Diagnosing Parametric Faults in Induction Motors with Nonlinear Parity Relations

Copyright @ IF AC Fault Detecti on. Supervision and Safety for Technical Processes. Budapest, Hungary. 2000

DIAGNOSING PARAMETRIC FAULTS IN INDUCTION MOTORS WITH NONLlNEAR PARITY RELATIONS Jouda Bouattour Tunisian University ISET Nabeul, Tunisia Janos Gertler and Yongtong Hu Electrical Engineering, George Mason University, Fairfax, VA 22030 USA [email protected]

Abstract. Induction motors are described by nonlinear differential equations which involve the inaccessible flux variables. Changes in the electrical parameters appearing in those equations (resistances and inductances) are indicative of faults such as overheating and short circuits in the winding. The motor is also subject to a disturbance in the form of load variations. We propose here a parity equation scheme, based on discretized versions of the nonlinear motor equations, to detect and isolate the parametric faults while being insensitive to the load variations. This is first done for the full set of equations, including the flux variables, then for a reduced set of equations from which the flux has been eliminated. Copyright @20001FAC

Keywords. Induction motor; Fault detection and diagnosis; Nonlinear parity relations; Parametric faults. the induction motor, including several papers submitted to this conference (Arnanz et ai, 2000; Lesecq and Barraud, 2000). Thus the induction motor may serve as a test-case for the comparison of various non linear techniques. The approach we propose here relies on non linear parity relations. These are obtained by rearranging the motor's bilinear input-output model equations . The parity relations are used to generate residuals which are decoupled from the load disturbance and are structured with respect to the various parametric faults. This is achieved by computing the algebraic derivatives of the primary residuals with respect to the parameters, which are themselves time-varying and nonlinear. This is the extension to nonlinear models of the idea proposed earlier for linear plants (Gertler et ai, 1985; Gertler and Kunwer, 1995).

INTRODUCTION Induction motors are extensively used in a wide variety of rotating machinery, ranging from large industrial drives (steel-mills, chemical plants, etc.) to public utilities (water pumps, etc.), all the way to household appliances (refrigerators, etc.). While some of their more frequent faults are mechanical (bearing), which may call for different diagnostic approaches, many of their potential faults are related to the electrical operation of the motor (such as local overheating and/or short circuits in the winding). These latter faults result in changes of the motor's basic electrical parameters, resistances and inductances, and thus may be detected and isolated by means of monitoring those parameters. The mathematical mode ling of the motor is well developed (Nasar, 1970; Keljik, 1995) . The usual model consists of five first-order differential (state) equations which are slightly nonlinear (bilinear) in the state-variables (. Among the five state variables, two are magnetic fluxes which are not measurable.

The residual design will first be done for the full set of motor equations which contain two flux variables as "observed" states. This is then followed by a more realistic design, in which the flux is eliminated from the set of equations; this results in a reduced set of equations which are of higher dynamic order and of higher polynomial degree than the original set.

Recently, there has been a growing interest in model-based diagnostic techniques developed for

971

STATE EQUATIONS OF THE MOTOR

(13)

We are considering a three-phase induction motor, with wound or cage rotor. The state variables of the motor are the currents and fluxes, together with the rotational speed. The inputs are the line voltages. The number of states and inputs in the model can be reduced by appropriate rotational transformations. The model we use employs the Clarke references (a - B references, fixed to the stator), in which flux, current and voltage are each represented by two projections (Comtet-Varga, 1997).

+ Tlf{ - g x 4 (t) +c[bx 2 (t) +X, (f)X 3 (f)) +U, (t)IL) X5

dX3 dt

f

dx4 dt

b L

m

=

x (t) - b x (t) 5

3

+

x, (t)x 2 (t)

The model described above contains five electrical parameters, namely TT =

-gx 4 (t)+c[bx 2 (f)+X, (t)X 3 (f)]+U, (t)/L.

