Inﬂuence of Selected Non-Ideal Aspects on Active and Reactive Power MRAS for Stator and Rotor Resistance Estimation

: Mathematical models of induction motor (IM) used in direct ﬁeld-oriented control (DFOC) strategies are characterized by parametrization resulting from the IM equivalent circuit and model-type selection. The parameter inaccuracy causes DFOC detuning, which deteriorates the drive performance. Therefore, many methods for parameter adaptation were developed in the literature. One class of algorithms, popular due to their simplicity, includes estimators based on the model reference adaptive system (MRAS). Their main disadvantage is the dependence on other machines’ parameters. However, although typically not considered in the respective literature, there are other aspects that impair the performance of the MRAS estimators. These include, but are not limited to, the nonlinear phenomenon of iron losses, the effect of necessary discretization of the algorithms and selection of the sampling time, and the inﬂuence of the supply inverter nonlinear behavior. Therefore, this paper aims to study the effect of the above-mentioned negative aspects on the performance of selected MRAS estimators: active and reactive power MRAS for the stator and rotor resistance estimation. Furthermore, improved reduced-order models and MRAS estimators that consider the iron loss phenomenon are also presented to examine the iron loss inﬂuence. Another merit of this paper is that it shows clearly and in one place how DFOC, with the included effect of iron losses and inverter nonlinearities, can be modeled using simulation tools. The modeling of the IM and DFOC takes place in MATLAB/Simulink environment.


Introduction
Mathematical models of the induction motor (IM) are needed for two purposes: the modeling of the machine itself and for the real-time high-performance control strategies, among which we can include field-oriented control (FOC), direct torque control (DTC), and model predictive control (MPC) [1]. The set of the parameters utilized by the mathematical models is defined by the IM equivalent circuit selection and model type [2][3][4].
The key parameters in the traditional T-equivalent circuit are the stator and rotor resistances, leakage inductances, and magnetizing inductance. Unfortunately, these parameters are not constant during the drive operation since they are affected mainly by the temperature rise and magnetic flux saturation [5]. For instance, the nonlinear magnetizing characteristics can be respected within the control algorithm using offline-measured data. However, the compensation of the resistances must be handled online since it depends on the machine's loading.
So far, numerous online identification methods of IM parameters have been proposed in the literature. These include model-based methods [6], recursive least-square algorithms (RLS) [7][8][9], model reference adaptive systems (MRAS) [10][11][12][13][14][15][16], signal injection (SI) techniques [17][18][19], state observers (SO) [20][21][22], and artificial intelligence (ANN) [23][24][25] methods. Typically, the greater the estimation accuracy and insensitivity to other machine parameters, the greater the algorithm complexity, which puts demands on hardware computational power and the experience of the implementation engineer. Therefore, due to • Effect of solver and sampling time selection. MRAS design is, in most cases, carried out in the continuous-time domain. However, the actual control algorithms are implemented on a discrete system: either a digital signal processor (DSP) or fieldprogrammable gate array (FPGA). Only a few papers consider the effect of discretization [40,41]. However, they are focused on a specific MRAS for speed estimation. • Effect of voltage-source inverter (VSI) nonlinearity. Most of the MRAS algorithms utilize directly or indirectly the stator voltage vector. As the real-time voltage measurement is hardware demanding and requires properly designed filters, the reference voltage (i.e., the input to the modulator) is usually utilized instead of the direct measurement. However, the fundamental output voltage of the commonly used IGBT inverters is distorted, mainly due to the inserted deadtime and finite semiconductor switching. • Effect of iron losses. Iron losses are a phenomenon that undoubtedly affects the IM flux, torque, speed, and parameter estimation [42][43][44]. However, most of the proposed MRAS estimators are based on IM equivalent circuits that do not consider iron losses.
This paper aims to examine and quantify the influence of the issues mentioned above on the performance of MRAS-type IM parameter estimators. The most popular reactive power MRAS (Q-MRAS) for the rotor resistance estimation and active power MRAS (P-MRAS) for the stator resistance estimation are selected as the candidates for the investigation. The presented results are based on simulations of the direct FOC (DFOC) of a 3.6 kW IM drive in the MATLAB/Simulink because, in a real drive, it is not possible to efficiently study the decoupled effects of various phenomena acting on the system. For machine modeling, the traditional T-equivalent circuit is utilized. The circuit is augmented with the fictitious iron loss resistance placed in parallel with the magnetizing branch to study the effect of iron losses. This paper is organized as follows: Section 2 shows full-order state-space models of IM with and without the included effect of iron losses that are used to model the machine itself. Section 3 then presents improved reduced-order models with the included effect of iron losses used in the DFOC model to assess the influence of iron losses on the parameter estimation. Furthermore, this section also introduces the numerical methods whose influence on the estimation accuracy is further examined in the simulations. In Section 4, the mathematical model of the VSI with the possibility of simulating the nonlinear behavior of the actual inverter is presented. Section 5 gives an overview of iron losses measurement, modeling, and implementation into the FOC and IM models. Section 6 demonstrates the derivation of the improved P-MRAS and Q-MRAS estimators that consider the influence of iron losses. Finally, the paper is concluded with Sections 7 and 8 dedicated to the results and discussion of the simulations.

