Design of Simplified Reduced Order model for Balanced Discrete Time Isolated Power System

C.S.Ravichandran¹     Dr.S.Subha Rani ²    Dr.V.Sundarapandian³

¹ Assistant Professor, EEE Department, Sri Ramakrishna Engineering College, Coimbatore – 641 022

e-mail: eniyanravi@gmail.com

² Assistant Professor,ECE Department,P.S.G.Colege of Technology,Coimbatore - 641 004

e-mail: ssrani61@yahoo.co.in

³ Professor and Head, ICE Department, SRM Institute of Science and Technology, SRM Nagar,

Kattankulathur - 603 203. e-mail: sundarsrm@yahoo.co.in

Abstract

This paper presents a simple procedure for obtaining a reduced order model in the discrete time domain of power system problem. In practical cases, the optimal control may require only restricted set of state variables. The problem of state space model reduction can be used to select the essential states in either the state vector or the output. The reduction procedure is simple and computer oriented. The proposed design is guaranteed to be stable if the original system is stable. The desired reduced order model yields good transient and steady state response matching.

Keywords: Reduced Order Model, Balanced realization, Essential State, Discrete Time Power System.

The power system is being complex, inherently high dimensional, multivariable and real time. For implementation of digital schemes, it is desirable to develop a discrete time model, which supports high speed digital signals. Discrete time system can be represented by difference equations of first order system or a transfer function in the z-domain. Fortmann and Williamson [1] were reconstructing vector linear functions of the state for multiple-output systems. An identification of the essential and non-essential modes of the system has been performed [2]-[5].The objective of this model order reduction is to obtain a reduced order approximant of a complex high order system that retains and reflects the important characteristics of the original system as closely as possible. The principal component approaches [6], [7] are elegant from the point of view of minimal realization; model reduction through subsystem elimination is not a well understood operation. The high dimensional power system uses a proposed procedure [8], [9] to identify the essential states of the linear dynamical system and hence used to indicate which state contains a significant contribution. The main feature of the proposed method is deriving the conditions for asymptotic tracking of these state vectors.

An n^th order linear discrete time system can be modelled by a state equation of the form.

                                                                                                                                              (1)

                                                                                                                                                                  (2)

where 'x' is the 'n' vector state of the system, 'u' is an 'r' vector of input signals and 'y' is the 'm' vector of system measurement output. Let (A, B, C) be an n^th order stable system that is both controllable and observable. The controllability (C_G) and observability (O_G ) gramians of the system (1),(2) are defined as

and                                                                                                     (3)

The gramians and are symmetric, positive definite and equal to a diagonal matrix. These gramians satisfy the following Lyapunov equations

and                                                                                             (4)

For computing the balance transformation matrix (T), such that the matrices andare both equal and diagonal, i.e. the system is internally balanced.

Let the following balanced transformation be applied to the system (1) and (2), where is an n-dimensional balanced state vector. The internally balanced system is obtained as

                                                                                                                                           (5)

                                                                                                                                                              (6)

Where the system matrices are defined by and. The system (5) has distinct eigen values and then the corresponding eigen vectors have been evaluated. The model transformation matrix is found so that the transformation

                                                                                                                                                                 (7)

results in the system (5), (6) being transformed into the following diagonal form

                                                                                                                                               (8)

                                                                                                                                                                  (9)

Where;and (10)

From (8), the plant (5), (6) has the following simple form in the new coordinates

                                                                                                                                      (11)

                                                                                                                                   (12)

                                                                                                                                    (13)

As ,, i.e. takes a constant value in the steady state.

The matrix is invertible. From (12), for large values of ,

,

i.e.,                                                                                                                                 (14)

Using (11),(12),(14), the reduced-order model in the -coordinates as

                                                                                                                                      (15)

                                                                                                                     (16)

By the linear change of coordinates (7), hence, it follows that

                                                                                                                                     (17)

                                                                                                                                   (18)

Using (14) and (18),

                                                                                               (19)

The matrix is invertible. The equation (19) may be simplified as

                                                                                                                     (20)

Define matrices and                                                                                   (21)

Using (21), the equation (20) may be simplified as

                                                                                                                                         (22)

Substituting (22) into (5), (6), the reduced-order model of the given linear plant as

                                                                                                                                    (23)

Where the matrices and are defined by

                                                                                                                                         (24)

Equation (23) represents a reduced order representation of the original system (1).

The design of power system problem for full state vector leads to a design problem with high dimensionality. The reduced order model provides a better transient response from the control of large power system.

3.1 Consider the discrete time model of single area load frequency control of power system [10] at sampling time interval of 0.1 sec as

Using (3), (4), the transformed balanced system is

The eigen values for this system are as

The eigen values are found to be essential eigen values of the original system. Using (23), the proposed reduced order model is

3.2 Consider the discrete time model of voltage regulator [11] at sampling time interval of 0.1 sec. The voltage regulator matrices are given by

Using (3), (4), the transformed balanced system is

The eigen values for this system are as

The eigen values are found to be essential eigen values of the original system.

Using (23), the proposed reduced order model is

The reduced order model were simulated for a given initial conditions using MATLAB. Figure (1) shows the step response of system state variables of original and reduced order system like frequency deviation of perturbed area are obtained for a power system. Figure (2) shows the step response of the original and reduced order systems are very close to each other by plotting them on the same graph. The given example demonstrates the fact that obtaining a low order model system by model transformation reduction method always tracks its essential state of the original system.

Figure 2. Step response of a voltage of power system regulator model

5. Conclusion

This paper proposes the novel reduced order memory less state space linear discrete time power system. The essential part of the state variables of the system was used for the reduction of discrete time systems to obtain stable low order models. The reduced order model has a close approximation to the higher order system and its response characteristics match closely with those of the original system. The computational algorithm is simple, systematic and applicable to any large scale systems with substantial merits over other conventional methods. The simulation results show the aptness of the proposed design method for a power system.

References:

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[8] Ravichandran, S., Subha Rani, S., "Observers for the Dominant States of Linear Difference system," Int.conference on Robotics Vision Informatics and Signal Processing, January 22-24, pp.467-471, Penang, Malaysia, 2003.

[9] Sundarapandian, V., Subha Rani, S., Ravichandran, S.,"Reduced order observer and stabilization of the dominant state of discrete time linear system," National Conference on Intelligent systems, Automation and Signal processing, March 1^st, pp.56-58, Chennai, 2003.

[10] Hadi Saadat, "Power System Analysis", Tata McGraw-Hill Publishing Company Limited, New Delhi, 2003.

[11] Gajic.Z., Lelic.M, "Modern Control System Engineering", Prentice Hall Int. Series in Systems and Control Engineering, New York, 1996.

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