Estimation of the Induction Motor Stator and Rotor Resistance Using Active and Reactive Power Based Model Reference Adaptive System Estimator

Featured Application

The presented parameters estimation method of an induction motor can be implemented in the applications of electric vehicles, traction systems and safety-critical drive systems. The method is based on signals which are easy to measure in the scalar and vector controlled drives.

Abstract

In this paper an induction motor parameters estimator, based on the Model Reference Adaptive System (MRAS), is presented and described. A comprehensive literature study on MRAS type parameters estimators for induction motors is also provided. The authors propose a novel PQ-MRAS estimator concept which enables the simultaneous calculation of stator and rotor resistances in the induction machine, which is its major advantage over previous investigations. The estimator employs the active (P) and reactive (Q) power of the machine which is calculated by the only measurable signals, such as stator voltage and current. The paper includes a detailed description of the proposed estimator. The PQ-MRAS was tested for various operating conditions of a drive system with the induction motor. The results obtained from computer simulation tests were verified on the laboratory set up with a DS1202 card.

1. Introduction

The estimation of induction motor (IM) parameters is an essential research topic in the investigations on electrical drive systems, especially in the case of drives requiring high operational performance and precision. Modern control methods, such as the Direct Field Oriented Control (DFOC) and the Direct Torque Control (DTC) require the information about the internal state variables of the motor. Such signals as an electromagnetic flux, torque (or rotational speed for fully sensorless drives) cannot be measured directly, therefore they are estimated [1,2,3]. A major fraction of state variable estimators requires accurate knowledge about machine parameters [1,4]. The most crucial parameters of the induction motor is stator winding resistance and rotor winding resistance which fluctuate due to temperature variations of the machine [5,6]. Other important parameters encompass inductances, but they are immune to temperature variations [5].

The information about motor parameters values can be used also for the condition monitoring of machines [7,8]. One of the used diagnostic methods is the assumption that the fluctuations of parameters can indicate a machine fault [9] (rotor bars rupture for squirrel-cage motors or stator windings short-circuit).

Recently, numerous methods for induction motor parameters estimation have been developed. In general, they can be subdivided into two main categories [5]:

• Off-line estimators—parameters are calculated at the standstill state of the machine (self-commissioning drives), these methods are often called parameters identification methods;
• On-line estimators—parameters are calculated during the normal operation of the machine. These methods are based mostly on: Kalman Filters [10], state observers [11], MRAS [5,12] or spectral techniques [5].

The paper discusses MRAS based estimators, therefore in the further part of the introduction a brief description of this estimation method is given. A typical MRAS estimator (Figure 1) includes two basic subsystems which calculate the same base signal in different ways [12] (the same base signal is expressed by two mathematical models):

• Reference model which should be independent of the estimated parameter;
• Adjustable model which is directly (sometimes indirectly) influenced by the estimated parameter.

The general idea of this estimation algorithm relies on the minimization of the error between a reference and adjustable model by an adaptation mechanism. If the estimated parameter is equal to its reference value, the error becomes minimized.

From the base signal standpoint, MRAS type parameter estimators can be classified into several major categories:

• Electromagnetic flux-based (F-MRAS)—proposed for the first time in [13] for speed estimation. An estimator of this type can be used in stator resistance [14] or rotor resistance estimation [15], due to the fact that it consists of two basic estimators of the flux: a voltage model and a current model. The former depends directly on the stator resistance value, therefore it is used as an adjustable model in F-MRAS stator resistance estimators (in that case the current model is the reference). The current model depends directly on the rotor resistance, hence it is used as an adjustable model in F-MRAS rotor resistance estimators (then the voltage model is the reference). The major shortcomings of F-MRAS estimators are that neither of the models is a measurable quantity; estimators directly depend on both resistances simultaneously, however, only one of them can be estimated.
• Reactive power-based (Q-MRAS)—proposed in [16] for rotor resistance estimation. This estimator is independent of stator resistance. It is implemented mainly in a synchronous reference frame (x-y), which requires the estimation of synchronous speed. The reference and adjustable models use internal signals from the control structure as inputs. Quite frequently, a simplified adjustable model is used (based on control method assumptions). This results in the strong dependence of estimator operation on the control system performance. In [17], an estimator was implemented in stationary reference frame (α-β), therefore it was based only on measurable signals—independently of the internal signals from a control scheme. In [18] Q-MRAS was proposed for speed estimation.
• Active power based (P-MRAS)—proposed in [19] for stator resistance estimation. Similarly to Q-MRAS, it is implemented only in the synchronous reference frame (x-y), which determines a strong reliance on the control structure performance. In [20] a semi-active power estimator is proposed to improve robustness on rotor resistance variation. In [21] it was used for rotor resistance estimation.
• X based (X-MRAS)—proposed in [22] for stator resistance estimation. Here X is a fictitious quantity based on stator voltage and current. It was implemented only in a synchronous reference frame. It can be used for synchronous speed estimation [22].
• PY based (PY-MRAS)—proposed in [23] for stator resistance estimation. It relies on the active power and Y which is a fictitious quantity (based on stator voltage and current). The major benefit of this estimator is that the adjustable model is independent of the rotor speed. However, only simulation results were presented.
• Electromagnetic torque (T-MRAS)—proposed in [21] for rotor resistance estimation. Its major shortcoming is the fact that both reference and adjustable models are non-measurable quantities. The reference model depends on the stator resistance.
• Back electromagnetic force (back-EMF MRAS)—proposed in [18] for speed estimation. In [24] it was proposed for the stator resistance estimation. Its major advantage is that pure integration is not required through the implementation of the estimator. The reference model depends directly on rotor resistance. In [25], an improvement of estimator stability was proposed. However, the estimator employs an internal signal from the control structure (implemented in the (x-y) reference frame).

To sum up, MRAS based estimators for induction motor parameters have been developed in the last few decades. As a result, it is possible to estimate the stator or rotor resistance of a machine. However, the research on a MRAS type estimator which enables the estimation of both resistances simultaneously is still insufficient. In [26], an F-MRAS type estimator was proposed for the simultaneous estimation of rotor and stator resistance, and also the estimation of motor rotational speed. The major drawback of the proposed approach is the fact that both (adjustable and reference) models are directly dependent on the estimated parameters. Therefore, incorrect estimation of at least one parameter may deteriorate the performance of the entire estimator. Additionally, both models of the proposed MRAS are not measurable signals, in consequence it is not possible to check how calculated signals correspond to their real values—estimated parameters are calculated using estimated signals.

This article presents a new PQ-MRAS estimator which enables the estimation of both resistances simultaneously. It is based on the active and reactive powers of the induction motor. The main benefits of proposed estimator are that reference models are calculated using only measurable signals, such as current, voltage and rotor speed. In the paper, the estimator was tested for different conditions through computer simulations as well as experimental tests.

It is necessary to point out that the approaches based on the Kalman Filter enable a simultaneous, highly accurate estimation of motor parameters and state variables [27,28]. However, this method is characterized by high computational requirements and a complicated design process. These are its major drawbacks compared to MRAS systems.

A MRAS estimator is a system which is easy to parametrize and can be quickly implemented on simple microprocessor systems. The main goal of the research was to develop an estimation system that is easier to tune than the commonly known estimators. It was assumed that only simple state variable simulators would be used to develop such a system. Complex observers or EKFs cannot be compared with such systems. All systems have their pros and cons.

The article is composed of several sections. In Section 2 a mathematical model of an induction motor as well as a detailed description of the PQ-MRAS estimator are presented [29]. Section 3 includes the explanation of the vector control structure for the induction motor, while Section 4 the results of simulation tests and experimental verification. Section 5 discusses the obtained results.

2. Mathematical Models of the Induction Motor and the PQ-MRAS Estimator

In this section the mathematical models of an induction motor and a PQ-MRAS estimator are presented. A detailed description of the proposed estimator is also provided.

2.1. Mathematical Model of the Induction Motor

The model of the three-phase squirrel cage induction motor can be expressed in the stationary reference frame (α-β) by the following equations [1]:

$$\frac{\mathrm{d}{\mathsf{\psi }}_{\mathbf{s}}}{\mathrm{d}\mathrm{t}}={\mathbf{u}}_{\mathbf{s}}-R_{\mathrm{s}}{\mathbf{i}}_{\mathbf{s}},$$

(1)

$$\frac{\mathrm{d}{\mathsf{\psi }}_{\mathbf{r}}}{\mathrm{d}\mathrm{t}}=-R_r{\mathbf{i}}_{\mathbf{r}}+\mathrm{j}{p}_b{\omega }_m{\mathsf{\psi }}_{\mathbf{r}},$$

