A modified neural learning algorithm for online rotor resistance estimation in vector controlled induction motor drives

Abstract

Online estimation of rotor resistance is essential for high performance vector controlled drives. In this paper, a novel modified neural algorithm has been identified for the online estimation of rotor resistance. Neural based estimators are now receiving active consideration as they have a number of advantages over conventional techniques. The training algorithm of the neural network determines its learning speed, stability, weight convergence, accuracy of estimation, speed of tracking and ease of implementation. In this paper, the neural estimator has been studied with conventional and proposed learning algorithms. The sensitivity of the rotor resistance change has been tested for a wide range of variation from -50% to+50% on the stability of the drive system with and without estimator. It is quiet appealing to settle with optimal estimation time and error for the viable realization. The study is conducted extensively for estimation and tracking. The proposed learning algorithm is found to exhibit good estimation and tracking capabilities. Besides, it reduces computational complexity and, hence, more feasible for practical digital implementation.

Keywords

neural networks / back propagation (BP) / rotor resistance estimators / vector control / induction motor

Introduction

Three phase induction motors are the most prominently used drives due to their reliability and cost. Indirect field oriented controlled (IFOC) induction motor drives are increasingly used in high performance drive systems. Parameter variation, in general, affects the performance of the IFOC drive [ 1]. Rotor resistance is the most significant parameter as it can vary largely during motor operation. Rotor resistance may vary up to 100% due to temperature, frequency and skew effects [ 2]. Online estimation of rotor resistance using thermal models and temperature sensors is not desirable as thermal models involve numerous complex mathematical computations and temperature sensors have to be mounted in different locations which may not be possible in all applications.

Researchers have proposed methods for identifying rotor time constant in IFOC drives [ 3]. In one class of method, estimation of rotor time constant is done using the spectral analysis techniques [ 4]. This group of methods is based on the measured response to a deliberately injected test signal or an existing characteristic harmonic in the voltage/current spectrum. Stator currents and voltages of the motor are sampled and the parameters are derived from the spectral analysis of these samples. The second classification of rotor resistance identification scheme used observer based techniques. Most of the methods have used the extended Kalman filter, which is a computationally intensive technique [ 5, 6]. The third group of online rotor resistance adaptation methods is based on principles of model reference adaptive scheme (MRAS). This is the approach that has attracted most of the attention due to its relatively simple implementation requirements [ 7]. In this scheme the flux vectors are estimated from two different directions. Basically, one of the systems used for estimation is independent of the rotor resistance. Thus, the error between the two estimators supplies correction to the rotor resistance which is usually a proportional or proportional integral controller. In almost all these systems, the rotor resistance adaptation is functional only in steady state condition and not active during dynamic states.

Hence, the need for new methods for rotor resistance estimation has motivated the use of artificial intelligent (AI) techniques in this area. A number of methods have been reported in literature for rotor resistance estimation. The neural network learning MRAS (NN-MRAS) based rotor resistance estimator is found to exhibit good dynamic performance [ 8]. It also demonstrates good fault tolerant capabilities and is computationally less intense. Further improvements are possible by using different learning algorithms. The training algorithm used has an effect on issues such as learning speed, stability, and weight convergence, accuracy of estimation, speed of tracking and ease of implementation. These issues remain as areas of research and comparison of training algorithm is, therefore, essential to arrive at the appropriate learning method [ 9– 11].

This paper envisages investigating the learning algorithms for rotor resistance estimation. A novel neural learning algorithm has been proposed which gives promising results. Besides, the vector control drive performance has been studied with the proposed neural estimator.

Structure of NN-MRAS rotor resistance estimator

This section presents a NN-MRAS rotor resistance estimator for the induction motor drive [ 12, 13]. A simple two layered feed forward neural network trained by back propagation (BP) technique is used for the real time adaptive estimation. Two models of the state variable estimation are used, one providing the actual induction motor output and the other giving the neural model output. The total error between the desired and actual state variables is then back propagated.

