1

Abstract-- This paper presents a novel speed & position

estimation algorithm for the PMSG based wind turbine using Adaptive Network-Based Fuzzy Inference System (ANFIS). The precise information of rotor speed and position is required to perform maximum power tracking in variable speed wind turbine, which is very difficult to estimate under varying operating conditions, as it may lead to parameter variation such as stator resistance, inductance and torque constant. To neutralize the effect of parameter variation, an ANFIS based adaptive model is developed to track the rotor position and speed accurately in the whole range of low to high speed under varying wind conditions. A detailed comparison of two different estimation techniques based on adaptive neuro-fuzzy system and sliding mode observers is also provided under varying speed with variable PMSG parameters. Both of the estimation techniques are realized and simulated in MATLAB/SPS environment. The simulation results under dynamic operating conditions are provided to demonstrate the effectiveness of proposed approach.

Index Terms-- Wind energy conversion system, Permanent magnet synchronous generator, Sensor-less control, Adaptive Neuro-Fuzzy Systems, Sliding mode observer.

I. INTRODUCTION

ecently the Permanent Magnet Synchronous Generator (PMSG) has received worldwide attention for wind power

generation, due to their small size, higher power density, high efficiency and higher torque to inertia ratio [1]-[2]. For the precise control of PMSG, the information about rotor speed and position is essentially required: either using sensors or estimator/observer. To avoid the additional sensor cost, complexity and the other associated problems, there has been significant interest in the sensor-less control of PMSG [3]-[5]. Moreover, the elimination of these sensors and their connecting leads increases the mechanical robustness and reliability of overall system. But the rotor speed and position typically requires the accurate knowledge of PMSG parameters, which may not be easily available or difficult to obtain, especially under varying weather and operating conditions. All these factors have made the sensorless control

This work was supported by the Govt. of India under National Overseas Scholarship (2005-2006). The authors would also like to thank High Commission of India, Ottawa, Canada for their timely financial support in carrying out this research work.

Mukhtiar Singh and A. Chandra are with Department de genie electrique, Ecole de Technologie Superieure, Universite du Quebec, 1100 rue Notre Dam, H3CIK3, Montreal, Quebec, Canada (smukhtiar_79@yahoo.co.in, ambrish.chandra@etsmtl.ca, phone: +1514-396-8870; fax: +1514-396-8694).

of PMSG more attractive. Several rotor speed and position estimation techniques have been reported in literature. In [6]-[8], the back-emf based rotor speed estimation is reported. This method works satisfactorily at higher speeds. However, the speed estimation becomes very difficult at lower speeds, due to small and distorted emf signal. Some state observer methods based on Extended Kalman Filter (EKF), Extended Luenburger Observer (ELO), and Sliding Mode Observer etc. have also been reported [9]-[10]. Most of them suffer due to complex computation, sensitivity to parameter variation and need of accurate initial conditions. However, the EKF has the advantage of estimating the parameters and speed simultaneously by considering them as state, but at the increased cost of computational burden [11]. The sliding mode observer is simple and offers a limited robustness against the parameter variation. However, sliding mode being a discontinuous control with variable switching characteristics has chattering problems, which may affect the control accuracy. Recently, some more advanced adaptive estimation techniques based on Artificial Neural Network (ANN) and Fuzzy Logic Control (FLC) have also been reported [12]-[14]. However, the estimation accuracy depends on no. of neurons and no. of fuzzy membership functions used for rule base.

In this paper, a novel ANFIS based position and speed estimation algorithm for PMSG has been proposed for wide range of speed operation. The ANFIS architecture has well known advantages of modeling a highly non-linear system, as it combines the capability of fuzzy reasoning in handling the uncertainties and capability of ANN in learning from the processes [15]-[16]. Thus, the ANFIS is used to develop an adaptive model of variable speed PMSG under highly uncertain operating conditions, which also automatically compensates the any variation in parameter such as inductance, resistance etc. An error gradient based dynamic back propagation method has been used for the on-line tuning of ANFIS architecture. A detailed comparison of simulation results is also provided with sliding mode observer based results under variable speed PMSG operation. Moreover, the simulation study is also carried out for the different value of PMSG resistance and inductance to validate the robustness of proposed algorithm against parameter variation.

The paper is organised as follows: The system under consideration with proposed estimation algorithm is discussed in section II. Simulation results are discussed in Section III. Finally section IV concludes the paper.