Induction Machine Equivalent Circuit
The IM traditional T-equivalent circuit can be augmented to include the effect of the iron losses ( Figure 1). In this case, the losses are considered load-independent. In Figure 1, the symbols ψ 1 , ψ 2 , and ψ m represent the stator, rotor, and magnetizing flux linkage space vectors, respectively; u 1 represents the stator voltage space vector; i 1 , i 2 , i m , and i Fe represent the stator, rotor, magnetizing, and equivalent iron loss current space vectors, respectively; R 1 , R 2 , and R Fe denote the stator, rotor, and equivalent iron loss resistance, respectively; ω k is the electrical angular speed of the general reference frame; ω is the rotor electrical angular speed; L m is the magnetizing inductance; and the symbol j represents an imaginary unit (j 2 = −1). A short-circuited rotor is considered; therefore, the rotor voltage equals zero. The stator inductance L 1 is defined as L 1 = L m + L 1σ , where L 1σ is the stator leakage inductance and the rotor inductance L 2 is defined as L 2 = L m + L 2σ , where L 2σ is the rotor leakage inductance. The superscript k denotes that the space vectors are expressed in an arbitrary reference frame. The two specific reference frames used in this paper are the stator-fixed (real and imaginary axis denoted as α and β, respectively) and rotor flux vector-attached (real and imaginary axis denoted as d and q, respectively) reference frames.

Full-Order State-Space Model with Included Effect of Iron Losses
For modeling IM with the included effect of iron losses, the full-order state-space model in the stationary αβ reference frame with the current space vector, magnetizing flux space vector, and rotor flux space vector components as state variables can be used [45]. The model is deductible from Figure 1 and can be expressed mathematically as where Energies 2021, 14, 6826 and where τ rσ = L 2σ /R 2 , The electromechanical torque and the motion equation, respectively, are considered in the form where p p is the number of pole-pairs, T is the electromagnetic torque, T L is the load torque, J is the moment of inertia, and the operator × denotes cross product.

Full-Order State-Space Model without Iron Losses
For the modeling of the machine without the effect of iron losses, it is convenient to use a model with the stator and rotor flux linkage vector components as state variables. The model in its state-space form can be written as [46] . where and where D = L 1 L 2 − L 2 m . The electromechanical torque is given by The equation of motion is the same as (7).

Reduced-Order Models for Rotor Flux Estimation Considering Iron Loss Effect
In the FOC strategies, the decoupled regulation of flux-producing and torque-producing current components is done in the synchronously rotating dq system where the electromagnetic quantities become DC values. This paper implements a DFOC where the transformation angle, i.e., the angle between the stationary αβ and rotation dq system, is calculated from the rotor flux linkage vector components. For this purpose, two IM reduced-order models can be used: the so-called IM current and voltage models. Conventionally, these models do not consider the effect of iron losses, contrary to the multiple known full-order models that are based on the respective modification (i.e., placement of the iron loss re- sistance) of the IM equivalent circuit. Since the reduced-order models that consider the iron losses are not often mentioned in the literature, their derivation is, for convenience, presented in the following subsections.

