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Bacterial Foraging Algorithm Based Multiobjective Optimal Design of Single Phase Transformer S. Subramanian* and S. Padma Abstract— Transformer is widely used in power conversion systems because of its function of isolation and voltage regulation. This implies the transformer manufacturing has direct impact in the design of the optimum transformers, i.e. the transformers that meets the specification with the minimum manufacturing cost and also improve performance. Numerous parameters involve in the transformer design, so this make the design problem more multifaceted and the proficient methods are necessitating for optimum design. In this article a multi objective based bacterial foraging algorithm has been developed for minimizing cost and improves the efficiency simultaneously. To verify the validity of the proposed method has been tested with sample transformer and numerical simulation results are compared to recent reports. The results show that the proposed method provides superior solution. Index Terms— Bacterial foraging algorithm, cost, efficiency, transformer design.

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1 INTRODUCTION

T

RANSFORMER is important equipment and commonly use in transmissions and distributions system and they are responsible for the stable and reliable power system. The failure of one transformer may cause significant effect for electrical utilities failure causes large outage and economic losses. So the transformer manufacturing industry must improve transformer efficiency and while reducing cost, since high quality, lower cost become key to survival [1]. In the past decades, several techniques are used in optimization procedure. The transformer design with the cost analysis has been reported in the literature. The authors developed a step by step procedure for the transformer design involved the material cost and labour cost [2]. The cost analysis involves with the material costs in detail and a labour analysis that sets forth a detailed list of labour operations necessary to construct the transformer being analyzed. To achieve accurate design with less time an economical manner using digital computers was discussed [3]. The circulating and rotational fluxes make large contribution to the total power loss in the transformer and it requires the optimal design of joint [4]. The joint design based on the knowledge about the localized flux distribution, both in the corners and in the limbs. A unified method has been developed for the design of electrical machines including power transformers [5]. Generally the transformer design problem has been formulated as single objective subjected to various ————————————————

• S.Subramanian, Professor of Electrical Engineering, Annamalai University, Annamalainagar-608002, Tamilnadu, India. • S.Padma, Assistant Professor in Electrical Engineering, Annamalai University, Annamalainagar-608002, Tamilnadu, India.

constraints. The objective function is either minimization of cost or maximization of efficiency where these are related with the production cost and performance respectively. Simulated Annealing (SA) technique [6] has been developed for optimal design of transformer to minimizing the cost. To achieve lower cost, reduced size and better operating performance of transformer using available materials economically in accordance with given specification was presented [7]. Several multiobjective techniques have been developed and implemented to solve various design problems. An algorithm belonging to the class of controlled random search algorithm [8] has been used for the solution of multiobjective optimization of induction motor design. The multiobjective optimization [9] has been applied to compare and investigate mathematical methodologies for electromagnetic devices.Multiobjective optimal design of induction motor for Electric Vehicle (EV) using a [10] modified Evolution Strategy (ES) has been reported. The Optimization process employing the weighting method and the best compromise solution search technique is suggested, to get around the trap of local minima. A new method multiobjective particle swarm optimization technique [11] was applied to transformer design optimization problem with competing objectives of cost and efficiency. Multiobjective optimal design of medium frequency (MF) transformers for full-bridge dc-dc converters using genetic algorithm [12] has been reported. The design problem employing minimizing the weight and the loss of the transformer while ensuring the satisfaction of a number of constraints. Artificial Neural Networks (ANNs) [13] has been applied to protection of three-phase power transformers, more over the design process and various design issues have been discussed

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including optimization of the pre- and post processing algorithms and the type of an ANN. Multiobjective optimal design of Medium Frequency (MF) transformers for isolated Switch Mode Power Supplies (SMPS) has been developed [14]. The design problem requires minimizing the weight and the loss of the transformer while ensuring the satisfaction of a number of constraints. The modern heuristic search technique, called Bacterial Foraging Algorithm (BFA) has been developed based on modelling of bacteria E.coli behaviour present in human intestine and it has been proven that is efficient [15]-[19] for various engineering optimization problems. In this paper, optimization of transformer design can be expressed as multiobjective problem. It differs from scalar optimization. The scalar optimization finding one global optimum while multiobjective optimization must find a set of solution ,which is called pareto set, as all the pareto solutions are equivalently important and all of them are the global optimal solutions.

