Electr Eng DOI 10.1007/s00202-017-0537-z
ORIGINAL PAPER
Power transformer optimal design (PTOD) using an innovative heuristic method combined with FEM technique Milad Yadollahi1 · Hamid Lesani1
Received: 6 November 2016 / Accepted: 29 April 2017 © Springer-Verlag Berlin Heidelberg 2017
Abstract Power transformer optimal design (PTOD) is a complex multi-objective optimization problem including numerous design variables that aim to design a transformer with minimal material costs. Due to large variables search space (SS), most optimization methods proposed in the literature for PTOD are prone to find local minimum instead of the global one. So, some design variables and constraints are neglected to reduce the SS size and alleviate the problem. They also suffer from a random nature which makes it impossible for them to explore all the SS. To prevail over the aforementioned problems, this work aims to propose a new heuristic algorithm, and also, some modifications to conventional PTOD procedure. Some promising features of the proposed method in comparison with the previously proposed methods are (1) it considers all design variables as a gene for transformer constructing chromosome (TCC), (2) before building the main TCC, it divides the PTOD algorithm into some sub-algorithms and starts to construct some sub-chromosomes with lower number of genes as parts of the main TCC, which allows us to detect improper genes based on a sensitivity analysis and design constraints, and ignore them, automatically narrowing the SS by simply ignoring its bad parts (the solutions which do not meet design constraints), and (3) not only it considers all technical and consumer constrains, but it also takes manufacturing constraints into Electronic supplementary material The online version of this article (doi:10.1007/s00202-017-0537-z) contains supplementary material, which is available to authorized users.
B
Milad Yadollahi
[email protected] Hamid Lesani
[email protected]
1
Department of Electrical and Computer Engineering, University of Tehran, P.O. Box 14395-515, Tehran, Iran
account. To verify the effectiveness of the proposed method in achieving the global minimum, it is used to design a 200 MVA and 15.75/400 KV power transformer. The validity of obtained optimal solution is further assessed by presenting comprehensive finite element method using JMAG software. Keywords Power transformer optimal design (PTOD) · Heuristic method (HM) · Finite element method (FEM) · Constructing cost (CC) · Short circuit impedance (SCI)
List of symbols S (MVA) VPT NLV NHV NReg D S (mm) Q S (cm2 ) B (Tesla) NL H (mm) W (mm) NPR NPA NP HW (mm) T (mm) LL (kw) NLL (kw) SCI (%)
Capacity Volt per turn Number of turns in low-voltage winding Number of turns in high-voltage winding Number of turns in regulating voltage winding Core diameter Cross section of core Flux density Number of layers Height of wire Width of wire Number of radial parallel wires Number of axial parallel wires Number of parallel wires in subdivided wires Height of winding Width of winding Load loss No-load loss Short circuit impedance
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1 Introduction Power transformer optimal design (PTOD) is one of the most important and interesting issues which have attracted attention in recent years [1,2]. Several works have been proposed in the literature to tackle the problem trying to achieve an optimum point of design by proposing different methods and/or by defining different objective functions with the aim of minimizing the total owning cost, final cost of power transformer including all the costs for the consumed copper and iron, the oil, the insulating paper and the transformer tank, reducing the transformer weight and size, and also increasing its efficiency by decreasing its losses in both no-load and fullload operations, or maybe a combination of these purposes [3–18]. In [3] only optimization of the overall dimensions of the winding and the core has been carried out regardless of saturation, leakage reactance and thermal limits for a single-phase distribution transformer. The power transformer design is a multi-disciplinary and multi-objective optimization problem considering electromagnetic, thermal and cost requirements of the design [3–6]. It is not possible to propose an accurate algorithm featuring a rapid convergence rate, without having a good knowledge and understanding about how different design variables affect design constraints and the cost function value. It also requires enough level of experience in the field of transformer manufacturing process because there are several manufacturing constraints which are linked together and need to be considered during the transformer design procedure. To sum up, the importance of PTOD is because of three main reasons: (1) having numerous design variables [7], (2) having large number of constraint including manufacturing constraint as both linear and nonlinear equations, (3) direct correlation between most of the variables and outputs. Most of the optimization methods presented in the literature focused on the optimal design problem for distribution transformers [8–11]. However, unlike the power transformers, distribution transformers are usually made up of only low- and high-voltage windings without the voltage regulation winding. Moreover, many of constraints in the PTOD including tensions caused by the short circuit forces are not considered in the distribution transformer. So, their optimal design is not as complex as that of power transformers and can be achieved by searching a considerably smaller SS in comparison with power transformer. These works also ignore some vitally important manufacturing constraints and choose only some main design variables to reduce the search space size, and accordingly, to speed up the convergence rate of optimization process [8–11]. The same is true about methods tackling the PTOD. Mixed Integer Nonlinear Programming (MINLP) is compared to Harmony Search (HS), Differential Evolution (DE) and Genetic Algorithm (GA) for the transformer design optimization problem in
