Real-time detection of stator resistance unbalances in three- phase drives
BHANU PRATAP SINGH
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCHOOL OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE
resistance unbalances in three phase drives
BHANU PRATAP SINGH
Master in Electric Power Engineering Date: September 22, 2020
Supervisor: Yixuan Wu, Arpit Gupta, Torbjörn Larsson Examiner: Luca Peretti
School of Electrical Engineering and Computer Science Host company: Volvo Cars Corporation
Swedish title: Realtids detektering av obalanser i statorsmotstånd i trefasiga enheter
Abstract
An estimated 30% of the faults in Induction Machine (IM) are related to its stator. These faults are mostly in the form of an Inter-Turn Short Circuit (ITSC) fault i.e., when two winding inside the stator of IM are shorted due to insulation failure. However, ITSC fault can be avoided by detecting them in advance and then scheduling the maintenance of the IM. This thesis studies two methods for detecting this incipient ITSC fault in a three-phase IM and then estimating the stator resistance unbalance due to the ITSC fault.
The first method is based on the asymmetry caused in the IM by the ITSC fault. As a result of this asymmetry, the negative sequence components of the stator voltages and the stator currents are generated inside the IM. A healthy IM also have these negative sequence components due to the manufacturing process and the supply voltage unbalances. The characteristics and the com- pensation methods of these negative sequence components in a healthy IM are discussed. The results show that after compensating the negative sequence components in a healthy machine, they can be used for detecting an ITSC fault and then to calculate the fault quantities as well as the stator resistance unbalances.
The second method for detecting an ITSC fault is based on analysing the stator resistance unbalances. A three-phase drive is used to inject DC volt- age in the stationary reference frame. The DC current generated by this DC voltage is measured and then by applying Ohm’s law stator phase resistances are calculated. In a healthy IM, the phase resistances are balanced. However, in case of ITSC fault in any of the phases, the phase resistance of that phase deviates from those of the other two phases which can be utilized for detecting ITSC fault.
Keywords:
DC voltage injection, Fault detection, Field-Oriented Control, Induction machine, Inter-Turn Short Circuit, Short circuit, Sequence, Stator resistance unbalance, Symmetrical componentsSammanfattning
Uppskattningsvis 30% av alla fel i induktionsmaskiner (IM) är kopplad till dess stator. Dessa fel är i huvudsak Inter-Turn Short Circuit (ITSC)-fel, dvs. två lindningar inom IM:ens stator blir kortsluta pga. ett isoleringsfel. Emellertid kan man undvika ITSC-fel genom att detektera dem i förhand och planera underhåll. Det här examensarbetet undersöker två metoder för att detektera ett förestående ITSC-fel i en tre-fas IM.
Den första metoden är baserad på asymmetrin i IM:er pga. ITSC-felet. Re- sultatet av den här asymmetrin är att en negativ sekvens genereras i IM:ens statorspänning och statorström. En oskadad IM kan också visa dessa negativa sekvenser pga. tillverksprocessen och statorspänningsobalanser. Egenskaper- na och kompensationsmetoderna för dessa negativa sekvenser i en oskadad IM kommer att diskuteras. Resultaten visar att efter kompenseringen av de nega- tiva sekvenserna i en oskadad IM, kan de användas för att detektera ITSC-fel och efteråt för att beräkna felstorheter och även statormotståndobalanser.
Den andra metoden för att detektera ITSC-fel är baserad på en undersök- ning av statormotståndobalanser. Ett tre-fas-drivsystem används för att injek- tera likspänning i den stationära referensramen. Likströmmen som följer av denna likspänning mäts och statorfasmotstånden beräkna efteråt med Ohms lag. I en oskadad IM är fasmotstånden balanserade. Däremot, när ett ITSC-fel uppstår i en fas, avviker fasmotståndet i den felaktiga fasen från de andra två fasernas, vilket kan användas för att detektera ITSC-fel.
Nyckelord:
likspänningsinjektion, feldetektering, fält orienterad reglering, induktionsmaskin, InterTurn Short Circuit, kortslut, sekvenser, statormotstånd obalans, symmetriska komponenterAcknowledgement
This master thesis has been carried out as a collaborative work between ART Power and Energy Division, Volvo Cars Corporation (VCC) and Electric Ma- chines and Drives group, KTH Royal Institute of Technology.
First, I would like to express my deep and sincere gratitude to VCC for pro- viding me with an opportunity to work on a master thesis for them. I especially want to thank my supervisors Arpit Gupta and Torbjörn Larsson from VCC for their continuous support and guidance throughout this journey. I really cherish their support during tough times and unforeseen delays.
I want to express my sincere gratitude to Yixuan Wu who has acted as my supervisor from KTH. I must say that he is the best supervisor and one of the best persons I have ever met. I want to thank him for having regular weekly meetings with me and providing precious guidance during those meetings. I also want to thank him for continuously believing in me and my work. This thesis wouldn’t have been possible without his support and guidance. I would like to say a special thank you to Giovanni Zanuso. I had the best experience of my life while working with him on building the test-bench. I am really thankful to him for allowing me to carry out the experiments on his test-bench. I also want to thank him for sharing all the valuable knowledge and also teaching me a lot about the experimental setups. I am extremely grateful to Associate Professor Luca Peretti. I want to express my thank to him for providing me with such a good master thesis topic. Additionally, I would also like to thank him for acting as my examiner.
I also want to thank everyone working in the Sustainable Power Labora- tory, KTH. It is because of their support that I have been able to finish my experiments in the lab during the tough times. I would like to thank my col- leagues in the lab. I really cherish the time we spent together in the lab and also the discussions that we had.