Comparative Study of Sliding Mode and ANFIS Based Observers for Speed & Position Sensor-

less Control of Variable Speed PMSG Mukhtiar Singh, Student Member, IEEE, Ambrish Chandra, Senior Member, IEEE

R

978-1-4244-5377-1/10/$26.00 ©2010 IEEE

2

II. PROPOSED SYSTEM AND CONTROL STRATEGY

It is extremely difficult to reproduce the exact model of PMSG for speed and position estimation, due to non-ideal rotor flux distribution, parameter variation and rapidly changing operating conditions. Therefore, in order to reject any such kind of external disturbances, an adaptive model is continuously tuned in parallel of reference PMSG shown in Fig. 1.

1S

2S

3S

4S

5S

6S

dcVBC

A

aV bV cV

aI

bI

cI*

aI *bI *

cI

1P 4P 3P 6P 5P 2P

α β α β

dcI

dcP

*rωˆrω

rθ*

qi* 0di =

iα iβ vα vβ

Fig. 1. Control Diagram of Proposed System Modeling of PMSG for Speed and Position Estimation:. The voltage equations of PMSG in stationary reference frame are as given below:

s sdiV R i L Edt

αα α α= − − + (1)

s s

diV R i L E

dtβ

β β β= − − + (2)

Where V�� are stator terminal voltages, Rs is stator resistance, Ls is stator inductance, i�� are output currents and E�� are back e.m.f.’s which can be given as:

sin( )cos( )

rr m

r

EE

Eα

α ββ

θω λ

θ−� � � �

= =� � � �� �� �

(3)

Here �r, �r and �m are rotor speed, rotor position and magnetic flux linkage respectively.On rearranging the equations (1) & (2) and re-writing them in matrix form, we have

/ 0 1 / 00 / 0 1 /s s s

s s s

i i V ER L Li V ER L Li

α α α α

β β ββ

� � � − −� � � � � �� � � �= + − �� � � � � � � �� � � � �− −� � � � � �� � � � � �� � �

�

� (4)

Using the same structure for adaptive model, the current estimator can be designed, if the accurate value of back e.m.f. is known in the equation given below:

ˆˆ ˆ. .( )i A i B V Eαβ αβ αβ αβ= + −� (5) Where the cap ‘ˆ’ indicates the estimated variables and

/ 00 /s s

s s

R LA

R L−� �

= � �−� �; 1 / 0

0 1 /s

s

LB

L−� �

= � �−� �;

ˆˆ

ˆ

ii

i

ααβ

β

� �� �=� �� �

��

�;

ˆˆ

ˆi

iiα

αββ

� �= � �� �� �

; VV

Vα

α ββ

� �= � �� �

; and ˆ

ˆˆ

EE

Eα

αββ

� �= � �� �� �

The estimated value of back e.m.f. in equation (5) can be expressed as given below:

ˆ ˆsin( )ˆ ˆˆ ˆcos( )

rr m

r

EE

Eα

αββ

θω λ

θ

� � � �−= =� � � �� � � �� �� �

(6)

From equation (6), it is evident that the estimated back e.m.f. carries the information of rotor position and speed, which can be obtained as:

1ˆˆ tan ˆr

EE

α

β

θ −�

= − � ��

andˆ1ˆ r

rd

p dtθω = ; Where p is the no. of

poles. On the basis of above discussed model of PMSG, the sliding mode observer and a novel adaptive neuro-fuzzy system are developed to estimate the rotor position and speed.

Sliding Mode Observer: The sliding mode observer (SMO) is based on stator current estimation, where the sliding mode surface is defined as

ˆ( )ˆ( )

S i iS

S i iα α α

β β β

� �−� �= = � �� �

−� �� � � � (7)

On subtracting the equation (4) from equation (5), the dynamic sliding mode equation can be obtained as

ˆ. .S A S B Eα β= +� (8)

Where, the equivalent control of sliding mode observer in equation (8) can be expressed as

ˆ ˆ( ) sin( )ˆ ˆˆ ˆ( ) cos( )

l rr m

l r

E k sign SE

k sign SEα α

αβββ

θω λ

θ

� � � �− −� �= = =� � � �� �−� � � �� � � �� �

(9)

In order to eliminate the high frequency components from the estimated e.m.f.s, the low pass filters (LPF) with sufficiently small time constant are used. On considering the effect of LPF, the equation (9) can be modified as

( )ˆˆ

ˆ( )

cl

c

cl

c

k sign SE s

EE k sign S

s

αα

α ββ

β

ωω

ωω

� �− ⋅� �� � +� �= =� �� �� �� � − ⋅� �+� �

(10)

Where �c is the cut-off frequency of the LPF, which should be designed properly according to the frequency of estimated currents. If the estimated currents are able to track the actual currents accurately, then it is obvious that the error trajectories will reach the sliding surface (S=0). According to Lyapunov’s stability criteria, the value of switching gain kl should be large enough to guarantee the stable operation of observer [10], with the condition

. 0TS S �=� (11)Due to the discontinuous nature of sliding mode observer, it

suffers from chattering problem at certain switching frequencies. To solve this problem, generally continuous functions, such as sigmoid function and saturation functions are used as shown in Fig. 2. But, it is very difficult to set the boundary limits and switching gain values under different operating conditions. The performance of sliding mode observer further deteriorates due to machine parameter variation with the passage of time.

