**4. Example Application and Verification of Results**

Let us present below the experimental results referring to the modelling of the real RIM with the following rated data: *P*<sup>n</sup> = 3.3 kW (rated power), *V*sn = 400 V (rated stator voltage), *I*sn = 9.5 A (rated stator current), *f*<sup>n</sup> = 50 Hz (rated frequency), *n*<sup>n</sup> = 940 rpm/min (rated rotational speed), Δ/*y* (winging connections), cos *ϕ*<sup>n</sup> = 0.89 (rated power factor), and *η*<sup>n</sup> = 0.87 (rated efficiency). Single-phase resistances for the stator and rotor are 2.9 Ω and 0.1 Ω, respectively.

Table 1 shows the measured results for the magnetisation characteristic obtained in the idle state. The value of the current *I*s0, obtained for the rated voltage *V*sn, was measured with a digital instrument, and is equal to 5.60 A. The uncertainty associated with this current was calculated using Equations (20) and (22), and is equal to 0.22 A.

Figure 6 shows the results obtained for the power loss in the stator iron Δ*P*Fe using the method shown in Figure 4 and Equations (23)–(30). The value of this power loss is 155.8 W, and the linear equation that approximates the measurement points is

$$P\_0 \left(V\_s^2\right) = a\_0 + a\_1 \cdot V\_s^2 = 41.5 + 9.74 \cdot 10^{-4} \cdot V\_s^2. \tag{41}$$


 *P*0 -

*V*s 2 = 3.52 W.

**Table 1.** Measured results for the magnetisation characteristic in the idle state.

The uncertainty associated with this approximation is *u*

**Figure 6.** Results for the power loss in the stator iron Δ*P*Fe.

The uncertainties associated with the parameters in the linear equation are *u*(*a*0) = 2.17 and *<sup>u</sup>*(*a*1) = 2.31·10−5, while the corresponding relative uncertainties are *<sup>δ</sup>*(*a*0) = 5.24% and *δ*(*a*1) = 2.38%.

The uncertainty and relative uncertainty associated with the power losses, obtained using Equations (31) and (32), respectively, are *u*(Δ*P*Fe) =2.18 W and *δ*(*P*Fe) = 1.40%. The measured results for the short-circuit state of the RIM are shown in Table 2. These results enabled us to determine the parameters included in the circuit model shown in Figure 3, using Equations (10)–(14).

**Table 2.** Measurement results for the short-circuit state.


The values of the voltages *V*sn, *V*s1, and *V*sm were determined using analogue voltmeters with an accuracy and measurement range of 0.5% and 400 V, respectively. The values of the quantities *K*V, *R*Fe, *I*Fe, *I*mg, *X*mg, *X*σs, and *C*s, calculated using Equations (3)–(9), are 4.167 V/V, 3.08 kΩ, 0.13 A, 5.59 A, 71.45 Ω, 2.23 Ω and 0.971, respectively. The corresponding uncertainties *u*(*KV*), *u*(*R*Fe), *u*(*I*Fe), *u* - *I*mg , *u*(*X*mg), *u*(*X*σs) and *u*(*C*s) are 0.001 V/V, 55 Ω, 0.020 A, 0.22 A, 14.23 Ω, 0.22 Ω and 0.0070, respectively.

Table 2 shows the measured results obtained for the short-circuit state. Based on these measurements, we can calculate the values of the parameters *R*k, *X*k, and *Z*<sup>k</sup> using Equations (11), (13) and (14), respectively; we can then determine the corresponding uncertainties using Equations (18)–(22). These measurements were made using a voltmeter and ammeter with accuracy and measurement ranges of 0.50%, 200 V and 0.50%, 10 A, respectively.

The values of the parameters *R*k, *X*k, and *Z*<sup>k</sup> and the associated uncertainties *u*(*R*k), *u*(*X*k) and *u*(*Z*k) are 36.90 Ω, 40.92 Ω, 17.68 Ω, 1.62 Ω, 2.36 Ω and 1.08 Ω, respectively.

Based on the above parameters, the coefficient *β* and the associated uncertainty *u*(*β*) were calculated using Equations (16) and (18) as 0.152 and 0.141, respectively. The value of the relative uncertainty *δ*(*β*) is 92.8%. The high value of this uncertainty was due to the significant complexity of Equation (16), which depends on eight indirectly measured quantities.


Table 3 shows the measured results for the torque-slip characteristic of the RIM.

