## 2 An Overview of Recent Advances in Design Optimization of Electromagnetic Devices
### Design Optimization Models in The Presence of Uncertainties
Second, generate a large amount of samples, like 10,000 samples (means 10,000 design schemes of SMES), from the distributions. Third, evaluate the SMES's performance quantities, such as \(E\) and \(B_{\textit{max}}\), for these 10,000 designs. Fourth, estimate the mean and standard deviation of these performance quantities. Then optimization algorithms can be applied to find the optimum solutions for this model. There are two main differences between the worst-case approach and DFSS approach. First, the worst-case multi-objective model is a minimax optimization problem. It uses the worst motor performance of a design under uncertainties as a measure of robustness. DFSS uses the sigma level as the measure of robustness. Second, the probability distribution functions of the uncertainty parameters are required for DFSS, while the worst-case approach only needs intervals for the uncertain parameters. In general, the computation cost of worst-case is higher than that of DFSS, as it is a minimax optimization problem. Moreover, the worst-case approach is typically more affected by modeling errors, as this quantity is estimated based on a single numerical result, while DFSS measures are determined by evaluating a significant number of design variations. In the case of hybrid uncertainties, the objective functions and constraints have the characteristics of both random and interval uncertainties. Both worst-case and DFSS should  
Figure 5: Probability density function of the standard normal distribution for short-term quality control and long-term quality control (with a 1.5 \(\sigma\) shift from the mean), with probabilities for three sigma levels. be considered in the optimization model, and the computation cost is huge. This kind of robust optimization has been investigated for a PM motor in our previous work. Polynomial chaos Chebyshev interval (PCCI) method was employed to improve its optimization efficiency [49]. In the context of Industry 4.0, robust design optimization has been an active and promising research topic in many fields recently, including electrical engineering, mechanical engineering, and civil engineering. A main driving force behind this is that robust design optimization is able to include the manufacturing data and product quality into the design problem. There are many research activities about the robust design analysis and optimization of different types of electromagnetic devices, such as SMES [21; 50; 51; 52; 53], and several types of electrical machines including high-temperature superconducting linear synchronous motor [55], and synchronous reluctance motors [56], and PM motors [57; 58; 59; 60; 61; 62; 63; 64; 65]. It is observed that there are more discussions on robust design optimization of PM motors than other types of electrical machines, due to the fact that there are many uncertainties for the PMs. These uncertainties will affect the performance of the PM motors and their reliability, e.g., considering potential demagnetization. This is also challenging for the mass-production of the PM motors. Recently, a special section on robust design and analysis of electric machines and drives was published in the IEEE Transactions on Energy Conversion. Both the robust design analysis and optimization of the motors and control systems are investigated by many authors and, correspondingly, a significant number of papers was published [66]. The outcomes will lay a solid foundation for the development of high-reliability electrical drive systems for many challenging applications, such as EVs and wind power generation.