Comparative study of field-oriented control of an induction machine with inter-turn short-circuit in healthy and faulty cases

. The vector control of an induction machine is among the most used controls, the inter-turn short-circuit fault is a very common fault in this type of machine. Vector controllers use current regulators that are highly insensitive to internal disturbances and can therefore mask the onset of a short-circuit fault. In this article we propose simulations allowing to study the behavior of the indirect field-oriented control when an inter-turn short-circuit fault appears. The primary aim of this article is a comparative study of the effect of this type of defect on estimated quantities such as rotor flux, slip and electromagnetic torque. The results of the simulation show that these estimated quantities can be used as fault indicators instead of the stator currents. The results of this comparative study also show that the ripple of the estimated rotor flux is a very good indicator of this type of fault in this type of regulator-machine association.


Introduction
Different control strategies exist for variable speed drives of induction machines.Vector controls allow precise regulation of the electromagnetic torque and/or angular speed of these machines.This combination of controller and machine is useful in many practical applications which require high speed and torque dynamics.The behavior of this association can be disturbed when an inter-turn short-circuit fault appears, which is one of the most frequent faults where stator currents represent the main indicators.In vector controls, current regulators with hysteresis or of the proportional and integral type are used, allowing great insensitivity to internal disturbances and can thus mask the short-circuit fault.A fault of this type, once masked, can worsen and thus present a danger not only for the regulator-machine association, but also for all the loads driven.
The objective of this article is to study the behavior of this type of controller-machine association in the presence of this type of fault.A comparative study is presented to allow a contribution in the correct choice of the indicators of this type of faults.
The study of the behavior of the induction machine presenting an inter-turn short-circuit requires a dynamic model in the Park domain.We base this modeling on articles in which this model is also validated by experimental results.It was in 2002 that RM Tallam [1] proposed a model for this type of defect, this model was used by Qian Lu in 2011 [2] then by A. BERZOY in 2015.The latter had published two articles ( [3] in 2015 and [4] in 2017) based on the model of an autotransformer charged by a resistor materializing the short-circuit and making it possible to limit the fault current for experimental reasons.
In 2019, I. IDRISSI [5] exploited the work of Qian Lu (2011) [2] and applied it to the diagnosis of the inter-turn short-circuit fault of the stator in double supply of an asynchronous wind turbine generator.
Many recent articles deal with vector controls in the case of healthy induction machines.In 2019, Najib El Ouanjli published a first review article [6] on the various improved direct torque control (DTC) techniques, then a second article [7] by this author on the implementation of DTC control.
In our study, we are based on the article [8] published in 2019 by Fang Xie.This author proposes an optimal speed and torque control of asynchronous motors for electric cars in the field weakening region.In 2015 Luis Amezquita-Brooks [9] had published an article which allowed us to improve the behavior of the decoupling regulators of the internal loops which makes it possible to improve the dynamics of the electromagnetic torque and consequently the behavior of the external speed regulator.
Our idea is mainly inspired by the article [10] published by Mateusz Dybkowski in 2019, this author discussed the fault-tolerant control in an induction motor driver with an inter-turn short-circuit (ITSC) of the windings of the stator, which is one of the most common internal faults of induction machines.He also showed that classical state variable estimators are sensitive to changes in induction motor parameters during stator winding failure, resulting in instability of direct field-oriented control (DFOC).
Our idea is also inspired by a series of works ( [11] in 2018 and [12] 2019) published by A. BERZOY and which are oriented towards the analysis and diagnosis of the impact of inter-turn short-circuit in the stator of asynchronous machines driven by direct torque control.
This article is structured as follows: Section 2 presents the dynamic modelling of the induction machine (IM) with (ITSC).Section 3 presents the studied control strategy (FOC).Sections 4 and 5 present the results of the simulation of the behavior of this association and the comparative study of the different indicators in the case of a fixed nominal mechanical load.Section 6 presents the simulation results in the case of a variable mechanical load.Section 7 provides conclusions.

