FEM Based Preliminary Design Optimization in Case of Large Power Transformers
Abstract
:1. Introduction
2. Proposed Methodology
2.1. Transformer Model for the Optimization
Algorithm 1 Transformer Model Evaluator 
function Evaluator(p) ▹ p means the independent design parameters, which generated by NSGAII within the given search space

2.2. Objective Function—Total Cost of Ownership
2.3. FEM Model
2.4. Ārtap
2.5. NSGAII
Algorithm 2 NSGA II 

2.6. Analytical Calculations
2.7. Power Criteria in Working Window
2.8. Regulating Winding Dimensions
Turn Voltage
2.9. Core Mass and NoLoad Loss Calculation
2.10. Geometric Programming
2.11. GP Based Embedded Winding Model
2.11.1. Eddy Losses in the Windings
2.11.2. Geometry
3. Results and Discussion
3.1. Validation of the Transformer Model
 ${D}_{c}=368$ mm is the core diameter,
 ${B}_{c}=1.57$ T is the flux density,
 ${h}_{s}=979$ mm is the height of the low voltage winding,
 $g=26.7$ mm is the main gap distance is,
 ${j}_{s}=3.02\frac{A}{\mathrm{mm}{}^{2}}$ is the current density in the LV winding,
 ${j}_{p}=3.0\frac{A}{\mathrm{mm}{}^{2}}$ is the current density in the HV winding,
 ${j}_{r}=1.86\frac{A}{\mathrm{mm}{}^{2}}$ is the current density in the REG.
3.2. Input Parameters of the Test Transformer
3.3. Discussion of the Results
4. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Quantity  Dimension  Variable 

Independent variables  
Core diameter  mm  ${D}_{\mathrm{c}}$ 
Flux density in the core  T  B 
Main insulation distance  mm  g 
Current density in the secondary coil  A/mm${}^{2}$  ${j}_{\mathrm{s}}$ 
Current density in the primary coil  A/mm${}^{2}$  ${j}_{\mathrm{p}}$ 
Current density in the regulating coil  A/mm${}^{2}$  ${j}_{\mathrm{r}}$ 
Height of the secondary winding  mm  ${h}_{\mathrm{s}}$ 
Dependent parameters (Analytical)  
Width of the working window  mm  s 
Core mass  t  ${M}_{\mathrm{c}}$ 
Radial thickness of secondary winding  mm  ${t}_{\mathrm{s}}$ 
Mean radius of secondary winding  mm  ${r}_{\mathrm{s}}$ 
Radial thickness of primary winding  mm  ${t}_{\mathrm{p}}$ 
Mean radius of primary winding  mm  ${r}_{\mathrm{p}}$ 
Radial thickness of regulating winding  mm  ${t}_{\mathrm{r}}$ 
Mean radius of regulating winding  mm  ${r}_{\mathrm{r}}$ 
No Load Loss  kW  ${P}_{\mathrm{nll}}$ 
Dependent parameters (FEM)  
Short circuit impedance  %  $SCI$ 
Maximum of radial flux density in LV  T  ${B}_{rs}$ 
Maximum of radial flux density in HV  T  ${B}_{rp}$ 
Maximum of axial flux density in LV  T  ${B}_{as}$ 
Maximum of axial flux density in HV  T  ${B}_{ap}$ 
Dependent parameters (GP subproblem)  
Number of turns in a winding  #  n 
Number of conductors in a turn  #  ${n}_{c}$ 
Number of axial turns  #  ${n}_{ax}$ 
Number of radial turns  #  ${n}_{rad}$ 
Copper area in one turn  mm${}^{2}$  ${A}_{cu}$ 
Copper volume in the winding  mm${}^{3}$  ${V}_{cu}$ 
Copper mass in the winding  kg  ${M}_{k}$ 
Optimal conductor height  mm  ${h}^{\ast}$ 
Optimal conductor width  mm  ${w}^{\ast}$ 
Dependent parameters (Complex)  
Load Loss  kW  ${P}_{\mathrm{ll}}$ 
Total Cost of Ownership  €  $TOC$ 
LV  HV  

