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INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 3, Issue 1, January-June (2012), © IAEME & TECHNOLOGY (IJEET) ISSN 0976 – 6545(Print) ISSN 0976 – 6553(Online) IJEET Volume 3, Issue 1, January- June (2012), pp. 145-155 © IAEME: www.iaeme.com/ijeet.html Journal Impact Factor (2011): 0.9230 (Calculated by GISI) I A E M E www.jifactor.com LARGE SCALE LINEAR DYNAMIC SYSTEM REDUCTION USING ARTIFICIAL BEE COLONY OPTIMIZATION ALGORITHM G.Vasu1 J. Nancy Namratha Assistant professor, Dept. of ElectricaAssistant professor, Dept. of Electrical Engineering, Engineering, S.V.P Engineering College ANITS College of Engineering Visakhapatnam, A.P , India Visakhapatnam, A.P, India E-mail: firstname.lastname@example.org E-mail: email@example.com V.Rambabu3 B.E Student, Dept. of Electrical Engg, ANITS College of Engineering Visakhapatnam, A.P, India E-mail: firstname.lastname@example.org ABSTRACT The authors proposes a method for Model order reduction of linear dynamic SISO system using an Optimization algorithm based on the intelligent foraging behaviour of honey bee swarm, called Artificial Bee Colony(ABC). The numerator and denominator polynomials of the reduced order model are computed by minimizing the Integral square error(ISE) pertaining to unit step input using ABC algorithm. The reduction procedure is simple, efficient and computer oriented. The proposed method guarantees stability of the reduced order model if the original higher order system is stable. The validity of algorithm is illustrated with the help of two numerical examples considered from the literature and the results are compared with other recently published reduction techniques to show its superiority. Keywords: Model Order Reduction, Artificial Bee Colony Optimization, Integral Square Error, Stability, Relative Mapping Errors. 145 International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 3, Issue 1, January-June (2012), © IAEME I INTRODUCTION The modelling of complex dynamic systems is one of the most important subjects in Engineering and Science. The mathematical procedure of system modelling often leads to higher order differential equations which are too complicated to use either for analysis or controller synthesis. So approximation procedures based on physical considerations (or) mathematical approaches are used to achieve simpler models for the original one. These reduction techniques are well-established part of the control system designer’s toolkit. At the forefront of these techniques have been those that deal with the linearized system models in both the time and frequency domains[1-4]. Each of the more established order reduction methods has its relative merits and is selected according to the system characteristics being approximated. Among them the frequency domain techniques are developed from the overwhelming amount of design information gathered by classical methods (e.g. Nyquist, Bode and Root-Locus plots), leading to the transfer function formation. This approach is still favoured by many designers and consequently the frequency domain order reduction methods continue to be of major importance. The most desirable properties of any order reduction techniques is that preserving stability of the original model in the reduced model and matching of time responses. In spite of several methods available, no single approach always gives the best results for all systems. Numerous methods of order reduction are available in the literature - which are based on the minimization of the Integral square error (ISE). However, a common feature in these methods is that the values of the denominator coefficients of the Reduced order model (ROM) are chosen arbitrarily by some stability preserving method such as Dominant pole, Routh approximation, Stability equation methods etc., and then the numerator coefficients of the ROM are determined by minimizing the ISE. Hewitt and Luss  suggested a technique, in which both the numerator and denominator coefficients are considered to be free parameters and are chosen to minimize the ISE in Impulse or Step responses. Recently, The Artificial Bee Colony Algorithm (ABC) appeared as promising evolutionary technique for handling the optimization problems, which is based on the intelligent foraging behaviour of honey bee swarm, proposed by Karaboga in 2005[7-8]. This swarm algorithm is very simple and flexible when compared to the other existing swarm based algorithms. It can be used for solving uni-model and multi-model numerical optimization problems. This algorithm uses only common control parameters such as colony size and maximum cycle number. It is a population based search procedure and can be modified using the artificial bees with time and the aim of the bees is to discover the places of food sources with high nectar amount and finally choose source with the highest nectar amount among the other resources. In the present work, ABC optimization algorithm is demonstrated for determining successful reduced order model for a given higher order linear dynamic continuous system based on the minimization of ISE between the Original higher order and Reduced order models pertaining to unit step input. The relative mapping errors between the original and reduced models are determined and time and frequency responses are plotted. The comparison between the proposed and the other well-known existing order reduction techniques is also shown. 