Global transformer design optimization using deterministic and non-deterministic algorithms Eleftherios I. Amoiralis Member, IEEE National Technical University of Athens 9 Iroon Polytechniou Street, 15780 Athens, Greece [email protected]Marina A. Tsili National Technical University of Athens 9 Iroon Polytechniou Street, 15780 Athens, Greece [email protected]Dimitrios G. Paparigas Independent Electrical/Electronic Manufacturing Professional [email protected]Abstract -- The present paper compares the application of two deterministic and three non-deterministic optimization algorithms to global transformer design optimization. Two deterministic optimization algorithms (Mixed Integer Nonlinear Programming and Heuristic Algorithm), are compared to three non-deterministic approaches (Harmony Search, Differential Evolution and Genetic Algorithm). All these algorithms are integrated in a design optimization software applied and verified in the manufacturing industry. The comparison yields significant conclusions on the efficiency of the algorithms and the selection of the most suitable for the transformer design optimization problem. Index Terms-- Transformers, Power transformers, Design optimization, Optimization methods, Algorithms, Artificial intelligence, Genetic algorithms, Heuristic algorithms, Software packages, Design methodology, Design for manufacture. I. INTRODUCTION In today’s competitive market environment, there is an urgent need for the transformer manufacturing industry to improve transformer efficiency and to reduce costs, since high-quality, low-cost products and processes have become the key to survival in the global economy. In optimum design of transformers, the main target is to minimize the manufacturing cost. Therefore, the objective function is a cost function with many terms, including material costs, labor costs, and overhead costs. These component costs, as well as the constraint functions, must be expressed in terms of a basic set of design variables [1]. Deterministic methods provide robust solutions to the transformer design optimization problem. In this context, the deterministic method of geometric programming has been proposed in [2] in order to deal with the design optimization problem of both low frequency and high frequency transformers. Furthermore, the complex optimum overall transformer design problem, which is formulated as a mixed- integer nonlinear programming problem, by introducing an integrated design optimization methodology based on evolutionary algorithms and numerical electromagnetic and thermal field computations, is addressed in [3]. However, the overall manufacturing cost minimization is scarcely addressed in the technical literature, and the main approaches deal with the cost minimization of specific components such as the magnetic material [4], the no-load loss minimization [5] or the load loss minimization [6]. Techniques that include mathematical models employing analytical formulas, based on design constants and approximations for the calculation of the transformer parameters are often the base of the design process adopted by transformer manufacturers [7]. Apart from deterministic methods, Artificial Intelligence techniques have been extensively used in order to cope with the complex problem of transformer design optimization, such as genetic algorithms (GAs) that have been used for transformer construction cost minimization [8] and construction and operating cost minimization [9][10], performance optimization of cast-resin distribution transformers with stack core technology [11], toroidal core transformers [12], furnace transformers [13], small low-loss low frequency transformers [14] and high frequency transformers [15]. GA is also employed for the optimization of distribution transformers cooling system design in [16]. Neural network techniques are also employed as a means of design optimization as in [17] and [18], where they are used for winding material selection and prediction of transformer losses and reactance, respectively. The comparison of deterministic and non-deterministic optimization algorithms is scarcely encountered in the relevant literature, as in [19] where GA and Simulated Annealing are compared to Geometric Programming for high-frequency power transformer optimization. It is therefore clear that global transformer optimization remains an active research area, since several approaches for its implementation have not yet been investigated. It must be noted that there is no single best optimization algorithm for all problems, this is called ‘no free lunch theorem’ [20]. Therefore, the purpose of the paper is to indicate a suitable optimization algorithm dedicated to this problem as well as to meet the demanding requirements of the industry. The present paper compares the application of two deterministic and three non-deterministic optimization algorithms to global transformer design optimization. The applied deterministic optimization algorithms are the Mixed Integer Nonlinear Programming (MINLP) and Heuristic Algorithm (HA), while
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Global transformer design optimization using … transformer design optimization using deterministic and non-deterministic algorithms Eleftherios I. Amoiralis Member, IEEE National
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Global transformer design optimization using deterministic and non-deterministic
adjusting rate (PAR); and the number of improvisations (NI),
or stopping criterion. The harmony memory is a memory
location where all the solution vectors (sets of decision
variables) are stored. Here HMCR and PAR are parameters
that are used to improve the solution vector, which are
defined in Step 3.
b) Initialization of Harmony Memory
In this step, the HM matrix with as many randomly
generated solution vectors as the HMS: 1 1 1 1
1 2 1
2 2 2 2
1 2 1
1 1 1 1
1 2 1
1 2 1
...
