## 2 Robust Design Optimization Methods and Challenges
### Robustness Methods and Measures  
[PERSON] published the first and widely used methodology [60] that can systematically deal with the process uncertainties and tolerances. This methodology was originally designed for design processes, not for the design optimization of electrical machines. It uses the orthogonal design array  
Figure 2: Coupled FEM model (**a**) and thermal camera image (**b**) of an induction brazing process design. The other three images show the possible errors during the brazing process: (**c**) dissolution, (**d**) porosity, and (**e**) gathering. together with the method of the design of experiments [61] to select the optimal design [46; 62; 63; 64]. This methodology is used for a wide range of applications, but it is possible to use the signal-to-noise ratio in a poor manner [65; 66; 67; 68; 62]. However, it is not easy to apply the traditional Taguchi method to design optimization problems [62]. This methodology was designed for discrete parameters, and it has difficulty handling wide and continuous parameters and a large number of constraints, which are essential during an electrical machine optimization process [46]. A wide range of papers deal with the sensitivity of a selected performance parameter of an electrical machine [69; 70; 71]. For instance, any mismatch between the motor and controller parameters will result in the deterioration of performance [8; 69; 72; 73; 74]. The thermal and cooling properties have a significant effect on their performance [75]. In the case of transformers, their leakage inductance and losses are non-linear functions of the excitation and geometry [7; 9; 14]. These examples are selected only to illustrate that a large number of sensitivity analyses should be considered during an electrical machine design. Generally, these sensitivity analyses consider the role of the uncertainties as a single variable problem. Various heuristics and artificial intelligence based methods were introduced to handle this kind of analysis, where the selected design variables are perturbed with the required tolerances to calculate the model sensitivities [36; 7; 76]. However, an electrical machine design problem generally means a non-linear optimization of many variables together, for which, if we improve one parameter, another one worsens. In the case of some simpler, transformer optimization methods, usually, more than 20 performance parameters are optimized [9]. The simplest methods directly use the sensitivity as the objective function of the optimization [62; 78], as [PERSON] [79] minimized this sensitivity information directly to obtain a robust solution, or [80], which used the second-order sensitivity to find robust solutions. However, these methods cannot be used for large-scale problems, such as complex electrical machine design tasks. Furthermore, the approximation of the objective function using the response surface methodology has been often used in robust optimization. [PERSON] and [PERSON] proposed a similar meta-modeling technique: propagating model based uncertainty [81; 82]. However, these methodologies have the risk of converging to a sensitive design because the sensitive part of the objective function can be ignored in the approximation process [62]. Generally, the design of an electrical machine is a multi-variable, multiobjective optimization task, where the designer has to select the best design from a set of contradictory objectives and parameters. In practice, it is hard to weight the importance of these different objectives. Therefore, a multiobjective formulation is one of the most accurate ways of handling these problems. The solution of the problem is not a single design, but a set of Pareto-optimal solutions. However, these objectives are not independent of a practical problem Figure 3. Here, none of the solutions is better than the others (Figure 4).