Finally, I would like to thank my parents who have constantly supported me during this journey. I want to thank them for always being positive about my career and studies during my entire master’s study.
Bhanu Pratap Singh Stockholm, Sweden
1 Introduction 1
1.1 Background . . . 2
1.2 Problem statement . . . 2
2 Theory 4 2.1 Analysis of an IM with ITSC fault . . . 5
2.1.1 Machine equations in abc reference frame . . . 6
2.1.2 Clarke transformation . . . 8
2.1.3 Two-phase representation of the IM with ITSC fault either in phase b or in phase c . . . 10
2.2 Symmetrical components equivalent circuits . . . 11
2.2.1 Symmetrical components representation of two-phase system . . . 14
2.3 Three-phase resistance monitoring using DC voltage injection 18 2.4 Field Oriented Control of IM . . . 21
2.4.1 Current Model . . . 23
2.4.2 Current Controller . . . 24
2.4.3 Speed Controller . . . 25
2.5 Signal processing . . . 26
2.6 Mechanical dynamics . . . 29
3 Methods 30 3.1 Analysing the negative sequence components . . . 30
3.1.1 Negative sequence equivalent circuit . . . 31
3.1.2 Calculation of µ, If and R0s. . . 34
3.2 DC voltage injection . . . 35
4 Technical System Description 37 4.1 Three-phase Induction Motor . . . 37
4.2 Current sensor . . . 41
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5 Results 42
5.1 Simulation . . . 42
5.1.1 Three-phase IM Simulink model . . . 42
5.1.2 Analysing the negative sequence components . . . 45
5.1.3 DC voltage injection . . . 51
5.2 Experimental . . . 54
5.2.1 Analysing the negative sequence components . . . 54
6 Discussion 66
7 Conclusions 70
Bibliography 72
List of symbols
¯iαβs,dc DC component of converter output current in stationary reference frame
¯isαβr [A]Rotor current space vector in stationary reference frame [A]
¯isαβs Stator current space vector in stationary reference frame [A]
¯isdqs Stator current space vector in synchronous reference frame [A]
¯
vαβ,dc DC component of converter output voltage in stationary reference frame [V]
¯
vαβinj Injected DC voltage space vector in stationary reference frame [V]
¯
vαβss Stator voltage space vector in stationary reference frame [V]
¯
vdqss Stator voltage space vector in synchronous reference frame [V]
Y¯np Negative sequence admittance due to IM asymmetry [S]
I˜f Fault current phasor [A]
I˜s+ Positive sequence component of the stator currents [A]
I˜s− Negative sequence component of the stator currents [A]
I˜s0 Zero sequence component of the stator currents [A]
I˜as Stator current phasor in phase a [A]
I˜bs Stator current phasor in phase b [A]
I˜cs Stator current phasor in phase c [A]
V˜s+ Positive sequence component of the stator voltages [V]
V˜s− Negative sequence component of the stator voltages [V]
V˜s0 Zero sequence component of the stator voltages [V]
V˜as Stator voltage phasor in phase a [V]
V˜a Supply voltage phasor in phase a [V]
V˜bs Stator voltage phasor in phase b [V]
V˜b Supply voltage phasor in phase b [V]
V˜cs Stator voltage phasor in phase c [V]
V˜c Supply voltage phasor in phase c [V]
V˜s,ref− Stator reference voltage negative sequence.
b Damping factor [N msec/rad]
viii
I Stator current amplitude in balanced condition [A]
If Amplitude of fault current.
if Instantaneous fault current [A]
Is− Amplitude of the stator current negative sequence.
iar Instantaneous rotor current in phase a [A]
ias,ac AC component of instantaneous stator phase current in phase a [A]
ias,dc DC component of instantaneous stator phase current in phase a [A]
Ias Stator current amplitude in phase a [A]
ias Instantaneous stator current in phase a [A]
ibr Instantaneous rotor current in phase b [A]
ibs,ac AC component of instantaneous stator phase current in phase b [A]
ibs,dc DC component of instantaneous stator phase current in phase b [A]
Ibs Stator current amplitude in phase b [A]
ibs Instantaneous stator current in phase b [A]
icr Instantaneous rotor current in phase c [A]
ics,ac AC component of instantaneous stator phase current in phase c [A]
ics,dc DC component of instantaneous stator phase current in phase c [A]
Ics Stator current amplitude in phase c [A]
ics Instantaneous stator current in phase c [A]
J Moment of inertia of IM [kg.m2] kic CC integral constant [Ω/sec]
kis SC integral constant [N m/rad] kpc CC proportional constant [H/sec]
kps SC proportional constan [kg.m2/sec].
LM Magnetizing inductance in inverse-Γ model of IM [H]
Lm Magnetizing inductance in T-model of IM [H]
Lr Rotor inductance in T-model of IM [H]
Ls Stator inductance in T-model of IM [H]
Lσn Nominal total leakage inductance in inverse-Γ model of IM [H]
Lσ Total leakage inductance in inverse-Γ model of IM [H]
Llr Per phase rotor leakage inductance in T-model of IM [H]
Lls Per phase stator leakage inductance in T-model of IM [H]
Lms Mutual inductance [H]
N Total number of turns in each phase Na Number of shorted turns in phase a Nb Number of shorted turns in phase b Nc Number of shorted turns in phase c np Number of pole pairs
R Sum of the stator and rotor resistance [Ω]
Ra Active resistance in CC [Ω]
RR Rotor resistance in inverse-Γ model of IM [Ω]
Rr Per phase rotor resistance in T-model of IM [Ω]
Rs Per phase stator resistance in a healthy IM [Ω]
R0s Stator resistance of phase with ITSC fault [Ω]
Ras Resistance of phase a [Ω]
Rbs Resistance of phase b [Ω]
Rcs Resistance of phase c [Ω]
s Slip of IM
V Stator voltage amplitude in balanced condition [V]
va Instantaneous value of the converter output voltage in phase a [V]
vb Instantaneous value of the converter output voltage in phase b [V]
vc Instantaneous value of the converter output voltage in phase c [V]
Vs− Amplitude of the stator voltage negative sequence.