...
. . . . .
. . . . .
. . . . .
...
...
N N
N N
HMS HMS HMS HMS
N N
HMS HMS HMS HMS
N N
x x x x
x x x x
HM
x x x x
x x x x
−
−
− − − −−
−
=
(4)
Static penalty functions are used to calculate the penalty
cost for an infeasible solution. The total cost for each solution
vector is evaluated using
( )
( )
2
1
2
1
( ) ( ) min[0, ( )]
min[0, ( )]
M
ii
i
P
ji
j
fitness X f X g x
h x
α
β
=
=
= +
+
∑
∑
� ��
�
(5)
c) Improvisation of a new harmony
A new harmony vector 1 2( , ,..., )
I
Nx x x x′ ′ ′=�
is generated,
based on three criteria: 1) memory consideration, 2) pitch
adjustment, and 3) random selection. Generating a new
harmony is called improvisation. According to memory
consideration, i-th variable 1 1
1( )HMS
i Ix x x= − The HMCR,
which varies between 0 and 1, is the rate of choosing one
value from the historical values stored in the HM, while (1-
HMCR) is the rate of randomly selecting one value from the
possible range of values, as shown in [28]:
1 2
( () )
{ , ,..., }HMS
i i i i i
i i i
if rand HMCR
x x x x x
else
x x X
end
<
′ ′← ∈
′ ′← ∈
(6)
where ()rand is a uniformly distributed random number
between 0 and 1 and iX is the set of the possible range of
values for each decision variable. For example, an HMCR of
0.85 indicates that HSA will choose decision variable value
from historically stored values in HM with 85% probability or
from the entire possible range with 15% probability. Every
component obtained with memory consideration is examined
to determine if pitch is to be adjusted. This operation uses the
rate of pitch adjustment as a parameter as shown in the
following:
( () )
()i i
i i
if rand PAR
x x rand bw
else
x x
end
<
′ ′= ± ∗
′ ′=
(7)
where bw is an arbitrary distance bandwidth for the
continuous design variable and ()rand is uniform
distribution between and 1.
d) Update of harmony vector
If the new harmony vector 1 2( , ,..., )
I
Nx x x x′ ′ ′=�
has better
fitness function than the worst harmony in the HM, the new
harmony is included in the HM and the existing worst
harmony is excluded from the HM.
e) Check of the termination criterion
The HSA is terminated when the termination criterion
(e.g., maximum number of improvisations) has been met.
Otherwise, steps 3 and 4 are repeated.
This methodology is recommended for:
• non-expert users
• definition of the range of input variables of the design
vector (refinement of the solution space)
• designs with specific technical requirements
2) Differential Evolution Algorithm
Differential Evolution (DE) is a parallel direct search method which utilizes NP D-dimensional parameter vectors
, , 1, 2,3,...,i G
x i NP= (8)
as a population for each generation G. NP does not change during the minimization process. The initial vector population
is chosen randomly and should cover the entire parameter space. As a rule, we will assume a uniform probability
distribution for all random decisions unless otherwise stated. In case a preliminary solution is available, the initial
population might be generated by adding normally distributed
random deviations to the nominal solution ,0nomx . DE
generates new parameter vectors by adding the weighted
difference between two population vectors to a third vector. Let this operation be called mutation. The mutated vector’s
parameters are then mixed with the parameters of another predetermined vector, the target vector, to yield the so-called
trial vector. Parameter mixing is often referred to as “crossover”. If the trial vector yields a lower cost function
value than the target vector, the trial vector replaces the target vector in the following generation. This last operation is
called selection. Each population vector has to serve once as the target vector so that NP competitions take place in one
generation. More specifically DE’s basic strategy can be described as
follows [31][32]: a) Mutation
For each target vector , ,
i Gx i a mutant vector is generated
according to
1 2 3, 1 , , ,( )i G r G r G r G
v x F x x+ = + ⋅ − (9)
with random indexes r1, r2, r3 {1, 2,3,..., }NP∈ , integer,
mutually different and F>0. The randomly chosen integers r1,
r2, and r3 are also chosen to be different from the running
index i, so that NP must be greater or equal to four to allow
for this condition. F is a real and constant factor ∈ [0, 2]
which controls the amplification of the differential variation
2 3, ,( )r G r G
x x− .
