Global transformer design optimization using … transformer design optimization using deterministic and non-deterministic algorithms Eleftherios I. Amoiralis Member, IEEE National

Global transformer design optimization using deterministic and non-deterministic

algorithms

Eleftherios I. Amoiralis Member, IEEE

National Technical University of Athens

9 Iroon Polytechniou Street, 15780 Athens, Greece

[email protected]

Marina A. Tsili National Technical University of

Athens 9 Iroon Polytechniou Street, 15780

Athens, Greece [email protected]

Dimitrios G. Paparigas Independent Electrical/Electronic

Manufacturing Professional [email protected]

Abstract -- The present paper compares the application of two deterministic and three non-deterministic optimization

algorithms to global transformer design optimization. Two

deterministic optimization algorithms (Mixed Integer Nonlinear

Programming and Heuristic Algorithm), are compared to three non-deterministic approaches (Harmony Search, Differential

Evolution and Genetic Algorithm). All these algorithms are

integrated in a design optimization software applied and verified in the manufacturing industry. The comparison yields significant

conclusions on the efficiency of the algorithms and the selection

of the most suitable for the transformer design optimization

problem.

Index Terms-- Transformers, Power transformers, Design

optimization, Optimization methods, Algorithms, Artificial intelligence, Genetic algorithms, Heuristic algorithms, Software

packages, Design methodology, Design for manufacture.

I. INTRODUCTION

In today’s competitive market environment, there is an

urgent need for the transformer manufacturing industry to

improve transformer efficiency and to reduce costs, since

high-quality, low-cost products and processes have become

the key to survival in the global economy.

In optimum design of transformers, the main target is to

minimize the manufacturing cost. Therefore, the objective

function is a cost function with many terms, including

material costs, labor costs, and overhead costs. These

component costs, as well as the constraint functions, must be

expressed in terms of a basic set of design variables [1].

Deterministic methods provide robust solutions to the

transformer design optimization problem. In this context, the

deterministic method of geometric programming has been

proposed in [2] in order to deal with the design optimization

problem of both low frequency and high frequency

transformers. Furthermore, the complex optimum overall

transformer design problem, which is formulated as a mixed-

integer nonlinear programming problem, by introducing an

integrated design optimization methodology based on

evolutionary algorithms and numerical electromagnetic and

thermal field computations, is addressed in [3]. However, the

overall manufacturing cost minimization is scarcely addressed

in the technical literature, and the main approaches deal with

the cost minimization of specific components such as the

magnetic material [4], the no-load loss minimization [5] or

the load loss minimization [6]. Techniques that include

mathematical models employing analytical formulas, based on

design constants and approximations for the calculation of the

transformer parameters are often the base of the design

process adopted by transformer manufacturers [7].

Apart from deterministic methods, Artificial Intelligence

techniques have been extensively used in order to cope with

the complex problem of transformer design optimization,

such as genetic algorithms (GAs) that have been used for

transformer construction cost minimization [8] and

construction and operating cost minimization [9][10],

performance optimization of cast-resin distribution

transformers with stack core technology [11], toroidal core

transformers [12], furnace transformers [13], small low-loss

low frequency transformers [14] and high frequency

transformers [15]. GA is also employed for the optimization

of distribution transformers cooling system design in [16].

Neural network techniques are also employed as a means of

design optimization as in [17] and [18], where they are used

for winding material selection and prediction of transformer

losses and reactance, respectively. The comparison of

deterministic and non-deterministic optimization algorithms is

scarcely encountered in the relevant literature, as in [19]

where GA and Simulated Annealing are compared to

Geometric Programming for high-frequency power

transformer optimization.

It is therefore clear that global transformer optimization

remains an active research area, since several approaches for

its implementation have not yet been investigated. It must be

noted that there is no single best optimization algorithm for

all problems, this is called ‘no free lunch theorem’ [20].

Therefore, the purpose of the paper is to indicate a suitable

optimization algorithm dedicated to this problem as well as to

meet the demanding requirements of the industry. The present

paper compares the application of two deterministic and three

non-deterministic optimization algorithms to global

transformer design optimization. The applied deterministic

optimization algorithms are the Mixed Integer Nonlinear

Programming (MINLP) and Heuristic Algorithm (HA), while

the three non-deterministic algorithms are Harmony Search

(HS), Differential Evolution (DE) (the use of both HS and DE

for transformer design optimization is introduced in this

paper) and Genetic Algorithm.