(5)

e (t+ 1) = x (f+ 1) - x (t) 2

with

U = [v ' VIS]

(7)

a

Y

=

[x"

b=Rr IL ,

c=L m IL,L.

f= 1 - c Lm

g=a+c g Lm

k=3p12

2

(16)

Disturbance decoupling calls for making the residuals insensitive to the only disturbance in the system, the load torque qU). Since only the first state-equation contains the load torque, the simplest (and only) way to achieve disturbance decoupling is by omitting the first primary residual. Thus there are four primary residuals for fault detection and isolation, so that

and a=R• IL •

2

- T [b Lmx4 It) - b x 2 (t) - x, It)x 3 (t))

(8)

X4 ' X5 )

(15)

Primary residuals are computed by simply rearranging each of the discretized state-equations. For example, from EQ. (11), the primary residual is

dx d: = -gX 5 (f)+c[bx 3 (t) -X, (t)X 2 (f))+u2 (t)/L.

(6)

[R• , L• , R, , Lr , Lm )'

We wish to design a diagnostic scheme which detects and isolates deviations in these parameters from their nominal value, while being insensitive to the load torque. First we will use the set of statespace equations, as if the flux projections were accessible variables. This affords significant flexibility in the design of residuals and serves as a demonstration of the approach. Then we will design residuals for the input-output equations which use only measurable variables.

(3)

x=[w,fP,fP",i,i,,] r 1I u a »

(14)

(f)

RESIDUALS FROM STATE EQUATIONS

(4)

f

X5

+ Tlf{ -gx 5 (f)+c[bx 3 (t) -X, (t)X 2 (t))+u 2 (t)/L.l

The model equations are as follows:

-- =

(t+ 1) =

(9) (10)

where p is the number of pole-pairs, J is the moment of inertia and q(f) is the load torque.

(17)

For reasons explained below (see disturbance decoupling), the first equation will be omitted from the set. The remaining four equations will be used in discretized form, as

Structured residuals will be designed to facilitate the isolation of the five possible faults. Structured residuals are sensitive to specific subsets of faults, so that the residual set returns a specific pattern in response to particular faults. To design structured residuals for parametric faults, we need the partial derivatives of the primary residuals with respect to those parameters:

(11 )

Sit) = deW I dTT (12)

972

I

(18)

A strongly isolating structure for the five faults requires five residuals; one possible set would be

where nO is the nominal value of the parameters. The partial derivatives, obtained from the state equations algebraically, are nonlinear functions of the inputs and state-variables. They are shown in Table 1. The primary residuals are then expressed as

" '4 '5 '8

"0

Structured residuals are generated from the primary ones by the transformation I

=

=

w I. (t) sW

w.I (t) SW IJ..n

I

[s'w

Si (t)] g

=

(20)

[0 ... 0 z . ]

(21)

Ig

I

= [0 . ..

0 z. ][s' (t) ~

Si

g

wr'

[s' (t)

s ~ (t)]

=

Rank

"'4

'5 '8

"0

(22)

s' (t)

'2"

'3 '4 '5 '6 '7 '8 '9

"0

0

0 0 0

1

0 0

1 1

0

1 1

1

0 0 1

L

R,

L,

Lm

1

1

0

0 0

0 0 0

0

0 0 0 0

•

1

0 0

1 1

0

1

1

0 0

0

1

R

L

•

R,

L,

Lm

1 1

1

0

0

1

0 0

0 0

1

0

0

1

0 0 0 0

•

"'2

'6 '8

1 1

INPUT-OUTPUT EQUATIONS

1

0

0

0

1

1 1

Define x 2.3-[x2

Rr

Lr

Lm

1 1 1 1

1

0

0 0 0

1

0 0

0 0 0

0 0 0

0 0 0

1 1

L

1 1 1

0 0

To obtain an input-output model for the motor, the unmeasurable flux variables x 2 and x have to be eliminated from the state equations. ~or a general treatment of nonlinear elimination in the context of residual generation, see the recent work by Comtet-Varga and Staroswiecki (1998). Here we will apply direct elimination using the discretized state model. The first state-equation will again be omitted, in order to decouple from the torque disturbance.

