# Design and Multi-Objective Optimization of Fiber-Reinforced Polymer Composite Flywheel Rotors
## 4 Optimization  
### Formulation and Characterization of the Optimization Problem  
The goal of the rotor optimization is to find optimal rotor designs which yield a higher TSE, lower costs and a faster production. These conflicting objectives are considered separately, thus implying the need for multi-objective optimization. In addition to these objectives, there are also three constraints that represent the three load cases for which the rotor must be designed. To formulate the optimization problem, the optimization variables as well as the design variables are defined. Design variables are externally specified variables. This study primarily considers flywheels for transportation applications such as in cars or trains where the construction volume is limited. The height and the outer radius of the flywheel are therefore design variables. The inner radius depends on the hub and shaft geometry which is typically determined by the used electrical machine and bearing system. FESS rotors are either built with an integrated or external electrical machine. While the external electrical machine uses a shaft, the integrated topology requires a central stator unit enclosed by the rotor. A major drawback of the external electrical machine is the necessary hub. High rotational velocities cause increasing radial forces in the rotor. Hence, the rotor will expand faster than the shaft. The hub must ensure a stable connection with the rim while compensating for the different growth of rim and shaft. The consequence is a considerable design challenge: if the hub is too rigid, it will detach under high radial stresses. Conversely, a too flexible hub does not allow transmitting the high forces from the rim to the shaft. An integrated topology avoids the aforementioned problems due to its different assembly. However, this advantage is associated with significant disadvantages. The air gap of (possible) magnetic bearings and of the electrical machine rises because of occurring strains with increasing rotational speed. Since an enlarged air gap requires increased magnetic flux to ensure the same power, energy efficiency is reduced. Further, the motor/generator is almost in direct contact with the composite materials. Due to the operation in a vacuum enclosure (to reduce substantial air drag losses), heat from electrical systems can only be dissipated by radiation. Thermal stresses on surrounding composite materials impose limitations because they reduce strength and may accelerate material creep processes. Since these disadvantages are difficult to overcome, the presented study is based on an external electrical machine configuration. Based on the above consideration, the rotor inner radius is considered to be a design variable. The optimization tries to increase the TSE by designing a multi-rim rotor. The number of rims highly increases the optimization time. Therefore, a maximum number of rims \(N_{\text{rim,max}}\) is also externally specified limiting the computation time. Calculating production times and costs requires information about the used fabrication equipment. Related information represents the last design variable. Results of the presented optimization tool are therefore only optimal for the selected fabrication facility configuration. Even though this restricts the validity of the results to a specific configuration, this procedure also highly simplifies the optimization problem. In practice, companies usually have machines in place that cannot be easily replaced due to high acquisition costs, and hence, the optimization tool and design variables would have to be adjusted accordingly. Optimization variables are optimized and returned by the algorithm. In this study, the optimization variables are the maximum angular velocity \(\omega_{\text{max}}\), the rim number \(N_{\text{rim}}\) as well as the rim thickness \(t_{i}\) (i.e., the radial rotor dimension), the radial displacement due to press fitting \(\delta_{i}\)and the material choice \(x_{\text{material},i}\) for each rim \(i\).