v0s Zero sequence component of stator voltage in stationary reference frame[V]
vas,ac AC component of instantaneous stator phase voltage in phase a [V]
vas,dc DC component of instantaneous stator phase voltage in phase a [V]
vas1 Instantaneous stator voltage across as1 in phase a [V]
vas2 Instantaneous stator voltage across faulty turns in phase a [V]
Vas Stator voltage amplitude in phase a [V]
vas Instantaneous stator phase voltage in phase a [V]
Va Supply voltage amplitude in phase a [V]
vbs,ac AC component of instantaneous stator phase voltage in phase b [V]
vbs,dc DC component of instantaneous stator phase voltage in phase b [V]
vbs2 Instantaneous stator voltage across faulty turns in phase b [V]
Vbs Stator voltage amplitude in phase b [V]
vbs Instantaneous stator phase voltage in phase b [V]
Vb Supply voltage amplitude in phase b [V]
vcs,ac AC component of instantaneous stator phase voltage in phase c [V]
vcs,dc DC component of instantaneous stator phase voltage in phase c [V]
vcs2 Instantaneous stator voltage across faulty turns in phase c [V]
Vcs Stator voltage amplitude in phase c [V]
vcs Instantaneous stator phase voltage in phase c [V]
Vc Supply voltage amplitude in phase c [V]
Vs,ref− Amplitude of the stator reference voltage negative sequence.
vsn Star point ’s’ voltage with respect to DC-link capacitor neutral point ’n’
[V]
X Transformation factor of µa/µb/µcinto stationary reference frame
Greek Symbols
αc Bandwidth of the CC αs Bandwidth of the SC
ψ¯sαβr Rotor flux linkage space vector in stationary reference frame [Vs]
ψ¯sαβs Stator flux linkage space vector in stationary reference frame [Vs]
ψ¯dqR Rotor flux linkage space vector in synchronous reference frame (Inverse- Γ model) [Vs]
ψ¯dqs Stator flux linkage space vector in synchronous reference frame [Vs]
µ Ratio between number of shorted turns and total turns in phase a µa Ratio between number of shorted turns and total turns in phase a.
µb Ratio between number of shorted turns and total turns in phase b µc Ratio between number of shorted turns and total turns in phase c ωe Supply frequency (Electrical) [rad/sec]
ωm Rotor mechanical speed [rad/sec]
ωr Rotor electrical speed [rad/sec]
φ Phase angle [rad]
ψ0s Zero sequence component of stator flux in stationary reference frame[Vs]
ψar Rotor flux linkages in phase a [Vs]
ψas1 Stator flux linkage in windings as1 of phase a [Vs]
ψas2 Stator flux linkage in faulty turns of phase a [Vs]
ψbr Rotor flux linkages in phase b [Vs]
ψbs2 Stator flux linkage in faulty turns of phase b [Vs]
ψbs Stator flux linkage in phase b [Vs]
ψcr Rotor flux linkages in phase c [Vs]
ψcs2 Stator flux linkage in faulty turns of phase c [Vs]
ψcs Stator flux linkage in phase c [Vs]
ψref Reference rotor flux [Vs]
τe Electromagnetic torque developed by IM [Nm]
τl Load torque [Nm]
θ1 dq-transformation angle [rad]
θi Phase angle betweenI˜asandV˜a(Electrical) [rad]
θr Angle of rotation [rad]
θv Phase angle betweenV˜asandV˜a(Electrical) [rad]
2.1 Stator windings of three-phase IM with ITSC fault in phase a
[13]. . . 6
2.2 Three-phase asymmetric or unbalanced system. . . 11
2.3 Symmetrical components of the phasors a, b and c. . . 12
2.4 Asymmetric three-phase phasors in the IM. . . 13
2.5 Steady-state sequence component equivalent circuits of an IM with ITSC fault in either of the three-phases [13]. . . 17
2.6 Single phase T-equivalent circuit of healthy three-phase IM . . 18
2.7 DC component equivalent stator circuit of IM. . . 20
2.8 Inverse-Γ model of the IM [14] . . . 22
2.9 Block diagram of IFO control of three-phase IM. . . 23
2.10 Block diagram of the CC with anti-windup. . . 24
2.11 Block diagram of the SC with anti-windup. . . 25
2.12 Hanning window function. . . 27
3.1 Negative sequence equivalent circuit [13]. . . 31
3.2 Block diagram for estimating three-phase resistances using DC voltage injection [10]. . . 36
4.1 Stator winding arrangement. . . 38
4.2 Connections inside the terminal box of IM. . . 39
4.3 Tapping diagram of IM. . . 40
4.4 Noise level in phase b and phase c current sensors. . . 41
5.1 Fault current (if) in the fault circuit due to ITSC fault in phase a. 43 5.2 Simulated zero-sequence stator voltage (voss ) in stationary ref- erence frame. . . 43
5.3 Oscillations in the electromagnetic torque and speed due to the occurrence of the ITSC fault at time t=2.5 sec (supply fre- quency = 50 Hz). . . 44
xii
5.4 Simulated negative sequence component of the stator current magnitude (Is−) at different CC bandwidths. [ITSC fault oc-
curs at t = 2 sec with µ = 8% in phase a and Rf = 1.5Ω.] . . . 45
5.5 Simulated magnitude of the positive and the negative sequence components of the stator current errors in FOC. [ITSC fault occurs at t = 2 sec with µ = 8% in phase a.] . . . 46
