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This work proposes an alternative strategy to the use of a speed sensor in the implementation of active and reactive power based model reference adaptive system (PQ-MRAS) estimator in order to calculate the rotor and stator resistances of an induction motor (IM) and the use of these parameters for the detection of inter-turn short circuits (ITSC) faults in the stator of this motor. The rotor and stator resistance estimation part of the IM is performed by the PQ-MRAS method in which the rotor angular velocity is reconstructed from the interconnected high gain observer (IHGO). The ITSC fault detection part is done by the derivation of stator resistance estimated by the PQ- MRAS estimator. In addition to the speed sensorless detection of ITSC faults of the IM, an approach to determine the number of shorted turns based on the difference between the phase current of the healthy and faulty machine is proposed. Simulation results obtained from the MATLAB/Simulink platform have shown that the PQ-MRAS estimator using an interconnected high- gain observer gives very similar results to those using the speed sensor. The estimation errors in the cases of speed variation and load torque are al mos t identical. Variations in stator and rotor resistances influence the per formance of the observer and lead to poor estimation of the rotor resistance. The results of ITSC fault detection using IHGO are very similar to the results in the literature using the same diagnostic approach with a speed sensor.

The induction motors (IM) have been widely used in various industrial applications requiring variable speed because they are simple in construction, low cost, very reliable, and very robust and require minimal maintenance [

With the goal to optimize the IM parameters estimation, several methods have been developed this last decade including off-line estimators and on-line estimators. Namely, the authors demonstrated in Refs. [

Some authors presented an active and reactive power based MRAS (PQ-MRAS) estimator of both stator and rotor resistances with significant results [

The scope of this paper falls into the faults detection under sensorless speed control technology. Facing these challenges previously established, the main contributions of this paper are three fold:

1) Minimizing the costs of implementation and maintenance by reducing the number of sensors.

2) Use of IHGO to estimate the rotor speed of the IM in the implementation of PQ-MRAS estimator,

3) Detect ITSC stator faults using simple estimated resistance derivation algorithms.

The observers used in this work are of the interconnected high-gain observer (IHGO) type. This paper proposes to combine the two methods where the IHGO is used to estimate speed rotor and PQ-MRAS for the estimation of stator and rotor resistances. The stator resistance estimated is utilized in faults detection algorithms, which is based on a simple assumption: inter-turn short circuit (ITSC) results in a sharp decrease of the estimated stator resistance [

The remaining parts of this paper are organized as follows: the mathematical model of the IM, the formulation of the IHGO, the PQ-MRAS estimator description, the direct field oriented control structure of IM and the fault diagnosis strategy are presented in section 2; the simulation results are presented in Section 3. The results comparison is presented in section 4 and finally conclusions and remarks are made in Section 5.

Test and validations of faults detection by parameter estimation require a model suited for fault modelling. For this purpose, it is presented, in this section, a model of IM dedicated to inter-turns short-circuit winding faults. The IGHO technique will make it possible to reconstruct the rotational speed of the rotor which will be immediately used through the PQ-MRAS technique to estimate the resistance of the stator and of the rotor using the measurement of the current and the voltage of the stator. In this section, the control structure and the diagnosis fault strategy are also presented.

The model used is that of [

1) the localization parameter θ c c k can takes only the three values: 0, 2π/3 or 4π/3, corresponding to the short-circuit on the stator phase a, b or c, respectively.

2) the detection parameter η c c k which allows to quantify the unbalance, is equal to the ratio between the number of inter-turn short-circuit windings ( N c c k ) and the total number of inter turns in one healthy phase ( N s ).

