Weight optimization of a dry type core form transformer by using ...

Energy Education Science and Technology Part A: Energy Science and Research 

2012 Volume (issues) 29(2): 1063-1072 

Weight optimization of a dry type 

core form transformer by using 

particle swarm optimization algorithm 

Huseyin Demir 1 , Ali Ozturk 2 , Leyla Kuru 3,* , Ersen Kuru 4 , 

1 Duzce University, Department of Electrical Education, 81100 Konuralp, Duzce, Turkey. 

2 Duzce University, Department of Electrical and Electronics Engineering, 81100 Konuralp, Duzce, Turkey. 

3 Duzce University, Department of Electrical and Electronics Engineering, 81100 Konuralp, Duzce, Turkey. 

4 Uzunmustafa M. Uzunmustafa C. 20/3, Duzce, Turkey. 

Abstract 

Received: 16 August 2011; accepted: 19 October 2011 

There are two imported parameters affecting the weight of a transformer. These are the iron part 

and the copper part of a transformer. Reducing the weight is the basic criteria to reduce the cost of a 

transformer. In this study, a partial swarm optimization (PSO) method is used to optimize the weight 

of a three phase core form dry type transformer. The total weight of the transformer (G T ), has been 

accepted as the goal function. The constraints are added to the fitness function as the penalty function 

to constrain fitness function values. The efficiency (η) and (Ls/a) has been taken as constraints. The 

fitness function has been obtained as a function of current density and transformer iron cross section 

convenience value. The critical values have been computed using the PSO method and the results 

have been evaluated. The results are compared with the results of the classical method, described in. 

The total weight which has been founded by using PSO is less than the weight which is founded with 

the classical method. The weight of a 1,5 KVA three phase core form dry type transformer, founded 

by using PSO, is 23.5547 kg whereas the weight, founded by classical method was 30.84 kg. It has 

been seen that intuitive PSO method is an alternative and better way to estimate the minimum weight 

values. 

Keywords: Dry type transformer; Particle swarm optimization; Algorithm 

©Sila Science. All rights reserved. 

1. Introduction 

In last decades there has been a great deal of interest on the applications of optimization 

algorithms to the engineering the problems [1-11]. Electrical energy is the most consuming 

energy type on the earth [12]. This energy is transmitted and distributed by transformers [1, 

2]. The aim of the transformer design is to completely obtain the dimensions of all the parts of 

the transformer based on the desired characteristics, available standards, and access to lower 

cost, lower weight, lower size, and better performance [13-15]. In this study, a partial swarm 

optimization method is used to optimize the weight of a three phase core form dry type 

_____________ 

* Corresponding author. Tel.: +90-535-305-0125; fax: +90-535-305-0100. 

E-mail address: leylakuru@hotmail.com (L. Kuru).

1064 H. Demir / EEST Part A: Energy Science and Research 29 (2012) 1063-1072 

transformer which is designed previously [2]. There are two imported parameters affecting the 

weight of a transformer. These are the iron part and the copper part of a transformer [1]. In section 

2, PSO method is explained. In section 3, the solutions of the problem are explained. In 

section 3.1 the solution of the problem is done by classical method and in section 3.2 the 

solution of the problem is done by PSO. The total weight of a three phase core form dry type 

transformer is obtained as a function of the total iron and the total copper weights of the 

transformer. This function has been accepted as the goal function. The constraints are 

added to the fitness function as the penalty function to constrain fitness function values. The 

efficiency (η) and (Ls/a) has been taken as the constraints. By using the given equations the 

fitness function has been obtained as a function of current density and transformer iron cross 

section convenience value. The minimum values for this function were computed using the 

PSO method and finally the results are given in the conclusion part. 

2. Partial swarm optimization (PSO) 

PSO is a population-based optimization algorithm developed for the first time by [16]. It 

was inspired by the moving school of fish and swarm inspection. Basically, PSO is an 

algorithm based on swarm intelligence. It is similar to the computational techniques based on 

development such as genetic algorithms (GA). While PSO is searching for the optimum in the 

search space, it uses the population which holds the possible solutions for the function to be 

optimized [17]. Thus, each individual is called a particle. Particles make up the population 

called a swarm. In PSO, however, each member of the swarm has a variable speed which 

changes based on the conditions and determines its movement in the search space. In addition, 

each member has a memory holding the best spot previously visited. Each particle adjusts its 

position toward the best position in the swarm by taking the advantage of its previous 

experience. PSO is basically based on the fact that the positions of the particles in the swarm 

Fig. 1. The flow diagram for PSO. 

are brought closer to the best position in the swarm. The speed of “bringing closer” happens 

in a random fashion and in most cases the particles in the swarm find better positions in their 

new movements. This process continues until it gets to the goal.