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[8]. In [9], an innovative method has been proposed for PTOD. This method employs continuous design variables, for which a movement step should be introduced by an operator, making it possible to change value of each variable and move over the SS. For high values of step, the local search ability of the algorithm is reduced, while for small steps, the algorithm suffers from a weak global search capability. So, it cannot be a certain way to reach the optimal response. Bacterial foraging algorithm is used by [10] for optimum design of single-phase transformer. It considers only four independent variables and two constraints to meet the requirement of the design. In [12] a hybrid method integrating numerical technique, decision trees and artificial neural networks is used for the selection of winding material in power transformer. Also recursive GA-FE method is used for the solution of transformer manufacturing cost minimization problem [13]. The geometric programming (GP) method is the most representative mathematical programming to optimize PTOD [14]. The GP method requires development of a mathematical model and cannot be easily combined with FEM technique for validation of results. Also, a parallel mixed integer programming-finite element method (MIP-FEM) technique as a mathematical programming method has been used for PTOD [15]. However, this technique is not often able to find the global optimum. In [16], a novel DE algorithm (NDE) based on truncated gamma probability distribution function is proposed for solving multi-objective optimization problems as the design of transformers. However, to achieve optimal performance with DE, time-consuming parameter tuning is essential as its performance is sensitive to the choice of the mutation and crossover values. Due to large number of design variables, it is difficult to achieve the global optimum point. In many of aforementioned methods, in order to limit the SS size, number of design variables is reduced, for example, design variables as the height and width of wires are ignored in [8] and [17]. The HA method assigns many alternative values to the design variables so as to generate a large number of alternative designs and finally to select the design that satisfied all the problem constraints with the optimum value of the objective function [9]. This method suffers from two main problems: (1) its accuracy is not validated, (2) it does not consider all design variables to reduce the computational power. To address the aforementioned problems, this paper proposes a new heuristic algorithm combined with FEM (HAFEM), in which the basic idea is to explore the entire feasible domain in a systematic way for a global minimum. Some promising features of the proposed method in comparison with the previously proposed methods are: • Not only considers technical and consumer constrains, but also takes manufacturing constraints into account. It
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•
•
•
• •
also considers more design variables in comparison with previously proposed works. It divides the PTOD algorithm into some sub-algorithms and starts to construct some sub-chromosomes with lower number of genes as parts of the main TCC. This allows the algorithm to omit improper genes based on design constraints, narrowing the SS by simply ignoring the solutions that do not meet design constraints. It uses discrete variables in order to achieve a high convergence rate and reduce computational power, while all continuous variables are considered as constraints. So, there is no need to define the movement step and the proposed method will be able to explore all over the SS to find global optimum. For continues independent variables, however, achieving the absolute optimum point will depends on the step change of each independent variable and if they are not selected properly, it will result in losing the global minimum point. As the proposed method finds the global optimum by systematically narrowing down the SS based on all consumer, technical, and manufacturing constraints and also based on electrical, magnetic, and heat transfer analyses, the number of iterations for a specific design case would be the same for every run of algorithm. As the proposed method narrows the SS, the algorithm required time and computational power are reduced. It can also be used to design other transformers like auto-transformers and single- or three-phase distribution transformers.
• The voltage regulating switch type and the voltage regulation percentage %VR , and also the number of positive and negative steps • The guaranteed full-load and no-load losses • The guaranteed short circuit impedance and its allowed tolerance • The core sheet type • The winding type in each of voltage levels • The conductor type in each winding • The minimum allowed insulation distances between the winding considering voltage levels (as illustrated in Fig. 1) • The prices of copper, core sheets, and oil to be used in objective function for costs 2.2 Objective function The aim of the PTOD is to design the transformer so as to minimize the transformer manufacturing cost, i.e., the sum of materials costs, subject to constrains imposed by international standards, transformer characteristics and manufacturing, technical and consumer constrains. In this paper, a heuristic method combined with FEM (HA-FEM) used to optimize the transformer design is based on the minimization of the cost of the transformer materials, according to the following equation:
Fitness function = min
4
c j f j (x)
j=1
The paper is organized as follows: Sect. 2 describes the mathematical formulation of the PTOD and the software developed to implement it. Section 3 presents the proposed transformer design optimization method, while an application of the proposed methodology to an actual transformer design case is rendered by Sect. 4. Finally, Sect. 5 concludes the paper.
where c j and f j are unit cost (e/kg) and the weight (kg) of jth main materials, respectively, including: (1) (2) (3) (4)
The windings copper Transformer core weight The transformer oil, The tank sheet steel of transformer.