Vα

Vβ

iαiα

iβiβ

Eα

Eβ

c

csω

ω+

c

csω

ω+

1ˆ

tan ˆEE

α

β

−� � �� 1 rd

p dtθ�

��

rθ

rω

Fig. 2. Sliding Mode Observer for Speed and Position Estimation.

3

ANFIS based Rotor Position and Speed Estimation: An ANFIS based on Takagi-Sugeno-Kang (TSK) method having 2 : 6 : 3 : 3 : 2 architecture with two inputs (estimated currents iα� and iβ

� ) and two outputs (estimated back e.m.f.’s Eα and

E β ) is used to develop the dynamic model of PMSG. While the errors between the actual and estimated currents ( ˆi iα β αβ αβξ = − ) are used to tune the precondition and consequent parameters as shown in Fig. 3.

Fig. 3. Schematic of ANFIS architecture The trapezoidal and triangular membership functions are used to reduce the computation burden as shown in Fig. 4 and their corresponding node equations are as given below:

1 1

1

11 1

1 1

1

1

( ) ( )

0

A B

X bX aX X b X ab a

X a

α βμ μ

� ≤� −�= = � �� −�� ≥�

(12)

2 2

22 2

2

2 2

1 0.50.5( ) ( )0 0.5

A B

X a X a bbX X

X a bα βμ μ

−� − − ≤�= = �� − ≥�

(13)

3 3

3

33 3

3 3

3

0

( ) ( )

1

A B

X aX a

X X a X bb a

X b

α βμ μ

� ≤� −�= = � �� −�� ≥�

(14)

where the value of parameters {ai, bi} changes according to the error and accordingly generates the linguistic value of each membership function [15]. Parameters in this layer are referred as premise parameters or precondition parameters.

Fig. 4. Fuzzy Membership Functions. To tune the ANFIS structure, gradient descent technique is used to reduce the error (usually a cost function given by the squared error), where the weights are iterated by propagating the error from output layer to input layer. This backward trip of such a calculation is termed as “back propagation” [16]. The training algorithm is completed in two stages, known as precondition parameter tuning and consequent parameter tuning, where the objective function to be minimized is defined as 2 2ˆ( )i iα β αβ αβξ = − (15)

(a). Precondition parameter tuning: The Precondition parameters are required to update the fuzzy membership functions in Layer1. To minimize the error function 2

αβξ by gradient descent method, the change in each precondition parameter must be proportional to the rate of change of the error function w.r.t. that particular precondition parameter, i.e.

2

1, 2, 3.i

i

AA

a ia

αξη ∂Δ = − =

∂ (16)

Where � is the constant of proportionality defined as the learning rate. Therefore, the new value of the consequent parameter is given as

( 1) ( ) 1, 2, 3.i i iA A Aa n a n a i+ = + Δ = (17)

or 2

( 1) ( ) 1, 2, 3.i i

i

A AA

a n a n ia

αξη ∂+ = − =

∂ (18)

Now the partial derivative term in equation (18) can be found by the chain rule of differentiation as follows:

1

1 1 1

2 21 1

1 1

ˆˆˆ ˆ

A

A A A

i Ea ai E

α α α α

α α

μξ ξ μ μμ μ μ

∂∂ ∂ ∂ ∂ ∂ ∂= ⋅ ⋅ ⋅ ⋅ ⋅

∂ ∂ ∂ ∂ ∂∂ ∂ (19)

(b). Consequent Parameter Tuning: To tune the consequent parameters in Layer 4, the following updated laws are developed:

2

0 00

( 1) ( ) 1, 2, 3.i i

i

ca n a n ia

α α αα

ξη ∂+ = − ⋅ =

∂ (20)

2

1 11

( 1) ( ) 1, 2, 3.i i

i

ca n a n ia

α α αα

ξη ∂+ = − ⋅ =

∂ (21)

where �c is the learning rate for consequent parameters. The derivative terms in equations (20)-(21), can be found by the chain rule as already discussed in case of precondition parameters as follows:

2 2

0 0

ˆˆ1, 2, 3.ˆ ˆ

i i

i E ia ai E

α α α αα α

α α

ξ ξ∂ ∂ ∂ ∂= ⋅ ⋅ =

∂ ∂∂ ∂ (22)

2 2

1 1

ˆˆ1, 2, 3.ˆ ˆ

i i

i E ia ai E

α α α αα α

α α

ξ ξ∂ ∂ ∂ ∂= ⋅ ⋅ =

∂ ∂∂ ∂ (23)

Similarly, the precondition and consequent parameter for second input can be found just by replacing � by � in above equations.