(2)

$${\mathbf{i}}_{\mathbf{s}}=\frac{L_r}{w}{\mathsf{\psi }}_{\mathbf{s}}-\frac{L_m}{w}{\mathsf{\psi }}_{\mathbf{r}},$$

(3)

$${\mathbf{i}}_{\mathbf{r}}=\frac{L_{\mathrm{s}}}{w}{\mathsf{\psi }}_{\mathbf{r}}-\frac{L_m}{w}{\mathsf{\psi }}_{\mathrm{s}}.$$

(4)

$${\mathit{t}}_e =\frac{3}{2}{p}_b\mathrm{I}\mathrm{m}\left\{{\mathsf{\psi }}_{\mathbf{s}}^{*}\cdot {\mathbf{i}}_{\mathbf{s}}\right\}$$

(5)

$$\frac{\mathrm{d}{\omega }_m}{\mathrm{d}\mathrm{t}}=\frac{1}{J}\left({\mathit{t}}_e-{\mathit{t}}_l\right)$$

(6)

where u_s = u_sα + ju_sβ—stator voltage vector, i_s = i_sα + ji_sβ—stator current vector, i_r = i_rα + ji_rβ—rotor current vector, ψ_r = ψ_rα + jψ_rβ—rotor electromagnetic flux vector, ψ_s = ψ_sα + jψ_sβ—stator electromagnetic flux vector, ω_m—rotational shaft speed, t_e—electromagnetic torque, t_l—load torque. R_s, R_r—stator, rotor resistances respectively; L_s, L_r, L_m—stator, rotor, magnetizing inductances, respectively; p_b—number of pole pairs, J—inertia of the shaft, w = L_sL_r − L_m².

It is necessary to point out that in the presented model, such phenomena as motor losses and magnetic saturation are not considered.

2.2. Mathematical Model of the PQ-MRAS Estimator

The proposed estimator consists of two subsystems:

• P-MRAS—based on the active power (P) of the machine; it is used to calculate the stator resistance.
• Q-MRAS—based on the reactive power (Q) of the machine; it is used to calculate the rotor resistance.

Each subsystem has its own independent adaptation mechanism based on a PI (proportional–integral) controller. The common part of P and Q-MRAS is the rotor flux estimator.

A reference model of the P-MRAS is given by:

$$P^{\mathrm{ref}}=\mathrm{R}\mathrm{e}\left\{{\mathbf{u}}_{\mathbf{s}}\cdot {\mathbf{i}}_{\mathbf{s}}{}^{*}\right\}=u_{s\alpha }{i}_{s\alpha }+u_{s\beta }{i}_{s\beta }$$

(7)

In an adjustable model, estimated stator voltage is used instead of the measured one. Based on Equations (1), (3) and (4), the estimated stator voltage vector can be expressed as:

$${\mathbf{u}}_{\mathbf{s}}{}^{\mathrm{e}\mathrm{s}\mathrm{t}}=R_{\mathrm{s}}{\mathbf{i}}_{\mathbf{s}}+\frac{L_m}{L_r}\frac{\mathrm{d}{\mathsf{\psi }}_r{}^{\mathrm{est}}}{\mathrm{d}\mathrm{t}}+\sigma L_{\mathrm{s}}\frac{\mathrm{d}{\mathbf{i}}_{\mathbf{s}}}{\mathrm{d}\mathrm{t}}$$

(8)

Finally, the formula of P-MRAS adjustable model is:

$$P^{\mathrm{adj}}=\mathrm{R}\mathrm{e}\left\{{\mathbf{u}}_{\mathbf{s}}{}^{\mathrm{est}}\cdot {\mathbf{i}}_{\mathbf{s}}{}^{*}\right\}=R_{\mathrm{s}}{}^{\mathrm{est}}\left(i_{s\alpha }{}^2+i_{s\beta }{}^2\right)+\sigma L_{\mathrm{s}}\left(i_{s\alpha }\frac{\mathrm{d}{i}_{s\alpha }}{\mathrm{d}\mathrm{t}}+i_{s\beta }\frac{\mathrm{d}{i}_{s\beta }}{\mathrm{d}\mathrm{t}}\right)+\frac{L_m}{L_r}\left(i_{s\alpha }\frac{\mathrm{d}{\mathsf{\psi }}_{ra}{}^{\mathrm{est}}}{\mathrm{d}\mathrm{t}}+i_{s\beta }\frac{\mathrm{d}{\mathsf{\psi }}_{r\beta }{}^{\mathrm{est}}}{\mathrm{d}\mathrm{t}}\right)$$