2 Mathematical formulation 2.3 Design variables 2.1 Input data The proposed method is implemented as a software which has a graphic user interface (GUI) by which the following input data are entered by the user for a specific design case before running the algorithm: • • • • •
The line voltage of each terminal The connection type of each winding in each voltage level The nominal power The nominal frequency The environment temperature and installation height
In our proposed method for PTOD, the number of design variables is a function of the winding numbers NW. Here, considering a transformer with regulating winding (NW = 4), the transformer constructing chromosome (TCC) X , which includes 17 genes of the discrete design variables, is divided into sub-chromosomes including VPT sub-chromosome x1 , low-voltage sub-chromosome x2 and high-voltage sub-chromosome x3 , and voltage regulation sub-chromosome x4 : X T = [[x1 ], [x2 ], [x3 ], [x4 ]]
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Fig. 1 Minimum insulation distances for 200 MVA transformer
in which: • x1 : The VPT sub-chromosome The genes forming this sub-chromosome are the NLV , NHV , and NReg . • x2 : The low-voltage winding sub-chromosome The genes forming this sub-chromosome are [H, B, NPA, NPR, NT]. • x3 : High-voltage winding sub-chromosome The genes forming this sub-chromosome are [H, B, NPA, NPR, NT]. • x4 : Voltage regulating winding sub-chromosome The genes forming this sub-chromosome are [H, B, NPA, NPR, NT]. All above variables are discrete variables. 2.4 Constraints Constraints are divided into dependent and independent constraints.
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• Independent constraints They only depend on one sub-chromosome. In other words, we do not need to determine the whole structure of transformer in order to calculate them. These constraints include: • • • •
Turn ratio error constrain Limitation of current density in each winding The winding temperature gradient Limitation of winding height that may be applied by manufacturer constraints
Employing independent constraint allows us to eliminate the inappropriate genes in each sub-chromosome. By this way, the search space will be converted to the discrete and limited space. Finally, those sub-chromosomes surviving from the first step of algorithm are combined together and form the overall structure of transformer. • Dependent constraints Unlike the independent constraints, the dependent design constrains are dependent to some sub-chromosomes. So, the overall structure of transforms need to be determined to calculate them. These constraints are: • Full-load losses • No-load losses
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• • • • •
No-load current Sound level constraint Upper and lower bounds of short circuit impedance The axial and radial forces of windings Constraints regarding the transformer ability to dissipate its heat to environment regarding its dimensions.
Top yoke
r1 Coil 1
Coil 2
Δ
Core
r2
x'
x
So, objective function is subject to: ⎤ g1 (load losses) ⎥ ⎢ g2 (noload losses) ⎥ ⎢ ⎢ g3 (short circuit impedance) ⎥ ⎥ ⎢ ⎥ ⎢ g4 (core flux density) ⎥ ⎢ ⎥ ⎢ g5 (short circuit forces) ⎥ c(x) = ⎢ ⎢ g6 (windings temperature rise) ⎥ ≤ 0 ⎥ ⎢ ⎥ ⎢ g7 (turn ratio error) ⎥ ⎢ ⎥ ⎢ g (transformer dimension) ⎥ ⎢ 8 ⎦ ⎣ g (hot spot factor) 9 g10 (winding current density) ⎡
r1
1 W = 2
lb ≤ x ≤ ub
3 Power transformer optimal design (PTOD) methodology Today, the advanced computers are used in complex structures of transformers in order to calculate the magnetic field by using numerical methods such as FEM. In addition to the advanced numerical methods, analytical methods are always important because of their advantages like high speed, no need for a large number of inputs, and acceptable accuracy. So, this paper also uses an analytical method for sensitivity analysis of short circuit impedance, which can be used to eliminate improper genes from TCC. 3.1 Sensitivity analysis of short circuit impedance in relation to winding dimensions in a specific voltage per turn (VPT) The balanced Ampere Turn (AT) in this method is shown in Fig. 2. So, the leakage inductance can be calculated by the following equations: If A is the magnetic vector potential, then ∇ × A = B and ∇ × H = J , so stored energy W is equal to:
r2
x
Fig. 2 AT balance between windings
in which:
where lb and ub are n ×1 matrices of lower and upper bounds on x, and C(x) is the nonlinear inequalities of the transformer design problem (consisting of the constraint functions gi for i = 1 : 10).
