**1. Introduction**

The ring induction motors are a group of electrical machines mainly used in electric drives with heavy starting. The electrical power of these machines varies over a wide range, from a fraction of a kilowatt to several megawatts, which allows for an extensive range of applications [1]. They are characterised by a simple structure, which implies low construction costs, easy operation and maintenance, and high levels of operational reliability [2].

One of the most important curves used to describe the RIMs is the torque-slip characteristic, which represents the relationship between the electromagnetic moment and the slip [3,4]. This is determined by the constant value of the stator supply voltage and the variable rotational speed of the motor, which is related to the slip by a simple and well-known formula. It is most often described using a simplified Kloss equation [3], which approximates the corresponding measured points with a high uncertainty value, however, most often in the case of very low-power machines [5]. An extended version of this equation [6–8], denoted here as EKE, was therefore developed based on the theory and practice of electric machines, for which much lower values of the approximation uncertainty of the mechanical characteristic of the motor can be obtained. This equation is necessary for the analytical determination of machine acceleration times [9], an analysis of the transient regimes [10], frequency control of inductive electric drives under conditions of overload [11], and for the testing of electric motors, for example in Tesla vehicles [12].

**Citation:** Tomczyk, K.; Makowski, T.; Kowalczyk, M.; Ostrowska, K.; Be ´nko, P. Procedure for the Accurate Modelling of Ring Induction Motors. *Energies* **2021**, *14*, 5469. https:// doi.org/10.3390/en14175469

Academic Editor: Armando Pires

Received: 5 August 2021 Accepted: 31 August 2021 Published: 2 September 2021

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Using a simplified and extended version of the Kloss equation, the value of maximum (critical) motor torque and the corresponding value of maximum (critical) slip can be determined [13]. For when the slip is equal to one, the value of the starting torque of the motor can be determined. Knowledge of the critical and starting torques is extremely important from the point of view of evaluating the mechanical properties of the RIM [14]. The first type of torque allows us to determine the possibility of short-term motor overload [15], while the second one represents the possibility of starting a drive system including both a motor and a generator [16].

All of the procedures for determining the TSC involve the accurate measurements of the corresponding points [17], and then their approximation with a minimum value of uncertainty [18–20]. The accurate determination of the maximum torque and slip is only possible by applying this approach to modelling. When the mathematical formula representing the measured points of the TSC is known, the satisfactory approximation results and the associated uncertainties can be obtained by applying the MC method [21–23]. This method should involve a pseudorandom number generator with a uniform distribution [24,25]. However, it is only possible to determine the maximum values of the torque and slip by applying the MC method. A third parameter related to the EKE also needs to be determined. This can be obtained by performing indirect calculations based on the equivalent circuit of the RIM [26–28] and by using the additional numerical method to obtain values of the power losses in the stator iron. Additionally, it should be emphasized that the MC method has so far been applied to the modelling of the RIMs, by analysis of their thermal behaviour and the detection of corresponding faults, based on the stator current measurements [29,30].

In Section 2 of this paper, we present a detailed discussion of the issues related to the determination of the equivalent circuit parameters corresponding to the RIM, based on measurements of the motor idle and short-circuit states. The use of polynomial approximation [31–33] to determine the power losses are also proposed as the additional numerical method. Section 3 describes the use of the MC method to model the RIM by using the TSC, while Section 4 presents an example of the use of MC-based modelling and the verification of the corresponding results.

The solution presented in this paper represents a new approach to modelling the RIM based on the measurement points of the TSC and the EKE. This is obtained by an application of the MC method and the polynomial approximation which allows us to determine the values of the maximum moment and slip with the associated uncertainties. Therefore, it is an example of accurate modelling of the RIM which is worked out according to the guidelines in [18,21] and can be applied in the field of the precise elaboration of measurement results. The lack of uncertainty analysis in the modelling of the RIMs can be considered a weakness of the research so far in the field of electrical motors.

The proposed method can be used for accurate modelling and appropriate control over the mechanical properties of the RIMs in order to ensure the correct operation of both single motors and complex electrical drive systems.
