UNIVERSITY OF LONDON
IMPERIAL COLLIDE OF SCIENCE AND TECHNOLOGY
Department of Electrical Engineering
TRANSIENT PHENOMENA IN LARGE INDUCTION
MOTORS IN A POWER SYSTEM
by
Swann Singh KALSI, B.Tech.(Hons.),M.Sc.(Eng.)
Thesis submitted for the degree of
Doctor of Philosophy in the
Faculty of Ensineering
LONDON, July 1970. 2
TO
DEEP AND ICARAM 3
ABSTRACT
As the size of the individual induction motor has increased significantly in the last few years, the behaviour of induction motors-in a power system under fault condition has assumed importance.
The two main problems considered are the immediate fault current, which determines the rating of switchgear and the subsequent stability of the system.
A standard method of determining the fault contribution made by 1 induction motors to the fault currents in a system is recommended . The basis of the proposals is that an induction motor can be treated in the same way as a synchronous machine. The recommendations are supported by a detailed theory of induction motor transients, in which accurate formulae are derived for the current following a sudden short circuit. The decpbar effectin the squirrel cage motors is theoretically simulated by two equivalent windings on the rotor. The proposed method is a simple one and is particularly useful for determining switchgear ratings in a mixed system containing both synchronous and induction machines, since it uses parameters of. the same kind for both, and permits existing procedures and method of computation to be used without change. The other problems studied on constant speed assumption are listed in Sect. 1.6.
The study has been extended to the transient stability of a large induction motor when it forms part of a composite system. A full mathematical derivation of the equations is given, together with a comparison with previously suggested methods. The analysis has been extended to a general multi-machine system also containing synchronous machines. A comparison between the test and calculated results with various methods of machine representation was made on a single model 4
induction motor, and on a simple system containing the motor and a micro-alternator, which is representative of a large generator. The model induction motor was especially designed to simulate a 1800 h.p. induction motor on a per-unit basis, and its main features are the low resistances and the use of a dcepbar cage winding.
The accurate representation of induction motors has been shown to be important in a system stability study for reliable and accurate results.
It is shown that the stray load losses have an important effect on the prediction of individual machine behaviour, and on the overall system• behaviour when the two types of machines are present.
At an early stage of the project, some work was carried out on synchronous machine stability2. This work, described in Part 1V, was a valuable preparation for the investigations as a whole and particularly for the combined system described in Chapter 11. 5
ACKTOWLEDGEnTITS
The work presented in this thesis was carried out under the supervision of Dr. B. Adkins, M.A., D.Sc. (Eng.), C.Eng., F.I.E.E. of the Electrical Engineering Department, Imperial College of Science and Technology, London. I wish to thank Dr. Adkins for his helpful guidance and constant encouragement.
I am grateful to the Science Research Council (U.K.) and the
Electrical Engineering Department, Imperial College of Science and
Technology, London, for the research assistantship under a grant from the S.R.C., and for the facilities, including the use of the IBM-70941I and CD6-6600 computers, to pursue the :•rork. The sincere appreciation is also extented to the S.R.C. for permission to register for the higher degree of the University. The author also thanks Messrs. E.E.-
A.E.I. Machines Ltd., Rugby, for the information on the large motors.
I also wish to thank Mr. A.J. Parsons of Messrs. Mawdsley's Ltd.,
Dursley and Mr. D.D. Stephen of Messrs. E.E.-A.E.I. Machines Ltd., Rugby for valuable discussions.