Current Model with Included Iron Losses
The model will be derived in an arbitrary reference frame and then concretized to αβ and dq reference frames. Using Figure 1, the rotor voltage equation and the rotor flux linkage vector equation, respectively, can be expressed as Substituting for the rotor current vector in (13) from (14) yields On the other hand, choosing the rotor flux linkage vector-attached dq reference frame, the steady-state expression for the rotor flux magnitude and slip frequency, respectively, can be expressed as where

Voltage Model with Included Iron Losses
The model will be derived in the αβ reference frame. According to Figure 1, the stator flux linkage vector can be expressed as Substituting for the rotor current vector in (14) from (19) and considering the statorfixed reference frame yields where σ = 1 − L 2 m /L 1 L 2 is the leakage factor. The stator flux linkage vector is obtained as

Implementation of Current and Voltage Model into Discrete System
The real-time motor control algorithm is implemented either on DSP or FPGA. Therefore, the continuous mathematical models must be solved numerically. The current model (16) represents a complicated set of coupled differential equations when resolved into the real and imaginary parts. Therefore, as mentioned in the introduction, one of the aims of this paper is to compare the influence of the solver and sampling time selection on the accuracy of the DFOC and MRAS algorithms. The following numerical methods are considered: the forward Euler method (FWEM), the trapezoidal rule (TR), and the fourth-order Runge-Kutta method (RK4). These approaches are well-known algorithms for the solution of ordinary differential equations (ODE); therefore, they will be described here only briefly.
Let us consider a first-order ODE in the form .
The most straightforward approach for numerically solving (22) is the forward Euler method given by the rule (a fixed step ∆t is assumed) [47] The method is first order, making it computationally undemanding but with limited accuracy and stability. An improvement can be achieved by using the trapezoidal rule: at the cost of one extra function evaluation, we improve the order of the method by one. The rule is given by the following formula [47] The last numerical approach considered in this paper is the popular fourth-order Runge-Kutta method, which can be summarized as [47]

Modelling of Inverter Nonlinearities
A proper inverter model is needed to assess the effect of the IGBT inverter nonlinearities on the DFOC and MRAS algorithms. By a nonlinear inverter behavior, we mean the semiconductor's finite turn-on and turn-off times, the voltage drop across the devices, and necessary protective time, i.e., dead-time inserted by the microcontroller or the transistor driver [48]. Since a 400 V IM drive is considered, the voltage drop across the devices will be neglected.
Let us consider the most common space-vector modulation (SVM) with a fixed switching period T PWM . By a simple graphical analysis, it can be concluded that the turn-on time T on increases and the turn-off time T off decreases the distortion given by the dead-time T dt [48]. Therefore, it is convenient to define the so-called effective dead-time as where it is assumed that the turn-on and turn-off times are only the function of the load current, and the symbols a, b, c denote the respective inverter leg. A direct inverter measurement can be performed to easily determine the dependence of the effective deadtime on the load current. The resulting characteristics can then be implemented as a look-up table or analytically approximated using the expression where c 1 , c 2 , m, and n are parameters. The measured and approximated dependence of the effective deadtime on the collector/load current of an IGBT module SKM100GB12T4 from SEMIKRON that is used later in the simulation model is depicted in Figure 2 The actual deadtime is selected as 2 µs. Let us assume that, within SVM, the standard up-down counters are utilized with the top value normalized to one and the period of the signal equal to T PWM . Moreover, suppose that the comparator, which compares the counter with the reference compare value, outputs a logical one when a match during up-count occurs and logical zero when a match during a down-count occurs. Then, to model the voltage distortion of an ideal inverter, the reference compare value d * x for the respective VSI leg (i.e., the SVM output) must be adjusted as where The following well-known expression can be utilized for the reconstruction of the actual phase voltage of a wye-connected machine: where u a , u b , and u c are the respective motor phase voltages and S a , S b , and S c are the logical switching variables (1-high-side switch in the respective inverter leg is on, 0-lowside switch in the respective inverter leg is on) from the comparator. Knowing the effective deadtime of each VSI leg, the nonideal inverter can be modeled using (32)-(34).