146 International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 3, Issue 1, January-June (2012), © IAEME II DESCRIPTION OF PROBLEM Consider an order linear dynamic system represented by: The objective is to find an order model that has a transfer function are scalar constants. The derivation of successful reduced order model coefficients for the original higher order model is done by minimizing the error index ‘E’, known as ISE, employing ABC and is given by : Where and are the unit step response of original and reduced order systems. III ARTIFICIAL BEE COLONY (ABC) ALGORITHM ABC is a population based optimization algorithm based on intelligent behaviour of honey bee swarm . In the ABC algorithm, the foraging bees are classified into three categories; Employed bees, Onlookers and Scout bees. A bee waiting on the hive for making decision to choose a food source is called an Onlooker and a bee going to the food source visited by it previously is named an Employed bee. A bee carrying out random search is called a Scout. The employed bees exploit the food source and they carry the information about the food source back to the hive and share information with onlookers. Onlooker bees are waiting in the hive at dance floor for the information to be shared by the employed bees about their discovered food sources and scouts bees will always be searching for new food sources near the hive. Employed bees share information about food sources by dancing in the designated dance area inside the hive. The nature of dance is proportional to the nectar content of food source just exploited by the dancing bee. Onlooker bees watch the dance and choose a food source according to the probability proportional to the quality of that food source. Therefore, good food sources attract more onlooker bees compared to bad ones. Whenever a food source is exploited fully, all the employed bees associated with it abandon the food source and become scout. Scout bees can be visualized as performing the job of exploration, where as employed and onlooker bees can be visualized as performing the job of exploitation. In the ABC algorithm, each food source is a possible solution for the problem under consideration and the nectar amount of a food source represents the quality of the solution 147 International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 3, Issue 1, January-June (2012), © IAEME which further represents the fitness value. The number of food sources is same as the number of employed bees and there is exactly one employed bee for every food source. At the first step, the ABC generates a randomly distributed initial population P (C=0) of SN solutions (food sources position), where SN denotes the size of population. Each solution (food sources) is a D-dimension vector. Here D is number of optimization parameters. After initialization, the population of the position (solution) is subjected to repeated cycles, of the search process of the employed bees, onlookers and scouts. The production of new food source position is also based on comparison process of food source’s position. However, in the model, the artificial bees do not use any information in comparison. They randomly select a food source position and produce a modification on the existing, in their memory as described in Eq.(5) provided that the nectar amount of the new source is higher than that of the previous one of the bee memorizes the new position and forgets the old position. Otherwise she keeps the position of the previous one. An onlooker’s bees evaluate the nectar information taken from all employed bees and choose a food source depending on the probability value associated with that food source , calculated by the following equation: Where is the fitness value of the solution ‘i’ evaluated by its employed bee, which is proportional to the nectar amount of food source in the position and SN. In this way, the employed bees exchange their information with the onlookers. In order to produce a new food position from the old one, the ABC uses following expression (5): Where and are randomly chosen indexes. Although ‘k’ is determined randomly, it has to be different from . is a random number between [-1, 1]. It controls the production of neighbour food source position around and the modification represents the comparison of the neighbour food positions visualized by the bee. Equation (5) shows that as the difference between the parameters of the decreases, the perturbation on the position decreases too. Thus, as the search approaches to the optimum solution in the search space, the step length is adaptively reduced. If its new fitness value is better than the best fitness value achieved so far then the bee moves to this new food source abandoning the old one, otherwise it remains in its old food source. When all employed bees have finished