Finally, I thank my colleagues who have contributed, directly or indirectly, to the success of the project. 6
TABLE OF CONTERS
Abstract 3
Acknowledgements 5
List of symbols 12
CHAPTE2 1 INIRODUCTION. 16
1.1 General 16
1.2 Problems associated with large induction motors 16
1.3 Review of the investigations 17
1.4 The model induction motor 19
1.5 Method of analysis 19
1.6 Problems studied and the new conclusions 20
PART I: THE LABORATORY EC4UIP1-EiT 26
CHAPTER 2 THE MODZI, INDUCTION MOTOR 27
2.1 Introduction 27
2.2 Comparison of large and small induction motors 29
2.3 Theory of deerbar induction motor 33
2.3.1 Basic design of an induction motor having retangular 33 deepbars.
2.3.1.1 A.C. impedance 35
2.3.1.2 General procedure for designing an induction motor 37 having rectangular deepbars
2.3.2 The design of the model induction motor 38
2.3.3 Inverted T-shared bar 70
2.3.4 Compound bars made of tapered and rectangular sections 43 2.3.4.1 Exact solution for estimatinn: the imnedonce of a L.5 tanered bar
2.3.4.2 An approximate solution for tho in edance of tancred •-vu
and comnosite bars 7
2.3.4.3 Step-by-step method of calculating the bar impedance 47
2.3.4.4 Circuit-analysis method of calculating the bar 49 impedance
2.3.4.5 Formulae for computing the other parameters 50
2.3.5 Comparison of results computed by various methods 50
2.3.6 The model induction motor set 62 2.4 Measurements for checking the computed results 68 2.4.1 No-load test 68 2.4.2 Locked rotor test 68 2.4.3 Torque speed characteristic 68 2.4.3.1 Calculation of the secondary impedance 73 2.4.4 Stray load losses 82
2.5 Conclusions 82 CHAPTER 3 OTHLa EXPERIMENTAL EQUIP= 85
3.1 The model motor system 85 3.2 Induction motor load simulation 87 3.3 Measurement of motor speed 87 3.4 Simulation of the transformer and the transmission line 91 impedances
3.5 Synchronous machine model 91 PART II: THE FAULT STUDIES OF LARGE INDUCTION roToRs 96 CHAPTER 4 INTRODUCTION 97 4.1 Review of investigations 97 4.2 The'object of the thesis 99 4.2.1 Fault currents in a system containing synchronous machines 99
4.2.2 Fault currents in a system containing induction motors 101
4.2.3 D7.termination of the sub-transient reactance or an induction 102 4
motor 8
4.2.3.1 The sudden short-circuit test 102
4.2.3.2 The standstill impedance test 102
4.2.3.3 Measurement of the frequency response characteristic 103
CHAPTIM 5 TRANSIENT THEORY OF A DEEP3AR INDUCTION MOTOR 105
5.1 Representation of a deepbar induction motor 105
5.2 Short circuit of an induction motor 107
5.2.1 Current and torque after a direct short circuit 107
5.2.2 Indirect short circuit 111
5.2.2.1 Terminal voltage after disconnection 111
5.2,2.2 Current and torque after a short circuit 112
5.3 Switching an induction machine to the supply 113
5.3.1 Machine running with trapped flux in the rotor 114
5.3.2 Machine running without trapped flux in the rotor 116
5.3.3 Machine initially at standstill 118
CHAPTER 6 EXPERIMENTAL PROCEDURE 121
6.1 Determination of transient parameters from the admittance 121 locus
6.2 Model test results and comparison with calculations 123
6.2.1 The direct short circuit test 125
6.2.2 The indirect short circuit test 125
6.2.3 Variable frequency impedance test 125
6.2.4 Variable speed impedance test 128
6.2.5 Approximate determination of operational admittance from 128 standard tests
6.2.6 Comparion of transient parameters determined by various 131 methods
CHAPTER 7 coNalusoN OF CMPUTED AND TEST RESULTS 1364
7.1 The model motor 136 9
7.1.1 Direct short circuit test 136
7.1.2 Electrical torque during the short circuit 138
7.2 Application of the method of calculation to the large machines 138
