Novel Advanced Artificial Neural Network-Based Online Stator and Rotor Resistance Estimator for Vector-Controlled Speed Sensorless Induction Motor Drives

Abstract

This paper presents a new approach for the online estimation of stator and rotor resistance of induction motors for speed sensorless vector-controlled drives, using feed-forward artificial neural networks with advanced adaptive learning rates. For the rotor resistance estimation, a neural network model based on rotor speed and stator currents is developed. The rotor flux linkages acquired from the voltage model are compared with the neural network model. The feed-forward neural network employs an adaptive learning rate as the function of the obtained error during training for quick convergence with minimal estimation error. A two-layered neural network model based on the stator voltage and current equations is developed for the stator resistance estimation. The d-q axes stator currents obtained from the developed model are compared with the acquired d-q axes stator currents. For the fast convergence with minimal estimation error, an adaptive learning rate as the function of error is adopted during training. Furthermore, the neural network estimates the induction motor’s speed. The simulation and experimental results justify that the developed algorithms track variation in the resistances quickly and precisely along with the speed as compared with the conventional constant learning rate algorithm, leading to reliable operation of the drive.

1. Introduction

Induction motors are extensively used motors in industrial environments due to their enormous advantages over similar categories of motors. Due to the vast advantages of excellent performance, rigorousness, and minimal maintenance, induction motors are the best alternative for DC motors. The speedy development in semiconductor and control technologies resulted in replacing DC motors with induction motors for similar applications in most industries. With the advancement in power electronics, the majority of speed control-related issues have been resolved. Nearly half of the energy generated in the world is consumed by electrical drives. Improvement in the energy efficiency of the drive systems is of great concern, where a minute improvement in energy efficiency reduces power loss and fuel burn [1]. Inverter-fed induction motors are extensively used for speed control where the stability of the drive is of more significant concern and is to be addressed together with the solutions to reduce energy consumption [2,3,4,5].

The scalar and vector control strategies are generally employed for the speed control of Induction Motor Drives (IMD); even though they are easy to implement, scalar-controlled drives suffer from inherent coupling effects and sluggish response, and are extremely prone to system instability. Despite the involvement of a complex control strategy, vector control or field orientation control has a wide range of advantages; a few are better dynamic response, higher efficiency over a wide range of speed, exploiting full torque capability at low speeds, and decoupled control of flux and torque [6]. With regard to field orientation control, accurate estimation of machine parameters is essential to achieving an excellent dynamic response as the slip relation is essential for the implementation of an indirect field-oriented scheme, which is a feed-forward technique, parameter sensitive especially to rotor resistance. The determination of rotor speed depends on the rotor flux estimation. The influence of rotor resistance on the estimation of rotor flux, which is the essential control parameter, is extremely significant [7]. For the speed estimation employing an induction motor model, accurate estimation of the motor parameters is essential. To achieve minimal speed estimation error and to eliminate the unstable operation of the drive, online stator resistance estimation is needed. With the rotor heating, rotor resistance can rise to 100%. It is highly undesirable to obtain rotor resistance value with the temperature sensor or thermal model [8].

Several rotor and stator resistance estimation algorithms are available in the literature. The most popular approaches are based on Extended Kalman Filters (EKFs) [9,10,11,12,13], the Model Reference Adaptive System (MRAS) [14,15,16], Sliding Mode Control [17,18,19], Luenberger Observer [20,21,22], and artificial intelligence-based techniques such as artificial neural networks [23,24] fuzzy—neural networks, etc. [25,26]. In recent days, several modifications have been proposed to the traditional methods with different combinations as available in the recent literature. The MRAS approach for the estimation of machine parameters is popular as the other traditional approaches are computationally intensive, with noise influence and complexity in practical realization.

Several MRAS-based techniques are available for the control of sensorless vector-controlled IMD, which are primarily based on the error minimization of the signals as Rotor flux error-based MRAS [27,28], Back-EMF error-based MRAS [29,30], Reactive power error-based MRAS [31,32,33], Stator current error-based MRAS [34,35], X-MRAS [36,37], and High-Frequency Injection Method (HFIM) based MRAS [38,39,40]. The conventional MRAS approach uses an integral or proportional-integral controller. Since the modern high-speed DSPs have made the implementation of computationally intensive tools such as Artificial Neural Network (ANN), fuzzy logic, etc. simple, modern advanced MRAS techniques incorporate ANN and fuzzy logic. The literature is available on the review of the performance of various MRAS-based techniques [41,42].

As ANNs are capable of approximating nonlinear functions with a great degree of accuracy, their application for identifying and controlling nonlinear dynamic systems is increased. The replacement of the adaptive model of conventional RF-MARS by neural networks in speed estimation of IMD reduces the computational efforts and is also immune to the effect of stator and rotor resistance variation on the system. A great improvement in the performance of the speed estimator is achieved, particularly at low speeds, both with open-loop and closed-loop conditions [43,44]. The online rotor and stator resistance estimation approaches are proposed, where a fixed learning rate is adopted during the estimation approach [45,46]. The selection of an inappropriate learning rate leads to output errors with slow convergence of the error during network training.

