In this paper, a computationally simple approach is proposed for order reduction of large scale system linear dynamic SISO and MIMO system using differential evolutionary (DE) optimization technique. The method is based on minimizing the integral square error (ISE) between the transient responses of original and reduced order models pertaining to step input. The reduction procedure is simple, efficient and computer oriented. Stability of the reduced order system is always assured in proposed method. The algorithm is illustrated with help of two numerical examples and results are compared with other well known reduction techniques to show its superiority.Keywords--SISO system; MIMO system; model order reduction; differential evolutionary optimization; integral square error. I INTRODUCTIONThe mathematical description of most physical systems leads to higher order differential equations which are difficult to use either for analysis or controller synthesis. It is hence useful and sometimes necessary to find the possibility of finding some equation of the same type that of lower order that may be considered to adequately reflect the dominant characteristics of the system under consideration. This will help to better understanding of the physical system, reduce computational complexity, reduce hardware complexity and simplify the controller design.Several reduction methods are available in literature for reducing the order of large scale linear single and multivariable systems in frequency domain [1][2][3][4][5][6][7][8][9]. Further several methods have also been suggested by combining the features of two different reduction methods [4][5][6]. Recently, numerous methods of order reduction are also available in the literature [3], [5,6] which are based on error minimization by optimization techniques. In spite of having several reduction methods, none always gives the satisfactory results for all the systems.In recent years, one of the most promising research fields has been "Evolutionary techniques", an area utilizing analogies with nature or social systems. Differential evolutionary is branch of Evolution algorithm (EA) developed by Rainer Storn and Kennith Price in 1995 for optimization problem [10]. The differential algorithm is a promising approach for engineering optimization problem. It is a heuristic approach mainly having three advantages; finding the true global minimum regardless of the initial parameters value, fast convergence, and using few control parameters [11]. The DE algorithm is a population based algorithm like genetic algorithm using similar operators; crossover, mutation and selection. The main difference in constructing better solutions is that GA relies on crossover while DE relies on mutation operation. Being simple, fast, easy to use, very easily adapted for integer and discrete optimization, quite effective in non linear constraint optimization including penalty functions and useful for optimizing multi model search spaces are the other important features of DE [12][13].In present work, the ...

In this paper, the authors propose an optimal IMC-PID controller design for the Load Frequency Control (LFC) of large-scale power system via model approximation method. The model approximation method uses the Enhanced Differential Evolution (EDE) algorithm to determine an optimal Reduced Order Model (ROM) for the considered large-scale power system by minimizing the performance measure called Integral Square Error (ISE) between their step responses. Later, the LFC design is carried out using an optimal ROM instead of processing with the large-scale power system model. Thus, this simplifies the design, reduces the computational efforts and also helps in determining the lower order controller. An optimal IMC design methodology is proposed by minimizing ISE between the actual output and the reference input responses of the large-scale power system using EDE algorithm. Further, PID controller gains are acquired by least square model matching with the optimal IMC transfer function. The proposed IMC-PID controller design allows a satisfied reference input tracking performance, robustness in disturbance rejection and improves the dynamic stability of the power system. The proposed method is validated by applying it to a single area power system of third-order SISO model and also extended to a centralized two-area thermal–thermal non-reheated power system of a seventh-order MIMO model. The simulation results and the comparison of error performance indices show the efficacy of the proposed method over the significant methods available in the literature.

In this article, the combination of stochastic search and conventional approaches are used to develop an optimal frequency-domain model order reduction method for determining the stable and accurate reduced-order model for the stable large-scale linear time-invariant systems. The method uses the enhanced particle swarm optimization with differentially perturbed velocity algorithm to determine the denominator polynomial coefficients of the reduced-order model, whereas the numerator polynomial coefficients of the reduced-order model are determined by using an improved multi-point Padé approximation method. The method generates an optimum reduced-order model by minimizing an objective function [Formula: see text], which is formulated using two functions. The first function, [Formula: see text], evaluates the measure of integral squared error between the step responses of the original system and the reduced-order model. And the second function evaluates the measure of retention of full impulse response energy of the original system in the reduced-order model. Therefore, by minimizing the objective function ‘ E’, the proposed method is guaranteed for preserving passivity, stability and the accuracy of the original higher order system in the reduced-order model. The proposed method is extended to the linear time-invariant multi-input multi-output system. In this case, an optimal reduced-order model is determined by minimizing a single objective function [Formula: see text], which is formulated by linear scalarizing of all the objective function [Formula: see text] components. The method is popular for preserving stability, passivity and accuracy of the original system in the reduced-order model. The validation of the method is shown by applying to a sixth-order single-input single-output hydropower system model as well as to the seventh-order two-area multi-input multi-output power system model. The comparison of the simulation results of integral squared error and impulse response energy values of the reduced-order models demonstrates the dominance of the proposed method than the existing reduction methods available in the literature.

A new computationally simple and precise model approximation method is described for large-scale linear discrete-time systems. By least squares matching of a suitable number of time moment proportionals and Markov parameters about [Formula: see text] of the original higher order system within the approximate model, stable denominator polynomial coefficients of the approximate model are determined. To improvise the accuracy of the approximate model, numerator polynomial coefficients are determined by minimizing the integral squared error (ISE) between the unit impulse responses of the original system and its approximate model. A matrix formula is formulated for evaluating numerator coefficients of the approximate model that leads to minimum ISE, and also for evaluating ISE. The efficacy of the proposed method is shown by illustrating three typical numerical examples employed from the literature, and the results are compared with many familiar reduction methods in terms of the ISE and relative ISE values pertaining to impulse input. Furthermore, time and frequency responses of the original system and the respective approximate model are plotted.

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