Bottom yoke
Ampere turn
1 B · H dV = 2
(∇ × A) · H dV (1)
∇ · (A × H ) = A · (∇ × H ) − H · (∇ × A)
(2)
Considering (1) and (2), one can write: 1 W = (A · (∇ × H ) − ∇ · (A × H ))Rd RdϕdZ 2 1 = (A · J )dV (3) 2 L leakage I 2 Iϕ 1 = (4) I B · dS = W = 2 2 2 So, the leakage inductance can be calculated by the help of the stored energy method: 1 L leakage I 2 2 μ0 |H |2 dV W = 2
W =
(5) (6)
V
The volume V is related to the whole of the infinite space, but regarding the boundary conditions, and given that, the core permeability coefficient is unlimited, so the magnitude of magnetic field strength resulted from leakage flux in the core window is negligible and its energy could be ignored. So, by having the magnetic field strength inside the core window the saved energy can be determined. The Ampere’s law is used to calculate the magnetic field strength as follows:
H · dl = N I
(7)
c
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According to Fig. 2, in the first winding area, the ampere turn increases linearly by an increase in the width of windings, so we have: H1 = N1 I1 ×
x T1 × L eq
In the second part, we change the HV winding width, while other dimensions are constant for a specific VPT (see Table 2; Fig. 5). In the third part, we change the first winding height, while other dimensions are constant for a specific VPT (see Table 3; Fig. 6). In the fourth part, we change the second winding height, while other dimensions are constant for a specific VPT (see Table 4; Fig. 7).
(8)
The magnetic field strength in the oil canal between two windings is constant, and it is equal to: H =
N 1 I1 L eq
(9)
3.2 Conclusion of sensitive analysis
Also, in the second winding area the ampere turn decreases linearly by a decrease in the width of windings, so we have: x N 1 I1 1− H2 = L eq T2
The sensitivity analysis of the SCI shows that in a specific VPT, the SCI decreases by each winding height increase, while other dimensions are constant. Additionally, by increasing each winding width while other dimensions are constant, the SCI increases. According to the aforementioned analysis, both maximum and minimum values of SCI are obtained for the short circuit impedance in each VPT as:
(10)
We can now calculate total stored energy in the system from (6). Since before the first winding and after the second winding the total AT is zero, we have: d 1 +T1
d2
|H1 | dV +
W = 0+
2
|H |2 dV
d1 +T1
d1 d 2 +T2
|H2 | dV + 0
+
2
(11)
We can now calculate leakage inductance L leakage by Eqs. (5) and (11). Then, one can obtain the imaginary part of the impedance at the frequency of 50 Hz as follows:
=
2π f × L leakage × 100 Z base
0.124 × S3ϕ × (2r1 + T1 ) ×
T1 3
+ (2r2 + T2 ) ×
To validate aforementioned analysis, a 250 MVA power transformer is modeled, as shown in Fig. 3, as an example in JMAG-designer 2014 environment, and its stored energy is calculated and is compared to that obtained by (11). The results are presented by Tables 1, 2, 3, and 4 and Figs. 4, 5, 6, and 7. In order to see how the windings dimensions affect the SCI%, initially, the LV winding width is increased keeping other dimensions fixed for a specific VPT (see Table 1; Fig. 4)
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T2 3
3 × VPT2 × L eq
After calculating L leak , X leak , SCI can be simply calculated as: 2 LL + (L leak × 2π f )2 (13) SCI = S
(14)
SCIMin α(HWMax , BWMin )
(15)
Thus, for a specific VPT, all SCI values for different arrangements of windings will be in this range. This result can be helpful in many ways. If the maximum and minimum SCI range calculated for a specific VPT has nothing in common with the range of guaranteed SCI, this VPT will be eliminated from the SS. So, by this method, many impermissible VPT can be eliminated,
d2
X leakage =
SCIMax α(HWMin , BWMax )
+
1 6
T12 − T22 + (2r2 − ) ×
(12)
resulting in a considerably smaller SS. On the other hand, by eliminating each VPT, all of the calculations regarding full-load and no-load losses, forces and tensions for all combinations of that VPT will be eliminated. Hence, the program running time and also the computer memory consumption rate are considerably reduced. In other words, for a specific VPT we have: If (SCIMax < minimum permissible of SCI) or (SCIMin > maximum permissible of SCI), then the VPT cannot satisfy the SCI constraint. Thus, we can calculate the maximum and minimum of SCI% for each VPT, and then, if it does not have any common range with the guaranteed area, all of that VPT designs should be eliminated and there is not any need to calculate other parameters and the SS is limited.
