Simplified Model of a Three-Phase Induction Motor for Fault Diagnostic Using the Synchronous Reference Frame DQ and Parity Equations

Proceedings, 2nd IFAC Conference on Proceedings, 2nd IFAC Conference Modelling, Identification and Controlon of Nonlinear Systems Proceedings, 2nd IFAC Conference on Available online at www.sciencedirect.com Modelling, Identification and Control of Nonlinear Systems Proceedings,Mexico, 2nd IFAC IFAC Conference on Guadalajara, June 20-22, 2018 Proceedings, 2nd Conference on Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Modelling, Identification and Control of Nonlinear Systems Modelling, Identification and Control of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Guadalajara, Guadalajara, Mexico, Mexico, June June 20-22, 20-22, 2018 2018

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IFAC PapersOnLine 51-13 (2018) 662–667

Simplified Simplified Model Model of of a a Three-Phase Three-Phase Induction Induction Motor Motor for for Fault Fault Diagnostic Diagnostic Using Using the the Simplified Model of a Three-Phase Induction Motor for Fault Diagnostic Using the Synchronous Reference Frame DQ and Parity Equations Simplified Model of a Three-Phase Induction Motor for Fault Diagnostic Using Simplified Model of a Three-Phase Motor for Fault Diagnostic Using the the Synchronous ReferenceInduction Frame DQ and Parity Equations Synchronous Reference Frame DQ and Parity Equations Synchronous Reference Frame Parity Equations Synchronous Reference Frame DQ DQ and Parity Equations Edgar Chulines, Marco A. Rodríguez, Iván and Duran, Rafael Sánchez