Reference  Model  Reference  Model  
Line voltage  kV  22  35  
Connection  kV  D  Y  
Phase Voltage  kV  22  20.23  
Number of turns  #  708  650  
Phase current  A  95.5  104  
Turn area  mm${}^{2}$  31.623  56.0  
Conductor height  mm  11.6  6.6  11.4  8.1 
Conductor width  mm  2.7  2.7  3  2.7 
Mean diameter  mm  437  436  578  572 
Winding width  mm  42.9  42.8  40.7  41.1 
Copper mass  kg  813  824  1071  1082 
Loss  kW  19.150  19.23  25.948  23.979 
Parameter  Dimension  Value  

Nominal power  MVA  31.5  
Frequency  Hz  50  
Connection group  Dyn1  
Number of phases  #  3  
Short circuit impedance  %  14.5  
Main gap  mm  37  
Sum of the end insulation  mm  150  
Phase distance  mm  37  
CoreInner winding distance  mm  20  
Core  Number of legs  #  3 
Flux density limit in columns  T  1.7  
Filling Factor  %  90  
Material Type  M1H  
Material Price  €/kg  3.5  
Low Voltage Winding  Line Voltage  kV  33 
Phase Voltage  kV  19.05  
BIL  kV  125  
AC  kV  50  
Copper filling factor  %  60  
Material and manufacturing price  €/kg  10  
High Voltage Winding  Line Voltage  kV  120 
Phase Voltage  kV  69.36  
BIL  kV  550  
AC  kV  230  
Copper filling factor  %  60  
Material and manufacturing price  €/kg  8.5  
Regulating Winding  Regulating range  %  $\pm 10$ 
Insulation  Fully insulated  
Regulated winding  High voltage  
Filling factor  %  65 
Parameter  Dimension  Lower Bound  Upper Bound 

${D}_{\mathrm{c}}$  mm  400  700 
B  T  1.4  1.7 
g  mm  37  70 
${j}_{\mathrm{s}}$  A/mm${}^{2}$  1.5  3.0 
${j}_{\mathrm{p}}$  A/mm${}^{2}$  1.5  3.0 
${j}_{\mathrm{r}}$  A/mm${}^{2}$  1.5  3.5 
${h}_{\mathrm{s}}$  mm  1200  2000 
Design Parameters  Dimension  Metaheuristic  NSGA2+GP 

Core data  
core diameter  mm  570  600 
flux density  T  1.64  1.58 
core mass  t  16.65  21.05 
turn voltage  V  83.6  89.3 
main gap  mm  37  58 
Low voltage winding  
inner diameter  mm  610  720 
winding height  mm  1003  1210 
winding width  mm  89  80 
turn number  #  228  214 
current density  A/mm${}^{2}$  2.35  2.02 
h*  mm    3.6 
w*  mm    2.5 
High voltage winding  
inner diameter  mm  861  1027 
winding height  mm  973  1170 
winding width  mm  107  110 
turn number  1579  1478  
h*  mm    8.1 
w*  mm    2.7 
current density  A/mm${}^{2}$  2.01  1.53 
Regulating winding  
inner diameter  mm  1149  1220 
winding height  mm  853  1025 
winding width  mm  10  10 
current density  A/mm${}^{2}$  2.7  2.71 
load loss  kW  114.9  88.3 
core loss  kW  13.2  17.82 
TOC  €  447,627  448,597 
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Orosz, T.; Pánek, D.; Karban, P. FEM Based Preliminary Design Optimization in Case of Large Power Transformers. Appl. Sci. 2020, 10, 1361. https://doi.org/10.3390/app10041361
Orosz T, Pánek D, Karban P. FEM Based Preliminary Design Optimization in Case of Large Power Transformers. Applied Sciences. 2020; 10(4):1361. https://doi.org/10.3390/app10041361
Chicago/Turabian StyleOrosz, Tamás, David Pánek, and Pavel Karban. 2020. "FEM Based Preliminary Design Optimization in Case of Large Power Transformers" Applied Sciences 10, no. 4: 1361. https://doi.org/10.3390/app10041361