(9)

It can be observed that the value of active power, calculated by the adjustable model, directly depends on stator resistance. Therefore, the role of stator resistance adaptation mechanism is the minimization of the error between the two models:

$${\epsilon }_{\mathrm{P}}=P^{\mathrm{ref}}-P^{\mathrm{a}\mathrm{d}\mathrm{j}},$$

(10)

$$R_{\mathrm{s}}{}^{\mathrm{est}}=K_{PR\mathrm{s}}{\epsilon }_P+K_{IR\mathrm{s}}{\displaystyle \int {\epsilon }_P\mathrm{d}t}$$

(11)

where K_PRs, K_IRs—coefficients of the proportional and integral terms, respectively.

Rotor flux components in (9) are derived from the current model estimator [1]:

$${\mathsf{\psi }}_r{}^{\mathrm{est}}={\displaystyle \int \left(\frac{R_r}{L_r}\left(L_m{\mathbf{i}}_{\mathbf{s}}-{\mathsf{\psi }}_r{}^{\mathrm{est}}\right)+j{p}_b{\omega }_m{\mathsf{\psi }}_r{}^{\mathrm{est}}\right)\mathrm{d}t}.$$

(12)

The reference model of the Q-MRAS is given by:

$$Q^{\mathrm{ref}}=\mathrm{Im}\left\{{\mathbf{u}}_{\mathrm{s}}\cdot {\mathbf{i}}_{\mathrm{s}}{}^{\ast }\right\}=u_{s\beta }{i}_{s\alpha }-u_{s\alpha }{i}_{s\beta },$$

(13)

whereas the adjustable model of this subsystem uses the estimated value of the stator voltage (8):

$$Q^{\mathrm{adj}}=\mathrm{Im}\left\{{\mathbf{u}}_{\mathrm{s}}{}^{\mathrm{est}}\cdot {\mathbf{i}}_{\mathrm{s}}{}^{\ast }\right\}=\sigma L_{\mathrm{s}}\left(i_{s\alpha }\frac{\mathrm{d}{i}_{s\beta }}{\mathrm{d}t}-i_{s\beta }\frac{\mathrm{d}{i}_{s\alpha }}{\mathrm{d}t}\right)+\frac{L_m}{L_r}\left(i_{s\alpha }\frac{\mathrm{d}{\mathsf{\psi }}_{r\beta }^{\mathrm{est}}}{\mathrm{d}t}-i_{s\beta }\frac{\mathrm{d}{\mathsf{\psi }}_{r\alpha }^{\mathrm{est}}}{\mathrm{d}t}\right).$$

(14)

In this model the dependence on the rotor resistance is implicit. It consists of the rotor flux components which are calculated by the current model (12) which directly depends on the value of this parameter. The rotor resistance adaptation mechanism minimizes the error between the reference and adjustable models:

$${\epsilon }_Q=\left|Q^{ref}\right|-\left|Q^{a\mathrm{d}j}\right|,$$

(15)

$$R_r{}^{\mathrm{est}}=K_{PRr}{\epsilon }_Q+K_{IRr}{\displaystyle \int {\epsilon }_Q\mathrm{d}t}.$$

(16)

where K_PRr, K_IRr—coefficients of the proportional and integral terms, respectively. The sign of the reactive power is changed due to the shaft rotational speed reversals, therefore in the adaptation mechanism the absolute values of Q^ref and Q^adj are used. In Figure 2 a block diagram of the PQ-MRAS estimator is shown.

3. Description of the Vector Control Structure for Induction Motor

During the tests, the induction motor was controlled by a closed-loop vector control structure known as the Direct Field Oriented Control—DFOC. A block diagram of the control structure is shown in Figure 3.

