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Leonardo Electronic Journal of Practices and Technologies

ISSN 1583-1078

Issue 11, July-December 2007

p. 19-36

Parametric Variations Sensitivity Analysis on IM Discrete Speed Estimation

Mohamed BEN MESSAOUD*and Abdennaceur KACHOURI

Electronic and Information Technology Laboratory, National Engineering School of Sfax,

Tunisia

[email protected](*corresponding author)

Abstract

Motivation:This paper will discuss sensitivity issues in rotor speed estimation

for induction machine (IM) drives using only voltage and current

measurements. A supervised estimation algorithm is proposed with the aim to

achieve good performances in the large variations of the speed. After a brief

presentation on discrete feedback structure of the estimator formulated from

d-q axis equations, we will expose its performances for machine parameters

variations.

Method: Hyperstability concept was applied to the synthesis adaptation low.

A heuristic term is added to the algorithm to maintain good speed estimation

factor in high speeds.

Results: In simulation, the estimation error is maintained relatively low in

wide range of speeds, and the robustness of the estimation algorithm is shown

for machine parametric variations.

Conclusions: Sensitivity analysis to motor parameter changes of proposed

sensorless IM is then performed.

Keywords

Induction Motor; Speed Estimator; Sensitivity; Parametric Variation;

Robustness

http://lejpt.academicdirect.org19
mailto:[email protected]:[email protected]

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Mohamed BEN MESSAOUD and Abdennaceur KACHOURI

Introduction

A high degree of sophistication of new control methods as vector control, adaptive or

variable structure control, is reached with the help of special measurement systems (state

observers, reconstruction of mechanical or electromagnetic variables).

During the last decay the speed control of induction machine (IM) requires the

knowledge of rotor speeds values, therefore in order to replace the mechanical sensors,

significant research effort has been devoted to the field of shaft-sensorless control of

induction motors. This research is interest on softwarebased methods of estimating rotor

speed of induction motors using electric measurement of the stator current and voltage. Direct

and indirect methods are developed to avoid magnetic or mechanical sensors mounted in the

motor [1, 2, 3]. It was observed that a speed estimation error can appears when one uses a flux

or state observers and then calculate the rotor speed [4,5].

Less error and less sensitivity on parameter variation are noted if one uses the Model

Reference Adaptive Systems (MRAS) [6] or sliding mode techniques [7, 8, 9].

Recently, neural identification method is applied to estimate motor speed; it seems to

be an interesting solution but it presents some problems in the case of reversal operation of

the motor [10].A novel parallel adaptive observer has been designed, starting from the series-

parallel Kreis-selmeier observer [11].

This paper deals with a new class for speed estimation of induction motor. The used

structure constitutes the feedback linear time varying structure in its discrete form. The

hyperstability of the loop are demonstrated and the stability is guaranteed.

The adaptation algorithm based on current quantities is deduced. The high-

performances of such estimator are shown in low speeds and when parameter changes, where

the most methods fail.In the objective of applicability of the algorithm in high speeds, adaptation low is

slowly modified by replacing the current by current error.

In simulation, the robustness of the proposed algorithm is checked for variations of the

stator and rotor parameters (resistances and inductances).

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Feedback structure of the estimator

In order to overcome the stability problem for low speed with parameter variations, we

present the structure of the inverse model of the machine in the form of feedback linear time

varying structure based on electrical equations described below.

Electrical Equations

Equations for induction motor can be expressed in the stationary d-q frame [6] as:

)ev(iRdtdiL SSSS

SS += (1)

somrmm i

Tr

1Jii

Tr

1

dt

di+=

where

=01

10J

(2)

dt

diLe mm= (3)

The signification of parameters and variables appear in appendix I.

Discretization

A discretized version suitable for digital implementation is developed, preserving the

characteristics of the original continuous-time procedure.

The discrete form of equations (1-2 and 3) is given by:

[ ])k(e)k(v)1(R

1)k(i.)1k(i mS

S

SS +=+ (4)

)k(iTr

L)k(i

Tr

1

Tr

1

L)k(e Sm

r

r

m +

=

m 2 2 r m

L Lor e (k) .I J i (k) i (k)

Tr Tr

= +

s

(5)

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)k(eL

T)k(i)1k(i mmm +=+ (6)

where

=

S

S

R

LTexp

Figure 1 illustrates the conventional adaptive structure. The proposed structure is

described by equations (4), (5) and (6). These equations constitute a feedback time varying

parameters system represented by the bloc diagram of the proposed speed estimator of the

figure 2. The input is the vector vso = [vsod, vsoq]T and the output is the estimated speed of

induction machine.

In counter part of conventional nonlinear structure, the present strategy presents a

dynamic which depends only on the values of a transition state matrix of the feedback system.