2 MULTIOBJECTIVE OPTIMIZATION The real word problems are multiobjective in nature as they often involve more than one goal. It is needed to optimize them simultaneously based on given critrria. To solve the multi objective problems, the weighting method is used. In case of a two-objective problem, objectives, f1 and f2 can be weighted using weighting values, w1 and w2 respectively, so that minimize: (1) f[ f 1 , f 2 ] = f(w1 , w2 ) = w1 f 1 + w2 (1/f 2 ) where, f1 is the cost (Rs) and f2 is the efficiency (%) We can divide the objective function by a positive number without altering the solution, after dividing (1) by w1, w2/w1 can be redefined as w. Then (1) can be written as follows: Minimize: (2) f(w) = f 1 + w( 1/f 2 ) Where w=[0, ∞] Because it is difficult to realize according to the w in total range, objective function is reformed for covering the total range. The final objective function is represented by: (3) f (w) = wf 1 ( x) + (1 - w)(1/ f 2 ( x)) Where: w= [0, 1] and x is a set of design variables and a convex set. (4) f(w) = U[f1 , f 2 ] The flowchart for multiobjective optimization of transformer is presented in Fig. 1. In scalar optimization there is usually one solution which gives the optimum value for the function, while in real vector optimization, problems there is no solution which gives the optimum value for all functions, but the final solution is taken from the non-dominant (Pareto) set. A point belongs to the Pareto set if no criterion can be improved without worsening at least one other criterion. In the weighting method, the Solution is taken from the non-dominant set according to the decision maker. The Decision Maker (DM) chooses the best compromise solution after making an implicit trade off between objectives. The method is

used interactively with DM. The DM gives Weights (w) which reflect the priority of objectives to the algorithm which will give the optimal solution.

Fig.1 Flowchart of multiobjective optimization of transformer design

3 DESIGN OF SINGLE PHASE TRANSFORMER – PROBLEM FORMULATION The optimization of single phase transformer design is formulated as multiple non commensurable objectives. Mathematically, the general non-liner multi variable constrained optimization problem can state as follows Find: (5) X = [X 1 , X 2 , X 3 … X n ] Such that F = f(X) is minimum subject to (1) , i=1, 2….n X
i

imax

and g i (X) < 0 , i=1,2,….m (6) Where X1, X2,……Xn are the set of independent design variables with their lower and upper bounds as Ximin and Ximax. F=f(X) is the objective function to be optimized and gi(X) are the constraints imposed on the design.

3.1 Design Variables The design procedure is based on the set of design variables and specification. There is very essential to select the design variables, because the objective function becomes very sensitive. Here the transformer design variables are chosen as consisting of 1. Maximum flux density (x1) Wb/m2 2. Current density in HV winding (x2) A/mm2 3. Current density in LV winding (x3) A/mm2 4. EMF per turn (x4) volts. 3.2 Constraints In the design problem imposed gj(X) is the set of m explicit constraints to make it feasible and practically acceptable, the constraints that have been used in this study are 1. Temperature rise. 2. Regulation. According to the selected design variables, the objective functions are calculated using the procedure described in [20].

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4 BACTERIAL FORAGING OPTIMIZATION TECHNIQUE The selection behaviour of bacteria tends to eliminate poor foraging strategies and improve successful foraging strategies. After many generations a foraging animal takes actions to maximize the energy obtained per unit time spent foraging. This activity of foraging led the researchers to use it as optimization process. The E.coli bacterium has a control system that enables it to search for food and try to avoid noxious substances. The bacteria distributed motion can model as the following four stages:

4.1 Swarming and Tumbling via Flagella (Ns) The flagellum is a left-handed helix configured so that as the base of the flagellum (i.e. where it is connected to the cell) rotate counter clockwise, as shown in Fig. 2, from the free end of the flagellum looking towards the cell, it produces a force against the bacterium pushing the cell. This mode of motion is called swimming. A bacterium swims either for maximum number of steps Ns or less depending on the nutrition concentration and environment condition. During clockwise rotation each flagellum pulls on the cell shown in Fig 2. So that the net effect is that each flagellum operates relatively independently of the others and so the bacterium “tumbles”. 4.2 Chemotaxics (Nc) A chemotaxis step is a set of consequence swim steps following by a tumble. A maximum of swim steps with a chemotactic step is predefined by Ns. The actual number of swim steps is determined by the environment. If the environment shows good nutrients concentration in the direction of the swim, the bacteria swim more steps. When the swim steps is stopped a tumble action takes place.

