Verification of the ANN

The verification of the ANN is done by comparing the calculated and the estimated fitness of individuals that arise in the context of the SA optimization following the construction of the network and are to be discarded due to a estimated value above the threshold. An example comparison is shown in Figure 6a. The mean deviation of the estimated from the calculated fitness of the discarded individuals in this case is 2.7%, but much of the estimation is more precise. However, some estimated fitness values have a relatively high deviation from the calculated fitness, which is a consequence of a large distance to the

closest individual in the database. Accordingly, the average precision can be increased if the allowed maximum distance is further reduced. It can also be seen that no individual was discarded whose fitness is better than the current optimum *f*(*x*min).

**Figure 5.** Comparison of the distribution of results for different optimization methods.

Reduction of the Solution Effort

By using the ANN, the types of fitness determination can be divided into three different variants:


In Figure 6b, the percentage breakdown of fitness determination of individuals in these three types is shown for the SA method for the optimization of the example machine. During the optimization, 150 simulations are performed using the TH-FEM to build the database of the ANN. In the SA stage, additional 1130 evaluations of individuals are performed. In each of the hybrid stages, 35 generations were evaluated using the ES method and 20 evaluations were calculated using the PS method. While a large proportion of individuals in the SA stage have invalid geometries, approximately 20.4% of the solutions can be discarded by neural networks and only 1.4% require time-consuming simulation in this use case. This results in a total of 166 simulations performed for the SA stage. Without the use of the ANN, all solutions that would otherwise be discarded due to the estimated fitness also require a simulation. The required simulations would thus increase to 397. In contrast to the SA optimization, the ANN only has a small impact on the solution effort of the ES-PS method. This is a consequence of the local convergence of this hybrid method, since it searches in a vicinity of the optimum where the estimated fitness values often do not exceed the required threshold. The simulation of each population in the 35 generations of the ES method is fully parallelized for all individuals. With the assumption that all valid individuals are electromagnetically simulated in the hybrid stage, the number of simulations performed by the ES method thus increases by 2 × 35 = 70 and by the PS method by 2 × 20 = 40. The performed total electromagnetic simulations using the ANN are 276 and without the ANN 507. This means a reduction of the simulation effort by 45.5%. This reduction is achieved assuming the same models in each stage. With the use of different models in the individual stages, this reduction factor will vary.

(**a**) Exemplary comparison of calculated and estimated fitness.

(**b**) Division of types of fitness determination for SA optimization.

**Figure 6.** Exemplary comparison of calculated and estimated fitness (**a**) and the breakdown of fitness determination (**b**).

## 4.3.2. IM Optimization Results

Starting from the cross-section of the rough designed machine shown in Figure 4b, the geometry optimization of the IM is performed. The cross section of the resulting machine geometry of the optimization is shown in Figure 4c. The basic design is similar to that of the reference machine. The optimized geometry differs primarily by two additional rotor bars, a rotor diameter larger by approximately 10% and a shortened active length of *l*Fe = 154 mm.

The evaluation of the quality of a solution is done with the fitness described in Section 2.6. It is calculated using the decision parameters defined problem-specifically in Section 4.1 by means of the TH-FEM simulation and related to the reference machine. The resulting fitness values of the roughly designed IM, the optimized machine, and the reference machine are presented in Table 5 and divided into the fractions of the volume as well as the mean losses over the WLTC 3. It can be seen that the optimization environment improves the fitness of the roughly designed IM by approximately 20%. Both the mean losses over the test cycle and the volume are lower in the optimized machine. Compared to the reference machine, the optimized geometry has a lower volume due to the shortened active length, but higher mean losses over the drive cycle. This results in a by 2.6% worse fitness. This is a consequence of the insufficient coverage of all possible degrees of freedom of the machine geometry by the seven optimization parameters, which leads to the fact that the reference machine cannot be completely reproduced. In addition, it is possible that the optimization method has not converged to the global minimum. As shown in Figure 5a, the optimum identified by the optimization environment has a dispersion of approximate 6%.

**Table 5.** Fitness values and decision parameters resulting from the TH-FEM.


For further verification of the optimized machine geometry, it is modeled by means of T-FEM simulations and compared with the reference machine. The operating maps of the total losses of the reference machine and the optimized machine resulting from the T-FEM are shown in Figure 7a,b. Here, the optimized geometry exhibits higher total losses, especially at the borders of the operating map, but the losses in the WLTC 3 driving cycle are of a similar order of magnitude to those of the reference machine.

