6

CHAPTER 2

EXISTING AND PROPOSED GRAPHICAL METHODS

2.1 DESCRIPTION OF PROBLEM FOR THE THESIS

It is important to describe the problem taken for the thesis. The

problem is stated as follows:

i) To propose a method for obtaining optimum values for the

variables simultaneously where the sum of variables make

the objective of the transformer considered to be minimum

while delivering its rated kVA.

ii) To understand that the conventional design is a trial and

error method where optimization is not possible.

iii) To solve the transformer problem by conventionally

available optimizing methods and comparing to satisfy.

iv) To use Random jumping method as a comparator and assess

the values of the variables.

v) To study the status of other variables for its use and

performance quantities, such as efficiency, regulation,

temperature rise, etc.,

7

2.2 TRANSFORMER EQUATIONS

The output in kVA of a single phase transformer is

-3Q = 2.22fB A A ×10m c i (2.1)

and for a three phase transformer is

33.33 10Q fB A Am c i (2.2)

The derivation of equations is given in Appendix 1.

2.3 VARIABLES

There are as many as 59 variables or parameters that appear at the

time of design of a transformer (Ramamoorthy 1987). The design objectives

mentioned previously can be achieved only if sufficient parameters are

available for optimization of an objective. Some of these parameters are

considered as free parameters since they are free to be chosen by the designer

so as to optimize the design. All other parameters are known and are called as

fixed parameters. The choice of which parameters are fixed and which are

free, is of course, made by the designer depending upon the requirement. The

authors have characterized two types of variables namely,

i) Electric and magnetic variables

ii) Geometry or structural variables

and approached for optimization by two different methods. One approach is to

fix the electrical and magnetic parameters and then choose the transformer

geometrical parameters to minimize an overall objective such as weight,

8

volume, losses or cost. An alternate approach is to start with an assumed core

geometry and then find values for the electrical and magnetic parameters

which minimize the design objective. But a new methodology is suggested in

this thesis for optimum design of a distribution transformer.

2.4 ASSUMPTIONS AND APPROXIMATIONS

It is recognized that in some designs, one or more assumptions are

used depending upon the experience of the designer to come out with simple

mathematical results. The following are some of the examples which are

made at the appropriate places in the design calculations.

The waveform of induced emf of a transformer is assumed as

sinusoidal. Hence the value for the factor in the emf equation is taken as 4.44.

Frequency of voltage waveforms is taken as 50Hz and fixed.

Induced primary and secondary emfs are approximately equal to the

designated terminal voltages respectively in order to make the calculation of

number of turns simple, since it is estimated by dividing the induced emf by

emf per turn.

The core loss is a sum of three loss mechanisms; hysteresis,

residual and eddy current (Petkov 1996). However, only hysteresis and eddy

current losses are considered towards the contribution for the core loss here as

a sum of two equations, one for hysteresis loss and another one for eddy

current loss as the residual loss due to residual magnetism, will be very small

and may be neglected. Winding or copper losses due to alternating current

are represented in the usual manner, namely a product of square of rms value

of current and resistance (I2R).

9

Parameters such as area, flux density associated with the core and

parameters of winding materials namely area of cross section, current density,

etc., are assumed as constants.

Judd (1977) Winding temperature rise is modeled as a linear

function of transformer losses of a given surface, available for heat

dissipation.

Say (1958) The magnetizing mmf is assumed to be very small and

hence the exciting current is neglected, since the two circuits of the

transformer links the common iron core. Anderson (1991) As the program

suggested by him in his paper is intended only for demonstration purpose, he

ignored the magnetizing current. Judd (1977) He stated in his paper that

exciting current can be neglected.

Primary and secondary current densities are identical.

The above assumptions and approximations which were used by

previous authors are reasonable in the design of transformers and lead to

simple mathematical results.

2.5 ERRORS

Errors i.e. differences from predicted and actual values are common

due to two factors. First, there were differences between the original given

specifications and the resulting measured specification parameter values due

to manufacturing process. Second, there were differences between computer

calculated and measured design parameter values due to approximation.

