Optimization Study on Oil Flow
and Temperature Distribution in Power
Transformer Windings
MARIA
HJALMARS
Master of Science Thesis
Stockholm, Sweden 2012
Optimization Study on Oil Flow
and Temperature Distribution in Power
Transformer Windings
MARIA
HJALMARS
2D1020, Master’s Thesis in Numerical Analysis (30 ECTS credits)
Degree Progr. in Engineering Physics 270 credits
Royal Institute of Technology year 2012
Supervisor at CSC was Jesper Oppelstrup
Examiner was Michael Hanke
TRITA-CSC-E 2012:023
ISRN-KTH/CSC/E--12/023--SE
ISSN-1653-5715
Royal Institute of Technology
School of Computer Science and Communication
KTH CSC
SE-100 44 Stockholm, Sweden
URL: www.kth.se/csc
Abstract
Optimization study on oil flow and temperature distribution in power transformer
windings.
The thermal design of a power transformer has to keep the temperature within limits according to
international agreement standards. Two of the limits are a maximum mean winding temperature and
a maximum Hotspot temperature.
The purpose of this study was to investigate methods to improve the winding thermal design/
properties to lower the Hotspot temperature without increasing the mean winding temperature.
Two methods have been used to investigate the discrete and continuous parameters affecting the oil
flow through the transformer. The first approach focused on heuristics derived from knowledge of
the physics and experience with cooling designs. The second approach was to handle the problem as
a global optimization problem solved by a genetic algorithm. The optimization was done using
predictions from a program used for cooling calculation. The optimization has to cope with failures of
the predictions and so the objective function needed to be well defined to not end up optimizing on
the programs bugs instead.
The first approach, heuristics derived, was inefficient. Finding improved theories without previous or
deeper knowledge of the oil flow in a transformer was difficult. Only one theory was found and
examined: to balance energy losses due to conductor resistance over all winding discs.
The genetic algorithm was successful; on all the designs tested it gave a lower Hotspot value without
increasing the mean winding temperature significantly. The challenge was to formulate the objective
function and to define what an optimal solution is.
The study has shown that a global optimization approach provided for better design solutions in
reasonable time and therefore has shown its potential for practical use. Both the methods could be
combined and used for further investigation.
Key words: transformer thermal design, oil flow, genetic algorithm
Sammanfattning
Optimeringsstudie över distributionen av oljeflöde och temperaturer i en
krafttransformatorslindning.
Den termiska designen av en krafttransformator måste hålla vissa gränser fastsatta av internationell
standard överenskommelser. Två av gränserna är en maxtemperatur på medellindningen och en max
Hotspot temperatur.
Syftet med studien var att hitta metoder som kan förbättra den termiska lindningsdesignen för att
minska Hotspot temperaturen utan att öka medellindningstemperaturen. Två metoder har använts
för att undersöka de diskreta och kontinuerliga parametrarna som påverkar oljeflödet genom en
transformator. Den första metoden var att hitta nya heuristiska antagande utifrån kunskap och
erfarenhet om fysiken bakom transformatorkylning. Den andra metoden var att se problemet som
ett globaltoptimeringsproblem, vilket kan lösas med hjälp av en genetisk algoritm. Optimeringen är
gjord på beräkningar från ett kylningsprogram uträkningar. Optimeringen ska kunna hantera fel i
antaganden gjorda av kylningsprogrammet, så objektsfunktion måste vara väl definierad så att
optimeringen inte blir på buggarna i programmet isstället.
Den första metoden, heuristiska, var ineffektiv. Att hitta nya förbättrande teorier utan tidigare
kunskap om oljeflödet i en transformator var svårt. Bara en teori hittades och undersöktes; att
balansera energiförlusterna i skivorna från motstånd i lindningsledarna.
Den andra metoden, genetisk algoritm, var lyckad; på alla designer testade gav den en lägre Hotspot
temperatur utan att öka medellindningstemperaturen märkbart. Utmaningen var att formulera en
objektfunktion och definiera vad en optimal lösning är.
Studien har visat att se problemet som ett globalt optimeringsproblem har gett bättre
designlösningar inom rimlig tid och har därför visat potential för praktisk användning. Båda
metoderna kan kombineras och användas för vidare undersökning.
Nyckelord: termisktransformatordesign, oljeflöde, genetiskalgoritm
Acknowledgement
I would like to thank my supervisor at KTH, Jesper Oppelstrup and Michael Hanke, for the help with
finishing the report and your knowledge. Also I would like to thank my supervisors at ABB, Jurjen
Kranenborg and Andreas Gustafsson for quick respond and your experience.
Östersund, April 2012
Maria Hjalmars
Table of Contents
1
Background...................................................................................................................................... 1
2
Inside a transformer ........................................................................................................................ 2
2.1 Cooling systems ....................................................................................................................... 3
2.1.1
ONAN/ONAF ........................................................................................................... 3
2.1.2
OFAF/OFWF ............................................................................................................ 4
2.1.3
ODAF/ODWF ........................................................................................................... 4
2.2 The Hotspot ............................................................................................................................. 5
2.3 Physics ..................................................................................................................................... 6
3
Thermal design ................................................................................................................................ 7
3.1 Purpose.................................................................................................................................... 7
4
Methods .......................................................................................................................................... 8
4.1 How the problem is solved today............................................................................................ 8
4.2 Methods to approach the problem ......................................................................................... 8
4.2.1
Network model ....................................................................................................... 9
4.2.2
Empirical search...................................................................................................... 9
4.2.3
Meta-heuristic / Global search methods................................................................ 9
4.2.3.1 Tabu search ............................................................................................. 9
4.2.3.2 Variable Neighborhood Search ............................................................. 10
4.2.3.3 Simulated Annealing.............................................................................. 10
4.2.4
Genetic Algorithms ............................................................................................... 10
4.2.4.1 Fitness value .......................................................................................... 12
4.2.4.2 Selection type ........................................................................................ 12
5
Application of the methods ........................................................................................................... 14
5.1 Network model ...................................................................................................................... 14
5.2 Empirical search .................................................................................................................... 14
5.2.1
Other hypotheses ................................................................................................. 15
5.3 Genetic algorithm .................................................................................................................. 16
5.3.1
Matlab GA ToolBox ............................................................................................... 16
5.3.2
Formulation of the problem for MATLAB GaTool ............................................... 16
5.3.2.1 Objective function ................................................................................. 17
5.3.2.2 Interface Matlab / Cooling program...................................................... 17
5.3.3
Parameter studies ................................................................................................ 18
5.4 What is an optimal solution?................................................................................................. 18
5.5 Bad solutions ......................................................................................................................... 19
6
Results ........................................................................................................................................... 20
6.1 Result Heuristic method ........................................................................................................ 20
6.2 Hypothesis 1 .......................................................................................................................... 20
6.3 Result Global optimization method ...................................................................................... 20
7
6.3.1
Case 1, ONAN/ONAF............................................................................................. 21
6.3.2
Case 2, OFAF ......................................................................................................... 22
6.3.3
Case 3, ONAF with extra losses ............................................................................ 22
Conclusions.................................................................................................................................... 23
7.1 Heuristic................................................................................................................................. 23
7.2 Genetic algorithm .................................................................................................................. 23
7.2.1
External program as objective function ............................................................... 25
7.2.2
Two load cases with one optimal set of thermal parameters .............................. 25
7.2.3
Local or global optima .......................................................................................... 26
7.3 Comparison of the two methods .......................................................................................... 26
8
Recommendations......................................................................................................................... 27
8.1 Application areas ................................................................................................................... 27
8.2 Improvements ....................................................................................................................... 27
Word list ................................................................................................................................................ 29
References ............................................................................................................................................. 30
1 Background
A transformer is a device that transfers electrical energy from one circuit to another. It can change
high voltage to low voltage and the other way around.