5.6 Negative sequence of error current in CC and stator reference voltage at different CC bandwidths. [ITSC fault occurs at t = 2 sec with µ = 8% in phase a and Rf = 1.5Ω.] . . . 47
5.7 Fault current If and µIf at differet CC bandwidths. [ITSC fault occurs at t = 2 sec with µ = 8% in phase a.] . . . 48
5.8 Fault current (If) at different ITSC fault levels with known value of Rf. [ITSC fault occurs at t = 2 sec with µ = 8% in phase a.] . . . 49
5.9 Fault current (If) with Rf equal to zero. [ITSC fault occurs at t = 2 sec with µ = 8% in phase a.] . . . 50
5.10 Sequence components of the actual versus reference voltages across the IM stator. [ITSC fault occurs at t = 2 sec with µ = 8% in phase a and Rf = 1.5 Ω.] . . . 51
5.11 Electromagnetic torque and rotor speed of IM in DC-injection method. [ITSC fault occurs at t = 7.5 sec with µ = 8% in phase a and Rf = 1.5 Ω.] . . . 52
5.12 DC voltages and DC currents in IM while using DC-injection method. . . 53
5.13 Resistance of faulty phase during ITSC fault at time t = 7.5 sec. [Rsof healthy IM is 0.7976 Ω. µ is 8.05% with Rf equal to zero.] . . . 54
5.14 Is−in a healthy IM. . . 55
5.15 Vs,ref− in a healthy IM. . . 56
5.16 µIf in a healthy IM at different load levels. . . 56
5.17 Is−due to the occurrence of ITSC fault at t = 2.5 sec. [ITSC fault occurs in phase a with µ = 8.05% and Rf = 1.5 Ω.] . . . . 57
5.18 Vs,ref− due to the occurrence of ITSC fault at t = 2.5 sec. [ITSC fault occurs in phase a with µ = 8.05% and Rf = 1.5 Ω.] . . . . 58
5.19 µIf due to the occurrence of ITSC fault at t = 2.5 sec.[ITSC fault occurs in phase a with µ = 8.05% and Rf = 1.5 Ω.] . . . . 59
5.20 IM slip for different ramp loads. . . 60
5.21 Sequence components and µIf during ITSC fault at t =0.5 sec after compensation of the sequence components from a healthy IM respectively. . . 61 5.22 Sequence components and µIf during ITSC fault at t =1.5 sec
at different fault levels. . . 62 5.23 Detection time with respect to fault level and load. . . 64 6.1 Variation in stator resistance with respect to fault level. [Stator
resistance in healthy state, Rs=0.7976 Ω.] . . . 66 6.2 Amplitude and phase angle of Is−in a healthy IM with respect
to slip at different type of loads. . . 68 6.3 Calculated fault current and fault percentage in a healthy IM . 69
2.1 Dependency of µ and X on faulty phase. . . 18
4.1 IM Specifications . . . 37
4.2 IM Parameters based on inverse-Γ model. . . 38
4.3 Tap notations . . . 39
4.4 Current Sensor (Sensitec CMS3050) Specifications . . . 41
5.1 Fault detection time in different cases with same cut-off values. 63 5.2 Calculated parameters and percentage errors. . . 65
xv
AC alternating current DC direct current DOL Direct On Line
DFT Discrete Fourier Transformation CC Current Controller
CM Current Model
FOC Field Oriented Control IM Induction Machine ITSC Inter Turn Short Circuit Ph-G Phase-Ground
Ph-Ph Phase-Phase SC Speed Controller
SCIM Squirrel Cage Induction Motors SVM Space Vector Modulator
LTI linear and time-invariant IFO Indirect Field Oriented
xvi
Introduction
Since the development of the Induction Machine (IM) in the 19thcentury, a lot of work has been done to improve its efficiency and on the control aspect of it.
IM has found various applications in different fields like automotive, process industries, traction, home appliances, etc and the reason behind this is its high durability, robustness, cost effectiveness and low maintenance. As mentioned in [1], approximately 85% of the electric machines used in industry are IM.
Due to the widespread application of these machines, it has become critical to assess their operating conditions, efficiency and reliability to avoid any critical failures.
The life cycle duration of the induction motor is in the range of 12-20 years [2]. According to the statistics, it is shown that annual downtime1of 0.5%-4%
is to be expected [3]. The life cycle and the failure rate can vary depending on the operating condition of the machine. Breakdown of the failures in Squirrel Cage Induction Motors (SCIM) from different sources is shown in [2]. As per this journal, on an average 30% of the faults in SCIM are related to stator winding, 47% are bearing faults, 6% are rotor faults and remaining are the other faults like shaft failure, coupling failure etc.
In order to avoid these failures, it has become essential to do the condition monitoring2of the machines regularly. Condition monitoring of the electrical machines can be performed either in offline or online mode. In offline mode, the machine parameters are analysed in the machine OFF condition and can only be performed during scheduled maintenance. Hence, this method has
1Downtime is the time duration during which the equipment is not able to perform its primary function.
