# FEM Based Preliminary Design Optimization in Case of Large Power Transformers
## 2 Proposed Methodology
### Transformer Model for the Optimization
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Quantity** & **Dimension** & **Variable** \\ \hline Independent variables & & \\ \hline Core diameter & mm & \(D_{\text{c}}\) \\ Flux density in the core & T & \(B\) \\ Main insulation distance & mm & \(g\) \\ Current density in the secondary coil & A/mm\({}^{2}\) & \(j_{\text{s}}\) \\ Current density in the primary coil & A/mm\({}^{2}\) & \(j_{\text{p}}\) \\ Current density in the regulating coil & A/mm\({}^{2}\) & \(j_{\text{r}}\) \\ Height of the secondary winding & mm & \(h_{\text{s}}\) \\ \hline Dependent parameters (Analytical) & & \\ \hline Width of the working window & mm & \(s\) \\ Core mass & t & \(M_{\text{c}}\) \\ Radial thickness of secondary winding & mm & \(t_{\text{s}}\) \\ Mean radius of secondary winding & mm & \(r_{\text{s}}\) \\ Radial thickness of primary winding & mm & \(t_{\text{p}}\) \\ Mean radius of primary winding & mm & \(r_{\text{p}}\) \\ Radial thickness of regulating winding & mm & \(t_{\text{r}}\) \\ Mean radius of regulating winding & mm & \(r_{\text{r}}\) \\ No Load Loss & kW & \(P_{\text{all}}\) \\ \hline Dependent parameters (FEM) & & \\ \hline 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 & \(B_{ap}\) \\ \hline Dependent parameters (GP sub-problem) & & \\ \hline 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 & \(h^{*}\) \\ Optimal conductor width & mm & \(w^{*}\) \\ \hline Dependent parameters (Complex) & & \\ \hline Load Loss & kW & \(P_{\text{ll}}\) \\ Total Cost of Ownership & € & \(TOC\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the optimized active part model. and optimized by the applied evolutionary algorithm (NSGA-II). The calculation of the dependent parameters are made in every iteration step, by the redefined evaluator function of the optimization framework (Artap [24]). This calculation consist of three main calculation steps: the solution of the analytical model, the FEM calculation and the embedded geometric programming based model. The structure of the implemented evaluator function is shown in Algorithm 1. The following subsections show the applied optimization framework [24] and explain the details of the different calculation steps (Figure 4). ```
functionEvaluator(p) \(\triangleright\) p means the independent design parameters, which generated by NSGA-II within the given search space
2: Evaluates the analytical expressions \(\triangleright\) determine the main geometrical design parameters
4:if The analytical solution is not feasible then returnTOC = inf
6: endif
8: Runs Agros2D - FEM calculation - Determine \(SCI\) from the magnetic energy
10: Determine \(B_{axp}\), \(B_{radp}\) and \(B_{axs}\), \(B_{rads}\) values
11:if Check SCI is False then returnTOC = inf
12:endif
13: Runs the GP based winding model calculates \(h^{*}\), \(w^{*}\) for both of the windings
14: calculate the load losses, TOC returnTOC
15:endfunction
```  
**Algorithm 1** Transformer Model Evaluator  
Figure 4: Structure of the realized methodology in the framework.