Edgar Chulines, Marco A. Rodríguez, Iván Duran, Rafael Sánchez Edgar Marco A. A. Rodríguez, Rodríguez, Iván Rafael Edgar Chulines, Chulines, Marco Iván Duran, Duran, Rafael Sánchez Sánchez Edgar Chulines, Marco A. Rodríguez, Iván Duran, Rafael Sánchez Universidad Autónoma del Carmen UNACAR, Departamento de Ingeniería Electrónica, Ciudad Universidad Autónoma del Carmen UNACAR, Departamento de Ingeniería Electrónica, Ciudad del Carmen Campeche México. C.P 24155 Universidad Autónoma del Carmen UNACAR, Departamento Ingeniería Electrónica, Ciudad Carmen Campeche México. C.Pde Universidad Carmen UNACAR, Departamento de Ingeniería Universidad Autónoma Autónoma del deldel Carmen UNACAR, Departamento de24155 Ingeniería Electrónica, Electrónica, Ciudad Ciudad del Carmen Campeche México. C.P 24155 E-mail: [email protected], [email protected] del Carmen Campeche México. C.P 24155 del Carmen Campeche México. C.P 24155 E-mail: [email protected], [email protected] E-mail: [email protected], [email protected] E-mail: E-mail: [email protected], [email protected], [email protected] [email protected] Abstract: This work is a simplified analysis of a nonlinear model for a three-phase induction motor in Abstract: This work is a simplified analysis of a nonlinear model for a three-phase induction motor in steady stateThis behavior toisdiagnose of additive andofparametric faults. Multiple faults are also considered using Abstract: work aa simplified analysis aa nonlinear model for aa three-phase induction motor in steady stateThis behavior tois diagnose of additive andof parametric faults. Multiple faults are also considered using Abstract: work simplified analysis nonlinear model for three-phase induction motor in Abstract: This work is a simplified analysis of a nonlinear model for a three-phase induction motor in the parity equation approach and some logical assumptions. The simplified analysis of the nonlinear model steady state behavior to diagnose of additive and parametric faults. Multiple faults are also considered using the parity equation approach and some logical assumptions. The simplified analysis of the nonlinear model steady state behavior to diagnose of additive and parametric faults. Multiple faults are also considered using steady state behavior to diagnose of parametric Multiple faults areofmotor, also considered using in aparity synchronous reference frame isadditive used andand matched withfaults. the linear model of d.c. then the model parity the equation approach and some logical assumptions. The simplified analysis the nonlinear in synchronous frame is used and matched with The the linear model of d.c.of then the model parity theaparity parity equationreference approach and the some logical assumptions. The simplified analysis ofmotor, thethe nonlinear model the equation approach and some logical assumptions. simplified analysis the nonlinear space is determined, and finally, residual equations are obtained. The performance of fault the detection in a synchronous reference frame is used and matched with the linear model of d.c. motor, then space is determined, and finally, the residual equations are obtained. The performance of the fault detection in aa synchronous synchronous reference frame is used used and matched matched with with the the linear linear model model of of d.c. d.c. motor, motor, then then the the parity parity in frame is and parity system isdetermined, validatedreference byand using the Matlab software. space is finally, the residual equations are obtained. The performance of the fault detection system isdetermined, validated byand using the Matlab software. space is finally, the residual equations are obtained. The performance of the fault detection space is determined, and finally, the residual equations are obtained. The performance of the fault detection system validated by usinginduction the Matlab Keywords: Fault diagnosis, motor, parity equation, modelling, residues, insulation © 2018,is (International Federation ofsoftware. Automatic Control) Hosting by Elsevier Ltd. All rights.. reserved. system is validated by the software. Keywords: Fault diagnosis, motor, parity equation, modelling, residues, insulation system isIFAC validated by using usinginduction the Matlab Matlab software. Keywords: Fault diagnosis, diagnosis, induction motor, motor, parity equation, equation, modelling, residues, residues, insulation.. Keywords: Keywords: Fault Fault diagnosis, induction induction motor, parity parity equation, modelling, modelling, residues, insulation insulation. 1. INTRODUCTION the Human-Machine Interface (HMI) is commonly expensive.  1. INTRODUCTION the Human-Machine Interface (HMI) is commonly expensive.  Although only the early detection time is not very necessary 1. INTRODUCTION the Human-Machine Interface is The Induction motor (IM) has been widely used in the industry Although only the early detection time is not veryexpensive. necessary 1. INTRODUCTION INTRODUCTION the Human-Machine Interface (HMI) (HMI) is commonly commonly expensive. 1. Human-Machine Interface (HMI) is commonly expensive. for incipient failures. The Induction motor (IM) has been widely used in the industry the Although only the early detection time is not very due to its simple construction, reliability, ruggedness, low cost, for incipient failures. Although only only the the early early detection detection time time is is not not very very necessary necessary The Induction motor (IM) has been widely used in the industry Although necessary due to its simple construction, reliability, ruggedness, low cost, The Induction motor (IM) has been widely used in the industry for incipient failures. The Induction motor (IM)Additionally, has reliability, been widely used in the industry and many applications. inruggedness, contrast tolow thecost, d.c. Also, an Adaptive Neuronal Network (ANN) online technique for incipient failures. due to its simple construction, for incipient failures. and many applications. Additionally, inruggedness, contrast tolow thecost, d.c. Also, an Adaptive Neuronal Network (ANN) online technique due to its reliability, due to its simple construction, reliability, ruggedness, low cost, motor, it simple can beconstruction, used in volatile or aggressive environment to only detect bearings fault has been used, Nandi et al. (2005, and many applications. in contrast to the d.c. Also, Adaptive Neuronal Network (ANN) online volatile or aggressive environment motor, it can be used inAdditionally, onlyan bearings fault has been used, Nandi et technique al. (2005, and many applications. Additionally, in contrast to the d.c. to Also, andetect Adaptive Neuronal Network (ANN) online technique and many applications. Additionally, in contrast to the d.c. Also, an Adaptive Neuronal Network (ANN) online technique since there are be no used problems with corrosion and spark, although with this technique seven faults has been detected but in offline motor, it can in volatile or aggressive environment to only detect bearings fault has been used, Nandi et (2005, since there are no problems with corrosion and spark, although with this technique seven faults has been detected but in offline motor, it can can be be used tends in volatile volatile or aggressive aggressive environment to onlyKolla detectetbearings bearings faultHowever, has been been the used, Nandi et al. al. (2005, motor, it used in or environment to only detect fault has used, Nandi et al. (2005, the reliability degree to decrease when the operating way, al. (2000). two previous works since there are no problems with corrosion and spark, although with this technique seven faults has been detected but in offline the degree tendswith to decrease when the operating way, Kolla et al. (2000). However, thedetected two previous works sincereliability there are are no problems with corrosion and spark, spark, although with this this technique seven faults has been been detected but in offline offline since there no problems corrosion and although with technique seven faults has but in conditions are extreme, which is magnified when the IM result in a high cost in storage, recursive operations, the reliability degree tends to decrease when the operating way, Kolla et al. (2000). However, the two previous works conditions are degree extreme, which is magnified when the IM result in aet cost in storage,the operations, the reliability to when the operating way, Kolla al. However, two previous works the reliability degree tends toa decrease decrease when the operating way, Kolla et high al. (2000). (2000). However, therecursive two delay. previous works operates a critical load,tends where failure could bewhen a risk to staff computational complexity andstorage, response time Although conditions are extreme, which is magnified the IM result in a high cost in recursive operations, operates a critical load, where a failure could be a risk to staff computational complexity and response time delay. Although conditions are extreme, which is magnified when the IM result in a high cost in storage, recursive operations, conditions are extreme, which is magnified when the IM result in a high cost in storage, recursive operations, safety, the environment and economy. for critical applications the first and second disadvantages are operates load, where aa failure could be aa risk to staff computational complexity response time delay. Although safety, theaaa critical environment and economy. for critical applications theand firstfault and tolerant second disadvantages are operates critical load, where failure could be risk to staff computational complexity and response time delay. Although operates critical load, where a failure could be a risk to staff computational complexity and response time delay. Although not important. In addition, for systems the third safety, the environment and economy. for critical applications the first and second disadvantages are A program of maintenance and specific attention must be not important. In addition, systems the third safety, the and critical applications the first and second are safety, the environment environment and economy. economy. for critical applications the for firstfault and tolerant second disadvantages disadvantages are drawback cannot be despised A program of maintenance and specific attention must be for not important. In addition, for fault tolerant systems the third given to be able to have a correct operation of the motors. This drawback cannot be despised not important. In addition, for fault tolerant systems the third A program of maintenance and specific attention must be not important. In addition, for fault tolerant systems the third given to be able to have a correct operation of the motors. This A program of maintenance and specific attention must be be despised A program of maintenance andmonitoring specific of attention mustThis be drawback can be done through permanent to detect incipient The stator cannot winding fault is one of the electrical faults that has drawback cannot befault despised given be able to have aa correct operation motors. drawback be despised can beto done through permanent monitoring tothe detect incipient The stator cannot winding is one of the electrical faults that has given to be able to have correct operation of the motors. This given to be able to have a correct operation of the motors. This faults and prevent them. Generally, the most frequent faults in reported short detection delay time using model-based can be done through permanent monitoring detect incipient The stator winding fault is one of the electrical that has faults and prevent them. Generally, the mostto frequent faults in reported short detection delay time using faults model-based can be done through permanent monitoring to detect incipient The stator winding fault is one of the electrical faults that has can be done through permanent monitoring to detect incipient The stator winding fault is one of the electrical faults that has IM are mechanical and these are related to the electrical methods, like the Winding Function Approach (WFA) faults and prevent them. Generally, the most frequent faults in reported short detection delay time using model-based IM are mechanical and these are related to the electrical methods, like the Winding Function Approach (WFA) faults and prevent them. Generally, the most frequent faults in reported short detection delay time using model-based faults and prevent them. Generally, the most frequent faults in reported short detection delay time using model-based operation of the motor such as local overheating and inter-turn Bianchini et al. (2011). Another method based on the model is IM are mechanical and are related to the electrical methods, like the Winding Function Approach (WFA) operation of the motor suchthese as Pineda-Sanchez