High Performance Parameter Observation of

2017 3rd International Conference on Electrical Information and Communication Technology (EICT), 7-9 December 2017, Khulna, Bangladesh

High Performance Parameter Observation of Induction Motor With Sensorless Vector Control Using Extended Kalman Filter Md. Samiul Haque Sunny1, Manash Mandal1, Eklas Hossain2, Md. Adbur Rafiq1 1

Khulna University of Engineering & Technology (KUET), Khulna-9203, Bangladesh 2 Oregon Tech, Department of Electrical and Renewable Energy, OR-97601, USA [email protected], [email protected], [email protected], [email protected] Abstract— Speed sensor less vector control of induction motor drive faces two major problems: speed estimation, and rotor flux observation. Because of the multiplication terms of state variables, the induction motor model is consisted of nonlinear state equations. To estimate the state variables of the motor model and gain the rotor flux and speed signals, a method is proposed in this paper using extended Kalman filter. Software programs are used to carry out extended Kalman filter (EKF) algorithm to estimate the rotor speed and fluxes. The obtained results prove that extended Kalman filter algorithm can estimate rotor speed and flux very accurately, and based on that, the speed sensor less drive system can have good static and dynamic performance. Keywords—Induction Motor, Sensorless Vector Control, Speed Estimation, Flux observer, Extended Kalman Filter.

I. INTRODUCTION For a large spectrum of industrial applications, to make the induction motor suitable, the speed is needed to be controlled; and the most common way of doing this is by vector control. But it requires a speed sensor on the shaft of the machine - which is quite tough. Moreover, being costly, the speed sensor reduces reliability and robustness of the induction motor. Thus, it has introduced a new area of research recently, and among the variety of solutions, sensorless vector control has become very popular as the industrial standard. As the examples of this new technology, artificial intelligence and neural networks come first. But they need offline calculations and under speed changes, they show weak performance. Also, the nonlinear model of the induction motor is in fifth order, containing unknown state variables and external inputs – which have made this sensorless control a challenging theoretical problem. Several approaches have been proposed in recent years to enhance the reliability of the system and lessen the cost by reducing or eliminating the speed and flux sensors from industrial induction motor drives. An auxiliary system is introduced as alternatives to transducers and encoders to estimate the state of the system, which is called an observer. An adaptive observer with one phase current sensor is shown in [1]. On-line trained multilayer feed forward neural networks with a fuzzy nonlinear observer is presented in [2]. Model reference adaptive system MRAS is shown in [3]. Sliding mode observers [4], Luerberger observer [5] are also proposed. These are the systems to reduce the number of the sensors. Based on the model of induction motor in the stationary reference frame, an adaptive observer is presented in [6]. Linear feedback is used to estimate 978-1-5386-2307-7/17/$31.00 ©2017 IEEE

the fluxes and the adaptive law estimates the speed here; the gains are designed using Lyapunov’s nonlinear stability method. Torque model reference adaptive schemes (TMRAS) are proposed in [7] to enhance the performance under low and zero speed conditions. Being able to take into account the model uncertainties and inherent non-linearities in last few years, the Extended Kalman Filter (EKF) is the most used observer for stochastic nonlinear systems such as induction motor drives. An extended Kalman filter is used in [8] to simultaneously estimate stator stationary axis parameters for direct torque controlled induction motor. A tuned EKF is proposed for estimation in [9] where particle swarm optimization is used to optimize the covariance matrices. Fig. 1 shows the list of categorized approaches for estimation. Spectral analysis is required for rotor slot harmonics methods which is a time-consuming task. It also provides a narrow band of speed control. And in case of frequency signal injection, the probing signals produces high frequency torque pulses and as a result, speed ripple occurs. On the other hand, model based methods are popular among researchers due to better performance at high speed, and simplicity. Multiple model extended Kalman filter is presented in [12] which obeys the Markov chain. The estimation is done here by mixing different model outputs with different weights. A robust reduced order extended Kalman filter is proposed in [13] for better estimation performance during speed and flux estimation of induction motor where the gross error on estimation accuracy is analyzed. An approach for rotor flux estimation is shown in [14]. Avoiding sixth order complexity is represented by a forth order descriptor type robust Kalman filter which improves the degree of robustness of the estimates. A fifth-order extended Kalman filter is used in [15] in order to reduce the sampling time as well as the process of the estimation when the system is running at a very low speed. To linearize the process of the extended Kalman filter, fast execution time is considered and no change has been made of conventional look up table of the direct torque control. In this paper, the rotor speed and the flux of the induction motor are estimated by a novel extended Kalman filter using the mathematical model of induction motor. Section II discusses about the nonlinear model of the induction motor. Section III presents the mathematical strategy of the extended Kalman filter. Estimation of Speed and flux of the induction motor using extended Kalman filter is proposed in section IV. Section V shows the estimation results with respect to actual values and the

