Stator Winding Short Circuit Fault Detection based on Uncertainty

Ellipsoid Intersection for Three Phase Induction Motors

Mohammed Obaid Mustafa and George Nikolakopoulos

Control Engineering Group, Division of Systems and interaction, Department of Computer Science,

Electrical and Space Engineering, Lule˚a University of Technology, SE- 97187 Lule˚a, Sweden

Keywords:

Three Phase Induction Motor, Fault Detection and Diagnosis, Set Membership Identiﬁcation, Uncertainty

Ellipsoid Intersection, Geometrical Analysis.

Abstract:

In this article a fault detection scheme for different percentage of stator winding short circuit in one phase of

three phase induction motors is presented. In the examined case, the induction motor in the faulty and healthy

case has been transformed in the two phase (q− d) model. The model has been identiﬁed by the utilization of

a Least Squares Set Membership Identiﬁcation (SMI) algorithm, where additional to the identiﬁed parameters,

conﬁdence intervals can be also calculated, based on a priori knowledge for the corrupting measurement

noise. The identiﬁed conﬁdence intervals in an µ–dimensional space can be represented as hyper–ellipsoids

having as a center the identiﬁed parameters’ vector. The novelty of this article stems from the proposal of a

fast and geometrical based scheme, which relies on the calculation of the distance among centers of hyper–

ellipsoids and the corresponding intersection in each iteration of the identiﬁcation procedure. Detailed analysis

of the proposed fault detection strategy, as also extended simulation results are being presented that prove the

efﬁciency of the suggested scheme.

NOMENCLATURE

, V

: Stator voltages Quadrature frame (V)

, i

: Stator currents Quadrature frame (A)

, r

: Resistance of stator’s and rotor’s winding (Ohm)

, L

: Stator’s and rotor’s self inductance (Henry)

: Mutual inductance (Henry)

: Rotor’s angular speed (rad/sec)

: Rotor’s speed (mechanical) (rad/sec)

: Supply angular frequency (rad/sec)

P : No. of poles pairs

J : Moment of inertia (Kg· m

)

: Load torque (Nm)

: Electromagnetic torque (Nm)

q : Quadrature axis frame

d : Direct axis frame

s : Stator quantities

r : Stator quantities

1 INTRODUCTION

Three phase induction motors are very important in

different applications in industrial and power sys-

tems (F. Jawad, 2009) as it is commonly known that

induction motors have good properties such as highly

reliability, require low maintenance, and have high

efﬁciency. Therefore, the condition monitoring of

these electrical machines has received considerable

attention in recent years as fault detection and diag-

nosis are very important to reduce the maintenance

cast and prevent downtimes for electrical drive sys-

tems (B. Mirafzal, 2006).

Induction motors have many type of faults, while

some of these fault happen in the stator or in the ro-

tor, such as short circuit in stator winding, broken ro-

tor bar, and bearing faults (Nandi and Toliyat, 2005).

Stator winding consists of coils of insulated copper

wire placed in the stator slots, while the stator wind-

ing faults start due to the insulation breakdown be-

tween two adjacent turns in a coil for the same phase;

this fault is usually called a turn–to–turn fault or inter-

turn short circuit. Therefore, this type of fault is very

serious as it will produce an extra heat, and it will cre-

ate an imbalance in the magnetic ﬁled of the machine.

In fact, the majority of these defects are due to a com-

bination of various stresses acting on the stator, which

can be classiﬁed into thermal, electrical, mechanical,

and environmental factors.

Until now, various scientiﬁc methods have been

proposed in the ﬁeld of fault detection and diagno-

206

Obaid Mustafa M. and Nikolakopoulos G..

Stator Winding Short Circuit Fault Detection based on Uncertainty Ellipsoid Intersection for Three Phase Induction Motors.

DOI: 10.5220/0004476802060212

In Proceedings of the 10th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2013), pages 206-212

ISBN: 978-989-8565-70-9

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

sis for induction motors such as: a) Artiﬁcial Neu-

ral Networks (Gaeid and Mohamed, 2010), b) Fast

Fourier Transform (Bachi et al., 2006), c) Time Step

Coupled Finite Element-State Space (Gaeid and Mo-

hamed, 2010), d) Motor Current Signature Analy-

sis (Aydin et al., 2011), e) Wavelets, and Complex

Park Vectors (Gaeid and Mohamed, 2010).

All of these methods base their operation on spec-

tral analysis of stator currents, stator voltages, and

electromagnetic torque (Aydin et al., 2011), while

are able to provide a faster detection and identiﬁca-

tion of the fault, but will lead to incorrect conclu-

sions in some cases, such as low load conditions, in

transient situations or in perturbed environments (i.e.