The resulting fitness values related to the transiently simulated reference machine as well as the resulting proportions of the decision parameters are shown in Table 6. The increase in fitness of the optimized geometry from 1.026 for the TH-FEM to 1.059 in the case ofaaT-FEM is thereby within the level of detail required by the model selection methodology. Since the deviation of the fitness values between the optimized geometry and the reference machine with approximately 6% is near the range of the assumed accuracy of the transient FEM of approximately 5%, the optimized geometry derived automatically from the rough design and the reference machine can thus be assumed to be similarly suitable solutions of the multiphysics problem.


**Table 6.** Fitness values and decision parameters resulting from the T-FEM.

### **5. Discussion and Conclusions**

The focus of this work is the development of a multi-stage optimization environment for the design of a IM. In the individual stages, the advantages of two stochastic and one deterministic optimization method are combined by successively applying SA, ES and PS. The search for the optimum starts in the SA stage in a global solution space and continues locally in the successive hybrid use of the ES and PS methods. In the first successive stage, significant optimization parameters are varied and less significant parameters are kept constant. In the second stage, significant parameters are then assumed to be constant and less significant parameters are varied. This successive implementation of the hybrid ES and PS method improves the convergence behavior in terms of a lower mean value, dispersion and variance. In addition, the reduction of the optimization parameters in the individual stages compared to a single-stage hybrid ES-PS method results in a reduction of the computational effort. Using the SA method as a global search performed before the successive hybrid optimization method also improves the convergence behavior in terms of lower mean and median, and lower dispersion and variance of the optimization results. A disadvantage, however, is an increased computational effort due to the additional introduction of another optimization stage. This disadvantage is compensated by the application of an indirect machine model in the form of an ANN. By the ANN individual parts of the objective function, which otherwise require a computationally expensive simulation, are estimated. As a result, the computational cost of the multi-stage optimization environment in the presented application can be reduced by 45% by using the ANN. This value is achieved when the same electromagnetic machine model is used in each stage. The use of different models leads to smaller reductions.

If a precise estimate via the ANN is not possible, direct machine models are used for electromagnetic computation of the IM in the optimization stages. Using a model selection approach in each stage different levels of detail can be considered and defined in each optimization step. Thus, a model of lower level of detail can be used in the global search and models of increasing level of detail can be used in the local search. This procedure results in a high degree of flexibility with respect to the accuracy and the solution effort of the optimization environment.

A methodical approach to parameter selection is used to determine the optimization parameters. For each geometry parameter, the sensitivities and elasticities are studied with respect to the output variables relevant to the optimization. The optimization parameters are sorted by their elasticities and the parameters with the greatest impact on the optimization problem are identified. Sorting by elasticities also allows a systematic division of the parameters into variable and constant values depending on the optimization level.

The optimization environment using the model and parameter selection procedure is applied to the design of a traction machine. The objective of the optimization is to minimize the design space while minimizing the mean electromagnetic losses over the WLTC. The optimization is performed using the TH-FEM of the IM determined by the model selection approach. The quality of the result is determined based on the fitness of the optimized machine and a reference machine. For comparison, both machines are then simulated again using the model with the highest level of detail, the T-FEM. The fitness value of the optimized machine is about 6% higher than that of the reference machine. Since this deviation is within the range of the assumed accuracy of the T-FEM of about 5%, it can be assumed that the optimized geometry automatically derived from the rough design and the reference machine are similarly suitable solutions for the multiphysics problem. Thus, the use of the presented optimization environment as a tool for the design of the machine is verified.

A further verification of the optimization environment with further machine designs for different applications still has to be performed. In addition, the individual machine models can be further improved with respect to their level of detail. Not integrated in the optimization is the simulation of external components, such as inverter or battery. In addition, active cooling of the machine is not considered in the current status. For further improvement, a more detailed thermal model of the machine can be considered.

Regarding the uncertainty of the parameters, the proposed method may be extended using neutrosophic statistics as future research, but this issue is not the content of the study presented here.

**Author Contributions:** Conceptualization, M.N. and A.K.; methodology, M.N. and A.K.; software, M.N. and A.K.; validation, M.N.; formal analysis, M.N.; investigation, M.N. and A.K.; writing original draft preparation, M.N.; writing—review and editing, M.N., A.K. and K.H.; visualization, M.N. and A.K.; supervision, K.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
