## 3 Model Complexity and Space Reduction Techniques for System-Level Optimization
### Surrogate Modeling
#### 3.1.1 Kriging Model  
The Kriging model is an interpolation method based on the Gaussian process that employs a stochastic approach to identify the relationship between optimization variables and objective functions using known data, most often extracted from the FEM simulations. A key feature of the Kriging model is its ability to represent the correlated error and the predictable variance in each simulation [32; 35]. Figure 6 illustrates the approximation of the input-output relationship using the Kriging model in comparison to the actual response. In addition, it visualizes the correlated errors (interpolation variance) previously discussed. In [30; 36], the authors employ the Kriging model, coupled with the NSGA-II algorithm, to establish a correlation between variables and optimization parameters to optimize the motor. In [36], they aim to mitigate the mutual coupling between two adjacent phases due to the presence of an even number of phases, which creates asymmetry in the current. To address this issue, they optimize the stator yoke and the controller through the turn-on angle, \(\theta_{on}\), and conduction width of the current pulse. While the Kriging model introduces slight deviations compared to the FEM results, it significantly reduces the computational time. The FEM would require on the order of 100,000 simulations; however, the surrogate-based approach reduces this to just 1125 simulations--about 1% of the original number,achieving a ~99% reduction. This translates to a near 89\(\times\) speedup in the runtime, cutting the total computational time from approximately 1389 h to just 15.6 h, saving around 1373 h in the process. This reduction in computational time highlights the significance of this method. The experimental results, alongside the simulation data, confirmed the effectiveness of the approach, showing a reduced torque ripple and improved current symmetry in the six-phase SRM design. The authors of [37] discuss the design of an SRM through system-level optimization, specifically Multi-Level Optimization (MLO). A regression-based sensitivity analysis is employed to classify parameters based on their significance, while the motor is modeled using the Kriging method. Shape optimization is applied via a Bezier curve to modify the rotor pole shape and enhance performance, the control optimization is conducted through the turn-on, \(\theta_{on}\), and turn-off, \(\theta_{off}\), angles, and, as an optimization algorithm, the NSGA-II is utilized. Validation is performed using the ANSYS Suite for the FEM, leading to minimal errors between the Kriging and FEM. Although losses slightly increase, the average torque and torque ripple are marginally optimized. Similarly, in [38], a local sensitivity analysis, along with the Kriging model, is used in conjunction with the NSGA-II algorithm for optimization, with the control parameters being the turn-on, \(\theta_{on}\), and turn-off, \(\theta_{off}\), angles. The model data are validated once again, and it is concluded that the errors between the Kriging and FEM are small, with a maximum error of 3.17% for the torque ripple, which is considered highly accurate and reinforces the significance of the Kriging model. Both the FEM and surrogate models are experimentally validated using a physical prototype of the SSRM drive system, where the measured data align closely with the simulations, and the surrogate predictions show errors below 1%. However, the entire analysis reduced the initial torque ripple from 93.64% to 47.84%, which remains relatively high, and, additionally, the overall losses increased. Despite these trade-offs, the method proves highly efficient, reducing the number of FEM evaluations by approximately 99.995% and achieving a speedup factor of nearly 20,300\(\times\) compared to a conventional FEM-based optimization approach. In [39], the authors focus on the design of a 12/10 SRM for EV applications, utilizing driving cycle data and the Dimension Reduction Optimization Method (DROM) to reduce the solution search space.