**1. Introduction**

Optimization methods are increasingly used in the electromagnetic design and redesign process of electrical machines. The selection of the electromagnetic machine model and the optimization methods has a great influence on the computational effort and the convergence of the optimization.

Optimization methods can be divided into deterministic and stochastic methods. A well-known deterministic method is Pattern Search (PS) [1,2]. Among the most commonly used stochastic methods are evolutionary algorithms such as Genetic Algorithms (GA) [3,4] or Evolution Strategy (ES) [5], Particle Swarm Optimizations [6], and Simulated Annealing (SA) [4]. Both deterministic and stochastic optimization methods are applied in the field of electric machine optimization. In addition, couplings of deterministic and stochastic methods, such as GA coupled with PS [7], are applied.

In the field of electrical machines, multi-objective optimization is often considered, which allows the machine to be optimized with respect to several objective functions. In these optimizations, stochastic methods such as GA or ES and the Design of Experience (DoE) are used. Other methods used are the sequential optimization method [8] and the multi-level or multi-stage optimization [3,9–11]. In some of these referred multi-stage optimizations, a successive two stage optimization using one optimization method is conducted. The optimization parameters are divided into significant and less significant parameters and the parameter groups thus defined are varied or kept constant depending on the current stage. Examples of such methods can be found in [3,9] using a GA and [10,11] using DoE as the optimization method. An overview and further literature on optimization methods in the field of electrical machines can be found in [12,13].

The machine models can be divided into direct and indirect machine models. Direct models include numerical ones such as the Finite Element Method (FEM) and analytical

**Citation:** Nell, M.; Kubin, A.; Hameyer, K. Multi-Stage Optimization of Induction Machines Using Methods for Model and Parameter Selection. *Energies* **2021**, *14*, 5537. https://doi.org/10.3390/ en14175537

Academic Editor: Ryszard Palka

Received: 30 July 2021 Accepted: 1 September 2021 Published: 4 September 2021

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ones such as the Equivalent Circuit Diagram (ECD) model. They differ in their range of values of the modeled effects, their level of detail and their computational effort. In the field of optimizing synchronous machines [14,15] and synchronous reluctance machines [16], machine modeling is often performed using the FEM. This is possible due to the negligible transient effects and the resulting lower computational effort compared to the consideration of the FEM in the optimization of an Induction Machine (IM) [17,18]. For the IM, transient effects can no longer be neglected without accepting a significant reduction of the level of detail, resulting in a high computational effort for the FEM. Therefore, lower order models like analytical ECD models are applied [19–22].

In machine optimization environments, a very high number of machine simulations can be required. In this case, the FEM and other analytical methods can lead to very high computational effort. To reduce this high computational effort indirect machine models like the Response Surface Model (RSM) [8], Kriging Model [8,10] or Artificial Neural Network (ANN) are used. These surrogate models replace the machine model and estimate the output parameters of the machine based on the input parameters.

This paper presents a multi-stage optimization environment for IM design optimization that combines the advantages of several of the described optimization methods. The methods used are SA by [23], ES by [24], and PS by [1]. While the stochastic SA method has good global convergence with low local convergence speed, the ES method is known for stable convergence in the local group [25]. PS, as a deterministic method, provides a tool for fast local convergence [26]. Both the application of the successive ES-PS optimization and the previously executed stage of SA improve the convergence behavior in this case. In all these stages, direct machine models are used for electromagnetic modeling. The successive ES-PS method also reduces the computational effort compared to the single-stage ES-PS method. The increased computational effort due to the use of SA is compensated for by the application of an indirect machine model in the form of an ANN. This leads to a multi-stage optimization environment that combines the advantages of deterministic and stochastic optimization methods and those of direct and indirect model building. The selection of electromagnetic machine models and optimization parameters in each stage is methodically performed using the model selection and parameter selection procedure approach presented in [27]. Thus, in each stage, the model can be adjusted according to the desired range of values and level of detail.

The presented optimization environment is exemplary used to design an IM as a traction drive for a small vehicle. The aim of the optimization is to minimize the losses occurring over a given driving cycle while at the same time minimizing the required installation space. Starting with a rough design of the machine, it is optimized using the optimization methods presented, considering geometric and thermal constraints. The resulting machine is compared with a reference machine, which is used as a benchmark. The simulation results of this example show a good robustness of the multi-stage optimization environment including SA, a successive ES-PS method and the use of an ANN. Compared to classical multi-stage optimizations using one optimization method and one type of machine model, as in [3,9–11], it shows an improved convergence behavior. It can therefore be used for the design process and the design optimization of IM.

#### **2. Optimization Environment and Optimization Methods**

The classical optimization methods can be categorized into stochastic and deterministic methods. While deterministic methods realize a fast convergence to a local optimum, stochastic methods enable the search for a global optimum. Furthermore, stochastic methods offer other advantages, including easy consideration of constraints and numerical stability avoiding the use of derivatives [25]. A detailed review of optimization methods in the context of electrical machines is presented in [12,28].

In this paper, with the SA, the ES and the PS, three different optimization methods are combined in one optimization environment, thus exploiting synergies. While the stochastic SA method exhibits good global convergence with low local convergence speed, the ES method is known for stable convergence in the local group [25]. PS, as a deterministic method, provides a tool for fast local convergence [26]. In the following, the multi-stage optimization environment used in this work is presented. First, the overall structure of the optimization environment is explained and then the individual parts of it are described in more detail.
