Sliding Mode Control for the Synchronous Generator

Based on the Lyapunov stability theorem and slidingmode control technique, a design of the nonlinear controller is proposed for the dual-excited and steam-valving control of the synchronous generators with matched and mismatched perturbations in this paper. By using some constant gains designed in the sliding surface function, the perturbations in the power system can be suppressed, and the property of asymptotical stability of the rotor angle and the voltage can be achieved at the same time.


Introduction
To achieve a high degree of reliability in the power systems, many works [1][2][3][4][5][6][7][8][9][10][11] have studied the stability of generators.In general, there are two ways to stabilize the generators: the excited control [1][2][3] and the steam-valving control [4][5][6][7].Using the excited control, Xie et al. [1] designed a linear matrix inequality (LMI) controller for a class of multimachine power systems with uncertain parameters to achieve the property of asymptotical stability.Galaz et al. [2] proposed a passivity-based controller and discussed the domain of attraction of the equilibria in power systems.Huang et al. [3] utilized a physical exact linearization method to design a controller for a dual-excited synchronous generators.For the steam-valving control, Zhang and Sun [4], Fu [5], Li et al. [6], and Li et al. [7] designed the adaptive backstepping controller for single machine infinite bus system in the presence of internal and external disturbances to achieve the property of asymptotical stability.
As for the systems with both steam turbine dynamics and the excited generator, Xi et al. [8] and Ma et al. [9] presented a novel nonlinear controller based on Hamiltonian energy theory steam for the turbine dynamics and single excited generator to achieve the property of asymptotical stability.The dual-excitation means the system has -axis and -axis field winding simultaneously.Each field voltage can be adjusted separately and hence the control objectives can be achieved more flexibly.Based on the passive lemma, Wang and Lin [10] designed the bounded passivity controller for the synchronous generators to achieve the property of asymptotical stability.Using the coordinated passivation technique, Chen et al. [11] designed backstepping controller for steam-valving and dual-excited synchronous generators to achieve the property of asymptotical stability.However, the perturbations were not considered in the works [10,11].
Sliding mode control (SMC) is well known to possess several advantages, for example, fast response, good transient performance, robustness of stability, and insensitivity to matched parameter variations and external disturbances [12,13].However, the property of asymptotical stability is in general hard to achieve by using the traditional SMC technique if the mismatched perturbations are presented in the systems [14].A lot of researchers applied SMC techniques to solve the tracking problems with mismatched perturbations [15][16][17].For example, Shieh and Shyu [15], Chen and Dunnigan [16], and Kwan [17] employed SMC techniques for an induction machine with an uncertain load torque; however, the mismatched perturbations considered in these works [15][16][17] belong to the unknown constants.
In this paper, we have proposed the nonlinear sliding mode controller for the dual-excited and steam-valving control of the synchronous generators with matched and 2 ISRN Applied Mathematics mismatched perturbations to achieve the property of asymptotical stability.Our proposed control scheme can be thought of as the extension work of [10,11], where no perturbations are considered in the works [10,11].Furthermore, the mismatched perturbations considered in this paper can be time varying.

System Model
Consider a machine power system with dynamic equations [11] and model uncertainties given by where .We further consider that the model perturbations   , 1 ≤  ≤ 5, may be applied in the power system (1) because the perturbation may come from the modeling errors, uncertainties, and disturbance in the control system.Then, (1) can be written as where  0 =    (0) [11], Remark 1.The assumptions of the mismatched perturbations  1 ( 1 ) and  2 ( 1 ,  2 ,  4 ,  5 ), not in the range of any control effort (,   ,   ), can be seen in some literatures [18,19].However, the stability analysis is not proposed in these works [18,19].  , 3 ≤  ≤ 5, are the matched perturbations.
Assumption 2. The upper bounds of the following vanished perturbations [20] where the system has been in the sliding mode.On the other hand, if the information of this upper bound is unknown, the adaptive mechanism can be used to estimate these parameters.

Design of the Sliding Surface
For tackling the perturbation in (3a)-(3e), the sliding surface  ≜ [ 1  2  3 ]  ∈  3 can be designed as where , where  6 is the known parameter defined in Appendix A. Theorem 3. Consider the perturbed power system (3a)-(3e).If the sliding surface function is designed as (7), the trajectory of state will reach zero asymptotically when the system is in the sliding mode.
Proof.Please see Appendix A.

Design of Controllers
According to (3a)-(3e), the robust controller can be designed as where

Conclusions
In this paper, a sliding mode controller has been successfully designed for the dual-excited and steam-valving control of the synchronous generators with perturbations.Even though the dynamics of the controlled systems are affected by the nonlinear perturbations, some constant gains designed in the sliding surface can effectively overcome these perturbations and achieve asymptotical stability.The proposed control scheme also demonstrates the robustness against the perturbations in the simulation.

A. The Dynamic of the System in the Sliding Mode
According to (3a) and ( 7), one can obtain the dynamics of  1 as Choose the first Lyapunov function candidate as  1 = (1/2) 2 1 .Using Assumption 2, the time derivative of  1 along the trajectory of (A.1) can be given by When the system is in the sliding mode,  = 0, from (8), it can be seen that  4 =  5 = 0 and Using (3b) and (A.3), the closed-loop reduced dynamics of  2 can be rewritten as From Assumption 2, (A.2), and (A.5), we can obtain the time derivative of  2 along the trajectory of (A.4) as )  2 = − 1 ( (A.7) Equation (A.6) implies that  and  1 will approach zero as  → ∞.From ( 7),  2 = ( + ) also reaches zero as  → ∞ because  = (−   1 ) approaches zero as  → ∞.Using (3a)-(3e), it is also noted that  2 (0, 0 ) also reaches zero as  → ∞ because  2 and    2 approach zero as  → ∞ in accordance with (A.3).Since  4 =  5 = 0 and  1 ,  2 ,  3 → 0 as  → ∞, the state trajectory of the state variable will approach zero asymptotically when the system is in the sliding mode.

B. The Proof of the Reaching Mode
From ( 7) and ( 8), one can obtain the time derivative of the sliding variable  as The lumped perturbations  can be assumed to satisfy the constraints See [12].
To prove that the sliding variable  will approach zero in a finite time, we define a Lyapunov function candidate as   = (1/2)  .By using (8)  The preceding equation indicates that the values of  will approach zero in a finite time.

5 Figure 1 :
Figure 1: Responses of the power angle .