Obviously it is not always possible to build a transformer with exactly the

10

calculated design parameter values. Errors are inevitable provided that the

variation in performance quantities is within limits. In addition, variations in

material properties and measurement tolerances can influence the measured

performance factor values. It is stated in the paper that differences between

specified and measured temperature rises tends to be less than 7 percent,

whereas differences between computed and measured VA capability tended to

be less than 5 percent (Judd 1977). However the other paper (Petkov 1996)

states that there is a 15 percent difference between the predicted and practical

values. It is clear that errors are common and it is to be seen that they are

within the acceptable limit.

2.6 EXISTING METHODS AND DEVELOPMENTS

There are number of variables, parts, performance quantities,

objectives for consideration in a transformer. Depending upon the nature or

requirement, authors have used different methodologies to arrive at an

optimization or solution. Some authors have developed softwares depending

upon the nature of problem.

Judd (1977) have demonstrated the design strategy which proposed

a technique which starts with an assumed core geometry and then finds values

of the electrical and magnetic parameters that maximize the volt-ampere (VA)

capacity or minimize loss. However this technique fell short of considering

secondary constraints such as high efficiency, low winding regulation and no-

load current. Some others have optimized the design of a transformer by

linking power loss in the winding and core with the values of the flux density

and frequency (Petkov 1996).

11

Hurley et al (1998) have presented an improved formulation and

solution of the minimum loss problem including high – frequency effects.

Although a procedure which assists in choosing a suitable core was given in,

the core dimensions were not treated as variables and secondary constraints

were still not accounted.

The papers (Judd et al 1977, Petkov 1996 and Hurley 1998) have

explained the optimization of performance quantities such as VA, efficiency

of a transformer. It has made use of two category of variables depending upon

the objective. They have used maxima/minima of mathematical method. The

optimum values for the variables are obtained successively one after another.

The optimum value of one variable is dependent upon the optimum value of

other variable.

Also the papers, Anderson (1991) and Ahmed Rubaai (1994) which

employ computer program are also indicated as samples for iterative method.

The methodology adopted in these papers is presented in Table 2.1 for

reference. The remarks are indicated in the Table itself.

12

Table 2.1 Transformer objectives and optimizing methods

S.

No.

Name of the

author

Title of the

PaperObjective Optimizing Method Symbols Remarks

1. Judd

(1977)

Design

optimization

of small low-

frequency

power

transformers

To maximize VA

(volt-ampere)with

constraint of

temperature rise

i) Pi/ J=0

ii) Jopt

iii) Substitute Jopt to get

Bopt

Pi – load power,VA

J-current density,

A/cm2

B- peak flux density,

Gauss

Calculus method.

Variables are

estimated

separately.

2. Petkov

(1996)

Optimum

design of a

High-power,

High –

frequency

transformers

To maximize VA

with temperature

rise as constraint and

to minimize loss

i) dPo2/dB(f=const)=0

ii) Boptpo (only for max.

VA and not for max.

Efficiency)

iii) Pomax

AND

i) dPt/dB(f=const)=0

ii) Bopteff

iii) Ptmin

Po-transferred power

of the core, VA

B-flux density, T

f-frequency, Hz

Boptpo-optimizing flux

density

For maximum core

VA rating,

Pomax-maximum VA

Pt-total transformer

loss

Bopteff-optimum flux

density for minimum

transformer loss

Ptmin-minimum

transformer loss

Calculus method.

The following

appear in the

paper. The

question now is

which one of the

derived optimum

value BOPTPO and

BOPTEFF should

be used in the

transformer

design?

13

Table 2.1 (Continued)

S.

No.

Name of the

author

Title of the

PaperObjective Optimizing Method Symbols Remarks

3. Hurley

(1998)

Optimized

transformer

design:

inclusive of

high

frequency

effects

To minimize loss

with rated VA as

constraint

i) For global minimum

(i.e. = )

dP/d(fBm)=0

ii) In general, (i.e. )

P/ Bm=0 for a fixed

value of frequency

, -material constants

P-total losses

f-frequency, Hz

Bm-maximum flux

density

Calculus method.

Losses are

estimated

separately.

4. O.W.Anderson

(1991)

Optimized

design of

Electric Power

Equipment

To explain the

optimization of an

objective with two

variables

Iterative method

- Requires an

initial set of

values for the

variables.

5. Ahmed Rubaai

(1994)

Computer

Aided

Instruction of

Power

Transformer

Design in the

Undergraduate

Power

Engineering

Class

To illustrate the

applications of the

computer program

for Transformer

Design to meet all

performance

requirements at

minimum cost.

