A novel NN based rotor flux MRAS to overcome low speed problems for rotor resistance estimation in vector controlled IM drives

Abstract

This paper presents a new neural network based model reference adaptive system (MRAS) to solve low speed problems for estimating rotor resistance in vector control of induction motor (IM). The MRAS using rotor flux as the state variable with a two layer online trained neural network rotor flux estimator as the adaptive model (FLUX-MRAS) for rotor resistance estimation is popularly used in vector control. In this scheme, the reference model used is the flux estimator using voltage model equations. The voltage model encounters major drawbacks at low speeds, namely, integrator drift and stator resistance variation problems. These lead to a significant error in the estimation of rotor resistance at low speed. To address these problems, an offline trained NN with data incorporating stator resistance variation is proposed to estimate flux, and used instead of the voltage model. The offline trained NN, modeled using the cascade neural network, is used as a reference model instead of the voltage model to form a new scheme named as “NN-FLUX-MRAS.” The NN-FLUX-MRAS uses two neural networks, namely, offline trained NN as the reference model and online trained NN as the adaptive model. The performance of the novel NN-FLUX-MRAS is compared with the FLUX-MRAS for low speed problems in terms of integral square error (ISE), integral time square error (ITSE), integral absolute error (IAE) and integral time absolute error (ITAE). The proposed NN-FLUX-MRAS is shown to overcome the low speed problems in Matlab simulation.

Introduction

For many years, the induction motor has been used in the industry due to its easy construction, maintenance and high robustness [ 1, 2]. Recently, great progress has been made in digital computing techniques. This shows that the path way for the field oriented control or vector control become more popular in industrial sectors to achieve high performance in adjustable speed control applications. Particularly, to achieve high performance, indirect vector control or field control is widely used in many industries [ 3– 5]. This is because indirect vector control uses slip frequency and rotor speed for field angle computation. Besides, the indirect vector control method does not require a flux controller. The calculation of the slip frequency depends on the rotor resistance (R_r). Due to heating, R_r may vary to a maximum value of 100%. To track this variation, many methods have been proposed which are broadly classified into MRAS and observer based techniques [ 6]. The Luenberger state observer is proposed for rotor resistance estimation [ 7]. The extended Kalman filter technique is proposed for rotor resistance estimation [ 8]. The Luenberger sliding mode observer is used for the estimation of rotor time constant [ 9]. The observer based techniques are complex and computationally rigorous [ 10]. On the other hand, the MRAS scheme is simple to implement and less intensive [ 10– 13]. The MRAS scheme with different state variables, namely, back emf, torque, real power, reactive power and rotor flux is proposed [ 14]. The famous MRAS scheme for the estimation of rotor resistance is the MRAS with rotor flux as the state variable [ 10, 11]. In this approach, voltage model equations (VME) independent of R_r, is used as the reference model. Current model equations (CME) dependent on R_r is used as the adjustable model. The proportional integral (PI) controller is used to minimize the error between the reference and adaptive model for R_r adaptation [ 11]. This scheme works in steady-state conditions and sometimes not in dynamic conditions [ 10].