In the stator faulty case, an additional shorted circuit winding B_{cc} appears in stator. This winding creates a stationary magnetic field H_{cc} oriented according to the faulty winding [_{cc} into inter-turn short-circuit winding at origin of a short-circuit flux φ_{cc}. Voltage and flux equations for faulty model of induction machine with global-leakage inductance referred to the stator can be written as

{ u s = [ R s ] i s + d d t ϕ s 0 = [ R r ] i r + d d t ϕ r 0 = R c c i c c + d d t ϕ c c ϕ s = [ L s ] i s + [ M s r ] i r + [ M s c c ] i c c ϕ r = [ L r ] i r + [ M r s ] i s + [ M r c c ] i c c ϕ c c = [ M s c c ] i s + [ M c c r ] i r + L c c i c c (1)

where u s represents the stator voltage, i s , i r and i c c are the stator, rotor, and short-circuit currents respectively. ϕ s , ϕ r and ϕ c c denotes the stator, rotor, and short-circuit flux respectively.

[ R s ] = [ R s 0 0 0 R s 0 0 0 R s ] and [ R r ] = [ R r 0 0 0 R r 0 0 0 R r ] are the stator and rotor resistances respectively while R c c = η c c ⋅ R s represents the short-circuit resistance with

η c c = Numberofinterturnsshort-circuitwindings Totalnumberofinterturnsinhealthyphase = N c c N s (2)

With respect to the stator and rotor, we define the following matrices for the inductors:

[ L s ] = [ L m + L f − L m 2 − L m 2 − L m 2 L m + L f − L m 2 − L m 2 − L m 2 L m + L f ] , [ L r ] = [ L m − L m 2 − L m 2 − L m 2 L m − L m 2 − L m 2 − L m 2 L m ] ; L c c = η c c 2 ( L m + L f )

[ M s r ] = L p [ cos ( θ ) cos ( θ + 2 π 3 ) cos ( θ − 2 π 3 ) cos ( θ − 2 π 3 ) cos ( θ ) cos ( θ + 2 π 3 ) cos ( θ + 2 π 3 ) cos ( θ − 2 π 3 ) cos ( θ ) ]

[ M s c c ] = η c c L p [ cos ( θ c c ) cos ( θ c c − 2 π 3 ) cos ( θ c c + 2 π 3 ) ] , [ M r c c ] = η c c L p [ cos ( θ c c − θ ) cos ( θ c c − θ − 2 π 3 ) cos ( θ c c − θ + 2 π 3 ) ]

[ M s r ] = [ M r s ] T , [ M c c s ] = [ M s c c ] T , [ M c c r ] = [ M r c c ] T

θ stand as the rotor angular position. L p and L f are, respectively, principal and global-leakage inductance referred to the stator.

To minimize the number of model variables, we use Concordia transformation, which gives ( α , β ) values of same amplitude as ( a , b , c ) ones. Thus, we define three- to two-axis transformation T 23 as

{ x _ α β r = P ( θ ) ⋅ T 23 ⋅ x _ r x _ α β s = T 23 x _ s (3)

where x _ α β is projection of x _ following α and β axis. The transformation matrix and the rotational matrix are defined respectively as follows:

T 23 = [ cos ( 0 ) cos ( 2 π 3 ) cos ( 2 π 3 ) sin ( 0 ) sin ( 2 π 3 ) sin ( 2 π 3 ) ] (4)

P ( θ ) = [ cos ( θ ) cos ( θ + π 2 ) sin ( θ ) sin ( θ + π 2 ) ] (5)

The short-circuit variables are localized on one axis, these projection on the two Concordia axis α and β is defined as

i _ α β c c = [ cos ( θ c c ) sin ( θ c c ) ] ⋅ i c c , ϕ _ α β c c = [ cos ( θ c c ) sin ( θ c c ) ] ⋅ ϕ c c (6)

Thus, (1) becomes

{ u _ α β s = R s i _ α β s + d d t ϕ _ α β s ϕ _ α β s = L m ( i _ α β s + i _ α β r + 2 3 η c c i _ α β c c ) + L f i _ α β s u _ α β r = 0 = R r ⋅ i _ α β r + d d t ϕ _ α β r − ω ⋅ P ( π 2 ) ϕ _ α β r ϕ _ α β r = L m ( i _ α β s + i _ α β r + 2 3 η c c i _ α β c c ) 0 = η c c R s i _ α β c c + d d t ϕ _ α β c c ϕ _ α β c c = 2 3 η c c L m Q ( θ c c ) ( i _ α β s + i _ α β r ) + ( 2 3 L m + L f ) η c c 2 i _ α β c c (7)

where ω is the rotor electrical frequency; L m = ( 3 / 2 ) L p the magnetizing inductance. Q ( θ c c ) is define as:

Q ( θ c c ) = [ cos 2 ( θ c c ) cos ( θ c c ) sin ( θ c c ) cos ( θ c c ) sin ( θ c c ) sin 2 ( θ c c ) ] (8)

If we neglect L f according to L m in short-circuit flux expression (7), we can write new flux equations as:

{ ϕ _ α β s = ϕ _ α β m + ϕ _ α β f = L m ( i _ α β s + i _ α β r − i ˜ _ α β c c ) + L f i _ α β s ϕ _ α β r = ϕ _ α β m = L m ( i _ α β s + i _ α β r − i ˜ _ α β c c ) ϕ ˜ _ α β c c = η c c Q ( θ c c ) ϕ _ α β m (9)

where

{ i ˜ _ α β c c = − 2 3 η c c i _ α β c c ϕ ˜ _ α β c c = 2 3 ϕ _ α β c c (10)

ϕ _ α β m and ϕ _ α β c c are magnetizing and leakage flux respectively. Then, the short-circuit-current equation becomes

i ˜ _ α β c c = 2 3 η c c R s Q ( θ c c ) d d t ϕ _ α β m (11)

According to this equation, the faulty winding B c c becomes a simple unbalanced resistance element in parallel with magnetizing inductance. The existence of localization matrix Q ( θ c c ) in (9) makes complex the state space representation in Concordia’s axis. In a large range of industrial application, voltage drop in R s and L f is neglected according to stator voltage u _ α β s then, we can put a short-circuit element Q c c in input voltage border (

It is much simpler to work in the rotor reference frame because we have only two stator variables to transform. Therefore, in state operation, all the variables have their pulsations equal to s ω s (where s is the slip and ω s is stator pulsation). Park’s

transformation is defined as

x _ d q = P ( − θ ) ⋅ x _ α β (12)

Afterward, the faulty model will be expressed under Park’s reference frame. Therefore, short-circuit current (11) becomes

i ˜ _ d q c c = 2 3 η c c R s P ( − θ ) Q ( θ c c ) P ( θ ) u _ d q s (13)

Fundamentally, in faulty case, an induction machine can be characterized by two equivalent modes. The common mode model corresponds to the healthy dynamics of the machine (Park’s model) whereas the differential mode model explains the faults.

We generalize this model by dedicating to each phase of the stator a short circuit element Q c c k to explain a possible faulty winding. So, in presence of several short circuits, each faulty element allows the diagnosis of a phase by watching the value of the parameter. This simple deviation allows to indicate the presence of unbalance in the stator.

Voltage and flux equations for faulty model with global leakage inductance referred to the stator can be written as:

- Park’s model (stator and rotor)

{ u _ d q s = R s i ′ _ d q s + d d t ϕ _ d q s + ω P ( π 2 ) ϕ _ d q s ϕ _ d q s = L f i ′ _ d q s + L m ( i ′ _ d q s + i _ d q r ) 0 = R r ⋅ i _ d q r + d d t ϕ _ d q r ϕ _ d q r = L m ( i ′ _ d q s + i _ d q r ) (14)

- Differential mode model (short-circuit currents)

i ˜ _ d q c c k = 2 3 η c c k R s P ( − θ ) Q ( θ c c k ) P ( θ ) u _ d q s , (15)

Resultant dq stator currents become

i _ d q s = i ′ _ d q s + i ˜ _ d q c c k = i ′ _ d q s + ∑ k = 1 3 i _ d q c c k (16)

Each stator phase is characterized by its faulty parameters ( η c c k , θ c c k ) , where k indicates one of the three stator phases.