b) Crossover
In order to increase the diversity of the perturbed parameter vectors, crossover is introduced. To this end, the trial vector:
, 1 1 , 1 2 , 1 , 1( , ,..., )i G i G i G Di G
u u u u+ + + += (10)
is formed where,
, 1
, 1
,
( ( ) ) ( )
( ( ) ) ( )
1, 2,..., .
ji G
ji G
ji G
u if randb j CR or j rnbr iu
x if randb j CR and j rnbr i
j D
+
+
≤ == > ≠
=
(11)
In (11) ( )randb j is the jth evaluation of a uniform random
number generator with outcome ∈ [0, 1]. CR is the crossover
constant ∈ [0, 1] which has to be determined by the user.
( )rnbr i is a randomly chosen index ∈ 1,2,…,D which
ensures that , 1i G
u + gets at least one parameter from, 1i G
v + .
c) Selection
To decide whether or not it should become a member of
generation G+1, the trial vector , 1i G
u + is compared to the
target vector ,i G
x using the greedy criterion1. If vector
, 1i Gu + yields a smaller cost function value than
,i Gx , then
, 1i Gx + is set to , 1i G
u + ; otherwise, the old value ,i Gx is retained.
This methodology is recommended for:
• expert users
• definition of the range of input variables of the design
vector (refinement of the solution space)
3) Genetic Algorithm
The Genetic Algorithm metaheuristic is traditionally applied
to discrete optimization problems. Individuals in the population are vectors, coded to represent potential solutions
to the optimization problem. Each individual is ranked according to a fitness criterion (typically just the objective
function value associated with that individual). A new population is then formed as children of the previous
population. This is often the result of cross-over and mutation operations applied to the fittest individuals [33].
In our case, 30 runs of the GA algorithm are performed
and the best solution is chosen as the optimum one. The
population type is bit string of size equal to 20. A random
initial population is created, that satisfies the bounds and
linear constraints of the optimization problem. Rank fitness
scaling is employed, scaling the raw scores based on the rank
of each individual, rather than its score. Stochastic uniform
selection function is used, which lays out a line in which each
parent corresponds to a section of the line of length
proportional to its expectation. The algorithm moves along
the line in steps of equal size, one step for each parent. At
each step, the algorithm allocates a parent from the section it
lands on. The first step is a uniform random number less than
the step size. As far as mutation and crossover functions are
concerned, the first one is adaptive feasible (it randomly
generates directions that are adaptive with respect to the last
successful or unsuccessful generation - a step length is chosen
along each direction so that linear constraints and bounds are
satisfied) and the second one is scattered (it creates a random
binary vector, selects the genes where the vector is a 1 from
the first parent, and the genes where the vector is a 0 from the
second parent, and combines the genes to form the child.).
The limit for the fitness function is set to 0.5, while the
maximum number of generations (iterations) is equal to 500.
This methodology is recommended for:
• expert users
• definition of the range of input variables of the design
vector (refinement of the solution space)
IV. RESULTS AND DISCUSSION
It is essential to find an optimum transformer that satisfies the
technical specifications and the purchaser needs with the
1 Under the greedy criterion, a new parameter vector is accepted if and
only if it reduces the value of the cost function.