Dynamic model of the IM with ITSC
We consider a three-phase asynchronous machine supplied by a balanced three-phase sinusoidal voltage source Vsabc=[vsa vsb vsc] t that generates currents Isabc=[isa isb isc] t .In this machine, an ITSC fault is considered in the stator winding shown in Figure 1, where the short circuit is shown in phase A having a total of n turns.In this figure, the current if crosses the short-circuit of the damaged turns with µ representing the short-circuit rate.Articles [1][2][3][4][5], which assume µ≠0 to allow inversion of the Lf matrix, detail the Park transformation of the machine in the dq basis, the following equations model this type of fault located in phase A.
The matrix [W] depends on the position of the reference frame in the transformation dq considered, we choose a fixed reference frame dq linked to the stator to model the machine and a rotating reference frame dq linked to the rotor flux for the oriented rotor flux control.ws and wr are the stator and rotor pulses.
The expressions of the different matrices are as follow.
and  

FOC strategy
The control strategy is based on vector control with orientation of the rotor flux described by the block diagram of figure 2 below.In this control strategy we propose the context of an induction machine associated with a pulse width modulation inverter and a speed sensor.The position of the inter-turn short-circuit fault is located in the first phase of the stator; its rate is variable without exceeding a limit value assumed to be intolerable for this type of regulatormachine association.
The current and speed regulators are of the proportional and integral type.Torque control, flux control and voltage decoupling are made to allow indirect field-oriented control [13,14].The parameter estimation block is governed by equations 5 and 6 which give the estimation of flux and rotor slip.
The torque and flux control block is governed by the following equations 9 and 10. ) Where: Equations 12 and 13 give the control law of the Proportional and Integral current and speed regulators.Where: Kcp, Kci, Ksp and Ksi represent proportional and integral actions of the current and speed regulators.

Simulation results with fixed nominal mechanical load
In this part of the simulation, the context of a machine-regulator association is chosen, the main parameters of which are grouped together in the following table 1.The adjustment of the parameters of the various regulators is made by compensation of the poles according to the parameters of the machine itself [15].A reference speed is used to allow a progressive start to reach the nominal mechanical angular speed of the machine which is close to 150rad/s.The objective of this scenario is to be able to study the behavior of this regulator-machine association during an inter-turn shortcircuit fault.
The following figures represent the simulation results with a short-circuit rate of 10% introduced into the winding of phase A of the induction machine at time t=2s.This instant E3S Web of Conferences 469, 00066 (2023) ICEGC'2023 https://doi.org/10.1051/e3sconf/202346900066represents the limit between the behavior in the healthy state and in the faulty state.Figure 3 gives the shape of the normalized values with respect to the nominal value of the stator currents; after start-up and in the healthy state of the machine, these quantities remain balanced and have root mean square values close to 1.In the faulty state the current in phase A increases and depends on the short-circuit rate which also affects the currents in the other two phases.These quantities are always considered as fault indicators since they increase with the short-circuit rate.

Fig. 3. Behavior of stator currents.
Figure 4 below gives the behavior of the real and estimated electromagnetic torque in the two cases considered.In the healthy state, the two torques real and estimated are equal; in faulty state, there is a large ripple in the real torque around the value of the nominal load torque of 10Nm, which can be dangerous for some loads.The torque estimation equation used in the healthy case is no longer valid in the defective state, a large error appears between these two pairs.This error can be used as a fault indicator since it increases considerably with the shortcircuit rate.
Figure 5 shows a correct behavior in the healthy state followed by a bad behavior of the rotor flux orientation in the faulty state.The estimated rotor flux varies little with the shortcircuit rate and becomes a poor fault indicator.
Figure 6 shows a correct behavior of the mechanical speed with respect to the reference in the two healthy and defective cases, the estimated slip and synchronism speed quantities are sensitive to the introduced short-circuit fault.
Figure 7 represents the effect of a 10% short-circuit rate on the estimated slip.

Comparative study of fault indicators
The purpose of this study is to show the interest of certain fault indicators in the study of the behavior of this controller-machine association.The short-circuit fault is introduced into the machine in steady state with a variable rate of 1% to 20%.Table 2 below groups together the results of the various indicators likely to be used in the diagnosis of this type of association.
The following results are given in the case of a fixed nominal mechanical load.The 20% to 0% ratio column presents the variation of the indicator from a healthy state to a fault of 20%.In open loop the current in the faulty phase is higher than the other currents but in closed loop the three currents are closer which shows that this indicator becomes less interesting in this association of regulator-machine.