The paper is organized as follows: Section II describes the

mathematical formulation of the transformer design

optimization problem and the software developed to

implement it. Section III provides a brief theoretical

background for the deterministic and non-deterministic

optimization algorithms. Section IV presents the results of the

application of the five algorithms to four different distribution

transformer ratings. Finally, Section V concludes the paper.

II. TRANSFORMER DESIGN OPTIMIZATION

The aim of transformer design optimization is the detailed

calculation of the characteristics of all transformer

components based on prescribed specifications, using

available materials economically to achieve lower cost, lower

weight, reduced size, and better operating performance.

The derivation of transformer designs is implemented with

the use of a software platform (TDO – Transformer Design

Optimization) developed to provide an integrated design,

simulation and visualization environment (Fig. 1) [21]. The

software package is based on advanced optimization

techniques, enabling computation of the optimal transformer

active and mechanical part configuration, analysis and

optimization of cooling system, mechanical design of tanks,

optimal selection between different core and winding

materials, losses and short-circuit impedance analysis as well

as economic evaluation in transformer management. The

integration of the software in an automated design

environment results to significant economic benefits for

transformer manufacturers, achieved without compromises to

the quality of the optimum designs, as far as conformity to the

design specifications and the rest of the performance

parameters are concerned.

TDO is used for:

• Technical calculations of wound core distribution

transformers (alternative core configurations such as stack

core or winding materials such as aluminum can be

considered) with fully parameterized input data and

specifications.

• Manufacturing Cost Minimization ensuring satisfaction of

all imposed constraints on the technical characteristics

• Optimal configuration of cooling system based on the

thermal calculations

• Total Owing Cost Minimization, incorporating the loss cost

to the design optimization

• Size Constrained Applications

• Customized Designs

• A number of different Standards such as NEMA-TP1 /

Department of Energy (DOE) Standard Efficiency, IEEE

and CENELEC can be used for the technical constraints.

• Economic analysis of the optimum designs

• Detailed Losses and Short-Circuit Impedance Evaluation

using advanced field analysis techniques (FEM-Finite

Element Method)

• Detailed thermal performance evaluation based on FEM

• Link to automated AutoCad routines in order to provide a

complete list of industrial drawings ready to be interpreted

by the production line engineers or to be embedded in a

complete study folder.

A. Transformer Input Data

In order to perform the design optimization process, the user must insert the main sixteen input parameters (using the

respective fields of the main TDO software screen) concerning the transformer technical characteristics. These

parameters are:

• Single-phase or three-phase transformer

• nominal power (in kVA): the rated power of the transformer

• guaranteed short-circuit impedance (in %): the guaranteed

value of short-circuit impedance.

• primary and secondary winding material: two choices are

provided, copper or aluminum

• primary and secondary line-to-line voltage (in V): the

nominal voltage of the primary and secondary winding

• primary and secondary winding connection type: three

choices are provided, delta, star or zig-zag (covering all the

possible connection types)

• primary and secondary winding conductor type: five choices

are provided: single or double circular or rectangular wire

and sheet.

• operating frequency (in Hz): two choices are provided,

namely 50 and 60 Hz

• type of magnetic material: three types of materials are

included in the software, namely M4, HiB and Fe-based

amorphous. The user can also insert their own material

through txt files.

• method for the determination of the windings cross-section:

the cross-section calculation is based on the current density,

which can be defined by three different methods, explained

in detail the next paragraph.

• guaranteed no-load and load losses: they can be defined

according to CENELEC standard or upon user selection.

The above sixteen input parameters in the Transformer

Input Parameters frame are sufficient for the derivation of the

rest of the transformer characteristics, since the software

implements calculations that define a significant number of

other electrical and mechanical data (e.g. guaranteed short-

circuit impedance, basic insulation level, windings insulation

type, number of cooling ducts, details of the tank and its

corrugated panels, etc.), while a certain number of design

constants are predefined based on the experience of

transformer design engineers in the manufacturing industry as

well as experimental data on a large amount of produced and

tested transformers.

However, it must be noted that the possibility to access and

modify these data is provided by the software, enabling more

expert users on transformer engineering to examine more

specialized designs. In total, the transformer designer has the

opportunity to change around one hundred input parameters,

based on the design specifications, in order to investigate and

finally reach the best possible result.

Fig. 1. TDO main screen [21].

B. Selection of the Current Density Input Method

One of the crucial design variables during the transformer

design optimization is the calculation of the conductors’

cross-section. The conductors’ cross-section derives from the

current density of the high voltage (HV) and low voltage

(LV) winding, which consist crucial design parameters,

dependent on the transformer rating and loss category.

