sensors

Article
Real-Time Detection of Incipient Inter-Turn Short Circuit
and Sensor Faults in Permanent Magnet Synchronous Motor
Drives Based on Generalized Likelihood Ratio Test and
Structural Analysis
Saeed Hasan Ebrahimi * , Martin Choux and Van Khang Huynh

Department of Engineering Sciences, University of Agder, 4879 Grimstad, Norway; martin.choux@uia.no (M.C.);
huynh.khang@uia.no (V.K.H.)
* Correspondence: saeed.ebrahimi@uia.no; Tel.: +47-462-47-096

Abstract: This paper presents a robust model-based technique to detect multiple faults in permanent
magnet synchronous motors (PMSMs), namely inter-turn short circuit (ITSC) and encoder faults. The
proposed model is based on a structural analysis, which uses the dynamic mathematical model of a
PMSM in an abc frame to evaluate the system’s structural model in matrix form. The just-determined
and over-determined parts of the system are separated by a Dulmage–Mendelsohn decomposition
tool. Subsequently, the analytical redundant relations obtained using the over-determined part of the
system are used to form smaller redundant testable sub-models based on the number of defined fault
terms. Furthermore, four structured residuals are designed based on the acquired redundant sub-
models to detect measurement faults in the encoder and ITSC faults, which are applied in different



levels of each phase winding. The effectiveness of the proposed detection method is validated by


an in-house test setup of an inverter-fed PMSM, where ITSC and encoder faults are applied to the
Citation: Hasan Ebrahimi, S.; Choux,
system in different time intervals using controllable relays. Finally, a statistical detector, namely a
M.; Huynh, V.K. Real-Time Detection
generalized likelihood ratio test algorithm, is implemented in the decision-making diagnostic system
of Incipient Inter-Turn Short Circuit
and Sensor Faults in Permanent
resulting in the ability to detect ITSC faults as small as one single short-circuited turn out of 102, i.e.,
Magnet Synchronous Motor Drives when less than 1% of the PMSM phase winding is short-circuited.
Based on Generalized Likelihood
Ratio Test and Structural Analysis. Keywords: fault diagnosis; inter-turn short circuit; sensor fault; structural analysis; generalized
Sensors 2022, 22, 3407. https:// likelihood ratio test; PM synchronous motor
doi.org/10.3390/s22093407

Academic Editor: Andrea Cataldo

Received: 14 March 2022 1. Introduction

Accepted: 26 April 2022
Permanent magnet synchronous motors (PMSMs) have gained popularity in industrial
Published: 29 April 2022
applications such as electric vehicles, robotic systems, and offshore industries due to their
Publisher’s Note: MDPI stays neutral merits of efficiency, power density, and controllability [1–3]. PMSMs working in such
with regard to jurisdictional claims in applications are constantly exposed to electrical, thermal, and mechanical stresses, resulting
published maps and institutional affil- in different faults such as electrical, mechanical, and magnetic faults [4]. Among these
iations. various faults, the stator winding inter-turn short circuit (ITSC) fault is considered as one
of the most common faults [5] due to the excessive heat produced by a high circulating
current in a few shorted turns of the stator winding [6]. Subsequently, this excessive heat
causes further insulation degradation and might lead to a complete machine failure [7]
Copyright: © 2022 by the authors.
if it is not detected and treated in time. Therefore, developing methods for monitoring
Licensee MDPI, Basel, Switzerland.
and detecting the ITSC fault in its early stages can substantially lower maintenance costs,
This article is an open access article
downtime of the system, and productivity loss.
distributed under the terms and
ITSC faults can be detected by signal-based, data-driven, and model-based tech-
conditions of the Creative Commons
Attribution (CC BY) license (https://
niques [8]. The first approach aims to detect fault characteristic frequencies in measured
creativecommons.org/licenses/by/
motor signals, namely, current, voltage, or vibration signals [9–11], being processed by
4.0/).
time–frequency signal analysis tools such as Fourier transform [12], matched filters [13],

Sensors 2022, 22, 3407. https://doi.org/10.3390/s22093407 https://www.mdpi.com/journal/sensors