III. EXTENDED KALMAN FILTER Linear quadratic estimation, also known as Kalman filter, is an algorithm that produces estimation of unknown variables using series of measurements over time, containing statistical noise and other inaccuracies. This basic Kalman filter is limited to a linear assumption. For more complex systems which are nonlinear in nature such as induction motor, the nonlinearity may occur from the process model or the observation model or for both. So, to handle the nonlinear functions of the systems, extended Kalman filter is introduced. And for extended Kalman filter the functions are of differentiable type. Two equations of Kalman filter are: =

+

=

Figure. 1. Speed estimation methods of sensorless systems. The Extended Kalman Filter approach falls under the ‘Machine Model Based Methods’ category.

performance of the extended Kalman filter. The conclusion is drawn in section VI. II. NONLINEAR MODEL OF INDUCTION MOTOR To transform nonlinear system for certain cases, reference frames are used in most controllers as a fundamental tool for development of equivalent circuits. In case of induction motor modelling, three reference frames are available. They are stationary or stator reference frame, rotor reference frame, and synchronously rotating reference frame. The state and mechanical motion equations [10] in – frame are: =−

+

+

+

(1) =−

+

+

+

(2)

(7)

is the The system must fit in these two equations. Here, measurement value. A, B, H are the matrices in general form. ut is the control signal and wt is the process noise and vt is the measurement noise. Next step is to determine initial values and necessary parameters. In the iteration process there are time update equations and measurement update equations. Time update equations are used to predict; on the other hand, measurement update equations are used for correction at each state. Equation for projecting the state ahead is, = ̅

+

(8)

Here, is the estimation of the original signal . Equation for projecting the error covariance ahead from initial estimates at t=0 is, ̅

=

+

(9)

For correction, the Kalman gain is computed by: ̅

+ ̅

(10)

Equation (1) is used to update the estimation by: =

−

(6)

+

=

−

+

̅

+

−

(11) ̅

The error covariance is updated and the outputs at t will be the input for t+1.

=

+

−

(3)

=

+

−

(4)

IV. SPEED AND FLUX ESTIMATION USING EXTENDED KALMAN

(5)

In sensorless vector control, speed is considered as an unknown state variable. From the motor mechanical equation and the original state equation, a new state equation is produced where the system input and output remain invariable. To estimate the states of the induction motor, the equations must be in discrete form because EKF is a recursive state estimation algorithm. So, according to the discrete formula, the discrete forms of the machine equations are:

= 1−

(13) ̅

FILTER

=

−

−

is the Where is the current component in and frame, is the rotor speed, is the stator rotor flux component, resistance, is the rotor resistance, and are the selfinductances of stator and rotor accordingly, and is the mutual inductance between stator and rotor. = is rotor time constant. The leakage coefficient is denoted by =

.

and

are

and

, here

axis component of stator

is the motor load torque, is the inertia and is the voltage. number of motor pole pairs. Rotor electrical angular velocity is represented by .

+1 = 1− +

+

+ +

(14)

+1 = 1−

+1 =

+

+

+

+

+1 =−

As the noise vector cannot be determined in advance, the values of the states are predicted by the deterministic equations.

(15)

+ 1−

The calculation of covariance matrix of predictive error is:

−

+ 1|

(16) +1 =−

+ 1−

(18)

=

∗

|

+

∗

(19)

+ (17)

Where: 1−

0

+

1−

0

=



+

−

0

− 0

0

Kalman gain K and the covariance matrix P of the predictive error is calculated then at the moment t+1. +1 =

+ 1|

∗

∗

+ 1| + 1 = 1 −

∗

+1 ∗

+ 1| ∗

∗

+ 1|

+ (21) (22)

And then the state estimation x(t+1) at moment t+1 is calculated. TABLE I shows the parameters of the induction motor model used in the simulation to get actual flux and speed. TABLE I.