Iterative method

- Requires an

initial design;

limits are to be

specified for the

variables.

14

The programming method of optimization involves more number of

steps and also the required data is estimated separately and not

simultaneously. The optimum values of variables are obtained one after

another for optimizing an objective.

Wu et el (1980) have addressed the problem of finding minimum

weight EI core and pot core transformer designs using the technique of

Lagrange multipliers.

Anderson (1991) have presented an optimizing routine, monica,

based on monte carlo simulation. In essence, the optimizing logic in monica

uses random numbers to generate a large set feasible designs, but in this case

to generate a response surface corresponding to the objective function. The

optimization was derived from the response of surface using classical

optimization theory of continuous variables. Geromel (2002) More recent

research considered the use of artificial intelligence techniques in power

transformer design: neural networks were used as an alternative modeling

strategy whereas genetic algorithms were employed in the search procedure.

Edvin Shehuet al (2005) describes a fitting algorithm suitable for

simultaneously approximating the real and imaginary parts of transformer

admittance curves. The algorithm follows a unique strategy to determine the

best initial guess. Rubaai A (1994) describes a single phase transformer

design suitable for classroom use. The scope of this design is limited to the

specification for the core configured transformers are designed in this paper.

A computer program is developed for the purpose of illustrating the design

procedure and demonstrating how it works. The objective is to meet all

performance requirements at minimum cost.

The transformer design procedures are improved by the blending of

traditional transformer design practice with the phenomenal speed and logic

15

adaptability of a modern computer such that a greater insight into transformer

design is possible. The scope of this computer programme covers single and

three phase transformer from 1 to 5000 W and utilizes 13 different kinds of

magnetic steel (Odessey P 1974). C.H.Yu et al (1993) describes several

important features of the development of software package and shows some

computed result for various three phase, three limb transformer cores.

A novel computer based learning frame work that has been

developed and applied for the online control and optimization of transformer

core manufacturing process is presented (Georgilakis P et al 1999). The

proposed frame work aims at predicting core loss of wound core distribution

transformers at the early stages of transformer construction. Moreover, it is

used to improve the grouping process of the individual cores by reducing iron

losses of assembled transformers. Three different automatic learning

techniques (namely decision trees, artificial neural networks and genetic

algorithms) are combined and their relevant features are exploited.

M. Krasl et al (2005) deals with the transformer losses especially

eddy current losses in windings. Nowadays, there is the stress on the

possibilities of lowering losses and optimization of transformer dimension.

Calculations were provided with using Finite Element Method (FEM).

Stadler A et al (2005) have explained the influence of the winding

layout on the core losses and also on the leakage inductance with air gapped

toroids. An analytical method is used in which the field distribution is

calculated by means of orthogonal expansion. Based on these results, some

design guidelines are derived in order to optimize these components.

The large number of variables and the fact that, in addition, their

multiple interrelations are not completely known make power transformer

design a quite involved task. Geromel L H (2002) presents a novel power

16

transformer design methodology using artificial neural networks (ANNS).

The methodology described allows the application of an ANN in certain

specific stages of the design.

Rabih A (2005) considers the transformer design optimization

problem. In its most general form, the design problem requires minimizing

the total mass (or cost) of the core and wire material while ensuring the

satisfaction of the transformer ratings and a number of design constraints. The

constraints include appropriate limits on efficiency, voltage regulation,

temperature rise, no-load current and winding fill factor. The design

optimization seeks a constrained minimum mass (or cost) solution by

optimally setting the

i) Transformer geometry parameters and

ii) The relevant electrical and magnetic quantities.

In cases where the core dimensions are fixed, the optimization

problem calls for a constrained maximum volt-ampere or minimum loss

solution. It shows that the above design problems can be formulated in

geometric programming (GP) format. The importance of the GP format stems

from two main features. First, GP provides an efficient and reliable solution

for the design optimization problem with several variables. Second, it

guarantees that the obtained solution is the global optimum. It includes a

demonstration of the application of the GP technique to transformer design. It

also includes a comparative study to emphasize the advantage of including the

transformer core dimensions as variables in design problem.

