# Design Optimization of the Magnet-Free Synchronous Homopolar Motor of a Subway Train
## 6 SHM Optimization Parameters and Procedure  
Optimization of the motor parameters is performed using Nelder-Mead method. The Nelder-Mead algorithm used for the optimization is well known [30] and is included in the basic MATLAB software package (function \"fminsearch\"). This method is a one-criterion optimization method that uses only one objective function. The conventional approach for implementing multi-criteria optimization using a one-criterion method is to include in the optimized function all optimization objectives as a sum or product with certain weight coefficients [22]. The cost function is composed taking into account the following targets listed in descending order of importance:  
1. Minimization of average losses <\(P_{\text{losses}}\)>;
2. Minimization of the maximum stator current magnitude during the entire operation cycle taking into account available voltage limit;
3. Minimization of the maximum symmetrized torque ripple along the entire operation cycle max (_TRsym_);
4. Minimization of the maximum non-symmetrized torque ripple along the entire operation cycle max(\(TR\)). A non-symmetrized torque ripple is produced by a single SRSC. A symmetrized torque ripple is produced by a SHM as a whole. Details of \(TR\) and _TRsym_ are given in [19]. According to the targets of optimization the cost function can be written as their product. The weights are the powers to which the multipliers are raised:  
\[ \begin{split} F&=<P_{loss,el}>\max(I_{i})^{0.7}\max( TRsym_{i})^{0.025}\max(TR_{i})^{0.01}f\Big{(}\frac{\max(B_{i})}{1.6[T]}\Big{)},\\ f(x)&=\left\{\begin{array}{l}1,x<1.6[T]\\ x^{3},\textit{otherwise}\end{array}\right.,\end{split}  \tag{9}\]  
where <\(P_{loss,el}\)> are the electric losses; \(I_{i}\) is the value of current in the \(i\)th operation point; _TRsym_, is the value of the symmetrized torque ripple in the \(i\)th operation point, \(R\) is the value of the non-symmetrized torque ripple in the \(i\)th operation point, \(B_{i}\) the magnetic flux density in the housing and the sleeve. The one-criterion Nelder-Mead method is applied in this study to optimize the SHM design. The Nelder-Mead method belongs to unconstrained optimization methods. The optimization constraints could be specified simply by assigning infinite values to the objective function when the constraint conditions are not met. However, this would lead to a rapid decrease in the volume of the simplex. For this reason, objective function (9) is modified by using the'soft constraints' with the penalty growing rapidly in the forbidden area (if \(B_{i}\geq 1.6\) T). Optimization was performed neglecting the discrete nature of some parameters. The number of turns in the section of power winding \(N_{sec}\) is assumed to be a real number; it is not checked that the width and height was selected from the standard [31]. The number of turns is selected according to \(V_{DC\ metal}=\max(V_{i}/k_{i})\)[23]. The number of parallel branches is assumed to be equal to 4. Figure 5 shows the geometric parameters of the SHM. The parameters fixed during the optimization and varied ones are given in Tables 2 and 3. Additionally, the areas of the cross-sections of the stator housing and the rotor sleeve are assumed to be equal since they transfer the same excitation flux. The shaft is assumed to be nonmagnetic or to have poor magnetic conductivity. All of these determine other geometric parameters of the rotor and stator lamination including the inner rotor lamination radius that is the radius of the rotor sleeve. For example, the outer radius of stator lamination is determined by the outer housing radius and the housing thickness.