Multiobjective H 2 / H ∞ control design with regional pole constraints for damping power system oscillations

This paper presents multiobjective H 2 /H ∞ control design with regional pole constraints for damping power system oscillations. The state feedback gain can be obtained by solving a linear matrix inequality (LMI) feasibility problem that robustly assigns the closed-loop poles in a prescribed LMI region. The proposed technique is illustrated with applications to the design of stabilizer for a typical single-machine infinite-bus (SMIB) power system. The LMI-based control ensures adequate damping for widely varying system operating conditions. The simulation results illustrate the effectiveness and robustness of the proposed stabilizer.


Introduction
Power systems are usually large nonlinear systems, which are often subject to low frequency oscillations when working under some adverse loading conditions. The oscillation may sustain and grow to cause system separation if no adequate damping is available. To enhance system damping, the generators are equipped with power system stabilizers (PSSs) that provide supplementary feedback stabilizing signals in the excitation systems. PSSs extend the power system stability limit by enhancing the system damping of low frequency oscillations associated with the electromechanical modes [1]. Many approaches are available for PSS design, most of which are based either on classical control methods [1][2][3] or on intelligent control strategies [4][5][6].
However, as power systems are large nonlinear systems, it is impossible for the system to always run at the preselected operating conditions. When the system is away from the specified operating point, the performance of the PSS will degenerate.
Power systems continually undergo changes in the operating condition due to changes in the loads, generation and the transmission network resulting in accompanying changes in the system dynamics. In other words, the stabilizer should be robust to changes in the power system over its entire operating range.
In the last few years, robust control technique has been applied to power system controller design to guarantee robust performance and robust stability, due to uncertainty in plant parameter variations. Some of those efforts have been contributed to design robust controllers for PSS and/or FACTS devices using H ∞ concept such as mixed-sensitivity [7][8][9][10];  -synthesis or structured singular value (SSV) [11] and H2 norm concept such as LQG [12]. Normally, the problem is formulated as a weighted mixed-sensitivity design and solved by a Riccati approach.
In addition, robust H ∞ design being essentially a frequency domain approach does not provide much control over the transient behavior and closed-loop pole location.
Robust pole placement stabilizer design using linear matrix inequalities (LMIs) has been presented in Ref. [13], where the feedback gain matrix is obtained as the solution of a linear matrix inequality expressing the pole region constraints for polytopic plants.
Design methods based on the H ∞ norm of the closed-loop transfer function have gained popularity, because unlike H2 methods (best known as LQG), they offer a single framework in which to deal both with performance and robustness. On the other hand, since an H2 cost function offers a more natural way of representing certain aspects of the system performance, improving the robustness of H2 based design methods against perturbations of the nominal plant is a problem of considerable importance for practical applications [14]. In the robust H2 approach, the controller is designed to minimize an upper bound on the worst-case H2 norm for a range of admissible plant perturbations. Thus, a combination of H2 control and H ∞ control, called multiobjective or mixed H2/H ∞ control that minimized the H2 norm of some closed-loop function subject to the H ∞ norm constraint of another closed-loop function. Khargonekar et al. [18] considered state-and output-feedback problems of multiobjective H2/H ∞ control and gave efficient convex optimization approach to solve the coupled nonlinear matrix Riccati equations.
With the development of numerical algorithms for solving linear matrix inequality (LMI) problems in the last 8 years, the LMI approach have emerged as a useful tool for solving a wide variety of control problems [16]. One of the advantages of linear matrix inequality (LMI) is mixing the time and frequency domain objectives. This paper proposes a multiobjective H2/H ∞ control design with regional pole constraints for damping power system oscillations base on linear matrix inequality.
The efficiency of an LMI-based design approach as a practical design tool is illustrated with case study, including a typical single-machine infinite-bus (SMIB) power system. The paper is organized as follows. A detailed description of the proposed design procedure is given in Section 2. Dynamic model of the power system is given in Section 3. In Section 4, simulation results are given for a typical single-machine infinite-bus (SMIB) power system to demonstrate the effectiveness of the proposed method. Conclusions are drawn in Section 5.

LMI-based multiobjective H2 /H ∞ controller design
Stability is a minimum requirement for control system. However, in most practical situations, a good controller should also deliver sufficiently fast and well-damped time responses. A customary way to guarantee satisfactory transients is to place the closed-loop poles in a suitable region of the complex s-plane.
This section discusses state feedback synthesis with a combination of multiobjective H2/H ∞ performance and pole assignment specifications. Here, the closed-loop poles are required to lie in some LMI region D contained in the left-half plane. Unconstrained multiobjective H2/H ∞ synthesis is considered in [18], where an LMI-based synthesis procedure is proposed. Excellent background material on LMI may be found in [15].

