*Article* **Approach for the Model and Parameter Selection for the Calculation of Induction Machines**

**Martin Nell \*, Alexander Kubin and Kay Hameyer**

Institute of Electrical Machines (IEM), RWTH Aachen University, 52062 Aachen, Germany; alexander.kubin@rwth-aachen.de (A.K.); kay.hameyer@iem.rwth-aachen.de (K.H.)

**\*** Correspondence: martin.nell@iem.rwth-aachen.de

**Abstract:** The solution of multiphysical problems in the field of electrical machines is a complex task that involves the modeling of a wide variety of coupled physical domains. Different types of models and solution methods can be used to model and solve the individual domains. In this paper a procedure for the methodical selection of the most suitable model for a given multiphysics task is presented. Furthermore, an approach for the selection of the most suitable variable machine parameters for a design optimization is presented. The model selection is presented on the basis of the electromagnetic calculation of an induction machine. For this purpose, models of different value ranges and levels of detail, such as analytical and numerical ones, are considered. The approach of the model selection is explained and applied on the basis of a coupled electromagnetic-thermal simulation of an exemplary induction machine. The results show that the model selection presented here can be used to methodically determine the most suitable model in terms of its value range, level of detail and computational effort for a given multiphysical problem.

**Keywords:** induction machine; electromagnetic models; model selection

**1. Introduction**

Complex tasks in the field of electrical machines usually comprise several physical domains and thus form a multiphysics problem. The aim of modeling and calculating multiphysics problems is to represent all physical effects of the individual domains that are relevant for the application. In a multiphysics problem, the individual domains can be independent or coupled. In [1], coupled problems are defined as those involving multiple domains and dependent variable sets that usually, but not necessarily, describe different physical effects, and where either no domain can be solved correctly independently of the other domains or none of the dependent variable sets can be eliminated explicitly. An example of such a coupled multiphysics problem is the thermal simulation of an operating point of an electrical machine. Here, among other things, the domains of electromagnetism and thermal are bidirectionally dependent on each other.

If electrical machines are not considered as an independent system, but as part of a system of several components from different fields of technology, the complexity of the coupled multiphysics problem increases. An example of such a system is an electric drive train. The individual components of the system, such as electric motor, gearbox or battery, can in turn be assigned to individual domains or divided into several domains.

Different models are used for modeling and calculating the individual components and their domains, depending on the required accuracy and the physical effects to be modeled. The individual models therefore differ in their value range and level of detail of the representation of the physical effects as well as in their computational or solution effort. They can be classified into the categories of empirical, analytical, lumped parameter, and numerical models. The exemplary classification in value range, level of detail and computational effort for models for the electromagnetic simulation of an Induction Machine (IM) used in this paper are shown in Figure 1. Analytical models such as the Fundamental

**Citation:** Nell, M.; Kubin, A.; Hameyer, K. Approach for the Model and Parameter Selection for the Calculation of Induction Machines. *Energies* **2021**, *14*, 5623. https:// doi.org/10.3390/en14185623

Academic Editor: Ryszard Palka

Received: 30 July 2021 Accepted: 2 September 2021 Published: 7 September 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Wave Model (FWM) or Harmonic Wave Model (HWM) have a smaller value range of modeled physical effects and a lower level of detail of individual effects than numerical models such as the Finite Element Method (FEM), but also a lower computational effort.

Examples of models, methods, and physical effects considered for solving multiphysics problems in the field of electrical machines are explained in [2]. There, a modular computational approach for the calculation, simulation, and design of electrical machines is presented, which has been and is being applied in several scientific papers.

For the computation and simulation of the aforementioned multiphysics and coupled problems, the question arises as to which models are most suitable for which domains. Suitability refers to the fact that the particular model can represent the desired effects at an appropriate level of detail and is computationally efficient. The respective suitability of a model is strongly problem-dependent, since different problems require the modeling of various effects with different degrees of accuracy. Due to its high computational effort, the selection of the most complex model in the respective domain is not always the means of choice, especially in the design process of the machine or the overall system. Furthermore, the sensitivity with respect to the coupling variables is a necessary requirement of the used models.

In view of the modular computational approach mentioned above, the question of efficient model selection may arise in many of the calculation columns listed. For example, which electromagnetic and thermal models can be used for the thermal calculation of the electrical machine and which electromagnetic and structural dynamics models can be used for the Noise Vibration Harshness (NVH) analysis?

Design and redesign processes of electrical machines are also performed with the help of multilevel mathematical optimization algorithms. Here, the choice of the model in the individual optimization stages is an important factor influencing the quality of the result and the computational efficiency. These parameters also depend on the choice of optimization variables.

In the publications mentioned, models are mostly used without evaluating their suitability or advantages and disadvantages for solving the problem in comparison with other models. A methodical selection of the used model is not considered in detail. Therefore, such a methodological approach to model selection is addressed in this paper and presented using the electromagnetic simulation of IM as an example.

**Figure 1.** Classification of IM models with respect to value range, level of detail, and computational effort.

By specifying a few parameters, such as the desired effects to be modeled and their level of detail, the method offers the possibility to automatically select the model that meets the requirements and has the lowest computational cost. The selection method is based on the analysis of an exemplary IM for a given power, torque, and speed range in certain operating points and the derived ranges of values, levels of detail, and degrees of freedom of each machine model. The model selection can then be applied to problems requiring simulations of machines in a similar power, torque and speed range, and similar geometric dimensions. The proposed model selection approach can also be used in machine optimization problems in which machines of similar power range but different designs are simulated to determine the appropriate models in individual optimization stages. Such an example is given in [3]. In such optimization environments, the choice of optimization parameters is crucial and can also be done methodically. For this purpose, an approach for parameter selection is presented in this paper, which ensures an efficient design and redesign optimization.

The paper is organized as follows. First, the models of an IM considered in the model selection are introduced. These include the FWM, the HWM according to [4–6], and an Extended Harmonic Wave Model (E-HWM) developed in this work to consider saturation, three analytical models, and the Time Harmonic Finite Element Model (TH-FEM) and Transient Finite Element Model (T-FEM), two numerical models. Afterward, the approach of the model selection is presented. The input parameters of the methodology and the procedure for the analysis of the value ranges, levels of detail, and computational efforts of the models are discussed. In the following, the parameter selection, which can be applied in optimization environments, is presented. Subsequently, the approach for the model selection is applied on the basis of an exemplary problem. As an example, a weakly coupled electromagnetic-thermal simulation of an IM is used to analyze the thermal operating behavior of the machine. The parameter selection in combination with the model selection is applied in [3] in an optimization of the machine design of an IM.
