The model used is that of [30], allowing the decoupling of two modes, the common and the differential modes. It is composed of an additional shorted winding in a three-phase axis. Figure 1 shows a three phase two-pole induction machine in case of a short-circuit winding at phase b. This fault induces in stator a new windings B c c short circuited and localized according to first phase by the angle θ c c = 2 π / 3   rad . θ c c k and η c c k are stator faults parameters where,

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 ).

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 Figure 4).

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 . [27] [28]. The dynamical equations used to describe this observer are [3]:

{ 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 [27] [28]:

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 [27] [28]:

{ 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 [27] [28].

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 [31]. In rotor flux-oriented strategy, the rotor flux orientation is defined as follows:

{ 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 Figure 5), the rotor flux ( ψ r ), the electromagnetic torque ( T e ) and the electrical synchronous stator ( ω s ) have to be estimated using the following equations:

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 Figure 6. The system is strictly related to the model-based fault detection approach [32] [33].

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.

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 (Figure 7(a));

- Mode 2: the open-loop mode takes independently of control structure but dependent of observer (Figure 7(b)).

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 Figure 8 shows that, during the variations of rotational speed the estimated resistances by Mode 2 (using IHGO) are the same with these estimated by Mode 1 (using speed sensor). In both cases, it is observed a maximum relative error of 0.33%. The explanation of this error can be found in the performance of the observer obtained by the reconstruction of the mechanical speed. From Figure 9, the results depict the estimations of stator and rotor resistances during the load torque variation. We clearly observe that, during the variations of load torque, the estimated resistances from Mode 2 (using IHGO) and Mode 1 (using speed sensor) shows the same loop in the steady states.

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 Figure 10).

Regarding the results from Figure 10, the following observations are made; the PQ-MRAS estimator using the speed sensor estimates well the nominal values of the stator and of the rotor resistances taking into account their variations [3]. Similar performance is obtained with the PQ-MRAS estimator using IHGO for stator resistance estimation but not true in the case of rotor resistance. The technique proposed by Bednarz [3] remains better in this case.

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 Figure 11 and Figure 12, for the three phase stator current ( i s a , i s b , i s c ) , it is observed that the stator current becomes unbalanced with the current intensity in phase “a” bigger than those in phases “b” and “c”. This increase occurs following a direct circular permutation. We also noticed that the greater the number of short-circuited turns, the greater the increase in currents. In addition to the current imbalance, we also notice an imbalance in the phase shifts shown in Figure 12. This phase shift is no longer equal to 120˚ (healthy operation). The curves Figure 13 shows oscillations of the short-circuit currents ( i c c a , i c c b , i c c c ) which increase when the ITSC fault increases. From Figure 14 and Figure 15 respectively the rotor speed ω and electromagnetic torque

( 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). Figure 16 and Figure 17 shows

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 (Figure 16(a)). Consequently, it results in the stator’s fault detector activation (Figure 16(c)) because the threshold value ε R s is exceeded by the modulus of estimated resistance derivative. The stator current increases due to ITSC and the amplitude level depends on the

fault severity (Figure 17).

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 Figure 16.

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. Figure 18 illustrates the reconstruction of the number of short-circuited turns.