ﬂuctuating load torque and unideal supply). From an-

other point of view Independent Component Analy-

sis (ICA) and Support Vector Machines (SVMs) have

been applied to detect and diagnose of induction mo-

tor faults (Widodo et al., 2007; A. Widodo, 2008)

as alternative approaches to the previous mentioned

cases.

The novelty and the main scientiﬁc contribution

of this article stems from: a) the combination of

the SM–identiﬁcation technique for calculating recur-

sively the motor parameters and the corresponding

uncertainty boundsbasedon theassumed noise levels,

with b) the proposal of a fault detection scheme based

on fundamental geometrical properties of the calcu-

lated parameter bounding uncertainty. More analyti-

cally, in this article the problem of fault detection is

transformed to the geometrical problem of represent-

ing uncertainty into µ–dimensional hyper–ellipsoid

spaces and examining basic properties of the result-

ing µ–dimensional spaces, such as center of ellipsoid,

ellipsoidal intersection, and distance from the centers.

The proposed scheme has a low complexity and as it

is going to be presented in the sequel, it can be di-

rectly transferred to real life implementations.

The rest of the article is being structured as it fol-

lows. In Section 2 the model derivation and simpli-

ﬁcation, for the healthy and the faulty cases is being

derived. In Section 3 the SMI scheme is being pre-

sented, followed by the proposed fault conditioning

framework in Section 4. Section 5 contains multiple

simulation results that prove the efﬁcacy of the pro-

posed methodology, while the conclusions are drawn

in the last Section 6.

2 INDUCTION MOTOR

MODELING

2.1 Healthy Case

The state space form for the three phase induction

motor can be represented as it follows: (Vas, 1992;

Mustafa et al., 2012b)













= A













+ B













(1)

where:

A =







−L

0 L

0 −L

0 L

0 −L

0 L

−w

δ −L







B =







0 −L

0 L

0 −L

−L

0 Ł

0 −L

0 L







with δ deﬁned as:

δ = L

− Lm

while the motor’s torque and angular speed, in case

that the fraction is being neglected, is denoted as:

P L

− i

]

dω

= T

− T

2.2 Stator Winding Short Circuit

Modeling

The focus of this research effort is on the stator faults

during short circuit between stator winding, which

happen in one phase of the motor. In the examined

case all the stator parameters are considered to be

identical when short circuit happens in the winding

of the three phase induction motor, while both sta-

tor’s resistance and inductance, as also the mutual in-

ductances between stator and rotor will be directly

affected. In the case of such a fault, the modiﬁed

(faulty) versions of the matrices A and B in equa-

tion (1), will be changed to A

∗

and B

∗

and become

as:

∗

= −R

∗

−1

∗

= L

∗

−1

StatorWindingShortCircuitFaultDetectionbasedonUncertaintyEllipsoidIntersectionforThreePhaseInductionMotors

207

while the inductance matrix in the faulty case will be-

come as (Chen and Zivanovic, 2009):

∗







0 L







The elements of L

∗

are being deﬁned as (Chen and

Zivanovic, 2009):

+ 1)L

(2g

+ 1)

= L

+ 1)L

, L

= L

+ L

= L

, L

= L

+ L

and

∗







∗

0 0 0

0 r

0 0

0 −ω

/ω

∗

rr −ω

/ω

∗

0 ω

/ω







with:

∗

= r

· r

s11

∗

= L

· L

s11

(2g

+ 1)

is the percentage of the remaining un–shorted sta-

tor windings in stator phase a (Chen and Zivanovic,

2009). These models will be utilized in the following

simulation studies for simulating the healthy and the

faulty induction motor operations.

3 SET MEMBERSHIP

IDENTIFICATION

The method of Weighted Recursive Least Squares

Set Membership Identiﬁcation has been utilized ef-

ﬁciently in the fault detection and diagnosis in both

cases of broken rotor bar and short circuit in stator

winding for the three phase induction motor (Mustafa

et al., 2012b). The objective of the Set Membership

Identiﬁcation technique (SMI) is the determination of

a feasible recursively identiﬁed uncertainty parame-

ter set that contains the nominal parameter vector and

is consistent with a linearly parameterizable model,

the measurement data and the a priori known bounded

noise–error. In the SMI approach, instead of identify-

ing directly the unknown parameter, the correspond-

ing uncertainty bounds that include the nominal value

are being calculated instead, while in each iteration

the center of the uncertainty interval is equal with the

current value of the identiﬁed parameter.