local overheating and inter-turn Bianchini et al. (2011). Another method based on the to model is IM are mechanical and these are related to the electrical methods, like the Winding Function Approach (WFA) IM are mechanical and these are related to the electrical methods, like the Winding Function Approach (WFA) short-circuit stator winding et al. (2011). the use of parity equations, which may be adequate detect operation of motor such local overheating Bianchini al. (2011). Another method based on model is short-circuit stator winding Pineda-Sanchez etand (2011). the use faults, of et parity equations, which maygeneral be adequate detect operation of the the motor such as asin local overheating andal.inter-turn inter-turn Bianchini et al.but (2011). Another method based on the the to model is operation of the motor such as local and inter-turn et al. (2011). Another method based on the model is These faults cause changes theoverheating basic parameters of the Bianchini several the residuals of the parity equation short-circuit stator winding Pineda-Sanchez et al. (2011). the use of parity equations, which may be adequate to detect These faults cause changes in the basic parameters of the several faults, but the residuals of the general parity equation short-circuit stator winding Pineda-Sanchez et al. (2011). the use of parity equations, which may be adequate to detect short-circuit stator winding Pineda-Sanchez et al. (2011). the use of parity equations, which may be adequate to detect motor faults such as resistances and inductances; and can be several must befaults, calculated and an accurate mathematic model of the These cause changes in basic of the but the of the general parity equation motor such as resistances andthe and can be must bemust calculated andresiduals anChan accurate mathematic model of the These faults faults cause changes in theinductances; basic parameters parameters of and the several faults, but the residuals of the general For parity equation These cause changes in the basic parameters of the several faults, the residuals of the general parity diagnosed in aas timely manner by determining the type, size system bebut obtained et al. (2006). thisequation reason, motor such resistances and inductances; and can be must be calculated and an accurate mathematic model of the diagnosed in aas manner by determining the type, be obtained et al.mathematic (2006). Formodel this reason, motor such resistances and inductances; and can be must be calculated and an accurate of motor such astimely resistances and inductances; and size can and be system must bemust calculated and anChan accurate mathematic model of the the location ofinthe fault asmanner well asby itsdetermining time of detection. this has mainly been applied to linear systems, where accurate diagnosed a timely the type, size and system must be obtained Chan et al. (2006). For this reason, location ofin the fault asmanner well asby itsdetermining time of detection. this has must mainly been applied to linear systems, where accurate diagnosed aa timely the type, size and system be obtained Chan et al. (2006). For this reason, diagnosed in timely manner by determining the type, size and system must be obtained Chan et al. (2006). For this reason, models are more readily available. However, since accurate location of the fault as well as its time of this has mainly been applied to linear systems, where accurate The mechanical vibration analysis is detection. the most popular models are more readily available. However, since location of as its has mainly been applied to linear systems, where accurate location of the the fault faultvibration as well well as asanalysis its time time of of detection. this has for mainly been applied to are linear systems, where accurate models nonlinear systems more difficult to obtain in The mechanical is detection. the most popular this models are more readily available. However, since accurate technique to detectvibration faults in the IM, Bianchini et al. popular (2011), models for nonlinear systems are more difficult to obtain in are more readily available. However, since accurate The mechanical analysis is the most models are more readily available. However, since accurate practice, in addition to the fact that the parity equation is technique to detect faults in the IM, Bianchini et al. (2011), The ismechanical mechanical vibration magnitudes analysis is is and thethemost most popular models for systems are more to obtain in The vibration analysis the popular this due to to detect the significant immunity to practice, in nonlinear addition to the uncertainties, fact that thedifficult parity equation is models for nonlinear systems are more difficult to obtain in technique faults in the IM, Bianchini et al. (2011), models for nonlinear systems are more difficult to obtain in sensitive to noise and model the parity equation this is due to the significant magnitudes and the immunity to technique to detect faults in the IM, Bianchini et al. (2011), practice, in addition the fact that parity equation is technique to detect faults inelectromagnetic the IM, Bianchini etimmunity al. (2011), external phenomena like the interference over sensitive to noise and to model uncertainties, the parity equation practice, ineasily addition to the fact that the the parity equation is this is due to the significant magnitudes and the to practice, in addition to the fact that the parity equation is cannot beto applied to uncertainties, nonlinear systems, Chan et al. external phenomena like the electromagnetic interference over this is due to the significant magnitudes and the immunity to sensitive noise and model the parity equation this is due to the significant magnitudes and the immunity to sensors that commonly are accelerometers; however, cannot be easily applied to nonlinear systems, Chan et al. sensitive to noise and model uncertainties, the parity equation external phenomena like the electromagnetic interference over sensitive to noise and model uncertainties, the parity equation (2006). be easily applied to nonlinear systems, Chan et al. sensors that commonly are accelerometers; however, external phenomena like electromagnetic interference over cannot external phenomena like the electromagnetic interference over (2006). accelerometers have verythe limited operation ranges. cannot be be easily easily applied applied to to nonlinear nonlinear systems, systems, Chan Chan et et al. al. sensors that commonly are accelerometers; however, cannot accelerometers have very limited operation ranges. sensors that commonly are accelerometers; however, sensors that have commonly are operation accelerometers; however, (2006). Generally, the diagnostic procedure is based on the heuristic (2006). accelerometers very limited ranges. (2006). Another technique widely used for the detection of mechanical Generally, the is based on the heuristic accelerometers have very ranges. accelerometers have very limited limited operation ranges. of diagnostic the processprocedure and observed analytical heuristic Another technique widely used foroperation the detection of mechanical knowledge the is based on heuristic and electrical faults in theused IMfor is the thedetection phase current spectral Generally, knowledge of diagnostic the processprocedure and observed analytical heuristic Generally, the diagnostic procedure is based on the the Another technique widely of mechanical Generally, the diagnostic procedure is based on the heuristic symptom, commonly represented in a signal called residual. In and electrical faults in the IM is the phase current spectral Another technique widely used for the detection of mechanical knowledge of the process and observed analytical heuristic Another technique widely used for the detection of mechanical analysis, Loparo et al. (2000) and Strangas et al. (2008). Other symptom, commonly represented in a signal called residual. In knowledge of the the process and knowledge observed analytical analytical heuristic and electrical faults in the IM is the phase current spectral knowledge of process and observed heuristic these last two cases, a priori of fault symptom analysis, Loparo et al. (2000) and Strangas et al. (2008). Other and electrical faults in the IM is the phase current spectral symptom, commonly represented in a signal called residual. In and electrical faults in(2000) theonIM isStrangas themotor phase current spectral techniques are based the phase current these last commonly two cases, a large prioridata knowledge of faultresidual. symptom symptom, commonly represented in signal called residual. In analysis, Loparo et al. and et al. (2008). Other symptom, represented in aa signal called In causalities, as well as, a bank is necessary, Isermann techniques are based on the motor phase current analysis, Loparo et al. (2000) and Strangas et al. (2008). Other these last two cases, a priori knowledge of fault symptom analysis, Loparo et al. (2000) Strangas et al.phase (2008). Other causalities, transformations, Mendes eton al.and (2006). Others works evaluate as well as, aaalarge data bank is necessary, Isermann these last two cases, priori knowledge of fault symptom techniques are based the motor current these last two cases, priori knowledge of fault symptom et al. (1995). However, in the diagnosis by using analytical transformations, eton al. (2006). Others phase works evaluate techniques are based the motor current causalities, as well as, a large data bank is necessary, Isermann techniques are Mendes based on the Cash motor phase current the neutral terminal voltage of(2006). IM, et al. (1998) or by et al. (1995). However, in the diagnosis by using analytical causalities, as well as, large data bank is Isermann transformations, Mendes et al. Others works evaluate causalities, as well as, aa values large data bank is necessary, necessary, Isermann symptoms with limit of measurement signals and the neutral terminal voltage of IM, Cash et al. (1998) or by transformations, Mendes et al. (2006). Others works evaluate et al. (1995). However, in the diagnosis by using analytical transformations, Mendes et al. (2006). Others works evaluate calculating system impedances, Klima et alet(2003). symptoms with limit values of measurement signals and et al. (1995). However, in the diagnosis by using analytical the neutral terminal voltage of IM, Cash al. (1998) or by et al. (1995). However, in the diagnosis by using analytical change detection have a minimal mathematical effort and calculating system impedances, Klima et al (2003). the neutral neutral terminal terminal voltage voltage of of IM, IM, Cash Cash et et al. al. (1998) (1998) or or by by change symptoms with limit values of measurement signals and the detection have a allows minimal mathematical effort symptoms with limit values of measurement signals and calculating system impedances, Klima et al (2003). symptoms with limit values of measurement signals and without data bank, which the simplicity and reliability The general problem of all these techniques is that the delay calculating system impedances, Klima et al al (2003). (2003). change data detection minimal and calculating impedances, et bank, have whichaaa allows the mathematical simplicity andeffort reliability The generalsystem problem of all theseKlima techniques is that the delay without detection have minimal mathematical effort and change detection have minimal mathematical effort and time of detection and localization of the fault is is that verythe longdelay and change without data bank, which allows the simplicity and reliability The general problem of all these techniques time of detection and localization of the fault very long and without data bank, which allows the simplicity and reliability The general problem of all these techniques is that the delay without data bank, which allows the simplicity and reliability The general problem of all these techniques is that the delay time and of the very and time of of detection detection and localization localization of the fault faultofis isAutomatic very long longControl) and Hosting by Elsevier Ltd. All rights reserved. time of detection and localization of the fault is very long and 2405-8963 © 2018, (International Federation Proceedings, 2nd IFAC IFAC Conference on 662 Proceedings, 2nd responsibility IFAC Conference on 662Control. Peer reviewIdentification under of International Federation of Automatic Modelling, and Control of Nonlinear Proceedings, 2nd IFAC Conference 662 10.1016/j.ifacol.2018.07.356 Modelling, Identification and Controlon of Nonlinear Proceedings, 2nd IFAC Conference on 662 Systems Proceedings, 2nd IFAC Conference 662 Modelling, Identification and Controlon of Nonlinear Systems Modelling, Identification and Control of Nonlinear Guadalajara, Mexico, June 20-22, 2018