The rotor flux of the induction motor estimated with a classical voltage model is the key input of the rotor resistance estimator. The flux estimated with this voltage model will be correct irrespective of the variations in R_r since voltage model equations are independent of R_r, which provides the desired state variable. Another state model is the neural model based on current model equations, which provides the actual output of the induction motor since the current model equations are dependent on rotor resistance of the induction motor.

The rotor resistance of an induction motor is estimated using the neural network system, as illustrated in Fig. 1. Two independent estimators are used to estimate the rotor flux vectors of the induction motor. The equation based on stator voltages and currents is called voltage model equation and is given in Eq. (1).

(1) \left[\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}{\lambda }_{d\mathrm{r}}}{\mathrm{d}t}}\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}{\lambda }_{q\mathrm{r}}}{\mathrm{d}t}}\hfill \end{array}\right]={\displaystyle \frac{L_{\mathrm{r}}}{L_{\mathrm{m}}}}\left\{\left[\begin{array}{c}\hfill v_{d\mathrm{s}}\hfill \\ \hfill v_{q\mathrm{s}}\hfill \end{array}\right]-\left[\begin{array}{cc}\hfill R_{\mathrm{s}}+s\sigma L_{\mathrm{s}}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill R_{\mathrm{s}}+s\sigma L_{\mathrm{s}}\hfill \end{array}\right]\left[\begin{array}{c}\hfill i_{d\mathrm{s}}\hfill \\ \hfill i_{q\mathrm{s}}\hfill \end{array}\right]\right\}.

The equation based on stator currents and rotor speed is called current model equations. The discrete current model equations are given in Eqs. (2) and (3).

(2) {\lambda }_{d\mathrm{r}}\left(k\right)=W_{\mathrm{1}}{\lambda }_{d\mathrm{r}}\left(k-\mathrm{1}\right)-W_{\mathrm{2}}{\lambda }_{q\mathrm{r}}\left(k-\mathrm{1}\right)+W_{\mathrm{3}}{i}_{d\mathrm{s}}\left(k-\mathrm{1}\right),

(3) {\lambda }_{q\mathrm{r}}\left(k\right)=W_{\mathrm{1}}{\lambda }_{q\mathrm{r}}\left(k-\mathrm{1}\right)+W_{\mathrm{2}}{\lambda }_{d\mathrm{r}}\left(k-\mathrm{1}\right)+W_{\mathrm{3}}{i}_{d\mathrm{s}}\left(k-\mathrm{1}\right).

The neural model represented by Eqs. (2) and (3) is shown in Fig. 2, where W₁, W₂, and W₃ represent the weights of the two layer neural network used to estimate rotor resistance, as given in Eq. (4)

(4) \{\begin{array}{c}\hfill W_{\mathrm{1}}=\mathrm{1}-{\displaystyle \frac{T}{T_{\mathrm{r}}}}\hfill \\ \hfill W_{\mathrm{2}}={\omega }_{\mathrm{r}}T\hfill \\ \hfill W_{\mathrm{3}}={\displaystyle \frac{T{L}_{\mathrm{m}}}{T_{\mathrm{r}}}}\hfill \end{array},

where T is the sampling period, T_r is the rotor time constant, ω_r is rotor angular velocity, {\lambda }_{d\mathrm{r}} and {\lambda }_{q\mathrm{r}} are d-axis and q-axis rotor fluxes, I_ds and I_qs are d-axis and q-axis stator currents, V_ds and V_qs are d-axis and q-axis stator voltages, {\lambda }_{\mathrm{r}}^{\mathrm{vm}}\mathrm{and}{\lambda }_{\mathrm{r}}^{\mathrm{nm}} are the rotor flux from the voltage model and neural model, and σ is called leakage coefficient. Among the three weights, W₂ is already known and W₁ and W₃ need to be updated. The weights between neurons, W₁ and W₃ are trained so as to minimize the energy function E. The energy function is given in Eq. (5).

(5) E={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left[{\lambda }_{\mathrm{r}}^{\mathrm{v}\mathrm{m}}-{\lambda }_{\mathrm{r}}^{\mathrm{n}\mathrm{m}}\right]={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left[{\mathit{ϵ}}_d{\left(k\right)}^{\mathrm{T}}{\mathit{ϵ}}_q\left(k\right)\right].