The Current Density Method comprises three flexible

approaches for the successful definition of the values of the

HV and LV winding current density (in A/mm2), denoted as

WCDHV and WCDLV, respectively.

At the first approach (Method a), the transformer designer

can define directly the value of the WCDHV and WCDLV. The

main drawback of this approach is that the transformer

designer should be quite experienced in order to correctly set

this value and direct the method to the optimal solution.

At the second approach (Method b), an interval with a set

of discrete cLV and cHV values for the LV and HV winding,

respectively, can be defined. In this case, the optimization

algorithm will be executed for all possible cLV.cHV

combinations of current density. Although this approach is

more time-consuming, it assures a global optimum design.

At the third approach (Method c), the designer can

increase the vector of the four design variables (explained at

the next subsection) into six. In particular, the correct

definition of the current density value is under the rules

(supervision) of the MINLP optimization method. In this way,

the transformer designer defines the initial, the upper and the

lower value of the WCDHV and WCDLV and the proposed

method finds an optimum transformer design, designating the

values of the six variables of the design vector.

Method b is recommended in general since it provides the

easiest method to examine a wide range of current densities

and result to the most efficient one. This feature is useful to

both inexperienced and experienced users, and is quite

important in case of designs with “difficult” specifications

(e.g. low guaranteed losses) as well as special designs.

C. Objective Function

By default, the objective of the transformer design

optimization is to minimize the cost of the transformer main

materials, according to the following equation:

8

1

min min( ) ( )j j

xj

Z x c f x=

= ∑ (1)

where cj and fj are the unit cost (€/kg) and the weight (kg) of

each component j of the eight main materials, namely:

• the primary and secondary winding material (Fig. 2)

• magnetic material (Fig. 2)

• insulating liquid

• insulating paper (Fig.2)

• duct strips

• tank sheet steel (Fig. 3)

• corrugated panels material (Fig. 3)

and x is the vector of the four design variables, i.e. the

number of low voltage turns, the magnetic induction

magnitude (B), the width of core leg (D) and the core window

height (G) (Fig. 2).

Fig. 2. Active part configuration

Fig. 3. Transformer tank.

However, upon user selection, the transformer loss cost

can also be integrated into the objective function enabling to

seek for the optimum design based on the total owning cost

(TOC), i.e., the transformer purchasing cost plus the

transformer operating cost:

( )

8

1

min min ( )1j j

j j

j

x x

c x CRM LC

TOC A DNLL B DLLM

=

⋅ + +

= + ⋅ + ⋅−

∑

(2)

where DNLL denotes the designed no-load loss (W), DLL

denotes the designed load loss (W), CRM denotes the cost of

the transformer remaining materials (€), LC denotes the labor

cost (€), M denotes the transformer sales margin (%), A

denotes the equivalent no-load loss cost rate (€/W), and B

denotes the equivalent load loss cost rate (€/W). Another

strong point of the proposed software is that the designer can

define the loss evaluation factors (A and B), either directly or

according to the IEEE Standard C57.120 [22].

III. OPTIMIZATION METHODOLOGIES

The structure of TDO software enables the combination of

the detailed design calculations described in Section II with

five different optimization methodologies and the comparison

of the respective results, as depicted in Fig. 4.

Fig. 4. Optimization Methodologies used in TDO.

A. Deterministic Methodologies

For the production of the optimum transformer design, two

deterministic methodologies can be used in TDO, as

described in the followings.

1) MINLP Methodology

The mixed integer nonlinear programming algorithm

seeks an optimum for the transformer design, defined by a

set of integer variables linked to a set of continuous variables

that minimize the objective function and meet the restrictions

imposed on the transformer design problem. These

restrictions are designated by the tolerances in the deviation

between the designed and guaranteed values of losses and

short circuit impedance [23], as well as manufacturing

constraints. The objective function variables, i.e. the design

variables are: the number of secondary winding turns, the

magnetic induction magnitude (B), the width of core leg (D)

and the core window height (G). Since the windings cross-

section is a major factor affecting the overall transformer

design, and it is linked to the windings current density, the

possibility to insert the primary and secondary winding

current density to the vector of the design variables is

provided by the software, increasing the number of design

variables from four to six.