2018 IFAC MICNON Guadalajara, Mexico, June 20-22, 2018

Edgar Chulines et al. / IFAC PapersOnLine 51-13 (2018) 662–667

663

From these equations relating the rotor in the axes dq of the magnetic flow links are obtained using the following equations in the stator voltages.

when a relative large change of their feature is obtained, Isermann et al (1997).

𝑖𝑖𝑞𝑞𝑞𝑞 = −

In this paper, a new approach is presented not only to detect faults, but to isolate faults for the nonlinear model of IM, matching it with the linear model of a d.c. motor considering the performance during the steady state behaviour. A set of simplified equations is obtained, that allow an easy implementation using modest digital processors or HMIs device, such as Arduino or Labview, respectively.

𝑖𝑖𝑑𝑑𝑑𝑑 =

𝛹𝛹𝑟𝑟 𝐿𝐿𝑟𝑟

𝑖𝑖 𝐿𝐿𝑟𝑟 𝑞𝑞𝑞𝑞

𝐿𝐿𝑚𝑚

−

(3)

𝑖𝑖 𝐿𝐿𝑟𝑟 𝑑𝑑𝑑𝑑

𝐿𝐿𝑚𝑚

(4)

Substituting the currents of the rotor (3) and (4) in (1) and (2) it obtained the following expressions 𝑉𝑉𝑞𝑞𝑞𝑞 = (𝑅𝑅𝑠𝑠 + 𝜎𝜎𝐿𝐿𝑠𝑠 𝜌𝜌)𝑖𝑖𝑞𝑞𝑞𝑞 + 𝜎𝜎𝜎𝜎𝑠𝑠 𝐿𝐿𝑠𝑠 𝑖𝑖𝑑𝑑𝑑𝑑 + 𝜔𝜔𝑠𝑠

2. NONLINEAR MODELLING OF INDUCTION MOTOR IN DQ REFERENCE FRAME

𝑉𝑉𝑑𝑑𝑑𝑑 = (𝑅𝑅𝑠𝑠 + 𝜎𝜎𝐿𝐿𝑠𝑠 𝜌𝜌)𝑖𝑖𝑑𝑑𝑑𝑑 − 𝜎𝜎𝜎𝜎𝑠𝑠 𝐿𝐿𝑠𝑠 𝑖𝑖𝑑𝑑𝑑𝑑 +

𝐿𝐿𝑚𝑚 𝐿𝐿𝑟𝑟

𝐿𝐿𝑚𝑚 𝐿𝐿𝑟𝑟

𝛹𝛹𝑟𝑟

𝜌𝜌 𝛹𝛹𝑟𝑟

(5) (6)

The IM is usually connected to an inverter for speed control applications. However, in the industry there are more noncritical applications where only constant operation in steady state is necessary. In this way, a steady state analysis for fault diagnosis in IM synchronous is important.

Where 𝜎𝜎 is the leakage coefficient and is obtained when the flow component produced by the stator current is constant in the stable state. So that the derivatives of the stator current in the d axis in the synchronous frame are.

The starting point to perform the analysis of the model of the IM in the synchronous reference frame is to initially deduce the transfer functions in the subsystems such as the mechanical and electrical part. For the mechanical and electrical parts to be coupled, there is a link between the current produced by the torque and the induced magnetic force.

The torque component produced by the stator current is the current in the q axis in the synchronous frame.

𝑖𝑖𝑓𝑓 = 𝑖𝑖𝑑𝑑𝑑𝑑

𝑝𝑝 · 𝑖𝑖𝑑𝑑𝑑𝑑 = 0

𝑖𝑖 𝑇𝑇 = 𝑖𝑖𝑞𝑞𝑞𝑞

It is also known that the rotor flow is given by. 𝛹𝛹𝑟𝑟 = 𝐿𝐿𝑚𝑚 𝑖𝑖𝑓𝑓

This link is implicit within the block diagram where the total current loop in the interior is independent of the mechanical part in the transfer function, Krishnan et al. (2001).

Replacing these values in the voltage equation. B dq idr ids if iqr iqs iT Is J Lm Lr Ls Rs Vqs Vds λr σ

 ωr ωs

Friction coefficient Park transformation dq d axis rotor current in the synchronous frame d axis stator current in the synchronous frames Flux-producing component of the stator current q axis rotor current in the synchronous frame q axis stator current in the synchronous frame Torque producing component of the stator current Stator current Moment of inertia Magnetizing inductance Rotor inductance Stator inductance Stator resistance q axis stator voltage in the synchronous frame d axis stator voltage in the synchronous frame Rotor flux linkage phasor Leakage coefficient Derivate Rotor electrical speed Slip speed

Where 𝐿𝐿𝑎𝑎 is.