Measuring, Modelling, and Compensation of Iron Losses
The last phenomenon examined in this paper is the influence of iron losses on the DFOC and MRAS performance. This section describes the integration of the iron losses measured on a real machine either into a machine or DFOC model. According to Figure 1, the equivalent iron loss current in an arbitrary reference frame can be expressed as where u k m is the voltage across the magnetizing (parallel) branch. The power dissipated in the fictitious iron loss resistance can be expressed as where i k Fe denotes the complex conjugate of the equivalent iron loss current. Substituting (35) into (36), the iron loss resistance is obtained as The iron loss current in the stationary system is obtained using (37) to substitute for the equivalent iron loss resistance in (35): Within the model of the machine, the voltage across the magnetizing branch can be obtained directly using the magnetizing flux vector time derivative. In the control algorithm, the voltage can be approximately calculated as Equation (39) supposes a steady-state operation and considers the fundamental wave only.

Iron Losses Measurement and Model Fitting
The iron losses can be obtained by a series of no-load tests at various fundamental supply frequencies. The separation procedure based on the IEC standard (IEC 60034-2-1: 2014 [49]) can then be used for the loss calculation [50]. For the measurement, the inverter is programmed to generate a fundamental voltage at a given frequency and magnitude that corresponds to the reference stator flux linkage vector magnitude (obtained from the voltage model). For the iron loss modeling, the following analytical function can be used [4] P where f s is the fundamental supply frequency, ψ 1 is the stator flux linkage amplitude, and κ, n, and R Fe0 are the model parameters. Figure 3 shows (40) fitted to the measured losses of a 3.6 kW IM drive (nameplate data and nominal model parameters are given in Table A1).

Improved MRAS Estimators with Included Effect of Iron Losses
The philosophy behind MRAS estimators is that the estimated parameter should somehow influence the adaptive model. In the case of the active power MRAS, the adaptive model directly depends on the stator resistance, making it suitable for the stator resistance estimation. Contrary to that, the adaptive model of the reactive power MRAS used for the rotor resistance estimation depends on the rotor resistance indirectly. By indirect dependence, we mean that the model utilizes quantities affected by the rotor resistance, i.e., the calculated slip speed and d and q-axis current components.
Since most of the MRAS algorithms for the parameter estimation proposed in the literature do not include the effect of iron losses, this section presents improved stator and rotor resistance estimators based on the active (P-MRAS) reactive (Q-MRAS) power.

Improved Reactive Power MRAS with Included Effect of Iron Losses
The reference model of the popular and widely used Q-MRAS for the rotor resistance (or inverse rotor time constant) estimation is given by [10][11][12][13][14] where i dq 1 denotes the conjugated current space vector. Using (20) transformed into the dq reference frame, the stator flux linkage vector can be obtained as The stator voltage equation in the dq reference frame can be written as By substituting (41) into (43) and considering the steady-state operation, we obtain Separating (44) into the real and imaginary parts, respectively, while considering that ψ 2d = L m (i 1d − i Fed ) and ψ 2q = 0, the adaptive model is finally obtained aŝ The synchronous speed is calculated as the sum of the measured speed and estimated slip speed (Equation (18)). The error for the rotor resistance adaptation mechanism is given by The estimated rotor resistance is then the output of the PI controller, i.e., where R 2(init) is the initial rotor resistance. The block diagram of the Q-MRAS estimator with the included iron loss effect is presented in Figure 4. Due to its dependency on the rotor resistance, the current model is used for the transformation angle calculation.