The online artificial neural network-based MRAS-based stator and rotor resistance estimator employing varying learning rates throughout the training as a function of obtained error during training is proposed. The adjustable learning rate based on the error function reduces the estimation time and improves the induction motor drive’s control quality. Both the stator and rotor resistance estimators, shown in Figure 1, make use of two state estimate variable models: one delivers the induction motor’s real output states, while the subsequent one provides corresponding neural network model states. The error obtained between the output states of the two models is back propagated by adjusting the weights of the neural network model until the error between the output states of the two models reaches the acceptable minimum value. The weights obtained at the end of the training are used to estimate the resistance values. The standard two-layer feed-forward neural network model and two-layer recurrent feed-forward neural network model are employed for the rotor and stator resistance estimations, respectively. The effectiveness of the developed rotor and stator estimators for the online estimation is justified by the simulation studies performed on the stator reference frame model of an induction motor using MATLAB-SIMULINK and presented. The motor parameters of a 3.7 kW induction motor are determined by performing the no-load and blocked rotor tests and developing the equivalent circuit. The performance of the developed algorithms for the online estimation of stator and rotor resistances, along with speed estimation, are tested experimentally on a 3.7 kW squirrel cage induction motor drive operated with rotor flux orientation controller. The experimental results demonstrate the capability of the proposed algorithm to estimate resistances along with speed precisely.

2. Artificial Neural Network-Based Rotor Resistance Estimator

The ANN-based MRAS structure for the rotor resistance estimation is presented in Figure 2.

The rotor flux vectors are estimated by two separate observers depicted as the voltage model and current model of the induction motor. The d- and q-axes rotor flux equations are derived by assessing the stator side of the equivalent circuit developed from the stationary reference frame model as in Equation (1), referred to as the voltage model equation [47,48].

$$\left[\begin{array}{c}\frac{d{\psi }_{dr}^s}{dt}\\ \frac{d{\psi }_{qr}^s}{dx}\end{array}\right]=\frac{L_r}{L_m}\left\{\left[\begin{array}{c}{v}_{ds}^s\\ v_{qs}^s\end{array}\right]-R_s\left[\begin{array}{c}{i}_{ds}^s\\ i_{qs}^s\end{array}\right]-\sigma L_s\left[\begin{array}{c}\frac{d}{dt}{i}_{ds}^s\\ \frac{d}{dt}{i}_{qs}^s\end{array}\right]\right\}$$

(1)

where $\sigma =1-\frac{L_m^2}{L_s L_r}$ is the leakage factor of the machine.

By assessing the rotor side of the equivalent circuit, the d-q axes rotor flux equations, also known as current or neural network model equation, is obtained, represented in Equation (2).

$$\overrightarrow{\frac{d}{dt}{\psi }_r^{s,im}}=\left[-\frac{1}{T_r}I+{\omega }_{r}J\right]\overrightarrow{{\psi }_r^{s,im}}+\frac{L_m}{T_r}\overrightarrow{i_{dq}^s}$$

(2)

where $T_r=\frac{L_r}{R_r}$ is the rotor time constant,

$$I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]; J=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right];\overrightarrow{i_{dq}^s}=\left[\begin{array}{c}{i}_{ds}^s\\ i_{qs}^s\end{array}\right];\overrightarrow{{\psi }_r^{s,im}}=\left[\begin{array}{c}{\psi }_{dr}^{s,im}\\ {\psi }_{qr}^{s,im}\end{array}\right];\overrightarrow{ {\psi }_r^{s,vm}}=\left[\begin{array}{c}{\psi }_{dr}^{s,vm}\\ {\psi }_{qr}^{s,vm}\end{array}\right]$$

The sample data model of Equation (2) is represented by Equation (3) as

$$\overrightarrow{{\psi }_r^{s,nm}\left(k\right)}=\left(W_{1}I+W_2 J\right)\overrightarrow{{\psi }_r^{s,nm}\left(k-1\right)}+W_3\overrightarrow{i_{dq}^s}$$

(3)

where $W_1=1-\frac{T_s}{T_r}$; $W_2={\omega }_r{T}_s$ and $W_3=\frac{L_m}{T_r}{T}_s$.

T_s is sampling time.