[Nm] 6.38 6.08 5.69 5.30 5.00 4.41 4.02 3.92 3.83 3.42

**Table 3.** Measured points for the torque-slip characteristic of the RIM.

The values of the parameters *T*maxi and *s*maxi were determined intuitively, as shown in Figure 5, as 7.00 Nm and 0.200, respectively. The values of the parameters *T*maxl, *T*maxh, *s*maxl, and *s*maxh were assumed in advance around the above parameters. These values define the draw ranges for the parameters *T*max and *s*max. The draw range for the coefficient *β* is determined based on the associated uncertainty *u*(*β*) by Equation (33), as follows:

$$
\beta\_1 = 0.152 - 0.141 = 0.011 \text{ and } \beta\_{\text{h}} = 0.152 + 0.141 = 0.293.
$$

A total of 2 × <sup>10</sup><sup>5</sup> MC trials were carried out using the pseudo-random number generator with a uniform distribution. Equations (34)–(40) were applied to the execution of the relevant numerical calculations using the MathCad 15 program, and the total computation time was 3 h and 24 min. The calculations were performed on a PC with the following parameters: Inter® Core™, Duo CPU E8400, processor ×64, 3.00 GHz, 4.00 GB RAM.

The minimum value Δ<sup>c</sup> min for the vector Δ<sup>c</sup> and the corresponding number of trials *m*min were 3.581 and 95360, respectively. The values of the parameters *T*maxopt, *s*maxopt, and *β*opt corresponding to quantity Δ<sup>c</sup> min are 7.3861 Nm, 0.19721, and 0.28927, respectively. The uncertainty *u*(MC) associated with the MC method is 0.032. The uncertainties *u*(*T*max), *u*(*s*max), and *u*(*β*), associated with the parameters *T*max, *s*max, and *β* are <sup>7</sup>·10−<sup>4</sup> Nm, 1.3·10−<sup>4</sup> and 1.82·10<sup>−</sup>4, respectively.

The relative uncertainty *δ*(*β*) associated with the coefficient *β* is *u*(*β*)/*β* =0.07%. Based on these results, it should be noted that the value of the uncertainty *δ*(*β*) was reduced from 92.8% (obtained from analytical calculations) to 0.07% (obtained using our MC-based procedure).

Figure 7 shows the results from our MC-based model of the RIM for the example of the TSC characteristic and using the EKE.

**Figure 7.** Results from our MC-based model of the RIM.

Figure 8 shows the distribution of the approximation uncertainty *u*(*T*) for the particular values of the slip *s*.

**Figure 8.** Distribution of the approximation uncertainty for the TSC characteristic.

The highest value of the approximation uncertainty was obtained for the slip s within the maximum slip value *s*max, as well as for the slip *s* with a value of about 0.85.

We now verify the implementation of our MC-based procedure by examining the influence of the number of MC trials on the value of the uncertainty *u*(MC). The results for the values of the parameters included in the EKE are given in Table 4.

**Table 4.** Results from our Monte Carlo procedure.


It can be seen from Table 4 that the values for the uncertainty *u*(MC) decrease as the number of MC trials increases. The values of the EKE parameters obtained for 2 × 105 MC trials (the lowest number of MC trials recommended by the corresponding guide), were assumed to represent the optimal solution to the modelling task for this example.

The results obtained in the section above show that the application of the MC method and the polynomial procedure in particular allows for a significant increase in the accuracy of the RIM modelling compared to other methods, which do not include analysis of the modelling uncertainty.

#### **5. Conclusions**

This paper presents a procedure that allows us to assess the accuracy of modelling of the RIMs on the example of the RSC and EKE, by applying the MC method. Based on the numerical simulations and calculations performed for an example of the RIM, it has shown that the effect of the uncertainty on the results of measurements is significant. The proposed method is based on the corresponding guidelines for the implementation of accurate measurements, and can significantly reduce the values of the uncertainties associated with the parameters in the ECE. For example, for the coefficient *β*, the application of our MC-based numerical modelling procedure reduced the value of corresponding uncertainty from 92.8% (analytical calculations) to 0.07% (MC-based procedure). It should also be emphasised that a further increase in the modelling uncertainty of both the TSC and the other parameters in the equivalent circuit of the RIM can be obtained by using more accurate measuring instruments, and by applying modern measurement techniques based on specialised computer software such as LabVIEW.

The solutions presented in this paper can be used in other applications in the field of electric machines and electric drives, for example in the precise determination of the characteristics of all types of motors, or for the development of accurate measurement reports. **Author Contributions:** Conceptualisation, K.T. and T.M.; data curation, T.M., M.K., K.O. and P.B.; writing—original draft, K.T.; formal analysis, T.M., M.K., K.O. and P.B.; methodology, K.T. and T.M.; writing—review and editing, K.T., T.M., M.K., K.O. and P.B.; software, K.T. and T.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was conducted at the Faculty of Electrical and Computer Engineering, Krakow University of Technology, and was financially supported by the Ministry of Science and Higher Education, Republic of Poland (grant No. E-3/2021).

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** MDPI Research Data Policies.

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