Improved Active Power MRAS with Included Effect of Iron Losses
The reference model of the P-MRAS is given by [10,29,30] Separating (44) into the real and imaginary parts, respectively, and substituting the resulting expression into (48) while considering that ψ 2d = L m (i 1d − i Fed ) and ψ 2q = 0, the adaptive model is obtained aŝ The error for the rotor resistance adaptation mechanism is calculated as The estimated stator resistance is then the output of the PI controller, i.e., where R 1(init) is the initial stator resistance. The block diagram of the modified P-MRAS estimator is presented in Figure 5. Here, the voltage model is selected for the transformation angle calculation due to its dependency on the stator resistance.

Simulation Results
As mentioned in the introduction, this paper aims to examine selected phenomena that impair the performance of motor control algorithms. These include inverter nonlinearity, iron losses, ODE solver type, and sampling time. The presented results are based on simulations in MATLAB/Simulink because:

•
It is not possible to fully separate the influence of the aforementioned adverse effects in an actual application. • Many additional nonlinearities and imperfections are present in a real system. • The exact system parameters are usually not known.

Simulation Setup Description
The principal block diagram of the Simulink model is depicted in Figure 6. The model permits to switch between the machine model with the iron losses (1)-(6) and without the iron losses (8)- (12). The simulated machine is a 3.6 kW IM whose nameplate data and nominal model parameters are given in Table A1. The iron losses are measured and implemented in accordance with Section 5. Both machine models are simulated using the ode4 solver with a fixed-step size equal to 1 µs. The inverter model is implemented using SVM in a linear mode with the possibility to simulate the inverter nonlinearity using the approach described in Section 4. The switching frequency is selected as 10 kHz. Again, the inverter is simulated using the ode4 solver with a fixed-step size 1 µs.
The block diagram of the considered DFOC scheme is depicted in Figure 7. The advantage of the presented DFOC scheme is the ability to estimate the stator and rotor resistance variation in the presence of iron losses using improved estimators. The disadvantage is that machine iron losses and nonlinear inverter characteristics need to be measured and implemented for proper functionality, and additional PI controllers have to be tuned. Moreover, in an actual application, the DFOC performance can be further improved by respecting the saturation of the main flux paths.
The FOC model is simulated as a triggered subsystem to mimic the fact that motor control algorithms are implemented on a discrete system. The type of the ODE solver (FWEM or TR) is selected by setting the appropriate integration method in the Discrete-Time Integrator blocks. The RK4 method is implemented using the MATLAB Function Block. Within FOC, it is also possible to switch between the voltage model and the current model, both either with or without the iron losses. Concerning the MRAS testing, the Q-MRAS is tested with the current model, and the P-MRAS is tested with the voltage model active.
The initial gains of the PI controllers within the FOC were calculated using the optimum modulus method. The obtained values were further adjusted to improve the controllers' performance. The gains of the adaptive PI controllers inside the MRAS estimators were tuned experimentally (the values are given in Table A2).
vantage is that machine iron losses and nonlinear inverter characteristics need to be measured and implemented for proper functionality, and additional PI controllers have to be tuned. Moreover, in an actual application, the DFOC performance can be further improved by respecting the saturation of the main flux paths. The FOC model is simulated as a triggered subsystem to mimic the fact that motor control algorithms are implemented on a discrete system. The type of the ODE solver (FWEM or TR) is selected by setting the appropriate integration method in the Discrete-Time Integrator blocks. The RK4 method is implemented using the MATLAB Function Block. Within FOC, it is also possible to switch between the voltage model and the current model, both either with or without the iron losses. Concerning the MRAS testing, the Q-MRAS is tested with the current model, and the P-MRAS is tested with the voltage model active.
The initial gains of the PI controllers within the FOC were calculated using the optimum modulus method. The obtained values were further adjusted to improve the controllers' performance. The gains of the adaptive PI controllers inside the MRAS estimators were tuned experimentally (the values are given in Table A2).