Higher voltage means a smaller current for the same power and a smaller current has less energy loss
due to the resistance in the conductors. When transporting energy long distance the losses should be
kept low. The first transformer was installed 1883 in London, by Gaulard and Gibbs. ASEA
manufactured their first transformer 1893, a three phase transformer [ref.5]. Figure 1 shows a three
phase power transformer. When higher voltage started to be transformed there was a need for a
better insulation material than air. Mineral-oil was then used as insulation material in the
transformers. A side effect of this was that the heat produced in the transformer was better
transported away and the transformer cooled down. An interest for cooling equipment began.
Winding
Figure 1, Picture over a three phase transformer.
1
2 Inside a transformer
The transformer can have different looks depending on the
voltage and power that will be transformed. This project is for
medium and large power transformer, but the techniques
investigated can be applied on fluid-cooled small transformers and
reactors too.
High Voltage
Low Voltage
The current is carried by copper conductors wound in a primary
and secondary winding around a core, figure 2; normally both the
primary and secondary winding is placed around the same core.
Usually there is a third winding, for correction if extra turns need
to be added to either the low or high voltage winding. The
conductors are heated by the resistive voltage drop suffered by
Figure 2, A schematic picture over
the current and the transformer will lose some of its performance.
the windings in a transformer.
Also in the top and bottom of the windings there will be more
losses due to eddy currents induced by stray magnetic field from the core. Such heating represents
losses which should be minimized.
The conductors wound
on the same height form
the discs. In figure 3, the
Duct height
duct height is the
Discs
channel height between
(Cooper threads)
two neighboring discs.
Figure 2, The first picture is over a winding and the discs and spacers. The second
Other
geometric
picture is a schematic picture of how the discs a represented in the report.
parameters such as the
vertical channel widths and placement of spacers can be adapted to influence the oil flow and the
temperature field.
Spacer
The conductor turns are isolated from each other by paper to avoid electrostatic discharge. The
paper ages faster at elevated temperatures which is another reason to control the winding
temperature. To guarantee that the insulation paper, figure 4, will last the life of the transformer at
all places in the winding, its
temperature must be kept below 98
Winding
C. There are different oils used for
Isolation
cooling, such as mineral, biochemical
paper
etc. Which oil that is used depends
on the design of the transformer or
clients request. Mineral oil has a
relatively low viscosity even at low
temperatures and the best thermal
conductivity of the oils in
consideration. The drawback is that it
is not environmentally friendly.
Collar
system
Figure 4. The top and bottom of a winding.
2
Figure 5, below is different cross sections for one winding. The flow of the oil inside one winding can
be directed or non-directed.
Figure 5. Schematic pictures over the oil flow in a winding. The first picture is over an axial oil flow. The second
is over an axial flow with an intermediate axial duct. The third is over a zig-zag oil flow.
Figure 5 to the left is an axial winding flow, figure 5 in the middle is an axial flow with an
intermediate axial duct and figure 5 to the right shows the effect of constrictions, indicated by the
green squares, on the flow path to make a zig-zag pattern. Such a pattern seems beneficial since it
tends to avoid locally stagnant oil channels.
There are different types of windings, high, low voltage, disc type, helical, and more. When
calculated on one winding the paper in between is seen as a thermal insulation and each winding is
calculated as a closed system.
2.1 Cooling systems
The load on a transformer can vary when in use, from going on full load case to half. Depending on
the load case the heat/losses that needs to get transported away can increase or decrease, also
different types of cooling system are used. The fans can either be on or off. Described below are
three common types of cooling system for a transformer ONAN/ONAF, OFAN/OFAF and
ODAN/ODAF. Depending on what type of cooling system the flow and pressure drop through the
channels will look differently. The N and F in the end stands for natural cooled (no fans) and fan
cooled.
2.1.1
ONAN/ONAF
In an ONAN/ONAF cooling system the flow is produced by natural convection, the density differences
of the oil is the driving force. The oil in the radiator can either be cooled by the air or by fans. If no
fans are used, radiation from the tank has a big impact on the cooling of the tank [ref. 9 chapter 4 ].
3
ONAN = Oil Natural Air Natural
ONAF = Oil Natural Air Forced
Tank
Radiator
Winding
Figure 3, A schematic picture over a ONAN/ONAF cooled
transformer
2.1.2
OFAF/OFWF
Pump
In this cooling system it is still natural
convection in the tank and windings that is
the driving force. The difference is a pump
that will enforce the cool bottom oil to be
along the height of the winding. It can be air
forced (AF) or water forced (WF) [ref.9
chapter 4].
Tank
Radiato
r
Winding
Windin
g
OFAF = Oil Forced Air Forced
OFWF = Oil Forced Water Forced
Figure 4, A schematic picture over an OFAF/OFWF cooled
transformer.
2.1.3
ODAF/ODWF
Pump
In this cooling system the convection is forced
into the tank. OD stands for oil directed, and AF
for air forced or WF for water forced [ref.9
chapter 4].
Tank
Radiator
ODAF = Oil Direct Air Forced
ODWF = Oil Direct Water Forced
Winding
Figure 5, A schematic picture over a ODAF/ODWF cooled
system.
4
2.2 The Hotspot
The hottest spot in the winding is called the Hotspot. It isn’t possible to measure where the Hotspot
will be, but it can be estimated by calculation. What can be measured are the temperatures at one
specific spot, in the top tank, and the mean winding temperature. The mean winding temperature is
indirectly given by measuring the resistance in the windings. The spot is measured by fiber optic
cables and the top tank oil by taking the temperature. Usually the fiber optics is positioned at the
estimated Hotspot.
The “Hotspot factor” H is defined to predict the highest temperature TP from the measurable
quantities, see Figure 11,
TP  TA  Hg r , g r  TQ  (TA  TD ) / 2
The Hotspot factor is usually in the range 0.9-2 with 1.3 a common value. It can be calculated in a
couple of ways depending on standard used, losses are often used for the calculation. Figure 9 and
the text is taken from IEC [ref. 9, Appendix B, p.26].
TA = the top oil temperature,
defined as the average of the tank
outlet oil and the tank oil pocket
temperatures.
TB = the oil temperature in the tank
at the top of the winding (assumed
= TA).
TC = the temperature of the average
oil in the tank.
TD = the oil temperature at the
bottom of the winding.
E = represents the bottom of the
tank.
