## 4 Selection of the Objective Function and Optimization Parameters
Since the same magnetic flux flows through the stator housing and the rotor sleeve, their cross sections are assumed to be equal. For this reason, a change in the thickness of the stator housing also leads not only to a change in the outer diameter of the stator lamination, but also to a change in the outer diameter of the rotor sleeve. Figure 6: Parameters of the SHG geometry. (**a**) Stator core parameters; (**b**) Rotor core parameters; (**c**) Armature winding parameters; (**d**) Parameters in the axial plane. The net fill factor of the armature slot with copper is assumed to be 0.8. The width \(w_{x}\) and the height \(w_{y}\) of the rectangular wire of the armature winding, which are taken into account when estimating its DC losses and eddy current losses, include the thicknesses of conductors and insulation [5]:  
\[b_{p}=w_{x}+a_{x};h_{p}=2\cdot(w_{y}+\Delta w)\cdot N_{sec}+a_{y}, \tag{7}\]  
where \(a_{x}=1.51\) mm, \(a_{y}=1.8\) mm, \(\Delta w=0.31\) mm take into account the thicknesses of the rectangular busbar insulation and the thickness of the slot insulation. When calculating losses in the excitation winding, eddy current losses are not taken into account. At the maximum torque, the angle between the center of the rotor tooth and the stator current vector (\"current angle\") is taken unchanged and equal to 0.1 electric radians as shown in Table 1. The current angle at maximum speed is also an optimization parameter as shown in Table 2.  
Figure 7 illustrates the order of calculations when calling the objective function \(F\) (6). The optimization procedure 'fminsearch'of MATLAB software, which implements the simplex gradientless Nelder-Mead method [35], is well known, and its details are described \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Characteristics**} & \multicolumn{1}{c}{**Values**} \\ \hline Length of all laminated stacks plus the axial clearances between them for & & \\ installing excitation coils \(L\), mm & & 150 \\ \hline Outer diameter of the stator housing, mm & & 370 \\ \hline Axial clearance between excitation coils and a rotor stack \(\Delta_{a}\), mm & & 15 \\ \hline Radial clearance between excitation coils and the rotor sleeve \(\Delta_{r}\), mm & & 12 \\ \hline Diameter of the rotor shaft, mm & & 40 \\ \hline Back iron width of stator lamination \(H_{ stator}\), mm & & 12 \\ \hline Back iron width of rotor lamination \(H_{ rotor}\), mm & & 9 \\ \hline Thickness of the stator wedge, \(\epsilon_{2}\), mm & & 1 \\ \hline Stator tooth tip height \(\epsilon_{1}\), mm & & 1 \\ \hline Current angle at 750 rpm, electrical radian & & 0.1 \\ \hline Grade of electrical steel & & 2412 \\ \hline Thickness of electrical steel, mm & & 0.35 \\ \hline \end{tabular}
\end{table} 
Table 1: Main unchangeable parameters of the generator. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Characteristics**} & \multicolumn{1}{c}{**Initial Design, \(x0\)**} & \multicolumn{1}{c}{**Optimized Design, \(x\)**} \\ \hline Housing thickness \(h\), mm & 15 & 11.07 \\ \hline Total stator stacks length \(L_{ stator}\), mm & 130 & 128.3 \\ \hline Stator slot depth \(h_{p}\), mm & 20 & 27.9 \\ \hline Stator slot width \(b_{p}\), mm & 5 & 6.37 \\ \hline Airgap width \(\delta\), mm & 2 & 0.91 \\ \hline Rotor slot thickness, \(\alpha_{1}\) & 0.5-\(t_{2}\)* & 0.481-\(t_{2}\)* \\ \hline Rotor slot thickness, \(\alpha_{2}\) & 0.6-\(t_{2}\)* & 0.689-\(t_{2}\)* \\ \hline Angle of field weakening at 3450 rpm, & & 0.6 & 0.93 \\ electrical radian & & & \\ \hline Current ratio ** @ 750 rpm & 8 & 14.54 \\ \hline Current ratio ** @ 3450 rpm & 8 & 9.76 \\ \hline \end{tabular} Notes: * \(t_{2}=360/6=60^{\circ}\) (mechanical) is the rotor tooth pitch; ** the current ratio is the ratio between the current in a layer of the armature winding and the excitation current.  
\end{table} 
Table 2: Initial and optimized vectors of the variable optimization parameters.