𝐿𝐿𝑎𝑎 = 𝜎𝜎𝐿𝐿𝑠𝑠 = (𝐿𝐿𝑠𝑠 −

With 𝑝𝑝 is a derivate

𝜔𝜔𝑠𝑠1 =

)

(8)

( )

𝑖𝑖𝑇𝑇 𝑅𝑅𝑟𝑟 𝑖𝑖𝑓𝑓

𝐿𝐿𝑟𝑟

(9)

The equation of the electric part of the motor is obtained by substituting 𝜔𝜔𝑠𝑠 in equation (2).

𝑉𝑉𝑞𝑞𝑞𝑞 = (𝑅𝑅𝑠𝑠 +

𝑅𝑅𝑟𝑟 𝐿𝐿𝑠𝑠 𝐿𝐿𝑟𝑟

+ 𝐿𝐿𝑎𝑎 𝜌𝜌) 𝑖𝑖 𝑇𝑇 + 𝜔𝜔𝑟𝑟 𝐿𝐿𝑠𝑠 𝑖𝑖𝑓𝑓

(10)

=

(11)

Where the torque produced by the stator current can be derived from the following equation. 𝐼𝐼𝑇𝑇 =

𝑉𝑉𝑞𝑞𝑞𝑞 −𝜔𝜔𝑟𝑟 𝐿𝐿𝑠𝑠 𝑖𝑖𝑓𝑓

𝑅𝑅 𝐿𝐿 𝑅𝑅𝑠𝑠 + 𝑟𝑟 𝑠𝑠 +𝐿𝐿𝑎𝑎 𝜌𝜌

Where

𝐿𝐿𝑟𝑟

𝑅𝑅𝑎𝑎 = 𝑅𝑅𝑠𝑠 +

Then, the stator equations are

𝑉𝑉𝑑𝑑𝑑𝑑 = (𝑅𝑅𝑠𝑠 + 𝐿𝐿𝑠𝑠 𝜌𝜌)𝑖𝑖𝑑𝑑𝑑𝑑 − 𝜔𝜔𝑠𝑠 𝐿𝐿𝑠𝑠 𝑖𝑖𝑞𝑞𝑞𝑞 + 𝐿𝐿𝑚𝑚 𝜌𝜌𝑖𝑖𝑑𝑑𝑑𝑑 − 𝜔𝜔𝑠𝑠 𝐿𝐿𝑚𝑚 𝑖𝑖𝑞𝑞𝑞𝑞

𝐿𝐿𝑟𝑟

𝜔𝜔𝑠𝑠 = 𝜔𝜔𝑟𝑟 + 𝜔𝜔𝑠𝑠1 = 𝜔𝜔𝑟𝑟 +

𝑝𝑝𝛹𝛹𝑟𝑟 = 0

𝑉𝑉𝑞𝑞𝑞𝑞 = (𝑅𝑅𝑠𝑠 + 𝐿𝐿𝑠𝑠 𝜌𝜌)𝑖𝑖𝑞𝑞𝑞𝑞 + 𝜔𝜔𝑠𝑠 𝐿𝐿𝑠𝑠 𝑖𝑖𝑑𝑑𝑑𝑑 + 𝐿𝐿𝑚𝑚 𝜌𝜌𝑖𝑖𝑞𝑞𝑞𝑞 + 𝜔𝜔𝑠𝑠 𝐿𝐿𝑚𝑚 𝑖𝑖𝑑𝑑𝑑𝑑

𝑖𝑖 𝑇𝑇 𝑅𝑅𝑟𝑟 ( ) 𝑖𝑖𝑓𝑓 𝐿𝐿𝑟𝑟

𝐿𝐿𝑚𝑚 2

(7)

Now the stator frequency is represented by.

The starting point for the deduction of the transfer functions applied to the IM, is to take as constant the flow link in the rotor. 𝛹𝛹𝑟𝑟 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶

𝐿𝐿𝑚𝑚 2 𝑖𝑖 𝐿𝐿𝑟𝑟 𝑓𝑓 = (𝑅𝑅𝑠𝑠 + 𝐿𝐿𝑎𝑎 𝜌𝜌)𝑖𝑖 𝑇𝑇 + 𝜔𝜔𝑠𝑠 𝐿𝐿𝑠𝑠 𝑖𝑖𝑓𝑓

𝑉𝑉𝑞𝑞𝑞𝑞 = (𝑅𝑅𝑠𝑠 + 𝐿𝐿𝑎𝑎 𝜌𝜌)𝑖𝑖 𝑇𝑇 + 𝜔𝜔𝑠𝑠 𝐿𝐿𝑎𝑎 𝑖𝑖𝑓𝑓 + 𝜔𝜔𝑠𝑠

NOMENCLATURE

(1)

𝐿𝐿𝑠𝑠

𝐿𝐿𝑟𝑟

𝑅𝑅𝑟𝑟

𝐾𝐾𝑎𝑎

1+𝑠𝑠𝑇𝑇𝑎𝑎

(𝑉𝑉𝑞𝑞𝑞𝑞 − 𝜔𝜔𝑟𝑟 𝐿𝐿𝑠𝑠 𝑖𝑖𝑓𝑓 )

𝐾𝐾𝑎𝑎 =

1

𝑅𝑅𝑎𝑎

𝑇𝑇𝑎𝑎 =

𝐿𝐿𝑎𝑎

𝑅𝑅𝑎𝑎

From this block, which converts the feedback voltage to torque current, then electromagnetic torque is written.

(2)

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2018 IFAC MICNON 664 Guadalajara, Mexico, June 20-22, 2018

𝜏𝜏𝑒𝑒 = 𝐾𝐾𝑓𝑓 𝑖𝑖 𝑇𝑇

Edgar Chulines et al. / IFAC PapersOnLine 51-13 (2018) 662–667

−𝑅𝑅𝑎𝑎 𝐿𝐿𝑎𝑎 𝐴𝐴 = 𝐾𝐾𝑡𝑡 𝑁𝑁𝑝𝑝 [ 𝐽𝐽

Where

(12)

Where the constant of torque is definite. 𝐾𝐾𝑓𝑓 =

3 𝑃𝑃 𝐿𝐿𝑚𝑚 2 2 2 𝐿𝐿𝑟𝑟

𝑖𝑖𝑓𝑓

(13)

The dynamic loads can be represented taking the electromagnetic torque and the torque of the load which is considered as friction in this particular case. + 𝐵𝐵𝜔𝜔𝑚𝑚 = 𝜏𝜏𝑒𝑒 − 𝜏𝜏𝐿𝐿 = 𝐾𝐾𝑓𝑓 𝑖𝑖 𝑇𝑇 − 𝐵𝐵𝑙𝑙 𝜔𝜔𝑚𝑚

