Journal of Applied Mathematics
Volume 2015 (2015), Article ID 709272, 9 pages
http://dx.doi.org/10.1155/2015/709272

Research Article

Piecewise Model and Parameter Obtainment of Governor Actuator in Turbine

Jie Zhao, Li Wang, Dichen Liu, and Jun Wang

School of Electrical Engineering, Wuhan University, Wuhan 430072, China

Received 13 December 2014; Revised 10 March 2015; Accepted 10 April 2015

Academic Editor: Georgios Sirakoulis

Copyright © 2015 Jie Zhao 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.

The governor actuators in some heat-engine plants have nonlinear valves. This nonlinearity of valves may lead to the inaccuracy of the opening and closing time constants calculated based on the whole segment fully open and fully close experimental test curves of the valve. An improved mathematical model of the turbine governor actuator is proposed to reflect the nonlinearity of the valve, in which the main and auxiliary piecewise opening and closing time constants instead of the fixed oil motive opening and closing time constants are adopted to describe the characteristics of the actuator. The main opening and closing time constants are obtained from the linear segments of the whole fully open and close curves. The parameters of proportional integral derivative (PID) controller are identified based on the small disturbance experimental tests of the valve. Then the auxiliary opening and closing time constants and the piecewise opening and closing valve points are determined by the fully open/close experimental tests. Several testing functions are selected to compare genetic algorithm and particle swarm optimization algorithm (GA-PSO) with other basic intelligence algorithms. The effectiveness of the piecewise linear model and its parameters are validated by practical power plant case studies.

1. Introduction

Power system analysis is essential for the system security and stability. For medium and long term stability analysis, such as the frequency control, the mathematical model of prime mover and its governor and the accuracy of the parameters are critical [1–4]. After the blackout accidents occurred in the Western United States in 1996, the World Electronic Circuits Council (WECC) in 2001 conducted two large units tripped disturbance tests in the Western United States grid, and the study found that the frequency response of original speed governor model did not match with the measured response process; thus a new thermal governor system model was established to improve the simulation accuracy [5]. To identify parameters and simulate the characteristics of the speed regulation system based on experimental test data is a fundamental work for the power system. Therefore, the study of prime mover and its governor simulation model and parameters identification for the power system is of importance to the theoretical and practical value.

Studies on parameter identification methods of steam turbine governor for different simulation cycles can be found in [6–11]. The fixed oil motive opening and closing time constants are adopted in the existing literatures and power system simulation software settings [9–12], but they are difficult to be applied to the nonlinear characteristics of some thermal power plant actuator valve’s fully open or close test. As for the nonlinear valve of hydroturbine, certain existing literature uses several sets of proportional integral derivative (PID) parameters to meet the different operating conditions [13]. However, PID parameters tuning based on intelligent algorithm is relatively complex, so a compromised parameter identification method under different working conditions is required to avoid the difficulty caused by the large number of PID parameters. For parameter identification methods, the classical methods [14] are only applicable to linear system whereas the intelligent algorithms [13, 15–19] are widely used due to robustness in presence of the input disturbance signals, such as input noises. Nevertheless, the measured data can be easily polluted by the noise, and the parameters need to be identified in time.

For different plants, the speed limit value of the oil motive movement is different through valve fully open and fully close test, so that to get the speed limit value is the key to simulate the characteristics of governor actuator with high precision, since speed limit is related to the oil motive opening and closing time constants. For some thermal power plants, the valve is nonlinear. As a result, piecewise speed limit is adopted, not only the maximum and minimum speed limit. The governor actuator piecewise linear model is established based on the movement principle of the oil motive.

The paper is organized as follows. In Section 2, a governor actuator piecewise linear model is established (hereinafter referred to as the improved model of servo and actuator system) using piecewise opening and closing time constants, including the main opening and closing time constants and the auxiliary opening and closing time constants. The effectiveness of genetic algorithm-particle swarm optimization (GA-PSO) algorithm is compared with other basic intelligence algorithms, through selection of multiple test functions in Section 3. Aimed at the nonlinear characteristics of actuator valve, the calculation method of main opening and closing time constants and their influence on PI parameter identification are discussed in Section 4 and the results and discussion are presented in this section as well. Conclusions are finally drawn in Section 5.