2Condition monitoring is the process of monitoring a parameter of condition in machinery (vibration, temperature etc.), in order to identify a significant change which is indicative of a developing fault [4].
1
a drawback that it cannot be used for continuous condition monitoring. The offline method is also expensive because it requires external testing and mea- suring equipment. In the online mode, the condition monitoring of different parameters can be performed even when the machine is running. By doing so, we can predict when the machine is required to undergo maintenance.
1.1 Background
As stated earlier in the breakdown of failures in SCIM, stator winding faults are the second most frequently occurring faults. Therefore, it becomes significant to investigate these faults to increase the reliability of an IM. Stator winding faults occur because of the short circuit inside the stator of the IM. The short circuit occurs due to the continuous ageing of the windings which can fail due to thermal stress, vibrations, voltage transients, etc [5]. The stator winding faults can be divided into Inter Turn Short Circuit (ITSC) fault, Phase-Phase (Ph-Ph) fault and Phase-Ground (Ph-G) fault. The ITSC fault results in the circulation of high fault current through the shorted turns. It increases the winding temperature which in turn results in the thermal degradation of the insulation of the neighbouring turns. Thus, the insulation of other phases are affected and can lead to Ph-Ph fault or Ph-G fault [5], [6]. Most faults start with the ITSC fault and then results in Ph-Ph fault or Ph-G fault [7], [6]. Therefore, it is highly essential to detect the ITSC fault in advance to avoid the machine failure.
A lot of work has been done in the last two decades on online condition monitoring of the IM. The detailed analysis and the comparison of the vari- ous condition monitoring methods can be found in [8] and [5]. An important aspect considered while selecting a method is to check if it is invasive or nonin- vasive. Usually, noninvasive methods are preferred because they require only current and voltage sensors which are usually available in the IM drive [6].
1.2 Problem statement
The objective of this thesis is to detect the ITSC fault in a three-phase IM drive.
As discussed in the previous section, noninvasive methods are investigated because they do not require any specific sensor and thus helps in keeping the cost of the IM drive minimum [9]. The first method is based on analysing a negative sequence component of the stator current and the voltage. As can be seen in Figure 2.1, the ITSC fault results in an asymmetry in the stator winding.
Due to this asymmetry, negative sequence current and voltage will be induced in the stator circuit. By analysing these negative sequence quantities, we can detect the occurrence of the ITSC fault and calculate the fault parameters along with the phase resistance of the faulty phase. This method has already been used for detecting and diagnosing the stator faults and can be found in [8], [5], [9]. The second method investigated in this thesis is based on injecting a direct current (DC) voltage in all the three phases using IM drive. From Figure 2.1, we can also see that the resistance of a faulty phase deviates from the other two due to the asymmetry. This method is based on measuring the deviation in stator resistance of the faulty phase from that of the other two phases. It is done by using Ohm’s law on the measured DC voltage and current to calculate the resistance of each phase [10], [11]. The calculated phase resistances are then analysed for any deviation to detect the ITSC fault.
Theory
The three-phase parameters can be transformed into the symmetrical com- ponents namely positive-, negative- and zero-sequence components [12]. In a symmetric three-phase system, like a balanced three-phase power supply, only positive sequence components are present. Negative and zero sequence com- ponents will be zero in this case. But, an asymmetric system can have both negative- and zero-sequence components along with the positive sequence component. In a star connected IM with star point not grounded, stator voltage has positive-, negative- and zero-sequence components due to the asymmetry created inside the IM by the ITSC fault. While, the stator current only has positive- and negative-sequence components. Stator current in the star con- nected IM can have zero-sequence component only if its star point is grounded.
In this thesis, only the star connected IM with star point not grounded will be studied.
A three-phase IM should be symmetric in a healthy condition. However, this is not the usual case and it always has some degree of inherent asymmetry present in it. This asymmetry is caused by the stator winding imbalances, supply voltage unbalances, rotor static eccentricities, iron saturation or other manufacturing imperfections [1]. Asymmetry will also be present during the stator current and voltage transients. As a result, a practical IM has the T- equivalent circuits in the positive-, negative- and zero-sequence. However, a star connected IM with star point not grounded will have the T-equivalent circuits only in the positive- and negative-sequence because the zero-sequence component of the stator current will be zero.
4
An occurrence of the ITSC fault in any one of the phases of the IM induces a fault current in the shorted turns. This fault current affects the symmetrical1 components of the stator current and voltage which is also shown in Figure 2.5. The change in the negative-sequence components of the stator current and voltage due to the fault current can be analysed to detect the ITSC fault. A detailed study about the theory behind the symmetrical components and their T-equivalent circuit during ITSC fault will be presented in this chapter.
This chapter also discusses the theory behind the monitoring of three- phase resistances of the IM using the DC voltage injection. Due to linear and time-invariant (LTI) T-equivalent circuit of the IM, shown in Figure 2.6, the DC and alternating current (AC) quantities can be separated. The DC quanti- ties can then be used to calculate the phase resistances by using Ohm’s law.
The sections in this chapter are divided as follows. Section 2.1 shows how to model the IM in a stationary reference frame with ITSC fault in any one of the phases. Section 2.2 explains modelling in terms of symmetrical compo- nents equivalent circuits. Section 2.3 explains the way to monitor the phase resistances using DC voltage injection. Section 2.4 discusses about the con- trol of IM. Section 2.5 explains the basics of signal processing and section 2.6 gives the mechanical dynamics of the IM.
2.1 Analysis of an IM with ITSC fault
The purpose of this section is to study and model the effects of the ITSC fault in phase a. Therefore, we will consider only the symmetric IM in our analysis.