Relative positions in the tank
H x gr
A
B
gr
P
C
Q
Calculated points
Measured points
D
Temperature in the tank
E
Figure 6,Picture over the temperature that can be measured
and calculated in a winding.
gr = the average winding to the
average oil (in tank) temperature difference at rated current.
TP = the hot-spot temperature
TQ = the average winding temperature determined by resistance measurement.
The Hotspot factor allows monitoring by measurable quantities of the aging of the isolation material
which affects the life cycle of the transformer. It is therefore very important to the transformer user
and is a key quantity in the evaluation of a transformer investment.
5
2.3 Physics
The interplay of two forces, the pressure head and the pressure drop, makes the oil flow through the
windings. The pressure head Figure 10, the driving force for oil movement, is created by the
buoyancy effect. The oil is heated up by the hot windings and therefore expands to a lower density.
Outside the windings the oil is
cooled by a cooling system, and
Tank
Tank
obtains higher density.
Cold oil
Warm oil
Warm oil
It follows that the hydrostatic
pressure difference over the cold oil
Cold oil
column is higher than that over the
warm column, and this pressure
difference must be balanced by the
friction created by the flow: the
Figure 7, A schematic picture over the pressured head
pressure drop. The wall friction
when the oil is forced between the discs in the windings slows down the flow, and when the resulting
pressure drop matches the pressure head the flow will reach equilibrium and stabilize.
Constrictions of the flow paths, such as decreased duct heights and channel widths and presence of
spacers increase the pressure drop.
6
3 Thermal design
The purpose of the thermal design is to keep the temperature within limits according to the
international agreement standards. For the cooling there are certain requirements guaranteed for
each designed transformer. Two of the requirements are a maximum mean winding temperature and
a maximum Hotspot temperature. If the Hotspot temperature is the limiting issue, then decreasing it
by even such a small amount as 1 or 2 degrees could save cooling equipment. A reduction in cooling
equipment could also decrease the size of the transformer which has its advantages. But the
decrease of the Hotspot value should not come at the expense of exceeding temperature constraints
in the surrounding windings. It is not optimal to improve one winding at the cost of worsening
another; the goal is to have the best overall transformer. The aging of the insulation paper limits the
temperature below 98 degrees for keeping its qualities throughout the lifetime of the transformer.
3.1 Purpose
The purpose of this study is to investigate methods to improve the winding thermal properties. The
thermal properties depend upon a large number of discrete parameters, like channel geometries,
duct heights, and constrictions from spacers and other structural details. The study is also done to
give deeper understanding of the role the winding design plays in cooling performance and winding
Hotspot prediction. We investigate optimization of the available geometric degrees of freedom of
the winding and the flow paths to minimize the Hotspot temperature and, if possible, decrease the
average winding temperature. Two methods have been investigated. The first focuses on heuristics
derived from knowledge of the physics and experience with cooling designs. The second method is to
handle the problem as a global optimization problem, where the Hotspot temperature is the
objective function and the geometric parameters (see above) are the variables to choose.
This study is applied to ABB’s type of transformer, and to respect ABB proprietary rights, the
examples presented in this report are somewhat generic and differ in some respects from the
production geometry. The requirements on the methods are that they should be applicable to all
types of cooled transformer as described above and to all winding types.
It was decided to consider only one winding at a time. The simplification is applied by prescribing the
values of bottom and top oil temperature in the tank in the optimization. The individual winding
theoretically sufficiently separated and that this approach is justified.
7
4 Methods
4.1 How the problem is solved today
Transformers are designed following the standards (ANSI, IEEE or IEC) and on the basis of customer
requirements. The winding design is done by a proprietary ABB design system based on a physicsbased model of the thermal behavior of a transformer. This model is proprietary and uses an inhouse cooling program that calculates the flow and temperature rise in a winding for a given
geometry. The cooling program was used in the study to predict the effects of parameter variation.
The program itself suggests an optimized set of the available geometrical degrees of freedom.
4.2 Methods to approach the problem
The heat flux from the windings to the tank depends upon many things. This problem can be
approached from different angles with previous knowledge, physical knowledge or by optimization
methods. The full problem can be attacked in several ways and methods. First the problem has to be
broken down to small pieces. Then we should figure out which parameters that could have an
influence on the oil flow in the windings. The parameters that have been looked into are:
 Geometry of the discs
 Duct spacers
 Losses
 Channel width, outer and inner
 Type of oil flow
The three methods initially chosen to analyze the problem were:



Development of simple network flow model for variable duct height
Methodical trial and error of new geometries for a single transformer case
Global optimization method
First a study was conducted to assess the influence of the different parameters on the heat flux by
analyzing one at a time. For this the small network model in Matlab was developed, to see how the
flows are affected by different duct heights and how they are placed in relation to each other. The
second method was an empirical study which took a standard design of a transformer and changed
parameters one a time to see the effect.
These methods were useful and provided more knowledge in the field and for testing hypotheses.
The third, most successful, approach was a global optimization method for combinatorial problems
requiring no pre-knowledge. This method sees the problem as a black box. This is a good start when
not to be locked in old thinking and find new heuristic rules and laws.
8
4.2.1
Network model
The Network model is a simple model over a small part of one winding.
Only the pressure drop is modeled. No load losses from transformer
are considered. The unknowns are the pressures at a number of nodes
connected by flow conduits. The pressures are calculated so the mass
flows satisfy the continuity equations. All the calculation was done in
Matlab. Figure 11 is a cut over a part of a winding of 4 discs, the dots
represents calculation points. The oil flow and heat flux is only counted
at the corners to not have a complex calculation.
4.2.2
Figure 8, A representation
of the nodes and discs for
calculation
Empirical search
Empirical search on one case by methodically changing one of the parameters at a time and seeing
how it affects the heat and oil flow in the winding.
4.2.3
Meta-heuristic / Global search methods
The problem with optimizing on the winding parameters can be seen as a combinatorial problem
with discrete values. The search space is huge, and some of the parameters can be continuous, such
as the geometry of the windings and other discrete such as presence of spacers. The continuous
parameters can be quantized to give a finite number of combinations. All in all, there is a finite but
huge number of feasible solutions, so is not possible to try them all.
To solve this kind of problems, local search methods aren't a good option compared to more global
methods. A local search method starts with a feasible solution and then makes small moves until it is
unable to obtain further improvement. Because small moves stay in the neighborhood of the starting
point it is unlikely that the solution found will be the global solution or even near a global solution to
the problem.
Three different types of “Meta-heuristic” methods will be discussed briefly, tabu search, variable
neighborhood search, and simulated annealing. A genetic algorithm is described next and was
chosen for application since it seemed the most suitable. It does not use a local search method for
moving and is therefore not bounded to the neighborhoods of one solution and a Matlab
implementation is available. It will be discussed in detail in sec. 4.2 4.