4.4 Elimiation and Dispersal (Ned) Elimination event may occur for example when local significant increases in heat kill a population of bacteria that are currently in a region with a high concentration of nutrients. A sudden flow of water can dispose bacteria from one place to another. The effect of elimination and dispersal event is possibly destroying chemotactic progress, but they also have the effect of assisting in Chemotaxis, since dispersal may place bacteria near good food sources. Flowchart for bacterial foraging algorithm is presented in the Fig. 3.

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IMPLEMENTATION OF MULTIOBJECTIVE OPTIMIZATION FOR TRANSFORMER DESIGN USING BFA

Optimal transformer design has need of choice of objective function that usually relate economic or performance features. For this reason we have chosen two conflicting objective functions that can affect the design optimization of transformers. Particularly: f1 is the cost (Rs) and f2 is the efficiency (%) Optimal transformer independent variables are finding by the proposed approach. In this design analysis each bacterium having four members: x1, x2, x3 and x4. If there are s numbers of bacteria in a population, then the dimension of population is s x 4. The algorithmic steps of proposed approach are summarized below. Step 1:

Step 2: Step 3: Step 4: Step 5: Step 6:

Step 7: Step 8:

Fig. 2. Swarming and Tumbling behaviour

4.3 Reproduction (Nre) After Nc chemotactic steps, a reproduction step is taken. Let Nre be the number of reproduction steps to be taken. It is assumed that half of the population members have sufficient nutrients so that they will reproduce with no mutations. For reproduction, the population is sorted in order of ascending accumulated cost accumulated cost represents that it did not get as many nutrients during its lifetime of foraging and hence, is not as “healthy” and thus unlikely to reproduce).Least healthy group of bacteria dies out and the other healthiest splits into two.

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Input the bacterial foraging parameters and independent variable, and also specify lower and upper limits of the variables. Generate the positions of the independent variable randomly for a population of bacteria. Evaluate the objective value of each bacterium. Modify the position of the variables for all the bacteria using the tumbling/swimming process . Perform reproduction and elimination operation. If the maximum number of chemotatic, reproduction and elimination-dispersal steps is reached, then go to step 7; otherwise, go to step4. Output the variable corresponding to the overall best bacterium Compute the operating performances of the transformer such as efficiency and regulation.

NUMERICAL SIMULATION RESULTS AND DISCUSSION

Simulation test have been performed to analyze the validity of the proposed approach. The results obtained by the proposed method are compared to conventional and multiobjective based PSO technique. The specification of sample transformer was presented in Table 1. In the present work the optimal design of single transformer when conflicting objectives are chosen as cost and efficiency. Optimization of transformer design objectives, minimizing the cost and maximizing efficiency separately using scalar optimization and the results was given in Table 2.

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TABLE 1 SAMPLE TRANSFORMER SPECIFICATIONS

Start Initialize all variables. Set all loop-counters and bacterium index i equal to 0. Increase elimination –dispersion loop counter l = l_+1 Print the results and stop

No Perform Elimination dispersal

l < Ned? Yes

Increase reproduction loop counter = k+1

k

No Yes Increase chemotactic loop counter j = j+1 No j
Set m=Ns

Yes Set Jlast = J(i,j+1,k,l) and swim (let the ith bacterium take a set of height C(i) along the direction of the same tumble vector ∆(i))

Fig. 3. Flowchart for BF

TABLE 2 DIMENSIONS OF SAMPLE TRANSFORMER WITH SINGLE OBJECTIVE FUNCTION Independent Variables

k
Perform Reproduction

Sample Transformer Capacity 500 KVA Phase Single phase Primary voltage 6600 V Secondary voltage 400 V Type Core Frequency 50 Hz

Minimising cost only

Maximizing efficiency only

1.44

1.45

2.08

2.21

2.28

2.07

16.4 71 18.4 98.96 1,41210

16.8 75 18.2 99 1,43356

Maximum flux density (Wb/m2) Current density in HVwinding (A/mm2) Current density in LV winding (A/mm2) EMF per turn (V) Temperature rise (0C) Regulation (%) Efficiency (%) Cost (Rs)

TABLE 3 COMPARISON OF CONVENTIONAL METHOD WITH PSO AND BFA FOR MULTIOBJECTIVE OPTIMAL DESIGN OF SAMPLE TRANSFORMER

Independent variables Maximum flux density (Wb/m2) Current density in HVwinding (A/mm2) Current density in LV winding (A/mm2) EMF per turn (V) Temperature rise (0C) Regulation (%) Efficiency (%) Cost (Rs)