(14)

+ 𝐵𝐵𝜔𝜔𝑟𝑟 = 𝐾𝐾𝑓𝑓 𝑖𝑖 𝑇𝑇 − 𝐵𝐵𝑙𝑙 𝜔𝜔𝑟𝑟

(15)

𝑗𝑗

𝑑𝑑𝜔𝜔𝑚𝑚

𝑗𝑗

𝑑𝑑𝜔𝜔𝑟𝑟

𝑑𝑑𝑑𝑑

−𝑅𝑅𝑎𝑎 ̇ 𝐿𝐿𝑎𝑎 𝐼𝐼 [ 𝑞𝑞𝑞𝑞 ] = 𝐾𝐾𝑡𝑡 𝑁𝑁𝑝𝑝 𝜔𝜔𝑟𝑟̇ [ 𝐽𝐽

Then

Where in terms of the electric speed of the rotor, it is derived from the multiplication of both sides by the pair of poles. 𝑑𝑑𝑑𝑑

𝑃𝑃 2

Where

Then the transfer function between the velocity and the torque produced current is.

𝜔𝜔𝑟𝑟 (𝑠𝑠) 𝐼𝐼𝑇𝑇 (𝑠𝑠)

=

Where

𝐾𝐾𝑎𝑎

1+𝑠𝑠𝑇𝑇𝑎𝑎

𝑃𝑃 𝐾𝐾𝑓𝑓 , 2 𝐵𝐵𝑡𝑡

𝐵𝐵𝑡𝑡 = 𝐵𝐵 + 𝐵𝐵𝑙𝑙 ,

𝑇𝑇𝑎𝑎 =

𝐽𝐽 𝐵𝐵𝑡𝑡

Once having the equations of the IM of the electrical and mechanical part, from the equations (7) and (8) the block diagram of the IM is made as shown in the Fig. 1. Ls.if B1 + Vqs

-

Ka 1+sTa

Te

iT Kf Iqs

P 2

Electrical

+

-

Where

Mechanical

0 𝑪𝑪𝑪𝑪 + 𝑪𝑪𝑪𝑪𝑪𝑪 ⋮ [𝑪𝑪𝑨𝑨𝒒𝒒−𝟏𝟏 𝑩𝑩

Fig. 1. Block diagram of IM model with constant rotor flux linkages This model is so similar to CD motor model obtained in Chan et al. (2006) and Höfling et al (1996). Although the main difference is that the input parameter is 𝑉𝑉𝑞𝑞𝑞𝑞 instead of the armature current.

A simple model of IM in steady state behaviour which is little similar to d.c motor model mentioned in Krishnan et al. (2001) is used in this work, because the fault detection based on parity equations for this model types is availability to detect several parameters, Chan et al. (2006).

2

𝑦𝑦(𝑡𝑡) 𝑪𝑪 𝑦𝑦̇ (𝑡𝑡) 𝑪𝑪𝑪𝑪 𝑦𝑦̈ (𝑡𝑡) = 𝑪𝑪𝑪𝑪𝟐𝟐 𝑥𝑥(𝑡𝑡) ⋮ ⋮ [𝑦𝑦 𝑞𝑞 (𝑡𝑡)] [𝑪𝑪𝑪𝑪𝑞𝑞 ] 0 0 𝑪𝑪𝑪𝑪 ⋮ 𝑪𝑪𝑨𝑨𝒒𝒒−𝟐𝟐 𝑩𝑩

0 0 𝑪𝑪𝑪𝑪 ⋮ …

… … … ⋱ 𝑪𝑪𝑪𝑪

𝑢𝑢(𝑡𝑡) 0 𝑢𝑢̇ (𝑡𝑡) 0 0 𝑢𝑢̈ (𝑡𝑡) ⋮ ⋮ 𝟎𝟎] [𝑢𝑢𝑞𝑞 (𝑡𝑡)]

𝒓𝒓(𝑡𝑡) = 𝑾𝑾𝑾𝑾(𝑡𝑡) − 𝑾𝑾𝑾𝑾𝑾𝑾(𝑡𝑡)

Based on known ways of theoretically modelling the structure of a linear mathematical model in the continuous time without considering disturbances (17) and (18), the state space representation obtained for the IM is displayed in (19) and (20). 𝑌𝑌(𝑡𝑡) = 𝑪𝑪𝑥𝑥(𝑡𝑡)

𝑃𝑃

0 ] 1

(19)

(20) (21)

(22)

(23)

Now, the residual vector based on state-space model for continuous time is given in (24) which is deduced in from the residual generation with parity equation for multiple-input multiple-output (MIMO) process with transfer functions and polynomial errors.

3. RESIDUAL GENERATION

𝑋𝑋̇(𝑡𝑡) = 𝑨𝑨𝑥𝑥(𝑡𝑡) + 𝑩𝑩𝑢𝑢(𝑡𝑡)

𝑁𝑁𝑝𝑝 =

0 𝑉𝑉𝑞𝑞𝑞𝑞 ][ ] 0 0

𝒀𝒀(𝑡𝑡) = 𝑻𝑻𝑥𝑥(𝑡𝑡) + 𝑸𝑸𝑸𝑸(𝑡𝑡)

ωr

1 B+sJ

0

−𝜓𝜓 1 𝐿𝐿𝑎𝑎 𝐼𝐼 [ 𝑞𝑞𝑞𝑞 ] + [𝐿𝐿𝑎𝑎 −(𝐵𝐵 + 𝐵𝐵1 ) 𝜔𝜔𝑟𝑟 0 ] 𝐽𝐽 1 0 𝐼𝐼𝑞𝑞𝑞𝑞 𝑌𝑌(𝑡𝑡) = [ ][ ] 0 1 𝜔𝜔𝑟𝑟

𝜓𝜓 = 𝐿𝐿𝑠𝑠 𝐼𝐼𝑑𝑑𝑑𝑑 ,

1 ] , 𝐶𝐶 = [ 0

0

Note that, the structure obtained in (19) is similar to DC motor model shown in Chan et al. (2006) and Höfling et al (1996) but no equal. An important difference is that the second term of 𝐼𝐼𝑞𝑞𝑞𝑞̇ in (19) the magnetic flow 𝜓𝜓 is defined as the relation between the stator inductance 𝐿𝐿𝑠𝑠 and the d axis stator current in the synchronous frames Ids. Another important difference is that the first term of 𝜔𝜔𝑟𝑟̇ ,the magnetic flow 𝜓𝜓 of d.c. motor model is defined as the relationship among poles number P, magnetic inductance Lm, rotor inductance Lr and fluxproducing component of the stator current If. A way to add redundancy in the equations at the same instant tis by introducing (17) in (18) with its respective derivatives like.

