## 2 An Overview of Recent Advances in Design Optimization of Electromagnetic Devices
### Design Optimization Models in The Presence of Uncertainties  
Theoretically, the performance of an electromagnetic device can be improved by optimizing the optimization model of (1) or its single-objective form. However, this kind of optimal design (mathematical optimum) often features a lower performance than expected after the practical manufacturing process, because there are many inevitable material diversities and uncertainties involved. For example, assume that the optimal height of PMs is 4 mm for an SPMSM after an optimization. Considering a batch production of this motor (for examples, 1000 motors) with this design scheme, the practical height should be around 4 mm, like 4.05 mm and 3.97 mm, after measurement. It normally follows a normal distribution, as indicated by some research work [15]. Therefore, the practical performance of this motor will be different from the theoretically optimized value. There are obvious variations in batch production. To improve the manufacturing quality of the motors and other electromagnetic devices, some quality control methods, like six-sigma quality control, can be applied. However, this requires a lot of resources which may be a burden for some companies. Alternatively, this problem can be investigated in the early stage of product development through robust design optimization [6; 42; 43; 44; 45; 46]. Figure 3 illustrates a comparison of deterministic and robust optimums, and their performance variations in the presence of uncertainties. As shown, there are two optima, indicated as deterministic and robust optimum. For rated conditions, the deterministic one is better than the robust one. However, when a variation \(\Delta x\) occurs, the performance of the deterministic design shows a significant degradation, while some designs likely will not fulfill the illustrated constraint regarding the maximum objective value. This will be regarded as a defect in practical quality evaluation. For example, considering the design optimization of a PM motor, the temperature rise in the winding shall be less than 70 \({}^{\circ}\)C. Then, normally, the deterministic design will have an optimum with a temperature rise of the exact 70 \({}^{\circ}\)C or very close to it, like, e.g., 69.7 \({}^{\circ}\)C. If any uncertainties happen during the manufacturing or operation, the practical temperature rise in the PM may exceed this limit. This may demagnetize the PMs and fail/damage the whole device. By contrast, the robust optimum can ensure the required quality of the device in batch production [6; 47]. That is why the popularity of robust design optimization is increasing compared to the conventional deterministic design optimization in many research fields, including the design optimization of electromagnetic devices. There are three popular approaches for the robust design optimization of electromagnetic devices, namely Taguchi parameter design, worst-case design, and design for six-sigma (DFSS) [47; 48; 49; 50; 51; 52; 53; 54]. Figure 4 shows a block diagram for the Taguchi parameter design method. In this method, the parameters are classified as two groups, control factors and noise factors. Some techniques, such as orthogonal array and signal-to-noise ratios, are then employed to determine the best combination of control factor levels so that the variation of this response is minimized in the presence of noise factors [54]. The Taguchi  
Figure 3: A comparison between the deterministic optimum and robust optimum in the presence of uncertainties for a function \(f(x)\), in which \(\Delta x\) and \(\Delta f\) stand for the variations of the parameter and objective, respectively. parameter design has been widely employed in many applications due to its efficiency and effectiveness. However, there are several drawbacks, e.g., it cannot effectively deal with the constraints in optimization models. Worst-case design and DFSS are able to handle both constraints and optimization objectives in a generic optimization model.