## 2 An Overview of Recent Advances in Design Optimization of Electromagnetic Devices
### Design Optimization Models in The Presence of Uncertainties
Regarding the worst-case approach, its multi-objective optimization model can be defined as  
\[ \begin{array}{rl}\min:&\left\{f_{wj}(\mathbf{x})=\max_{\xi\in U(\xi)}f_{i}( \mathbf{x},\xi),\ i=1,2\ldots,p\right\}\\ \text{s.t.}&g_{wj}(\mathbf{x})=\max_{\xi\in U(\xi)}g_{i}(\mathbf{x},\xi)\leq 0,j=1,2,\ldots,m\\ &U(\xi)=\left\{\xi\in\mathbb{R}^{k}\Big{|}|\xi-\xi_{n}|\leq\Delta_{\xi}^{\xi} \right\}\end{array}  \tag{4}\]  
where \(\xi\) and \(\xi_{n}\) are vectors representing the actual and nominal values of noise factors, respectively, and \(U(\xi)\) represents the uncertainty range of these parameters, \(\Delta\xi\) is a vector for the limit of uncertainty range, \(\mathbb{R}\) stands for real coordinate space, \(k\) is dimension, subscript \(w\) in the objective, and constraints means the worst case. For the DFSS approach, its multi-objective optimization model can have the form as  
\[ \begin{array}{rl}\min:&\left\{F_{i}\Big{[}\mu_{f}(\mathbf{x}),\ \sigma_{f}(\mathbf{x})\Big{]},\ i=1,2,\ldots,p\right\}\\ \text{s.t.}&g_{j}\Big{[}\mu_{f}(\mathbf{x}),\ \sigma_{f}(\mathbf{x})\Big{]} \leq 0,\ \ j=1,2,\ldots,m\\ &\mathbf{x}_{i}+n\alpha_{\mathbf{x}}\leq\mu_{\mathbf{x}}\leq\mathbf{x}_{u}-n \alpha_{\mathbf{x}}\\ &\text{LSL}\leq\mu_{f}\pm n\sigma_{f}\leq\text{USL}\end{array}  \tag{5}\]  
where \(\mu\) and \(\sigma\) are the mean and standard deviation, respectively, \(\mu_{\mathbf{x}}\) and \(\sigma_{\mathbf{x}}\) are the mean and standard deviation, respectively, the mean and standard deviation of \(\mathbf{x}\), respectively, LSL and USL are the lower and upper specification limits, respectively. \(n\) is the sigma level, and it is defined as 6 in many applications. The value of \(n\) can be equivalent to a probability of a normal distribution, as shown in Figure 5. Six-sigma level (\(n=6\)) has been widely adopted in industry as it can provide good reliability for both short-term quality control (equivalent to statistic values) and long-term quality control (with considerations of uncertainties by shifting mean with 1.5 \(\sigma\)). It is equivalent to 0.002 (or a per cent of pass 99.9999998%) for short-term quality control and 3.4 defects per million opportunities (DPMO) for long-term quality control [6; 14; 48]. Figure 4: Block diagram of a product/process design in Taguchi method considering control factors (design variables) and noise factors (uncertainties). As an example, the robust optimization model of the investigated SMES with deterministic optimization model (2) can be defined as  
\[\min: \left\{ \begin{array}{l}f_{1}(\mathbf{x})=\mu(B_{\textit{stray}})+ \sigma(B_{\textit{stray}})\\ f_{2}(\mathbf{x})=\mu(|E-180|)+\sigma(|E-180|)\\ \end{array} \right.\] (6) s.t. \[\left\{ \begin{array}{l}g_{1}(\mathbf{x})=\mu\Big{(}|B_{\textit{ max}}|-\min(\frac{54-|f_{1}|}{6.4})\Big{)}+6\sigma\Big{(}|B_{\textit{max}}|-\min( \frac{54-|f_{1}|}{6.4})\Big{)}\leq 0\\ g_{2}(\mathbf{x})=\mu\Big{(}R_{1}+\frac{d_{1}}{2}+\frac{d_{2}}{2}-R_{2}\Big{)} +6\sigma\Big{(}R_{1}+\frac{d_{1}}{2}+\frac{d_{2}}{2}-R_{2}\Big{)}<0\\ \end{array} \right.\] \[\mathbf{x}_{l}+6\sigma_{\mathbf{x}}\leq\mu_{\mathbf{x}}\leq\mathbf{ x}_{u}-6\sigma_{\mathbf{x}}\]  
where \(\mu\) and \(\sigma\) are the mean and standard deviation, respectively. In the implementation, each design optimization in \(\mathbf{x}\), like \(R_{1}\) (radius of the inner solenoid) can be assumed to follow a normal distribution with two parameters, a mean (the nominal value of a design) and a standard deviation (one third of its manufacturing tolerance) [6; 14]. In the implementation of the robust optimization, the evaluation processes of nominal performance quantities, like \(E\), \(B_{\textit{max}}\), and \(B_{\textit{stray}}\) are same as those applied for solving (2), like FEA. The main difference between (2) and (6) is that some extra information (\(\mu\) and \(\sigma\)) is needed in (6). To obtain the required data sets, Monte Carlo method can be applied with four main steps. First, assume that each parameter in \(\mathbf{x}\) follows a normal distribution.