Recently, applications of artificial intelligence (AI) techniques and usefulness in power electronics and drives are gaining a lot of momentum [ 15, 16], which is an active area of research. The fuzzy logic technique is employed to estimate rotor resistance in indirect vector controlled IM drives [ 17]. The artificial neural network for rotor resistance estimation is proposed [ 10, 18– 21]. A two layer NN based rotor flux estimator is used as the adaptive model in MRAS with rotor flux as the state variable for R_r estimation [ 10, 18, 19]. This scheme, named as “FLUX-MRAS,” is attracting and popular because it exhibits good dynamic performance and is computationally less rigorous [ 10]. In this scheme, the R_r is one of the weights of the two layer neural network. A back-propagation (BP) learning method is adapted to train the two layer neural network online to update the value of R_r so as to minimize the error between the reference and adaptive model. Although this scheme estimates R_r very well for medium and higher speed ranges, it fails to estimate at low speed. This is because the flux estimator designed based on VME is used as the reference model in this MRAS. The VME fails to give a correct flux estimation at low speeds due to integrator and stator resistance variation problems [ 22, 23]. To address the problems of integrator, a number of methods are proposed [ 24– 27] which are computationally rigorous and time consuming in real time. The stator resistance (R_s) of the motor depends on the temperature which varies during motor operation and makes the R_s of the motor vary from the nominal values. To track the variation in R_s, various techniques are proposed. The Luenberger observer [ 28], sliding mode observer [ 29] and extended Kalman filter [ 8] are proposed for stator resistance estimation. Various AI based techniques, namely, fuzzy logic, artificial neural network, neuro-fuzzy techniques are employed to track the variation in stator resistance [ 18, 19, 30– 32]. The use of the additional estimator would make the drive system complex. Hence, in this paper, a novel MRAS scheme which addresses both R_s variation and integrator drift problems at low speed is proposed. The neural network trained offline with the data comprising R_s variation is proposed for flux estimation to solve low speed problems of flux estimator modeled using VME. Various neural network architectures can be used to design the NN based flux estimator [ 33– 38]. In this paper, the cascade neural network (CC-NN) is chosen to design the NN based flux estimator because the flux estimator designed based on this architecture has resulted in the compact model [ 36– 38]. The offline trained CC-NN based flux estimator is used as the reference model instead of the voltage model in the FLUX-MRAS. Thus, two neural networks are employed, the first being online trained two layer neural network used as the adaptive model, the second being offline trained neural network used as the reference model. The existing NN based MRAS techniques for R_r estimation uses NN only for the adaptive model. But in the proposed MRAS scheme, the NN model is used both for the adaptive model and for the reference model. The idea of employing two neural networks in the MRAS scheme is inspired from Ref. [ 5] in which the idea is applied for rotor speed estimation. The same idea is applied to estimate R_r which is new and different from other MRAS techniques for R_r estimation. The use of two neural networks forms a new scheme named as “NN-FLUX-MRAS” for R_r estimation. The performance of the novel scheme is compared with the conventional scheme for low speed problems extensively through simulation.

Vector control of induction motor

The vector control is preferred for high dynamic performance of IM drives, because in vector control, both the magnitude and phase alignment of vector variables are controlled and kept valid for steady-state as well as transient conditions [ 22]. In vector control, the control of an induction motor is transformed similar to a separately excited DC motor by creating independent channels for flux and torque control and hence decoupled control is achieved [ 22]. This results in a fast transient response. Thus this method is a better option than the scalar control (In scalar control, only magnitude alone is controlled and the inherent coupling effect gives poor and sluggish dynamic response.) to obtain the desired dynamic performance. Some of the important key features of vector control [ 22] are as follows:

1) It provides fast transient response similar to the DC machine because torque control does not affect the flux.

2) The frequency is not directly controlled as in scalar control. In vector control, the frequency is controlled indirectly with the help of the unit vector.

3) It does not have instability problem when it crosses the operating point beyond the breakdown torque as in the case of scalar control.

The schematic diagram of vector control using indirect rotor flux field orientation with rotor resistance estimator is presented in Fig. 1. The reference speed is compared with the measured speed and the error signal is given to the PI controller. The speed error signal is processed and the torque command is generated. Using the torque command, the corresponding reference torque producing component of stator current is generated. Using flux as the reference, corresponding reference flux producing component of stator current is generated. Using PI controllers, the current error signals are processed and the corresponding two command voltages are generated. Using the d-q transformation, the two command voltages in synchronous frame are transformed to common reference voltages in stationary frame. The reference signal is compared with the carrier signal to generate the pulses for triggering the three phase DC-AC converter.

In the indirect vector control, slip frequency and rotor speed are used for field or flux angle computation. The equation for field angle ( {\theta }_e ) and slip frequency ( {\omega }_{\mathrm{sl}} ) is presented in Eqs. (1) and (2).