Equation (3) can be represented as

$$\overrightarrow{{\psi }_r^{s,nm}}=W_1{X}_1+W_2{X}_2+W_3{X}_3$$

(4)

where $X_1=\left[\begin{array}{c}{\psi }_{dr}^{s,nm}\left(k-1\right)\\ {\psi }_{qr}^{s,nm}\left(k-1\right)\end{array}\right]$; $X_2=\left[\begin{array}{c}-{\psi }_{qr}^{s,nm}\left(k-1\right)\\ {\psi }_{dr}^{s,nm}\left(k-1\right)\end{array}\right]$; $X_3=\left[\begin{array}{c}{i}_{ds}^s\left(k-1\right)\\ i_{qs}^s\left(k-1\right)\end{array}\right]$.

Based on Equation (4), a two-layer neural network is developed, presented in Figure 3. X₁, X₂, and X₃ represent the inputs to the network, and W₁, W_2, and W₃ represent the network weights. Since W₂ is independent of the rotor resistance term, during the training of the network, weights W₁ and W₃ are required to be updated.

The square function of the rotor flux error between the reference model and the neural network model is

$$E_r=\frac{1}{2}{\in }_r^2\left(k\right)=\frac{1}{2}{\left\{\overrightarrow{{\psi }_r^{s,vm}}-\overrightarrow{{\psi }_r^{s,nm}}\right\}}^2$$

(5)

During training, weights W₁ and W₃ are updated to obtain the minimum square function as

$$W_1\left(k\right)=W_1\left(k-1\right)+{\eta }_1\Delta W_1\left(k\right)$$

(6)

$$W_3\left(k\right)=W_3\left(k-1\right)+{\eta }_3\Delta W_3\left(k\right)$$

(7)

Using the generalized delta rule, the weight adjustment factors ∆W₁(k) and ∆W₃(k) are evaluated as

$$\Delta W_1\left(k\right)=-\frac{\partial E_r}{\partial W_1}={\left[\overrightarrow{{\psi }_r^{s,vm}}\left(k\right)-\overrightarrow{{\psi }_r^{s,nm}}\left(k\right)\right]}^T\overrightarrow{{\psi }_r^{s,nm}}\left(k-1\right)$$

(8)

$$\Delta W_3\left(k\right)=-\frac{\partial E_r}{\partial W_3}={\left[\overrightarrow{{\psi }_r^{s,vm}}\left(k\right)-\overrightarrow{{\psi }_r^{s,nm}}\left(k\right)\right]}^T\overrightarrow{i_{dq}^s}\left(k-1\right)$$

(9)

With experience gained from the repeated training of the network, learning rates η₁ and η₂ are selected to obtain a minimum error.

The adaptive learning rate algorithm is developed, where the adaptive learning rate substitutes the constant learning rate throughout the training, for quick convergence and reduced estimate error. During each training iteration, weights are updated based on the modified learning rate to reduce the error E_r. The learning rate is updated during training determined by the product of weight adjustment factors at k and (k − 1) iteration given by φ_i(k) = ΔW_i(k) × ΔW_i(k − 1).

By using the bipolar sigmoid function as given in Equation (10), the learning rate for the kth iteration during learning is computed.

$$f\left({\mathsf{\phi }}_i\left(k\right)\right)={\alpha }_i\frac{1-e^{-s\ast {\mathsf{\phi }}_i\left(k\right)}}{1+e^{-s\ast {\mathsf{\phi }}_i\left(k\right)}}$$

(10)

where s is the steepness parameter, which may vary between −1 and +1; ${\alpha }_i$ is the positive constant.

Considering the learning rate of (k − 1) iteration, the updated learning rate for the kth iteration is obtained from

$${\eta }_i\left(k\right)={\eta }_i\left(k-1\right)\ast \left[1+f\left({\mathsf{\phi }}_i\left(k\right)\right)\right]$$

(11)

With the updated learning rate as obtained from Equation (11), the weights W₁ and W₃ updated are as per Equations (6) and (7).

At the end of the training, rotor resistance R_r-est is estimated by considering either W₁ or W₃ as per Equations (12) or (13) as

$$R_{r-est}=\frac{L_r{W}_3}{L_m{T}_s}$$

(12)

$$R_{r-est}=\frac{L_r\left(1-W_1\right)}{T_s}$$

(13)

The d and q axes rotor fluxes estimated from the voltage model depend on the stator resistance Rs as given in Equation (1). To minimize the error in rotor resistance estimation, an online stator resistance estimator is developed and is presented in Section 3.