2. Governor Actuator Mathematical Model and Parameter Identification Steps

2.1. Governor Actuator Mathematical Model

Electrohydraulic servo and actuator are essential, one of the most important factors influencing the dynamic characteristics of the speed control system and the ability of primary frequency speed control [20, 21]. The actuator amplifies and converts control signal from the regulator to the opening of the valve, to control steam flow into the turbine. The mathematical model is shown in Figure 1, where , , and are the proportional, integral, and derivative multiple coefficients of the integrated amplifier module, respectively, which is commonly proportional or proportional plus integral part. and are the upper and low limits of the integrated amplifier module, respectively. VELopen and VELclose denote the rapid opening and rapid closing coefficient, respectively. and denote the oil motive opening and closing time constants, respectively. and are the maximum and minimum valve opening value, respectively. is the oil motive stroke feedback link time constant, usually taken as 0.02 seconds.

Figure 1: Servo and actuator system model.

When the valve moves sharply, the servo valve output reaches its limit position, and the oil motive has the maximum opening or closing regulating speed. The output value demonstrates linearity. However, nonlinear valve cases exist and the model in Figure 1 is no longer applicable. Thus an improved model of servo and actuator system is shown in Figure 2, where and also are the opening and closing time constants, respectively, and for the distinction, and are defined as the main opening and closing time constants, respectively. and are defined as the auxiliary opening and closing time constants, respectively. The piecewise opening and closing valve points are determined according to the actual condition.

Figure 2: Improved model of servo and actuator system.

As the model shown in Figure 2, the oil motive operates by applying the main opening and closing time constant when the valve responds to a small disturbance; under big disturbances, the oil motive soon reaches its maximum adjusting speed. In other words, oil motive firstly operates linearly by the main opening and closing time constant. When it is close to the steady state, it acts with the auxiliary opening and closing time constant. So the model can be applied to both linear and nonlinear valves. Parameters that need to be identified in this model are the main and auxiliary piecewise opening and closing time constants of the oil motive, PI parameters, namely, and .

2.2. Parameter Identification Steps

The current existing method to obtain the oil motive opening and closing time constants is based on the fully open or close test curves of the actuator [22]. When the valve operates nonlinearly, taking the fully open test as an example, the diagram using different segmented curves to obtain the oil motive opening time constant is shown in Figure 3.

(a)The whole segment: the valve moves from fully closed to fully open.(b)The front segment: the valve moves from fully closed to about 75 percent opening.(c)The middle segment: the valve moves from about 25 percent opening to 75 percent opening.

Figure 3: Opening time constant calculation diagram.

The blue curve in Figure 3 is the measured actuator fully open curve and the dotted line uses the piecewise opening time constants. If the main opening time constant is obtained from the front segment curve data of fully open test, the expression is shown as

If the main closing time constant is obtained from the front segment curve data of fully close test, the calculation method is

where and are constant coefficients. The whole or middle segment curve of the fully open/close test can also be applied to calculate and , and the expressions are similar to (1)-(2).

Assuming that the piecewise opening and closing valve points are and , and are generally defined as

where , , , and in (1) to (3) are the corresponding time of the curves in the valve opening change process.

The procedure to obtain parameters of the modified actuator model is described as follows.

Step 1. Process the selected input valve instruction and the output valve opening data with the method of wavelet denoising.

Step 2. Calculate the main opening and closing time constants through the linear segment of the fully open or close test curve.

Step 3. Identify and using small disturbance experimental tests data of the valve.

Step 4. Determine the piecewise opening and closing valve points and the auxiliary opening and closing time constants through fully open or close test curve.

Step 5. Check actuator parameters.

The actuator parameter identification process of servo and actuator is shown in Figure 4.

Figure 4: The parameter identification process of servo and actuator.

3. The Parameter Identification Method Based on GA and PSO Algorithm

3.1. The Parameter Identification Method

An improved particle swarm optimization (IPSO) algorithm and GA-PSO algorithm are presented to identify parameters. Particle swarm optimization [17–19] is a kind of bionic algorithm to solve optimization problems. The location and speed of particle should be constantly updated. The concept of inertia weight is introduced to revise the speed update equation to improve search in the global scope. To further enhance the search ability, the learning factors and are linearly changed and the convergence factor is introduced:

Then the location update equation is

To improve the performance of the initial solution, chaos method is used to generate initial particle population. The two-dimensional chaotic map is

Genetic algorithm (GA) [9] is also a kind of optimization method simulating natural biological evolution mechanism applied to many optimization problems. However, it has its limitations such as low efficiency and premature convergence. So a combined GA-PSO algorithm is used to improve optimization performance, and the algorithm process is shown in Figure 5.

Figure 5: The algorithm process of GA-PSO.

The algorithm process of GA-PSO is divided into four parts: Part for particle initialization; Part for genetic algorithm initialization; Part for extremum comparison; Part for the particle position and speed update. Individual fitness value is determined by objective function and the following criterion function is used as fitness function

where is the number of sampling points, is the th simulation output value, and is the th measured output value.