(1) {\theta }_e={\displaystyle \int ({\omega }_{\mathrm{sl}}+{\omega }_r)}dt,

(2) {\omega }_{\mathrm{sl}}={\displaystyle \frac{L_m{R}_r}{{\phi }_{\mathrm{ref}}{L}_r}}{{\displaystyle i}}_{\mathrm{bs}}^e.

It is obviously observed from Eqs. (1) and (2) that the calculation of the slip frequency depends on R_r. If R_r deviates from the nominal value, it will affect the accuracy of slip frequency and hence the field angle computation. Therefore, decoupled control is lost which is the important feature of vector controlled induction motor drives. Hence, this necessitates the need for the online estimation of rotor resistance.

FLUX-MRAS

The FLUX-MRAS with two layer neural network as adaptive model is demonstrated in Fig. 2. In this scheme, the VME are used as the reference model. The two layer neural network is used as the adjustable model with R_r expressed as one of its weights. The error between the reference model and the adjustable model is back propagated to adjust the weights of the adaptive model to estimate the R_r of an induction motor drive. The error is minimized using back propagation (BP) with the momentum learning method.

The VME to estimate rotor flux are used as the reference model because it is independent of R_r and the same are presented in Eqs. (3) and (4).

(3) {{\displaystyle \phi }}_{\alpha r}^s={\displaystyle \frac{L_r}{L_m}}\left[{\displaystyle \int ({{\displaystyle v}}_{\alpha s}^s-R_s{{\displaystyle i}}_{\alpha s}^s)}dt-\left({\displaystyle \frac{L_s^{\mathrm{2}}{L}_r-L_s{L}_m^{\mathrm{2}}}{L_r{L}_s}}\right){{\displaystyle i}}_{\alpha s}^s\right],

(4) {{\displaystyle \phi }}_{\beta r}^s={\displaystyle \frac{L_r}{L_m}}\left[{\displaystyle \int ({{\displaystyle v}}_{\beta s}^s-R_s{{\displaystyle i}}_{\beta s}^s)}dt-\left({\displaystyle \frac{L_s^{\mathrm{2}}{L}_r-L_s{L}_m^{\mathrm{2}}}{L_r{L}_s}}\right){{\displaystyle i}}_{\beta s}^s\right].

The CME for rotor flux estimation are presented in Eqs. (5) and (6).

(5) {\displaystyle \frac{d{{\displaystyle \phi }}_{\alpha r}^s}{\mathrm{d}t}}={\displaystyle \frac{R_r{L}_m}{L_r}}{{\displaystyle i}}_{\alpha s}^s-{\displaystyle \frac{R_r}{L_r}}{{\displaystyle \phi }}_{\alpha r}^s-{\omega }_r{{\displaystyle \omega }}_{\beta r}^s,

(6) {\displaystyle \frac{\mathrm{d}{{\displaystyle \phi }}_{\beta r}^s}{\mathrm{d}t}}={\displaystyle \frac{R_r{L}_m}{L_r}}{{\displaystyle i}}_{\beta s}^s-{\displaystyle \frac{R_r}{L_r}}{{\displaystyle \phi }}_{\beta r}^s-{\omega }_r{{\displaystyle \omega }}_{\alpha r}^s,

The discretized form of CME can be obtained using backward Euler’s rule [ 39] which is given in Eqs. (7) and (8).

(7) {{\displaystyle \phi }}_{\alpha r}^{\mathrm{\text{'}s}}(t\mathrm{)\; =}{w}_1{{\displaystyle \phi }}_{\alpha r}^{\mathrm{\text{'}s}}(t-\mathrm{1)}-w_2{{\displaystyle \phi }}_{\beta r}^{\mathrm{\text{'}s}}(t-\mathrm{1)+}{w}_3{i}_{\alpha s}^s(t-\mathrm{1),}

(8) {{\displaystyle \phi }}_{\beta r}^{\mathrm{\text{'}s}}\mathrm{(t)\; =}{w}_1{{\displaystyle \phi }}_{\beta r}^{\mathrm{\text{'}s}}(t-\mathrm{1)+}{w}_2{{\displaystyle \phi }}_{\alpha r}^{\mathrm{\text{'}s}}(t-\mathrm{1)+}{w}_3{i}_{\beta s}^s(t-\mathrm{1)}.