minimum manufacturing cost. The HA, MINLP (deterministic
group) and HS, DE, GA (non-deterministic group) optimization algorithms are applied for the design
optimization of four 20-0.4kV three-phase distribution transformers, of 160kVA, 400kVA, 630 kVA and 1000 kVA
rating. Tables I, II, III and IV compare the respective optimization results. In addition, Figs 5-8 illustrate the
comparison of the guaranteed versus designed short-circuit impedance, as well as the total guaranteed versus designed losses for each examined transformer rating based on the five
different optimization method results. To be more precise, spider charts are used to compare and evaluate each algorithm
performance for each transformer design, based on two important characteristics: short-circuit impedance and total
losses. In each spider graph, the blue polygon represents the guaranteed values and the red straight dotted line polygon
shows the designed values (final results from each optimization method). It must be noted that the maximum
permissible deviation between the guaranteed and designed
values is equal to ± 10% in the case of short-circuit
impedance and +10% in the case of total losses. Tables I-IV show the results of the five optimization
algorithms based on the rated power. In particular, the first four lines of each Table show the optimum values of the
design vector, the next four lines depict the guaranteed losses and short-circuit impedance, the next four present the
designed losses and short-circuit impedance, and finally the last two lines refer to the cost analysis.
Regarding Table I, DE shows to have the best performance in comparison with the other algorithms in terms of cost.
However, during the decision making process, values of technical specifications can influence our final choice. In this
case, HA and GA have quite good behavior concerning the total losses and the short-circuit impedance (Fig. 5), and the
first algorithm (HA) dominates to the second one (GA) due to the lower total losses and lower cost of the respective optimal
transformer. As a result, HA seems to be the best possible selection.
In the case of the 400 kVA transformer (Table II) HS is the lowest cost solution, however the results of HA or MINLP are
more efficient in terms of technical performance (better total losses) (Fig. 6).
In the case of the 630 kVA transformer (Table III) HA provides the most efficient solution in terms of cost. As far as
losses are concerned, the HA solution exhibits slightly higher losses compared to MINLP but an improved short-circuit
impedance value (Fig. 7). The non-deterministic algorithms correspond to optimal designs of higher cost and losses but
better short-circuit impedance results (especially the HS algorithm).
Finally, in the case of 1000 kVA transformer (Table IV) HA and HS provide the best solutions which are very close in
cost and performance characteristics.
TABLE I
Comparison of the Optimization Algorithms for the 160 KVA transformer.
Characteristics 160kVA
of the optimum transformer
design
MINLP HA HS GA DE
Low voltage turns 31 29 32 31 31
D (mm) 161 204 183 194 199
G (mm) 207 206 228 228 229
B (Gauss) 16570 16090 16692 16009 16693
Guaranteed Fe
losses (W)
300 300 300 300 300
Guaranteed Cu
losses (W)
2350 2350 2350 2350 2350
Total guaranteed
losses (W)
2650 2650 2650 2650 2650
Guaranteed Short-
Circuit Impedance
(%)
4 4 4 4 4
Designed Fe
Losses (W)
344 330 342 323 345
Cu Losses (W) 2404 2369 2374 2358 2450
Total designed
losses (W)
2748 2699 2716 2681 2795
Short-Circuit
Impedance (%)
4.34 3.91 4.13 3.9 4.21
Cost (€) 2243 2241 2297 2331 2215
Cost
Classification
3 2 4 5 1
3.6
3.7
3.8
3.9
4
4.1
4.2
4.3
4.4
MINLP
HA
HSGA
DE
Guaranteed Usc (%)
Designed Usc (%) (a)
2550
2600
2650
2700
2750
2800
MINLP
HA
HSGA
DE
Total guaranteed losses (W)
Total designed losses (W) (b)
Fig. 5. Comparison of the guaranteed and designed short-circuit impedance
(a), and total guaranteed and designed losses (b) for each 160 KVA
transformer design based on the five different optimization results.
TABLE II
Comparison of the Optimization Algorithms for the 400 KVA transformer.