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Fig. 3 Magnetic flux strength in the core window Table 1 SCI (%) changes according to the first winding width changes
HW1
T1
HW2
T2
SCI (%)
Stored energy by the JMAG-designer (J)
Error percent (%)
1874
43.5
1848
113.5
12.32
16,294
0.27
1874
63.5
1848
113.5
13.1
17,115
0.2
1874
67.5
1848
113.5
13.26
17,320
0.16
1874
71.5
1848
113.5
13.42
17,534
0.33
1874
75.5
1848
113.5
13.58
17,742
0.71
1874
83.5
1848
113.5
13.9
18,159
0.21
1874
87.5
1848
113.5
14.07
18,380
0.54
1874
91.5
1848
113.5
14.23
18,588
0.15
1874
95.5
1848
113.5
14.4
18,815
0.61
1874
99.5
1848
113.5
14.56
19,020
0.16
1874
103.5
1848
113.5
14.73
19,241
0.1
1874
123.5
1848
113.5
15.56
20,327
0.64
1874
135.5
1848
113.5
16.07
20,996
0.35
3.3 Implementing proposed algorithm The proposed algorithm includes 8 steps described as follows. • Step 1 Constructing the first sub-chromosome x1 (The VPT sub-chromosome)
This sub-chromosome includes the high-voltage, low-voltage, and voltage regulation turns. Regarding the input information of the program, in this step, we form a list of VPTs and their turn ratio error in all taps is less than permissible value. The turn ratio error can be defined as:
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Table 3 SCI (%) changes according to the first solenoid height changes
Table 4 SCI (%) changes according to the second winding height changes
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HW1
T1
HW2
1874
83.5
1848
1874
83.5
1874
83.5
1874
T2
SCI (%)
Stored energy by the JMAG-designer (J)
Error percent (%)
83.5
13.07
17,279
0.22
1848
93.5
13.35
17,649
0.12
1848
105.5
13.68
17,967
0.55
83.5
1848
107.5
13.74
17,990
0.41
1874
83.5
1848
109.5
13.79
18,205
0.16
1874
83.5
1848
111.5
13.85
18,291
0.19
1874
83.5
1848
113.5
13.9
18,353
0.37
1874
83.5
1848
115.5
13.96
18,431
0.27
1874
83.5
1848
117.5
14.03
18,519
0.75
1874
83.5
1848
119.5
14.07
18,571
0.18
1874
83.5
1848
121.5
14.13
18,650
0.42
1874
83.5
1848
123.5
14.19
18,737
0.66
1874
83.5
1848
133.5
14.47
18,903
0.35
1874
83.5
1848
143.5
14.75
19,462
0.2
HW1
T1
HW2
T2
SCI (%)
Stored energy by the JMAG-designer (J)
Error percent (%)
1612
83.5
1848
113.5
15.12
19,955
0.31
1663
83.5
1848
113.5
14.69
19,440
0.76
1716
83.5
1848
113.5
14.49
18,993
0.47
1768
83.5
1848
113.5
14.29
18,769
0.22
1821
83.5
1848
113.5
14.09
18,600
0.16
1874
83.5
1848
113.5
13.9
18,353
0.18
1980
83.5
1848
113.5
13.54
17,888
0.28
2032
83.5
1848
113.5
13.37
17,661
0.34
2084
83.5
1848
113.5
13.2
17,448
0.62
2137
83.5
1848
113.5
13.04
17,230
0.55
2190
83.5
1848
113.5
12.87
16,911
0.25
2295
83.5
1848
113.5
12.56
16,601
0.33
2506
83.5
1848
113.5
11.98
15,857
0.15
HW1
T1
HW2
T2
SCI (%)
Stored energy by the JMAG-designer (J)
Error percent (%)
1874
83.5
1237
113.5
16.45
21,675
0.62
1874
83.5
1421
113.5
15.59
20,566
0.4
1874
83.5
1607
113.5
14.81
19,550
0.45
1874
83.5
1680
113.5
14.52
18,973
0.91
1874
83.5
1735
113.5
14.31
18,799
0.52
1874
83.5
1790
113.5
14.1
18,625
0.1
1874
83.5
1846
113.5
13.9
18,438
0.55
1874
83.5
1901
113.5
13.71
18,201
0.15
1874
83.5
1975
113.5
13.47
17,765
0.48
1874
83.5
2066
113.5
13.17
17,490
0.36
1874
83.5
2160
113.5
12.88
16,919
0.25
1874
83.5
2342
113.5
12.35
16,220
0.17
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17
SCI 16 (%)
SCI 16 (%)
15 15
14 14
13 12
13
11
12
10 40
60
80
100
120
11 1100
140
Width of LV winding , T1
1300
1500
1700
1900
2100
2300
2500
Height of HV Winding , HW2 (mm)
Fig. 4 SCI (%) changes according to the first winding width changes
Fig. 7 SCI (%) changes according to the second winding height changes
15 14.8
SCI (%) 14.6
Considering the lower and upper limits of each variable, we save all sub-chromosomes which meet the following constraints:
14.4 14.2 14 13.8 13.6 13.4 13.2 13 12.8 60
80
100
120
140
160
Width of the HV winding, T2
Fig. 5 SCI (%) changes according to the first winding height changes
• The lower and upper bounds of the current density. • Maximum value of the skin effect, and eddy current stemmed from the leakage fluxes components in the core window space which can create a hot spot in the winding ac ≤ permissible value). ( PPdc • The winding temperature gradient • The fraction of (wire height/wire width). (It is a vitally important manufacturing constraints, violating from which can cause serious damages to insulating paper in the production process).