Figures 1 and 2 illustrate the difference between conventional adaptive structure and the

proposed structure.

Figure 1. Diagram of conventional Adaptive speed estimator

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Figure 2. Diagram of proposed speed estimator

Hyper-stability of feedback structure

In this section, we briefly review the use of hyperstability concept [12] to the synthesis

of adaptation low. The hyperstability analysis of nonlinear systems requires a linear time

invariant discrete transfer function H(z) in the feedforward path, and a nonlinear block in the

feedback one (figure 3).

Figure 3. Configuration of feedback system

22S I.z

)1(R

1

)z(H

=

Hyperstability Theorem[13]:

The non linear feedback system of the figure 3 is hyperstable if:

The linear time invariant discrete matrix H(z) is real positive; i.e.

o all poles of elements of H(z) lies in unitary circle

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o the matrix H(z)+H(z*) is a positive semi definite Hermitian matrix for allz=1,

the star indicates the complex conjugate.

In the non-linear feedback part, the following inequality of Popov (7) holds; i.e.

2

oS

Tkk

0k

)k(I).k(me1

=

=

(7)

Synthesis of adaptation low

By substituting equations (4), (5) in inequality (7), one can write:

[ ] [ ] ( )

=

= +++

1kk

ok

2

osdmqsqmdrrmqsqmdsd

2

sq

2

sd iiiiTiiiiii (8)

Under steady state and the following approximation:

0)ii(i)ii(iiiiiii mqsqsqmdsdsdmqsqmdsd2

sq

2

sd +=++ (9)

Equation (7) becomes:

( )=

=

1kk

ok

2

osdmqsqmdr iiii (10)

Lets take the Integral adaptation law as:

=

+=+=+k

0i

rrr )i()0()k()k()1k( (11)

Without loss of generality, letting r(0)=0 and replacing (11) in (10), yields:

( ) =

= =

1kk

0k

k

0i

2

osdmqsqmd )i(iiii (12)

Using the relation (13),

2

c

2

cx

2

1cx

2

1cxx

2k

0k

22

k

2k

0k

k

0k

k

k

0i

ik

11 1

+

+=

+

== ==

(13)

One obtains the particular solution for as follows

)k(i)k(i)k(i)k(i)k(sdmdsqmd = (14)

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where is a positive and constant.

Finally, the Integral adaptation law is deduced:

)iiii()k()1k( sdmqsqmdrr +=+ (15)

where is any positive real.

Taking into account the adaptation mechanism (15), Popov inequality (7) is hold and

the hyperstability is guaranteed for the nominal parameters and in the unloaded motor case.

Performance analysis

The speed estimation algorithm described in Equation (15) is tested in the wide range

of speed and torque variations and the machine parameter variation is also considered to

evaluate the performance of the algorithm.

Simulation conditions

To achieve the following simulation results, Matlab- Simulink software is used to

simulate the hardware and the software parts.

The simulation block diagram is represented in figure 4 where the ideal voltage

inverter is used and the Open loop speed control is applied.

The voltage and current measurement quantities constitute the inputs of the algorithm

to estimate the motor speed.

Figure 4. Simulation scheme

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The motor is trained by electrical frequency s, therefore the correspondent trajectory

of the motor speed is deduced. The reference speed of the motor is changed at different time

instant as illustrated in table 1.

Table 1. Reference speed variation.

Scale of speed Low speed Middle speed Nominal speed Stop position

Time range (s) 0-1 2-2.5 3-3.5 4.5-5

Reference speed value (rad/s) 10 50 150 0

The figure 5 illustrates the detailed diagram of simulation using the Matlab- simulink

blocks.

Figure 5. Matlab- simulink diagram for simulation

Simulation Results

The proposed estimation algorithm generates the estimated speed for a power motor of

1.5 kW, the rated torque 7 Nm and the rated speed 1420 rpm. The open loop control is applied

for the motor, in order to give the profile of the measured speed.

Figures 6 and 7 evaluate the performances of the algorithm in the cases of unloaded

motor and for +50% rotor resistance variations.

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Remarks

In unloaded torque case, the figure 6(a) presents the evolution of motor speed and

estimating one for different values (low and high speeds). Simulation reveals negligible

steady state errors as illustrated in figure 6 (b).

For unloaded machine the speed is estimated perfectly [14].

Figure.6. Typical parameters: unloaded motor (a) measured and estimated speed.

(b) error on speed estimator

Figure.7. Deviation of +50 % in Rs; unloaded motor (a) measured and estimated speed.

(b) error on speed estimator

Sensitivity of the algorithm to parameter variation

To evaluate the influence of the parametric variation to estimated speed, we introduce

the following performances indexes expressed in percent:

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The Percent of Root-mean-square Difference (PRD):

[ ]

[ ]2N

1i

mot

N

1i

2

estmot

)i(

)i()i(

100(%)PRD

=

=

= (16)

And the steady state error s,:

mot

estmot

ts lim100(%)

=

(17)

where motis the actual speed of the motor and est is its estimation.