Characteristics
of the optimum transformer
design
400kVA
MINLP HA HS GA DE
Low voltage turns 19 19 21 20 21
D (mm) 243 250 213 238 213
G (mm) 267 268 309 309 308
B (Gauss) 16822 17000 16986 16657 17036
Guaranteed Fe
losses (W)
610 610 610 610 610
Guaranteed Cu
losses (W)
4600 4600 4600 4600 4600
Total guaranteed
losses (W)
5210 5210 5210 5210 5210
Guaranteed Short-
Circuit Impedance
(%)
4 4 4 4 4
Designed Fe
Losses (W)
691 701 693 689 699
Cu Losses (W) 4670.00 4676.0
0
4824 4758 4692
Total designed
losses (W)
5361 5377 5517 5447 5391
Short-Circuit
Impedance (%)
4.24 4.23 4.09 3.8 4.2
Cost (€) 4402 4383 4449 4494 4539
Cost
Classification
2 1 3 4 5
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
4.3
MINLP
HA
HSGA
DE
Guaranteed Usc (%)
Designed Usc (%) (a)
4500
4700
4900
5100
5300
5500
MINLP
HA
HSGA
DE
Total guaranteed losses (W)
Total designed losses (W) (b)
Fig. 6. Comparison of the guaranteed and designed short-circuit impedance
(a), and total guaranteed and designed losses (b) for each 400 KVA
transformer design based on the five different optimization results.
TABLE III
Comparison of the Optimization Algorithms for the 630 KVA transformer.
Characteristics of
the optimum
transformer design
630kVA
MINLP HA HS GA DE
Low voltage turns 14 14 15 15 17
D (mm) 279 292 260 286 244
G (mm) 291 296 336 330 392
B (Gauss) 16106 16150 16080 16038 16467
Guaranteed Fe
losses (W)
860 860 860 860 860
Guaranteed Cu
losses (W)
6500 6500 6500 6500 6500
Total guaranteed
losses (W)
7360 7360 7360 7360 7360
Guaranteed Short-
Circuit Impedance
(%)
4 4 4 4 4
Designed Fe
Losses (W)
989 989 983 951 978
Cu Losses (W) 5238 5284 5386 5457 5730
Total designed
losses (W)
6227 6273 6369 6408 6708
Short-Circuit
Impedance (%)
4.4 4.32 4.03 4.24 4.24
Cost (€) 7109 7084 7167 7241 7260
Cost
Classification
2 1 3 4 5
3.8
3.9
4
4.1
4.2
4.3
4.4
MINLP
HA
HSGA
DE
Guaranteed Usc (%)
Designed Usc (%) (a)
5500
6000
6500
7000
7500
MINLP
HA
HSGA
DE
Total guaranteed losses (W)
Total designed losses (W) (b)
Fig. 7. Comparison of the guaranteed and designed short-circuit impedance
(a), and total guaranteed and designed losses (b) for each 630 KVA
transformer design based on the five different optimization results.
TABLE IV
Comparison of the Optimization Algorithms for the 1000 KVA transformer.
Characteristics of
the optimum
transformer design
1000kVA
MINLP HA HS GA DE
Low voltage turns 15 13 13 13 12
D (mm) 265 285 276 316 292
G (mm) 384 338 338 338 276
B (Gauss) 17321 16800 16741 16534 16699
Guaranteed Fe
losses (W)
1100 1100 1100 1100 1100
Guaranteed Cu
losses (W)
10500 10500 10500 10500 10500
Total guaranteed
losses (W)
11600 11600 11600 11600 11600
Guaranteed Short-
Circuit Impedance
(%)
6 6 6 6 6
Designed Fe
Losses (W)
1249 1264 1265 1196 1265
Cu Losses (W) 10478 9703 9675 9936 9462
Total designed
losses (W)
11727 10967 10940 11132 10727
Short-Circuit
Impedance (%)
6.32 5.57 5.57 5.72 6.53
Cost (€) 9001 8785 8786 8918 8777
Cost
Classification
5 1 2 4 3
5
5.2
5.4
5.6
5.8
6
6.2
6.4
6.6
MINLP
HA
HSGA
DE
Guaranteed Usc (%)
Designed Usc (%) (a)
10200
10400
10600
10800
11000
11200
11400
11600
11800
MINLP
HA
HSGA
DE
Total guaranteed losses (W)
Total designed losses (W) (b)
Fig. 8. Comparison of the guaranteed and designed short-circuit impedance
(a), and total guaranteed and designed losses (b) for each 1000 KVA
transformer design based on the five different optimization results.
In non-deterministic methodologies, as well as in
deterministic methodologies, the design vector definition is crucial in order to meet some desired performance objective.