16
SCI (%) 15
Finally, for the sub-chromosomes which meet the aforementioned constrains, we can calculate the winding dimensions and then sort them according to their dimensions.
14
• Step 3 Constructing the third sub-chromosome x3 (The high-voltage winding sub-chromosome)
13 12 11 10 1500
1700
1900
2100
2300
2500
Height of LV Winding, HW1 (mm)
Fig. 6 SCI (%) changes according to the second winding width changes
Turn ratio error =
Ph VLV Ph VHV
− Ph VLV Ph VHV
NLV NHV
(16)
• Step 2 Constructing the second sub-chromosome x2 (The low-voltage winding sub-chromosome)
We should go over through all the steps of Step 2 for the highvoltage winding as well. This stage output is the dimensions of allowable high-voltage windings sorted according to their dimensions. • Step 4 Constructing the fourth sub-chromosome x4 (The voltage regulating sub-chromosome) We should go over through all the steps of step 2 for the highvoltage winding as well. This stage output is the dimensions of allowable high-voltage windings sorted according to their dimensions. • Step 5 In this step, we calculate the maximum and minimum values of the SCI, respectively, according to the maximum and
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Start
X1= [ NLV, NHV, NReg ]
Calculation of VPT and turn ratio error in any taps Yes Turn Ratio Error is violated? No j = NW , i = i +1
The sub-chromosome is rejected
Xi =
Independent constraints
n = n +1
Maximum and minimum Permissible current density is violated?
Yes
No Permissible winding temperature rise is violated?
Yes
No Eddy current (hot spot factor ) is violated? Yes No
Yes
Calculation and sort of winding dimensions
n < NXi
Yes
No
i
Yes
Sensitive analyses
Calculation of maximum and minimum SCI
Permissible SCI > SCImax OR Permissible SCI < SCImin
No 2
Fig. 8 Flowchart of the heuristic algorithm combined with FEM for the PTOD
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2 The solution is rejected
Calculate LL, NLL, SCI, Short Circuit forces and tank dimensions
No Yes Permissible deviation between designed And specified load loss is violated?
No Yes
Dependent constraints
Permissible deviation between designed And specified no-load loss is violated?
No Yes
Permissible deviation between designed And specified short circuit impedance is violated?
No Yes Permissible deviation between designed And maximum winding and oil temperature rise is violated?
No Yes Tension of the radial and axial forces is violated?
No Calculation of material weight and Transformer cost
Yes End
Is the k’th solution validated by FEM ?
No
Go to the next acceptable solution (k=k+1)
Electromagnetic 3D FEM Fig. 8 continued
minimum dimensions of each VPT, and then they can be compared with guaranteed values of SCI. • Step 6 The survivor sub-chromosomes from the previous steps are now combined to form the TCC for each allowable VPT.
• Step 7 Considering all dependent constrains for each TCC, we can obtain the allowable design solutions and sort according to the desired objective function.
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The flowchart of the proposed optimization method for the PTOD, shown in Fig. 8, is composed of 4 sub-chromosomes and 8 parts. Figure 9 shows the 3-D model of the threephase, core type power transformers, consisting of the low-voltage (LV), high-voltage (HV) and regulating voltage windings.
4 Result and discussion of proposed algorithm In this section, to verify the effectiveness of proposed approach for solving the optimization problem, it is used to optimally design a power transformer parameters of which are presented in Table 5. We further investigate the results of an optimum design of transformer in order to show the efficacy of the algorithm. The chosen objective function is cost of transformer material. To demonstrate the superiority of the proposed method, the results obtained from this algorithm are compared with to those of MINLP methods, GA method, and DE method [8].