4.2.3.1 Tabu search
The principle is to keep track of bad solutions so they will be tried again with small probability. The
method has the same structure as for local search methods, but to avoid cycling backwards and try
to find new optima it uses memory. One short time memory, tabu lists, for not cycling backwards
when a local optimum has been reached, and a long term memory with attractive components to use
9
when searching for new optima. The short time memory is limited in quantity and only keeps track of
the last moves, not the whole solution. When moved forward in a neighborhood, the tabu list
contains the last steps that where taken in that neighborhood. After some iteration former forbidden
moves could be accepted. There can be several tabu-lists kept at the same time, with varying lengths.
There are modifications to this method and it can be performed in many different ways, see e.g.
Chapter 2, ref. 4.
4.2.3.2 Variable Neighborhood Search
The basic concept is to search in different neighborhood structures, and the global optimum should
be the best solution found in all types of neighborhoods. There are three facts used [ref.4, Chapter
3.2, p.63]:

A local minimum w.r.t. one neighborhood structure is not necessarily so for another.

A global minimum is a local minimum w.r.t. all possible neighborhood structures.

For many problems, local minima w.r.t. one or several neighborhood structures are relatively
close to each other.
A neighborhood in this problem could be the change of duct height; another neighborhood could be
the change of channel width.
4.2.3.3 Simulated Annealing
This is a local search based method whose name is taken from the process of physical annealing
when creating a defect free crystal by slow cooling.
The method avoids getting stuck in local optima by allowing moves which worsen the objective
function with a certain probability. Improved solutions are always accepted, while only a fraction of
non-improving solutions are accepted. The probability for a solution with a worse objective function
value is depending on a temperature parameter (same as in physical annealing). Normally the
temperature is decreased in each iteration so, with time, solutions stay close with higher probability,
[ref. 4, chapter 1].
4.2.4
Genetic Algorithms
The method was first used by John Holland 75 [ref 4, chapter 5, page 109], but is known that the
method has been discussed in the sixties. Genetic algorithm is also known as evolutionary computing
or evolutionary algorithms. It has been shown that it could be useful for NP-hard problems like the
traveling salesman problem [ref. 3, page 20]. It is also widely used in different areas and also in
artificial intelligence [ref. 3, page 1].
The task is to find the decision variables x in the discrete search space A that minimize the objective
function [ref. 4, chapter 5.2, page 111]. In this case the decision variables are the winding
10
parameters and the objective function is the Hotspot temperature as predicted by the ABB cooling
program. The discrete search space is the set of all possible combinations of the available winding
geometry parameters.
The fitness function is sometimes only another name for the objective function, but it could also
withhold more information than the objective function. For example the fitness function could also
take in to account how well the solution works for the problem, it could mean not only counting the
Hotspot value but also how well the solution works for the whole transformer.
The basic concept behind Genetic Algorithm is the same principle as the evolution of nature, the
survival of the fittest. The Genetic algorithm is a controlled evolution. It decides who will breed, have
“clones” of good individuals and a controlled mutation which make it more deterministic and goal
orientated. Genetic algorithm is a copy of the nature’s way to breed, but adapted to mathematical
optimization problems.
In this problem, an individual represents a specific setting of thermal winding parameters. A “gene”
carries a part of the total set of design parameter.
The reproduction of new individuals can be made in three ways, see figure 12.



Elite count, no changes of the individual for the next generation (This is done for not losing
the fittest genes).
Cross-over, as normal reproduction in nature. Two individuals create a new individual/child
by crossing their genes.
Mutation, as the name suggests one gene that is changed in some way.
Elite
Cross-over
Mutation
Figure 9, the three types for reproduction.
The algorithm begins with a start population – generation 1 - at a fixed size, let’s say 20 individuals
which reproduce to make generation 2, etc. Stopping criteria terminate the algorithm when
(hopefully) the global optimum is reached. Stopping criteria could include the number of
generations, or a specific time limit, when the fittest individual has been the same for a certain
number of generations etc.
11
Regarding population size there are questions about the efficiency of the algorithm depending on the
size Handbook of Metaheuristics [ref. 4, chapter 5.2 ] discusses this problem with the conclusion that
there is a minimum size of population. For a problem with 50 variables (in this case the number of
different thermal parameters) the minimum population size would be 17 individuals.
Then there is the fitness scaling and fitness selection which will decide who of the parents from one
generation will get to reproduce to the next generation.
4.2.4.1 Fitness value
Fitness scaling is different methods to change the fitness value1 from badly distributed values to a
better range of the values. Fitness scaling is done to get a better result out of the genetic algorithm,
and to be able to use the selection methods.
The fitness scaling can change the spread in the raw score, objective value. If the raw score has a big
spread, it could be better to make it smaller and the opposite if there is no spread to increase the
difference in the fitness values. This is done because of the selection will decide the proportion of
each solution corresponding to one fitness value that will be parent for the next generation. If the
spread is small, the best solutions will all have the same probability to be parent and opposite, if the
spread is big, one solution will be parent to all individuals for the next generation.
The Matlab gatool has five fitness scaling functions included,





Rank
Proportional
Top scales
Shift linear
Custom
For this problem there is no need to make an own fitness scaling any of them above can be used.
There was no test which of the fitness scaling that worked best because a satisfy result was given
with the one chosen, rank. Descriptions of the methods are explained at Mathworks homepage
reference 10.
4.2.4.2 Selection type
This method selects the individuals that will reproduce for the next generation. The selection
methods are more or less stochastic. The Matlab gatool has seven different types of selection,
 Stochastic uniform
 Remainder
 Uniform
1
Fitness value is the value of the objective function or the how fit a solution is.
12




Roulette
Shift linear
Tournament
Custom
Same as for fitness in this problem there is no need to formulate an own selection method.
Descriptions of the methods are explained at Mathworks homepage 10.
13
5 Application of the methods
5.1 Network model
The Network model is a simple model of the flow between a certain
numbers of discs. This study focused on one geometrical parameter the
duct height and the changes of it. What could be learned from this model
were only assumptions. Figure 13 is a schematic picture over the nodes
used for calculations, same figure as 11.
5.2 Empirical search
Step 1
Figure 10, same as figure
11, a representation of
the nodes and discs for
calculation
An empirical search was done to find new heuristic
rules that can be used for the thermal winding design. The suggested
distribution was used as a starting point for the empirical search. To keep the
same pressure head in the tank in order not to worsen another winding the
same number of “constrictions” was used.
Step 2
Step 3
The method used is sequential trial-and-error, by moving one constriction one
step at the time (see figure 14), starting with the top constriction and moving
it one step upwards until it worsens the solution. Then back off one step and
start moving the second constriction, etc. When the third was at is optimal
placement the rest of the constrictions were evenly distributed over the
remaining discs. This method can even be seen as a brute force algorithm,
which could be looked into further.
It was conjectured before that the flow should be more zig-zag at the top of
the winding to decrease the Hotspot value. The result of the empirical studies
confirmed the conjecture about Hotspot temperature and flow pattern.
The next question is if it is possible to infer heuristic rules from these results.
Which of the parameters could be used for making that rule? The only thing
that separates the middle discs from the top is the extra loss in the top discs.
Hypothesis one, below, is to aim for the same amount of losses between each
pair of constrictions. It was noticed during the empirical study that reasonable
solutions must have certain properties, such as a minimum velocity. Also some difference in the oil
flow of inner and outer vertical channels could be noticed. The value of the minimum velocity
probably depends on the design and loss for the transformer.
