## 2 An Overview of Recent Advances in Design Optimization of Electromagnetic Devices
### Deterministic Design Optimization  
A generic optimization model of the following form can be defined to the multi-objective optimization of electromagnetic devices. \[ \begin{array}{ll}\min:&\{f_{i}(\mathbf{x}),\ i=1,2,\ldots,p\}\\ \text{s.t.}&g_{j}(\mathbf{x})\leq 0,j=1,2,\ldots,m\\ &\mathbf{x}_{l}\leq\mathbf{x}\leq\mathbf{x}_{u}\end{array}  \tag{1}\]where \(p\) and \(m\) are the numbers of objectives, \(f_{i}(\mathbf{x})\), and constraints, \(g_{j}(\mathbf{x})\), respectively. \(\mathbf{x}\) is a vector of design parameters, and \(\mathbf{x}_{l}\) and \(\mathbf{x}_{u}\) are vectors of the lower and upper boundaries of \(\mathbf{x}\). This model will be simplified as a single-objective problem if \(p\) is equal to 1. The detailed forms of \(\mathbf{x}\), \(f(\mathbf{x})\), and \(g(\mathbf{x})\) depend on the specific type and application of an electromagnetic device. Figure 1 illustrates three popular applications. They are a SMES, a high-frequency transformer, and a surface-mounted permanent magnet synchronous motor (SPMSM). Figure 1: Design optimization illustrations of several electromagnetic devices, (**a**) a topology of superconducting magnetic energy storage (SMES) with two solenoids, (**b**) SMES design structure and optimization parameters, (**c**) a high-frequency transformer with Litz-wire windings, (**d**) design structure and optimization parameters for the high-frequency transformer, (**e**) a topology of an outer-rotor surface-mounted permanent magnet synchronous motor (SPMSM), (**f**) SPMSM design structure and optimization parameters. SMES is a grid-enabling device for power systems as it can store and discharge a large amount of electricity/power almost instantaneous. SMES stores the power in terms of magnetic energy by its superconducting coils. Nowadays, high penetration of renewable energy sources, like wind and solar, are integrated into the power system worldwide. They will affect the power quality and stability due to their intermittency. This is one of the main challenges of integrating renewable energy sources in the smart grid. SMES is a promising technology to address this challenge [19; 20; 21]. The common shapes of superconducting coils are solenoid (Figure 1a) or toroidal. The solenoid type is simple, robust, and cost-effective. For the design optimization of solenoid-type SMES, there are several parameters, such as the dimensions of the solenoids and currents. Figure 1b illustrates an optimization structure of an SMES based on a benchmark problem (TEAM problem 22) in ICS. For this example, eight parameters, \(\mathbf{x}=[R_{1},h_{1},d_{1},J_{1},\)\(R_{2},h_{2},d_{2},J_{2}]\), where \((R,h,d)\) and \(J\) are the dimension and current density of the solenoid, respectively, subscript 1 and 2 mean the inner solenoid and outer solenoid, respectively. These parameters will be optimized to minimize the mean stray fields (\(B_{stray}\)) while keeping the total stored energy (\(E\)) close to 180 MJ. The optimization model can be defined as  
\[ \begin{array}{rl}\min:&\left\{\begin{array}{l}f_{1}(\mathbf{x})=B_{stray} \\ f_{2}(\mathbf{x})=|E-180|\\ \end{array}\right.\\ \text{s.t.}&\left\{\begin{array}{l}g_{1}(\mathbf{x})=\left|B_{max}\right|- \min\left(\frac{54-|J_{1}|}{6.4}\right)\leq 0\\ \par g_{2}(\mathbf{x})=R_{1}+\frac{d_{1}}{2}+\frac{d_{2}}{2}-R_{2}<0\\ \par\mathbf{x}_{l}\leq\mathbf{x}\leq\mathbf{x}_{u}\end{array}\right.\end{array}  \tag{2}\]  
In the model, \(B_{stray}\) is estimated by the magnetic fields on 21 points with the same space along lines \(a\) and \(b\), as shown in Figure 1b. The first constraint is related to the superconductivity of the SMES, where the maximal magnetic field (\(B_{max}\)) is limited to a value determined by the current density of two coils. This optimization problem can be converted to a single-objective problem (minimizing the mean stray fields only) by considering the requirement of stored energy through a constraint [22; 23; 24; 25].