The change in weight updates have been derived for all the neural algorithms which are coded as m-File in Matlab. The voltage model equations are implemented in simulink model file. The neural model estimator is updated at a sampling frequency of 10 kHz, so the sampling period for on-line rotor resistance estimator is T = 0.0001 s.

Conventional and proposed learning algorithms for NN-MRAS based R_r estimator

Rotor resistance estimation using BP with momentum factor

The task of training the network can be thought of in terms of an optimization problem. The variables to manipulate are the various weight factors while the variable which is to be minimized is the NN output error.

The momentum coefficient in this approach encourages the direction and relative magnitude of the previous change. This mechanism can enable the training equation to work through local minimums [ 14].

The update laws for this algorithm are given through Eqs. (6) to (9) which are based on the standard BP learning rule.

(6) \mathrm{\Delta }{W}_1\left(k\right)=\eta \left[{ϵ}_d\left(k\right){\lambda }_{d\mathrm{r}}^{\mathrm{vm}}\left(k-1\right)+{ϵ}_q\left(k\right){\lambda }_{q\mathrm{r}}^{\mathrm{vm}}\left(k-1\right)\right],

(7) W_{\mathrm{1}}\left(k\right)=W_{\mathrm{1}}\left(k-1\right)+\mathrm{\Delta }{W}_{\mathrm{1}}\left(k\right)+\alpha \mathrm{\Delta }{W}_{\mathrm{1}}\left(k-1\right),

(8) \mathrm{\Delta }{W}_{\mathrm{3}}\left(k\right)=\eta \left[{ϵ}_d\left(k\right)I_{d\mathrm{s}}\left(k-\mathrm{1}\right)+{ϵ}_q\left(k\right)I_{q\mathrm{s}}\left(k-\mathrm{1}\right)\right],

(9) W_{\mathrm{3}}\left(k\right)=W_{\mathrm{3}}\left(k-1\right)+\mathrm{\Delta }{W}_{\mathrm{3}}\left(k\right)+\alpha \mathrm{\Delta }{W}_{\mathrm{3}}\left(k-1\right).

Equations (6) and (8) give the change in weights. The final update laws are Eqs. (7) and (9), where η is the learning rate and α is momentum constant.

The rotor resistance can be calculated from either W₁ or W₃ from Eqs. (10) and (11).

(10) R_{\mathrm{r}}={\displaystyle \frac{L_{\mathrm{r}}{W}_{\mathrm{3}}}{L_{\mathrm{m}}T}},

(11) R_{\mathrm{r}}={\displaystyle \frac{L_{\mathrm{r}}}{T}}\left(\mathrm{1}-W_{\mathrm{1}}\right).

Rotor resistance estimation using quick prop

The quick prop is another heuristic technique to accelerate the convergence rate of gradient descent technique by employing a dynamically changing momentum factor. The acceleration rate is determined by the successive difference between the slope values. This is similar to Newton’s method since it also considers the second order gradient. The update equations for this algorithm are given in Eqs. (7) and (9). The momentum factor α is to be substituted as in Eq. (12).

(12) \alpha ={\displaystyle \frac{\delta E/\delta W_i\left(k\right)}{\delta E/\delta W_i\left(k-\mathrm{1}\right)-\delta E/\delta W_i\left(k\right)}}.

where i is the weight subscript, and k is the iteration. The advantage of quick prop is fast convergence as first and second derivatives are used. Another advantage is that it involves less intense computations compared to BP with variable learning rate but a bit complex when compared with the learning technique of BP with momentum factor.

Rotor resistance estimation using BP with variable learning rate

This learning strategy BP with variable learning rate (VLR) (BP with VLR) works with the adaptive learning rate which helps in fast convergence irrespective of the start values. This is one of the heuristic methods to accelerate the convergence rate of the gradient descent algorithm. The step size is decided based on the error. The learning rate η and the momentum factor α are decided based on the mean squared error (MSE).

The update laws for this algorithm are the same as that for the previous case except that the learning rate is not fixed but changes as per the above criteria. The advantages of BP with VLR are dynamically changing learning rate and fast convergence.