This method involves the independent control of the flux and torque of the machine by two decoupled control paths [6]. The implementation of this control method requires the transformation of the stator current vector to the synchronous reference frame (x-y) rotating with the speed of the rotor flux vector. The rotor flux is controlled by the i_sx component of the stator current, whereas the electromagnetic torque is controlled by the i_sy stator current component. The reference frame transformation requires the information about the actual position of the rotor flux vector. During the simulation and experimental tests, the current model was used for the rotor flux vector estimation [1]:

$${\mathsf{\psi }}_r{}^{\mathbf{i}}={\displaystyle \int \left(\frac{R_r}{L_r}\left(L_m{\mathbf{i}}_{\mathbf{s}}-{\mathsf{\psi }}_r{}^{\mathbf{i}}\right)+j{p}_b{\omega }_m{\mathsf{\psi }}_r{}^{\mathbf{i}}\right)\mathrm{d}t}.$$

(17)

Instead of the current model, the voltage model can be used in the vector controlled IM drives for rotor or stator flux estimation:

$${\mathsf{\psi }}_r{}^v=\frac{L_r}{L_m}{\displaystyle \int ({\mathbf{u}}_{\mathbf{s}}-R_{\mathrm{s}}{\mathbf{i}}_{\mathbf{s}})\mathrm{d}t}-\frac{w}{L_m}{\mathbf{i}}_{\mathbf{s}},$$

(18)

This estimator directly depends on the stator resistance value. Therefore, it can be used to evaluate P-MRAS operation in the closed-loop mode.

Voltage vector components were calculated using control signals and measured DC bus voltage from a Voltage Source Inverter (VSI):

$$u_{s\alpha }=\frac{2}{3}{u}_{\mathrm{d}}\left(S_A-\frac{1}{2}\left(S_B+S_C\right)\right),$$

(19)

$$u_{s\beta }=\frac{\sqrt{3}}{3}{u}_d\left(S_B+S_C\right).$$

(20)

The PQ-MRAS estimator can operate in two modes (Figure 4):

• Open-loop mode—independently of the control structure;
• Closed-loop mode—coupled with the control structure; the estimated parameters are injected to the control structure (flux estimator).

4. Simulation and Experimental Results

This section includes the results of both the simulation tests and experimental verification of the PQ-MRAS estimator. The estimator was examined for different conditions of the drive operations, such as: load torque and speed variations (including speed reversals). The rated data of the tested induction motor are summarized in

Table A1. IM Data.

Symbol	Rated Data	Value	Unit
PN	Power	1.1	kW
UN	Stator voltage	400	V
IN	Stator current	2.8	A
nN	Mechanical speed	1360	rpm
tN	Torque	7.7	Nm
Rs	Stator resistance	5.9	Ω
Rr	Rotor resistance	4.5	Ω
Ls	Stator inductance	451.0	mH
Lr	Rotor inductance	451.0	mH
Lm	Magnetizing inductance	424.4	mH
J	Inertia	0.0143	kg∙m2
pb	Pole pairs	2	-
η	Efficiency	0.79	-

(Appendix A). During the simulation and experimental tests, the values of adaptation mechanism coefficients were the same (Appendix B). State variables from the drive, such as: rotational speed, electromagnetic torque, rotor flux and stator current are presented along with internal signals from the PQ-MRAS estimator: active and reactive powers, estimated stator and rotor resistances. Each of the signals and state variables is normalized to its nominal value.

4.1. Simulation Results

Simulation tests were performed using MATLAB/Simulink software (version 2018b, The MathWorks, Natick, MA, USA). During the simulations, the estimator was tested for the open-loop as well as closed-loop operation modes. In a simulation model, effects such as signal offsets, measurements noise and inverter nonlinearity were not taken into consideration.

4.1.1. Operation of the PQ-MRAS in Open-Loop Mode

The main goal of the first test was to check the general performance of the PQ-MRAS. The results are presented in Figure 5 and Figure 6. It can be observed that both estimated parameters tracked real values with high accuracy.

Even during speed reversal variations the estimated parameters were negligible (maximum estimation error for stator resistance increased to 2%). During the second test, the robustness of the estimator to load torque variations was considered. The obtained results (Figure 7 and Figure 8) show that both parameters were calculated with high accuracy even for load variations.

The aim of the next analysis was to check how the values of adaptation mechanism coefficients impact the estimator performance. Figure 9 shows a comparison of estimator signals for different sets of coefficients: nominal values (Appendix B), nominal values multiplied by 5 and nominal values divided by 5. The coefficient values have a major impact on the dynamics of the estimator, nevertheless it operated in a stable manner.