H. Demir / EEST Part A: Energy Science and Research 29 (2012) 1063-1072 1065 

Assuming that the search space is found with D dimensions, the position of the i th particle 

in the swarm can be expressed in a D dimensional vector as in (X i =(x i1 ,x i2 ,…..,x iD ) T ) and its 

speed can be expressed in a D dimensional vector as in (V i =(v i1 ,v i2 ,…..,v iD ) T ). In additon, the 

best position of the particle ever visited can be expressed in a D dimensional vector as in 

(P i =(p i1 ,p i2 ,…..,p iD ) T ). . In the following swarm expressions numbered 1 and 2, g denotes the 

index number of the best particle, and the upper scripted numbers denote the iteration 

numbers. 

V id n+1 =wv id n +c 1 r 1 n (p id n -x id n )+ c 2 r 2 n (p gd n -x id n ) (1) 

X id n+1 =x id n +v id 

n+1 

(2) 

Here, d = 1, 2, . . .,D, and i = 1, 2, . . . , PS, where PS = the size of the population in the 

swarm. R 1 and r 2 are randomly selected values between 0 and 1. n denotes the iteration 

number. x id and v id denote the position and the speed values, respectively. w is the inertia 

weight and C1 and C2 are for the scaling factors. Fig. 2 depicts the flow diagram for PSO 

[18]. 

3. The solution of the problem 

The weight of the three phase core type transformer is determined first by the classical 

method which was described in [2] previously. This method is a method in which known 

equations have been calculated with some assumptions to find the transformer weight. In the 

second method, the PSO algorithm is suggested. PSO is such a method that is searching for 

the optimum in the search space and uses the population which holds the possible solutions 

for the function to be optimized. 

3. 1. The Solution of the problem by classical method 

The weight, calculated by the classical method was described in [2]. In that classical 

method, C; transformer iron cross section convenience value, and s; current density, were 

accepted as constants. The results were evaluated regarding to these accepted values.. 

The iron and copper parts of a transformer are the two basic parameters affecting the 

weight of a transformer. The total weight (G T) (kg) can be expressed as the sum of total iron 

weight (G fe ) (kg) and total copper weight (G cu ) (kg). G T is the objective function of the PSO 

algorithm. The total iron weight can be expressed as the yoke weight (G fej ) and the three legs 

(G feb ) weight. The total copper weight is the sum of the total of primary winding weight (G cu1 ) 

and the total of secondary winding weight (G cu2 ). 

G cu = G cu1 + G cu2 (3) 

G fe = G feb + G fej (4) 

G T= G cu + G fe (5) 

G cu1, G cu2, G feb, G fej are calculated as follows: 

G cu1 =3.10 -5 .δ cu .w 1 .q 1 .L m1 (6)


G cu2 =3.10 -5 .δ cu .w 2 . q 2 .L m2 (7) 

G feb =3.10 -3 .δ fe .q fe.L s (8) 

G fej =3.10 -3 . δ fe .q fej.2(2M+0,8D) (9) 

δ cu , δ fe are the specific copper and iron weight respectively. w 1, w 2 are the primary and 

secondary turns of the windings respectively. L m1 (cm), L m2 (cm) are the average length of the 

primary and secondary windings respectively. q 1 (mm 2 ),q 2 (mm 2 ), q fe (cm 2 ), q fej (cm 2 ) are the 

primary winding cross section, secondary winding cross section, iron cross section from the 

leg and the iron cross section from the upper-lower part of the core where; 

q fe=C. 

10 3 S 

(10) 

3f 

q fej=1,1. q fe (11) 

q 1 = s 

I 1 

(12) 

I 

q 2 = 2 

(13) 

s 

U 

1 

w 1 = 

8 

3.4,44.f. .10 

U 

2 

w 2 = 

8 

3.4,44.f. .10 

(14) 

(15) 

L m1= π.D m1 (16) 

L m2= π.D m2 (17) 

D m2 =D+2(Δ 2 +δ 2 )+a 2 (18) 

D m1 =D m2 +a 2 +2(Δ 1 +δ 1 )+a 1 (19) 

D m1 (cm), D m2 (cm) are the average cross section of the primary and secondary windings 

respectively. U 1, U 2 (V) is the primary and secondary voltage respectively and f is the 

frequency. 