PARAMETERS FOR INDUCTION MOTOR MODELLING

Parameter name

Value

Phase Supply voltage

Number of poles

Unit

3

-

220

Volt

3

-

Inductance of stator ( )

0.1004

Henry

Inductance of rotor ( )

0.0969

Henry

Mutual inductance (

0.0915

Henry

)

1.294

Ohm

Stator resistance ( )

1.54

Ohm

0.095

Nmsec

80

radsec

13

Nm

Rotor resistance (

)

Inertia ( ) Maximum speed ( Rated torque ( ) Magnetic field ( )

)

0.005

− 1−



−

0



−

(20)

− −

0 0

1

machine equations, a system is built to fit the extended Kalman filter equations. By time update and measurement update, the projections of state parameters are done. The covariance matrix and the Kalman gain is calculated for the iteration of the extended Kalman filter. The estimation performance depends on the system parameters, covariance matrices, state and measurement noises. As the covariance matrices are not known for containing the weight factors, they were first tuned to get maximum performance from the extended Kalman filter during the estimation. Estimation of the constant speed of the induction motor is shown in Fig. 2. By changing the load torque, the speed can be varied during the simulation. To check the performance of the EKF estimation of speed at varying speed, load torque of the model is changed and the speed estimation is presented in Fig. 3.

Nmsecrad

V. RESULT AND DISCUSSION With the defined parameters and the model equations, a Simulink model of induction motor is built; and with proper external inputs, the simulation is done. The actual speed is calculated from the simulation, and calculation of the current component gave the rotor flux amount. After discretizing the

Figure. 2. Estimated constant speed of induction motor. The estimated speed appeared very close to the actual speed.

To measure the performance of the EKF estimation, statistical analysis is done with calculating the cross correlation between the actual and the estimated values of speed and flux. High degree of symmetry in Fig. 6 and Fig. 7 shows the high correlation of speed and flux with negligible noise respectively. The mean square error of the EKF estimation is compared with the result of [11]. Back propagation (BP), uncorrelated real time recurrent learning (URTRL), and correlated real time recurrent learning (CRTRL) conducted to verify the performance of the method are shown in Fig. 8. And with minimum mean square error, EKF shows the highest performance.

Figure. 3. Estimated speed changing the load torque of the model. It appeared to accurate in predicting the actual speed with respect to varying load torques.

Estimated rotor flux at both alpha and beta axis along with the actual rotor flux are calculated from the current component of the simulated induction motor model by extended Kalman filter. Results and parameters are shown in Fig. 4 and Fig. 5 accordingly.

Figure. 6. Normalized cross correlation between actual and estimited speed. High degree of symmetry indicates the noise is negligible.

Figure. 7. Normlized Cross correlation plot between estimated and actual flux. High correlation factor is observed. Figure. 4. Actual and estimated α axis rotor flux. Though the estimation showed some in predicting the initial flux, it became accurate as steady state approached.

Figure. 5. Actual and estimated β axis rotro flux. The estimation was accurate for steady state flux prediction, though it exhibited error at the beginning.

Figure. 8. Mean square error comparison. Extended Kalman Filter (EKF) demonstrated the least amount of mean square error, and therefore, it is the superior performer.

VI. CONCLUSION Rotor speed and flux of an induction motor have been estimated in this paper using extended Kalman filter. A mathematical model of induction motor has been modeled with nonlinear equations. With suitable machine parameters, a Simulink model of the induction motor has been designed. Applying Field orientation control to the induction motor model, the stator current and voltage has been observed. These results have been used in the software program of the extended Kalman filter to estimate the motor parameters. For this, the induction motor’s nonlinear equations are converted into state variable equations at the alpha beta frame. After discretizing the motor state equations, the deterministic equations have been predicted to calculate the covariance matrix of prediction error. Kalman gain has been calculated from above results and the rotor speed and flux of the induction motor have been estimated. Results have been compared with the actual value of the speed and flux and quality has been analyzed with statistical analysis. REFERENCES [1]

[2]

[3]

[4]

[5]