Some authors give importance for the losses in a transformer and

developed methods for the estimation of the same. Enokizona M et al (1999)

have developed a new expression for hysteresis and introduced into a finite

element formulation and applied to transformer core model. Georgilakis P S

17

et al (1998) tried artificial neural network (ANN) approach to predict and

classify distribution transformer specific iron loss, ie losses per weight unit.

Moses J has produced an algorithm to estimate the iron loss of power

transformers from quantification of the contributions of the effect of joints,

rotational and harmonic flux, stress, inter-laminar flux and core geometry.

Ilo et al (1996) states that the most important properties of

transformer cores are losses and noise.

The power transformer is one of the most important equipment in a

power system. Optimum design of a transformer involves determination of

design parameters of a power transformer when a chosen objective is

optimized, simultaneously satisfying a set of constraints. Padma S et al (2006)

has proposed Simulated Annealing (SA) techniques for Optimization of three

phase Power Transformer Design (OPTD). The initial cost of transformers

viz. material cost of stampings and cost of copper used for windings is chosen

as the objective that is to be minimized. The method yielded a minimum, the

computation time and cost of active material are much reduced when

compared with conventional design results. The efficiency of transformer is

found to improve with application of the algorithm.

The problem taken by the authors are of different kind. Some have

tried to optimize the VA, optimize loss. Some authors have tried to minimize

the core loss, eddy current loss in windings. Rabih (2005) seeks a constrained

minimum mass (or cost) solution by optimally setting the two types of

variables. The authors have used and or developed different methods to solve

the problem. They use calculus method, ANN, GP, GA, SA, etc., techniques

for optimization.

The above papers dealing optimization uses either calculus of

mathematical method where optimum values for the variables are obtained

18

one after another with a dependence between them or iterative method,

needing initial values for the variables.

The problem for this thesis is stated in sec 2.1 namely to propose a

method for obtaining optimum values for the variables simultaneously where

the sum of variables make the objective of the transformer considered to be

minimum while delivering its rated kVA.

Hence the problem of this thesis is slightly different from others.

2.7 TWO VARIABLE APPROACH & PROPOSED GRAPHICAL

METHOD

Optimization uses the most effective value of variables, whereas

optimum is the best value of the variable for practical purposes, as said

earlier.

Iron and copper are the two active materials of a transformer. The

objective namely, cost, volume, weight or loss is directly related to iron and

copper.

2.7.1 Controlling Factor for Optimum Design

(Sawhney 2001) Transformers are to be designed to make one of

the aforesaid quantities or objectives as minimum. In general, the objectives

are contradictory and it is normally possible to satisfy only one of them. All

these quantities vary with the ratio,

r ATm (2.3)

If a high value of r is chosen, the flux becomes large and

consequently a large core cross section is needed which results in higher

19

volume, weight and cost of iron and also gives a higher iron loss. On the other

hand owing to decrease in the value of AT, the volume, weight and cost of

copper required decreases and also the copper losses decrease. Thus it may be

said that r is a controlling factor for the above mentioned quantities.

Depending upon the need, some variables are treated as fixed and other

variables as free in an equation.

The controlling factor for single phase and three phase transformers

are

2B Am irAc

(2.4)

and4B Am ir

Ac (2.5)

respectively.

2.7.2 Conditions for Optimum Design

(Sawhney 2001) The conditions for optimum design are derived in

Appendix 1.

2.7.2.1 Minimum Cost

(Say 1958), (Sawhney 2001) The cheapest transformer in first cost

is that in which the aggregate cost of material is a minimum (neglecting any

variations in construction cost).

The cost is concerned with the transformer active materials, namely

iron and copper.

20

Hence for a given value of flux density and current density, i.e., the

magnetic and electrical parameters, the condition for minimum cost is

Cost of iron = Cost of copper

i.e. C Cci (2.6)

2.7.2.2 Minimum Volume

(Sawhney 2001) Similar to minimum cost, the condition for

minimum volume can be obtained as

Volume of iron = Volume of copper

i.e. U Uci (2.7)

2.7.2.3 Minimum Weight

(Sawhney 2001) The condition is

Weight of iron = Weight of copper

i.e. G Gci (2.8)

2.7.2.4 Minimum Full Load Loss

(Sawhney 2001) Since the losses are governed by magnetic and

electric parameters, the flux density and current density are assumed as

variables and the structural parameters are considered as fixed parameters.