For example, the exploring of the geometrical core
parameters can be ensured through careful planning, and thus
the quality of transformer design can be established during the definition of initial design vector values. However, the input
data of the design vector are deployed randomly, in the case of the non-deterministic methods. As a result, there is
possibility of little control over investigating the entire solution space. Therefore, deterministic methods are often
pursued for only a selected subset of the design vector with the aim of in-depth searching of the solution space.
According to the above results, HA provides the best
solution both in terms of cost and operating performance (especially on total losses). Heuristic algorithm does not
guarantee optimal, or even feasible, solution and is often used with no theoretical guarantee. Despite this main disadvantage,
heuristic evaluations still perform an important role in the transformer design, and if implemented properly can provide
powerful results. Based on the case studies that have been carried out, it
should be noted that since the non-deterministic methods or stochastic methods use random processes, an algorithm run at
different times can generate different transformer designs. Therefore, a particular transformer study needs to be run
several times before the solution is accepted as the global optimum. On the contrary, MINLP and HA belong to
deterministic methods which are able to find the global minimum, but by an exhaustive search. In this case, in order
to avoid huge calculations, stochastic methods can provide us with the first suboptimal solution, and afterwards, HA can be
used in order to finalize our decision. Since no other method gives an absolute guarantee of finding the global minimum in
a finite number of steps, HA technique becomes important. Despite the fact that five different optimization techniques
were investigated in order to find the most economic transformer design with respect to a sequence of mechanical
and electrical constraints, the transformer manufacturing factories declare that a relative near-optimal solution (to the
optimal one) is often preferred and finally is chosen to be constructed. Under these conditions, it is obvious that the
criterion of cost is not the only factor which should be taken into account at the final decision but also the transformer
specifications of each optimum design, such as the no-load and load losses and the short circuit impedance, are vital
aspects. Based on the above-mentioned fact, HA and MINLP become important methods, since they can store a wide range
of several optimum solutions with different technical specifications, especially the HA.
It must be pointed out that the tuning of non-deterministic algorithms has derived through comparison of various
combinations and choice of the best one between them (in order to exclude the possibility that they are not properly
tuned, thus they cannot converge to the global optimum as the deterministic algorithm). The methods of stochastic nature
fail to find the global optimum due to the fact that the optimality of the solution provided by them cannot be
guaranteed and multiple runs may result to different suboptimal solutions [33], with a significant difference
between the worst and the best one. On the other hand, deterministic methods provide more robust solutions to the
transformer design optimization problem and are more
suitable for the search of global optimum.
V. CONCLUSION
In the present paper, comparison of deterministic and non-deterministic optimization methods has been carried out in
order to achieve optimal global transformer design. The design optimization has been carried out with the use of an
integrated software platform, which has been experimentally verified and integrated in the automated design process of
several transformer manufacturing industries. The combination of the proposed methods is very effective
because of its robustness, its high execution speed and its ability to effectively search the large solution space. The
ability to locate the global optimum is illustrated by the application to a wide spectrum of actual transformers, of
different power ratings. The development of user-friendly software based on the combination of these methods provides
significant improvements in the design process of the manufacturing industry.
According to the results, HA provides the best solution both in terms of cost and operating performance. The
methods of stochastic nature fail to find the global optimum due to the fact that the optimality of the solution provided by
them cannot be guaranteed and multiple runs may result to different suboptimal solutions, with a significant difference
between the worst and the best one. On the contrary, MINLP and HA belong to deterministic methods which are able to
find the global minimum, but by an exhaustive search. It is however pointed out that the goal is not only to find the most
economic transformer, but a design that meets the technical specifications with the less possible deviation from the
guaranteed values. In this context, the criterion of cost is not the only factor which should be taken into account at the final
decision but also the transformer specifications of each optimum design.
REFERENCES
[1] E. I. Amoiralis, M. A. Tsili, A. G. Kladas, “Transformer design and
optimization: a literature survey,” IEEE Trans. Power Del., vol. 24, no.
4, pp. 1999-2024, Oct. 2009.
[2] R. A. Jabr, “Application of geometric programming to transformer