Rotor resistance estimation using proposed constraint based BP

In this algorithm, the BP learning technique is used to update the weight W₃ while the weight W₁ is found from the value of W₃. Here the initial transients are reduced in the estimation, which also shows excellent tracking performance. The update laws for the constraint based BP algorithm are given in Eqs. (9) and (13).

(13) W_{\mathrm{1}}\left(k\right)=\mathrm{1}-\left(W_{\mathrm{3}}\left(k\right)/L_{\mathrm{m}}\right).

Of all these methods the proposed constraint based BP is concluded to be the most suitable learning algorithm for neural based rotor resistance estimator. Though the BP with VLR provides comparable performance with BP with constraint, the complex computation involved in VLR increases the time taken for estimation.

Significance of estimation time on drive performance

The indirect field oriented control or vector control presented here is rotor flux oriented control. Figure 3 illustrates the complete schematic of indirect field oriented control for induction motor drives [ 15]. The torque command is generated as a function of the speed error signal, generally processed through a PI controller. The torque and flux command are processed in the calculation block. The three phase reference current generated from the functional block is compared with the actual current in the hysteresis band current controller and the controller takes the necessary action to produce PWM pulses. The PWM pulses are used to trigger the voltage source inverter to drive the induction motor.

The drive performance is analyzed with various estimators in which the estimation error is kept constant at 1% and the time of estimation is varied. Similarly, with the same operating conditions, the steady state analysis of the torque and flux response of the drive can be done by having the estimation time as constant with various estimation errors. The performance is studied with a constant estimation time of 120 ms. The torque and the flux responses for the above conditions are analyzed. It is seen that as the estimation error increases, the steady state error also increases. The more detailed analysis is given in Tables 1 and 2.

The sensitivity of the rotor resistance change has been tested with three load conditions namely, 100%, 50% and 25% of rated load with a fixed speed reference. The rotor resistance is changed from -50% to+50% of their nominal value for the respective load condition. Table 1 provides the result of the vector controlled drive without estimator. Table 2 provides the stability analysis of drive scheme with online rotor resistance estimator. The estimator is operated with a fixed estimation time of 120 ms and with various estimation errors. It is obvious that as the estimation time and the estimation error (E_e) are increased, the drive performance is being deteriorated. However it is quiet appealing to settle down with the maximum allowable estimation time and estimation error, so that the drive performance is satisfactory. Thus from the results it can be concluded that the performance of the drive is satisfactory with a maximum estimation time of 120 ms and an estimation error of 5%.

Simulation results and discussion

The induction motor has been modeled using T-model equations in Matlab/simulink to incorporate the variations in rotor resistance as in the practical case. The simulation results of the rotor resistance estimation using various learning strategies are studied for the following cases.

1) With 40% step change in rotor resistance.

2) With 40% ramp change in rotor resistance.

3) With 40% trapezoidal change in rotor resistance.

The simulation studies are conducted for the same cases with rotor resistance estimator trained using the quick propagation learning algorithm. From the results, it can be observed that this learning algorithm also performs the similar BP with momentum factor. To further improve the performance, BP with VLR is used and the results are presented in Fig. 4.

The performance of the neural based rotor resistance estimator using proposed learning algorithm is demonstrated in Fig. 5. Comparing Fig. 4 and Fig. 5, it can be seen that the time taken for estimation is almost same as that taken by the BP with variable learning rate for step changes and the computation involved in this strategy is also less rigorous. Besides, the performance for tracking is superior compared to other learning algorithms. Thus, it can be concluded that the proposed learning algorithm, namely constraint based BP, is the most suitable for neural based rotor resistance estimator.

The voltage model equations were implemented in simulink model file and neural network representing the current model was implemented as an m-file. The neural model estimator is updated at a sampling frequency of 10 kHz, so the sampling period for online rotor resistance estimator is T = 0.0001 s.

The above simulation results are consolidated in a tabular form in Table 3. From the results, the actual and estimated value of rotor resistance, accuracy of estimation in terms of percentage error and time taken for estimation are tabulated for all learning techniques for the three cases.