The q−d model of the induction motor is being trans-

formed into an ARMA system, which can be de-

scribed in a generic form as:

(t) = Φ

(t)

(t) +e

(t) (2)

where

(t) is the identiﬁed parameter vector set and

the subindex j represents the current set that can be

selected as one from: [qs, ds]. Moreover θ

(t) con-

tains the corresponding coefﬁcients of the selected

ARMA model and can be deﬁned in the general case

as:

(t) = [F

j,1

(t),.. ., F

n,1

(t), T

j,1

(t),.. .,T

m,1

(t)] (3)

the regression vector Φ

(t) is formulated as:

(t) = [−y

(t − 1), ... ,−y

(t −n), .. . ,

(t +m− n−1),. .. ,u

(t −n)]

and the adopted ARMA model representation of the

transfer function of induction motor will be denoted

as:

j,1

m−1

+ T

j,2

m−2

+ ....... + T

j,m

+ F

j,1

n−1

+ ....... + F

j,n

The identiﬁed parameters presented in equation (3),

are directly related with the motor parameters (resis-

tance and inductance of the stator and rotor), while

more details about the mathematical relationship be-

tween these parameters of induction motor in the

healthy case could be found in (Mustafa et al., 2012b;

Mustafa et al., 2012a), where n, m ∈ Z

are the orders

of the numeratorand denominatorforeach considered

transfer function and µ = n + m.

In equation (2) the additive measurement noise is

assumed to be bounded by γ

∈ ℜ

as:

||e

(t)||

≤ 1,∀ t

The core of the SMI technique is based on the

Weighted Recursive Least Squares (WRLS) with a

forgetting factor (λ) for identifying the

motor’s

parameters and can be formulated by the following

double recursions (Guastafsson, 2001) in the sample

instance t and for the MIMO case j as:

(t) =

(t −1) + K

(t)(y

(t) −Φ

(t)θ

(t −1))

(t) = P

(t)Φ

(t) = P

(t − 1)Φ

(t)(λ

+Φ

(t)P

(t −1)φ

(t))

−1

(t) = (I − K

(t) Φ

(t))P

(t − 1)/λ

(t) = y

(t) −Φ

(t)θ

(t −1)

(t) = Φ

(t)P

(t −1)Φ(t)

In the SMI approach the uncertainty description is

evolving with the time, as the better the knowledge of

the parameters is, the smaller these bounds are, while

ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics

208

as a fundamental property in SMI (Deller, 1989a),

these bounds cannot be less than the assumed or com-

puted range of noise corrupting the measurements.

For calculating the upper and lower boundary of the

identiﬁed parameters, the uncertainty bounds σ

(t),

should be computed in every iteration and will be-

come as:

(t) =

diag(P

(t)) (4)

with the covariance matrix denoted as

(t) = P

(t)

−1

and P positive deﬁnite (Deller,

1989b; Le et al., 2008).

4 FAULT DETECTION

CONDITIONING

Based on the SMI scheme, presented in Section 3 and

on the uncertainty bounds in Eq. (4), for every iden-

tiﬁed parameter a hyper ellipsoid can be deﬁned hav-

ing as center the current values of the identiﬁed pa-

rameters from Eq. (2). In the following analysis, the

notation for these identiﬁed values will be simpliﬁed

to (Kurzhanskiy and Varaiya, 2008):

(t) = [q

(t) ... q

n+m

(t)] (5)

and the ellipsoid will be considered as the set ε(q, σ)

in R

n+m

with center q and shape matrix σ as:

ε(q,σ) = {x ∈ R

| h(x− q),σ

−1

(x− q)i ≤ 1} (6)

where σ is positive deﬁnite, hx,σxi > 0 for all nonzero

x ∈ R

n+m

and with h·,·i to denote the inner product.

The support function of a set X ⊆ R

n+m

and the sup-

port function of the ellipsoid in Eq. (6) will become

as:

ρ(l| X) = sup

x∈X

hx,li (7)

ρ(l| ε(q,σ)) = hl,qi + (hl, σli)

(8)

Therefore, it is useful to give an alternative deﬁnition

of an ellipsoid using the expression Eq. (8) as:

ε(q,σ) = {x ∈ R

| hl, xi ≤ hl, qi+ (9)

hl,σli

} for all l ∈ R

n+m

where σ is positive semideﬁnite, hx,σxi ≥ 0 for all

nonzero x ∈ R

n+m

(Kurzhanskiy and Varaiya, 2008).