(16)

𝐾𝐾𝑎𝑎 =

−𝜓𝜓 1 𝐿𝐿𝑎𝑎 , 𝐵𝐵 = [𝐿𝐿𝑎𝑎 −(𝐵𝐵 + 𝐵𝐵1 ) 0 ] 𝐽𝐽

(24)

An important condition to satisfy that both of the first and second term of (24) are zeros is that WT=0, Höfling et al (1996) where W is called the null space of T and can be obtained by proposing the greater number of zeros possible at the rows, taking into account that the lines are linearly independent. In our case of study, the W matrix obtained by the IM is (25).

(17) (18)

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Where

𝑅𝑅𝑎𝑎 𝜓𝜓 −𝛼𝛼 𝛽𝛽 𝑾𝑾 = [ 𝛾𝛾 0 0 𝛾𝛾

𝛼𝛼 = 𝐾𝐾𝑡𝑡 𝑁𝑁𝑝𝑝 ,

𝐿𝐿𝑎𝑎 0 0 𝐽𝐽 𝛿𝛿 0 0 𝛿𝛿

Edgar Chulines et al. / IFAC PapersOnLine 51-13 (2018) 662–667

0 0 𝐽𝐽𝐿𝐿𝑎𝑎 0

𝛽𝛽 = 𝐵𝐵 + 𝐵𝐵1 ,

0 0 ] 0 𝐽𝐽𝐿𝐿𝑎𝑎

𝛾𝛾 = 𝜓𝜓𝜓𝜓 + 𝑅𝑅𝑎𝑎 𝛽𝛽, 𝛿𝛿 = 𝐿𝐿𝑎𝑎 𝛽𝛽 + 𝐽𝐽𝑅𝑅𝑎𝑎

It is important to remember that the detection of these faults are only in steady state behavior which allows to propose heuristically a pair of fixed thresholds close to the nominal current of the IM, this condition is shown in Fig. 2 and is represented in the expression (29) in ideal condition.

(25)

Table 1. Fault detection matrix

(26)

𝑟𝑟1 (𝑡𝑡) = 𝑅𝑅𝑎𝑎 𝐼𝐼𝑞𝑞𝑞𝑞 (𝑡𝑡) + 𝜓𝜓𝜔𝜔𝑟𝑟 (𝑡𝑡) + 𝐿𝐿𝑎𝑎 𝐼𝐼𝑞𝑞𝑞𝑞̇ (𝑡𝑡) − 𝑉𝑉𝑞𝑞𝑞𝑞 𝑡𝑡

r1 I

r2 0

r3 0

r4 I

Rr

I

0

0

I

Ls

I

0

I

I

Lr

I

I

I

I

B

0

I

I

I

Bl

0

I

I

I

(27)

Iqse

I

I

I

ωr

I

I

Vqse

I

0

additive

𝑟𝑟3 (𝑡𝑡) = 𝛾𝛾𝐼𝐼𝑞𝑞𝑞𝑞 (𝑡𝑡) + (𝐿𝐿𝑎𝑎 𝛽𝛽 + 𝐽𝐽𝑅𝑅𝑎𝑎 )𝐼𝐼𝑞𝑞𝑞𝑞̇ (𝑡𝑡) + 𝐽𝐽𝐿𝐿𝑎𝑎 𝐼𝐼𝑞𝑞𝑞𝑞̈ (𝑡𝑡) − 𝛽𝛽𝑉𝑉𝑞𝑞𝑞𝑞 𝑡𝑡 − 𝐽𝐽𝑉𝑉𝑞𝑞𝑞𝑞̇ (𝑡𝑡)

faults Rs parametric

By the assumption that in the healthy operation the parameter do not change, 𝑟𝑟(𝑡𝑡) = 0, then, a fault is detected when 𝒓𝒓(𝑡𝑡) ≠ 0. The residuals obtained by the IM are. ̇ 𝑟𝑟2 (𝑡𝑡) = −𝛼𝛼𝐼𝐼𝑞𝑞𝑞𝑞 (𝑡𝑡) + 𝛽𝛽𝜔𝜔𝑟𝑟 (𝑡𝑡) + 𝐽𝐽𝜔𝜔𝑟𝑟 (𝑡𝑡)

665

̇ + 𝐽𝐽𝐿𝐿𝑎𝑎 𝜔𝜔𝑟𝑟 (𝑡𝑡) ̈ 𝑟𝑟4 (𝑡𝑡) = 𝛾𝛾𝜔𝜔𝑟𝑟 (𝑡𝑡) + [𝐿𝐿𝑎𝑎 𝛽𝛽 + 𝐽𝐽𝑅𝑅𝑎𝑎 ]𝜔𝜔𝑟𝑟 (𝑡𝑡) − 𝛼𝛼𝑉𝑉𝑞𝑞𝑞𝑞

Note that during the steady state, the derived of 𝑥𝑥(𝑡𝑡) is zero, and 𝑉𝑉𝑞𝑞𝑞𝑞 𝑡𝑡 = 𝑉𝑉𝑞𝑞𝑞𝑞 , thus, the residual can be simplified, this is so suitable when the fault type is incipient; considering that is the most common in the electrical machines, then, the residuals can be reduced as.

Table 2. Insolation of fault in Rs and Ls Failure in Is

𝐹𝐹𝑎𝑎

0

𝐹𝐹𝑏𝑏

0

𝐹𝐹𝑐𝑐

0

0

I

0

I

0

0

I

I

0

I

0

0

0

I

I

0

I

I

I

I

I

0

I

I

I

Where “I” represents a significant change which can be positive or negative

0

Failure Healthy operation Failure in phase c Failure in phase b Failure in phase b, c Failure in phase a Failure in phase a, c Failure in phase a, b Failure in phase a,b,c

𝑟𝑟1 (𝑡𝑡) = 𝑅𝑅𝑎𝑎 𝐼𝐼𝑞𝑞𝑞𝑞 (𝑡𝑡) + 𝜓𝜓𝜔𝜔𝑟𝑟 (𝑡𝑡) − 𝑉𝑉𝑞𝑞𝑞𝑞 𝑟𝑟2 (𝑡𝑡) = −𝛼𝛼𝐼𝐼𝑞𝑞𝑞𝑞 (𝑡𝑡) + 𝛽𝛽𝜔𝜔𝑟𝑟 (𝑡𝑡) 𝑟𝑟3 (𝑡𝑡) = 𝛾𝛾𝐼𝐼𝑞𝑞𝑞𝑞 (𝑡𝑡) − 𝛽𝛽𝑉𝑉𝑞𝑞𝑞𝑞

𝑟𝑟4 (𝑡𝑡) = 𝛾𝛾𝜔𝜔𝑟𝑟 (𝑡𝑡) − 𝛼𝛼𝑉𝑉𝑞𝑞𝑞𝑞

(28)

Likewise, as in the d.c. motor model, if an additive fault occurs, all residuals except the decouple one are deflected as shown in Table 1. This supports strongly to locate the sensor faults, and thus this fault types are easy detectable. When a parametric fault occurs on Rs or Rr there is no a considerable increase in r3, thus, a null value can be considered to simplify the fault detection matrix. Finally, a simple way to distinguish the fault is by using classical limit-values detectors with a suitable tuning, taking into account the behaviour of residual obtained in Fig. 2.

