Stabilization for Damping Multimachine Power System with Time-Varying Delays and Sector Saturating Actuator

This paper studies the stabilization problem for dampingmultimachine power systemwith time-varying delays and sector saturating actuator. The multivariable proportional plus derivative (PD) type stabilizer is designed by transforming the problem of PD controller design to that of state feedback stabilizer design for a system in descriptor form. A new sufficient condition of closedloop multimachine power system asymptomatic stability is derived based on the Lyapunov theory. Computer simulation of a twomachine power system has verified the effectiveness and efficiency of the proposed approach.


Introduction
To cope with the increasing demand for quality electric power, excitation control, power system stabilizer (PSS), and other power system controllers are playing important roles in power system stability and maintaining dynamic performance.Conventional PSS is mainly designed based on a linear model and considered one operating point.Recently, to interconnect large energy pools connecting neighboring electric grids together and transmit bulk energy during peak times of load demand can satisfy the growing demand for energy [1].But it introduces some modes of electromechanical oscillations and frequency deviations within the range of 0.2-2 Hz in the power system which will make power system more complicated [2,3].A conventional PSS cannot guarantee to have the best performance.Hence, a variety of control strategies have been used to obtain PSS, such as lead-lag controller [4], variable structure controller [5][6][7], robust controller [8], PID controller [9][10][11], and fuzzy logic controller [10].Most of the controllers are nonlinear.Some researchers have designed the PSS by using searching algorithms such as genetic algorithms [10,12], particle swarm optimization [13,14], and chaotic optimization algorithm [15,16].But these algorithms are hard to program and are not sure to find the optimum solutions.
It is well known that the amplitude of the controller is always bounded in the real world [17].So it is very necessary that the actuator saturation is taken into consideration.Timedelay is very common in power systems which can be a source of instability of performance degradation [18].Multimachine power system with time-varying delay and sector saturating actuator [19] is a complex interconnected large-scale system that is composed of many electric devices and mechanical components with a better description of real world.The state feedback control problem for such a system is addressed by [19] based on the LMI methods.However, the conditions in [19] are conservative because of the amplifying technique to deal with the nonlinear terms in the conditions.Moreover, we usually cannot find the state feedback controller to satisfy demand when system becomes more complex.
The purpose of this paper is to design a PD controller for damping multimachine power systems with time-varying delay and sector saturating actuator.Under a descriptor transformation, the problem of PD type controller design is transformed into the state feedback controller design for a descriptor system.Then, a new sufficient condition is derived for the admissible of the descriptor system based on the Lyapunov theory.Compared with the existing LMI methods in [19], our method introduces more relax matrix variables.Therefore, it is less conservative.Compared with some of the

Problem Formulation and Preliminaries
Consider -machine power system with time-varying delays and input constraints which is described by the interconnection of  subsystems as follows: where is the control input vector to the actuator, and   () is the control input vector to the plant.
(  (),   ( −   ()) is the nonlinear function vector characterizing the interconnection between th generator and th generator with where   () is the time-varying delay and satisfied The nominal system matrices are represented as follows: In this modeling, the single-machine infinite bus is modeled by Heffron-Phillips model which is shown in Figure 1.
The nonlinear saturation function   () is considered to be inside sector (  , 1) and is shown in Figure 1, where 0 ⩽   ⩽ 1.
The control law for a PD controller is Substituting ( 4) into (1), we have Taking the inverse of the left-hand side of (5), we obtain Properly selecting the controller gains   and   , so that the closed-loop systems are stable, then we have the PD controller design.It is obvious from (6) that the PD controller is nonlinear.Some researchers have designed such a controller with the aid of searching algorithms [10].A huge amount of computation burden is foreseeable.In the following, we introduce a new state variable   () = [   () ẋ   ()]  , then system (1) with controller (4) is transformed into the following PD control system: Figure 1: Heffron-Phillips model for single-machine power system connected to infinite bus along with SSSC series in the transmission line.where where The following definition and lemmas will be useful in this paper.
Definition 1 (see [20]).(i) Descriptor system is said to be regular and impulse-free, if pair (, ) is regular and impulse-free.
(iii) System ( 10) is said to be admissible if it is regular, impulse-free, and stable.

Main Results
In this section, we will give the following condition for system (9).
Theorem 4. The delay descriptor system ( 9) is admissible with 𝑁), such that the following inequalities hold: where  is a fixed scalar which satisfies 0 <  < 1.
Proof.Firstly, we prove that system (9) with PD gain matrices   is regular and impulse-free.System (9) can be rewritten as where From ( 14), it is easy to see that sym That is where  = diag{ 1 , . . .,   }.Since  is descriptor, there exist nonsingular matrices  and  such that Suppose Equation ( 19) yields sym{ 22   22 } < 0, which implies that the pair (, Ã+ BK ) is regular and impulse-free.By Definition 1, system ( 16) is regular and impulse-free.It also shows that system ( 9) is regular and impulse-free.
Remark 7.Both Theorem 4 and Corollary 6 are LMIs.The solutions of   ,   are obtained, and the corresponding controller gain matrices are derived as   =    −  .Our method is a deterministic method which can be solved easier than some of the nondeterministic methods, such as the genetic algorithm [10] and particle swarm optimization [13].

Simulation
In this section, a two-machine infinite bus example system is chosen to show the effectiveness of the proposed method, which is shown in Figure 3.The system parameters used in the simulation are as follows: (49) If we set  1 =  2 = 0.3 and á 1 = á 2 = 0.5, so  1 =  2 varies between 0.4 and 0.9.The upper bound of delay  = 5 and τ * = 0.5.The method in [19] fails to find a state feedback controller for this system.According to Theorem 4, the PD controller can be solved as (50) The close-loop state trajectories of generator 1-2 are shown in Figure 4.

Conclusion
In this paper, a decentralized PD control scheme has been proposed to deal with the time-delay multimachine power system with sector saturating actuator.A sufficient condition of closed-loop system asymptomatic stability is presented in terms of LMIs, which can be solved easily by LMI toolbox.Then, a sufficient condition of state feedback control is also obtained which is less conservative than that in [19].A twomachine infinite bus system is considered as an example, and the simulation result shows the effectiveness of proposed method.

Nomenclature
: Constant of either 1 or 0 and   = 0 means that th generator has no connection with th generator   : th row and th column element of nodal susceptance matrix at the internal nodes after eliminated all physical buses, in pu   : Inertia constant for th generator, in seconds

Figure 3 :
Figure 3: Connecting diagram of two-machine system.
(5)ark 5. Theorem 4 provides a PD control method for system(1).An LMI based criterion is obtained by transforming a regular system into the state feedback stabilizer design for a descriptor system.It is worth noting that if   ,   ,   , , and  are replaced with   ,   ,   , , and   , the state feedback controller can be solved by the following corollary.It is obvious that Theorem 4 has wider range of application.The delay system(5)with   = 0 is stable with   =    −