Influence of VSI Nonlinearities on P-MRAS and Q-MRAS Estimators
This simulation series aims to examine the influence of the inverter nonlinearity on the accuracy of the MRAS estimators. The simulation setup is as follows:

Influence of VSI Nonlinearities on P-MRAS and Q-MRAS Estimators
This simulation series aims to examine the influence of the inverter nonlinearity on the accuracy of the MRAS estimators. The simulation setup is as follows:

•
The sample time of the FOC model (triggered subsystem) is selected as 100 µs (synchronized with PWM).

•
The rotor flux is set to a nominal value.
Within FOC control strategies, it is common that a reference voltage (i.e., the input to the modulator) is utilized instead of a measured voltage. However, when this approach is used, inverter nonlinearities should be compensated appropriately. The influence of the nonideal inverter on the estimation of the stator flux linkage vector α component using (21) is depicted in Figure 8. As expected, the nonlinearities become more significant at small speeds and light loads (Figure 8a). At higher speeds and loads (Figure 8b), the relative influence of the distorting voltage vector decreases. Figure 9 shows the influence of the nonlinear inverter behavior on the stator resistance ( Figure 9a) and rotor resistance (Figure 9b) estimation. The resistances are presented in a per-unit system (indicated by lower-case letters), with the base impedance selected as the ratio of the machine nominal phase voltage and current.  In the case of the stator resistance, the voltage nonlinearity significantly impacts the estimation accuracy because the distorted voltage is utilized not only by the voltage model but also by the P-MRAS reference and adaptive models. The wrong estimate during the low-speed and light-load operation is given mainly by the relatively low ratio of the fundamental and distorting voltage vector. At higher speeds, the relative influence of the stator resistance on the FOC performance decreases, which also impairs the estimator's performance.
The rotor resistance estimation depicted in Figure 9b is, overall, much less sensitive to the voltage distortion because the stator voltage influences the estimator only through the Q-MRAS reference and adaptive models. Like the P-MRAS, the resulting estimates are influenced by the voltage nonlinearity mainly during low-speed and high-speed lowload operations.

Influence of Iron Losses on P-MRAS and Q-MRAS Estimators
The simulation setup is as follows:

•
The inverter is bypassed, i.e., the machine is supplied by a sinusoidal voltage. • The FOC model makes it possible to switch between voltage, current, and synchronous speed estimation with and without iron losses.

•
The sample time of the FOC model (triggered subsystem) is selected as 10 µs.

•
The rotor flux is set to a nominal value. Figure 10 shows the influence of the iron losses on the stator resistance ( Figure 10a) and rotor resistance (Figure 10b) estimation. The iron losses are a non-linear phenomenon whose influence on the flux estimation at different speeds and applied load torques is described by complicated functions [42]. Combining the FOC with MRAS creates a complex system where it is complicated to express the influence of the iron losses on the whole system performance by explicit analytical expression. Overall, it can be stated that the iron losses definitely and not negligibly affect the parameter estimation process and that the influences of the speed and load are interconnected. However, in the case of the P-MRAS, the impact of the iron losses on the estimation accuracy is relatively lower compared to the inverter nonlinearity.

Influence of Discretization and Sampling Time on P-MRAS and Q-MRAS Estimators
The last examined phenomenon is the effect of the discretization and sampling time on the relative error of the estimate. The simulation setup is as follows:

•
The inverter is bypassed, i.e., the machine is supplied by a sinusoidal voltage.

•
The sample time of the FOC model (triggered subsystem) is varied from 10 µs to 300 µs.

•
The FWEM and TR are selected by specifying an integration method in the Discrete-Time Integrator blocks used in the DFOC model. The RK4 method is implemented manually using the MATLAB Function block.