Sensors 2022, 22, 3407 2 of 22

Hilbert–Haung transform [14], wavelet transforms [15], and Cohen distributions [16]. These
signal-based methods face challenges of real-time implementations due to the computa-
tional burden, and missing fault characteristic signals does not guarantee that the machine
is healthy [8]. The data-driven approach such as an artificial neural network (ANN) [17]
and Fuzzy systems [18] requires a lot of historical data to train models and classify lo-
calized faults. Historical data is restricted in industry and producing a lot of historical
data in healthy and faulty conditions is costly and time-demanding [19]. Alternatively,
model-based techniques have been proposed to detect ITSC faults [20–22]. Among them,
the finite element method (FEM)-based models have been widely used due to the accuracy
and convenience of taking into account physical phenomena, e.g., saturation. FEM models,
known as time-demanding and computationally heavy ones, require deep knowledge
of the system, e.g., detailed dimensions and material characteristics. Other model-based
methods that use mathematical equations to model a motor’s behavior have been reported
to have challenges regarding validity when experiencing abnormal conditions such as
internal faults [8]. To address the mentioned challenges, structural analysis is proposed
as an alternative solution for detecting ITSC faults in electrical motors. The structural
analysis algorithm has been well studied and developed in the literature [23–25] and ap-
plied to different structures. The structural analysis approach has been able to successfully
detect faults in automotive engines [26–28], hybrid vehicle [29], and battery systems [30].
In [31,32]. The algorithm has successfully been applied on PMSM electric drive systems to
detect sensor faults such as voltage, current, encoder, and torque sensors. In our previous
study [33], it was proposed that the algorithm can be used on an electric drive system to also
detect common physical faults in PMSMs such as ITSC and demagnetization, and residual
responses were obtained by simulation. However, in previous studies, this algorithm has
not been implemented in real-time diagnosis of an industrial PMSM for detection of ITSC
faults. Implementing a structural analysis technique on a PMSM and drive, this paper aims
to achieve the following contributions:
• Detection of both internal motor faults and external measurement faults, namely ITSC
and encoder faults;
• Detection of the lowest level of ITSC fault, with one shorted turn in stator phase
winding;
• Early detection of an ITSC fault, i.e., considering a lower fault current in the degrada-
tion path as compared to shorted turns;
• Modeling of the noise in drive system measurement signals with unknown amplitude
and variance.
This paper presents a systematic fault diagnosis methodology based on structural
analysis for detecting multiple faults in PMSM drives, namely ITSC faults and encoder
fault. To achieve this, a healthy dynamic mathematical model of PMSM is defined in the
abc reference frame based on the dynamic constraints, measurements, and derivatives. To
model an ITSC fault in any phase, specific fault terms are added to the three-phase flux and
voltage equations. These fault terms include the deviations in the voltage, flux, and currents
of the stator winding caused by an ITSC fault, since a part of winding is shorted; hence,
three-phase voltage and flux signals are subjected to changes. In addition, fault terms are
added to the dynamic model to take into account the encoder faults, resulting in errors of
the angular speed and angle measurements. Subsequently, the analytical redundant part of
the structural model is extracted and divided into minimally over-determined sub-systems
from which three sequential residuals are obtained based on the error in the current signal
of each phase. Furthermore, a resultant residual is formed in the αβ frame to achieve a
better demonstration of different ITSC fault levels. Finally, a generalized likelihood ratio
test is developed to detect the faults in the resultant and encoder residuals under unknown
noise parameters assumptions, i.e., unknown amplitude and unknown variance.
Sensors 2022, 22, 3407 3 of 22

2. Modeling Inter-Turn Short-Circuit Fault

The studied PMSM consists of distributed three-phase windings on the stator and
PMs on the rotor. Each phase winding contains several coils in parallel, being formed by
wrapping bundles of wires together. The wire insulation of the stator windings might
be degraded over time under electrical, mechanical, and thermal stresses, which may
eventually lead to electrical faults such as an inter-turn short circuit (ITSC), a phase to
ground short circuit (PGSC), and a phase to phase short circuit (PPSC).The stator ITSC fault
is considered the most common electrical fault [34] and usually occurs in a few shorted
turns. The degraded path among the shorted turns is provided by a nonzero resistance of
the faulty insulation, leading to a circulating fault current. This circulating fault current
results in copper losses and excessive heat in the shorted turns since only a few turns are
involved, and the current-limiting impedance is low. The insulation might further degrade
and even propagate to nearby turns. This might cause other critical faults such as a PGSC
fault, a PPSC fault, and even a complete failure. Therefore, monitoring and detecting the
ITSC fault in early stages would reduce costs and downtime caused by the machine failure.
To model ITSC faults in a PMSM, it is necessary to know how the motor signals and
parameters are affected by the different levels of the fault. The schematic of a PMSM stator
winding under ITSC faults with different levels in each phase is shown in Figure 1. The
level of fault in abc phases is denoted by µ a , µb , and µc , respectively, which are defined
by the ratio of the number of shorted turns to the total number of turns per abc phase
winding. In a healthy condition, each phase winding of a PMSM has a resistance of Rs and
an inductance of Ls . In the presence of ITSC faults in each phase, the phase winding is
split into a faulty part with µRs and µLs , and a healthy part with (1 − µ) Rs and (1 − µ) Ls
resistance and inductance values. As a result, there is not only mutual inductance between
healthy and faulty parts in each phase winding, but also between the faulty winding with
other phase windings [35]. In addition, the degraded resistance of the insulation in each
phase is denoted by R a f , Rb f , and Rc f , while the circulating fault currents are i a f , ib f , and
ic f , respectively. To detect an incipient ITSC fault, the resistance of the degraded path
should be higher than the resistance of the shorted turns [36]. This is due to the fact that an
ITSC fault forms gradually over time and starts with a low current circulating through the
degraded path.

Figure 1. Schematic of PMSM stator windings under ITSC faults.

Sensors 2022, 22, 3407 4 of 22

3. Structural Analysis for PMSM with ITSC and Encoder Faults

Structural analysis aims to extract the analytic redundant relations (ARRs) of a sys-
tem based on the mathematical equations that describe the system’s dynamic [23,37]. A
structural analysis algorithm relies on redundancy in a system (a redundant part of the
complex system) and yields residuals for fault detection and isolation (FDI) based on ARRs.
Assuming that a model M has outputs z and inputs u, a residual is extracted by eliminat-
ing all the unknown variables, i.e., substituting an unknown variable with its equivalent
obtained value through a redundant path. Therefore, it leads to a relation that contains
only the known variables r (u, z) = 0 which is known as an ARR if the observation z is
consistent with the system model [23]. As a result, this residual’s response will maintain
a zero value under a null hypothesis (nonfaulty case) H0 and a nonzero value under an
alternative hypothesis (faulty case) H1 as follows:

H0 : r (u, z) = 0 (1)
H1 : r (u, z) 6= 0

This methodology is especially effective for fault diagnosis of complex systems where a
prior deep knowledge of the whole system is neither needed nor affordable in terms of
computational burden and processing time. Instead, a small redundant part of the system is
selected and processed to obtain smaller redundant subsystems that can be used in forming
residuals for detecting each predefined fault. First, the structural model of a redundant
system is formed and represented by an incidence matrix with variables as columns and
equations as rows. The variables are categorized as unknown variables, known variables,
and faults, while the equations are categorized as dynamic equations, measurements,
and differential equations. Each row of the incidence matrix connects an equation to
the corresponding variables if they are present in that specific equation. Next, the just-
determined and over-determined parts of the system are separated by rearranging the rows
and columns in a way to form a diagonal structure that is known as Dulmage–Mendelsohn
(DM) decomposition. Using the analytic redundant part of this structure and based on the
degree of the redundancy, several smaller sets of ARRs are identified. These smaller sets are
called minimally over-constrained sets and have one degree of redundancy, holding exactly
one more equation than the number of variables. Subsequently, a fault signature matrix is
formed to demonstrate which fault can be detected or even discriminated. Finally, specific
diagnostic tests (residuals) are designed to detect faults. Here, a structural analysis of a
PMSM experiencing independent ITSC faults in each phase is presented, and diagnostic
tests are proposed to detect and discriminate them. Figure 2 shows the modeling diagram
of a faulty PMSM and the drive system where measurements are acquired by sensors and
faults are located inside the motor.
Sensors 2022, 22, 3407 5 of 22

Figure 2. Modeling diagram of PMSM and drive system.

3.1. PMSM Mathematical Model

The dynamic equations of a faulty PMSM in an abc frame with ITSC faults present
in three phases are represented by equations e1 − e9 as shown in (2), where v a , vb , and vc
are the stator phase voltages; i a , ib , and ic are the stator phase currents; λ a , λb , and λc are
the stator phase fluxes; Te is the electromagnetic torque; TL is the load torque; ωm is the
rotor’s angular speed; θ is the electric angular position; R a , Rb , and Rc are the stator phase
resistances and L a , Lb , and Lc are the stator phase inductances; λm is the flux produced by
rotor PMs; p is the pole pairs; J is the rotor inertia, and b is the friction coefficient.
As discussed in Section 2, an ITSC fault splits the phase winding into a faulty part with
resistance and inductance of µRs and µLs and a healthy part with resistance and inductance
of (1 − µ) Rs and (1 − µ) Ls . The changed resistance and inductance of the winding have
direct correlation with voltage equations and flux equations. Under a healthy condition,
the model of PMSM, especially e1 –e6 , have no fault terms. Therefore, any changes in the
inductance will affect both voltage and flux equations (e1 –e6 ) directly, and any changes in
the resistance will affect only voltage equations (e1 –e3 ) directly. Here, f va and f λa are added
to the corresponding equations of the healthy PMSM to account for the ITSC fault in phase
a. Similarly, f vb , f vc , f λb , and f λc terms are added to account for ITSC faults in phases b and
c, respectively. These fault terms are shown in red in (2).

dλ a
e1 :v a = R a i a + + f va
dt
dλ
e2 :vb = Rb ib + b + f vb
dt
dλc
e3 :vc = Rc ic + + f vc
dt
e4 :λ a = L a i a + λm cos θ + f λa
e5 :λb = Lb ib + λm cos (θ − 2π/3)+ f λb (2)
e6 :λc = Lc ic + λm cos (θ + 2π/3)+ f λc
e7 :Te = − pλm [i a sin θ + ib sin (θ − 2π/3)
+ ic sin (θ + 2π/3)]
dωm 1
e8 : = ( Te − bωm − TL )
dt J
dθ
e9 : = pωm
dt
Sensors 2022, 22, 3407 6 of 22

The known variables consist of the motor signals, which are measured for both control
purposes and fault diagnosis. Thus, in addition to the three-phase currents and angular
position, i.e., yia , yib , yic , and yθ that are necessary for the control system. Three-phase
voltages, i.e., yva , yvb , and yvc , are also measured to complete the diagnostic system. Equa-
tion (3) shows these known variables, where f θ and f ω fault terms are also added to account
for speed and angle measurement error.

m1 : yva = v a m4 : yia = i a m7 : yθ = θ + f θ
m2 : yvb = vb m5 : yib = ib m8 : yωm = ωm + f ωm (3)
m3 : yvc = vc m6 : yic = ic

In addition, since the dynamic model of PMSM includes five differential constraints in
the abc frame, these are needed to be defined as unknown variables. Equation (4) shows
the differential constraints for the structural model.
dλ a d dωm d
d1 : = (λ a ) d4 : = ( ωm )
dt dt dt dt
dλ d dθ d
d2 : b = (λb ) d5 : = (θ ) (4)
dt dt dt dt
dλc d
d3 : = (λc )
dt dt

3.2. Structural Representation of the PMSM Model

The structural model of the PMSM with ITSC and encoder faults is obtained based
on the redundant dynamic model in (2)–(4), as shown in Figure 3. The incidence matrix
contains 22 rows, representing the nine defined equations in (2), the eight measured known
variables in (3), and the five differential constraints of unknown variables as shown in (4).
The columns of the matrix are subdivided into three groups of unknown variables, known
variables, and faults. The known variables are obtained directly from the measurements,
while the unknown variables can be calculated based on the known variables. The faults
considered in the structural model are variations in phase voltage and flux to represent
ITSC faults in each phase.

Figure 3. PMSM structural model.