The linear change in rotor resistance due to gradual variation in temperature can be considered as ramp change and sudden change in rotor resistance due to any rotor bar breakage because excessive temperature can be considered as step change. The simulation results are taken for step, ramp and trapezoidal changes in rotor resistance using all the learning algorithms. A quick view of all the neural algorithms is presented in Table 4 in terms of complexity, accuracy and speed of estimation. It can be seen from the above results that the performance factors such as speed of estimation and percentage error are found to be better with the proposed learning method, namely the constraint based BP. Hence this can be the appropriate learning method for neural network based rotor resistance estimator.

Though the BP with VLR provides comparable performance with BP with constraint, the complex computation involved in VLR shows its limitation on digital implementation [ 16]. The significance of estimating time of the rotor resistance also supports the proposed constraint based BP.

Enhanced performance of vector controlled drive with proposed NN-MRAS based R_r estimator

The indirect field oriented control induction motor drives have been used for numerous industrial applications. Instead of observing the machine flux, the correct field orientation control is obtained by a feed forward slip control in an IFOC based drive. With the use of a shaft encoder, the rotor flux oriented control (RFOC) can be accomplished with low cost and high performance.

The drive performance is analyzed with the reference speed of 100 rad/s, reference flux linkage of 0.9 Wb, torque 7.5 N·m from 0 s to 2 s and then it is reduced to 5 N·m at 2 s and the rotor resistance is given a step change 40% at 2.5 s. Figure 6 depicts the torque response of the drive system where the torque increases by 19.5% from its command value. These torque transients are highly unendurable in high performance drive schemes. In addition, the flux response displayed in Fig. 6 portraits that rotor flux rises by 22.77% from its nominal value. Though theoretically the flux linkages can increase up to 20%, practically it is not possible due to magnetic saturation effects. Moreover, another adverse effect on the drive performance due to rotor resistance change is that the decoupled control is lost which is the most desirable feature of the vector controlled drive.

Thus from the above results, it is evident that the rotor resistance change of 40% (step) is having a great influence on the torque and flux response of the drive. The actual torque of the motor (T_e) deviates from the reference torque ( T_{\mathrm{e}}^{*} ) which is generated by the speed controller. The rotor flux ( {\lambda }_{\mathrm{r}} ) increases from the command value of 0.9 to 1.1 which is an increase of 22.22%. Though theoretically the rotor flux can increase up to 20%, practically it is not permissible due to magnetic saturation effects.

Thus from the above observation, it can be concluded that the response of the drive scheme is not satisfactory when the rotor resistance changes and the controller cannot adapt to the changing R_r. To give acceptable results, the drive system demands the knowledge of exact value rotor resistance which can be provided with online estimator. Without estimator, the drive performance has been brought down to an intolerable level.

From the results, it can be observed that the performance of the drive is enhanced with online estimator as it retains the decoupled control. The enhanced results of the drive scheme can be observed from the results displayed in Fig. 7. The responses shown in Fig. 7 can be concluded as follows. The actual torque of the motor (T_e) tracks the reference torque ( T_{\mathrm{e}}^{*} ) which is generated by the speed controller. The rotor flux ( {\lambda }_{\mathrm{r}} ) tracks the command value. The ratings of the induction motor used for simulation studies are given in Table 5.

Conclusions

The performance of the IFOC drive scheme has been studied for various operating conditions with and without online rotor resistance estimator. Without R_r estimator, the instantaneous torque control is lost, the rotor flux increases from the command value and the decoupled control is also lost. The effect of estimation time on the drive system performance is studied. The maximum allowable estimation time and estimation error, for the satisfactory drive performance is evaluated. The sensitivity of the rotor resistance change on the stability of the drive is presented. It is quiet interesting to settle down with the maximum allowable estimation time and estimation error, so that the drive performance is satisfactory. Thus from the results, it can be concluded that the performance of the drive is satisfactory with the maximum estimation time of 120 ms and an estimation error of 5%.

The neural based rotor resistance estimator has been trained with various learning strategies and the results are compared in terms of time taken for estimation, accuracy of estimation and complexity. The proposed constraint based BP is found to be the most appropriate learning strategy for the neural based rotor resistance estimator as it tracks faster and is computationally less rigorous.

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