R

0 0 0 0 0 0

1

0

(23)

Assuming no rank defect, the following ten residual structures may be obtained from the motor state equations (with the Os indicating decoupling):

•

1 1

Alternatively, one may design a subset from the six residuals not affected by the rank-defect; this will be strongly isolating for the first four faults but would not even detect the fifth:

for h =g, then (22) is not solvable, or for any other column outside s', then there is an excess decoupling . In either case, the intended residual structure is not attainable.

•

Lm

•

The response to the additional faults cannot be specified but, for the desired structure to occur, they must be nonzero. If, however, Rank

Lr

R

where s' contains the columns of S which belong to the faults from which ' . is decoupled and Si is the column of the fault w(th the specified respon~e zIg. . The transformation is then computed w .W

Rr

•

However, an analysis of the S matrix reveals that it does have a permanent rank defect for this system. Namely, the first four columns are linearly dependent (but no subset of those columns is). This implies that no residual structure may be implemented which contains three Os in the first four columns. This eliminates the residuals '4' '7' '9' " 0 ' Without these, it is not possible to design a strongly isolating set for the five faults. One may settle with partially weak isolation; for example, the above set of five residuals would become

where w . W is a time-varying vector. This is so chosen that the residual,.I (t) is decoupled from the selected faults. With m primary residuals, and provided that SW has no column-rank defects, any residual may be decoupled from m-1 faults, with one nonzero fault-response specified . Then w.W

L

•

(19)

'. (t)

R

0 0

1

1

1

0 0

0 0

0

1

1

0 0

973

where

equations (11) and (12) may be written as x

2.3

(t) = K(t-1)x

2.3

+

(t-1)

and equations (13) and (14) as x

4.5

(t) = L(t-1)x

2. 3

+

(t-1)

A,

1

Primary residuals are computed from (30) and (31) by taking the difference of the left and right-hand sides. With two model equations, there is only one possible structure for the enhanced residuals:

Substituting these into (24):

X

45

X • 4 5

'2"

(28)

(t+ 1) - N(t)]

Finally, solving for

+

'3 '4 '5

M(t-1)

(t+ 1);

+ L(t){K(t-1)L- (t-1)[x • (t) - N(t-1)] ' 4 5

+

M(t-1)}

Expanding (29)' with the help of the MATLAB symbolic Toolbox, yields the following result 4

4

+

L

0 1 1 1 1

1 0 1 1 1

(TlfL. )u , (t)

•

R,

L,

Lm

1 1 0 1 1

1 1 1 0 1

1 1 1 1 0

The enhanced residuals are computed according to equation (22). For the transformation, the partial derivatives of the primary residuals (that is, of the right-hand sides of (30) and (31» are needed. These have been found algebraically. The derivatives are rather extensive and therefore are not shown here. They contain the measurable variables (in a nonlinear fashion) and are thus time-varying. Now the S(t) matrix of partial derivatives does not exhibit any column-rank defect and thus all of the envisioned residual structures are implementable.

(29)

(t+ 1) = N(t)

x (t+ 1) = (1-Tglf)x (t)

R

•

4.

(34)

(27)

L- (t)[X • (t+ 1) - N(t)] 4 5

= K(t-1)L -1 (t-1)[x 5 (t) - N(t-1)]

(33)

b 2 + x 21 (1) (1-Tb)x 7 (t) - Tx 1 (t-1)b

and, by shifting,

L- 1 (t)[X4• 5

(1-Tb)b + Tx 1 (t-1)x 1 (t) (t)

A2 It) = -----;:-----;;:--'--b 2 + x 21 (1)

(26)

=

(32)

m

(25)

N(t-1)

where the matrices K(t-1), L (t-1), M(t-1), N(t-1) contain state-variables and inputs. From (25),

X • (t) 2 3

T2 beL If

Ao

(24)

M(t-1)

(30)