III. SIMULATION RESULTS AND DISCUSSION

The performance of proposed ANFIS based position and speed estimation for sensor less control of PMSG is demonstrated through the extensive simulation results as shown in Fig. 5. to Fig. 7. Besides this, a detailed comparison is also provided w.r.t. the SMO based results under variable speed operation, with different value of PMSG parameters. The variable torque is applied to the PMSG and accordingly its speed and position are estimated. The traces of PMSG voltage, position, speed and speed errors are discussed in simulation results.

In Fig. 5., the simulation results with nominal value of PMSG parameters are shown. The Fig. 5(a) shows the traces of 3–phase PMSG voltage waveforms, where the variable frequency of these waveforms indicates the variable speed operation of PMSG. In Fig. 5(b), the traces of actual speed, ANFIS speed, and SMO speed are shown. The Fig. 5(c) shows

4

the actual position, ANFIS position and SMO position. The errors between actual and estimated speed based on ANFIS and SMO estimators are shown in Fig. 5(d)., where the ANFIS estimator shows better performance than SMO estimator. The simulation results with 10% and 25% variation in PMSG resistance and inductance are shown in Fig. 6. and Fig. 7. respectively. The performance of SMO deteriorates with the variation in PMSG parameters, while the ANFIS estimator shows promising immunity to any such kind of variation as evident from Fig. 6(d) and Fig. 7(d).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-50

0

50

Vge

n(V

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0

10

20

30

Gen

.Spe

ed(r

ad/s

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

5

10

Gen

.Pos

ition

(rad

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-5

0

5

Spe

ed E

rror

(rad

/s)

Actual SpeedANFIS Speed

SMO Speed

ANFIS Speed ErrorSMO Speed Error

Actual Position ANFIS PositionSMO Position

(b)

(c)

(d)

(a)

Fig. 5. Results with nominal PMSG parameters

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-50

0

50

Vge

n(V

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0

10

20

30

Gen

.Spe

ed(r

ad/s

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

5

10

Gen

.Pos

ition

(rad

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

-5

0

5

Spe

ed E

rror

(rad

/s)

Actual SpeedANFIS Speed

SMO Speed

Actual Position ANFIS Position

SMO Position

ANFIS Speed Error

SMO Speed Error

(a)

(b)

(c)

(d)

Fig. 6. Results with 10% variation in PMSG resistance and inductance

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-50

0

50

Vge

n(V

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0

10

20

30

Gen

.Spe

ed(r

ad/s

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

5

10

Gen

.Pos

ition

(rad

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-20

0

20

Spe

ed E

rror

(rad

/s)

Actual SpeedANFIS Speed

SMO Speed

Actual Position ANFIS PositionSMO Position

ANFIS Speed Error

SMO Speed Error

(a)

(b)

(c)

(d)

Fig. 7. Results with 25% variation in PMSG resistance and inductance

IV. CONCLUSION

In this work, a novel ANFIS based PMSG speed and position estimating algorithm is developed, especially for low speed operating region. A systematic ANFIS structure is presented to develop a dynamic model of PMSG, as it does not require the explicit mathematical model of plant and automatically takes care of parameter variations as well. In order to achieve adaptive model under different operating conditions, an on-line weight tuning algorithm is developed using dynamic error back propagation method. The proposed algorithm accurately estimates the position and speed under variable speed PMSG operation with parameter variation. A detailed comparison of simulation results based on ANFIS estimator and SMO estimator is provided, where ANFIS estimator shows better performance and immunity against parameter variation.

REFERENCES

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[14] A. A. Adam, and K. Gulez, “Fast response adaptive fuzzy logic controller for sensorless direct torque control of PMSM with minimum torque ripple,” COMPEL-27, vol. 2, pp. 534-550, March-2008.

[15] Mukhtiar Singh, and A. Chandra, “Adaptive Neuro-Fuzzy Control of Renewable Interfacing Inverter to Maintain Smooth Power Flow and Non-Linear Unbalanced Load Compensation Simultaneously,” in Proc. IEEE EPEC’09, Oct. 22-33, 2009, Montreal, Quebec, Canada.

[16] J. Shing and R. Jang, “ANFIS: Adaptive-Network-Based Fuzzy Inference System,” IEEE Trans. On Systems, Man, and Cybernetics, vol. 23, no. 3, May 1993.