For simulation, it is necessary to write the faulty model in state-space representation. So, the fourth-order state-space representation of the IM with a winding fault is given by:

{ x ˙ _ ( t ) = A ( ω ) x _ ( t ) + B u _ ( t ) y _ ( t ) = C x _ ( t ) + D u _ ( t ) (17)

where x _ ( t ) = [ i ′ d s i ′ q s ϕ d r ϕ q r ] T is the state-space vector; u _ ( t ) = [ u d s u q s ] T , y _ ( t ) = [ i d s i q s ] T are the systems inputs-outputs. The matrices of Equation (17) are defined as follows:

A ( ω ) = [ − a ω b ρ r b ω − ω − a − b ω b ρ r R r 0 − ρ r 0 0 R r 0 − ρ r ] ; B = [ b 0 0 0 0 b 0 0 ] ; C = [ 1 0 0 0 0 1 0 0 ]

D = [ 2 3 R s ∑ k = 1 3 η c c k P ( − θ ) Q ( θ c c k ) P ( θ ) 0 0 0 ] ;

The inner parameters are as follows:

a = R s + R r L f , b = 1 L f , ρ r = R r L m and η c c k = N c c k N s

P ( θ ) = [ cos ( θ ) − sin ( θ ) sin ( θ ) cos ( θ ) ] ; Q ( θ c c k ) = [ cos 2 ( θ c c k ) cos ( θ c c k ) sin ( θ c c k ) cos ( θ c c k ) sin ( θ c c k ) sin 2 ( θ c c k ) ] .

i d s , i q s are stator current components in ( d , q ) frame while ϕ d r , ϕ q r are rotor flux linkages current in ( d , q ) frame. θ is the rotor position; Q ( θ c c k ) matrix depending on short-circuit angle θ c c k . R s , L f , R r and L m are respectively, stator resistance, global leakage inductance referred to the stator, rotor resistance and magnetizing inductance.

The PQ-MRAS estimator method that consists to the association of P-MRAS and Q-MRAS estimators has been proposed by Bednarz and al. in Ref. 3. The P-MRAS estimator is based on active power of IM and is used to estimate the stator resistance while the Q-MRAS estimator is based on reactive power used to estimate the rotor resistance. Each subsystem has its own independent mechanism based on a proportional-integral (PI) controller. The PQ-MRAS allows the simultaneous estimation of stator and rotor resistances (see

In the P-MRAS estimator, the active power reference is given by:

P r e f = u s α i s α + u s β i s β (18)

and the adjustable model is given by:

P a d j = Re { u s e s t ⋅ i s ∗ } = R s e s t ( i s α 2 + i s β 2 ) + σ L s ( i s α d i s α d t + i s β d i s β d t ) + L m L r ( i s α d ψ r α e s t d t + i s β d ψ r β e s t d t ) (19)

In order to minimize the error between the reference and adjustable models using the adaptive mechanism, the resistance of the stator can be evaluated by the following equation:

R s e s t = K P R s ε P + K I R s ∫ ε P d t (20)

with

ε P = P r e f − P a d j (21)

In the Q-MRAS estimator, the reactive power reference can be defined as follows:

Q r e f = u s β i s α − u s α i s β (22)

and the adjustable model is given by:

Q a d j = σ L s ( i s α d i s β d t − i s β d i s α d t ) + L m L r ( i s α d ψ r β e s t d t − i s β d ψ r α e s t d t ) (23)

In the case of Q-MRAS estimator, the resistance can be calculated by the equation:

R r e s t = K P R r ε Q + K I R r ∫ ε Q d t (24)

with

ε Q = | Q r e f | − | Q a d j | (25)

In this model, the current model for different axes ( α , β ) used for the rotor flux components is obtained from:

d ψ r α e s t d t = R r L r ( L m i s α − ψ r α e s t ) − p ω ψ r β e s t (26)

d ψ r β e s t d t = R r L r ( L m i s β − ψ r β e s t ) + p ω ψ r α e s t (27)

In the IHGO technique, the IM is seen as an interconnection of two subsystems, where each of the subsystems satisfies some required properties of observability. The observer used is constructed on the basis of electrical values including stator currents ( i s α , i s β ) and stator voltages ( u s α , u s β ) and returns the mechanical values including the rotor flux ( ψ r α , ψ r β ) , speed rotor ω and load torque T l . [