+ Ao [bx (t-1) +X, (t)x (t-l)) 4 5

SIMULATION STUDIES + [A, (t)b+A

2

(t)x , (t-l))

Simulation studies have been performed with the discretized state-space model. The model was run with T = O. 1ms, to maintain model accuracy, but the residuals were computed at every tenth sample. The input voltage was 220 V 50 Hz. A small random noise was added to the "measured" outputs. Step-faults were applied in mid-experiment, one at a time, to the five electrical parameters, with the following relative magnitudes:

· [x (t) - (1-Tglf)x (t-1) - (TlfL. )u , (t-1)] 4 4

+

[A2 (t)b - A, (t)x 1 (t-l))

· [x (t) - (1-Tglf)x (t-1) - (TlfL. )u (t-1)] 5 5 2 X5 (t+ 1)

= (1-Tglf)x 5 (t) +

(TlfL. )u (t) 2

(31)

+ A [bx (t-1) -x, (t)x (t-1)] o 5 4

.R = 2%

+ [A, (t)b+A

AR, = 0.15% AL , = 0.1% AL m = 0.05%

2

•

(t)x , (t-1)]

AL = 0.1%

•

The enhanced residuals were designed in the strong isolation structure. However, due to the rank-defect in the derivative matrix, two of the five residuals appeared in degenerate structure (as expected). Figure 1 shows the simulation plots; they follow the layout of the structure matrix, the

· [x (t) - (1-Tglf)x (t-1) - (TlfL. )u (t-1)] 5 5 2 - [A (t)b - A, (t)x 1 (t-l)) 2

· [x 4 (t) - (1-Tglf)x 4 (t-1) - (TlfL. )u 1 (t-1)]

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rows of subplots corresponding to enhanced residuals while the columns to faults. Note that the very small fault sizes indicate the high fault sensitivity of the algorithm. Also, they kept small the error of linearization introduced when approximating the effect of parameter changes with first derivatives.

process Symp. (Budapest, Hungary). Comtet-Varga, G. (1997). Surveillance des Systemes Non Lineaires. Application aux Machines Asynchrones. Ph.D. Thesis, University of Lille. Comtet-Varga, C. and Staroswiecki, M. (1998). Analytic redundancy relations for fault detection and isolation in algebraic dynamic systems. Internal report, LAIIL, University of Lille.

Residuals were also designed from the input-output model, in strong isolation structure. The statespace model served as a plant emulator to provide "measurement" data. Figure 2 shows the results.

Gertler, J., Singer, D. and Sundar, A. (1985). A robustified linear fault isolation technique for complex dynamic systems. IFAC/IFIP Symp. on Digital Computer Applications to Process Control (Vienna, Austria).

CONCLUSION This work has demonstrated the use of nonlinear parity relations for the isolation of parametric faults, in the context of the nonlinear model of an induction motor. Structured residuals were generated first from the state-space model, which contains inaccessible variables, and then from a nonlinear input-output model derived analytically from the state equations. In both cases, the design of the structured residuals relied on the derivatives of the primary residuals with respect to the parameters. These derivatives, obtained analytically, are nonlinear functions of state and input variables.

Gertler, J. and Kunwer, M. (1995). Optimal residual decoupling for robust fault diagnosis. Int. J. of Control, Vol. 61, 395-421 . Keljik, J. (1995). Electric Motors and Motor Controls. Delmar Publishers. LesecQ, S. and Barraud, A. (2000). Off-line diagnosis using error membership set estimation: application to induction motor. IFAC Safe process Symp. (Budapest, Hungary). Nasar, S.A. (1970). Electromagnetic Energy Conversion Devices and Systems. Prentice-Hall.

A major difficulty of model-based techniques in a nonlinear framework is the need for algebraic manipulation of complex relationships. This may be circumvented by an "empirical" approach in which the nonlinear model is used as a plant emulator, to generate data to which empirical models are fitted by systems identification methods. This approach may be used to obtain the input-output model, and also to approximate the derivatives with respect to the parameters. For the latter, the model has to be run not only with the nominal parameters but also with deviated ones (one parameter changed at a time); then empirical nonlinear models may be fitted to the observed deviations in the "plant" outputs. This technique has been used in the past, in a linear framework (Gertler and Kunwer, 1995)' but has yet to be tested for nonlinear models. Note that even linear models are nonlinear functions of the physical parameters so the difference should not be really substantial.

