#### Abstract

A single unit infinite system of the excitation and steam valve control was proposed based on Barrier Lyapunov theory of restrictive log type. The input amplitude constraint of the steam valve control was considered, and the coordinated nonlinear backstepping controller was designed by switching mechanism. At the same time, the generator rotor effect was considered to be an external unknown large disturbance on the output system, and the conservativeness of the simple estimates for the upper and lower bounds and scaling disturbance was reduced by Minimax. The Minimax method also ensured that the output of the controller and the power angle were within the prescribed range and inhibited the system output effect of disturbance as much as possible. Finally, simulation results of the generator disturbance of mechanical power in the single unit infinite system show that the control scheme effectively improves the transient stability of the dynamic processes of power systems.

#### 1. Introduction

Turbine generator excitation control and steam valve control are two key methods for improving the stability of power systems. The rising speed of the excitation current is limited to the constant of the field winding, and excitation control is limited to the maximum value of the excitation current (the maximum value of the excitation current of a generator is generally 1.8 up to 2.0 times its rated current). Requiring excessive maximum value excitation current will increase the manufacturing costs of generators [1], and relying just on excitation control to enhance system stability results in limited improvement.

A prime mover using nonlinear control for opening the steam valve is often effective for further improving the transient stability of power systems. In view of the large power reheat turbine generator sets used in power systems, mechanical hydraulic governors are increasingly being replaced by power-frequency electrohydraulic governors. The capacity and load adaptability of the primary frequency are improved by the opening steam valve control in the reheat turbine generator set. Thus, an increasing number of engineering researchers are focusing on the stability of power systems.

Many advanced control methods have been recently applied to the excitation and steam valve controller design of generators. Reference [2] described a nonlinear model of power systems that conducts precise linearization of state variable feedback. In this model, the nonlinear controller of the opening steam valve was designed by differential geometry method, but for the state variables feedback exact linearization method was applied based on the exact state space model; the method was not robust to uncertainty of the parameters and models.

The nonlinear robust controller designed by adaptive backstepping was discussed in [3]. Although the internal and external disturbances of the system were considered, disturbance was just binding scaling, and the bound properties of the control variable were not considered. The nonlinear dispersion steam valve controller was designed by limiting the control variable for multimachine systems, which was difficult to achieve in [1] because of excessive computations. The adaptive backstepping method was used to design a nonlinear adaptive robust controller that simultaneously considered the input constraints of the main valve and the fast valve [4]. The steam valve controller was designed without considering the coordinated excitation and valve control. A design was proposed for a nonlinear adaptive robust controller based on adaptive backstepping in which coordinated excitation and valve control were simultaneously considered under input constraints [5]. However, disturbance was simply limited to the scaling of estimate bounds without consideration of state constraints. Although the state constraints of the main steam valve were considered, the impact of disruption perturbation was not considered in the design processes of [6–9].

Modern power systems are very complex, and they contain a wide variety of devices. These devices differ in physical properties and operating regions. Thus, the different constraints of these various devices should be considered in the automation of power systems [10]. Existing research results were obtained by common Lyapunov function, and the switching mechanism was introduced to obtain a better solution by establishing the constraints of control variable input. However, these studies did not indicate any limit to the state constraints for the system. The control problem for a class of constrained systems was solved by constructing a log type of symmetric Barrier Lyapunov Function [11]. A class of control problems with strictly limited output feedback nonlinear systems was studied in [12]. In the present paper, we propose a new type of log BLF (Barrier Lyapunov Function) to expand the range of the initial selection. This method is convenient and easy for the controller design process.

In addition, this paper introduces the system constraints of the status rotor running angle by constructing Barrier Lyapunov Function to establish the state bounds of the system. First, a nonlinear controller is designed by backstepping. Then, external disturbance is combined with Minimax to reduce the conservativeness of simple scaling disturbances. By using parameter-mapping mechanism, the adaptive law is designed by the upper and lower bounds of the parameters. Then, we estimate the uncertain parameters of the system and establish the coordinated excitation and valve control model. The switching mechanism is introduced to deal with the input amplitude constraint problems of the main steam and fast steam valve control in the system. To satisfy the system’s state constraints, the controller is designed with backstepping which is nonlinear and simultaneously combined with relevant constrained signal and constraint values. The construction of a restrictive log type of Barrier Lyapunov Function guarantees that the state power angle set will be within the prescribed range. The Simulation results reveal that amplitude of the state response curve for the system is small, inhibitory effect of disturbance is good, and the limited control inputs are within the set range.

#### 2. Establishment of the Coordinated Excitation and Valve Control Model

The dynamic equations of generator excitation winding are as follows:where is the excitation voltage, is the time constant of the excitation winding, is the axis transient voltage, is the no load electromotive force of generator, is given the excitation voltage of the stable operation, is the infinite bus voltage, is the equivalent reactance between and , is the equivalent reactance between and , and and are the -axis reactance and transient reactance, respectively.