{ d ψ r α d t = R r L m L r i s α − R r L r ψ r α − p ω ψ r β d ψ r β d t = R r L m L r i s β − R r L r ψ r β + p ω ψ r α d i s α d t = L r M u s α − ( L r R s M + L m 2 R r M L r ) i s α − L m R r M L r ψ r α + L m M p ω ψ r β d i s β d t = L r M u s β − ( L r R s M + L m 2 R r M L r ) i s β − L m R r M L r ψ r β − L m M p ω ψ r α d ω m d t = 1 J ( T e − T l ) d T l d t = 0 (28)

where ( ψ s α , ψ s β ) and ( ψ r α , ψ r β ) are the stator flux and rotor flux respectively. ( i s α , i s β ) and ( i r α , i r β ) represents respectively the stator and rotor currents. ω is the rotor speed. These parameters (flux, current and speed) are dynamics parameters of the system. The inputs control parameters are stator voltages ( u s α , u s β ) . Among these parameters, we have a measurable ( i s α , i s β , i r α , i r β , ω ) and non-measurable ( ψ s α , ψ s β , ψ r α , ψ r β ) . Despite the dynamic parameters of the system we can also define another elements that helps in descriptions of the all system such as the, load torque T l , stator and rotor resistances ( R s , R r ) , stator, rotor and mutual inductances ( L s , L r , L m ) . p, J, T e denotes respectively the number of pole pairs, the moment of inertia and the electromagnetic torque. M = L r L s − L m 2 is a reduce constant parameter.

Considering the states involved in the implementation of the IHGO, Equation (28) can be rewritten in terms of two interconnected subsystems as [

d d t ( i s α ω T l ) = ( 0 b p ψ r β 0 0 0 − 1 J 0 0 0 ) ( i s α ω T l ) + ( − γ i s α + a b ψ r α + m 1 u s α m ( i s β ψ r α − i s α ψ r β ) 0 ) (29)

d d t ( i s β ψ r α ψ r β ) = ( − γ − b p ω a b 0 − a − p ω 0 p ω − a ) ( i s β ψ r α ψ r β ) + ( m 1 u s β a L m i s α a L m i s β ) (30)

with a , b , γ , m and m 1 defined by: a = R r L r , b = L m M , γ = L r R s M + L m 2 R r M L r , m = p L m J L r , m 1 = L r M .

The two subsystems (29) and (30) can be represented in the following compact interconnected forms:

{ X ˙ 1 = A 1 ( u , y , X 2 ) X 1 + g 1 ( u , y , X 1 , X 2 ) y 1 = C 1 X 1 (31)

{ X ˙ 2 = A 2 ( u , y , X 1 ) X 2 + g 2 ( u , y , X 1 , X 2 ) y 2 = C 2 X 2 (32)

where A 1 ( u , y , X 2 ) , A 2 ( u , y , X 1 ) , g 1 ( u , y , X 1 , X 2 ) and g 2 ( u , y , X 1 , X 2 ) are:

A 1 ( u , y , X 2 ) = ( 0 b p ψ r β 0 0 0 − 1 J 0 0 0 ) , g 1 ( u , y , X 1 , X 2 ) = ( − γ i s α + a b ψ r α + m 1 u s α p L m J L r ( i s β ψ r α − i s α ψ r β ) 0 ) ,

A 2 ( u , y , X 1 ) = ( − γ − b p ω a b 0 − a − p ω 0 p ω − a ) , g 2 ( u , y , X 1 , X 2 ) = ( m 1 u s β a L m i s α a L m i s β ) ,

The states vectors X 1 and X 2 and the matrices C 1 , C 2 , u and y are defined by:

X 1 = ( i s α ω T l ) , X 2 = ( i s β ψ r α ψ r β ) , C 1 = C 2 = ( 1 0 0 ) T , u = ( u s α u s β ) , y = ( i s α i s β )

