**2. Theoretical Basis for the Accurate Modelling of the RIM**

The EKE is defined by the following formula:

$$T = \frac{T\_{\text{max}} \left(2 + \beta \cdot s\_{\text{max}}\right)}{\frac{s}{s\_{\text{max}}} + \frac{s\_{\text{max}}}{s} + \beta \cdot s\_{\text{max}}},\tag{1}$$

where *T*, *T*max, *s*, and *s*max are the motor torque, maximum torque, slip, and maximum slip, respectively. The coefficient *β* is expressed as follows:

$$\beta = 2 \frac{R\_{\text{s}} C\_{\text{s}}}{R\_{\text{r}} K\_{V} \bar{\omega}} \, \text{} \tag{2}$$

where *R*s, *R*<sup>r</sup> , and *KV* denote the stator resistance, rotor resistance, and voltage ratio, respectively [6–8]. The voltage ratio is calculated by the formula:

$$K\_V = \frac{1}{2} \left( \frac{V\_{\text{s1}}}{V\_{\text{rm}}} + \frac{V\_{\text{sm}}}{V\_{\text{rm}}} \right) ,\tag{3}$$

where *V*s1 is the stator supply volge, which is lower than the rated voltage *V*sn, while *V*rm is the maximum voltage produced between any two rotor phases, and *V*sm is the maximum voltage produced between any two stator phases when the rotor is supplied by the voltage *V*rm [27]. The coefficient *C*s is defined by:

$$C\_s = \frac{X\_{\rm mg}}{X\_{\rm cs} + X\_{\rm mg}},\tag{4}$$

where *X*mg and *X*σ<sup>s</sup> denote the magnetisation reactance and the stator phase leakage reactance, respectively, and are calculated using the following formulae:

$$X\_{\rm mg} = \frac{V\_{\rm Sn}}{I\_{\rm mg}} \tag{5}$$

and

$$X\_{\rm crs} = (V\_{\rm Sn} - K\_V \cdot V\_{\rm rm}) / I\_{\rm s0} \tag{6}$$

where *I*mg is the magnetisation current, *I*s0 is the stator current under idle conditions, and *V*rm is the voltage induced in the rotor [6–8]. Figure 1 shows a circuit model of the RIM, which is valid when the measurements are made in the idle state.

**Figure 1.** Circuit model of the RIM in the idle state.

The magnetisation current is

$$I\_{\rm mg} = \sqrt{I\_{\rm s0}^2 - I\_{\rm Fe}^2} \,\,\,\,\tag{7}$$

where *I*Fe denotes the current corresponding to the power losses in the stator iron, and is calculated using the simple formula:

$$I\_{\rm Fe} = \frac{V\_{\rm sn}}{R\_{\rm Fe}}.\tag{8}$$

Here, *R*Fe denotes the resistance of the iron, and is defined by the following equation:

$$R\_{\rm Fe} = \frac{3V\_{\rm sn}}{\Delta P\_{\rm Fe}}^2,\tag{9}$$

while Δ*P*Fe denotes the power losses in the stator iron.

The complete equivalent circuit of the RIM is shown in Figure 2, where *R* r and *X* σr denote the rotor phase resistance and the reactance transformed to the stator side.

**Figure 2.** Complete equivalent circuit of the RIM.

The quantities *R*Fe, *I*Fe, *X*mg, and *I*mg, given in Equations (5), (7)–(9) are calculated based on the complete phase equivalent diagram (the left side of Figure 2), omitting the voltages related to the stator resistance and the stator winding leakage reactance.

The values of the parameters *R* <sup>r</sup> and *X* <sup>σ</sup><sup>r</sup> can be determined based on the circuit model of the RIM in the short-circuit state, as shown in Figure 3, where *V*sk denotes the current in this state.

**Figure 3.** Circuit model of the RIM in the short-circuit state.

The transverse branch of the circuit model shown in Figure 2 is omitted in Figure 3, due to the significant value of its impedance compared to the impedance of the longitudinal branch. The short-circuit current *I*sk is also assumed to be equal to the rated stator current *I*sn [6–8].