3.2. The Characteristic of Different Algorithms

The test functions of Griewank, Ackley, Schwefel, Shubert, and Schaffer [23, 24] are used to compare the optimizing performance of four algorithms (i.e., the basic PSO algorithm, IPSO algorithm, GA algorithm, and GA-PSO algorithm). The five test functions are named as to , respectively.

For each test function, the algebra of four algorithms is 70, and the maximum iteration step is 100. For the functions with the extreme point of zero, the tolerance of error is set within , and the optimization of four algorithms is tested for 50 times. For each algorithm, the optimal value obtained by per optimization program running has little difference. The average runtime on optimization program costs, average optimal value, and standard deviation of four algorithms are shown in Table 1.

Table 1: The average running time, average optimal value, and standard deviation of the four algorithms.

Function name
PSO
IPSO
GA
GA-PSO
Theory extreme

Average runtime (s)
0.115
0.092
0.453
0.199
0

Average optimal value
0.0049
0.0040
0.0125
0.0019

Standard deviation
0.0071
0.0066
0.0188
0.0039

Average runtime (s)
0.166
0.115
0.501
0.230
0

Average optimal value

0.162

Standard deviation

0.5932

Runtime (s)
0.164
0.170
0.487
0.336
−837.9

Average optimal value
−822.55
−814.28
−837.96
−834.41

Standard deviation
42.70
55.55
0
20.51

Average runtime (s)
0.114
0.122
0.508
0.318
−186.7

Average optimal value
−180.28
−183.46
−181.50
−184.48

Standard deviation
11.43
7.99
10.85
3.84

Average runtime (s)
0.156
0.122
0.465
0.241
0

Average optimal value
0.006
0.004
0.016
0.002

Standard deviation
0.007
0.006
0.034
0.004

is equal to 6.49 × 10⁻¹¹.

In Table 1, the average optimal value is displayed instead of the best and the worst optimal values of four algorithms. The results show that GA algorithm performs better for certain functions (e.g., , namely, the Schwefel function) but the running time is longer. GA-PSO performs best overall with shorter running time.

Take the Schaffer function as an example to give evolving optimization curve. Under the condition of given precision, the fitness value curves of the four identification algorithms are shown in Figure 6.

Figure 6: Curves comparison of fitness value.

The vertical axis in Figure 6 is the logarithm of fitness value. It shows that IPSO algorithm and GA-PSO algorithm can obtain higher accuracy with less algebra.

By comparing several algorithms of five test function optimization results, the optimizing performance of these algorithms ranks from good to bad as follows: GA-PSO algorithm, IPSO algorithm, GA algorithm, and the basic PSO algorithm.

4. Results and Discussion

GA-PSO algorithm which has better parameters identification stability is used to identify the parameters of two different turbine governor actuators with linear and nonlinear characteristics, respectively, and the governor actuator piecewise linear model is verified. Two thermal power plants (i.e., Xiaolongtan power plant and Qujing power plant) with reheat steam turbine in Yunnan province of West China are adopted to test the improved model. The rated power of these two power plants is 300 MW. With the field static tests of turbine, actuator fully open or close wave record can be got, as well as the input valve instruction and output valve opening data under small disturbance experimental tests of the valve.

4.1. The Verification of Piecewise Model

Xiaolongtan power plant with linear valve is studied to verify the improved servo and actuator system model. Fully open wave record of Xiaolongtan power plant is shown in Figure 7.

Figure 7: Servo and actuator fully open wave record of Xiaolongtan power plant.

The valve fully open curve is approximately linear in Figure 7, as well as the fully close test curve. After analyzing each valve movement speed from the fully open and close tests, the valves with similar movement speed are selected. As a result, the main opening and closing time constants are seconds and seconds, and the piecewise opening and closing valve points are approximate to be 0. Therefore the auxiliary opening and closing time constants are set as seconds and seconds.

Take the average valve opening value of the six valve (i.e., GV1 to GV6) curves as the simulation compared valve opening value. The identification results by GA-PSO algorithm are , under small disturbance experimental tests with valve opening ranging from 50% to 55% and valve opening ranging from 50% to 45%. The valve opening simulation compared with the measured data is shown in Figure 8.

(a)

(b)

Figure 8: Valve opening simulation compared with the measured curve.

Figure 8 shows that the results identified by GA-PSO algorithm coincide with the measured data. For Xiaolongtan power plant, since the piecewise opening and closing valve points are set to 0, the simulation results of the improved servo and actuator system model (see Figure 2) are similar to the servo and actuator system model in Figure 1.