Thus, the excitation equation is where is the rotor angle of generator, is the rotor speed of the generator, is the generator synchronous speed, is the inertia of generator rotor, is the uncertain parameter of the system model, that is, damping coefficient which cannot be accurately measured, is external disturbance of the generator rotor, is external disturbance of the generator transient potential, and and are considered external disturbances of the system.

The model is based on the excitation control model of power system. The proposed model is relatively close to the actual model because of two factors for consideration. One is the uncertainty derived from the damping coefficient which is difficult to measure in the actual system. The second is the influence of external disturbances both on generator potential transient speed and on transient voltage.

Nowadays, middle reheat is widely used by large-capacity turbine generator sets, and its physical structure is shown in Figure 1.

The high-pressure cylinder and the middle-pressure cylinder are equivalent to an inertia link. The equivalent time constants are and . The equivalent power distribution coefficients are and , with mechanical power output of and , respectively. The total mechanical power output of the prime mover is , which is equal to the mechanical power output of the high-pressure cylinder and the middle-pressure cylinder expressed as and . Figure 2 shows the block diagram of the value control system transfer function.

From Figure 2, we can determine practical valve control system differential equations as follows:where and are the time constants of the high-pressure cylinder and the main valve oil motive, respectively; and are the time constants of the low-pressure cylinder and the regulating valve oil motive, respectively; and are the mechanical power outputs of the high-pressure cylinder and the middle-pressure cylinder, respectively; is the sum time constant of the high-pressure cylinder and the main valve controller; is the sum time constant of the low-pressure cylinder and the fast valve controller; and are the power distribution coefficients of the high-pressure cylinder and the low-pressure cylinder and are 0.7 and 0.3, respectively; and are the control input of the main valve and the fast valve, respectively; and are the min and max control input of the main valve, respectively; and are the min and max control input of the fast valve, respectively; and are the external disturbances of the main valve and the fast valve, respectively.

According to (3) and (4), we can obtain the system equations of the main valve and the fast valve as follows.

The main valve control equation is

The fast valve control equation is

The control model of the main valve and the fast valve contains the uncertain parameters , external disturbances , , and , and input constraint bounds of the main valve and the fast valve control.

In summary, the excitation and the main valve equations are as follows:

The excitation and the fast valve equations are as follows:

The two switching subsystems are the excitation and main valve control system and the excitation and fast valve control system. These systems are very similar and therefore we can only choose coordinated control of the excitation and main valve for instructing the controller design process.

The coordinated control model of the excitation and main valve is as follows:where and are the upper and lower bounds of , respectively, , , , , , , , , , , , is the regulating output, and are nonnegative weighting coefficients that account for the weighted proportion between and , with , is the uncertain parameter that is difficult to accurately measure, and is an uncertain constant.

#### 3. Excitation and Valve Controller Design of State and Input Constraints

Lemma 1 (see [12]). *A Barrier Lyapunov Function is a scalar function , defined with respect to the system on an open region containing the origin, that is continuous, positive definite, has continuous first-order partial derivatives at every point of , has the property as approaches the boundary of , and satisfies along the solution of for and some positive constant .*

Lemma 2 (see [12]). *For any positive constants , let and be open sets. Consider the system , where and is piecewise continuous in . Suppose that there exist functions : and : , continuously differentiable and positive definite in their respective domains, such that**where and are class functions. Let , and belong to the set . If the inequality holds , then remains in the open set .*

*Step 1. *Considering to be a virtual controller and choosing , , let , ; we obtain the common Barrier Lyapunov Function of in the first step as follows:ThenClearly, if , we have .

*Step 2. *
We consider the second equation of (10), our task in this step is to make . By augmenting , the Lyapunov function is choosen as . Satisfying dissipation inequality for arbitrary disturbance, we define the energy function as The performance function isbecauseBy substituting (16) into (14), we getTake the first partial derivative of for and let ; then . Therefore, we can obtain Take the second partial derivative of for ; then we get . We know that has a maximum value for . According to the proof presented in the previous chapter, we know that can get the maximum for . Thus, we can conclude that the performance reaches the maximum. Clearly, is the greatest degree of disturbance impact on the system.

By substituting (18) into (17), we getwhere , , and .

Let

We can get the virtual control where , and we use the estimated as . Then

*Step 3. *
Considering the unknown parameter in the second equation of (10), ; , and are, respectively, the upper and lower bounds of . Augmented , the function of a common Lyapunov function of subsystem 1 and subsystem 2, is where is a mapping mechanism, is the estimate of , is the adaptive law of gain, , , and .