Based on the circuit model shown in Figure 3, we have:

$$R\_{\mathbf{k}} = R\_{\mathfrak{k}} + R\_{\mathfrak{r}}' = R\_{\mathfrak{k}} + R\_{\mathfrak{r}} K\_{V}^{\ \ \bar{z}} \tag{10}$$

and

$$X\_{\rm k} = X'\_{\rm or} + \mathbb{C}\_{\rm s} X\_{\rm ors} = X'\_{\rm or} + \frac{X\_{\rm mg} X\_{\rm ors}}{X\_{\rm mg} + X\_{\rm ors}} \tag{11}$$

while

$$X'\_{\sigma\mathfrak{r}} = X\_{\sigma\mathfrak{s}} K\_V^{\mathfrak{r}}.\tag{12}$$

Based on Ohm's law, we obtain:

$$Z\_{\rm k} = \frac{V\_{\rm sk}}{I\_{\rm sn}},$$

and applying the impedance triangle gives

$$R\_{\mathbf{k}} = \sqrt{Z\_{\mathbf{k}}^2 - X\_{\mathbf{k}}^2}.\tag{14}$$

By transforming Equation (10), we obtain the rotor resistance

$$R\_{\rm r} = (R\_{\rm k} - R\_{\rm s}) / K\_{\rm V}^{\cdot^2} \tag{15}$$

We can express the coefficient *β* in terms of the measured values of the parameters of the RIM. We can obtain this relation by substituting Equations (3)–(13) into Equation (2) to give [1,6–8]:

$$\beta = \frac{2R\_s}{\left(\frac{\gamma\_{2\uparrow 2}}{\gamma\_4} + 1\right) \left[\sqrt{\left(\frac{V\_{\rm sh}}{I\_{\rm on}}\right)^2 - \left(\frac{\gamma\_{1\uparrow}}{8 \cdot I\_{\rm o}} + \frac{V\_{\rm m} \cdot \gamma\_3}{\gamma\_4 + \gamma\_{2\uparrow}\gamma\_3}\right)^2} - R\_s\right]}\tag{16}$$

where the auxiliary parameters are

$$\begin{aligned} \gamma\_1 &= \frac{V\_{\rm Si}}{V\_{\rm rm}} + \frac{V\_{\rm sm}}{V\_{\rm rm}}, \ \gamma\_2 = \sqrt{I\_{\rm s0}^2 - \left(\frac{\Delta P\_{\rm Fe}}{\overline{\lambda}V\_{\rm sm}}\right)^2},\\ \gamma\_3 &= 2V\_{\rm sn} - \gamma\_1 V\_{\rm rm}, \ \gamma\_4 = 2V\_{\rm sn}I\_{\rm s0}. \end{aligned} \tag{17}$$

The uncertainties associated with the quantities given in Equations (3)–(16) can be calculated using the formula:

$$u(\mathbf{x}) = \sqrt{\sum\_{j=1}^{l} \left[\frac{\partial \mathbf{x}}{\mathbf{x}\_{j}} u(\mathbf{x}\_{j})\right]^{2}}\,\mathrm{}\,\tag{18}$$

where *x* denotes the quantity under consideration, and *J* is the number of indirect quantities necessary to determine the value of *x*. The relative uncertainty associated with the quantity *x* is defined by the equation:

$$
\delta(\mathbf{x}) = \mu(\mathbf{x}) / \mathbf{x}. \tag{19}
$$

For both analogue and digital measuring instruments, the uncertainty *u*(*x*) is determined by the formula:

$$
\mu(\mathbf{x}) = \Delta(\mathbf{x}) / \sqrt{3} \; , \tag{20}
$$

where Δ(*x*) is the absolute error, while the value of the denominator results from the probability density function of a uniform distribution, which is valid for both analogue and digital instruments.

For analogue instruments, the error is determined on the basis of the static accuracy class *κ*, according to the equation:

$$
\Delta\_{\mathbf{a}}(\mathbf{x}) = \frac{\kappa Y\_{\mathbf{m}}}{100\%} \tag{21}
$$

where *Y*<sup>m</sup> denotes the measurement range for the quantity to be measured. For digital instruments, the following formula is usually applied:

$$
\Delta\_{\mathbf{d}}(\mathbf{x}) = a\mathbf{y} + c\mathbf{Y}\_{\mathbf{m}}\,\prime \tag{22}
$$

where *Y* denotes the value of the quantity to be measured, and *a* and *c* are constant parameters that are typical for the particular instrument and are included in the corresponding datasheet.

The power losses in the stator iron Δ*P*Fe are determined using a graphical method for the rated stator voltage *V*sn, as shown in Figure 4.

The quantity *P*<sup>0</sup> is the active power consumed by the motor during idling, and is equal to the sum of the losses in the stator iron Δ*P*Fe, and the mechanical power losses Δ*P*m.