Introduction of linear matrix inequality
A wide variety of problems in control theory and system can be reduced to a handful of standard convex and quasi-convex optimization problems that involve linear matrix inequalities (LMIs), that is constraints of the form [15]: is convex, and need not have smooth boundary.
When the matrices i F are diagonal, the LMI is just a set of linear inequalities. Nonlinear (convex) inequalities are converted to LMI form using Schur complements. The basic idea is as follows: In other words, the set of nonlinear inequalities Eq. (3) can be represented as the LMI Eq. (2).
Two standard LMI optimization problems are of interest: (1) LMI feasibility problem. Given an LMI , the corresponding LMI feasibility problem is to find or determine that the LMI is infeasible.
(2) Semi-definite Programming problem (SDP). An SDP requires the minimization of a linear objective subject to LMI constraints: where c is a real vector, and F is a symmetric matrix that depends affinely on the optimization variable x . This is a convex optimization problem. constraint, the signal 2 z is the performance associated with the H2 criterion. The state space representation of the controlled system can be written as follows: where all the matrices are constant real matrices of appropriate dimension. The illustration of the controlled system is shown in Fig.1.
Let w z T  and w z T 2 be the closed-loop transfer matrices from the generalized disturbance w to the performance output  z and 2 z , respectively: The goal of multiobjective H2/H ∞ control is to find an internally stabilizing controller K that minimizes the 2 H performance, and places the closed-loop poles in some LMI stability region D that will be explained in the next subsection. In this subsection, pure H2 and H ∞ synthesis are not given. For proofs and more details, see [19,20].
We are now ready to give tractable necessary and sufficient conditions for solving the following multiobjective H2/H ∞ problem: The optimization problem above is not yet convex because of the products KP arising in terms like P A cl . So, defining the variables Y = Y T = P, L = K Y and W = W T and using Schur's complement it is possible to rewrite the problem above as the LMI problem min ) (W trace (11) s.t.

LMI formulation for regional pole constraints
In the synthesis of control systems, meeting some desired transient where the characteristic function

Hermitian matrices).
The location of the closed-loop poles of in Eq. (6) concern with the performance of the closed-loop system, i.e., the stability, the decay rate, the maximum overshoot, the rise time and settling time. Therefore, it is interesting work for control engineers to design the control gain K such that the closed-loop poles of lie in a suitable subregion of the left half plane. The interesting region for control purposes is the set as shown in Fig. 2 , the above LMIs are equivalent to

Multiobjective H2/H ∞ control with regional pole constraints
The combination objectives of robust multiobjective H2/H ∞ control with regional pole constraints can be characterized as follows: The Lyapunov shaping paradigm for multi-objective design provides a greater flexibility than single-objective optimal design techniques such as H ∞ synthesis or H2-norm technique.

Dynamic model of the power system
In this study, a single-machine infinite-bus (SMIB) power system [21] as shown in Fig. 3 is considered. The static fast exciter is shown in Fig. 4. Block diagram of conventional power system stabilizer (CPSS) used for comparison is shown in Fig. 5. Neglecting the effect of damper winding, stator transient and resistance, the synchronous machine together with its excitation system is modeled using the following 4 th order non-linear dynamic equations: It can be seen that this model is non-linear. To permit analysis and control of the

Simulation results
A typical single-machine infinite-bus (SMIB) power system [21] is chosen for analysis of the proposed controller. The machine data and the exciter data are shown in Tables 1 and 2, respectively. The data for CPSS constants is given in Table. 3.   Table 3 CPSS constants  For evaluation purposes, the performance of the system with the proposed controller was compared to the CPSS and H2 control (without LMI stability region).
A small disturbance of 10% step increase in the reference voltage ( ref V ) was applied to the SMIB power system at four different operating conditions. The system eigenvalues and damping ratios of electromechanical modes at various operating conditions are given in Table 5. Note that the damping ratio as shown in Table 5 is written in the brackets. It is clear that the system stability is greatly enhanced with the proposed stabilizer. It can also be seen that all eigenvalues and damping ratios with the proposed stabilizer lie in an LMI region of S. Simulation results shown in Figs 6-9 illustrate the performance and robustness of the proposed PSS under different operating conditions. It can be seen that the proposed PSS yields the better dynamic performance, it is less sensitive to changes in the system parameters and more robust against model uncertainties.  as shown in Fig. 10 and Fig. 11, respectively. Fig. 11 shows both CPSS and H2 control fail to stabilize the system with disturbance (b), the proposed PSS provide good damping characteristics and system is stable under this disturbance. It is clear that the proposed PSS exhibits better damping properties and guarantees robust stability of the power systems.

Conclusions
This paper has presented the design of multiobjective H2/H ∞ control with regional pole constraints for damping power system oscillations. The required state feedback gain has been obtained by solving a linear matrix inequality (LMI) feasibility problem that robustly assigns the closed-loop poles in a prescribed LMI region. The performance of the proposed stabilizer on a SMIB power system is seen to be robust over a wide range of operating conditions. Finally, simulation results show the effectiveness and robustness of the proposed stabilizer to enhance the damping of low frequency oscillations.