this process, they share the fitness information with the onlookers, each of which selects a food source according to probability given in Eq. (4). With this scheme, good food sources will get more onlookers than the bad ones. Each bee will search for better food source around neighbourhood path for a certain number of cycles (limit), and if the fitness value will not improve then that bee becomes scout bee. IV RELATIVE MAPPING ERRORS The relative mapping errors of the original model relative to its Reduced model are expressed by means of the relative integral square error criterion, which are given by  : 148 International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 3, Issue 1, January-June (2012), © IAEME Where, and are the impulse and step responses of original system, respectively, and and are that of their approximants. In this paper, both the relative mapping errors ‘I’ and ‘J’ are calculated and plotted with respect to time for the proposed reduction algorithm. These relative mapping errors are also compared in the tabular form for the proposed reduction algorithm and the other well- known existing order reduction techniques. Fig 1 Flow chart of ABC algorithm V NUMERICAL EXAMPLES Two numerical examples are chosen from the literature to show the flexibility and effectiveness of the proposed reduction algorithm than other existing methods, and the response of the original and reduced models are compared. 149 International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 3, Issue 1, January-June (2012), © IAEME Example 1: Consider an eighth-order system  described by the transfer function as: To obtain lower order model for given higher order model ABC optimization algorithm is employed. The objective function ‘E’ defined as Integral square error between the responses given by Eq.(3) is minimized by ABC. In the present study, a population size of SN=50, and maximum number of cycles ( have been used. nd Finally the successful reduced 2 order model employing ABC algorithm is obtained as given in equation below: A comparison of the proposed algorithm with the other well known existing order reduction techniques for a second-order reduced model is given in Table I. Figure 5(a)-(c) Presents diagrams of step, impulse and frequency responses of and , respectively. TABLE I COMPARISON OF REDUCED ORDER MODELS Method of Reduction Reduced Models I J Proposed method 0.002145 0.000501 PSO  0.001781 0.000706 BFO  0.009214 0.036183 PSO and Eigen Spectrum  0.046305 0.025086 Pole Clustering and Pade  0.008349 0.009590 Mukherjee 0.089933 0.038810 et al. Pal 0.729677 1.126099 Chen et al 1.031795 4.918133  Safonov et al.  0.001361 0.004016 150 International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 3, Issue 1, January-June (2012), © IAEME 151 International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 3, Issue 1, January-June (2012), © IAEME Example 2: Consider an Eighth order system transfer function taken from : By applying the proposed algorithm, the following reduced second-order model is obtained: A comparison of the proposed algorithm with the other well known existing order reduction techniques for a second-order reduced model is given in Table II. Figure 5(d)-(f) Presents diagrams of step, impulse and frequency responses of and, respectively. TABLE II COMPARISON OF REDUCED ORDER MODELS Method of Reduced Reduction Models: I J Proposed Algorithm 0.00413 0.02947 S.N.Sivan andam etal 0.03112 0.06408  152 International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 3, Issue 1, January-June (2012), © IAEME C.S.Hsieh 0.55945 0.86299 et al  R.Prasad 0.9327 2.95833 et al  Y.Shamash 0.88419 2.32537 [ 21] 153 International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 3, Issue 1, January-June (2012), © IAEME V CONCLUSION In this paper, an Optimization algorithm for reduction of large scale linear dynamic SISO systems has been presented. The reduction algorithm is based on minimization of the Integral Square Error by ABC Optimization Technique pertaining to unit step input. The proposed method generates better approximation for a higher order linear dynamic system. The Relative step and impulse mapping errors between the original and reduced order models are determined and plotted with respect to time. A comparison of these mapping errors for the Proposed reduction method and existing reduction techniques are also shown in TABLE I and II, from which it is clear that proposed method is superior over the other existing techniques. The results show that the proposed method leads to good stable reduced models for linear dynamic systems. REFERENCES  R. Genesio and M. Milanese, “A note on the derivation and use of reduced order models”, IEEE Trans. Automat.Control, Vol. AC-21,No. 1, pp. 118-122, February 1976.  M. Jamshidi, Large Scale Systems Modelling and Control Series, NewYork, Amsterdam, Oxford, North Holland,Vol. 9, 1983.  G.Parmar, R.Prasad and S.Mukherjee, “Order reduction of linear dynamic systems using stability equation method and GA”, World Academy of science engg. and tech. Vol-26 ,2007.  C.B. Vishwakarma , R. 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