3. Artificial Neural Network-Based Stator Resistance Estimator

The equations corresponding to voltage and current models used in rotor resistance estimation [7], represented as Equations (1) and (2), are used to obtain the d and q axes currents of the induction motor as

$$\frac{d}{dt}{i}_{ds}^s=\frac{L_m}{\sigma L_s{L}_r{T}_r}{\psi }_{dr}^s+\frac{L_m}{\sigma L_s{L}_r}{\omega }_r{\psi }_{qr}^s-\frac{L_m^2}{\sigma L_s{L}_r{T}_r}{i}_{ds}^s+\frac{1}{\sigma L_s}{v}_{ds}^s-\frac{1}{\sigma L_s}{R}_s{i}_{ds}^s$$

(14)

$$\frac{d}{dt}{i}_{qs}^s=\frac{L_m}{\sigma L_s{L}_r{T}_r}{\psi }_{qr}^s-\frac{L_m}{\sigma L_s{L}_r}{\omega }_r{\psi }_{dr}^s-\frac{L_m^2}{\sigma L_s{L}_r{T}_r}{i}_{qs}^s+\frac{1}{\sigma L_s}{v}_{qs}^s-\frac{1}{\sigma L_s}{R}_s{i}_{qs}^s$$

(15)

Equations (14) and (15) are expressed in discrete form as

$$i_{ds}^{s*}\left(k\right)=W_4{i}_{ds}^s\left(k-1\right)+W_5{\psi }_{dr}^s\left(k-1\right)+W_{6 }{\psi }_{qr}^s\left(k-1\right)+W_7{v}_{ds}^s\left(k-1\right)$$

(16)

$$i_{qs}^{s*}\left(k\right)=W_4{i}_{qs}^s\left(k-1\right)+W_5{\psi }_{qr}^s\left(k-1\right)-W_{6 }{\psi }_{dr}^s\left(k-1\right)+W_7{v}_{qs}^s\left(k-1\right)$$

(17)

where $W_4=\left[1-\left(\frac{T_s}{\sigma L_s}\right)R_s-\left(\frac{L_m^2}{L_r{T}_r}\right)\left(\frac{T_s}{\sigma L_s}\right)\right]$; $W_5=\left(\frac{L_m}{L_r{T}_r}\right)\left(\frac{T_s}{\sigma L_s}\right)$; $W_6=\left(\frac{L_m}{L_r}\right)\left(\frac{T_s}{\sigma L_s}\right){\omega }_r$; $W_7=\frac{T_s}{\sigma L_s}$.

A two-layer recurrent neural network, as shown in Figure 4, is developed from Equations (16) and (17) for the stator current estimation. The * indicates the estimated values in neural network model.

The weights W₅, W₆, and W₇ are obtained from machine parameters, sampling interval T_s, and rotor speed ω_r. During the network training, only W₄ is adjusted, as other weights are independent of stator resistance. The standard back-propagation algorithm is adopted for the training of the neural network.

The square function of the stator current error between the reference model and the neural network model is

$$E_s=\frac{1}{2}\overrightarrow{{\in }_s^2}\left(k\right)=\frac{1}{2}{\left\{i_s\left(k\right)-i_s^{*}\left(k\right)\right\}}^2$$

(18)

To obtain the minimal error square function E_s, Weight W₄ is updated during training as

$$W_4\left(k\right)=W_4\left(k-1\right)+{\eta }_{4 }\Delta W_4\left(k\right)$$

(19)

Based on the repeated training performed on the network, with trial and error, the initial learning rate η₄ is selected. The weight adjustment factor ∆W₄(k) is obtained as

$$\Delta W_4\left(k\right)=-\frac{\partial E_s}{\partial W_4}={\left[\overrightarrow{i_s\left(k\right)}-\overrightarrow{i_s^{*}\left(k\right)}\right]}^T\ast \overrightarrow{i_s^{*}\left(k-1\right)}$$

(20)

The constant learning rate is replaced by an adaptive learning rate based on the bipolar sigmoid function. The learning rate is updated during the training as

$${\eta }_4\left(k\right)={\eta }_4\left(k-1\right)\ast \left[1+f\left({\phi }_i\left(k\right)\right)\right]$$

(21)

where φ_i(k) = ΔW₄(k) × ΔW₄(k − 1).

The function $f\left({\phi }_i\left(k\right)\right)$ is determined by adopting the activation function as explained for the rotor resistance estimation in Section 2.

Considering the adjustment weight W₄ obtained at the end of the training, the stator resistance is estimated as

$$R_{s-est}=\left\{1-W_4-\frac{T_s}{\sigma L_s}\frac{L_m^2{R}_r}{L_r^2}\right\}\frac{\sigma L_s}{T_s}$$

(22)

The stator resistance is thus estimated by the ANN-based MRAS estimator, as shown in Figure 5.

The speed of the sensorless drive is estimated as shown in [8] given below as

$${\omega }_{r-est}={\omega }_{r-est}\left(k-1\right)+\frac{{\eta }_{sp}}{T_s}\left\{\begin{array}{c}\left[{\psi }_{qr}^{s,vm}\left(k\right)-{\psi }_{qr}^{s,im}\left(k\right)\right]{\psi }_{dr}^{s,im}\left(k-1\right)\\ -\left[{\psi }_{dr}^{s,vm}\left(k\right)-{\psi }_{dr}^{s,im}\left(k\right)\right]{\psi }_{qr}^{s,im}\left(k-1\right)\end{array}\right\}$$

(23)

where η_sp is the learning rate for rotor speed estimation, and T_s is the sampling time.