Given two hyper–ellipsoids ε(q

,σ

) and ε(q

,σ

the distance between them is:

dist(ε(q

,σ

),ε(q

,σ

)) = max

hl,li=1

(−ρ(−l|ε(q

,σ

))

−(ρ(l|ε(q

,σ

))

= max

hl,li=1

(hl,q

i −(hl,σ

li)

−hl,q

i −(hl,σ

li)

)

The intersection between two ellipsoids can be also

transformed to a distance problem, which can be

cast as QuadraticallyConstrained QuadraticProgram-

ming (QCQP) problem as: (Kurzhanskiy and Varaiya,

2008):

dist(ε(q

,σ

),ε(q

,σ

)) = minh(x

− y

),(x

− y

)i (10)

subject to

h(q

− x

),σ

−1

− x

)i ≤ 1

h(q

− y

),σ

−1

− y

)i ≤ 1

with x

, x

, y

∈ ℜ

points on the hyper–ellipsoids

surfaces for the healthy and the faulty case respec-

tively, with the following conditions:

dist(ε(q

,σ

),ε(q

,σ

))







> 0: no common points

= 0: one common point

< 0: ellipsoids intersect

The proposed fault detection scheme is based only on

geometrical properties of the structured parameters’

uncertainty and more speciﬁcally is based on the on-

line and fast computation of the recursively computed

centers of ellipsoids and the corresponding distances

between the centers. In all the performed geometri-

cal calculations two ellipsoids are being considered:

a) the nominal and converged ellipsoid of the error

free identiﬁed motor model, denoted as ε

, and b) the

iteratively computed uncertainty ellipsoid ε after pa-

rameters’ convergence and the deﬁnition of the previ-

ous nominal ellipsoid ε

. In Figure 1, two illustrative

instances representing bounding ellipsoid uncertainty

from the case of the healthy and faulty motor’s opera-

tion are presentedin a projectioninto a 2–dimensional

frame, to allow a straight forward and comprehensive

geometrical representation, without any loss of gener-

ality.

Figure 1: Projection of ellipsoid intersection in X-Y plane.

In this Figure, the general case of intersection and

shifting of centers between the two ellipsoids has been

StatorWindingShortCircuitFaultDetectionbasedonUncertaintyEllipsoidIntersectionforThreePhaseInductionMotors

209

presented also. while the distance L(t) ∈ ℜ

n+m

be-

tween the centers has been denoted as:

L(t) = [(q

− q

)

+ (q

− q

)

+ .... +(q

n+m

− q

n+m

)

]

In the case of a stator winding short circuit, the val-

ues of the identiﬁed parameters and the correspond-

ing uncertainty bounds will drift from their converged

values, which has a direct effect and change on the

center and the direction of the bounding uncertainty

µ–dimensional hyper ellipsoids at each sample time

(iteration) and thus a fault detection scheme can be

directly established based on a straight forward in-

spection of geometrical properties for the bounding

ellipsoids as it is being presented in Figure 2.

The initialization phase of the proposed fault de-

tection algorithm assumes that the SMI scheme has

converged in identifying the nominal parameters of

the adopted model for the induction motor and thus

the corresponding bounds on the parametric uncer-

tainty intervals have been also converged. In the se-

quel the SMI scheme continuous the online identiﬁ-

cation of the parameters, while constructing the se-

quential bounding uncertainty µ–D ellipsoids. Upon a

short circuit fault the upcoming online calculated µ–D

bounding ellipsoids will present a change in their vol-

ume and an intersection will occur with the nominal

deﬁned converged µ–D ellipsoid volume, which is a

direct indication of a fault. In case that there is no in-

tersection the fault detection algorithm will continue

to operate, while checking the existence of such an in-

tersection. The parametric drift after the event of the

fault will also producea corresponding drift on the el-

lipsoid’s center and thus the distance among ellipsoid

centers are being also tracked by calculating the value

of L. In case that this distance is not close to 0, then in

combination with the ellipsoid intersection, these two

conditions will lead to a fault detection instance.

Figure 2: Flowchart of fault detection conditions.

5 SIMULATION RESULTS

The suggested scheme for fault detection is being

evaluated on a model of three phase induction motor

having the parameters depicted in Table 1. The pre-

sented results examine the application of the proposed

geometrical analysis on two cases of short circuit sta-

tor winding with 2%, and 5% faults.

Table 1: Induction Motor Parameters.