At any fraction x of full load, the condition for minimum loss or

for maximum efficiency can be obtained as 2P x Pci

21

Hence for minimum full load loss (i.e., x=1), the condition is

Full load iron loss = Full load copper loss

i.e. P Pci (2.9)

The above conditions hold good for both single and three phase

transformers.

2.7.3 Objective Functions

Taking minimum cost design, for example, the total cost of a

transformer is the sum of cost of iron and cost of copper. Similarly the total

minimum of other objectives can be written as the sum of the corresponding

quantities. In equation form, they may be written as,

(for minimum cos )T

C C C tci (2.10)

(for minimum volume)T

U U Uci (2.11)

(for minimum weight )T

G G Gci (2.12)

(for minimumT

P P Pcifull load loss) (2.13)

It is seen that the objective functions are having only two variables.

The objective functions have linear straight line characteristics. The objective

functions can be modified depending upon the available known parameters.

2.7.4 Constraint Equation

As the transformer is to deliver rated kVA, the output equation is

taken as constraint equation. Accordingly the constraint equations for single

22

and three phase transformers are as given in Equations (2.1) and (2.2)

respectively.

Now depending upon the objective, the constraint equation is

modified.

2.7.4.1 Minimum Cost/Area

The cost of the material is given by the product of specific cost,

density, area and length of the material. Out of these four quantities, specific

cost and density are known and fixed. The other two parameters are structural

parameters and they are free to vary. Out of these two, lengths are assumed to

be constant, whereas area is treated as free parameter. This is advantageous,

since the output constraint equation is already in terms of area only. However,

the area may also be treated as constant and lengths may be taken as variable

parameters.

Accordingly, the objective function for total minimum cost is

written in terms of area as,

T 1 i 2 cC = K A + K A (2.14)

where K1 and K2 are calculable constants. If area alone is to be optimized,

the objective function is

TA A Aci

(2.15)

The constraint equation for a single phase transformer is rewritten

as

32.22 10Q fB A A kVAm c i

23

Considering that the magnetic and electric parameters are given

and fixed, the above equation may be modified as

3102.22

QA Aci f Bm

(2.16)

which is the constraint equation.

The corresponding equation for a three phase transformer is

3103.33

QA Aci f Bm

(2.17)

2.7.4.2 Minimum Volume

The constraint equation for a single phase transformer in terms of

volume may be written as

32.22 10U Uc iQ f Bm L l

mt i

Assuming that the structural parameters namely Lmt

and li

are

known and fixed and also magnetic and electric parameters are given, the

above equation may be written as

310

2.22

Q L lmt iU Uci f Bm

(2.18)

and is the constraint equation for minimum volume.

24

The corresponding equation for a three phase transformer is

310

3.33

Q L lmt iU Uci f Bm

(2.19)

2.7.4.3 Minimum Weight

The constraint equation for a single phase transformer in terms of

weight may be written as

32.22 10GGc iQ f Bm g L g lc mt i i

After transforming the variables, the constraint equation for

minimum weight is

310

2.22

Q g L g lc mt i iG Gci f Bm (2.20)

The corresponding equation for a three phase transformer is

310

3.33

Q g L g lc mt i iG Gci f Bm (2.21)

2.7.4.4 Minimum Full Load Loss

In the output constraint equation, Bm and are related to iron and

copper losses respectively and hence they are treated as free independent

variables. At the same time, the structural parameters Ac and Ai

are assumed

as known and fixed. Frequency is also known. The task is to modify the

25

constraint equation such that Pi and Pc appear as variables in place of

Bm and respectively.

After modification,

2 2 26

629.57 10

Q K t f L lc mt ih sP Pci f A Ac is

(2.22)

Equation (2.22) is the constraint equation for minimum full load

loss of a single phase transformer.

The corresponding equation for a three phase transformer is

2 2 26

644.3556 10

Q K t f L lc mt ih sP Pci f A Ac is

(2.23)

The modification is shown in Appendix 2.

Since some of the variables appear in constraint equations are non-

linear in nature, the constraint characteristic will also be non-linear in shape,

in particular in parabolic shape.

The constraint equations are presented in Table 2.2 for quick

reference.

26

Table 2.2 Constraint equations of a transformer

S.

No.