Figure 11, A picture
over the discs an each
step taken for one
iteration.
Hypothesis 1:
A zig-zag flow distribution with equal amount of losses in each “leg” is beneficial for Hotspot
temperature.
14
There is a total loss for the whole winding and then there are some extra losses in
the top discs. The hypothesis could be summarized in that same mass flow should
carry away the same amount of heat.
20
19
18
Example:
17
Figure 15 is over a winding with 20 discs. The winding has a loss of 5,8 kW, which
gives each disc a loss of 0.29 kW. The top four discs number 17-20 have extra losses
0.08, 0.11, 0.11 and 0.14 for a total of 6.24 kW. There are 4 sections. Each section
should have the same amount of losses, which gives each section a total loss of
(6.24-0.29)/4 = 1,4875 kW since disc number one isn’t counted for in any section.
Total loss of:
Loss per section:
Loss per disc:
Top discs:
Section 1, disc 2 – 6:
Section 2, disc 7 – 11:
Section 1, disc 12 – 16:
Section 2, disc 17 – 20:
6.24 kW
1.4875 kW
0.29 kW
0.37 kW, 0.40 kW, 0.40 kW, 0.43 kW
1.45 kW
1.45 kW
1.45 kW
1.6 kW
16
15
14
13
12
11
10
9
8
7
6
5
4
The result from this thesis is discussed at Conclusions, heuristic 7.1.
3
2
1
5.2.1
Other hypotheses
Figure 12, Section
representation for
one winding.
There are other heuristic approaches that could be tried. Suggestion for
parameters, rules that can be taken in account is the same parameters as mentioned in the Methods
4.2.
What we want to accomplish with the geometry variation is to keep the same disc temperature for
the whole winding. A non- increasing temperature in the whole winding is not possible, but maybe in
the topmost discs? Figure 16, the graph, shows how a temperature profile with no rise in the last 10
% of the winding could look for lowering the Hotspot value.
Temperature
Winding temperature
Oil temperature
Disc bottom-top
10 % of the winding
Figure 13, A representation over the winding and
the temperature rise for the oil and the winding,
and no rise in the last top 10% of the winding.
15
5.3 Genetic algorithm
5.3.1
Matlab GA ToolBox
The Matlab global optimization toolbox (figure 14) has a
tool called gatool for genetic algorithm problems. The
toolbox contains functionalities for population creation,
fitness value, selection types, mutation, crossover and
stop criteria and much more. The toolbox is simple to use
and has a description to all the functions on the sidebar.
For this specific problem I choose to write my own
function, for the simple reason that handling the
constraints would be easier.
The objective function is called the fitness function in the
toolbox. The settings can be changed during the run of
the algorithm, even the functions in use. The changes will
be applied on the next generation.
Figure 14, A picture over Matlabs gatool window.
A more detailed guideline for gatool is found in the Mathworks homepage [ref.11].
5.3.2
Formulation of the problem for MATLAB GaTool
To use the gatool in Matlab the thermal winding problem had to be adjusted to the genetic
algorithm. The interaction of the different modules is shown in the figure below.
The first generation is randomly chosen from the search space of feasible settings of the thermal
winding parameters.
The start population creates the individuals for the first generation.
The mutation and cross over functions replace the default procedures to comply with the constraints
set to the problem.
The set of thermal parameters is screened by the feasible module before being sent to the cooling
program for verification that the parameters follow the constraints.
The objective function and the API are the connection between Matlab and the cooling program. The
output from the cooling program is evaluated in the objective function by inspection of its Hotspot
value and how well the cooling is for the entire transformer.
16
Start population
Mutation
Matlab
API
Feasible
Elite
Cooling
Program
Cross-over
Objective function
2nd –last generation
1st generation
Figure 15, A representation of how the different modules/codes in genetic algorithm is connected. Green represent
Malab codes, Pink codes written for the specific problem, blue the existing cooling program used for calculation.
5.3.2.1 Objective function
In genetic algorithm the word fitness function is used for objective function. How well a function
works depends on other circumstances than only measuring the objective value, like if it is feasible or
is it worsens something else. For this specific problem a worsen solution would give a decrease in
Hotspot temperature of 1 degree, but increase the mean winding temperature with 1 degree. This
wouldn’t be a great solution, the objective function would give a lower value but how well, fit, the
solution is wouldn’t improve, but rather worsen.
For the thermal winding problem the fitness function will be how well a certain geometry variation
works for the whole transformer.
Many geometry variations are infeasible, for several reasons. A number of constraints should be
fulfilled: In a zig-zag pattern each leg should contain at least 4 discs. Other constraints are specific for
the different transformer types. In some designs there are extra spacers at fixed placement due to
isolation constrictions.
5.3.2.2 Interface Matlab / Cooling program
The cooling program is a stand-alone program with an input file and an output file which can be
executed by Matlab system commands to compute fitness values. The tricky question is to choose
the fitness function - what to search for and what characterizes a good solution are questions that
must be answered. This will be discussed further in section 5.4.
17
5.3.3
Parameter studies
Before running the genetic algorithm on the thermal winding problem, there was a test case to
investigate what settings to be used while running the genetic algorithm. The parameter settings
tried were the population size, elite count, fitness value and selection type. The test case was a
simplification of the real thermal winding problem, the objective function was continuous and
smooth.
Because the test case was good-natured compared to the real problem, the only thing said about the
settings was not to use Uniform settings [ref. 10] (this is also written in the Matlab guidelines). In the
test case the other settings performed equally well and couldn’t therefore be separated.
While running on the real case some settings worked better than others, but no study was made of
which gave the best result. This because the result was satisfying and other more important
questions arose.
5.4 What is an optimal solution?
This problem arose after running the genetic algorithm on the winding configuration problem. The
need for definition of an optimal solution becomes obvious when you have to formulate the
constraints for the algorithm. There may be difficulties in implementing exactly the geometry
suggested, and the predictions are inaccurate. Therefore the solution should also be stable, in the
sense that small changes to parameters should not change the result too much – the mathematician
would say that the prediction of temperatures from geometry must be a well-posed problem. If we
would only optimize on the
Hotspot value the result could
Red line =
Temperature
end up in a solution with a big
Predicted
temperature peak in the
Max Hotspot temperature
temperature rise
bottom (figure 16), several of
Dotted line = Real
small
peaks
or
other
temperature rise
undesirable
features.
As
shown in figure 20 a small
peak in the bottom could
correspond to a big peak in be
Oil discs bottom-top
very sensitive.
Figure 16, A picture over a badly distributed temperature rise in a winding and
Therefore some solutions
should be avoided like:


what the effect of a temperature peak in the bottom (red line) could be in reality
(dotted line).
Solutions sensitive to small changes in the thermal design. Example an extra spacer is taken
away.
Peaks of temperatures in the bottom, cause they are a sign of high losses in the bottom and
low velocities that could be sensitive to numerical miscalculation or small changes in the
design.