F. R. Salmasi and T. A. Najafabadi, “An Adaptive Observer With Online Rotor and Stator Resistance Estimation for Induction Motors With One Phase Current Sensor,” IEEE Transactions on Energy Conversion, vol. 26, no. 3, pp. 959–966, Sep. 2011. B. Karanayil, M. F. Rahman, and C. Grantham, “Stator and rotor resistance observers for induction motor drive using fuzzy logic and artificial neural networks,” IEEE Transactions on Energy Conversion, vol. 20, no. 4, pp. 771–780, Dec. 2005. P. Rao, J. Nakka and R. Shekar, "Sensorless vector control of Induction machine using MRAS techniques", International Conference on Circuits, Power and Computing Technologies (ICCPCT), 2013. A. Ammar, A. Bourek and A. Benakcha, "Modified load angle Direct Torque Control for sensorless induction motor using sliding mode flux observer",4th International Conference on Electrical Engineering (ICEE), 2015. H. Talhaoui, A. Menacer, A. Kessal and R. Kechida, "Fast Fourier and discrete wavelet transforms applied to sensorless vector control induction motor for rotor bar faults diagnosis", ISA Transactions, 53(5), 2014, pp.1639-1649.

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

M. Comanescu, "A hybrid adaptive observer for the speed and flux estimation of induction motors," 2016 IEEE Applied Power Electronics Conference and Exposition (APEC), Long Beach, CA, 2016, pp. 28552860. doi: 10.1109/APEC.2016.7468269 A. N. Smith, S. M. Gadoue and J. W. Finch, "Improved Rotor Flux Estimation at Low Speeds for Torque MRAS-Based Sensorless Induction Motor Drives," in IEEE Transactions on Energy Conversion, vol. 31, no. 1, pp. 270-282, March 2016. doi: 10.1109/TEC.2015.2480961 R. Demir, M. Barut, R. Yildiz, R. Inan and E. Zerdali, "EKF based rotor and stator resistance estimations for direct torque control of Induction Motors," 2017 International Conference on Optimization of Electrical and Electronic Equipment (OPTIM) & 2017 Intl Aegean Conference on Electrical Machines and Power Electronics (ACEMP), Brasov, 2017, pp. 376-381. doi: 10.1109/OPTIM.2017.7974999 S. Allaoui, K. Chafaa, Y. Laamari and B. Athamena, "Induction motor state estimation using tuned Extended Kalman Filter," 2015 4th International Conference on Electrical Engineering (ICEE), Boumerdes, 2015, pp. 1-5. doi: 10.1109/INTEE.2015.7416676 Zhang Wei and J. J. Luo, "Speed and rotor flux estimation of induction motors based on extended kalman filter," The 6th International Conference on Networked Computing and Advanced Information Management, Seoul, 2010, pp. 157-160. M. A. Rafiq, N. K. Roy and B. C. Ghosh, "Three algorithms for learning artificial neural network: A comparison for induction motor flux estimation," 2009 12th International Conference on Computers and Information Technology, Dhaka, 2009, pp. 355-360. Z. Yin, G. Li, Y. Zhang, J. Liu, X. Sun and Y. Zhong, "A Speed and Flux Observer of Induction Motor Based on Extended Kalman Filter and Markov Chain," in IEEE Transactions on Power Electronics, vol. 32, no. 9, pp. 7096-7117, Sept. 2017. doi: 10.1109/TPEL.2016.2623806. Z. Yin, C. Zhao, J. Liu and Y. Zhong, "Research on Anti-Error Performance of Speed and Flux Estimator for Induction Motor Using Robust Reduced-Order EKF," in IEEE Transactions on Industrial Informatics, vol. 9, no. 2, pp. 1037-1046, May 2013. doi: 10.1109/TII.2012.2222422. F. Alonge, F. D'Ippolito and A. Sferlazza, "Sensorless Control of Induction-Motor Drive Based on Robust Kalman Filter and Adaptive Speed Estimation," in IEEE Transactions on Industrial Electronics, vol. 61, no. 3, pp. 1444-1453, March 2014. doi: 10.1109/TIE.2013.2257142. I. M. Alsofyani and N. R. N. Idris, "Lookup-Table-Based DTC of Induction Machines With Improved Flux Regulation and Extended Kalman Filter State Estimator at Low-Speed Operation," in IEEE Transactions on Industrial Informatics, vol. 12, no. 4, pp. 1412-1425, Aug. 2016. doi: 10.1109/TII.2016.2571682.