# FEM Based Preliminary Design Optimization in Case of Large Power Transformers
## 2 Proposed Methodology
### Core Mass and No-Load Loss Calculation  
Similarly to the metaheuristic method in [27], in the case of a three phase three legged core, the core mass can be calculated by the following formulas:  
\[M_{c}=M_{leg}+M_{yoke}+M_{corner}, \tag{7}\]  
\[M_{corner}=R_{c}^{3}\cdot\lambda_{c}\cdot\pi\cdot\rho_{fe}\cdot\zeta, \tag{8}\]  
\[M_{column}=R_{c}^{2}\cdot\lambda_{c}\cdot\pi\cdot\rho_{fe}\cdot(EI_{TOP}+EI_{ BOT}+h_{in}), \tag{9}\]  
\[M_{yoke}=R_{c}^{2}\cdot\lambda_{c}\cdot\pi\cdot\rho_{fe}\cdot(4\cdot s+2\cdot p _{d}+6\cdot R_{c}), \tag{10}\]  
where \(M_{corner}\) is the mass of the corners of the core, \(M_{leg}\) is the mass of the leg, \(M_{yoke}\) is the mass of the yoke. \(\lambda_{c}\) is the filling factor of the core, it depends on the quality of the applied electrical steel and the construction technology. \(\zeta\) is a technology dependent factor for the core volume calculation, \(\rho_{fe}\) is the density of the electrical steel. \(EI_{TOP}\) and \(EI_{BOT}\) are the end insulation distances, between the bottom and the top of the yoke and the inner winding, \(p_{d}\) is the phase insulation \(h_{in}\) represents the height of the inner winding and \(s\) represents the width of the working window. The \(h_{in}\) winding is used as a reference height in the model as in the metaheuristic method based optimization [27]. The height of the outer and the regulating windings are taken into consideration by a simple multiplication of one factor. Hysteresis (\(P_{chyst}\)) and eddy current losses (\(P_{ceddy}\)) cause together the core-losses:  
\[P_{nll}=P_{ceddy}+P_{chyst}, \tag{11}\]In high quality electrical steels, the hysteresis and eddy current losses contribute about equally to the total loss. Eddy current loss, occurring on account of eddy currents produced due to induced voltages in laminations [3; 27; 44]. Where hysteresis loss is a function of the area of hysteresis loop:  
\[P_{ch}=k_{1}\cdot f\cdot t^{2}\cdot B_{p}^{n} \tag{12}\]  
where \(k_{1}\) is a material dependent empirical factor, \(B_{p}\) is the peak value of the flux density and \(n\) is the Steinmetz constant, which depends on the lamination and the operating flux [3]:  
\[P_{ceddy}=k_{2}\cdot f^{2}\cdot t^{2}\cdot B_{c}^{2} \tag{13}\]  
where \(k_{2}\) is a material dependent factor, \(f\) is the frequency, \(t\) is the lamination thickness and \(B_{c}\) is the inductance. These equations describes the theory of the loss generation in the magnetic core. However, this optimization model uses measurement results to determine the core losses. Every manufacturer provides a loss-curve from their steels, where the loss is a function of the induction in W/kg units. These practical formulas are the results of measurements, which are made by an Epstein-apparatus and they are take the hysteresis and eddy losses into account. The applied loss function is fitted to the applied electrical steel data (M1H [45]) and approximated by a polynomial expression [9; 14; 27; 46]:  
\[p_{nll}=a_{0}+\sum_{i,j}a_{i}\cdot B_{c}^{a_{i}}, \tag{14}\]  
where the fitted constants are \(a_{0}\), \(a_{i}\) and \(a_{j}\) and \(p_{nll}\) is the specific loss at the given magnetic flux density in \(W/kg\). The effect of the applied technology: lamination, joints, cooling ducts in the core and the corner losses are taken by the building-factor(\(f_{b}\)) into consideration, which typical value is between 1.1-1.4 [9]:  
\[P_{nll}=M_{c}\cdot f_{b}\cdot p_{nll}. \tag{15}\]  
The value of the applied building factor is 1.2 in the calculations.