4. Results and Discussion

The machine parameters of the induction motor with 3.7 kW are determined in the laboratory by performing the no-load and blocked rotor tests. With the obtained parameters, the performance of the proposed algorithm for online stator and rotor resistance is evaluated by the simulation and with the hardware setup, as presented in the following sections, followed by a discussion of the results.

4.1. Tests for Identification of Machine Parameters

The resistance of stator windings is evaluated by the volt-amp test. The no-load, locked rotor tests are performed to determine equivalent circuit parameters. The experiments are performed using the machine specified by the nameplate information provided in

Table 1. Induction motor information.

Power (HP)	3.7 kW
Current (Amps)	7.5
Terminal voltage (Volts)	415
Frequency (Hz)	50
Poles	4
No-load speed (RPM)	1498
Stator connection	Delta

. Figure 6 illustrates the experimental setup employed in the laboratory for the determination of machine characteristics.

4.1.1. Stator Winding Resistance Estimation

The low voltage DC is applied to one of the phase windings of three-phase winding, as shown in Figure 7. Calculating DC resistance requires taking the data from both an ammeter and voltmeter. The AC resistance was computed by multiplying a constant value of 1.25 to the previously determined DC resistance.

Table 2. Experimental observations on stator winding resistance estimation.

Sl No.	Applied Voltage (volt)	Current (amp)	DC Resistance (ohm)	Average dc Resistance (ohm)	Stator ac Resistance (ohm)
1	6.82	1.5	4.55
2	9.34	2	4.67	4.57	5.71
3	11.21	2.5	4.484

displays the results of the DC resistance measurement findings. Temperature variation is neglected in the estimation of stator resistance.

4.1.2. No-Load Test

The voltages at rated value and frequency are applied during the no-load test or light-load test. The circuit arrangement for the no-load test is shown in Figure 8. The input current, power input, and applied voltages are noted. As the input power factor is less than 0.5, low power factor watt meters are used, and one of the wattmeter shows a negative reading. Total power input corresponds to stator copper and iron losses without any applied load. No-load power factor, core loss resistance, magnetizing current, and inductances are determined from the measurements obtained. The observations obtained during no-load tests are presented in

Table 3. Observations of No-Load Test.

Voltage Vas (volt)	Ia (amp)	Ib (amp)	Ic (amp)	No-Load Current Io (amp)	W1 (watts)	W2 (watts)	Power Input Poc (watts)	Speed (rpm)
415	4.15	4.1	4.1	4.12	−672	1080	408	1498

.

4.1.3. Locked Rotor or Short Circuit Test

With locking the rotor to prevent rotation, applied voltage to the stator is increased until rated current flows to the stator. With the voltage, current, and power input readings obtained, rotor resistance and leakage inductance are calculated by developing the equivalent circuit. The results that were obtained from the examination are shown in

Table 4. Locked rotor test observations.

SC Voltage Vsc (volt)	SC Current Isc (Amp)	W1 (watts)	W2 (watts)	SC Power Input Psc (watts)
82	7.5	−11	510	499

.

4.1.4. Locked Rotor or Short Circuit Test

The exact equivalent circuit (IEEE) model of an induction motor was built based on the findings made during the blocked rotor and no-load tests, as illustrated in Figure 9.

Table 5. Induction motor parameters obtained from DC test, no-load, and blocked rotor tests.

Rs (ohm)	Rr (ohm)	Ls = (Lls + Lm) (Henry)	Lr = (Llr + Lm) (Henry)	Lm (Henry)
5.7	4.11	0.5634	0.5634	0.5379

presents an estimation of the values for the machine parameters based on the results of the tests performed on the induction motor using the equivalent circuit model [49,50].

4.2. Modeling Analysis

The block diagram of the induction motor drive with integrated online rotor and stator resistance estimator with speed estimation is shown in Figure 10. The induction motor drive is operated with a rotor flux orientation controller. A multi-level low-pass filter is used for the estimation of voltage model fluxes from the measured stator voltages and currents [51,52]. The induction motor parameters are obtained from no-load and blocked rotor tests performed in the laboratory as presented in Section 4.1. Induction motor parameter details used in the simulation study are summarized in

Table 6. Induction motor parameter details.

Parameters	Values
Rated power	3.7 kW
Rated frequency	50 Hz
Rated voltage	415 V
Rated current	7.5 A
Number of poles	4
Type of stator connection	Delta
No-load speed	1498 rpm
Rotor resistance	4.11 Ohm
Stator resistance	5.7 Ohm
Rotor inductance (Lr)	0.5634 H
Stator inductance (Ls)	0.5634 H
Magnetizing inductance (Lm)	0.5379 H
Moment of inertia (Jr)	0.01542 kg-m2

. Simulations are performed individually on the rotor resistance estimator, stator resistance estimator, and on an integrated system as shown in Figure 10. The performance of the developed algorithms is evaluated by simulation using MATLAB/Simulink software (R2023b, MathWorks, Natick, MA, USA).