Fig. 9 Perspective view of the transformer FEM model Table 5 Desired parameters of power transformer Capacity
200 MVA
Phase
Three phase
Primary voltage
15.75 KV
Secondary voltage
420 KV
Type
Core
Frequency
50
Vector group
YNd
Installation level
1000 m
Ambient temperature
48 C
4.1 Results of the proposed method (HA-FEM)
◦
Tap changer
On-load (±10%, ±10 step)
Simple wire unit cost (e/kg)
8
Subdivided wire unit cost (e/kg)
9
Fe unit cost (e/kg)
2.1
Oil unit cost (e/kg)
2
Tank material unit cost(e/kg)
1
• Step 8 The best rendered solution is validated by FEM technique. Magnetic FEM is used to calculate transformer parameters such as SCI, short circuit forces, and eddy current loss in the winding and NLL. If validation of the best solution fails, the solution is rejected, and then the next best solutions sorted by a specific fitness function should be validated until a valid solution is found. This enhances the accuracy of the proposed method and eliminates the possibility of infeasible optimum designs. Figure 8 shows the whole flowchart of the proposed optimization method for the PTOD, in which N xi denotes the number of possible states for ith sub-chromosome: N xi = NLi × Hi × Wi × NPAi × NPRi × NPi
123
(17)
After running the program, a user-friendly interface of software like what is shown by Fig. 10 is appeared. As can be seen from this figure, the proposed algorithm finds and sorts all feasible solutions rapidly. Table 6 shows acceptable solutions sorted by manufacturing cost, while Table 7 shows the number of acceptable results for each VPT. The most important and critical step in designing a power transformer is to select the proper VPT. This parameter is directly proportional to the core cross section. Hence, the core diameter is increased for an increase in the VPT for a fixed flux density. On the other hand, the VPT is inversely proportional to the number of winding turns, meaning that the number of turns is increased for a decrease in VPT. Furthermore, this parameter is inversely proportional to the square of transformer SCI percent. According to the aforementioned results, the appropriate selection of the VPT is very important both economically and technically. The main advantage of the proposed algorithm could be seen clearly in this example. The VPT below 207 has the higher SCI in comparison with the guaranteed values, so they cannot present a proper solution and have to be eliminated from the cycle before performing the memory-intensive calculations of LL and NLL, and short circuit forces for these solutions. The SS will be more restricted and the time needed to run the program is considerably reduced, and less computer memory will be occupied. Table 8 shows the allowable VPTs according to the turn ratio error.
Electr Eng
Fig. 10 The output results of developed software
Table 6 Acceptable solutions sorted by manufacturing cost Number
VPT
LL (kw)
NLL (kw)
SCI (%)
Magnetic steel (kg)
Copper (kg)
Oil (kg)
Tank (kg)
Manufacturing cost ($)
1
242.308
494.8
94.2
13.42
79,715
20,696
72,501
11,859
520,057
2
246.094
493.1
95.1
13.2
80,902
20,668
72,222
11,875
521,828
3
266.949
488.6
101.6
13.22
86,379
19,084
73,423
12,145
521,830
4
242.308
499.5
94.8
13.33
80,263
20,745
73,101
11,903
522,750
…
…
…
…
…
…
…
…
…
…
27735
271.552
390.8
106
12.98
95,181
28,456
82,377
13,151
648,195
27736
271.552
391
104.5
13.1
95,972
28,336
82,386
13,165
648,674
27737
276.316
387.9
105.5
13.37
95,259
28,584
82,161
13,201
649,025
27738
276.316
384.6
106.8
13.06
95,895
28,431
82,656
13,220
649,910
The best result is validated by FEM using JMAG software. As an example, Fig. 11 and Table 9 show short circuit force’s both vectors and magnitudes for the best output result. The best result validated by FEM can now be used to compare the proposed HA-FEM with other methods.