Table 2 shows that the estimated speed is not affected by the parameter variations,than it is obvious that the proposed algorithm gives satisfactory results.

Table 2. Performances of the PI estimation for parametric deviation in the case of unloaded

machine

steady state error (%)Parameter PRD (%)

mot = 10 rad/s mot = 50 rad/s mot = 150 rad/sRated 0.45 -0.0032 -0.02 -0.18

0,5 Rs

1,5 Rs

0.97

0.56

-0.0548

-0.0045519

-0.0364

-0.0214

-0.18

-0.180,5 Lls1,5 Lls

0.29

0.46

-0.0032419

-0.00317

-0.02

-0.0204

-0.17

-0.19

0,5 Rr1,5 Rr

1.05

0.56

-0.00395

-0.0044

-0.009

-0.0294

-0.08

-0.28

0,5 Llr1,5 Llr

0.52

0.47

-0.0032

-0.00324

-0.02

-0.02

-0.18

-0.18

0,5 M

1,5 M

0.51

0.70

-0.0059

0.0154

-0.0244

-0.0187

-0.2

-0.18

Load effect on the proposed algorithm

The behavior of the speed estimator of induction motor with a mechanical load equal

to 200% of its nominal value is checked.

Figure 8 shows the influence of the load in the case of nominal parameters and the

figure 9 for +50% variations on the stator resistance referring to its nominal value.

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Figure 8. Effect of the load torque on the estimation algorithm: (top) measured and estimated

speeds; (bottom) estimation error

Figure 9. Effect of deviation +50% Rs in the case of loaded motor: (top) measured and

estimated speeds; (bottom) estimation error

In the following, table 3 presents the quadratic error and the steady state error for low

and high speeds with the machine parameters variations.

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Table 3. Performances of the PI estimation for parametric deviation for the case of loaded

machine

steady state error (%)parameter PRD (%)

mot = 10 rad/s mot = 50 rad/s mot = 150 rad/snominal 7.8 -0.876 -2.63 -9.37

0,5 Rs1,5 Rs

7.23

8.7

-0.64

-1.38

-2.52

-2.81

-8.52

-10.5

0,5 Lls1,5 Lls

7.4

8.2

-0.85

-0.9

-2.52

-2.74

-8.87

-9.91

0,5 Rr1,5 Rr

3.9

11.9

-0.43

-1.32

-1.3

-4.0

-4.47

-14.7

0,5 Llr1,5 Llr

7.8

18.7

-0.876

-0.876

-2.63

-2.63

-9.3

-9.46

0,5 M

1,5 M

8.6

7.56

-2.11

-0.63

-3.12

-2.5

-10.4

-9.05

Referring to the table 3, the simulation results show the dependence of estimation

error and the speed. We note that the relative error increases until 10% in the nominal speed

as shown in figure 8.

Discussion

The analysis of the tables 2 and 3 shows that the parameter variations practically do

not affects the estimated speed. However, the machine load is the preponderant factor which

affects the estimated speed. It is to be noted that the motor load is not considered in the

algorithm. Therefore, the influence of the load appears clearly in high speed.

Modified algorithm

To overcome the error introduced by the load, we introduce a correction signal

depending on the load in the adaptation low. The current components i sare replaced by the

current error sin the high speed and the adaptation mechanism is described by (18):

)iiii()iiii()k()1k( sodmqsodmdsdmqsqmdrr +=+ (18)

where:

is any positive integer parameter which increases with the speed;

iso, isare measured and calculated currents.

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The input vector of the estimation algorithm becomes [vsodvsoqisodisoq]T .

Parameter Sensitivity and Simulation Results

To study the influence of parameter deviation on the performance of the modified

estimation speed algorithm, we will take a variation of 50% of each machine parameter.

Unloaded motor TL=0

Table 4 represents the performance indexes evaluated earlier for different values of the

desired speed and for different variations on motor parameters.

Table 4. Performance of the modified algorithm for parameter variations in the unload motor

caseSteady state error (%) depending on motParameter PRD (%)

10 rad/s 50 rad/s 150 rad/s

Nominal 2.11 -0.0032 0.0992 0.196

0,5 Rs1,5 Rs

2.33

1.82

-0.0546

-0.0046

-0.769

0.846

0.0069

0.383

0,5 Lls1,5 Lls

2.24

2.03

-0.0032

-0.0032

0.135

0.065

0.218

0.176

0,5 Rr1,5 Rr

2.70

1.65

-0.0039

-0.0044

0.115

0.098

0.301

0.098

0,5 Llr1,5 Llr

2.21

2.6

-0.0032

-0.0032

0.097

0.101

0.196

0.196

0,5 M

1,5 M

2.89

2.51

-0.0059

0.0154

2.60

-0.301

1.13

0.052

Referring to table 4, it is obvious that the modified algorithm gives satisfactory results

in the case of unloaded motor. The estimation error does not access 0.3% in most cases.