By assumptions the signals ( u , X 1 ) and ( u , X 2 ) are regularly persistent and known for the subsystems (29) and (30) respectively, considering that the functions g 1 ( u , y , X 1 , X 2 ) and g 2 ( u , y , X 1 , X 2 ) are globally Lipschitz with respect to X 1 and X 2 and uniformly with respect to ( u , y ) , the observers of subsystems (29) and (30) are defined by [

{ Z ˙ 1 = A 1 ( Z 2 ) Z 1 + g 1 ( u , y , Z 1 , Z 2 ) + ( Γ S 1 − 1 C 1 T + B 2 ( Z 2 ) ) ( y 1 − y ⌢ 1 ) + ( K C 2 T + B 1 ( Z 2 ) ) ( y 2 − y ⌢ 2 ) S ˙ 1 = − θ 1 S 1 − A 1 T ( Z 2 ) S 1 − S 1 A 1 ( Z 2 ) + C 1 T C 1 y ⌢ 1 = C 1 Z 1 (33)

{ Z ˙ 2 = A 2 ( Z 1 ) Z 2 + g 2 ( u , y , Z 1 , Z 2 ) + S 2 − 1 C 2 T ( y 2 − y ⌢ 2 ) S ˙ 2 = − θ 2 S 2 − A 2 T ( Z 1 ) S 2 − S 2 A 2 ( Z 1 ) + C 2 T C 2 y ⌢ 2 = C 2 Z 2 (34)

where

Z 1 = ( i ^ s α ω ^ m T ^ l ) , Z 2 = ( i ^ s β ψ ^ r α ψ ^ r β ) , B 1 ( Z 2 ) = k m Λ 1 ψ ^ r α , B 2 ( Z 2 ) = k m Λ 2 ψ ^ r β , Λ 1 = ( 0 0 1 ) , Λ 2 = ( 0 0 − 1 ) , K = ( − k c 1 0 0 − k c 2 0 0 0 0 0 ) , K = ( 1 0 0 0 1 0 0 0 α )

The matrices A 1 ( Z 2 ) and A 2 ( Z 2 ) as well as the field vectors g 1 ( u , y , Z 1 , Z 2 ) and g 2 ( u , y , Z 1 , Z 2 ) are defined by:

A 1 ( Z 2 ) = ( 0 b p ψ ^ r β 0 0 0 − 1 J 0 0 0 ) ; A 2 ( Z 2 ) = ( − γ − b p ω ^ a b 0 − a − p ω ^ 0 p ω ^ − a )

g 1 ( u , y , Z 1 , Z 2 ) = ( − γ i ^ s α + a b ψ ^ r α + m 1 u s α m ( i ^ s β ψ ^ r α − i ^ s α ψ ^ r β ) 0 ) ; g 2 ( u , y , Z 1 , Z 2 ) = ( m 1 u s β a L m i ^ s α a L m i ^ s β )

In the observers, θ 1 , θ 2 , k , k c 1 , k c 2 and α are positives constants. S 1 and S 2 are positive symmetric matrices obtained by solving the Lyapunov equations [

During the estimation of the stator and rotor resistances, the IM have to be in controlled. In the present work, the controller implemented is the vector control and more precisely the direct rotor flux-oriented control (DRFOC). The principle of vector control is to equate the behavior of the asynchronous machine with that of a DC machine, i.e. a linear and decoupled model, which allows a linear and decoupled control between the flux and torque of IM [

{ ψ r q = 0 ψ r d = ψ r (35)

The constraint (31) imposed on the quadrate rotor flux component allows that, the rotor flux is controlled by the direct component of the stator current i s d , whereas the electromagnetic torque is controlled by the quadrature component of the stator current i s q .

