## 2 An Overview of Recent Advances in Design Optimization of Electromagnetic Devices
### Deterministic Design Optimization
Please note that there are no analytical expressions to show the relationship between design parameters and performance quantities of many electromagnetic devices, for example, the relationship between the parameters \(\mathbf{x}\) and \(E\) in (2). Thus, finite element analysis (FEA) method is widely employed to calculate the magnetic field distribution. For example, Figure 2 shows a design scheme of SMES and its magnetic field distribution by using FEA method (can be done in several software like \(ANSYS\)). Due to the symmetry, only the part above x-axis is given. As shown, the maximal magnetic field (indicated as MX) is around 4.27 T. Other performance measures like the energy can be obtained based on the results for the magnetic field. If a parameter, like radius of the inner solenoid, is changed, the corresponding magnetic field and the values of \(E\), \(B_{max}\), and \(B_{stray}\) should vary as well. Thus, FEA and model link the design parameters and performance quantities. Figure 2: The magnetic field distribution for a design scheme of SMES with current density of 20 MA/m\({}^{2}\), (**a**) the design scheme (symmetric about the x-axis), and (**b**) the corresponding magnetic field obtained from \(ANSYS\). Figure 1c shows a prototype of a high-frequency transformer with Litz-wire windings and a magnetic core made of nanocrystalline films. High-frequency transformers have many potential and promising applications, including in the power systems and wireless power transmission systems [26; 27; 28]. For the design optimization of a high-frequency transformer, there are many objectives, such as minimizing the loss and volume. Regarding the design optimization parameters, dimensions (as shown in Figure 1d) and core materials (like nanocrystalline or amorphous) can be considered [29; 30; 31; 32]. Detailed optimization models can be referred to these works as well. The third example is a PM motor. PM motors have been widely used in industry and transportation, such as hybrid electric vehicles [33; 34; 35; 36; 37]. The design optimization of electrical machines, including MP motors, is very challenging in many situations due to the consideration of multi-physics analysis. Figure 1e shows the topology of an outer-rotor SPMSM. This kind of machine has been used in many applications as well, like in EVs. In our previous work, it is designed as an in-wheel motor for an EV to achieve four-wheel-drive performance [38; 39]. This motor has many parameters to optimize, like the dimensions shown in Figure 1f. In addition, the material of PMs and winding parameters (like the number of turns and winding diameter) can be investigated. Popular optimization objectives are maximizing the output power, average torque, and efficiency, and minimizing the cost and torque ripple. Furthermore, as this motor is used as the in-wheel motor, the operating condition should be considered. There are two major challenges for in-wheel-motors, the unsprung weight and cooling [40; 41]. The unsprung mass is the weight of all components that are not supported by the suspension, including the wheels with motors, tires, and brakes. As the EV travels up and down over various bumps, potholes, and debris, excessive unsprung weight would cause serious vibration. The weight of in-wheel motors must be minimized (e.g., through topology optimization, will be discussed in Section 2.3.5) for smooth drive performance and better vehicle reliability and durability. Cooling is a critical issue for safe operation of high torque density in-wheel motors due to the limited and sealed space in the wheels. Therefore, accurate multi-physics analysis is required, including the electromagnetic, thermal, and mechanical analysis. Based on these considerations, an optimization model of this motor can be defined as  
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