1
The simulation model of governor.The mathematical model of the generator uses 1-order model; the transfer function is(14)GSS=1TaS+eg.In the formula,Ta is the unit inertia time constant; eg is the generator load self-regulation coefficient.


### 3.1.2. Simulation Results and Analysis

Simulation step is 0.01 s andH=164 m. The simulation was carried out in small and large disturbance with 5-order matrix model of hydraulic turbine and IEEE proposed turbine model. When the disturbance is small, the frequency change process is more consistent, but the absolute position of the servomotor is not the same. Then the simulations for 25% and 50% load rejection results are shown in Figures 2 and 3. The solid lines indicate simulation results of 5-order turbine matrix model and the dashed line represents simulation results using IEEE recommendation model. The blue lines are the prototype test results. The simulation results in Tables 2 and 3 show that when the disturbance is large, the simulation results of the two models are quite different. First, the maximum frequency of 5-order matrix and prototype is obviously different with IEEE proposed turbine model. Besides, the simulation results using 5-order turbine matrix model coincide better with the prototype test results in the law of gate opening movement.Table 2
The result of 25% load rejection.
Variable
5-Order matrix
Prototype
IEEE
Frequency maximum (Hz)
52.50
52.51
51.96
Time of maximum frequency (s)
9.21
9.25
7.62
Frequency minimum (Hz)
49.99
49.98
49.52
Time of minimum frequency (s)
20.82
21.03
15.73
Time of stabilization (s)
42.13
41.35
48.23
Time of servomotor reversing (s)
13.22
13.24
10.09Table 3
The result of 50% load rejection.
Variable
5-Order matrix
Prototype
IEEE
Frequency maximum (Hz)
58.20
58.18
57.12
Time of maximum frequency (s)
9.98
10.01
10.02
Frequency minimum (Hz)
49.98
49.96
48.85
Time of minimum frequency (s)
37.83
37.80
23.72
Time of stabilization (s)
46.85
47.32
51.24
Time of servomotor reversing (s)
24.78
24.76
18.54Figure 2
Simulation comparison of 25% load rejection.Figure 3
Simulation comparison of 50% load rejection.


## 3.1.1. The Model and Parameters

The main parameters of a hydropower station are as follows: rated headHr=216 m, rated speed nr=150 r/min, rated flow Qr=360.3 m3/s, rated power Pr=714 MW, the maximum of the opening amax=28.5 mm (model), head loss coefficients are 0.0098 and 8.402×10-4, and their water flow inertia time constants are 1.3 s and 0.65 s, respectively. The time constant of the surge tank Tj=482 s; impedance hole loss coefficient f0=0.0049. The governor adopts the parallel PID microcomputer governor which is consistent with the actual governor control law and structure; the simulation model is shown in Figure 1. The parameters are set according to the actual governor parameters, permanent speed droop bp = 0.04, proportional gain Kp = 4.5, integral gain Ki = 0.1, differential gain Kd = 2, differential time constant Tn = 0.2, and servomotor time constant Ty = 6.39.Figure 1
The simulation model of governor.The mathematical model of the generator uses 1-order model; the transfer function is(14)GSS=1TaS+eg.In the formula,Ta is the unit inertia time constant; eg is the generator load self-regulation coefficient.


## 3.1.2. Simulation Results and Analysis

Simulation step is 0.01 s andH=164 m. The simulation was carried out in small and large disturbance with 5-order matrix model of hydraulic turbine and IEEE proposed turbine model. When the disturbance is small, the frequency change process is more consistent, but the absolute position of the servomotor is not the same. Then the simulations for 25% and 50% load rejection results are shown in Figures 2 and 3. The solid lines indicate simulation results of 5-order turbine matrix model and the dashed line represents simulation results using IEEE recommendation model. The blue lines are the prototype test results. The simulation results in Tables 2 and 3 show that when the disturbance is large, the simulation results of the two models are quite different. First, the maximum frequency of 5-order matrix and prototype is obviously different with IEEE proposed turbine model. Besides, the simulation results using 5-order turbine matrix model coincide better with the prototype test results in the law of gate opening movement.Table 2
The result of 25% load rejection.
Variable
5-Order matrix
Prototype
IEEE
Frequency maximum (Hz)
52.50
52.51
51.96
Time of maximum frequency (s)
9.21
9.25
7.62
Frequency minimum (Hz)
49.99
49.98
49.52
Time of minimum frequency (s)
20.82
21.03
15.73
Time of stabilization (s)
42.13
41.35
48.23
Time of servomotor reversing (s)
13.22
13.24
10.09Table 3
The result of 50% load rejection.
Variable
5-Order matrix
Prototype
IEEE
Frequency maximum (Hz)
58.20
58.18
57.12
Time of maximum frequency (s)
9.98
10.01
10.02
Frequency minimum (Hz)
49.98
49.96
48.85
Time of minimum frequency (s)
37.83
37.80
23.72
Time of stabilization (s)
46.85
47.32
51.24
Time of servomotor reversing (s)
24.78
24.76
18.54Figure 2
Simulation comparison of 25% load rejection.Figure 3
Simulation comparison of 50% load rejection.


## 3.2. Case 2

The main parameters of a hydropower station are as follows: rated headHr=73 m, rated speed nr=150 r/min, rated flow Qr=180.2 m3/s, rated power Pr=115 MW, actual operating head H=71.5 m, the maximum of the opening amax=34 mm (model), and water flow inertia time constant Tw=1.47 s. The governor model adopts the parallel PID microcomputer governor which is consistent with the actual governor control law and structure.Initial condition isH=73 m and P=90 MW, with large net mode; the frequency disturbance is ±0.2 HZ.Using the characteristic matrix to establish the 5-order hydroturbine matrix model, the simulation result of primary frequency regulation is shown in Figure4. The black line indicates the power of the prototype experiment result, and the blue line represents the simulation power according to the law of gate opening of the prototype.Figure 4
The power output using prototype and 5-order matrix models in primary frequency regulation mode after frequency disturbance.Then load regulation test is also simulated, the initial condition isH=71.5 m and P=115 MW, with large net mode, and the range of load regulation is 0~100%; the result of simulation is shown in Figure 5.Figure 5
The power regulation using prototype and 5-order matrix models in large net.From the cases, the 5-order matrix model has similar transition process and steady-state value to the prototype. So the 5-order matrix model can also coincide well with the prototype in the condition of the primary frequency modulation.


## 4. Conclusion

The characteristic of the hydraulic turbine has a strong nonlinearity with the change of the operating conditions. In this paper, the turbine characteristic is modeled with 5-order matrix, and the detailed calculation method is given. The simulation indicates that the method can represent the strong nonlinearity of turbine and has better convergence and fast calculation speed. The calculated result is well identical to the results of prototype experiment. So the model can be used in the calculation of the transient process of the hydraulic turbine governing system and the dynamic simulation of the power system. It can meet the needs of power system dynamic simulation and stability analysis better.



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