3.3. Analytical Redundancy of the Model

To detect specific faults in a redundant system, faults must first be introduced to the
model, and then a proper diagnostic test containing the considered fault is selected. A
Sensors 2022, 22, 3407 7 of 22

diagnostic test is a set of equations (or consistency relations) extracted from the system
model, in which at least one equation is violated in case of the presence of a considered
fault. A system model is called a redundant model if the system model consists of more
equations than unknown variables. Assuming that model M = (C, Z ) contains constraints
(equations) C and variables Z, let unknown variables X be the subset of all variables Z in
model M (X ⊆ Z). The degree of redundancy of the model M is defined as:

ϕ( M ) = |C | − | X | (5)

where |C | denotes the number of equations, and | X | is the number of unknown variables
contained in the model M. According to bipartite graph theory, any finite dimensional
graph such as M = (C, Z ) can be decomposed into three sub-graphs as follows [37]:
• M− : structurally under-determined part of the model M, where fewer equations than
unknown variables lie, and the degree of redundancy is negative ϕ( M ) < 0.
• M0 : structurally just-determined part of the model M, where equal equations and
unknown variables lie, and the degree of redundancy is zero ϕ( M ) = 0.
• M+ : structurally over-determined part of the model M, where more equations than
unknown variables lie, and the degree of redundancy is positive ϕ( M ) > 0.
For diagnostic purposes, the over-constrained sub-graph is the interesting part because
it contains the important redundancy that is necessary for detecting a fault. According
to [38], a fault is structurally detectable if the equation that contains the fault variable lies
in the over-determined part of the whole model (e f ∈ M+ ). To obtain these sub-graphs, a
canonical decomposition of the main structure graph (M) is required. An example of this
canonical decomposition is shown in Figure 4, where three canonical sub-models of the
model M are obtained as {M− , M0 , M+ }.

Figure 4. Canonical decomposition of the structure graph M.

This canonical decomposition is achieved after the rows and the columns of the main
structural graph (structural model incidence matrix) are rearranged so that the matched
variables and constraints appear on the diagonal. Therefore, having a decomposition tool
that analyzes the redundancy of the structural model and forms this diagonal structure is
very beneficial. Dulmage–Mendelsohn (DM) is a key decomposition tool that is applied on
a structural model directly and obtains a unique diagonal structure by a clever reordering
of equations and variables [39]. Figure 5 shows the DM decomposition for the PMSM
structural model, where the analytic redundant part is expressed in the bottom-right
part containing all the faults. Since this part includes redundancy and can be monitored,
diagnostic tests can be designed with the set of ARRs in M+ . As a result, if a fault is defined
in the model and is supposed to be detected by the diagnosis system, a residual that is
sensitive to the presence of that fault must exist.
Sensors 2022, 22, 3407 8 of 22

Figure 5. DM decomposition for PMSM structural model.

4. Diagnostic Test Design

This section presents the procedure of designing diagnostic tests for ITSC and encoder
faults. First, the over-determined part of the structural model is separated into smaller
redundant subsystems where faults are observable, and then the sequence of obtaining
residuals for the detection of each fault is explained.

4.1. Minimal Testable Sub-Models

According to the definition given by [38], an equation set M is a TES set if:
1. F ( M ) 6 = ∅.
2. M is a proper structurally over-determined set.
0 0
3. For any M ) M where M is a proper structurally over-determined set, it holds that
0
F( M ) ) F( M)
where F ( M ) is the set of faults that influence any of the equations in M. A TES M is a
minimal test equation support (MTES) if there exists no subset of M that is a TES holding
the degree of redundancy of one. Following the algorithm in [38], the structural model is
subdivided into efficient redundant MTES sets. Each MTES set contains a group of ARRs
that together hold the degree of redundancy of one, meaning that there is only one equation
more than the number of variables involved. In addition, they are obtained in a way that the
effect of faults is considered. This reduces computational complexity significantly without
reducing the possible diagnosis performance as compared to structurally over-determined
(MSO) sets. Figure 6 shows all the MTES sets found for the considered structural model here,
where each row of the matrix connects the corresponding MTES to the equations involved.
Figure 7 shows the signature matrix of MTES sets, indicating which fault terms are
included in each MTES. MTES1 includes f θ and f ω fault terms that can be used for detecting
a rotor’s speed and angle measurement error. MTES2 and MTES3 contain f vc and f λc fault
terms for detecting an ITSC fault in phase c. MTES4 − MTES6 contain f vb and f λb fault
terms for detecting an ITSC fault in phase b. Similarly, MTES7 − MTES10 can be used
for detecting an ITSC fault in phase a since it has f va and f λa fault terms. If a MTES set
containing the information of changes in voltage and flux of a phase winding is found, it
can be used to form a residual that is sensitive to the presence of an ITSC fault in that phase.
Sensors 2022, 22, 3407 9 of 22

Figure 6. MTES sets.

Figure 7. Fault signature matrix of MTES sets.

4.2. Diagnosability Index

An important criterion for selecting MTES sets is to satisfy diagnosability requirements.
This includes detectability of any single fault as well as isolability between any two faults.
Here, an index for the proper selection of MTES sets that are suitable to be used in sequential
residual generators is introduced. Zhang [40] proposed a diagnosability index that is aimed
at achieving the maximum degree of diagnosability for each residual by comparing the
distance between the fault signature matrices of MTES sets:

n −1 n
1
mD = 
n+1
 ∑ ∑ D ( Vf i , Vf j ) (6)
i =0 j = i +1
2

where D (Vf0 , Vf j ) stands for the distance between the fault signature of f j and the healthy
case and measures the detectability of fault f j . D (Vf i , Vf j ) is the distance between two
fault signatures and is defined as the Hamming distance [41] between the two fault signa-
ture strings:

S
D ( Vf i , Vf j ) = ∑ | Vf i − Vf j | (7)
n =1

4.3. Sequential Residuals for Detecting ITSC and Encoder Faults

This section presents the sequence of deriving four residuals (R1 − R4 ) based on the
obtained MTES sets. These residuals aim to detect ITSC faults in any of the phase windings
as well as encoder measurement faults. To form residual R1 that is sensitive to an ITSC fault
in phase a winding, an MTES set should be chosen that contains f va and f λa fault terms.
Sensors 2022, 22, 3407 10 of 22