The effect of inherent asymmetries and the unbalances will be discussed in the later sections. Figure 2.1 shows the stator windings with ITSC fault across the turns as2 in phase a [13]. Additionally, the fault current if and the fault resistance Rf are also defined.
In Figure 2.1, let N be the total number of turns in each phase out of which Naturns in phase a winding are shorted. µadenotes the ratio between Naand N , µa = Na/N. In the following derivations, it is assumed that the leakage inductance of the shorted turns is proportional to µa[13]. Therefore, the leak- age inductance of the shorted turns as2will be µaLlsand that of the remaining winding as1 will be (1 − µa)Lls.
1The positive-, negative- and zero-sequence components when taken separately are re- ferred as the symmetrical components. It is because they represent the symmetrical sets.
as1
as2
ias
if
ics ibs
s
Rf
Vf
Vas
b c
a
Figure 2.1: Stator windings of three-phase IM with ITSC fault in phase a [13].
2.1.1 Machine equations in abc reference frame
This sub-section describes the symmetric IM circuit equations in the abc refer- ence frame. The equations will then be used for modelling the IM in Simulink in the stationary reference frame. The equations derived in this section are based on the work done in [13]. The stator and rotor equations of the IM can be stated in the abc reference frame as in (2.1).
vs = Rsis+dψs dt 0 = Rrir+dψr
dt (2.1)
The quantities in (2.1) are defined as followed:
vs = [vas1 vas2 vbs vcs]T is = [ias (ias− if) ibs ics]T ir = [iar ibr icr]T
ψs = [ψas1 ψas2 ψbs ψcs]T
= Lssis+ Lsrir ψr = [ψar ψbr ψcr]T
= LsrTis+ Lrrir.
The inductance matrices (Lss, Lsr, Lrr) are:
Lss = Lls
1 − µa 0 0 0
0 µa 0 0
0 0 1 0
0 0 0 1
+ Lms
(1 − µa)2 µa(1 − µa) −1−µ2 a −1−µ2 a µa(1 − µa) µ2a −µ2a −µ2a
−1−µ2 a −µ2a 1 −12
−1−µ2 a −µ2a −12 1
Lsr = Lms
(1 − µa) cos (θr) (1 − µa) cos (θr+ 2π/3) (1 − µa) cos (θr− 2π/3) µacos (θr) µacos (θr+ 2π/3) µacos (θr− 2π/3) cos (θr− 2π/3) cos (θr) cos (θr+ 2π/3) cos (θr+ 2π/3) cos (θr− 2π/3) cos (θr)
Lrr =
Llr+ Lms −Lms/2 −Lms/2
−Lms/2 Llr+ Lms −Lms/2
−Lms/2 −Lms/2 Llr+ Lms
.
The stator and the rotor phase resistances used in (2.1) are:
Rs = Rs
1 − µa 0 0 0
0 µa 0 0
0 0 1 0
0 0 0 1
Rr = RrI3x3.
Adding the voltages across as2and as1, we will get (2.2).
vas = vas1 + vas2 = (1 − µa)Rsias+ µaRs(ias− if) + d(ψas1 + ψas2)/dt
= Rsias+ d(ψas1 + ψas2)/dt − µaRsif (2.2) Combining (2.1) with (2.2) for the ITSC fault case, we get (2.3).
v0s= Rsi0s+dψs0/dt + µaA1if
0 = Rrir+dψr/dt (2.3)
The quantities in (2.3) are defined as:
v0s= [vas vbs vcs]T i0s= [ias ibs ics]T
ψ0s= [(ψas1 + ψas2) ψbs ψcs]T
= L0ssi0s+ Lsr0 ir+ µaA2if ψr = [ψar ψbr ψcr]T
= L0srTi0s+ Lrrir+ µaA3if A1 = −[Rs 0 0]T
A2 = [−(Lls+ Lms Lms/2 Lms/2)]T
A3 = −Lms[cos θr cos (θr+ 2π/3) cos θr− 2π/3]T. The inductance matrices (Lss, Lsr) will be modified into:
L0ss =
Lls+ Lms −Lms/2 −Lms/2
−Lms/2 Lls+ Lms −Lms/2
−Lms/2 −Lms/2 Lls+ Lms
L0sr= Lms
cos θr cos (θr+ 2π/3) cos (θr− 2π/3) cos (θr− 2π/3) cos θr cos (θr+ 2π/3) cos (θr+ 2π/3) cos (θr− 2π/3) cos θr
.
The voltage and the flux equations for the shorted turns as2 are given by (2.4) [13].
vas2 = µaRs(ias− if) + dψas2/dt = Rfif ψas2 = −µaA2T
i0s− µaA3T
ir− µa(Lls+ µaLms)if (2.4)
2.1.2 Clarke transformation
As described in [14], the IM equations in the three-phase system can be trans- formed into an equivalent two-phase system using complex space vector. The two-phase system representation is done in a complex plane having perpen- dicular axes. These axes are denoted as α and β axes where α−axis is real and β−axis is imaginary. The α−axis is aligned with the phase a of the three- phase system and the β−axis is perpendicular to it. Thus, we can represent the two-phase system using complex space vector as:
¯
vαβss = vsαs+ jvsβs.
This transformation from the three-phase system to the two-phase system is known as Clarke transformation [15]. Superscript ’s’ in the complex space vector denotes the stationary reference frame. Therefore, on applying Clarke transformation on equation (2.3) we get (2.5).