Based on the measured points for the active power *P*0, it is easy to determine the linear characteristic *P*<sup>0</sup> = *f* - *V*s 2 and the associated uncertainty. This can be done by applying the polynomial method, using the formula:

$$P\_0 \left(V\_s^2\right) = a\_0 + a\_1 \cdot V\_s^2 + \varepsilon,\tag{23}$$

where *a*<sup>0</sup> and *a*<sup>1</sup> are the polynomial coefficients, and *ε* denotes the error of approximation.

**Figure 4.** Graphical method for determining the power losses Δ*P*Fe.

Estimates *<sup>a</sup>* of the polynomial coefficients are obtained using the following matrix equation:

$$\tilde{\mathbf{A}} = \left(\boldsymbol{\Phi}^T \boldsymbol{\Phi}\right)^{-1} \boldsymbol{\Phi}^T \boldsymbol{\Lambda},\tag{24}$$

where

$$\begin{aligned} \boldsymbol{\Phi} &= \begin{bmatrix} 1 & \begin{pmatrix} V\_{\sf s}^2 \end{pmatrix}\_0 \\ \vdots & \vdots \\ 1 & \begin{pmatrix} V\_{\sf s}^2 \end{pmatrix}\_N \end{bmatrix}' \\ \boldsymbol{\Lambda} &= \begin{bmatrix} \varepsilon\_0 & \varepsilon\_1 & \dots & \varepsilon\_N \end{bmatrix}^T \end{aligned} \tag{25}$$

and *N* denotes the number of measured points for the characteristic *P*<sup>0</sup> = *f* - *V*s 2 .

The uncertainty of approximation is denoted as the error *ε*, and is given by the following formula:

$$\mu\left(P\_0\left(V\_s^2\right)\right) = \sqrt{\frac{\left(\Phi\tilde{\mathbf{A}} - \mathbf{A}\right)^T \left(\Phi\tilde{\mathbf{A}} - \mathbf{A}\right)}{N - 3}}\tag{26}$$

The standard uncertainty associated with the coefficients *a*<sup>0</sup> and *a*<sup>1</sup> is

$$
\mu(a\_i) = \mu\left(P\_0\left(V\_s^{\;2}\right)\right)\sqrt{\Theta\_{i,i\;\prime}}\tag{27}
$$

where

$$
\Theta = \left(\Phi^T \Phi\right)^{-1},
\tag{28}
$$

and *i* = 0, 1, 2 [29–31].

The relative uncertainties associated with the coefficients *a*<sup>0</sup> and *a*<sup>1</sup> are calculated as follows:

$$
\delta(a\_i) = \frac{\mu(a\_i)}{a\_i} 100\%. \tag{29}
$$

The values of the power losses in the stator iron Δ*P*Fe and the associated uncertainty *u*(Δ*P*Fe) are calculated using the expressions:

$$
\Delta P\_{\text{Fe}} = P\_0 \left( V\_{\text{sn}}^{\text{-}2} \right) - P\_0(0) \tag{30}
$$

and

$$
\mu(\Delta P\_{\rm Fe}) = \mu(a\_0) + \mu(a\_1) P\_0 \left(V\_{\rm sn}\right) \,. \tag{31}
$$

The corresponding relative uncertainty is:

$$
\delta(P\_{\rm Fe}) = \frac{\mu \left(\Delta P\_{\rm Fe}\right)}{\Delta P\_{\rm Fe}} 100\%. \tag{32}
$$

The procedure for determining the values of the parameters included in Equation (1) and the corresponding uncertainties is discussed in detail in the section below.

#### **3. Monte-Carlo-Based Modelling of the RIM**

We now present the example of the application of the MC method in the accurate modelling of the RIM, which involves determining the corresponding parameters of the EKE and the associated uncertainties. This procedure is based on an intuitive method of determining the parameters *T*maxi and *s*maxi for the possible ranges of variability of the parameters *T*max and *s*max, which are included in the EKE [21–24]. Figure 5 shows the typical TSC which describes the RIM for its motor work and covers the stable range of this characteristic. Figure 5 also shows examples of the variability ranges of the parameters *T*max and *s*max.

**Figure 5.** The typical TSC for the motor work of the RIM.

The index 'i' denotes the intuitive values of both parameters, while the indexes 'h' and 'l' represent the high and low assumed values of these parameters. The high and low values are assumed in advance to ensure that the estimated values of the parameters *T*max and *s*max are within these selected ranges. A suitable selection of these ranges constitutes the first step in this method.