As can be seen in the fault signature matrix in Figure 7, MTES7 − MTES10 can be used for
forming such a residual because these four MTES sets all contain f va and f λa fault terms.
Among them, an MTES set is preferred that contains a lower number of fault terms because
it will be more isolated and less influenced by other faults. MTES7 and MTES8 contain
three fault terms, while MTES9 and MTES10 contain four fault terms. Therefore, either
MTES7 or MTES8 should be chosen, and MTES8 is preferred due to a lower number of
involved equations (MTES8 contains six equations, while MTES7 contains eight equations)
which leads to less complexity, as seen in Figure 6. MTES4 − MTES6 can be used for
forming residual R2 because they contain f vb and f λb fault terms. Among them, MTES5
is preferred because it contains a lower number of fault terms compared to MTES6 and
a lower number of equations compared to MTES5 . Similarly, MTES3 is chosen to form
residual R3 that is sensitive to an ITSC fault in phase-c winning and contains a lower
number of equations compared to MTES2 , given the fact that both contain f vc and f λc
fault terms. To form residual R3 that is sensitive to an encoder fault (angular velocity and
position measurements), an MTES set is preferred that contains both f θe and f ωm , and the
only MTES set that contains such fault terms is MTES1 . The combination of these four
MTES sets, i.e., MTES1 , MTES3 , MTES5 , and MTES8 yield a high diagnosability index as
m D = 1.88, and this maximizes the chance of discrimination of each fault from others. The
sequential residuals are obtained as follows:
1. R1 : MTES8 is used for deriving R1 based on the error between calculated and mea-
sured current of phase a winding, i.e m4 in (3):

m4 : R1 = i a − y i a (8)

And the sequence of obtaining these variables is as follows:

SV1 : λ a = λ astate
m7 : θ = y θ
m1 : v a = y v a (9)
1
e4 : i a = (λ a − λm cos θ )
La
e1 : dλ a = v a − R a i a

where λ astate is a state variable and updated at each time-step as follows:

Z
e17 : λ astate = dλ a dt (10)

2. R2 and R3 follow the same procedure mentioned for R1 based on the error between
calculated and measured currents of phase b and phase c using MTES5 and MTES3 ,
respectively.
3. R4 : MTES1 is used for deriving R4 based on the error between the calculated and
measured shaft’s angular speed, i.e m8 in (3):

m8 : R4 = ω m − y ωm (11)

and the sequence of obtaining the unknown variable, ωm , is as follows:

m7 : θ = y θ
dθ d
d5 : = (θ ) (12)
dt dt
dθ
e9 : ω m = p
dt
(13)
Sensors 2022, 22, 3407 11 of 22

5. Experiments and Results

The proposed diagnostic method is implemented and validated through an in-house
experimental setup in this section. First, ITSC faults were applied to the phase windings of
a four-pole PMSM, as shown in Figure 8. Each phase winding of the motor has two coils
in series, each of which has 51 turns with three parallel branches. For phase a, one of the
turns was short circuited, or about a 1% fault level. For the phases b and c, three and five
turns were short circuited, resulting in almost 3% and 5% fault severity, respectively. The
connection wires to these extra taps in the phase windings were taken out of the motor and
connected to 100 mΩ resistors (similar to R f Figure 1) both to limit the short circuit current
and to simulate the winding insulation degradation, as shown in Figure 9. Furthermore,
controllable relays were placed between winding taps and fault resistors to activate or
deactivate the fault. The faulty motor was mechanically coupled to a generator as a variable
load and an incremental encoder to measure the rotor’s angle and velocity. The motor was
driven by a Watt&Well DEMT 3-ph voltage source inverter, which had embedded voltage
and current sensors, being fed by a Keysight N8949A dc supply. In addition, a dSpace
MicrolabBox control unit was used as a real-time interface device for implementing both
control strategy and data acquisition from Matlab/Simulink with a sampling time of 50µs.
The parameters of the studied PMSM are listed in Table 1.
To test the residual responses and effectiveness of the diagnostic system, the motor was
driven from stationary to nominal speed, i.e., 1500 rpm, and kept in a steady-state condition.
During the operation of the motor, the encoder and ITSC faults were applied at different
time intervals using controllable relays. At t = 1–3 s, the encoder measurement fault was
applied with a 1 rad/s error. At t = 4.471–7.238 s, the ITSC fault in phase a was applied
which had 1% fault severity (one shorted turn in phase a winding); at t = 9.613–12.76 s,
the ITSC fault in phase b appeared with 3% fault severity (three shorted turn in phase b
winding); at t = 15.6–18.41 s, the ITSC fault in phase c with 5% fault severity (five shorted
turn in phase c winding) was applied on the motor.
The residual responses for the mentioned faults were obtained and are shown in
Figure 10. Before the faults were applied, the motor was operating in a healthy mode
(t = 0–1 s), and all the residuals remained averagely zero (neglecting the noise). This is
because there was no error between the measured signals and the calculated ones used
in each residual. First, when the encoder fault appeared, R4 obtained a nonzero dc value,
and it went back to average zero as soon as the fault disappeared. When the ITSC fault in
phase a was applied, R1 was directly affected and obtains/ed a higher oscillating value.
Due to mutual induction of the fault current, this fault was also observable in R2 and R3 . In
addition, the controller response had a role in the increase of other phase currents. Since a
part of the winding was gone, more Iq was required to keep the motor speed constant at 1500
rpm. The same logic can be used for ITSC faults in phases b and c as the residuals obtain
higher oscillating values. The behavior and response of the residuals during each ITSC
fault, can be used as the ground for detection of faults in the PMSM. This is implemented
using signal processing–detection theory and explained in the following section.
Sensors 2022, 22, 3407 12 of 22

Figure 8. Applied ITSC faults on a PMSM.

Figure 9. Experimental Setup for control and diagnosis of a PMSM.

Table 1. Parameters of a PM synchronous motor.