Fi2, Fi3 and Fi4 indicators
These indicators represent the average estimated electromagnetic torque, its peak-to-peak ripple and its ripple rate.The Fi3 indicator is the best among these three indicators and it exceeds the currents in its sensitivity to this type of fault.

Fi5, Fi6 and Fi7 indicators
These indicators represent the average estimated rotor flux, its peak-to-peak ripple and its ripple rate.Fi5 is of no interest for this type of fault but the Fi6 and Fi7 indicators are very sensitive to this type of fault.Fi6 is the best indicator among all other indicators for this type of fault.

Fi8, Fi9 and Fi10 indicators
These indicators represent the average estimated synchronism speed, its peak-to-peak ripple and its rate.Fi9 and Fi10 indicators have better sensitivity compared to Fi8.

Fi11, Fi12 and Fi13 indicators
These indicators represent the average estimated slip speed, its peak-to-peak ripple and its ripple rate.The Fi12 indicator has better sensitivity compared to Fi11 and Fi13.
In this comparative study, we note a great sensitivity to this type of fault in the indicators linked to the undulation of the estimated rotor flux, the average value of which remains insensitive to this type of fault.Figure 8 gives a graphical representation of the influence of the short-circuit rate on these different indicators.In this figure we wanted to divide all these indicators by their values corresponding to the healthy state of the machine.We did not want to take into consideration the stator currents because they are more useful in open loop and they intervene in closed loop in the estimation of the other indicators.

Effect of variable mechanical load
In this step, we are interested in the behavior of our regulator-machine in the presence of a load couple formed by a moment of inertia of 0.2kgm 2 and a dry friction of 10Nm disturbed by a sinusoidal undulation of 2Nm in amplitude and 30rad/s.It is in steady state, at instant 1s, that this charge will be introduced.The following figure 9 shows the effect of a variable mechanical load on the chosen fault indicator, it can be seen that the effect of the fault is masked by the effect of the mechanical load.The diagnosis of this type of fault must be made before the introduction of this real load after the engine has started with its own inertia, its own viscous friction and its own dry friction.

Conclusions
The simulation results demonstrated that the oriented rotor flux regulator of an induction machine is very effective in improving the dynamic behavior of torque and speed.In addition, the regulator settings, when selected, provide good performance over a wide operating range even in the event of an inter-turn short-circuit fault.
In this context, the stator currents become bad fault indicators.The estimated rotor flux ripple can be used to replace the stator currents for real-time diagnosis of the inter-turn shortcircuit fault.The fault indicators presented in this article remain useful in steady state on a nominal mechanical load with no ripple greater than 1% so as not to mask the ripple due to the fault.
This type of regulator-machine diagnosis allows preventive maintenance for inter-turn short-circuits not exceeding 20%.For higher short-circuit rates, this command cannot be launched and therefore a diagnosis at zero speed is essential before starting regulation.

Fig. 1 .
Fig. 1.Representation of the fault in phase A. Equations 1 and 2 relate the different voltages to the different fluxes and currents in the rotating frame dq.vf, if and Φf are respectively the voltage, current and flux in the short-circuit turns.vf=0, if , Φf and the [Rf] and [Lf] matrices depend on the position of the fault (phase A, B or C) and the short-circuit rate µ.The expression of the electromagnetic torque Tef represented by equation 3 shows that the short-circuit rate µ introduces a fault torque as indicated in articles[3][4][5].The mechanical equation 4 depends on the characteristic Tef (wr) which can be modeled by a moment of inertia J, an external load torque TL and a coefficient of friction kloss.

Fig. 2 .
Fig. 2. Block diagram of the FOC strategy.The voltage compensation block is governed by equations 7 and 8 which give the direct and quadratic voltage decoupling equations making it possible to dissociate the flux and torque control to obtain an induction machine model identical to that of the naturally decoupled direct current machine.The orientation of the rotor flux makes it possible to write: Φqr=0 and Φdr= Φr.

Table 1 .
Parameters of the controller-machine association.

Table 2 .
Comparison of different indicators.

Fi0 and Fi1 indicators
These indicators represent the RMS values of the three stator currents normalized with respect to the nominal current In respectively in open and closed loops.The simulation results show that the stator currents are much more sensitive to this type of fault in open loop than in closed loop.