Initial values

PSO

BFA

1.5

1.47

1.43

2.75

2.62

2.22

2.75

2.53

2.19

17.9 80 19.7 98.93 1,45030

17.1 76 18.8 98.95 143125

16.7 73 18.3 98.98 142544

By observing the results, there is some variation in design variables. It is evident that the geometry of the transformer minimizing the cost is not coincident with the geometry of maximizing the efficiency. For this reason the multiobjective optimization is applied to optimize the transformer design. The results of proposed multi objective based BFA are shown in Table 3. From the

obtained proposed results clearly indicate that there is

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slightly increase in efficiency due to reduction in losses resulting from reduction in flux density, current density, also overall reduction in losses will lead to lower temperature rise. More over winding height is also reduced, so size of the transformer is reduced. This proves the cost of the transformer is considerably reduced compared to multiobjective based PSO. The resulting reduction in cost will makes it very efficient for transformer user. The parameters of the BFA used for the simulation studies are summarized in Table 4. Fig. 4 shows the variations of efficiency with iteration for sample transformer. Fig. 5 shows the variations of cost with iteration for sample transformer. TABLE 4 PARAMETER SELECTED FOR BFA Parameter

Value

Number of bacterium (s) Number of chemotatic steps (Nc) Swimming length (Ns) Number of reproduction steps (Nre) Number of elimination and dispersal events (Ned) Depth of attractant (dattract) Width of attractant (ωattract) Height of repellent (hrepellant) Width of repellent (ωrepellant) Probability of elimination-dispersal events (Ped)

20 10 4 4

Fig. 4. Variations of efficiency with iterations

5 0.1 0.2 0.1 10 0.02

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CONCLUSIONS

Transformer design is composite task that contain many design variables. These design variables related to the performance of transformer. Hence a more important is give to transformer design optimization. In this article, multiobjective based bacterial foraging algorithm has been developed for optimal design of single phase transformer. In this design optimization, the chosen objectives are cost and efficiency. The proposed method provides best compromise solution from a pareto optimal solution. To verify the effectiveness of the proposed method has been tested with sample transformer. The results reveal that multiobjective based BFA is efficient for optimal design of transformers and also suited for economical problems.

ACKNOWLEDGEMENT The authors are grateful to the authorities of Annamalai University for providing all facilities to carry out this work.

NOMENCLATURE C (i) - Step size i - Bacterium number j - Counter for chemotactic step J(i, j, k, l) - Cost at the location of ith bacterium Jcc - Swarm attractant cost J ihealth - Health of bacterium i k - Counter for reproduction step l - Counter for elimination- dispersal step m - Counter for swimming locomotion - Maximum number chemotactic steps Nc Ned - Number of elimination- dispersalevents Nre - Maximum number of reproduction steps Ns - Maximum number of swims P - Dimension of the optimization problem Ped - Probability of occurrence of eliminationdispersal events s - Population of the E. coli bacteria θi (j, k, l) - Location of the ith bacterium at jth chemotactic step, kth reproduction step, and l the eliminationdispersal step ωattract - Width of attractant ωrepellant - Width of repellent hrepellent - Height of repellent dattract - Depth of attractant LV - Low voltage winding HV - High voltage winding

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S.Subramanian is born in Maduari, Tamilnadu, India. He received the B.E degree in Electrical and Electronics Engineering and the M.E degree in Power Systems with distinction from Madurai Kamaraj University, Madurai, India in the year 1989 and 1990 respectively. He received the Ph.D degree in Electrical Engineering from Annamalai University, Annamalainagar, India in the year 2001. He published 130 research articles in various referred international journals, national journals, international conferences and national conferences. He guided five PhD students to his credit. His area of interest includes power system operation and control, design analysis of electrical machines, power system state estimation and power system voltage stability studies. Dr. S. Subramanian is a Senior member in IEEE, Fellow of Institution of Engineers (India) and a member of various professional bodies such as System Society of India, India Society for Technical Education and Indian Science Congress Association. His biography has been included in MARQUIS who is who in the world, MARQUIS who is who in Engineering and International Biographical Centre (IBC), UK. He received the best teacher award in recognition of his research contributions in Annamalai University. S.Padma is working as an Assistant Professor in the Department of Electrical Engineering, Annamalai University. She received her BE (Electrical) (2000) degree and ME (power system) (2005) degree from Annamalai University. She is doing PhD work in design analysis of electrical machines.