Symbol Parameter Value Unit

Vdc Rated dc bus voltage 280 V
Is Rated rms phase current 5 A
Tout Rated Output Torque 7 N·m
ns Rated speed 1500 rpm
Rs Phase resistance 0.8 Ω
Ls Stator inductance 8.5 mH
J Rotor inertia 0.0026 kg·m2
b Rotor damping factor 0.00382 N·m·s/rad
λm Flux linkage of PMs 0.3509 Wb-turn
p Pole-pairs 2
Sensors 2022, 22, 3407 13 of 22

0.5

R1 (A)
0

-0.5

-1
0 5 10 15 20
time (s)

0.5
R2 (A)

-0.5

-1
0 5 10 15 20
time (s)

0.5
R3 (A)

-0.5

-1
0 5 10 15 20
time (s)

1
R4 (rad/s)

-1

-2
0 5 10 15 20
time (s)
Figure 10. Residual responses in abc phases.

6. Diagnostic Decision
Using the residual responses, a diagnostic decision making system was designed to
detect the ITSC faults based on statistical signal processing–detection theory. While R4 can
be directly used to detect encoder faults, a combination of R1 –R3 is required to effectively
detect ITSC faults. The R1 –R3 residuals obtained in the previous section, are designed
based on abc frame voltage equations e1 –e3 in (2), and an ITSC fault in any phase creates
unbalance in the residual output. Before designing the statistical detector and to form a
better index that obtains a nonzero dc value in case of an ITSC fault, the residuals in the
abc frame are taken into an αβ frame using the power invariant Clarke transformation
as follows:
Sensors 2022, 22, 3407 14 of 22

 r " # R 
1
− −√21  1 

Rα 2 1 √2
= 3 R2 (14)
Rβ 3 0 2 − 23 R
3

The absolute value of the resultant is calculated:

Rr = | Rα + jR β | (15)

Figure 11 shows the absolute value of the resultant residual in an αβ frame where
ITSC faults in all phases are more obvious compared to abc residuals R1 –R3 .

Figure 11. Resultant residual response in an αβ frame.

In implementing a structural analysis, the goal was to form residuals that have a zero
value in a healthy scenario and a nonzero value in a faulty scenario. However, derivatives,
integrals, and even uncertainties in the dynamic model affect the calculation of unknown
variables and cause the variable output signal to be a little bit distorted. In addition,
phenomena such as environmental noise and switching noise affect the signals. These lead
to a residual output signal that fluctuates around zero instead of having a perfect signal
that holds the absolute zero value in a healthy scenario. Even in a faulty scenario, the
residual signal fluctuates around a nonzero value as seen in Figure 11. Therefore, extra
signal processing is required to deal with model uncertainties and environmental noise and
to be able to distinguish and isolate the indicator signal from noise. Here, a generalized
likelihood ratio test (GLRT) is proposed to deal with such model uncertainties and also
to provide the ground for calculating and setting thresholds based on the probabilities of
detection and false alarms in a formulated and scientific manner.

6.1. Generalized Likelihood Ratio Test

GLRT is a composite hypothesis testing approach that can be used for detecting a
signal in realistic problems [42]. It is noted that GLRT does not require prior knowledge of
the unknown parameters such as mean (µ) and variance (σ2 ) values in a probability density
function (PDF) of a signal. GLRT deals with unknown parameters by replacing them with
their maximum likelihood estimates (MLEs). If data x have the PDF p( x; θˆ0 , H0 ) under a
null hypothesis H0 and p( x; θˆ1 , H1 ) under alternative hypothesis H1 , the GLRT decides
H1 if:

p( x; θˆ1 , H1 )
LG ( x) = >γ (16)
p( x; θˆ0 , H0 )

where θˆ1 is the MLE of θ1 assuming H1 is true, θˆ0 is the MLE of θ0 assuming H0 is true, and
γ is the threshold.
Sensors 2022, 22, 3407 15 of 22

6.2. Design of Test Statistic Based on Generalized Likelihood Ratio Test

Before going through the design process, it is beneficial to know the PDF of the
measurement noise signal. This gives us enough knowledge to make the assumptions
that are close to our realistic problem. Using the first part (t = 0–1 s) of the resultant
residual in Figure 11, the PDF of the noise signal in a noise-only hypothesis is obtained and
shown in Figure 12. The PDF of the noise signal in Figure 12 is very close to the PDF of a
white Gaussian noise (WGN), thus it can be reasonably modeled with a WGN probability
distribution function.

30
Residual
25
WGN
20
PDF (x)

15

10

0
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06
x
Figure 12. Comparison of PDF of residual and WGN.

To design a realistic detector, it is assumed that the arrival time of the fault is completely
unknown. Furthermore, the PDF is not completely known, meaning that the parameters
mean µ and variance σ2 are to be estimated using MLE. The noise in the resultant residual
during operation in a healthy condition is modeled as WGN. Since the resultant residual
(Rr ) obtains a nonzero dc value when ITSC faults appear, the data are considered as only
noise under nonfaulty hypothesis H0 , and an added dc level value to the noise under faulty
hypothesis H1 . Thus, the detection problem becomes as follows:

H0 : x [ n ] = w [ n ] n = 0, 1, ..., N − 1 (17)
H1 : x [ n ] = A + w [ n ] n = 0, 1, ..., N − 1

where A is unknown amplitude with −∞ < A < ∞, and w[n] is WGN with unknown
positive variance 0 < σ2 < ∞. The GLRT decides H1 if:

p( x; Â, σˆ1 2 , H1 )
LG ( x) = >γ (18)
p( x; σˆ0 2 , H0 )

where  and σˆ1 2 are the MLE of parameters A and σ12 under H1 , and σˆ0 2 is the MLE of the
parameter σ02 under H0 . By maximizing p( x; A, σ2 , H1 ), parameters  and σˆ1 2 are obtained
as follows [43]:
N −1
1 1
p( x; A, σ2 , H1 ) = N
exp[−
2σ2 ∑ ( x [ n ] − A )2 ]
(2πσ2 ) 2 N =0
∂p( x; A, σ2 , H1 )
= 0 ⇒ Â = x̄ (19)
∂A
N −1
∂p( x; A, σ2 , H1 ) 1
∂σ12
= 0 ⇒ σˆ1 2 =
N ∑ ( x [ n ] − A )2
N =0
Sensors 2022, 22, 3407 16 of 22

which results in:

1 N
p( x; Â, σˆ1 2 , H1 ) = N
exp(− ) (20)
(2π σˆ1 ) 2 2 2

Similarly, by maximizing p( x; σˆ0 2 , H0 ), σˆ0 2 is obtained as follows:

N −1
1 1
p( x; σ2 , H0 ) = N
exp(−
2σ2 ∑ x2 [n])
(2πσ2 ) 2 N =0
N −1
∂p( x; σ2 , H 0) 1
∂σ02
= 0 ⇒ σˆ0 2 =
N ∑ x2 [n] (21)
N =0

which results in:

1 N
p( x; σˆ0 2 , H0 ) = N
exp(− ) (22)
(2π σˆ0 ) 2 2 2

Therefore, (18) becomes:

σˆ0 2 N
LG ( x) = ( )2 (23)
σˆ1 2

which is equivalent to:

σˆ0 2
2lnLG ( x ) = Nln (24)
σˆ1 2

From (19) and (21), σˆ1 2 can intuitively be obtained as follows:

N −1 N −1
1 1
σˆ1 2 =
N ∑ ( x [ n ] − A )2 =
N ∑ (x[n] − x̄)2
N =0 N =0
N −1 N −1
1 1
=
N ∑ (x[n]2 − 2x[n]x̄ + x̄2 ) = N ∑ x [n]2 − x̄2
N =0 N =0
2 2
= σˆ0 − x̄ (25)

which yields:

x̄2
2lnLG ( x ) = Nln(1 + ) (26)
σˆ1 2

x̄2 x̄2
Since ln(1 + ) is monotonically increasing with respect to , an equivalent and
σˆ1 2 σˆ1 2
normalized test statistic can be obtained as follows:

x̄2
T (x) = > γ0 (27)
σˆ1 2

The GLRT has normalized the statistic by σˆ1 2 which allows the threshold to be de-
termined. Since the PDF of T ( x ) under null hypothesis H0 does not depend on σ2 , the
threshold is independent of the value σ2 [42].
Sensors 2022, 22, 3407 17 of 22

6.3. GLRT for Large Data Records

As N −→ ∞, the asymptotic PDFs of x̄ will converge to normal distributions under
both hypotheses as follows:
(
N (0, σ2 ) under H0
x̄ ∼ (28)
N ( A, σ2 ) under H1

and therefore:
(
x̄ N (0, 1) under H0
∼ (29)
σ N ( Aσ , 1) under H1

Squaring the normalized statistic in (29) will lead to the modified test statistic T ( x )
in (27) which produces a central chi-squared distribution under H0 and a noncentral
chi-squared distribution under H0 , with one degree of freedom:
(
x̄2 X12 under H0
T (x) = 2 ∼ 0 2 (30)
σ X 1 (λ) under H1

where λ is the noncentrality parameter and is calculated as [42]:

A2 x̄2
λ= 2
= 2 (31)
σ σ
It was shown in (30) that T ( x ) has a noncentral chi-squared distribution with one
√ freedom, and it is equal to the square of random variable x in (29), therefore
degree of
x ∼ N ( λ, 1). Thus, the probability of a false alarm (PFA ) can be obtained as:

PFA = Pr { T ( x ) > γ0 ; H0 }
p p
= Pr { x > γ0 ; H0 } + Pr { x < − γ0 ; H0 }
p
= 2Q( γ0 ) (32)

where Q( x ) is the right-tail probability of random variable x. Thus, the threshold can be
obtained as follows:
PFA 2
γ 0 = [ Q −1 ( )] (33)
2
Similarly, the probability of detection PD can be obtained as follows:

PD = Pr { T ( x ) > γ0 ; H1 }
p p
= Pr { x > γ0 ; H1 } + Pr { x < − γ0 ; H1 }
p √ p √
= Q( γ0 − λ) + Q( γ0 + λ)
P √ P √
= Q( Q−1 ( FA ) − λ) + Q( Q−1 ( FA ) + λ) (34)
2 2

6.4. GLRT Test on Residual Response

For the case study, the statistical detector should be designed in a way that it is able
to detect even the smallest ITSC fault (<1%). Therefore, the noncentrality parameter λ is
calculated based on the implementation of (31) on the resultant residual at t = 4.471s–7.238 s
when the motor is experiencing the lowest ITSC fault level in phase a winding and yields
λ = 6.78. Using this value, the threshold and receiver operating characteristics (ROC) of
the detector is obtained based on (32)–(34) and shown in Figure 13. The PFA values here are
for the lowest ITSC fault level in phase a, which means other ITSC faults in phases b and c
Sensors 2022, 22, 3407 18 of 22

have lower PFA values. Using PFA = 2%, the threshold is obtained as γ0 = 5.41, and this
results in PD = 60.93% for ITSC in phase a. Furthermore, the probability of detection for
ITSC faults in phases b and c and the encoder fault are calculated PD = 98.13%, PD = 100%,
and PD = 99.65%, respectively.
The test statistics were implemented on the resultant residual as shown in Figure 14.
The values x̄2 and σˆ1 2 were calculated using a moving window (FIFO register) with the
length of N = 10, 000, which runs through the resultant residual over time. Figure 14a
shows the output of test statistic on resultant residual along with the threshold of γ0 = 5.41
while Figure 14b shows the output of the test statistic on R4 . The test statistic’s output value
is compared with the threshold value over time, and if it exceeds the threshold, the fault
alarm is tripped accordingly. Figure 15 shows the detector’s logical output value which
attains a low value in a healthy condition and a high value during a faulty case. This proves
that the detector has successfully detected all the faults that are fairly close to expected
values of PD , while experiencing no false alarm.