MINLP method enables non-expert users to design an

optimum transformer with the least possible knowledge, by

providing default input values for the design vector (initial

value, lower and upper value) and the solution space

according to the transformer nominal rating and voltage

[24][25]. These values are based on the constructional

experience on a wide spectrum of various distribution

transformer ratings. This methodology is recommended for:

• non-expert users

• definition of the range of input variables of the design

vector (refinement of the solution space)

• designs with specific technical requirements

2) Heuristic Methodology

This algorithm is based on implementation of the design

calculations for discrete values of the design variables (in

contrast to the MINLP algorithm, where the design variables

can variate among a continuous range of values). Each

combination of the discrete values of the design vector

corresponds to a candidate solution [26]. For each one of the

candidate solutions, it is checked if all the specifications

(limits) are satisfied, and if they are satisfied, the

manufacturing cost is estimated and the solution is

characterized as acceptable. On the other hand, the candidate

solutions that violate the specification are characterized as

non-acceptable solutions and are rejected by the algorithm.

Finally, among the acceptable solutions, the transformer with

the minimum manufacturing cost is selected, which is the

optimum transformer. However, all of the acceptable

solutions are stored and listed by manufacturing cost,

providing the user the ability to select anyone of them,

display it to the main form and investigate its characteristics.

Giving nLV different values for the turns of the low voltage

(LV) coil, nD values for the core’s dimension D, nFD tries for

the magnetic induction (flux density), nG different values for

the core’s dimension G, cLV different values for the LV

winding current density and cHV different values for the HV

winding current density, the total candidate solutions,

Niterations, are calculated from the following equation:

Niterations = nLVnDnFDnGcLVcHV (3)

This methodology is recommended for:

• expert users

• refinement of the optimum solutions provided by MINLP

• direct transformer design based on given input values of the

design vector

It must be noted that the definition of the discrete values of

the design variables often requires prior experience in

transformer design. Selecting a small number of iterations and

incorrect values might result to candidate solutions that do not

satisfy the imposed constraints. In case of non-expert users, it

is recommended that the MINLP algorithm is used for an

initial optimization run and the optimum value of the design

vector is used in order to decide the discrete values of the

design variables of the heuristic algorithm.

B. Non-Deterministic Methodologies

Apart from the aforementioned deterministic methodologies,

three non-deterministic methodologies can also be used in

TDO, as described in the followings [27].

1) Harmony Search Algorithm

The Harmony Search algorithm (HS) is a new

metaheuristic population search algorithm proposed by Geem

et al. [28]. HS was derived from the natural phenomena of

musicians’ behavior when they collectively play their musical

instruments (population members) to come up with a pleasing

harmony (global optimal solution). This state is determined

by an aesthetic standard (fitness function).

The HS is simple in concept, less in parameters, and easy

in implementation. It has been successfully applied to various

benchmarking, and real-world problems like traveling

salesman problem [29]. The main steps of HS are as follows

[28][30].

Step 1) Initialize the algorithm parameters.

Step 2) Initialize the harmony memory.

Step 3) Improvise a new harmony.

Step 4) Update the harmony memory.

Step 5) Check the termination criterion.

These steps are described in the next subsections.

a) Initialization of algorithm parameters

The algorithm parameters are: the harmony memory size

(HMS), or the number of solution vectors in the harmony

memory; harmony memory considering rate (HMCR); pitch

adjusting rate (PAR); and the number of improvisations (NI),

or stopping criterion. The harmony memory is a memory

location where all the solution vectors (sets of decision

variables) are stored. Here HMCR and PAR are parameters

that are used to improve the solution vector, which are

defined in Step 3.

b) Initialization of Harmony Memory

In this step, the HM matrix with as many randomly

generated solution vectors as the HMS: 1 1 1 1

1 2 1

2 2 2 2

1 2 1

1 1 1 1

1 2 1

1 2 1

...

...

. . . . .

. . . . .

. . . . .

...

...

N N

N N

HMS HMS HMS HMS

N N

HMS HMS HMS HMS

N N

x x x x

x x x x

HM

x x x x

x x x x

−

−

− − − −−

−

=

(4)

Static penalty functions are used to calculate the penalty

cost for an infeasible solution. The total cost for each solution

vector is evaluated using

( )

( )

2

1

2

1

( ) ( ) min[0, ( )]

min[0, ( )]

M

ii

i

P

ji

j

fitness X f X g x

h x

α

β

=

=

= +

+

∑

∑

� ��

�

(5)

c) Improvisation of a new harmony

A new harmony vector 1 2( , ,..., )