Fig. 2. Dry type core form transformer core


θ (Maxwell) is the magnetic flux ; θ =q fe .B. B (Tesla) is the flux density. C is the transformer 

iron cross section convenience value and is considered as the first variable for PSO. 

5.9 < C < 10.6 

s is the current density value and is considered as the second variable for PSO. 

2.2 < s < 3.5 

Δ 1 , δ 1 (cm) are the primary insulator coil thickness and oil canal respectively. Δ 2 , δ 2 (cm) 

are the secondary insulator coil thickness and oil canal respectively. a 1 , a 2 (cm) are the 

thickness of the primary and secondary windings. I 1 , I 2 (A) are the primary and secondary 

current, S (VA) is the complex power; 

S= 3 .U.I (20) 

M (cm) is the length between the axis of the legs, D (cm) is the transformer core diameter, Ls 

(cm) is the height of the window, a (cm) is the width of the window and can be calculated as; 

M=0,851.D+a (21) 

q 

0,677. 

fe 

D= 2 (22) 

2.w1I 

L s = 

A 

s 

1 

(23) 

a= 

4.w .q1 

100.k .Ls 

1 

(24) 

cu 

k cu is the window copper fill factor and is obtained from Fig. 3. A s (A/cm) is the specific 

ampere turn and is obtained from Table 1. 

Fig. 3. The window copper fill factor of three phase core form dry type transformer. 

Table 1. Specific amper turn of three phase core form dry type transformer 

S (kVA) 0,5 1 2 4 8 10 15 20 25 30 

As (A/cm) 80 90 100 110 123 127 135 142 147 150


Efficiency η can be calculated as; 

η = 

S 

S 

p k 

(25) 

p cu is the the total copper loss and p fe is the total iron loss. The sum of these are p k (watt). 

p k =p cu +p fe (26) 

The copper loss of the primary winding is p cu1 whereas the copper loss of the secondary 

winding is p cu2, the iron loss of the yoke is p fej and the iron loss of the legs is represented as 

p feb . r 1 , r 2 are the primary and secondary winding resistances respectively. 

p cu1 =3.I 2 .r 1 (27) 

p cu2 =3.I 2 .r 2 (28) 

B 

p fej = p 10 .ξ 2 .( j ) 2 .G fej (29) 

10000 

B 

p feb = p 10 .ξ 2 .( ) 2 .G feb 10000 

(30) 

p cu = p cu1 + p cu2 (31) 

p fe = p fej + f feb (32) 

p 10 is an additional loss factor whereas ξ 2 is the loss factor. r 1 and r 2 are the primary and 

secondary winding resistances respectively. 

B j =B/1.1 (33) 

The weight of a 1,5 KVA three phase core form dry type transformer founded by classical 

method was 30.84 kg. The values for this result are given in Table 2. 

Table 2. The values of a three phase core type transformer using classical method. 

WEIGHT OPTIMIZATION RESULTS OF A DRY TYPE CORE FORM TRANSFORMER BY CLASSICAL METHOD 

Variables and Other Values Symbol Unit Classical Method Results 

Iron cross section convenience value C cm 2 *joule -1/2 9,48 

Current density value s A/mm 2 2,2 

Window width a mm 58 

Primary winding cross section q1 mm 2 1,79 

Secondary winding cross section q2 mm 2 3,58 

Transformer core diameter D cm 7,5 

Primary winding turn w1 turn 174 

Secondary winding turn w2 turn 87 

Primary winding weight Gcu1 kg 2,91 

Secondary winding weight Gcu2 kg 2,16 

The Three legs weight of the Transformer Gfeb kg 9,92 

The yoke weight of the Transformer Gfej kg 15,85 

Total Weight of the Transformer Gtotal kg 30,84 

Efficiency η % 93 

4. The solution of the problem by PSO 

In this study, transformer iron cross section convenience value, and s; current density 

have been taken as variables whereas in the classical method, C and s, were accepted a 

constants. The total weight G T is the objective function. The minimum values were found


with the PSO algorithm for the objective function.. This function is obtained as a function of 

two variables C and s using the above equations. The circumstances where the minimum 

values were considered as appropriately functional are shown in Eq. 2. However, the copper 

and iron weight equations are obtained according to references [2]. If there are constraints, 

those constraints are added to the fitness function as the penalty function to constrain fitness 

function values. C (cm 2 *joule -1/2 ) is the transformer iron cross section convenience value and 

s is the current density value are considered as the first and second variable for PSO. The 

commonly accepted boundary values for C and s of three phase core type transformers are 5.9 

< C < 10.6; 2.2 < s < 3.5. These boundaries are accepted in this study. Efficiency and Ls/a 

are constraints, those constraints are added to the fitness function as the penalty function to 

constrain fitness function values. In this study, since solutions are evaluated for constant 

weight function values, the efficiency and Ls/a must be constant as general usage. This 

situations has been taken as constraints. 

