## 3 Methodology and Theoretical Background
### Designing LSHT Machines via Multi-Objective Optimization  
The design optimization of electrical machines involves the trade-off of multiple objectives. A multi-objective optimization framework can be employed to identify the best solutions within a complex design space. Typically, a Pareto front is used to visualize the best trade-off solutions for the optimization problems. The initial stage of the optimization process involves the parametrization of the machine model to enable its automation. In this study, the model's parameters are defined based on geometric data and dimensions, with constraints applied to ensure the feasibility of the proposed designs. Typical performance objectives are considered in this paper, including maximizing the electromagnetic torque, efficiency, and power factor, while minimizing the torque ripple. Additionally, factors such as cost reduction, material constraints, or restricted volume may be also considered. Each objective is mathematically formulated to facilitate quantitative evaluation during the optimization process. Design constraints, including limitations on size, material properties, or manufacturing feasibility, are also explicitly defined. The objectives and constraints defined for this analysis are summarized in Table 1 for all considered LSHT topologies. The considered current density range falls within the typical recommendations found in the literature for passive-cooled machines [59], though it can be adjusted if a more advanced thermal solution is applied. Phase voltage is selected following a typical available low-voltage source of 380 V\({}_{\text{rms}}\) and a star connection. The volume constraint was chosen to match the dimensions of a standard 12-pole 45-kW SCIM. Rated losses were estimated through a preliminary thermal evaluation considering the maximum heat dissipation of \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Symbol** & **Units** & **PMSM** & **SynRM** & **SCIM** \\ \hline Mean & T\({}_{\text{mean}}\) & Nm & maximize & maximize & maximize \\ Torque & T\({}_{\text{rp}}\) & \% & minimize & minimize & minimize \\ Ripple & \({}_{\text{r}}\) & \% & maximize & maximize & maximize \\ Efficiency & \(\eta\) & \% & maximize & maximize & maximize \\ Power & \(\cos\phi\) & - & maximize & maximize & maximize \\ Factor & \({}_{\text{rms}}\) & A/mm\({}^{2}\) & \(3-5\) & \(3-5\) & \(3-5\) \\ Current & \({}_{\text{rms}}\) & A/mm\({}^{2}\) & \(3-5\) & \(3-5\) & \(3-5\) \\ Density & \({}_{\text{Phase}}\) & V\({}_{\text{ph, rms}}\) & V & \(=220\) & \(=220\) & \(=220\) \\ Voltage & - & m\({}^{3}\) & \(=0.05\) & \(=0.05\) & \(=0.05\) \\ Rated losses & - & W & \(\cong\)2500 & \(\cong\)2500 & \(\cong\)2500 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Objectives and constraints defined for the optimization process. a 0.05 m\({}^{3}\) case in passive-cooling scenarios, which was also cross-referenced with catalog data for off-the-shelf low-speed motors. To solve the optimization problem, an appropriate multi-objective optimization algorithm must be selected. Popular choices include genetic algorithms (GAs), particle swarm optimization (PSO), and the non-dominated sorting genetic algorithm II (NSGA-II). The algorithm selection is guided by the specific problem characteristics and the desired balance between computational efficiency and solution quality. However, in machine learning and optimization, it can be concluded from the well-known no-free-lunch-theorem that for certain types of problems the averaged computational cost for finding a solution over all problems of that type is the same for any optimization technique [60]. For this analysis, a multi-objective genetic algorithm (MOGA) is selected, also adopted and preferred in other related works [61]. The number of initial samples is set to correctly map the input parameters, and the number of designs evaluated per iteration is selected as a fraction of this initial population.