I

Nx x x x′ ′ ′=�

is generated,

based on three criteria: 1) memory consideration, 2) pitch

adjustment, and 3) random selection. Generating a new

harmony is called improvisation. According to memory

consideration, i-th variable 1 1

1( )HMS

i Ix x x= − The HMCR,

which varies between 0 and 1, is the rate of choosing one

value from the historical values stored in the HM, while (1-

HMCR) is the rate of randomly selecting one value from the

possible range of values, as shown in [28]:

1 2

( () )

{ , ,..., }HMS

i i i i i

i i i

if rand HMCR

x x x x x

else

x x X

end

<

′ ′← ∈

′ ′← ∈

(6)

where ()rand is a uniformly distributed random number

between 0 and 1 and iX is the set of the possible range of

values for each decision variable. For example, an HMCR of

0.85 indicates that HSA will choose decision variable value

from historically stored values in HM with 85% probability or

from the entire possible range with 15% probability. Every

component obtained with memory consideration is examined

to determine if pitch is to be adjusted. This operation uses the

rate of pitch adjustment as a parameter as shown in the

following:

( () )

()i i

i i

if rand PAR

x x rand bw

else

x x

end

<

′ ′= ± ∗

′ ′=

(7)

where bw is an arbitrary distance bandwidth for the

continuous design variable and ()rand is uniform

distribution between and 1.

d) Update of harmony vector

If the new harmony vector 1 2( , ,..., )

I

Nx x x x′ ′ ′=�

has better

fitness function than the worst harmony in the HM, the new

harmony is included in the HM and the existing worst

harmony is excluded from the HM.

e) Check of the termination criterion

The HSA is terminated when the termination criterion

(e.g., maximum number of improvisations) has been met.

Otherwise, steps 3 and 4 are repeated.


• non-expert users



• designs with specific technical requirements

2) Differential Evolution Algorithm

Differential Evolution (DE) is a parallel direct search method which utilizes NP D-dimensional parameter vectors

, , 1, 2,3,...,i G

x i NP= (8)

as a population for each generation G. NP does not change during the minimization process. The initial vector population

is chosen randomly and should cover the entire parameter space. As a rule, we will assume a uniform probability

distribution for all random decisions unless otherwise stated. In case a preliminary solution is available, the initial

population might be generated by adding normally distributed

random deviations to the nominal solution ,0nomx . DE

generates new parameter vectors by adding the weighted

difference between two population vectors to a third vector. Let this operation be called mutation. The mutated vector’s

parameters are then mixed with the parameters of another predetermined vector, the target vector, to yield the so-called

trial vector. Parameter mixing is often referred to as “crossover”. If the trial vector yields a lower cost function

value than the target vector, the trial vector replaces the target vector in the following generation. This last operation is

called selection. Each population vector has to serve once as the target vector so that NP competitions take place in one

generation. More specifically DE’s basic strategy can be described as

follows [31][32]: a) Mutation

For each target vector , ,

i Gx i a mutant vector is generated

according to

1 2 3, 1 , , ,( )i G r G r G r G

v x F x x+ = + ⋅ − (9)

with random indexes r1, r2, r3 {1, 2,3,..., }NP∈ , integer,

mutually different and F>0. The randomly chosen integers r1,

r2, and r3 are also chosen to be different from the running

index i, so that NP must be greater or equal to four to allow

for this condition. F is a real and constant factor ∈ [0, 2]

which controls the amplification of the differential variation

2 3, ,( )r G r G

x x− .

b) Crossover

In order to increase the diversity of the perturbed parameter vectors, crossover is introduced. To this end, the trial vector:

, 1 1 , 1 2 , 1 , 1( , ,..., )i G i G i G Di G

u u u u+ + + += (10)

is formed where,

, 1

, 1

,

( ( ) ) ( )

( ( ) ) ( )

1, 2,..., .

ji G

ji G

ji G

u if randb j CR or j rnbr iu

x if randb j CR and j rnbr i

j D

+

+

≤ == > ≠

=

(11)

In (11) ( )randb j is the jth evaluation of a uniform random

number generator with outcome ∈ [0, 1]. CR is the crossover

constant ∈ [0, 1] which has to be determined by the user.

( )rnbr i is a randomly chosen index ∈ 1,2,…,D which

ensures that , 1i G

u + gets at least one parameter from, 1i G

v + .

c) Selection

To decide whether or not it should become a member of

generation G+1, the trial vector , 1i G

u + is compared to the

target vector ,i G

x using the greedy criterion1. If vector

, 1i Gu + yields a smaller cost function value than

,i Gx , then

, 1i Gx + is set to , 1i G

u + ; otherwise, the old value ,i Gx is retained.