Equations (7) and (8) can be represented as two layer linear neural network (Fig. 3) and used as the adaptive model. It contains four input nodes. The input signals to these input nodes are past values of estimated a-axis and b-axis rotor fluxes and stator currents. There are two output nodes which represent the present values of estimated a-axis and b-axis rotor fluxes. The connections between the nodes are represented by the weights w₁, w₂, and w₃. w₂ is kept constant while w₁ and w₃ are variable weights and are proportional to R_r. The rotor speed required for the adaptive model is obtained using a speed sensor.

The adaptive weights w₁ and w₃ are adjusted to minimize the error function (Eq. (9)). The BP learning method with momentum is used to update the weights as it is simple for online learning. The algorithm is coded in Matlab/m-file. The appropriate choice of a and h will produce the best results. The learning rate and momentum factor are chosen as 0.00005 and 0.0000002, respectively. The change in weights is given in Eqs. (10) and (11) with the chosen learning rate. The weights of NN are updated as expressed in Eqs. (12) and (13) with the momentum factor [ 18, 19]. The sampling time is chosen as 100ms.

(9) E\mathrm{=0}\mathrm{.5}{(\epsilon (t))}^2\mathrm{=0}\mathrm{.5}{({\phi }_r^s(t)-{{\displaystyle \phi }}_r^{\mathrm{\text{'}s}}(t))}^2,

(10) \Delta {\omega }_1(t\mathrm{)\; =}\alpha [{\epsilon }_{\alpha }(t){{\displaystyle \phi }}_{\alpha r}^s(t-\mathrm{1)\; +}{\epsilon }_{\beta }(t){{\displaystyle \phi }}_{\beta r}^s(t-\mathrm{1)}],

(11) \Delta w_3\mathrm{(t)\; =}\alpha [{\epsilon }_{\alpha }\mathrm{(t)}{{\displaystyle i}}_{\alpha s}^s(t-\mathrm{1)\; +}{\epsilon }_{\beta }(t){{\displaystyle i}}_{\beta s}^s(t-\mathrm{1)],}

(12) w_1(t\mathrm{)=}{w}_1(t-\mathrm{1)+\Delta }{w}_1(t\mathrm{)+}\eta \Delta w_1(t-\mathrm{1),}

(13) w_3(t\mathrm{)=}{w}_3(t-\mathrm{1)+\Delta }{w}_3(t\mathrm{)+}\eta \Delta w_3(t-\mathrm{1),}

where

{{\displaystyle \phi }}_r^s(t\mathrm{)=}{[{\phi }_{\alpha r}^s(t\mathrm{),}{{\displaystyle \phi }}_{\beta r}^s(t)]}^T,

{\epsilon }_{\alpha }(t\mathrm{)=}{{\displaystyle \phi }}_{\alpha r}^s(t\mathrm{)-}{{\displaystyle \phi }}_{\alpha r}^{\mathrm{\text{'}s}}(t\mathrm{),}

{\epsilon }_{\beta }(t\mathrm{)=}{{\displaystyle \Psi }}_{\beta r}^s(t\mathrm{)-}{{\displaystyle \Psi }}_{\beta r}^{\mathrm{\text{'}s}}(t).

R_r can be obtained either from w₁ or w₃ from Eqs. (14) and (15).

(14) R_r=\left({\displaystyle \frac{L_r{w}_3}{L_m{T}_s}}\right),

(15) R_r={\displaystyle \frac{L_r}{T_s}}(1-w_1).

An offline trained CC-NN based flux estimator used as a reference model

The offline trained NN based flux estimator to be used as the reference model is designed using cascade architecture. In the cascade architecture, one neuron per hidden layer is chosen. The detailed description of CC-NN architecture and advantages of using one neuron per hidden layer is presented in Refs. [ 36– 38, 40– 44]. The CC-NN is depicted in Fig. 4.

The induction motor drive system with indirect vector control is built using Matlab/simulink. The sinusoidal pulse width modulation technique is used. The carrier frequency is chosen as 10 kHz. The CC-NN based flux estimator with inputs and outputs is shown as a block diagram in Fig. 5.