For the implementation of the bloc diagram of DRFOC (see

ψ r = ∫ ( R r L r ( L m i s d + ψ r ) ) d t (36)

T e = p L m L r ψ r i s q (37)

ω s = p ω + R r L r L m i s q ψ r (38)

The compensator terms e s d and e s q are defined by:

{ e s d = ω s σ L s i s q + L m R r L r 2 ψ r e s q = − ω s σ L s i s d − L m L r p ω ψ r (39)

An ITSC fault detection algorithm is discussed in the paper. He cooperates with estimator of resistance. A general idea of a detection system is illustrated in

It assumes, that ITSC results in a sharp decrease of the estimated stator resistance. Based on this hypothesis, the detection method is proposed, which uses a derivative of the estimated stator resistance:

If | d d t R s e s t | ≥ ε r s then 1 else 0

where: ε r s threshold of a stator windings fault detection.

If the modulus of the derivative is equal of greater to the threshold the detector sends the logical 1, which indicates on occurring of ITSC.

Simulation tests were performed in the MATLAB/Simulink environment. The simulation results of the PQ-MRAS estimation of stator and rotor resistances with IHGO and the ITSC stator faults detection are presented in this section. In this work, the specific parameters of the induction motor used for simulations are given in

Model parameter | |
---|---|

Parameters | Values |

Output power | P n = 1.1 kW |

Stator voltage | U s = 220 V |

Stator resistance | R s = 9.8 Ω |

Rotor resistance | R r = 5.3 Ω |

magnetizing inductance | L m = 0.5 H |

global leakage inductance referred to the stator | L f = 0.04 H |

Inertia shaft | J = 12.5 × 10 − 3 kg ⋅ m 2 |

Pole number | p = 2 |

Number of stator turns per phase | N s = 464 |

Symbol | Value |
---|---|

K P R s | 10 |

K I R s | 20 |

K P R r | 2 |

K I R r | 0.25 |

For the model described with our proposed technique, the results of the two modes of PQ-MRAS estimator are presented. First of all, the two modes of operation of the PQ-MRAS simulator with IHGO are presented, followed by the simulation results obtained for each mode.

1) Operation modes of the PQ-MRAS simulator with IHGO

The two modes are:

- Mode 1: the open-loop mode takes independently of control structure and observer (

- Mode 2: the open-loop mode takes independently of control structure but dependent of observer (

2) Simulation results

In order to improve the PQ-MRAS estimator performance, tree groups of

simulations will be carried for each mode. For the first group, simulation is carried out considering constant load torque with variable speed of rotor. The second group consider the constant speed of rotor with variable load torque. In the third group, speed and load torque are constant but with variations of stator’s and rotor’s resistances for different values of adaptive mechanism. In open-loop mode, the stator and rotor resistances estimated are not used. The control structure and IHGO depend on the nominal values of stator and rotor resistances. The results are presented in Figures 8-10.

The results of

The results obtained by mode 2 show sensitivity to variations of the load torque taken as disturbances. This sensitivity is reflected in a very oscillating transient regime due to the behavior of the observer which no longer faithfully reconstructs the quantities that are used in the PQ-MRAS estimation during this laps of time. In addition, these results also show the robustness of the estimation using the observer with respect to a variation in load torque. Except for these transient periods, the two approaches (with and without a speed sensor) have almost equal estimated values.

Considering now the variations of the resistance of the rotor and of the stator simultaneously, the results of simulation for both resistances increased up to 150% of their nominal values (insert the value), which was related to the changes of

machine windings temperatures (see

Regarding the results from

1) Impact of ITSC Fault

The IM drive is considered to operate respectively in healthy and faulty mode. In order to observe the influence of the faults on the IM performances, IM starting at nominal voltage, with a sinusoidal three-phase balancing, takes place without load. The simulation for all the experiment is carried for a period of 10 seconds in the following conditions:

- At the time t = 1 s the machine is subjected to a nominal load torque T l = 5 N ⋅ m ;

- At the time t = 3 s, 4 s, 5 s, 6 s, 7 s and 8 s a short-circuit of N c c a = 2 , 3 , 4 , 5 , 6 and 7 windings are introduced on phase “a”.

From the results showing in

( T e ) variations give rise to the ITSC application that causes fluctuation in the rotor speed at high frequency.