¯
vαβss = Rs¯isαβs+ p ¯ψsαβs− 2
3XµaRsif 0 = Rr¯isαβr+ (p − jωr) ¯ψsαβr vs0s = −1
3µaRsif + pψs0s (2.5)
Here, p represents d/dt operator and, ωr= pθr. For ITSC fault in phase a:
X = 1.
On transforming the stator and rotor flux linkages, ψ0s and ψr, into the stationary reference frame we get (2.6).
ψ¯αβss = Ls¯isαβs+ Lm¯isαβr− 2
3XµaLsif ψ0ss = −1
3µaLlsif
ψ¯αβrs = Lr¯isαβr+ Lm¯isαβs− 2
3XµaLmif (2.6) The inductance parameters in (2.6) are defined as:
Lm = 3 2Lms Ls= Lls+ Lm Lr = Llr+ Lm.
The expressions given in (2.4) can also be written as in (2.7).
vas2 = µaRs(ias− if) + pψas2 = Rfif
ψas2 = −µaLls(ias− if) + µaLm(isas+ isar −2
3µaLmif) (2.7) Transforming ias and iar into the stationary reference frame by applying Clarke transformation, (2.7) can be written as (2.8).
vas2 = µaRs(isαs+ i0s− if) + pψas2 = Rfif
ψas2 = −µaLls(isαs+ i0s− if) + µaLm(isαs+ isαr− 2
3µaLmif) (2.8)
2.1.3 Two-phase representation of the IM with ITSC fault either in phase b or in phase c
For a symmetric IM with ITSC fault in phase b, (2.5)-(2.6) will remain same except for the two changes as mentioned below:
1. µawill be replaced by µbwhich is defined as µb =Nb/N. Nbhere denotes the number of faulty turns in phase b.
2. X in this case will be equal to −12 + j
√ 3 2 .
Therefore, for ITSC fault in phase b, (2.5) will be written as in (2.9).
¯
vαβss = Rs¯isαβs+ p ¯ψsαβs− 2
3XµbRsif v0ss = −1
3µbRsif + pψs0s
0 = Rr¯isαβr+ (p − jωr) ¯ψαβrs (2.9) The stator and rotor flux linkages in (2.6) will be given as in (2.10).
ψ¯sαβs= Ls¯isαβs+ Lm¯isαβr− 2
3XµbLsif ψ0ss = −1
3µbLlsif
ψ¯αβrs = Lr¯isαβr+ Lm¯isαβs− 2
3XµbLmif (2.10) The voltage and the flux equations for the shorted turns will be given by (2.11).
vbs2 = µaRs(ibs− if) + pψbs2 = Rfif
ψbs2 = −µaLls(ibs− if) + µaLm(isbs+ isbr− 2
3µaLmif) (2.11) On transforming ibs and ibr into the stationary reference frame, we can write (2.11) as given by (2.12).
vbs2 = µbRs(−1 2isαs+
√3
2 isβs+ i0s− if) + pψbs2 = Rfif ψbs2 = −µbLls(−1
2isαs+
√3
2 isβs+ i0s− if) + µbLm(−1
2isαs+
√3
2 isβs−1 2isαr+
√3
2 isβr− 2
3µbLmif) (2.12)
Similarly, for ITSC fault in phase c, the stator and rotor voltage equations will be as in (2.5) and the flux equations will be as in (2.6) where:
1. µais replaced by µc=Nc/N. 2. X is −12 − j
√ 3 2 .
The voltage and the flux equations for the faulty turns will then be as in (2.13).
vcs2 = µaRs(ics− if) + pψcs2 = Rfif
ψcs2 = −µaLls(ics− if) + µaLm(iscs + iscr −2
3µaLmif) (2.13) On transforming icsand icr into the stationary reference frame, (2.13) will be written as (2.14).
vcs2 = µcRs(−1 2isαs−
√3
2 isβs+ i0s− if) + pψcs2 = Rfif ψcs2 = −µcLls(−1
2isαs−
√3
2 isβs+ i0s− if) + µcLm(−1
2isαs−
√3
2 isβs− 1 2isαr−
√3
2 isβr− 2
3µcLmif) (2.14)
2.2 Symmetrical components equivalent cir- cuits
a c
b
Figure 2.2: Three-phase asymmetric or unbalanced system.
As mentioned in the introduction of this chapter, the ITSC fault creates an asymmetry in the IM. This asymmetry results in the unbalances in the IM’s three-phase quantities, like the stator current and voltage, thus creating an asymmetric system. Figure 2.2 represents a general three-phase asymmetric system. Such asymmetric system can be solved by transforming it into the sets of symmetrical components i.e., positive-, negative- and zero-sequence com- ponents, using Fortesque transformation [12]. Therefore, we get three sym- metrical sets of phasors with the positive-sequence components rotating in the direction of the actual phasors. The same applies to the zero-sequence com- ponents. Contrary, the negative-sequence components turns oppositely. The symmetrical components for the asymmetric system in Figure 2.2 are shown in Figure 2.3. Now, considering the three-phase supply voltage phasors shown in Figure 2.4 are represented as in (2.15).
V˜a = Va, V˜b = Vbe−j2π/3, V˜c= Vce−j4π/3 (2.15) The stator voltage in each phase is the difference between the supply volt- age of the corresponding phase and the voltage at the star point s of the IM, as can be seen in the Figure 2.1 for phase a. Thus, the stator voltage and the stator current phasors, also shown in the Figure 2.4, are defined as in (2.16).
a c
b
(a) Positive sequence.
a
c b
(b) Negative sequence.
a c b
(c) Zero sequence.