18

Velocities and temperatures with great variance next to each other, because this could be a
sign that the solution will be sensitive to a small change in the design.
Note that we are optimizing on a prediction made by a physics-based computer program. The
program is very accurate for conventional designs with well-behaved flow solutions. But the genetic
algorithm will try also unconventional designs for which the flow solver may not be able to produce a
solution or gives unphysical “solutions”. So the optimizer must cope with unreasonable and missing
predictions. Imposition of a priori constraints and filtering of the predictions are necessary, and what
the constraints must be is usually discovered only in the process of trying to run the optimization.
Also the result of the optimization will not be better than the accuracy of the calculations done in the
cooling program.
Some precautions are taken to guarantee the optimal solution. To avoid optimization on numerical
artifacts there are some constraints the solution needs to fulfill:



The Hotspot value must be above the oil temperature in the winding.
Velocities must be in a good range.
No negative velocity must appear.
These are the conditions used for the cases tried, there could be other constraints set that might be
more efficient to rule out bad solutions.
An optimal solution should have a linear rise of the temperatures from bottom to top. The change in
bottom and top oil should be as similar to the solution already calculated for keeping the hydrostatic
pressure in the transformer.
5.5 Bad solutions
How to handle bad solutions another question that might need to be answered before running the
genetic algorithm. What is a bad solution? That might be the first question to ask. Is it the opposite of
a good solution or something else? Is it even of big importance for the result or could it be left
unhandled.
In this problem bad solutions are defined by the constraints set up in the fitness function and given a
fixed fitness value. If the prediction program fails to produce a solution, it was counted as a very bad
solution. But if the velocities are close to acceptable minimum speed the solution seems close to ok
and it wasn’t given ranked so bad. The constraints set could be bad assumptions, for example the
distribution that causes the program to crash could be excellent except for one place which is the
black sheep.
Is it possible to decide how bad a solution is or should the solutions be sorted by their failure to the
constraints. In this case the bad solutions should be handled, because they will give a lower fitness
value than the feasible solutions.
19
6 Results
6.1
Result Heuristic method
The first result relates to the suggested geometry from the cooling program, in particular, the
number of extra flow constrictions. The hypothesis was tried on standard medium power
transformer, with an ONAN cooling and some extra losses in the top discs. The number of guide rings
suggested was 9 rings. The result is compared with different number of guide rings, but they are all
equally distributed over the winding. The pressure drop over the winding should stay intact for all
other solutions. If the gradient of the bottom and oil temperature are bigger than for 9 oil guide rings
the pressure drop in this winding will be higher and that will implicate on the overall pressure drop in
the tank. Therefore the difference in pressure shouldn’t be changed causing the transformer to be
better at one place worse at another.
The number of guide rings tried was 7, 9, 11 and 13, using less or more won’t be feasible. Nine guide
rings had the lowest Hotspot and also the best pressure drop (as it was the reference design), this
case shows the suitability of the logic in the cooling program.
The hypothesis that the cooling program uses the right amount of guide rings was tried on a couple
of cases, with same result or some more guide rings could have been used.
6.2 Hypothesis 1
Hypothesis:
A temperature distribution with equal losses in each section (between two constraints) is beneficial
for hotspot temperature.
The whole hypothesis is built on the assumption that there is a sizeable difference in losses for the
discs. In reality there will seldom be a difference great enough to make this hypothesis work, though
combined with other hypothesis it could be used.
The hypothesis was tried on a standard design, but with changes in losses and number of constraints.
The first design tried on was a standard 30MVA design with no or little extra losses in the top disc.
The result from this design was exactly the same as the already existing.
The second case was the same as the first only this time there were more extra losses in the top
discs. The top discs had from 1.2 - 2.1 times the loss of other discs. In this case the hypothesis
worked well, the Hotspot temperature was significantly reduced.
6.3 Result Global optimization method
The genetic algorithm was tried on three cases and with a better result for all of them. It took
approximately 15 minutes runtime to reach within 0.5 C of the optimal solution but 2 hours to finish
20
the run. The time to finish could be decreased by either changing the stopping criteria to a higher
function tolerance2 or smaller stall generation3 or by improving the algorithm.
The settings in Matlab for the genetic algorithm were the same at all cases
Settings:






Population size:
60
Elite count:
6
Cross factor:
0.7
Fitness scaling:
Rank [ref. 10 ]
Selection type:
Stochastic uniform [ref. 10]
The rest on default
The file functions settings were:


Mutation file: this had the probability of add/subtract with 25 % and to move one oil guide
ring 75 %. There was only one move of oil guide ring at the time, not double or more.
The weighting of the “wrong” messages was relative the marginal value for the transformer
design.
Figure 17, Result window for each generation , a black dot for
best individual and blue mean individual.
Figure 17 shows the genetic algorithm after one run. The black dots on the bottom represent the
fittest individual in each generation the blue above the mean fitness of a generation
6.3.1
Case 1, ONAN/ONAF
The first case was an ONAN/ ONAF cooling system, the ONAF has a higher load on it and therefore
more losses than in the ONAN case. They are both the same design only driven on full load or half
load. When the transformer is driven on half load there is no need for fans to cool the transformer in
2
3
Function tolerance specifies the minimum tolerance for the objective function.
Stall generation is maximum number of generations with no change in the objective function.
21
comparison on full load when fans are used. The problem is to find a configuration which fits both
cooling systems, even though the optimal configuration could be distinctive from each other. This
because the oil flow through the windings will not be the same with or without fans. The
optimization was done on the cooling system that had the worst Hotspot value, as long none of them
was over the guaranteed values for the transformer.
After running the algorithm the Hotspot value of the worst cooling case that the algorithm was
optimizing on decreased some, but the other cooling systems Hotspot value increased a bit. Though
in total it was a small decrease, there wasn’t a big difference between the original solution and the
new one, but the result was still better for the genetic algorithm than the existing. To be added is
also that the case was already optimal, before running the algorithm on it. Below are two graphs
over the temperature rise in the winding from the bottom disc to the top disc.
6.3.2
Case 2, OFAF
In this case it is an oil distribution for a transformer cooled by an OFAF system. This case with neither
extra loss in the top nor any extra spacers is a simplified case of a pumped cooling system. In this
case the Hotspot had a significant decrease. As for the ONAN/ONAF case this also placed the
constraints closer in the top, otherwise evenly distributed over the winding.
6.3.3
Case 3, ONAF with extra losses
In this case it is a transformer cooled by an OFAF system, it is the same as the other OFAF case only
this is with extra losses and spacers. The spacers are placed in two groups of a couple of discs with
higher distance. This case had the same winding type and had a fixed top and bottom configuration.
The Hotspot decreased with satisfaction for this case too. As for the others it was a closer placement
in the top discs, and some precaution taken to the extra spacers. The same number of constraints
was used for both the cooling program and the genetic algorithm.
22
7 Conclusions
From the perspective of finishing the study the genetic algorithm should have been the starting point
and continued by looking at the result to create new heuristics rules. Though there are some positive
inputs to first familiarize with the problem, it was to help for formulating the functions and
constraints used in the genetic algorithm.