The stator resistance, rotor resistance, and speed are estimated as presented in the following steps:

• With the sampled terminal voltages and currents and by using the developed neural network model as presented in Section 3, stator resistance is estimated online.
• Using the obtained stator resistance, rotor fluxes are estimated. Using the neural network model as presented in Section 2, rotor resistance is determined.
• The speed is evaluated by using the obtained stator resistance, rotor resistance, and rotor fluxes as presented in Section 3.

The results of the simulation are presented in the following sections.

4.2.1. Simulation Results of the Performance of the Rotor Resistance Estimator and Analysis

The initial actual resistance values of rotor and stator windings were obtained from the various laboratory tests and are shown in Table 5. To evaluate the ability of the proposed algorithm to track the variation in the rotor resistance, the simulations were performed with the wide range of actual rotor and stator resistance values under the various conditions such as fixed stator resistance and speed, fixed stator resistance and varying speed, and varying stator resistance and speed. Simulations are performed by considering the actual rotor resistance values of 10, 15, 20, 25, 40, 50, 75, 90, and 100 percent step rises from its initial value under the condition of fixed stator resistance and speed. Also, arbitrarily stepped rises in rotor and stator resistance values are considered to evaluate the performance of the algorithm with both variable speed and stator resistance.

The training of the neural network is performed in all the above conditions with the parameters as presented in

Table 7. Training parameters of neural network for rotor resistance.

Parameter	Value
The initial learning rate for weight W1	2.4 × 10−4
The initial learning rate for weight W3	10 × 10−6
Steepness factor of bipolar sigmoid function s1 and s3	1
α1, α3	0.1
Sampling time Ts	2 ms

. These parameters are selected based on a trial-and-error process by performing repeated training trials.

The simulation results (testing) are presented in

Table 8. Estimated rotor resistance with percentage error in estimation and number of iterations for fixed stator resistance and speed.

Actual Rotor Resistance	Wr = 307 Radian/Second			Wr = 310 Radian/Second
	Estimated Resistance	Percentage Error	Number of Iterations	Estimated Resistance	Percentage Error	Number of Iterations
4.11	4.11	0	1	4.11	0	1
4.521	4.54	−0.4242	6	4.568	−1.033	6
4.7265	4.761	−0.7391	8	4.59	3.36	6
4.932	4.954	−0.4538	10	4.982	−1.01	9
5.1375	4.995	2.765	10	5.141	−0.06854	10
5.754	5.631	2.14	14	5.736	0.3159	13
6.165	6.335	−2.754	9	6.188	−0.3811	15
6.576	6.642	−1.008	10	6.464	1.709	16
7.1925	7.154	0.5389	12	7.129	0.8871	18
7.809	7.826	−0.219	16	7.756	0.68	19
8.22	8.345	−1.525	17	8.194	0.3145	20

,

Table 9. Estimated rotor resistance with percentage error in estimation and number of iterations for fixed stator resistance and varying speed.

Actual Rotor Resistance	Speed Wr (rad/s)	Estimated Rotor Resistance	Percentage Error	Number of Iterations
4.11	313	4.11	0	1
4.521	312	4.476	0.9889	5
4.7265	311	4.705	0.4627	7
4.932	310	4.958	−0.5192	9
5.137	309	5.072	1.283	10
5.754	308	5.661	1.62	14
6.165	308	6.08	1.385	16

and

Table 10. Estimated rotor resistance with percentage error in estimation and number of iterations for varying stator resistance and speed.

Actual Rotor Resistance	Actual Stator Resistance	Speed (rad/s)	Estimated Resistance	Percentage Error	Number of Iterations
4.11	5.7	313	4.11	0	1
4.521	6.27	312	4.472	1.077	5
4.726	6.55	311	4.812	−1.817	8
4.932	6.84	310	4.95	−0.3642	9
5.137	7.125	309	5.165	−0.5305	11
5.754	7.98	308	5.537	3.78	14
6.165	8.55	308	6.06	1.705	17

and Figure 11, Figure 12 and Figure 13. A stepped increase in the actual rotor resistance, speed, and stator resistance is presented for the simulation purpose to evaluate the performance which is not true under the actual operating conditions. The percentage estimation error in rotor resistance in each case is also shown.