11,266 input/output data are collected for different operating conditions. It is understood from the literature that R_s may change to the maximum value of 50% [ 18, 19]. The data are collected with the maximum of 50% change in R_s. The CC-NN is trained using the data with R_s variation to obtain a robust CC-NN based flux estimator. The induction motor is modeled using dynamic d-q model Equations [ 22] to incorporate variation in R_s. The transfer function for hidden neurons is chosen as the hyperbolic tangent sigmoid function while pure linear function is used for output neurons.

The CC-NN is trained offline in Matlab/m-file with the collected data sets using the Levenberg Marquardt algorithm. For the LM algorithm, the initial constant value µ is chosen as 0.001. The µ increase factor is chosen as 10. The µ decrease factor is chosen as 0.1. The weights and biases are initialized using the Nguyen Window algorithm. The desired average squared error of 0.00000188876 is achieved with 13 hidden neurons after training the NN for about 5000 epochs on Athlon Processor (AMD) LE-1640, 2.60 GHz. The offline trained CC-NN model is employed as the reference model in the FLUX-MRAS to form the new “NN-FLUX-MRAS” scheme (Fig. 6) to estimate the value of R_r.

Simulation results and discussion

The performance of NN-FLUX-MRAS is compared with FLUX-MRAS for the integrator drift and R_s variation problem. The performance comparison is conducted extensively using Matlab simulations. The main objective of this paper is to study the effect of low speed problems on the performance of R_r estimator. The study of low speed problems are done by operating the drive at a very low speed of around 9.54 r/min under half the rated load condition. To avoid instability problem at low speed, the drive is not fully loaded.

Comparison of NN-FLUX-MRAS and FLUX-MRAS for integrator drift problem

The performance of NN-FLUX-MRAS scheme is compared with the conventional FLUX-MRAS for the integrator drift problem. To demonstrate the integrator drift problem, a DC bias of various percentages of the peak current is superimposed to the phase b stator current at 5 s. The 100% step change in R_r is also effected at 2.5 s. The sample results for 2% DC bias are presented. The estimated a-axis rotor flux using the VME and CC-NN model along with the actual flux is shown in Fig. 7 and Fig. 8, respectively. The estimated b-axis rotor flux using VME and CC-NN model along with actual flux is demonstrated in Figs. 9 and 10, respectively. It is observed that as soon as the DC bias is superimposed to the current at 5 s, the fluxes computed from the VME drift from the actual fluxes, become unstable, and fails to estimate. But the fluxes computed from the CC-NN model track the actual fluxes even in the presence of DC bias and shows stable performance. This is due to the absence of integrator and the presence of limiting hyperbolic tangent sigmoid function in the NN [ 45, 46].

The actual and estimated R_r using FLUX-MRAS and NN-FLUX-MRAS is presented in Fig. 11. It is obviously seen that at 5 s, as soon as the DC bias is superimposed to the current signal, the R_r computed from the conventional scheme deviates from the actual value and becomes unstable. But the R_r computed from the novel scheme exhibits stable performance and attempts to track the actual value satisfactorily even in the presence of DC bias.

The performance of NN-FLUX-MRAS and FLUX-MRAS is also compared in terms of various performance indices, namely, ISE, ITSE, IAE, ITAE. The formula to compute these performance indices are presented in Eqs. (16) to (19). These errors are computed between the actual and estimated rotor resistance for various percentages of DC bias. The results obtained is consolidated and presented in Table 1. For better understanding, the sample bar chart of FLUX-MRAS and NN-FLUX-MRAS for ISE is displayed in Fig. 12. It is observed that the error increases with the increase in DC bias for both the schemes. But the increase in the error of FLUX-MRAS is greater and becomes unstable. However, the increase in the error of NN-FLUX-MRAS is marginal and exhibits stable performance.