• expert users



3) Genetic Algorithm

The Genetic Algorithm metaheuristic is traditionally applied

to discrete optimization problems. Individuals in the population are vectors, coded to represent potential solutions

to the optimization problem. Each individual is ranked according to a fitness criterion (typically just the objective

function value associated with that individual). A new population is then formed as children of the previous

population. This is often the result of cross-over and mutation operations applied to the fittest individuals [33].

In our case, 30 runs of the GA algorithm are performed

and the best solution is chosen as the optimum one. The

population type is bit string of size equal to 20. A random

initial population is created, that satisfies the bounds and

linear constraints of the optimization problem. Rank fitness

scaling is employed, scaling the raw scores based on the rank

of each individual, rather than its score. Stochastic uniform

selection function is used, which lays out a line in which each

parent corresponds to a section of the line of length

proportional to its expectation. The algorithm moves along

the line in steps of equal size, one step for each parent. At

each step, the algorithm allocates a parent from the section it

lands on. The first step is a uniform random number less than

the step size. As far as mutation and crossover functions are

concerned, the first one is adaptive feasible (it randomly

generates directions that are adaptive with respect to the last

successful or unsuccessful generation - a step length is chosen

along each direction so that linear constraints and bounds are

satisfied) and the second one is scattered (it creates a random

binary vector, selects the genes where the vector is a 1 from

the first parent, and the genes where the vector is a 0 from the

second parent, and combines the genes to form the child.).

The limit for the fitness function is set to 0.5, while the

maximum number of generations (iterations) is equal to 500.


• expert users



IV. RESULTS AND DISCUSSION

It is essential to find an optimum transformer that satisfies the

technical specifications and the purchaser needs with the

1 Under the greedy criterion, a new parameter vector is accepted if and

only if it reduces the value of the cost function.

minimum manufacturing cost. The HA, MINLP (deterministic

group) and HS, DE, GA (non-deterministic group) optimization algorithms are applied for the design

optimization of four 20-0.4kV three-phase distribution transformers, of 160kVA, 400kVA, 630 kVA and 1000 kVA

rating. Tables I, II, III and IV compare the respective optimization results. In addition, Figs 5-8 illustrate the

comparison of the guaranteed versus designed short-circuit impedance, as well as the total guaranteed versus designed losses for each examined transformer rating based on the five

different optimization method results. To be more precise, spider charts are used to compare and evaluate each algorithm

performance for each transformer design, based on two important characteristics: short-circuit impedance and total

losses. In each spider graph, the blue polygon represents the guaranteed values and the red straight dotted line polygon

shows the designed values (final results from each optimization method). It must be noted that the maximum

permissible deviation between the guaranteed and designed

values is equal to ± 10% in the case of short-circuit

impedance and +10% in the case of total losses. Tables I-IV show the results of the five optimization

algorithms based on the rated power. In particular, the first four lines of each Table show the optimum values of the

design vector, the next four lines depict the guaranteed losses and short-circuit impedance, the next four present the

designed losses and short-circuit impedance, and finally the last two lines refer to the cost analysis.

Regarding Table I, DE shows to have the best performance in comparison with the other algorithms in terms of cost.

However, during the decision making process, values of technical specifications can influence our final choice. In this

case, HA and GA have quite good behavior concerning the total losses and the short-circuit impedance (Fig. 5), and the

first algorithm (HA) dominates to the second one (GA) due to the lower total losses and lower cost of the respective optimal

transformer. As a result, HA seems to be the best possible selection.

In the case of the 400 kVA transformer (Table II) HS is the lowest cost solution, however the results of HA or MINLP are

more efficient in terms of technical performance (better total losses) (Fig. 6).

In the case of the 630 kVA transformer (Table III) HA provides the most efficient solution in terms of cost. As far as

losses are concerned, the HA solution exhibits slightly higher losses compared to MINLP but an improved short-circuit

impedance value (Fig. 7). The non-deterministic algorithms correspond to optimal designs of higher cost and losses but

better short-circuit impedance results (especially the HS algorithm).

Finally, in the case of 1000 kVA transformer (Table IV) HA and HS provide the best solutions which are very close in

cost and performance characteristics.

TABLE I

Comparison of the Optimization Algorithms for the 160 KVA transformer.

