H∞ Control Design of Power System Based on Generalized Model

The generalized model power system is a complex interconnected system with complex structure and many external disturbance factors, which makes the stability of the system difficult to control. In order to guarantee the stability of the generalized model power system under the condition of external disturbance, more and more effective stability controller design methods are sought. Based on ricarty method and LMI method, the design conditions of H ∞ controller for the generalized power system composed of N subsystems are studied. At the same time, it also provides an effective way and method for analyzing the stability of similar systems.


Introduction
Conventional Power Systems consist of power plants, substations, transmission lines, and load centers. With the development of power industry, regional interconnection is gradually formed and generalized model power system is formed. The generalized model power system is often composed of many subsystems which are interrelated. The dynamic stability problem easily occurs after the interconnection, which seriously affects the whole operation reliability of power system. Therefore, for the generalized model power system, improving the stability of the whole system is the basic condition to ensure the normal operation of the system.
In recent years, the application of h ∞ control theory in time-varying systems [1], Nonlinear Systems [2], stochastic systems [3] and discrete systems [4] has become one of the research hotspots. Among h ∞ control methods, Algebraic Riccati equation Method [5] and Lmi method [6] are easy to grasp and widely used. So far, the domestic and foreign literatures mainly focus on the general power system model to analyze the system stability [7][8][9][10] . By using the Lyapunov equation and Riccati equation of singular systems, the conditions for the existence of decentralized state feedback to stabilize the closed-loop system structure with certain robust stability margin are given in reference [11]. Literature [12] applies sliding mode variable structure method and improved particle swarm optimization algorithm to load power control in interconnected power systems. For the problem of low frequency oscillation in wide area interconnected power systems, a decentralized damping controller is designed for the linearized model of interconnected power systems to improve the stability of the systems. The above literatures only focus on the stability analysis and controller design of general power system. Lmi Method is used to realize the mean square asymptotic stability analysis and controller design for General Power System Model in reference [14]. In this paper, according to the characteristics of the generalized model power system, the H ∞ controller design of the generalized model power system is studied by establishing a reasonable mathematical model and using Riccati method and LMI method respectively, the correctness of the two methods is verified by the same number of examples.

Problem description
Generalized Model Power System is a class of generalized systems, which is composed of several generalized subsystems. The system model can be expressed as Compared with the general generalized system, the model of the system is more complex because of the addition of interconnected subsystems. To stabilize the closed-loop system, the state controller is constructed as follows: Then the system (1) can be represented as a compact system (2) EH are no eigenvalues on the imaginary axis。

Controller design
Riccati method is using the standard tools to sovle the Riccati algebraic inequality equations. Getting the H ∞ controller.
For the system(5) 1 1 12 The state feedback controller is selected replace (6) Here are , , , E A B is stable and pulse-controlled。 ,is the orthogonality condition.
If ② found，then an allowable state feedback matrix is obtained According to the theorem 1，There is a matrix X ,satisfying equation (10) T  T  T  T  T   T  T  T  T  T  T  T  T   T  T  T  T  T   T  T  T  T  T  T  T  T   T  T  T  T  T  T Here is If there is an invertible matrix X make (8) is true，and K is given by(9)，then 0 M = and ( 10 ) is true ， According to the theorem 1,the system ( 7 )

An example analysis of Riccati method
In order to prove the effectiveness of this method, a generalized power system composed of secondorder subsystems is given, with the following parameters: can be detected and pulse view，According to (2)and (3)

H∞ Controller design based on LMI method
Lyapunov matrix inequality is a linear matrix inequality(LMI)，and Riccati matrix inequalities and quadratic matrix inequalities contain quadratic terms ,they are not linear matrix inequalities。Using LMI method to transform the Lyapunov matrix inequality and Riccati matrix inequality into ①There is a state feedback matrix K, such that the system (7) If the proposition ②is true, then the permissible state feedback matrix is If there is a feedback matrix K, then ( ) 2 , E A B K + is admissible . According to theorem 1, there exists an invertible matrix V that satisfies the following inequality: For equation (14), left-multiply matrix T T and right-multiply matrix T , where 1 0 0 (14) to (12) is true. ② → ① If there is invertible matrix X and matrix Y satisfy the inequality (12). And K is given by equation (13). The inverse process of ① → ② can obtain equation (14), and then according to theorem 1, the system (7) is admissible, and () zw T s   ＜ .Theorem 3 is true.

Example simulation of LMI method
In order to prove the effectiveness of this method, a generalized power system composed of secondorder subsystems is given, with the following parameters the same to section 3.2,and , , E A C  = ， ,the interconnected second-order system is sorted out into equation (15). According to equation (12), LMI is used to solve (15)