The variation in learning rates, weight adjustment factors, resistance value, and convergence of squared error during the training of the neural network is shown in Figure 14. As the iteration progresses, the estimated error between the fluxes decreases, giving rise to an increase in learning rates for the succeeding iterations, followed by a larger rate of weight adjustment. This results in quicker convergence of the error to the acceptable minimum value. Thus, the adaptive learning rate results in a quick estimation of the resistance with minimal error. As the iteration progresses, the squared error decreases, reaches the minimum value, then again increases, as shown in Figure 14c. The iteration is stopped when the error converges to the minimum value.

4.2.2. Simulation Results of Performance of Stator Resistance Estimator and Analysis

The performance of the proposed back-propagated recurrent neural network with a bipolar sigmoid-based adaptive learning rate algorithm for tracking the variation in stator resistance is evaluated for 10, 15, 20, 25, 40, 50, 75, 90, and 100 percent rises from its nominal value under the various conditions as with fixed rotor resistance and speed, fixed rotor resistance and variable speed, and variable speed and rotor resistance. The training of the neural network is performed in all the above conditions with the parameters as presented in

Table 11. Training parameters of neural network for stator resistance.

Parameter	Value
The initial learning rate for weight W4	0.001
Steepness factor of bipolar sigmoid function s4	0.01
α4	4
Sampling time Ts	4 ms

. These parameters are selected based on a trial-and-error process by performing repeated training trials.

The simulation results are presented in

Table 12. Estimated stator resistance with percentage error in estimation and number of iterations for fixed rotor resistance and speed.

Actual Stator Resistance	Wr = 307 rad/s			Wr = 310 rad/s
	Estimated Resistance	Percentage Error	Number of Iterations	Estimated Resistance	Percentage Error	Number of Iterations
5.7	5.7	0	1	5.7	0	1
6.27	6.325	−0.8841	5	6.154	1.857	6
6.55	6.519	0.4765	7	6.561	−0.1756	8
6.84	6.742	1.437	7	6.926	−1.25	10
7.125	6.994	1.838	8	6.99	1.901	10
7.98	7.932	0.6067	11	7.838	1.778	13
8.55	8.415	1.578	12	8.558	−0.091	15
9.12	8.955	1.807	13	9.091	0.3181	16
9.975	9.684	2.916	15	9.819	1.566	17
10.83	10.5	3.068	16	10.63	1.848	18
11.4	10.79	5.366	16	10.94	4.078	19

,

Table 13. Estimated stator resistance with percentage error in estimation and number of iterations for fixed rotor resistance and varying speed.

Actual Stator Resistance	Speed Wr (rad/s)	Estimated Stator Resistance	Percentage Error	Number of Iterations
5.7	313	5.7	0	1
6.27	312	6.307	−0.5941	8
6.55	311	6.615	−0.9972	9
6.84	310	6.92	−1.166	10
7.125	309	7.085	0.566	11
7.98	308	7.956	0.3065	12

and

Table 14. Estimated stator resistance with percentage error in estimation and number of iterations for varying rotor resistance and speed.

Actual Stator Resistance	Actual Rotor Resistance	Speed Wr (rad/s)	Estimated Stator Resistance	Percentage Error	Number of Iterations
5.7	4.11	313	5.7	0	1
6.27	4.727	312	6.306	−0.5775	6
6.55	5.138	311	6.616	−1.009	11
6.84	5.343	310	6.74	1.456	10
7.685	5.548	309	7.633	0.6825	14
8.55	6.165	308	8.18	4.21	14

and Figure 15, Figure 16 and Figure 17. A stepped increase in the actual rotor resistance, speed, and stator resistance is presented for the simulation purpose to evaluate the performance which is not true under the actual operating conditions.

As the iteration progresses during the training, the learning rate increases due to the decrease in error between the actual stator current and the estimated current from the neural current model, leading to weight adjustments at a larger rate in the succeeding iterations. Hence, quicker convergence of the error to an acceptable value is obtained, as presented in Figure 18a. As the iteration progresses, squared error falls initially, converges to minimum value, and then increases, as shown in Figure 18b.

4.2.3. Speed Estimation with Integrated Online Rotor and Stator Resistance Estimation

The simulation results of the integrated system are shown in Figure 19. The online rotor and stator resistance estimator tracks the variation in the rotor and stator resistance precisely with minor errors. With online rotor and stator resistance estimation, actual drive speed is measured accurately, which closely follows the set reference speed within the error of 0.16%.

4.3. Experimental Evaluation and Results

The proposed rotor and stator resistance, together with the speed estimation algorithm, was verified on an induction motor drive with a rotor flux orientation scheme [49], as shown in Figure 20 and Figure 21. The DSpace DS1104 controller board residing in the PC is employed for experimenting. The experiment is performed on a 3-phase induction motor with the rating as detailed in Table 6. The induction motor is driven with an IGBT inverter operated with a 5 kHz switching frequency. The induction motor is loaded through a rigidly coupled DC motor of 3 kW for torque control with the help of a 4Q rectifier. The proposed stator and rotor resistance estimation blocks are implemented with a sampling time of 2000 µs; a sampling time of 200 µs is employed for current and flux controllers and a 2000 µs sampling time for the speed estimator. For position and speed feedback, an encoder with 5000 pulses/cycles is used.