Fig. 2. Ideal current for fixes threshold close to nominal current of IM 𝑓𝑓𝑛𝑛 = { Where

1→ 1→ 0→

𝐼𝐼𝑛𝑛 > 𝐼𝐼ℎ𝑛𝑛+ 𝐼𝐼𝑛𝑛 < 𝐼𝐼ℎ𝑛𝑛− 𝐼𝐼ℎ𝑛𝑛− > 𝐼𝐼𝑛𝑛 > 𝐼𝐼ℎ𝑛𝑛+

(29)

n represents phase a, b or c Additionally, multiple failures for only these two parameters are involved. The above is valid for a three-phase IM connected in star and neutral terminal connected to ground, this is because the affected phase current will unbalance the sum of the zero current in the neutral point n (Fig. 4). Then, the excess current will flow to earth without affecting the phases in healthy condition. In Fig. 3 shown the simulation in Matlab software of three-phase IM connected in star with and neutral terminal connected to ground with parametric variation.

Table 1 shows the parameters associated with the diagnosis of electrical faults in the IM model using parity equations based on the reference frame dq for its linearization. However, this information does not identify the damaged phase associated with the electrical parameter. Therefore, it is proposed to analyze in particular the stator currents to accurately identify the damaged element. On the other hand, it is important to mention that, for practicality during corrective maintenance, it is not necessary to know the damaged phase come from the rotor. The Table 2 shows the Insolation of fault in Rs and Ls to identify the affected phase in the fault diagnosis in Rs or Ls.

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Fig. 3. Three-phase IM connected in star with and neutral terminal connected to ground: (a) Rs nominal value -50%, (b) Rs nominal value +50%, (c) Rr nominal value -50%, (d) Rr nominal value +50%, (e) Lr nominal value -50%, (f) Lr nominal value +50%, (g) Ls nominal value -50%, (h) Ls nominal value +50%, (i) B nominal value -50%, (j) B nominal value +50% and (29) obtained in this work, which allow to obtain tables 1 and 2, are extremely simple to implement in any digital processor since the operators are addition, subtraction and multiplication without recursion. The main problem is that the parameters of the IM must be known a priori. REFERENCES Bianchini, C., Immovilli, F., Cocconcelli, M., Rubini, R., & Bellini, A. (2011). Fault detection of linear bearings in brushless AC linear motors by vibration analysis. IEEE Transactions on Industrial Electronics, 58(5), 16841694. Cash, M. A., Habetler, T. G., & Kliman, G. B. (1998). Insulation failure prediction in AC machines using lineneutral voltages. IEEE Transactions on Industry Applications, 34(6), 1234-1239. Chan, C. W., Hua, S., & Hong-Yue, Z. (2006). Application of fully decoupled parity equation in fault detection and identification of DC motors. IEEE Transactions on Industrial Electronics, 53(4), 1277-1284. Höfling, T., & Isermann, R. (1996). Fault detection based on adaptive parity equations and single-parameter tracking. Control Engineering Practice, 4(10), 13611369. Isermann, R. (1995, June). Model base fault detection and diagnosis methods. In American Control Conference, Proceedings of the 1995 (Vol. 3, pp. 1605-1609). IEEE. Isermann, R. (1997). Supervision, fault-detection and faultdiagnosis methods—an introduction. Control engineering practice, 5(5), 639-652. Klima, J. (2003). Analytical investigation of an induction motor drive under inverter fault mode operations. IEE Proceedings-Electric Power Applications, 150(3), 255262. Kolla, S., & Varatharasa, L. (2000). Identifying three-phase induction motor faults using artificial neural networks. ISA transactions, 39(4), 433-439. Krishnan, R. (2001). Electric motor drives: modeling, analysis, and control. Prentice Hall.

Fig. 4. Three-phase IM connected in star with and neutral terminal connected to ground Then with the above matrix of table 1 can be seen that the fault detection probability for Rs, Rr, Ls, B, Bl and Vqse are 50% and only for Lr, Iqse and ωr are 100%, but according to table 2, Rs and Ls the fault detection probability is 100%. 4. CONCLUSIONS The linearization of IM nonlinear model using the synchronous reference frame dq is valid only when the IM is working in stable state operation. In this way, it is convenient to consider heuristically a diagnostic dead time at start-up and fixed current thresholds, according to the stabilization time and input nominal current respectively. Since the IM model is matched to the d.c.. motor model the analysis allows to assure the existence of parity space and therefore to obtain the advantages of the fault detection for this system type. Add a pair of fixed thresholds to the stator current and use Table 2, it allows to insolation the affected phase for Lsn and Rsn. In addition, the detection of multiple faults to these parameters is obtained naturally, as long as the IM is connected in star and the neutral terminal connected to ground. The expression (28) 666

2018 IFAC MICNON Guadalajara, Mexico, June 20-22, 2018

Edgar Chulines et al. / IFAC PapersOnLine 51-13 (2018) 662–667

Loparo, K. A., Adams, M. L., Lin, W., Abdel-Magied, M. F., & Afshari, N. (2000). Fault detection and diagnosis of rotating machinery. IEEE Transactions on Industrial Electronics, 47(5), 1005-1014. Mendes, A. M., & Cardoso, A. M. (2006). Fault-tolerant operating strategies applied to three-phase inductionmotor drives. IEEE Transactions on Industrial Electronics, 53(6), 1807-1817. Nandi, S., Toliyat, H. A., & Li, X. (2005). Condition monitoring and fault diagnosis of electrical motors—A review. IEEE transactions on energy conversion, 20(4), 719-729. Pineda-Sanchez, M., Riera-Guasp, M., Roger-Folch, J., Antonino-Daviu, J. A., Perez-Cruz, J., & PuchePanadero, R. (2011). Diagnosis of induction motor faults in time-varying conditions using the polynomialphase transform of the current. IEEE Transactions on Industrial Electronics, 58(4), 1428-1439. Strangas, E. G., Aviyente, S., & Zaidi, S. S. H. (2008). Time–frequency analysis for efficient fault diagnosis and failure prognosis for interior permanent-magnet AC motors. IEEE Transactions on Industrial Electronics, 55(12), 4191-4199.

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