During the next test, it was assumed that both resistances increased up to 150% of their nominal values, which was related to the changes of machine windings temperatures. It can be seen that both resistances were estimated correctly (Figure 10). Due to rotor resistance variations, the estimation error of the rotor flux can be observed (Figure 11), because the current model is directly influenced by this parameter (17). The inaccurate calculation of the rotor flux vector results in the coupling of flux and torque control paths, which consequently results in the deterioration of drive performance.

4.1.2. Operation of the PQ-MRAS in Closed-Loop Mode

During further tests, the estimator was operating in the closed-loop mode. Both estimated parameters tracked their real values (Figure 12).The estimated value of the rotor resistance was injected into the control system (to the rotor flux estimator—to the current model) at t = 12 s. It resulted in the elimination of the flux error estimation and the decoupling of the control paths (Figure 13).

A similar study was performed with a voltage model [1] instead of the current model as the estimator of the rotor flux (18). This estimator directly depends on the stator resistance value (current model depends on rotor resistance). Therefore, it can be used to evaluate P-MRAS operation in the closed-loop mode. The estimated stator resistance was injected into the control system at t = 11 s, which resulted in the improvement of the rotor flux estimation accuracy (Figure 14 and Figure 15).

4.2. Experimental Results

The experimental verification of the PQ-MRAS estimator was carried out on the laboratory setup shown in Figure 16. The setup consisted of two induction motors coupled by a clutch—one motor was controlled by the DFOC algorithm, the other one worked as a load. The control algorithm and PQ-MRAS were implemented in a rapid prototyping unit dSPACE MicroLabBox (dSPACE Inc., Wixom, MI, USA) through the MATLAB/Simulink software (version 2015b, The MathWorks). The drive system was controlled in real-time by an application created in the ControlDesk software (version 5.4, dSPACE Inc., Wixom, MI, USA).

During the experiments, the estimator was tested for the open-loop operation mode. The rotor flux vector was estimated by the current model, whereas an electromagnetic torque was calculated by:

$$t_e{}^{est}=\frac{3}{2}{p}_b\frac{L_m}{L_r}\mathrm{I}\mathrm{m}\left\{{\mathsf{\psi }}_{\mathbf{r}}^{\mathbf{i}*}\cdot {\mathbf{i}}_{\mathbf{s}}\right\}.$$

(21)

During the first test, the general performance of the PQ-MRAS was analyzed. It can be observed that the obtained results correspond with the simulation results (Figure 17 and Figure 18). It should be noted, however, that neither of the estimated parameters exactly matched its nominal values. Nevertheless, the estimation error for rotor resistance was about 5% and for stator resistance it was about 10%.

The main aim of the second test was to check estimator robustness to load torque variations (Figure 19 and Figure 20). The values of both estimated resistances fluctuated according to load variations but in limited ranges (3% for rotor resistance, 5% for stator resistance).

Subsequently, a speed reversal test was carried out (Figure 21 and Figure 22). The estimator operated stably, yet the fluctuations of estimated parameters were noticeable—especially during transients (about 5% for rotor resistance, ±10% for stator resistance).

During the last test, the influence of the values of adaptation mechanisms coefficients on the estimator operation was analyzed. The coefficients were changed it the same way as during the simulation test. A noticeable influence on the dynamics was visible, however, the estimator operated stably for each of the sets of parameters (Figure 23).

5. Discussion

Based on the presented results, it can be concluded that the experimental results correspond with those obtained through simulations. It should be noted that the estimation errors of resistances were greater for experiments than for simulations, however, they did not exceed 10%. These discrepancies resulted from the simplifications of the simulation model.

Nevertheless, the main contributions of the presented research as regards MRAS estimators are:

• simultaneous estimation of stator and rotor resistance;
• estimator requires only measurable input signals (independence of the control structure).

Additional attributes of the presented estimator include:

• stable estimation during speed reversals (including zero-speed range) and load torque variations;
• estimation of parameters during steady-states and transients;
• variations of adaptation mechanism coefficients impact only subsystem dynamics; the estimator works stably.

Further research directions will focus on employing an analytical method to adaptation mechanism coefficients setting. The impact of other parameters, such as inductances, on the estimator performance will be also taken into the consideration.