4.2 The proposed algorithm in comparison with other methods In this section, the proposed method is compared to DE [16], GA method [8], and MINLP method [8]. For the GA, it was found that a population size of 120 chromosomes and a num-
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Electr Eng Table 7 Number of acceptable results for each of VPT VPT
Number of acceptable solutions
Best result
Worst result
Reason
…
…
…
…
SCI > 14.5 %
199.367
0
-
-
SCI > 14.5 %
201.923
0
-
-
SCI > 14.5 %
204.545
0
-
-
SCI > 14.5 %
207.237
0
-
-
SCI > 14.5 %
215.753
0
-
-
LL > 565
218.75
0
-
-
LL > 538
221.831
0
-
-
LL > 504
225
0
-
-
Temperature rise of HV winding is violated
238.636
900
20
19,875
242.308
4637
1
25,262
246.094
5813
2
27,643
266.949
9433
3
27,728
271.552
5263
3508
27,736
276.316
1692
7051
27,738
297.17
0
-
-
NLL > 111
…
…
…
…
SCI < 12.5
Table 8 The allowable voltage per turn according to the turn ratio error VPT
LV turn HV turn HV turn HV turn Step turn Upper tap
Middle tap
Lower tap
Fig. 11 Short circuit force vectors, and magnetic flux lines in the JMAG-designer Software
Turn ratio error
Table 9 The short circuit forces’ magnitudes Magnitude (KN)
… 199.367
79
1336
1216
1096
12
0.143
201.923
78
1321
1201
1081
12
0.019
204.545
77
1306
1186
1066
12
0.15
207.237
76
1290
1170
1050
12
0.293
215.753
73
1234
1124
1014
11
0.245
218.75
72
1218
1108
998
11
0.112
221.831
71
1203
1093
983
11
0.082
225
70
1188
1078
968
11
0.212
238.636
66
1116
1016
916
10
0.162
242.308
65
1101
1001
901
10
0.037
246.094
64
1086
986
886
10
0.196
266.949
59
998
908
818
9
0.12
271.552
58
983
893
803
9
0.084
276.316
57
968
878
788
9
0.277
297.17
53
896
816
736
8
0.219
…
ber of 80 generations provide very good result for optimization. The results comparison of the optimization algorithms for sample transformer is given in Tables 10 and 11.
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Radial forces (KN)
Axial forces (KN)
LV winding
29,146
29,140
587
HV winding
36,663
36,658
540
The results presented in Table 10 are also shown by graphical diagram in Fig. 12, where it can be deduced that the cost function in our proposed HA-FEM is reduced by 3.7, 4.8, and 8.8% in comparison with GA [8], DE [16], and MINLP [8], respectively. As previously stated, total manufacturing cost is selected as cost function. After sorting the results, it can be seen that better results inherit higher VPT. The explanation is that, since the copper is more expensive than other materials, better results are those with lower windings’ turns, that is, the results with higher VPT. On the other hand, as VPT is inversely proportional to the SCI, solutions with very high VPT values result in SCIs lower than guaranteed SCI values and are automatically omitted. So, the proposed algorithm has to make a compromise between high VPTs and SCIs.
Electr Eng Table 10 Comparison of the optimization algorithms for 200 MVA transformer
5 Conclusion
Parameter
In this paper, an innovative heuristic algorithm has been proposed to search all over the SS and achieve the global optimum point in the PTOD. All of variables have been selected discrete, and all the guaranteed constrains, international standards, and manufacturing limitations have been considered in the design process. According to the sensitivity analysis, programming techniques, and modifications to design process, the search space is limited by systematically ignoring its bad parts (the solutions which do not meet design constraints) and the program execution speed is increased, enabling it to effectively search the large solution space. On the other hand, the user has access to all allowable designs in the software output according to the design constrains, so we do not need to run the program again for other objective functions. The output results are finally validated by FEM. An exhaustive comparative study is carried out and shows that the cost function in our proposed HA-FEM is reduced by 3.7, 4.8, and 8.8% in comparison with GA, DE, and MINLP, respectively.
Best solution of MINLP
DE
GA
Proposed algorithm
NLV
57
59
64
65
DS
1023
1015
968
960
NLLV
2
2
2
2
HLV
7.4
5.7
5.5
7.7
BLV
1.6
1.5
1.8
1.4
PACLV
3
4
4
3
PRCLV
1
1
1
1
NTLV
51
39
35
39
HHV
8.8
8.3
9.8
9.8
BHV
2.6
2.5
2.5
2.4
PACHV
1
1
1
1
PRCHV
2
2
2
2
NTHV
1
1
1
1
HReg
4.6
4.5
4
4
BReg
2
2.1
2.3
2.2
NTReg
9
9
11
11
References Table 11 Comparing output results Parameter
Best solution of MINLP
DE
GA
Proposed algorithm
Magnetic steel (kg)
90,290
90,101
82,114
79,715
Copper (kg)
22,695
19,054
21,820
20,696
Oil (kg)
75,827
81,591
74,494
72,501
Tank (kg)
12,339
12,853
12,094
11,859
LL (kw)
432.6
503
481.4
494.8
NLL (kw)
106.5
106.1
97
94.2
Material cost ($)
570,425
546,242
540,046
520,057
GA
The proposed algorithm
580000 570000 560000 550000 540000 530000 520000 510000 500000 490000
MINLP
DE
Fig. 12 The graphical diagram comparing our proposed HA-FEM method with DE, GA, and MINLP
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