First constraint: 0.9 < efficiency < 1 

Second constraint: 2 < Ls/a < 4.5 

In this study a weight optimization of a 1.5 KVA three phase dry type core form transformer 

has been done by using PSO algorithm. The values taken for this study are: 

U 1 =220V, U 2 =110V, F=50H, B=11000 Gauss, T=75 0 . The resulting objective equation F 1 

and efficiency are expressed as a function of C and s; 

F 1 (C, s)=G T (36) 

Efficiency: η= 

1500 

1500 

p k 

(35) 

This problem solving with PSO was taken by some PSO values as pulsation size = 100, 

C1= 2, C2= 2.1, w = 0.73, and n = 200. Solution of the PSO algorithm was used to obtain the 

minimum weight value of the three phase dry type core form transformer. 100 populations 

and 200 iteration have been done by PSO. Fig. 4 illustrates the transformer total weight 

fitness functions values with PSO. Fig. 5 illustrates the Transformer efficiency fitness 

functions values with PSO. The weight of a 1.5 KVA three phase core form dry type 

transformer, founded by using PSO, is 23.5547 kg. The values evaluated are given in Table 3. 

Table 3. The results evaluated of a three phase core type transformer by PSO 

WEIGHT OPTIMIZATION RESULTS OF A DRY TYPE CORE FORM TRANSFORMER BY PSO 

Variables and Other Values Symbol Unit PSO Method Results 

Iron cross section convenience value C cm 2 *joule -1/2 6.62 

Current density value s A/mm 2 2.76 

Window width a mm 45.9 

Primary winding cross section q1 mm 2 1.43 

Secondary winding cross section q2 mm 2 2.85 

Transformer core diameter D cm 6.25 

Primary winding turn w1 turn 248 

Secondary winding turn w2 turn 124 

Primary winding weight Gcu1 kg 2.94 

Secondary winding weight Gcu2 kg 2,08 

The Three legs weight of the Transformer Gfeb kg 9,848 

The yoke weight of the Transformer Gfej kg 8,67 

Total Weight of the Transformer Gtotal kg 23,5547 

Efficiency η % 92.3


Fig. 4. Transformer total weight fitness functions values with PSO. 

4. Conclusion 

Fig. 5. Transformer efficiency fitness functions values with PSO. 

In this study a weight optimization of a 1.5 KVA three phase dry type core form 

transformer has been done by using PSO algorithm. 

Table 4. The values evaluated of a three phase core type transformer by PSO 

compared with classical method 

WEIGHT OPTIMIZATION RESULTS OF A DRY TYPE CORE FORM TRANSFORMER 

Variables and Other Values Symbol Unit Classical Method Results PSO Method Results 

Iron cross section convenience value C cm 2 *joule -1/2 9,48 6,62 

Current density value s A/mm 2 2,2 2,76 

Window width a mm 58 45,9 

Primary winding cross section q1 mm 2 1,79 1,43 

Secondary winding cross section q2 mm 2 3,58 2,85 

Transformer core diameter D cm 7,5 6,25 

Primary winding turn w1 turn 174 248 

Secondary winding turn w2 turn 87 124 

Primary winding weight Gcu1 kg 2,91 2,94 

Secondary winding weight Gcu2 kg 2,16 2,08 

The Three legs weight of the Transformer Gfeb kg 9,92 9,848 

The yoke weight of the Transformer Gfej kg 15,85 8,67 

Total Weight of the Transformer Gtotal kg 30,84 23,5547 

Efficiency η % 93 92,3


In Table 4, the results are compared with the results of the classical method, described in 

[1]. The total weight which has been founded by using PSO is less than the weight which is 

founded with the classical method in [1]. The weight of a 1.5 KVA three phase core form dry 

type transformer, founded by using PSO, is 23.5547 kg whereas the weight, founded by 

classical method was 30.84 kg. The results show that PSO algorithm which can directly reach 

the minimum values is an alternative and better way to estimate the minimum weight values. 

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