# FEM Based Preliminary Design Optimization in Case of Large Power Transformers
## 3 Results and Discussion
### Discussion of the Results  
The main goal of this task is to verify that the resulted TOC of the proposed algorithm can find the global optimum of the equation system. The result of the cost optimization are the TOC and the key-design parameters of the transformer (Figure 6). The key-design parameters are good for a design study and a cost estimation, but these parameters can not determine one and only final transformer design. Therefore, the result of the proposed method (Table 5) are compared with the results of a previously validated, metaheuristic based optimization method. The metaheuristic method uses the combination of the method of branch and bound and geometric programming to find the \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameter** & **Dimension** & **Lower Bound** & **Upper Bound** \\ \hline \(D_{\text{c}}\) & mm & 400 & 700 \\ B & T & 1.4 & 1.7 \\ \(\delta\) & mm & 37 & 70 \\ \(j_{\text{s}}\) & A/mm\({}^{2}\) & 1.5 & 3.0 \\ \(j_{\text{p}}\) & A/mm\({}^{2}\) & 1.5 & 3.0 \\ \(j_{\text{r}}\) & A/mm\({}^{2}\) & 1.5 & 3.5 \\ \(h_{\text{s}}\) & mm & 1200 & 2000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The applied bounds for the independent parameters. \begin{table}
\begin{tabular}{l l c} \hline \hline  & **Parameter** & **Dimension** & **Value** \\ \hline 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 \\ Core-Inner winding distance & mm & 20 \\ \hline \multirow{4}{*}{Core} & Number of legs & \# & 3 \\  & Flux density limit in columns & T & 1.7 \\  & Filling Factor & \% & 90 \\  & Material Type & & M1H \\  & Material Price & \(\in\)/kg & 3.5 \\ \hline \multirow{4}{*}{Low Voltage} & Line Voltage & kV & 33 \\  & Phase Voltage & kV & 19.05 \\  & BIL & kV & 125 \\  & AC & kV & 50 \\  & Copper filling factor & \% & 60 \\  & Material and manufacturing price & \(\in\)/kg & 10 \\ \hline \multirow{4}{*}{High Voltage} & Line Voltage & kV & 120 \\  & Phase Voltage & kV & 69.36 \\  & BIL & kV & 550 \\  & AC & kV & 230 \\  & Copper filling factor & \% & 60 \\  & Material and manufacturing price & \(\in\)/kg & 8.5 \\ \hline \multirow{4}{*}{Regulating} & Regulating range & \% & \(\pm 10\) \\  & Insulation & & Fully insulated \\ \cline{1-1}  & Regulated winding & & High voltage \\ \cline{1-1}  & Filling factor & \% & 65 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: List of the optimization model input parameters. optimal solution of an analytical transformer model. It was shown in a previous article [27], that the usage of this geometric programming based solver guarantees that this method finds the global optima of the optimization task [14; 27]. The physical correctness of the results of the metaheuristic method were validated by FEM. However, the robustness and the precision of the used analytical formulas is about 5% lesser, compared to the FEM based calculations [14; 27]. This difference explains the relatively small difference between the optimal values of the two TOCs. This difference is relatively small and acceptable, lesser than 1%. Figure 7 illustrates the convergence of the algorithm. During the optimization, the NSGA-II algorithm was generated 100 individuals for every 100 generations. As it can be seen on Figure 7, after 80 th iteration the algorithm was found the optimal solution. However, the shapes of the two resulted transformer designs are very different. The main reason of this difference is the non-linearity of the transformer design optimization problem and the differences between the mathematical representation of the problem. These significant differences in the key-design parameters indicates that it is important to use the proposed, extended model to find more robust solutions. Figure 6: Results of the optimization process, namely optimal key-design parameters and magnetic flux distribution in window of optimized, 31.5 MVA, 132 kV/33 kV power transformer.