Characteristics 160kVA

of the optimum transformer

design

MINLP HA HS GA DE

Low voltage turns 31 29 32 31 31

D (mm) 161 204 183 194 199

G (mm) 207 206 228 228 229

B (Gauss) 16570 16090 16692 16009 16693

Guaranteed Fe

losses (W)

300 300 300 300 300

Guaranteed Cu

losses (W)

2350 2350 2350 2350 2350

Total guaranteed

losses (W)

2650 2650 2650 2650 2650

Guaranteed Short-

Circuit Impedance

(%)

4 4 4 4 4

Designed Fe

Losses (W)

344 330 342 323 345

Cu Losses (W) 2404 2369 2374 2358 2450

Total designed

losses (W)

2748 2699 2716 2681 2795

Short-Circuit

Impedance (%)

4.34 3.91 4.13 3.9 4.21

Cost (€) 2243 2241 2297 2331 2215

Cost

Classification

3 2 4 5 1

3.6

3.7

3.8

3.9

4

4.1

4.2

4.3

4.4

MINLP

HA

HSGA

DE

Guaranteed Usc (%)

Designed Usc (%) (a)

2550

2600

2650

2700

2750

2800

MINLP

HA

HSGA

DE

Total guaranteed losses (W)

Total designed losses (W) (b)

Fig. 5. Comparison of the guaranteed and designed short-circuit impedance

(a), and total guaranteed and designed losses (b) for each 160 KVA

transformer design based on the five different optimization results.

TABLE II


Characteristics

of the optimum transformer

design

400kVA

MINLP HA HS GA DE


D (mm) 243 250 213 238 213

G (mm) 267 268 309 309 308

B (Gauss) 16822 17000 16986 16657 17036

Guaranteed Fe

losses (W)

610 610 610 610 610

Guaranteed Cu

losses (W)

4600 4600 4600 4600 4600

Total guaranteed

losses (W)

5210 5210 5210 5210 5210

Guaranteed Short-

Circuit Impedance

(%)

4 4 4 4 4

Designed Fe

Losses (W)

691 701 693 689 699

Cu Losses (W) 4670.00 4676.0

0

4824 4758 4692

Total designed

losses (W)

5361 5377 5517 5447 5391

Short-Circuit

Impedance (%)

4.24 4.23 4.09 3.8 4.2

Cost (€) 4402 4383 4449 4494 4539

Cost

Classification

2 1 3 4 5

3.5

3.6

3.7

3.8

3.9

4

4.1

4.2

4.3

MINLP

HA

HSGA

DE

Guaranteed Usc (%)


4500

4700

4900

5100

5300

5500

MINLP

HA

HSGA

DE






TABLE III


Characteristics of

the optimum

transformer design

630kVA

MINLP HA HS GA DE


D (mm) 279 292 260 286 244

G (mm) 291 296 336 330 392

B (Gauss) 16106 16150 16080 16038 16467

Guaranteed Fe

losses (W)

860 860 860 860 860

Guaranteed Cu

losses (W)

6500 6500 6500 6500 6500

Total guaranteed

losses (W)

7360 7360 7360 7360 7360

Guaranteed Short-

Circuit Impedance

(%)

4 4 4 4 4

Designed Fe

Losses (W)

989 989 983 951 978

Cu Losses (W) 5238 5284 5386 5457 5730

Total designed

losses (W)

6227 6273 6369 6408 6708

Short-Circuit

Impedance (%)

4.4 4.32 4.03 4.24 4.24

Cost (€) 7109 7084 7167 7241 7260

Cost

Classification

2 1 3 4 5

3.8

3.9

4

4.1

4.2

4.3

4.4

MINLP

HA

HSGA

DE

Guaranteed Usc (%)


5500

6000

6500

7000

7500

MINLP

HA

HSGA

DE






TABLE IV


Characteristics of

the optimum

transformer design

1000kVA

MINLP HA HS GA DE


D (mm) 265 285 276 316 292

G (mm) 384 338 338 338 276

B (Gauss) 17321 16800 16741 16534 16699

Guaranteed Fe

losses (W)

1100 1100 1100 1100 1100

Guaranteed Cu

losses (W)

10500 10500 10500 10500 10500

Total guaranteed

losses (W)

11600 11600 11600 11600 11600

Guaranteed Short-

Circuit Impedance

(%)

6 6 6 6 6

Designed Fe

Losses (W)

1249 1264 1265 1196 1265

Cu Losses (W) 10478 9703 9675 9936 9462

Total designed

losses (W)

11727 10967 10940 11132 10727

Short-Circuit

Impedance (%)

6.32 5.57 5.57 5.72 6.53

Cost (€) 9001 8785 8786 8918 8777

Cost

Classification

5 1 2 4 3

5

5.2

5.4

5.6

5.8

6

6.2

6.4

6.6

MINLP

HA

HSGA

DE

Guaranteed Usc (%)


10200

10400

10600

10800

11000

11200

11400

11600

11800

MINLP

HA

HSGA

DE






In non-deterministic methodologies, as well as in

deterministic methodologies, the design vector definition is crucial in order to meet some desired performance objective.