20

15
Threshold

10

0
0 0.01 0.02 0.03 0.04 0.05
P FA

ITSC a
PD

0.5
ITSC b
ITSC
c
Encoder
0
0 0.01 0.02 0.03 0.04 0.05
P FA

Figure 13. Threshold and ROC for low values of PFA .

Sensors 2022, 22, 3407 19 of 22

60

50

Motor Fault Index

40

30

20

10

0
0 5 10 15 20
time (s)
(a)

50
Encoder Fault Index

40

30

20

10

0
0 5 10 15 20
time (s)
(b)

Figure 14. Test statistic for ITSC fault and encoder faults.

1
Denc
Diagnostic Decision

0.8
DITSC

0.6 f enc
f ITSC
a
0.4
f ITSC
b

0.2 f ITSC
c

0
0 5 10 15 20
time (s)
Figure 15. Timing of actual faults ( f enc , f ITSCa , f ITSCb , f ITSCc ) versus diagnostic system’s decision
(Denc , D ITSC ).

7. Discussion
Some remarks can be withdrawn regarding the presented methodology and the ob-
tained results. First, structural analysis for detecting ITSC and encoder faults was success-
fully implemented on the in-house setup including the PMSM and the drive system, and
the residuals were formed based on ARRs. Second, a GLRT-based detector was designed to
effectively detect the changes in the residuals even with unknown noise parameters. Third,
Sensors 2022, 22, 3407 20 of 22

a scientific threshold was calculated based on the probability of a false alarm (PFA ) and the
probability of detection (PD ). The suggested combination method is very effective for the
fault detection since it can detect the lowest level of ITSC fault, i.e., one single shorted turn
(<1%) in the stator winding. On the other hand, using a Clarke transformation disabled
the diagnostic system to isolate the ITSC faults in different phases, and using a moving
window with the length of N = 10, 000 over the test statistics causes a delay in detection of
the faults. These small demerits were found when testing the diagnostic method under the
smallest ITSC fault.
In previous studies, a GLRT-based detector has been implemented for stator imbal-
ancefault detection in induction motors [44]. The noise parameters were also considered
unknown, and therefore, they have been replaced with their MLEs. Moreover, a threshold
was calculated based on PFA = 0.1% and PD , which makes the diagnostic system experi-
ence fewer false alarms. However, the first fault level that the system can detect is 25%
of stator-phase resistance, which is a quite high level of fault severity. As a result, the
system would go into severe imbalance from the time that the fault appears until the time
the diagnostic system detects it. In our case, even if the PFA was chosen as 0.1%, the PD
for ITSC in phase a would be 24.61%, the PD for ITSC in phase b would be 86.8%, the PD
for ITSC in phase c would be 99.99%, and the PD for the encoder fault would be 95.86%.
Thus, the diagnostic system still detects the smallest fault, even with PFA = 0.1%. However,
knowing that a slightly higher probability of a false alarm is not that irritating (PFA = 2%),
a better probability of detection is achieved (PD = 60.93%) in our study based on setting
a lower threshold. Other studies with different methods have also chosen a higher level
of fault as the starting point. A Kalman filter for detection of ITSC in PM synchronous
generators has been implemented in [45], which can successfully detect fault levels as low
as 8%. In addition, a combination of an extended Park’s vector approach with spectral
frequency analysis was introduced in [46] which could successfully detect three shorted
turns in synchronous and induction motors.

8. Conclusions
This paper presents a novel method for real-time and effective detection of incipient
ITSC and encoder faults in the PMSM. Structural analysis was employed to form the
structural model of the PMSM. The Dulmage–Mendelsohn decomposition tool was used
to evaluate the analytical redundancy of the structural model. The proposed diagnostic
model was implemented on industrial PMSM, ITSC, and encoder faults were applied to the
system in different time intervals, and residuals responses were obtained. Subsequently, a
GLRT-based detector was designed and implemented based on the behavior of the residuals
during healthy (only noise) and faulty (noise + signal) conditions. To make the GLRT-based
detector effective to deal with such a realistic problem, the parameters such as mean µ
and variance σ2 in the probability density function of the noise signal were considered
to be unknown. By replacing these unknown parameters by their maximum likelihood
estimates, a test statistic was achieved for the GLRT-based ITSC and encoder fault detector.
Following this step, a threshold was obtained based on choosing the probability of a false
alarm PFA and the probability of detection PD for each detector based on which decision
was made to indicate the presence of the fault. The experimental results show that the
designed GLRT-based detector is able to efficiently detect even small ITSC and encoder
faults in the presence of noise, proving the effectiveness of this diagnostic approach.

Author Contributions: Conceptualization, S.H.E.; Data curation, S.H.E.; Investigation, S.H.E.;

Methodology, S.H.E.; Resources, M.C.; Software, S.H.E.; Supervision, M.C. and V.K.H.; Valida-
tion, S.H.E.; Visualization, S.H.E.; Writing—original draft, S.H.E.; Writing—review & editing, M.C.
and V.K.H. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Sensors 2022, 22, 3407 21 of 22

Data Availability Statement: The data presented in this study are available on request from the
corresponding author. The data are not publicly available due to ongoing research within Ph.D. program.
Conflicts of Interest: The authors declare no conflict of interest.

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