We select the energy function The performance function iswhere and the derivative of formula (23) isThe derivatives of formula (21) are By substituting from (26) to (28) into (24), we get Take the first partial derivative of for and and make them equal to 0; thenThereforeTake the second partial derivative of for and ; then we can get and . We know that has a respective maximum value for and . Therefore, because and can make reach the maximum value, we can also make the performance reach the maximum. Clearly, and are really the greatest degrees of disturbance impact on the system.

By substituting (31) into (29), we getLet The controller functions are By substituting (34) into (32), we getThe adaptive law followswhere Then, we can obtain Let ; we get , taking integral from both sides as follows:Satisfying the dissipation inequality, the closed-loop system (10) is stable asymptotically.

To avoid duplication, we can design coordinated excitation and valve control and just provide the design results as follows:where and . We design the adaptive law as followswhere

Theorem 3. *For the closed-loop system (10), under the condition of Lemma 1, satisfying the initial condition as follows and , the error variable is always in the compact set as follows:*

*Proof. *Due to , we get . For , the state variable satisfies , acoording to Lemma 2. Because of the inequality , we can get .

#### 4. Design of Switching Law

The switching law is designed to ensure that the amplitude controller constraints are not destroyed in this section. Given the existence of a common Lyapunov function in subsystems, subsystems 1 and 2 are stable, and switching systems are stable under arbitrary switching laws. Thus, switching law can be designed in any form to avoid frequent switching systems. Thus, we design the hysteresis switching law as follows [13].

*Hysteresis Switching Law.* Consider , for ; if , , then holding . If , but or , then we get ; similarly, if , but or , then we obtain , where is the switching signal , showing that system runs subsystem 1, the main valve control system; consider , showing that system runs subsystem 2, the fast valve control system. and are, respectively, the opening minimum and maximum of the main valve or the quick steam valve. The hysteresis switching law ensures that the system switches subsystem 2, after the system runs for some time, even if the control input satisfies subsystem 1, to avoid frequent switching and continue running subsystem 2, until the constraints are reached.

#### 5. Simulation Analysis

The system parameters are selected as , , , , , , , and . We choose the stable operation points of the system as , , , , , , and .

The following are simulations for two kinds of stem valve opening degree conditions: within the limit and exceeding the limit.

Mechanical power only appears 10% of the perturbation in a certain period of time (6 s to 6.1 s) and then returns to the known initial value, that is, , and the main valve control amplitude does not exceed the limit,

Figures 3–6 show the state response curves of the system. The solid line denotes the response curve of the controller designed by Barrier Lyapunov, and the dotted line represents the response curve without considering state constraints. Figure 3 shows the condition of main valve control within the limit, in which the state response curve of is constrained. Visibly, the state is within the constraints. Moreover, the oscillation of the amplitude is also small for the other states. As shown in Figures 7, 9, and 10, which show the control input curve, the designed controller gain becomes larger with Barrier Lyapunov Function. The main valve control input does not exceed the limited amplitude, and the switching signal is keeping the value 1, as shown in Figure 8.

Mechanical power only appears 20% of the perturbation in a certain period of time (6 s to 6.1 s) and then returns to the known initial value, that is, , and the main valve control exceeds limited amplitude. Consider

Figures 11 to 14 show the state response curve of the system. The solid line denotes the response curve of the controller designed by Barrier Lyapunov, and the dotted line represents the response curve without considering state constraints. Figure 11 shows the condition of main valve control exceeding the limit, in which the state response curve of is constrained. Visibly, the state is within the constraints. Moreover, the oscillation of amplitude is also small for the other states. As shown in Figures 15, 17, and 18, which show the control input curve, the designed controller gain becomes larger with Barrier Lyapunov Function. When the main valve control input exceeds the limited amplitude, the switching signal changes from 1 to 2, as shown in Figure 16. That is, when the main steam valve is overrun, its control input is 0, and the fast valve starts to work. The valve control inputs are within the set range, without constraints destroyed.

#### 6. Conclusion

In this paper, the coordinated excitation and valve controller of the steam turbine generator is designed by Barrier Lyapunov Function and backstepping method. Simulation results show that the generator power angle did not exceed the allowable value and the excitation and steam valve control gain is limited and within a reasonable range. With just requiring the initial power angle to satisfy certain conditions, we can conclude that the constrained state does not exceed the limit set by the constraint in the system. Thus, the design of the controller reduces the extent strictly required by the initial conditions. Furthermore, we can design a more effective controller and improve the efficiency of the power system.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to thank the anonymous reviewers for their helpful and insightful comments for further improving the quality of this work. This work is supported by the Fundamental Research Funds for the Central Universities, under Grant N130404022, and the National Natural Science Foundation of China, under Grants 61304021 and 61233002, and Specialized Research Fund for the Doctoral Program of Higher Education, under Grant 20020145007.