For example, the exploring of the geometrical core

parameters can be ensured through careful planning, and thus

the quality of transformer design can be established during the definition of initial design vector values. However, the input

data of the design vector are deployed randomly, in the case of the non-deterministic methods. As a result, there is

possibility of little control over investigating the entire solution space. Therefore, deterministic methods are often

pursued for only a selected subset of the design vector with the aim of in-depth searching of the solution space.

According to the above results, HA provides the best

solution both in terms of cost and operating performance (especially on total losses). Heuristic algorithm does not

guarantee optimal, or even feasible, solution and is often used with no theoretical guarantee. Despite this main disadvantage,

heuristic evaluations still perform an important role in the transformer design, and if implemented properly can provide

powerful results. Based on the case studies that have been carried out, it

should be noted that since the non-deterministic methods or stochastic methods use random processes, an algorithm run at

different times can generate different transformer designs. Therefore, a particular transformer study needs to be run

several times before the solution is accepted as the global optimum. On the contrary, MINLP and HA belong to

deterministic methods which are able to find the global minimum, but by an exhaustive search. In this case, in order

to avoid huge calculations, stochastic methods can provide us with the first suboptimal solution, and afterwards, HA can be

used in order to finalize our decision. Since no other method gives an absolute guarantee of finding the global minimum in

a finite number of steps, HA technique becomes important. Despite the fact that five different optimization techniques

were investigated in order to find the most economic transformer design with respect to a sequence of mechanical

and electrical constraints, the transformer manufacturing factories declare that a relative near-optimal solution (to the

optimal one) is often preferred and finally is chosen to be constructed. Under these conditions, it is obvious that the

criterion of cost is not the only factor which should be taken into account at the final decision but also the transformer

specifications of each optimum design, such as the no-load and load losses and the short circuit impedance, are vital

aspects. Based on the above-mentioned fact, HA and MINLP become important methods, since they can store a wide range

of several optimum solutions with different technical specifications, especially the HA.

It must be pointed out that the tuning of non-deterministic algorithms has derived through comparison of various

combinations and choice of the best one between them (in order to exclude the possibility that they are not properly

tuned, thus they cannot converge to the global optimum as the deterministic algorithm). The methods of stochastic nature

fail to find the global optimum due to the fact that the optimality of the solution provided by them cannot be

guaranteed and multiple runs may result to different suboptimal solutions [33], with a significant difference

between the worst and the best one. On the other hand, deterministic methods provide more robust solutions to the

transformer design optimization problem and are more

suitable for the search of global optimum.

V. CONCLUSION

In the present paper, comparison of deterministic and non-deterministic optimization methods has been carried out in

order to achieve optimal global transformer design. The design optimization has been carried out with the use of an

integrated software platform, which has been experimentally verified and integrated in the automated design process of

several transformer manufacturing industries. The combination of the proposed methods is very effective

because of its robustness, its high execution speed and its ability to effectively search the large solution space. The

ability to locate the global optimum is illustrated by the application to a wide spectrum of actual transformers, of

different power ratings. The development of user-friendly software based on the combination of these methods provides

significant improvements in the design process of the manufacturing industry.

According to the results, HA provides the best solution both in terms of cost and operating performance. The

methods of stochastic nature fail to find the global optimum due to the fact that the optimality of the solution provided by

them cannot be guaranteed and multiple runs may result to different suboptimal solutions, with a significant difference

between the worst and the best one. On the contrary, MINLP and HA belong to deterministic methods which are able to

find the global minimum, but by an exhaustive search. It is however pointed out that the goal is not only to find the most

economic transformer, but a design that meets the technical specifications with the less possible deviation from the

guaranteed values. In this context, the criterion of cost is not the only factor which should be taken into account at the final

decision but also the transformer specifications of each optimum design.

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Documents

Global transformer design optimization using … transformer design optimization using deterministic and non-deterministic algorithms Eleftherios I. Amoiralis Member, IEEE National