(16) \mathrm{ISE=}{\displaystyle \underset{0}{\overset{t}{\int }}{e}^2}dt,

(17) \mathrm{ITSE=}{\displaystyle \underset{0}{\overset{t}{\int }}t{e}^2}dt,

(18) \mathrm{IAE=}{\displaystyle \underset{0}{\overset{t}{\int }}\left|e\right|}dt,

(19) \mathrm{ITAE=}{\displaystyle \underset{0}{\overset{t}{\int }}t\left|e\right|}dt.

Performance comparison of NN-FLUX-MRAS and FLUX-MRAS for R_s variation problem

The performance of the both the schemes is compared for R_s variation. The performance of both the schemes is studied for various percentage step change in R_s. At 2.5 s, 100% step change in R_r is effected. The step change in R_s is also effected at 5 s. The sample results for the 50% change in R_s are presented.

The estimated a-axis rotor flux using the VME and CC-NN model along with the actual flux is exhibited in Figs. 13 and 14, respectively. The estimated b-axis rotor flux using the VME and CC-NN model along with the actual flux is shown in Figs. 15 and 16, respectively. It is seen that, as soon as the 50% step change in R_s is applied at 5 s, the fluxes computed from the VME deviate from the actual fluxes and fails to estimate. But the fluxes computed from the CC-NN model trained with parameter variation track the actual fluxes even in the presence of parameter variation. Thus the NN-flux estimator trained with R_s variation estimates flux without the need for R_s estimator leading to a low cost solution.

The actual and estimated R_r using FLUX-MRAS and NN-FLUX-MRAS is presented in Fig. 17. It is clearly noticed that as soon as the step change in stator resistance is applied at 5 s, the R_r computed using the FLUX-MRAS oscillates between 9.94 (ohms) and 16.18 (ohms) which is not permissible for high performance drives. But the R_r computed using the novel scheme tracks the actual flux very well even in the presence of R_s variation.

The performance of NN-FLUX-MRAS and FLUX-MRAS is again compared in terms of various performance indices, namely, ISE, ITSE, IAE, and ITAE. The results obtained for various percentages of R_s is consolidated and presented in Table 2. For better understanding, the sample bar chart for FLUX-MRAS and NN-FLUX-MRAS for ISE is shown in Fig. 18. It is observed that the error increases with the increase in R_s for FLUX-MRAS. But the error is almost constant and independent of R_s variation for the proposed NN-FLUX-MRAS which validates the robustness of the proposed NN-FLUX-MRAS to R_s variation.

The effect of step change in R_s on R_r estimation is conducted in the above investigation. The step change is the worst case condition. In an actual drive system, R_s and R_r vary slowly with time. The slow variation is studied by varying R_r and R_r in trapezoidal fashion as indicated in Fig. 19. The estimation of R_r using FLUX-MRAS and NN-FLUX-MRAS is presented in Fig. 20. It is again observed that the R_r computed from the FLUX-MRAS deviates from the actual value. But the R_r estimated from the NN-FLUX-MRAS tracks the actual value very well even in the presence of slow variation in R_s.

Conclusions

This paper presented a novel NN-FLUX-MRAS proposed to overcome low speed problems for R_r estimation in IFOC IM drives. The NN-FLUX-MRAS used two neural networks, namely, online trained adaptive neural network model and offline trained reference neural network model. The offline trained NN model was designed using the CC-NN architecture to achieve compact flux estimator. The robust CC-NN model was obtained by training the CC-NN with the data comprising R_s variation. This would make the drive system less complex by avoiding the need for separate estimator to track the variation in R_s.

The NN-FLUX-MRAS was compared extensively with FLUX-MRAS for major low speed problems, namely, the integrator drift and R_s variation problem. The performance of NN-FLUX-MRAS and FLUX-MRAS was investigated extensively with various percentages of DC bias and stator resistance variation. The NN-FLUX-MRAS was shown to estimate R_r even in the presence of DC bias and R_s variation at low speed. The NN-FLUX-MRAS were robust to R_s variation, the reference model was free from the integrator and computationally less rigorous. It can be concluded that the proposed R_r estimator offers a good solution for the low speed problems and reduces the complexity for vector controlled IM drives.

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