The parameter *β* and the associated uncertainty *u*(*β*) are determined based on the procedure discussed in Section 2. The low and high values, *β*<sup>l</sup> and *β*h, are determined as follows:

$$
\beta\_{\rm l} = \beta - \mathfrak{u}(\beta), \ \beta\_{\rm h} = \beta + \mathfrak{u}(\beta). \tag{3.3}
$$

The second step in our MC-based procedure involves the choice of the type of pseudorandom number generator. Taking into account the analogous probability of the occurrence of the optimal value of the estimates *<sup>T</sup>*max, *s*max, and *<sup>β</sup>* for any value from the above intervals, we are justified in choosing the pseudorandom number generator with a uniform distribution. The above estimates should accurately map the parameters *T*max, *s*max, and *β*, which requires an approximation of the TSC with minimal uncertainty.

In the third step, we determine the number of MC trials. According to the recommendations given in the guide [20], the optimal number of trials should be greater than <sup>10</sup>4/(<sup>1</sup> − *<sup>v</sup>*), where *<sup>v</sup>* denotes the coverage probability.

In the fourth step, the following matrix is determined

$$\mathbf{Y} = \begin{bmatrix} \ & T(\mathbf{s}\_0)\_0 & \dots & T(\mathbf{s}\_0)\_{M-1} \\ \vdots & \ddots & \vdots \\ \ & T(\mathbf{s}\_{N-1})\_0 & \dots & T(\mathbf{s}\_{N-1})\_{M-1} \end{bmatrix} \tag{34}$$

based on Equation (1), where *N* and *M* denote the number of measured points for the TSC and the number of MC trials, respectively [22–25]. The matrix **Ψ** is determined by substituting the values of the parameters *T*max*m*, *s*max*m*, and *β<sup>m</sup>* into Equation (1), as obtained for a sequence of MC trials *m* = 0, 1, ... , *M* − 1. The value of each slip *sn* is substituted into Equation (1) for each MC trial *m*, where *n* = 0, 1, ... , *N* − 1. In the fifth step, the matrix of approximation errors for the measured points of the TSC is determined as follows:

$$\mathbf{A}\_{\mathbf{c}} = \begin{bmatrix} \tilde{T}(\mathbf{s}\_0)\_0 & \dots & \tilde{T}(\mathbf{s}\_0)\_{M-1} \\ \vdots & \ddots & \vdots \\ \tilde{T}(\mathbf{s}\_{N-1})\_0 & \dots & \tilde{T}(\mathbf{s}\_{N-1})\_{M-1} \end{bmatrix} \tag{35}$$

where *T*(*sn*)*<sup>m</sup>* = *T*(*sn*)*<sup>m</sup>* − *T*(*sn*). The next rows of the matrix **Δ**<sup>e</sup> correspond to the approximation uncertainties obtained for each value of the slip *sn*.

The sixth step in our MC-based modelling process involves the determination of the vector

$$\Delta\_{\mathbf{c}} = \sum\_{\mathbf{n}} \left(\Delta\_{\mathbf{c}}\right)^{2},\tag{36}$$

in which each element is the sum of the squared errors calculated for each column of the matrix **Δ**<sup>e</sup> [22–25].

In the next step, the minimum value - **Δ**c min of the vector **Δ**<sup>c</sup> and the corresponding number of trials - *m*min are determined. The parameters *T*maxopt, *s*maxopt, and *β*opt corresponding to the value **Δ**<sup>c</sup> min are assumed to represent the optimal solution to the MC-based model. These parameters correspond to the estimates *<sup>T</sup>*max, *s*max, and *<sup>β</sup>* as defined above.

We then determined the uncertainty associated with the MC method using the following formula:

$$
\mu(\text{MC}) = \sqrt{\frac{1}{M(M-1)} \sum\_{m=0}^{M-1} \left[ \mathbf{A}\_{\text{cm}} - \overline{\mathbf{A}}\_{\text{c}} \right]^2} \tag{37}
$$

where ¯

$$\stackrel{\cdots}{\Delta\_{\mathbb{C}}} = \frac{1}{M} \left( \sum\_{m=0}^{M-1} \Delta\_{\mathbb{C}m} \right). \tag{38}$$

The last step in our MC-based procedure involves the determination of the uncertainties associated with the parameters *T*max, *s*max, and *β* using the formula:

$$\mu(\mathbf{x}) = \sqrt{\frac{1}{M(M-1)} \sum\_{m=0}^{M-1} \left[ \mathbf{x}\_{\text{ff}} - \overline{\mathbf{x}} \right]^2} \tag{39}$$

where

$$
\overline{\mathfrak{X}} = \frac{1}{M} \sum\_{m=0}^{M-1} \mathbf{x}\_m. \tag{40}
$$

The last two formulae are valid for all the parameters in the MC model.
