## 2 An Overview of Recent Advances in Design Optimization of Electromagnetic Devices
### Optimization Methods
#### 2.3.3 Multilevel and Space Reduction Optimization Strategies  
Surrogate models can be used to improve the optimization efficiency or reduce the computation cost of low-dimensional electromagnetic design problems, for example, in case a total of five dimensions is not exceeded. Its efficiency is not good for high-dimensional optimization problems, such as the optimization of SMES with 8 parameters (Figure 1b) and the optimization of a PM motor with 10 parameters and FEM (Figure 1f). Therefore, appropriate optimization strategies should be considered. For this purpose, three optimization strategies, namely multilevel optimization, space reduction optimization, and sequential optimization strategies have been proposed in our previous work for both deterministic/robust and single- or multi-objective optimization problems of electromagnetic devices [6; 14; 47; 97; 98; 100].  
For the multilevel optimization strategy, a high-dimensional optimization problem is converted into several low-dimensional optimization problems by using sensitivity analysis techniques, such as local sensitivity and analysis of variance. Considering the optimization of SMES with 8 parameters, a three-level structure can be defined as: Level 1 (3 parameters of \([R_{1},h_{1},d_{1}]\)), Level 2 (2 parameters of \([J_{1},J_{2}]\)), or Level 3 (3 parameters of \([R_{2},\)\(h_{2},d_{2}]\)). To implement the optimization, a sequential optimization process, Level 1-Level 2-Level 3, will be conducted, as shown in Figure 6. This process should be repeated until a convergence criterion is met (for example, the relative error of the objectives between two iterations are no more than a given value \(\varepsilon\) like 1%). This kind of optimization strategy will decrease the computation cost, as the optimization of each level is a low-dimensional problem which can be done effectively by a surrogate model. For example, if each factor needs 5 levels in a DoE technique, then Level 1 requires 125 points, Level 2 requires 25 points, and Level 3 requires 125 FEM points, resulting in a total of 275 points for one loop of the optimization. If three optimization loops are needed, a total of 825 points are required for multilevel optimization. This is much smaller than the samples (125 \(\times\) 125 \(\times\) 25 = 390,625) required by developing a model for 8 input parameters.  
Figure 6: A three-level optimization flowchart for SEMS, in which 8 optimization parameters are allocated to three subspaces X1, X2, and X3.  
For the sequential optimization strategy, it uses a sophisticated strategy to sample the most important variants in a small subspace (instead of the initial big design space) by using some space reduction and moving techniques. According to the design examples on SMES and PM motors with soft magnetic composite cores, it can be found that the computation cost of FEA has been reduced significantly by using these strategies. These and improved optimization strategies (like new and improved sensitivity analysis methods) have been successfully applied to the design optimization of other PM motors [35; 38; 101].