Requirement

or Objective

Constraint equation

Single phase Three phase

1 Full load loss

2 2 26

629.57 10

Q K t f L lc mt ihP Pci f A Ac i

s

s

2 2 26

644.3556 10

Q K t f L lc mt ihP Pci f A Ac i

s

s

2 Area310

2.22Q

A Aci f Bm

3103.33

QA Aci f Bm

3 Volume

310

2.22

Q L lmt iU Uci f Bm

310

3.33

Q L lmt iU Uci f Bm

4 Weight

310

2.22

Q g L g lc mt i iG Gci f Bm

310

3.33

Q g L g lc mt i iG Gci f Bm

27

2.7.5 Equivalent Optimization Problem of Transformer

Singiresu S Rao (1996) Optimization is the act of obtaining the

best result under given circumstances. In design, construction and

maintenance of any engineering system, engineers have to take many

technological and managerial decisions at several stages. The ultimate goal of

all such decisions is either to minimize the effort required or to maximize the

desired benefit. Since the effort required or the benefit desired in any practical

situation can be expressed as a function of certain decision variables,

optimization can be defined as the process of finding the conditions that give

the maximum or minimum value of a function. Since the objective function

and constraint equation is obtained from the conditions of optimum design,

the transformer optimum design problem may be viewed as an equivalent

optimization problem now.

2.7.5.1 Statement of an Optimization Problem

An optimization or a mathematical programming problem can be

stated as follows:

Find

1

2

n

x

x

x

which minimizes ( )f X (2.24)

subject to the constraints

( ) 0, 1,2,...,g X j mj

(2.25)

( ) 0, 1,2,...,l X j pj

(2.26)

28

where X is an n – dimensional vector called the design vector, ( )f X is

termed the objective function, ( )g Xj

and ( )l Xj

are known as inequality

and equality constraints respectively. The number of variables n and the

number of constraints m and/or p need not be related in any way. The

problem stated above is called a constrained optimization problem.

Some optimization problems do not involve any constraint and can

be stated as:

Find

1

2

3

n

x

x

x

x

which minimizes ( )f X (2.27)

Such problems are called unconstrained optimization problems

(Singiresu S Rao 1996). Conventional optimizing methods are given in

Appendix.3.

2.7.6 Proposed Graphical Method

From the statement of optimization problem, it is clear that the

problem should have an objective and constraint. For a transformer design

problem, the objectives may be minimization of cost, volume, weight or loss.

The necessary objective functions are already given in Equations (2.10) to

(2.13). The functions are linear straight line functions and the number of

variables is only two.

The constraint equations of a transformer are given in Table 2.2.

They are non-linear equations. The equation can be simplified since other

parameters namely electric and magnetic and or geometrical parameters are

29

known depending upon the type of problem such that a value for the product

of two similar variables are obtained. The ratings, frequency, material

constants are also known. The product of two variables is an integer which is

known as constraint equation.

Since the constraint equation and objective function are only in

terms of two variables, simple analytical graphical method is suggested. The

methodology for optimum total full load loss design, for example, is

explained as under.

First the constraint characteristic is drawn as detailed hereunder.

For all assumed values of full load iron loss, Pi, the full load copper loss, Pc

is found from constraint equation and tabulated. A graph is drawn between Pi

and Pc as shown in Figure 2.1. The shape of the graph will be parabola.

Figure 2.1 Graphical method

30

Then the objective function graph is drawn. It is a linear straight

line graph. For a series of assumed total full load loss, points for Pi and

Pc are obtained and tabulated. A graph is drawn for each assumed total full

load loss. There are number of straight line graphs as shown in Figure 2.1 for

each assumed total full load loss. In this process one straight line becomes

tangent to the constraint curve. The tangent point gives the optimum values

for the variables Pi and Pc . It is evident that for full load loss conditions,

Pi= Pc .

The values for both the variables are obtained quickly, easily and

simultaneously. The values are optimum and unique which is a special feature

of this analytic graphical method. The sum of these two variables give the

optimum value for the objective considered, the total full load loss for the

present.

2.8 VALIDITY OF PROPOSED GRAPHICAL METHOD

In order to understand the validity of the proposed graphical

method, one single phase and one three phase transformer problem is taken

and solved by both conventional method and graphical method in subsequent

chapters. Optimality is further verified by popularly available Lagrange

multiplier and Random jumping methods.