What could be learned from the three methods used ? The network model wasn’t as useful as hoped,
but it gave pointers like, it is not always the best way to put extra spacers where the velocities are
the lowest. Not said that it should always be avoided, but there are cases they are better placed
somewhere else. This could be something to take a closer look at, when not to put spacers where the
velocity is low.
For the heuristic and genetic algorithm the results were more satisfactory. The heuristic didn’t solve
the problem, but it helped analyze the results and confirmed assumptions made. The genetic
algorithm did solve the problem with satisfaction, and then there is always the question of how well
it solved it.
Another question, what are the criteria if a method has worked or was a waste of time? If it didn’t
solve the problem is it still a waste of time or has it given answer to other question, show faults in
the model. For example, should the three constraints used in today’s model be extended with more
constraints? This will decrease the search space but also avoid distributions that will cause the
winding to collapse.
7.1 Heuristic
This is more a verification method of heuristics rules taken from assumptions based on knowledge or
guesses. To try to find new heuristic rules, there have to be knowledge and creativity behind or wild
guesses. Sometimes wild guesses can rule out hypothesis or give ideas on new ones, but still it can be
time consuming without any result.
The heuristic rule with equal losses in each section makes no improvement, if not the extra losses are
much greater than the load losses which they rarely are in reality. The positive with the equal losses
is, it is easy to apply and could with the use of other heuristic rules give a new algorithm to
implement instead of the existing algorithm. A modification of this hypothesis could be taking the
worst case scenario, the disc with the highest loss in one section represent the whole sections losses.
7.2 Genetic algorithm
The genetic algorithm has worked and in reasonable time given a better solution to the problems
tried than the existing solution. It is a good tool when optimizing on problems with several local
optima, not a clear neighborhood and a non-continuous objective function. It could be applied to
many different types of problem, but is not good for all problems. Obviously more accurate results
could be found for smooth optimization problems by local search methods.
23
The formulation and running of the algorithm produced interesting questions, like what are we
looking for? Better definition of what constitutes a good solution is required for more consistent
results.
The results found will never be better than the program used for the prediction of effects of
variations. For example, if there are bugs or other problems causing the fitness function to give a
good value to a bad solution, it is not easy to optimize on the problem. The prediction should be
feasible and useful. It could also be the other way around, that the program throws away feasible
solutions. Details of the numerical solution procedure could cause feasible solutions to be out of
range for what is possible to calculate.
In this specified problem there isn’t a clear line what is a feasible solution which makes it more
difficult. To deal with this problem and to set up good constraints the algorithm was tried out with
small population sizes repeatedly until the right criteria were found. When the trials appear to be
working, the algorithm was assumed to be correctly formulated and used.
There will always be the question if good solutions are thrown away. Starting with lax constraints and
gradually making them harder (or by reformulating the constraints so they fit the problem better)
should give well defined constraints.
Seeing to the time-run of the algorithm the solution doesn’t change more than 0.5 degrees in fitness
value the last 1.5 hour of the run. What is causing the fitness value to not improve that much in the
end of the run and can it be fixed?
One reason that finding the last 0.5 degrees to the optimal solution takes long time could be the
search space. It is wide and no strong gradient in one direction, which makes it difficult to know in
what direction it should continue. A single feature of a configuration is ”good” if it cooperates well
with the rest of the configuration. So the position of a constraint improves or worsens the Hotspot
value depending on the configuration as a whole. When there is a large number of almost equally fit
configurations the optimization tends to stagnate. One thing that can be done is to speed up the
mutation, expose the distributions to more radical changes and hope to get closer the optimum in
that way. In this case it could be to let the mutation file treat larger units of the configuration at the
time.
In the OFAF case the top 200 individuals have a Hotspot value within 0.110 C. What does it tell us
about the solution? Firstly the optimal solution is flat and stable, the search space is huge. The good
news is that the optimal solution is stable, between these solutions the patterns were very similar.
What can be learned about the configurations, some new heuristic rules? First in all three cases the
distance between constraints in the top are smaller than in the rest of the winding. That was
conjectured before, and the optimization study strengthens the conjecture. A new observation was
a tendency to have a smaller distance between the inner channels than the outer. The smaller
distance in the inner channels can be a result of the pressure being higher and causing the oil to only
pass through fewer channels and for keeping the flow at a good rate through each vertical channel.
Using the genetic algorithm created more questions than it answered. In one of the cases the
suggestion given by genetic algorithm had two more constraints than what is used today. Is it only
because the difference in the method use to find the optimal solution, for the existing cooling
24
program tries to use the minimum number of oil guide rings while the genetic algorithm only look at
the Hotspot value.
The other thing is to focus on the bad solutions and investigate what causes the cooling program to
malfunction. This information can be useful to create new constraints.
7.2.1
External program as objective function
The genetic algorithm and gatool works really well for optimization problems on external programs,
but the genetic algorithm will not give a better answer than what the external program is able to
calculate. For example you want to optimize on the Hotspot value, but you might end up optimizing
on the faults of the program, because some bugs in the code will give a better solution than the
feasible solutions.
Also a given problem could be more complex than seen at first sight; the optimization might cause
side effects not thought of. So there is a need to review the result.
The most difficult problem is not the optimization, but to formulate a good problem by an objective
function, to set the constraints and define what a good solution is. If possible, one should analyze the
suggested problem formulation for existence of feasible solutions and convergence.
7.2.2
Two load cases with one optimal set of thermal parameters
In some cases there will be a question of finding one optimal solution for two cooling system as in
Case 1, due to different cooling system and load cases on the same design. A transformer are
sometimes driven on half the load case it is dimensioned for and therefore the fans used for cooling
can be turned off causing a distinct pressure drop and oil flow in the winding. The geometric
parameters have to be the same for the both cooling systems even though the optimal solution for
them could be different due to the pressure drop.
The question is which cooling system to optimize on? Or should the optimization be on both of them
combined? The goal with the optimization is to keep the property of the insulation and the
constraints set on the design. So this means that as long the Hotspots for the both load cases are
kept under the constraints, the highest Hotspot value should be the value to optimize on.
Case nr 1 is a transformer that has two optima’s, one when half load and no fans ONAN are in use
and the second when the transformer is going on full power ONAF. In Case 1 the cooling system
ONAF with the highest Hotspot value was the limiting value and the cooling system optimized on.
This gave a result were the Hotspot value for the ONAF system decreased some and for the ONAN
case increased. Though this isn’t always true, there could be a decrease for both systems.
25
7.2.3
Local or global optima
If the algorithm is run with larger populations, reaching a global optimum is more likely, though for
the genetic algorithm there is a span where the algorithm works best.
Doing a statistic search on the genetic algorithm by running it on the same problem several times
and see if it reaches the same optimal solution each time could also show if the optimal solutions is a
global optimum. Though there can’t be any certainty, but the probability of having the global
optimum increase.
The cases have been tried with 10 times the population size, and have in these cases reached the
same optimal solution. They have also been tried a couple of times with the same settings, the result
wasn’t perfect. In some cases it is exactly the same solution, but in less than half of them there is a
difference within 0.05 degrees of the Hotspot.