4.2. The Influence of Different Calculated Main Opening Time Constants on PI Parameter Identification

Qujing power plant with nonlinear valve is also studied to verify the improved servo and actuator system model. The actuator fully open wave record of Qujing power plant is shown in Figure 9.

Figure 9: Servo and actuator full open recorded diagram of Qujing power plant.

It can be seen from Figure 9 that the former period of fully open test curve is approximately linear, but it turns flat when being close to steady-state value. The fully closed test curve presents similar trend. Given , in (1) to (2), the front, the middle, and the whole segment curve data of valve fully open test are selected to calculate the main opening time constant. Similarly, the main closing time constant is obtained. The results of different methods (see Figure 3) calculating the main opening and closing time constants are shown in Table 2.

Table 2: Main opening and closing time constant calculation results.

Calculated
The whole segment
The front segment
The middle segment

2.54
1.89
2.14

2.35
1.58
1.91

Table 2 shows that since the fully open test curve is nonlinear, the main opening and closing time constant calculation results are different when using different curve data.

The valve opening data ranging from 50% to 55% under small disturbance is selected to discuss the influence of different main opening and closing time constant calculation results on PI parameter identification. The identified PI parameters using GA-PSO algorithm are shown in Table 3.

Table 3: Identification results using GA-PSO algorithm.

Parameters
Range
 s
 s
 s

0~100
9.52
6.83
8

0~100
10.73
7.7
8.87

has been calculated out already. As shown in Table 3, the PI parameter identification results are varying as the calculated main opening time constant changes. In conclusion, the PI parameters identified depend on the calculation results of the main opening and closing time constant. The higher the calculated is, the higher the identified is.

Taking the three groups of parameters in Table 3 into simulation model, the valve opening results are shown in Figure 10.

(a)

(b)

Figure 10: Valve opening simulation and measured contrast.

In Figure 10, the parameters of simulated curve 1 are , . The parameters of simulated curve 2 are , , and the parameters of simulated curve 3 are , . With small disturbance experimental tests contrast, the simulation results of three groups of parameters are consistent with the measured data.

4.3. Actuator Comparison under Large Disturbance Test

As shown in Figure 7, the measured fully open or close test curve is nonlinear, so a certain group of oil motive opening and closing time constants can not match with the measured curve. The main and auxiliary piecewise opening and closing time constants are adopted. The specific plan is as follows.

(a)When valve fully opens, the oil motive opening time switches from 1.89 seconds to 4 seconds at the valve opening of 0.64.(b)When valve fully closed, the oil motive closing time switches from 1.58 seconds to 5.7 seconds at the valve opening of 0.3.

For instance, for Qujing power plant, the parameters are seconds, seconds, seconds, seconds, , and . Actuator PI parameters are , . The comparison of measured and simulation data with and without piecewise oil motive opening/closing time constants is shown in Figure 11.

(a)

(b)

Figure 11: Valve fully open and close simulation comparison.

The simulation results using the actuator piecewise linear model with main and auxiliary piecewise opening and closing time constants coincide with the measured curve (see Figure 11). In conclusion, when the valve is nonlinear, adopting main and auxiliary piecewise opening and closing time constants is not able to influence the simulation of valve under small disturbance. At the same time, simulation under large disturbance is more consistent with the measured curve. Thus, it can solve the problem of nonlinearity of the valve. But using fixed oil motive opening/closing time constants, the difference between the simulation and the measured curve under large disturbance is significant.

5. Conclusion

The GA-PSO parameter identification algorithm is studied and the effectiveness of GA-PSO algorithm is validated by multiple test functions and simulation results. Aimed at actuators in certain power plants whose valve fully open and fully close tests are nonlinear, three segmented curves (i.e., the whole segment curve, the front segment curve, and the middle segment curve) are selected to calculate the main opening and closing time constants and their effect on the PI parameter identification is studied. The conclusion is that the higher the calculated value of the main opening and closing time constant is, the higher the proportion link of actuator is.

Although the existing servo and actuator system model can meet the simulation need under small disturbance of the valve, it has large error with tests under large disturbance. The actuator piecewise linear model with main and auxiliary piecewise opening and closing time constants increases the feasibility of this model, which can fit the cases when the valve is nonlinear. The actual power plant cases verify the reliability of the established governor actuator piecewise linear model and its parameters are effective.

Conflict of Interests

The authors have declared that no conflict of interests exists.

Acknowledgments

This work is funded by the National Natural Science Foundation of China (51307123) and China Postdoctoral Science Foundation (2013M531736).

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