2) ITSC faults detection and isolation

An ITSC in phase A of stator windings were simulated (moments of turns short and their amounts are marked on figures as arrows).

transients of the estimated stator resistance, signals from fault detector and stator current components. It can be observed, that the estimated stator resistance sharply decreases according to short circuits (

fault severity (

It is evident that the value of the stator resistance varies as a result of heating (or cooling) of the machines’ winding. It may result in incorrect performance of the fault detector. Therefore, other tests were carried out where a linear increase of the stator resistance was simulated up to 120% of the nominal value. Results are shown in

It can be noticed, that the derivative value doesn’t exceed assumed threshold. It can be concluded that proposed stator fault detector is immune to thermal effects which are occurred in the machine. A study of the data shows that the number of short-circuited turns and the difference between the phase current of the healthy and faulty machine are related by expression of the form:

N c c j = k ( i s j f a u l t y − i s j h e a l t h y ) (40)

where k = 40.128 is a coefficient of proportionality obtained from the results of the simulations.

Let’s point out some promising results obtained in the literature using the similar approaches. Bednarz and Dybkowski [

remarks can be considered (a) the authors completely ignored the rotor resistance; (b) the complexity of the algorithm, in addition the authors fail to reconstruct efficiently the speed of the IM. An emblematic comparison can be made by viewing the

The estimation of induction motor parameters is a major area of research for sensorless drive, condition monitoring and fault diagnosis. It is proposed in this work an alternative strategy to the use of a speed sensor in the implementation of the PQ-MRAS estimator in order to calculate the rotor and stator resistances of an induction motor. The simulation results obtained from MATLAB/Simulink platform showed that the PQ-MRAS estimator using interconnected high-gain observer gives very similar results to those using the speed sensor. The estimation errors in the cases of speed and load torque variation are almost identical. The variations in stator and rotor resistances influence the performance of the observer and lead to a wrong estimation of the rotor resistance. This problem of the robustness of the observer with respect to rotor and stator resistances in the case of PQ-MRAS estimator provides a perspective for future work so that the observer can definitively and efficiently replace the physical speed sensor in the implementation of PQ-MRAS estimator. The derivation approach is exploited for ITSC fault detection in IM in order to show the impact of such fault on IM behavior; healthy and faulty IM models are established. Different fault scenarios and operation conditions are studied. Simulation results obtained are presented to highlight the performance and validity of the developed scheme. The analysis of the results allows the identification of the affected phase (by the faults) and show the sensitivity to ITSC fault occurrence. The system relies on the hypothesis that stator windings faults results in a sharp change of internal parameters’ values of the motor, thus the observation and analysis of the results of the simulations allowed us to highlight the influence of the defects and thus giving the possibility of isolating them which deserves recognition as a profound contribution.

The authors declare no conflicts of interest regarding the publication of this paper.

Teguia, J.B., Kenne, G., Kammogne, A.T.S., Fouokeng, G.C. and Nanfak, A. (2021) The Detection of Inter-Turn Short Circuits in the Stator Windings of Sensorless Operating Induction Motors. World Journal of Engineering and Technology, 9, 653-681. https://doi.org/10.4236/wjet.2021.93046

PQ-MRAS

Active (P) and reactive (Q) power Model Reference Adaptive System

IM

Induction Motor

DRFOC

Direct Rotor Field Oriented Control

P

Active power

Q

Reactive power

u s

Stator voltage vector

i s

Stator current vector

i r

Rotor current vector

ϕ s

Stator electromagnetic flux vector

ϕ r

Rotor electromagnetic flux vector

ω

Rotational shaft speed

T e

Electromagnetic torque

T l

Load torque

R s

Stator resistance

R r

Rotor resistance

L s

Stator inductance

L r

Rotor inductance

L m

Magnetizing inductance

p

Number of pole pairs

J

Shaft inertia

P e s t

Estimation error of the active power

Q e s t

Estimation error of the reactive power

K P R s

Coefficient of the proportional term in the stator resistance adaptation mechanism

K I R s

Coefficient of the integral term in the stator resistance adaptation mechanism

K P R r

Coefficient of the proportional term in the rotor resistance adaptation mechanism

K I R r

Coefficient of the integral term in the rotor resistance adaptation mechanism