Pole Numbers 4 r

0.0616 per unit

Input Voltage 240V r

0.0753 per unit

Frequency 50Hz J 0.00155 Kg.m

0.019 per unit L

0.019 per unit

0.01833 per unit

The identiﬁed parameters and uncertainty bounds in

the healthy and faulty case of 2% short circuit are pre-

sented in ﬁgures 3 and 4, where the fault occurred

at the 10

sample time instance and the change in

the motor parameters are obvious because of this sta-

tor fault. In Figure 5 the iterative evolution of

0 0.5 1 1.5 2 2.5 3

x 10

qs,1

Short circut

0 0.5 1 1.5 2 2.5 3

x 10

−10

qs,2

0 0.5 1 1.5 2 2.5 3

x 10

−10

qs,3

0 0.5 1 1.5 2 2.5 3

x 10

−5

Sample time

qs,4

Figure 3: SMI based identiﬁed parameters for T

qs,i

and cor-

responding uncertainty bounds.

the ellipsoid volume (bounding uncertainty), during

identiﬁcation in the healthy case and for a projection

based on the selected triplet (center of ellipsoid) of

qs,1

, T

qs,2

, T

qs,3

is being presented at different time

indexes. From the obtained results it can be observed

that the ellipsoids volume is monotonically decreased

without any intersections taking place at the different

time instances. This result and the presented bound-

ing uncertainty are in fully accordancewith the results

presented in Figure 5.

In Figures 6 and 7 the uncertainty ellipsoids in the

nominal case and after the occurrence of the fault at

different sample times for the same case of consider-

ing the triplet of T

qs,1

, T

qs,2

, T

qs,3

identiﬁed parame-

ters, for both the cases of 2% and 5% of stator wind-

ICINCO2013-10thInternationalConferenceonInformaticsinControl,AutomationandRobotics

210

0 0.5 1 1.5 2 2.5 3

x 10

−2

qs,1

0 0.5 1 1.5 2 2.5 3

x 10

−2

qs,2

0 0.5 1 1.5 2 2.5 3

x 10

−0.5

0.5

qs,3

0 0.5 1 1.5 2 2.5 3

x 10

−0.1

0.1

Sample time

qs,4

Figure 4: SMI based identiﬁed parameters for F

qs,i

and cor-

responding uncertainty bounds.

ing short circuit correspondingly are being presented.

−5

−30

−20

−10

−0.06

−0.04

−0.02

0.02

0.04

0.06

0.08

0.1

qs,1

qs,2

qs,3

@10

nominal case

@20000

@50000

Figure 5: Ellipsoid in healthy case at different sample time.

The centers of ellipsoids and the corresponding uncer-

tainty bounds will drift from their converged healthy

value at each sample time. The ellipsoids intersec-

tion between the nominal case and the faulty case has

been easily tracked in the proposed geometrical ap-

proach to fault detection, while this drift will evolve

after fault’s occurrence. In Figure 8 it is also pre-

sented the geometrical drift in the bounding ellipsoids

and the corresponding intersection, for the case where

a different triplet have been considered of a center

qs,1

, F

qs,2

, T

qs,4

for the case of 2% short circuit.

Finally the distance L in the case that the whole

(µ)-dimensional hyper ellipsoid is being considered

during the healthy operation and after the fault oc-

currence, for the cases of 2%, and 5% short circuit

are being presented in Figure 10. As it can be pre-

sented, the hyper distance will be increased after the

fault occurrence and it will become greater than zero

as shown in ﬁgures (9).

Figure 6: Ellipsoid interaction between healthy and faulty

case (2% short circuit).

Figure 7: Ellipsoid interaction between healthy and faulty

case (5% short circuit).

Figure 8: Ellipsoid interaction between healthy and faulty

case (2% short circuit).

StatorWindingShortCircuitFaultDetectionbasedonUncertaintyEllipsoidIntersectionforThreePhaseInductionMotors

211

0 0.5 1 1.5 2 2.5

x 10

Sample time

Distance L

5% short circuit

2% short circuit

Fault

occurrence

Figure 9: The distance L in healthy and faulty case (2%, 5%

short circuit).

6 CONCLUSIONS

In this article a fault detection scheme for different

percentageof stator winding short circuit in one phase

of three phase induction motors is presented. The mo-

tor’s model was identiﬁed by the utilization of a Least

Squares Set Membership Identiﬁcation (SMI) algo-

rithm, where additional to the identiﬁed parameters,

conﬁdence intervals have been calculated, These in-

tervals in an µ–dimensional space can be represented

as hyper–ellipsoids having as a center the identiﬁed

parameters’ vector and thus a geometrical fault detec-

tion scheme has been proposed, which relied on the

calculation of the distance among centers of hyper–

ellipsoids and the corresponding intersections. Ex-

tended simulation results were presented that proven

the efﬁciency of the suggested scheme.

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