Why the genetic algorithm doesn’t produce identical solutions for every run could be bugs in the
code that prevents the algorithm to reach the whole search space or other misses.
7.3 Comparison of the two methods
The two methods aren’t a contradiction of each other, heuristic rules can be used when formulating
the constraints for the genetic algorithm and observation done while running the genetic algorithm
could be formulated to heuristic rules and verified.
To do an empirical search before using an optimization method was good for getting an
understanding of the problem and its complexity. In this problem it helped me understand how the
search spaced looked and how to formulate the mutation and cross function, it also gave a better
understanding what is a good solution.
The heuristic method is more a verification of design rules and is often used to see if new heuristics
rules will work. Heuristics rules are used when formulating an algorithm to create a transformer
design, they are built on knowledge of what will work. The problem with heuristics rules are that
they are often general and will not look at new details which sometimes can oversee small variances
that need special handling. In these cases there is better to use methods that aren’t bound to
assumptions taken on a general model.
The genetic algorithm can be used as a new method for finding optimal solution; it isn’t bound to a
specific design and will therefore find the optimal solution for every new type of transformer. It has
its advantages when new or special design transformers are going to be designed and optimized.
It would have been good to have more time to explore the solutions generated by the genetic
algorithm and see if any new heuristic rules could be found and tried.
26
8 Recommendations
The recommendations are for using genetic algorithm, to find new heuristic rules will be left for the
persons continuing with the problem. The method genetic algorithm can be used on a wide range of
optimization problems. Before using genetic algorithm I would suggest to:
 Have a well-defined fitness function
 A clear description what a feasible solution is
 See that the mutation function and crossover function covers the whole search space
8.1 Application areas
The genetic algorithm can be used on different optimization problems, though some problems are
faster solved with simpler methods or other approaches than the genetic algorithm. For the thermal
winding configuration problem the result and genetic algorithm could be very useful in areas as:
 First as an alternative to the already existing algorithm
 To find new heuristics rules
 To find new constraints/no-no’s
 To learn more about the cooling program
 To be used on difficult designs
As it was meant for in the beginning, it could be added as an extra tool to the design program for
optimization. To find new heuristic rules, a good start could be to let the algorithm run on some
cases and see if there are any similarities between the solutions suggested by the genetic algorithm.
The use of running multi geometric winding setup on an external program can also indicate if there
are bugs in the code not thought of that need to be fixed. Also the bad solution could be interesting,
if there is combination of geometric parameters that should be avoided. For example not to have a
all extra spacers in the bottom discs.
All this suggestions are good to combine with the knowledge and design rules already existing.
8.2 Improvements
There are many things that could be improved to the code written for the genetic algorithm, below I
have listed some improvements area:



Mutation file
Flexibility of the code
Cluster
The mutation file could be improved or more drastically. Today it only changes one geometrical
parameter at time, it could be changes so there are several of changes for each mutation.
Flexibility to the code should be added. Today it is only for one type of design, all parameters have to
be changes inside the code. The parameters for each new design should be taken in as an input, not
changed in the code.
27
The genetic and similar search algorithms are easily parallelizable, indeed “embarrassingly parallel”.
Most of the time is spent in running the prediction code for different parameter sets and these
computations are independent.
28
Word list
High-voltage winding: the winding having the highest rated voltage [ref.8, chapter 3.3.4, p.10].
Hotspot factor: a dimensionless factor to estimate the local increase of the winding gradients due to
the increase of additional loss and variation in the liquid flow stream [ref.9, chapter 3.15, p.8].
Hotspot winding temperature: the hottest temperature of the winding conductors in contact with
solid insulation or insulating liquid [ref.8, chapter 3.13, p.8].
Load loss: the absorbed active power at a rated frequency and reference temperature, associated
with a pair of windings when rated current (tapping current) is flowing through the line terminals of
one of the windings, and the terminals of the other winding are short-circuited. Further windings, if
existing are open-circuited [ref.8, chapter 3.6.3, p.15].
Low-voltage winding: the winding having the lowest rated voltage [ref.8, chapter 3.3.5, p.10].
No-load loss: the active power absorbed when a rated voltage (tapping voltage) at a rated frequency
is applied to the terminals of one of the windings, the other winding or windings being open-circuited
[ref.8, chapter 3.6.1, p.15].
Phase winding: the assembly of turns forming one phase of a three phase winding [ref.8, chapter
3.3.3, p. 10].
Power Transformer: a static piece of apparatus with two or more windings which, by
electromagnetic induction, transforms a system of alternating voltage and current into another
system of voltage and current usually of different values and at the same frequency for the purpose
of transmitting electric power [ref.8 ,chapter 3.1.1,p.8].
Temperature rise: the difference between the temperature of the part under consideration and the
temperature of the external cooling medium [ref.9, chapter 3.3, p.6].
Total losses: the sum of the no-load loss and the load loss [ref.8, chapter 3.6.4, p.16].
Winding: the assembly of turns forming an electrical circuit associated with one of the voltages
assigned to the transformer [ref.8, chapter 3.3.1, p.8].
29
References
1. Carlson, Åke (95-08-09) Power Transformer Design Fundamentals, ABB Transformers,
Ludvika.
2. Del Vecchio Robert M., Poulin Betrand, Feghali Pierre T., Shah Dilipkumar M., Ahuja Rajendra
(2002) With Applications to Core-Form Power Transformers , Taylor &Francis
3. Engberg, Peter, Genetiska algoritmer, Luleå tekniska Universitet, systemvetenskap D-nivå,
URL: http://pure.ltu.se/ws/30977852/LTU-SHU-EX-053233-SE.pdf
4. Genderau, Michel; Potvin, Jean-Yves, Handbook of Metaheuristics (2010) 2nd edition, ISBN
978-1-4419-1663-1, Chapter 2-5
5. Heathcote, J. Martin, Ceng, FIEE, The J & P Transformer Book, a practical technology of the
power transformer, Edition 12th, p. 13-17
6. Holland, John H (92-07) "Genetic Algorithms," Scientific American, URL:
http://www.econ.iastate.edu/tesfatsi/holland.GAIntro.htm
7. Kokkoras F., Paraskevopoulos K., International Hellenic University
http://rad.ihu.edu.gr/fileadmin/labsfiles/decision_support_systems/lessons/genetic_algorith
ms/GAs.pdf
8. International Standard IEC 60076-1 Edition 3.0 IEC
9. International Standard IEC 60076-2 Edition 3.0 IEC
10. http://www.mathworks.se/help/toolbox/gads/f6010dfi3.html
Titel: Using the genetic algorithm Date: 2011/12-2012/03
Content: The mathworks homepage on global optimizations tools
11. http://www.edc.ncl.ac.uk/highlight/rhjanuary2007g02.php/
Title: Roulette wheel selection
Date: 2012-01-07 15.13
Content: Newcastle university site about the selection method roulette wheel for genetic
algorithm.
30
TRITA-CSC-E 2012:023
ISRN-KTH/CSC/E--12/023-SE
ISSN-1653-5715
www.kth.se