Figure 2.3: Symmetrical components of the phasors a, b and c.
˜ Va
˜ Vb
˜ Vc
˜ Vas
˜ Vbs
˜ Vcs
˜ Ias
˜ Ibs
˜ Ics
θv θi 120o
120o
120o
Figure 2.4: Asymmetric three-phase phasors in the IM.
V˜as= Vase−jθv, ˜Vbs = Vbse−j(θv+2π/3), ˜Vcs = Vcse−j(θv+4π/3)
I˜as= Iase−jθi, ˜Ibs = Ibse−j(θi+2π/3), ˜Ics = Icse−j(θi+4π/3) (2.16) Vas, Vbs and Vcs are the phase voltage amplitudes and Ias, Ibs and Ics are the phase current amplitudes. θv and θi are the phase angles by which the stator voltage and current lags behind the reference respectively. It should be noted that the phasors in Figure 2.4, representing a three-phase system, are asym- metric only because of having different amplitudes in the three phases. The phase angle in between the subsequent phasors is kept 2π/3 so as to simplify the case. By using Fortescue transformation, the stator voltage and current phasors can now be transformed into the symmetrical components. The trans- formation of the unbalanced stator voltage and current in a three-phase system into the symmetrical components is shown in (2.17) and (2.18) respectively [12].
V˜as+ V˜as− V˜as0
= 1 3
1 ¯a ¯a2 1 ¯a2 ¯a
1 1 1
V˜as V˜bs V˜cs
(2.17)
I˜as+ I˜as− I˜as0
= 1 3
1 ¯a ¯a2 1 ¯a2 ¯a
1 1 1
I˜as I˜bs I˜cs
(2.18)
Here, ¯a = ej2π/3and ¯a2 = ej4π/3. In (2.17),V˜as+,V˜as− andV˜as0 are the phase a phasors of the positive-, negative- and zero sequence components of the stator
voltages respectively. And in (2.18),I˜as+,I˜as− andI˜as0 are the phase a phasors of the positive-, negative- and zero sequence components of the stator currents respectively. To get the phase b phasors of the positive and negative sequence components, we multiply their corresponding phase a phasors by ¯a2 and ¯a re- spectively. Similarly, for getting the phase c phasors of the positive and the negative sequence components, we multiply their corresponding phase a pha- sors by ¯a and ¯a2 respectively.
Figure 2.2 in a symmetric condition will have equal amplitudes in all the three phases and the phase angles in between any two subsequent phases will be equal to2π/3. Thus, Figure 2.2 in this condition will be similar to its positive sequence shown in Figure 2.3a. The other symmetrical components will be zero. Similarly, in a symmetric IM, the three phases shown in Figure 2.4 are balanced. Therefore, the amplitude of the stator voltage and current phasors in the three phases will be equal. The stator voltage and current phasors in (2.16) can now be written as in (2.19).
V˜as= V e−jθv, ˜Vbs = V e−j(θv+2π/3), ˜Vcs = V e−j(θv+4π/3)
I˜as= Ie−jθi, ˜Ibs = Ie−j(θi+2π/3), ˜Ics = e−j(θi+4π/3) (2.19) Using (2.19) in (2.17) and (2.18), we get phase a of the symmetrical com- ponents of the stator voltage and the current phasors in the balanced condition and are shown in (2.20).
V˜as+ V˜as− V˜as0
=
V e−jθv
0 0
I˜as+ I˜as− I˜as0
=
Ie−jθi
0 0
(2.20)
2.2.1 Symmetrical components representation of two-phase system
Till now, transformation of the asymmetric three-phase system into the sets of symmetrical components have been studied. However, the stator and the rotor equations defined in the sections 2.1.2 and 2.1.3 are in the two-phase stationary reference frame. Therefore, this section shows how to transform an asymmetric two-phase system into the sets of symmetrical components. Later, the symmetrical components will be used to build the positive- and negative- sequence of the T-equivalent circuit.
Stator voltage in (2.16) can be written in the sinusoidal form as in (2.21).
vas = Vascos(ωet − θv)
vbs = Vbscos(ωet − θv− 2π/3)
vcs = Vcscos(ωet − θv− 4π/3) (2.21) Using Euler’s transformation, the sinusoidal voltage in (2.21) can be writ- ten as a sum of two complex exponential. For example, ˜vaswill be written as in (2.22).
vas = 1
2[Vasej(ωet−θv)+ Vase−j(ωet−θv)] (2.22) The amplitude and phase angle in (2.22) can be combined to write the equation in the phasor form as
V˜as= Vase−jθv.
Therefore, (2.22) can now be written as in (2.23).
vas= 1 2
V˜asejωet+1 2
V˜as∗e−jωet (2.23)
Similarly, (2.23) for phase b and phase c voltages will be written as in (2.24).
vbs = 1 2
V˜bsejωet+1 2
V˜bs∗e−jωet vcs = 1
2
V˜csejωet+1 2
V˜cs∗e−jωet (2.24) In (2.24), the quantities are defined as:
V˜bs = Vbse−j(2π/3+θv) V˜as= Vcse−(j4π/3+θv).
Applying the Clarke transformation on (2.21) transforms it into the two- phase system. Now using the Euler’s representation, the two-phase system can further be written as shown in (2.25).
¯ vαβs = 2
3(vas+ ¯avbs+ ¯a2vcs)
= 1
3( ˜Vas+ ¯a ˜Vbs+ ¯a2V˜cs)ejωet+1
3( ˜Vas∗ + ¯a ˜Vbs∗ + ¯a2V˜cs∗)e−jωet (2.25)