•
The rotor flux is set to a nominal value. Figure 11 shows the influence of the discretization method and sampling time on the resulting relative error of the stator resistance estimation. As expected, the influence of the discretization method grows with the increasing sampling time. Furthermore, the relative error is much more significant at higher speeds because of the higher fundamental and sampling frequency ratio. Since the order of the trapezoidal method is only one order higher than the Euler method, it should be a preferred choice.  Figure 12 shows the results for the Q-MRAS estimator. Here, because of the current model, the RK4 solver is also tested. The results and conclusions for the TR and FWEM are the same as in the case of Figure 11. The RK4 method can increase the estimation accuracy, especially at higher sampling times. Moreover, it is supposed to maintain higher numerical stability during fast transients. If hardware with sufficient computational power, such as FPGA or DSP with FPGA, is used in real applications, the FOC performance can be increased by oversampling. However, this might not always be the most cost-effective solution. The sampling time is often tied to the PWM period in medium-power drives operating with the switching frequencies around 10 kHz, which, according to Figures 11 and 12, can deteriorate the accuracy of the MRAS-type estimators.

Discussion
This paper investigated three actual application phenomena on the accuracy of the popular Q-MRAS and P-MRAS rotor and stator resistance estimators. The studied adverse effects include inverter nonlinearity, iron losses, ODE solver type, and sampling time selection. The main results and contributions of the paper can be summarized as follows:

•
If not adequately accounted for, the inverter nonlinearity has a key influence on the parameter estimation in the case of the P-MRAS and voltage model. The Q-MRAS is also affected by the inaccurate voltage evaluation, but relatively much less.

•
The influence of iron losses on the stator and rotor resistance estimation accuracy is investigated using improved reduced-order IM models and improved Q-MRAS and P-MRAS estimators with the included iron losses. The simulation results show that, if not adequately compensated, the nonlinear phenomenon of iron losses impairs the estimation process and leads to inaccurately identified parameters. The error can become quite significant and is comparable for both the stator and rotor resistance estimation. • Numerical method and sampling time selection can also affect the MRAS-based parameter estimation. When oversampling is impossible, and the drive control system operates with the sampling time around 100 µs, it is recommended to use at least the trapezoidal rule because the improvement compared to Euler's method is significant. The difference between the Runge-Kutta fourth-order method and the trapezoidal rule is not so pronounced. However, the RK4 method is expected to exhibit better numerical stability.
In the case of the inverter nonlinearity and iron losses, the operating conditions, i.e., the applied load and the rotor speed, represent a complicated mixed influence on the stator and rotor estimation accuracy. The analytical description of the observed would represent a highly complex task due to the nonlinearity of the whole system.

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

List of Symbols
The following symbols are used in the paper: i 1 , i 2 , i m , i Fe stator, rotor, magnetizing, and iron loss current space vector (A) u 1 , u m stator and magnetizing voltage space vector (V) ψ 1 , ψ 2 , ψ m stator, rotor, and magnetizing flux linkage space vector (Wb) L 1 , L 2 , L m stator, rotor, and magnetizing inductance (H) L 1σ , L 2σ stator and rotor leakage inductance (H) R 1 , R 2 , R Fe stator resistance, rotor resistance, equivalent iron loss resistance S a , S b , S c logical switching variables (-) T on , T off , T dt , T eff turn-on time, turn-off time, dead-time, and effective dead-time (s) T PWM , T s PWM period, sampling time (s) U DC DC-link voltage (V) d * x , d x , d dist(x) reference, adjusted, and distorting compare value (-) p p number of pole-pairs (-) Ω mechanical rotor speed (rad·s −1 ) J moment of inertia (kg·m 2 ) P, P Fe active power, iron losses (W) Q reactive power (VAr) T, T L , T n motor electromechanical torque, load torque, nominal torque (Nm) σ leakage factor (-); σ = 1 − L 2 m /(L 1 L 2 ) ω, ω s , ω sl , ω n electrical rotor speed, synchronous speed, slip speed, nominal speed (rad·s −1 ) * reference value a, b, c notation of stator phases d, q real and imaginary axis of the synchronous reference frame k notation of a general reference frame α, β real and imaginary axis of the stator-fixed reference frame Appendix A