Overloaded motor: 200 % of rated load

The behavior of the modified speed estimator of the induction motor is checked with a

mechanical load equal to 200% of its nominal value.

Figures 10 and 11 illustrate the measured and the estimated speeds when the stator

resistance varies in the range of -50% to +50% of Rs of nominal value.

The simulation result shows that it is not possible clearly to distinguish between

measured and estimated speed. Thus, the estimation steady state error is less than 0.3 rad/s for

rated speed.

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Figure 10. Loaded motor: 200% of nominal torque (a) Measured and estimated speed

for -50% variation on Rs (b) Estimation error

Figure 11 (a) Measured and estimated speed for +50% variation on Rs and for 200% motor

nominal load; (b) Estimation error

The table 5 summarizes the performance of the modified estimator in the case of over

loaded motor and for the parameter variations.

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Table 5. Performance of the modified algorithm for parameter variations in the overload

motor case

Steady state error (%)Parameter PRD (%)

mot = 10 rad/s mot = 50 rad/s mot = 150 rad/sNominal 2.08 -0.876 0.427 0.0344

0,5 Rs1,5 Rs

2.63

1.74

-0.640

-1.38

-1.22

1.96

0.191

-0.30

0,5 Lls1,5 Lls

2.24

1.99

-0.855

-0.897

0.617

0.237

0.453

-0.424

0,5 Rr1,5 Rr

4.96

3.96

-0.434

-1.32

1.79

-0.857

4.52

-4.88

0,5 Llr1,5 Llr

2.13

2.06

-0.876

-0.876

0.404

0.446

0.0344

0.017

0,5 M

1,5 M

5.03

2.35

-2.12

-0.63

12.4

-0.53

1.09

-0.133

The analysis of the table 5 shows the efficiency of the proposed algorithm with respect

to the variations of Rs, Ls and Lr for all range of speeds; in fact the relative error doesn't

access 0.3%.

Thus, the robustness of the proposed algorithm for all range of speeds is guaranteed.

On the other hand, its sensitivity to the variation of Rr is acceptable for the high speeds; it is

in the order of 4%. The only case where a relatively error appeared is the case of the reduction

of 50% of M for the middle speeds (these error remains limited). Consequently, the

robustness of the modified algorithm to parametric variation is shown for all the range of

speeds.

Conclusions

The feedback structure of estimation speed algorithm is presented. It is fairly general

and would seem to be the natural extensions to nonlinear adaptive structure case of estimation

speed of induction machines.

It is to be noted that the advantages of the previous structures are believed maintained.

In this paper, the discrete form adapted to the implementation purpose is developed and the

stability analysis is performed using the hyper stability theory.

In the case of unloaded motor, simulation results show the robustness of the algorithm

to the motor parameter variations (Rs, Ls, Rr, Lr and M).

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For torque load, it is shown that estimation errors that are not present in the previous

case occur for high speeds. The presence of load disturbs the estimated speed.

The high-speed problems are remedied by a careful choice of standard relation. This is

done by adding the term 'stator current components', which depending on the torque.

Finally, simulation examples are considered to illustrate the advantages that can be

gained by using the modified algorithm. There was proved that the proposed adjustable low is

able to estimate the proper values of the rotor speed even in the case of parameter and speed

errors.

The estimation speed algorithms have proven to be a powerful tool in order to give the

real induction motor speeds. Special attention must be designed when the mutual inductance

decreases.

Appendix I

L = M2/Lr= Equivalent mutual inductance

= 1-M2 /LsLr = leakage factor.

Electrical variables

vs= [vsdvsq]T= stator voltage vector.

is= [isdisq]T= stator current vector.

ir= [irdirq]T= rotor current vector.

im=[imdimq]T= magnetizing current vector.

Electrical parameters

Rs= 4.58 , Ls= 253 mH = stator resistance and inductance

Rr= 4.58 , Lr= 253 mH = rotor resistance and inductance

Tr= Lr/Rr= rotor time constant inductance

M = 242.3 mH = mutual inductance

Lls= Ls-M = stator leakage inductance

Llr= Lr-M = rotor leakage inductance

Mechanical variable and parameters

r= rotor electric angular velocity (150 rad/s rated)

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F = 0.0026 kgm2/s = friction coefficient.

J = 0.023 kgm2= moment the inertia

Matrix notation

Inn= nn identity matrix

= orthogonal rotation matrix

=01

10J

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