TABLE 1. Partial derivatives s .. = de./drr . //

S2' = 0 S22 =

0 0

S3' =

0

5

32

/

/

=

S23 = (Lm x 4 - x 2 )lL,

S33 = (L m X5

S24 = (R,X2 - Lmx4 )IL,2

S34 = (R,x 3 - Lmx5 )IL,2

-

X3 )lL,

S35 = R,x5 IL ,

S25 = R,x 4 1L, S4' = - x41L.

S42 = [gx 4 - e(bx2 +x,X3 ) - u,IL. - eLmx;VL. S43 = (ex 2 - eL m x 4 )lL, S44 = [2caL m x 4 - 2cax 2 - CX , X3 - cLmx;VL, S45 = - 2acx 4

ACKNOWLEDGEMENT This work has been supported by the US Agency for International Development and by NSF under Grant #CTS-9714891.

+

(bx 2 +x,x 3 )IL,L.

+

2ex;

55' = - x5 1L •

5 52 = [gx 5 - c(bx 3 - X, X ) - u 2 1L. - cLmx; VL. 2

S53 = (CX 3 - cL m x 5 )lL,

REFERENCES

S54 = [2caL m X5

Arnanz, R., de Miguel, L.J. and Peran, J.R. (2000), Model-based diagnosis of AC motors. IFAC Safe-

S55 = - 2acx 5

975

+

+ 2cax 3 + ex, X2 (bx 3

-

-

X I x 2 )lL,L.

eLm x; VL,

+

2ex;

x 10"

As

• 10"

La

\ 10"

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4

4

\ 10"

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..

-Sc=J \ 10.5 3.S 4

\ 10"

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4

X 10"

Lt

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3.S

4

-Se] \ 10" 3.S ..

\ 10"

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3.S

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x 10"

At

X 10"

lm

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:c=J :c=J :c=J :CJ1 ~CJ ~C=J ~~ ~:cn :CJi :c=J :c=J :c=J -S~ -S~ ~e=] ~:c=J :c=n :DI :c=J :c=J -S~ -S~ ~C=J ~CJ ~:CJ :CJI :5] ~CJ -S~ ~ ~:C=J :c=J :c=J :c=J :CJ1 ~CJ -S~ ~:C=J

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3.S

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4

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i

10.5

3.S

-S C J \ 10" 3.S 4

4

-Se=] 3 3.S 4

3

3.5

X 10.5

lm

..

Figure 1. Fault isolation with the state-space model

X

10"

Ra

X

10.5

La

X

~:e=] :cr==l -se=] -S~ \ 10"

3.5

..

~:cr==l -1~ \ 10.5

3.5

..

~:~

\ 10-3

3.S

10.5

X 10.5

At

Lt

:C1 :Crl :c-l

-Sc==1

..

\ 10"

3.5

4

-S~ ~Qd \ 10-3

3.5

4

:c=J :0=1 :5] -1c=J -1~ -1 \ 10.5

3.5

..

\ 10.5

3.5

..

\ 10"

3.5

\ 10"

3.5

4

:~

-1~

..

\ 10"

3.5

..

:cn :e=] :01 :C1

-S~ ~~ -se=] ~~ ~CJ::d

·.:E!J .:E!J:EJ .:B:@ \ 10"

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..

\ 10.5

3.5

4

~:5] :CI -S

-S c = : : J 3

3.S

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3

3.S

..

\ 10.5

3.S

4

3

3.S

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\ 10.5

3.5

..

\ 10"

3.5

..

:5] :Cr=1 :c=J -5~

-S

3

3.S

-SCJ

..

Figure 2 . Fault isolation with the input-output model

976

3

3.S

..