The ability of the proposed rotor resistance estimator to track the actual rotor resistance is evaluated by conducting the temperature rise test at an ambient temperature of 25 °C, where the motor is loaded with 6.4 N-m and driven at 1480 rpm. The results obtained for the estimated stator and rotor resistances collected from the experiment for nearly an hour are presented in Figure 22. As the motor runs with the load, the temperature of the stator and rotor winding rises, leading to a rise in resistance values. The d-axis rotor flux linkages collected from the voltage model and neural network model at the end of the heat run test are presented in Figure 23. The estimated d-axis rotor flux by the neural network model precisely tracks that of the voltage model and current model within an error of 1.2%.

The performance of the stator resistance estimator is evaluated by adding a resistance of 2.8 ohms per phase in series with the stator windings of the induction motor. The addition of 2.8 ohm raises the stator resistance value from 5.7 ohm to 8.5 ohm, as shown in Figure 24. A torque of 6.4 Nm was applied with the motor running at 1480 rev/min. The estimated stator resistance tracks the actual stator resistance precisely and converges closely to 8.46 ohm with an error of 0.48%, as shown in Figure 24 with the proposed adaptive learning rate algorithm. With the conventional constant rate learning algorithm, an error of 1.9% is obtained on stator resistance estimation. With the proposed adaptive learning rate algorithm, the stator resistance estimation converges within 150 ms as compared to the conventional constant learning rate algorithm of around 250 ms. A pulsation of 0.35% is obtained in stator resistance estimation with the proposed algorithm as against 1.56% with the conventional constant learning rate approach.

The tracking of the d-axis stator current by the neural network model during the online estimation of the stator resistance is shown in Figure 25. The estimated current by the neural network model closely follows the d-axis measured stator current owing to the online training of the neural network.

The results obtained for stator resistance estimation from the experiment are compared with the developed model by applying a step change in stator resistance of 2.8 ohms without making any changes in Rr. The simulation is carried out simultaneously with the conventional and the proposed adaptive learning rate algorithm. The results obtained from the model simulation, as shown in Figure 26, show close agreement with the experimental results shown in Figure 24. With the proposed adaptive learning rate algorithm, the estimated resistance converges at 8.53 ohm with an error of −0.35%, whereas, with the conventional constant learning rate algorithm, the estimated stator resistance converges at 8.624 ohms with an error in the estimation of −1.46%.

The experiments were performed to evaluate the performance of the proposed stator and rotor estimator along with the speed sensorless operation. To evaluate the effect of variation in the stator resistance on the speed estimation, a 2.8 ohm resistance per phase is added to the stator windings. The addition of 2.8 ohm raises stator resistance from its nominal value of 5.7 ohm to 8.5 ohm. The drive is operated with RFOC. When the stator resistance estimator block is off, the speed drops to 1472 rpm, as shown in Figure 27. With the stator resistance estimator block on, the estimated speed was closely matching with the measured speed.

4.4. Discussion

The results of the simulations and hardware, as presented in the preceding sections, point out that

• As the number of iterations needed for the estimation of stator and rotor resistances is reduced with the proposed algorithm, the quick estimation of stator and rotor resistances is obtained as compared to the conventional constant learning rate approach. For the step change in the stator resistance, with the proposed approach, estimated stator resistance converges with the minimum error of estimation within 150 ms as compared to 250 ms obtained from the conventional constant learning rate approach [7].
• The proposed algorithm ensures the reduction in error of estimated resistances. For the step change in stator resistance, with the proposed approach, the estimation error is 0.48%, which is superior to the conventional constant learning rate approach of around 2%.
• The proposed algorithm tracks the stator resistance smoothly with the pulsation in estimated resistance within 0.35%, which is better than the constant learning rate or other function-based learning rate approach [8].
• The smooth estimation of speed is obtained with the developed algorithm where pulsation is within 0.4% which is much lower than that obtained by a conventional constant learning rate algorithm [24].

5. Conclusions

This paper presents an advanced back-propagation neural network with error-function-based adaptive learning rate algorithms for the online estimation of stator and rotor resistances used in speed sensorless vector-controlled IMD. As compared with the constant learning rate, the suggested back-propagation algorithm for the feed-forward network performs superiorly with an adjustable learning rate defined by a bipolar sigmoid function. The results of the simulation and hardware experiments justify the ability of the proposed algorithm to track the variation in stator and rotor resistances quickly and precisely, leading to the estimation of the speed close to the real speed. The proposed algorithm has the potential to improve the control quality of the vector-controlled speed sensorless IMD. Further, the developed algorithm could be used to evaluate the drive’s performance in different operating states.