7.3 Indirect short circuit test 143 7.4 Transient current and torque following the sudden connection 148 of a machine to the supply
7.4.1 Machine initially running with trapped flux 148
7.4.2 Machine initially running without trapped flux 152 7.4.3 Machine initially at .standstill 152
CHAPTER 8 CONCLUSIONS 158 PART III: TRANSIENT STABILITY OF POWER SYSTEMS CONTAINING BOTH 160 SYNCHRONOUS AND INDUCTION MACHINES
CHAPTER 9 INTRODUCTION 161
9.1 General 161
9.2 Influence of digital computer on system studies 162
9.3 The past work 162
9.4 The object of the thesis 163
CHAPTER 10 STUDY OF A SINGLE INDUCTION MACHINE SYSTEM 165 10.1 General 165
10.2 Mathematical derivations 165
10.2.1 Accurate representation 165 10.2.2 Approximate representation 167
10.2.2.1 Method A (pli, and s terms neglected) 167
10.2.2.2 Method B (pyl term neglected) 168
10.2.2.3 Method C (Steady state equivalent circuit) 168
10.3 Comparison of computed and test results 169
10.3.1 Effect of rotor trapped flux 170
10.3.2, Three phase short circuit at full load 170 10
10.3.3 Effect of stray load losses 173
10.3.4 Open circuit fault at full load 175
10.4 Numerical integration techniques 175
10.5 Conclusions 182
CHAPTER 11 MULTI MACHINE SYSTEM STUDIES 184
11.1 System under investigation 184
11.2 Synchronous machine representation 186
11.2.1 Alternator equations 186
11.2.2 Accurate representation 188
11.2.3 Approximate representations 189
11.2.3.1 Method A (ppd, pTcl, and s terms neglected) 189
11.2.3.2 Method B (pY and ptP qterms neglected) d 190
11.2.3.3 Method C (Damping neglected) 191
11.3 System network representation 192
11.3.1 Transformation from the machine axis (d, q) 192 variables to the system ( cc, p axes) variables
11.3.2 Accurate representation of the network 193
11.3.3 Approximate representation of the network 194
11.4 The method of analysis for the system under investigation 194
11.4.1 Method-X 195
11.4.1.1 Method-Xi (Accurate) 195
11.4.1.2 Method-X2(APproximate) ' 201
11.4.2 Method-Y 205
11.4.3 Method-Z 208
11.5 Application of the method of calculations to a general 211 n--machine system.
11.6 Conclusions 2111 APTIM 12 SUGGESTIONS FOR TEE FUTURE WORK 217 rPART 1V: OTHER STUDIES ON THE SYNCRHONOUS MACHINES 219
CHAPTER 13 STUDIES ON THE ALTERNATORS 220
13.1 Quadrature axis excitation studies 220
13.2 Design of a D.W.R. micro-alternator 231
Appendices 236
References 277 12
LIST OF SYMBOLS
Any symbol which does not appear in this list
will be defined or specified separately in
the text. vd , vq : direct - and quadrature - axis voltages v , i : instantaneous phase voltage and current a a ia : direct = and quadrature - axis currents d V : bus voltage b V : peak voltage and current m Ira I : steady complex axis voltage and current V10' 10 V I : transient complex axis voltage and current 1 , 1 V' , V" : voltage behind transient and sub-transi6nt reactances
V' I' : change in primary voltage and current ' V td'vtq : d-and q-axis components of terminal voltage of alternator } and bus voltage vbd'vbq V : complex axis voltage-and current of the alternator gaga V ,I : complex axis voltage and current of the motor ma ma V , VR : components of the bus voltage a : complex voltage and current on the c:, f3 axes for altermtor ✓ $ Is I : complex voltage and current on the a , p axes for motor ✓ m V : complex voltage on the c, p axes
X, X', X" synchronous, transient, sub-transient reactance and admittance y, yt , yet
Xi,X2,X3 : leakage reactances and inductances L L L 2' L , X : magnetising inductance and reactance m m • L X field leakage inductance and reactance f ' f 13
X : armature resistance and leakage reactance a a R X
R b Xb : resistances and reactances of the tie line R X c c R X f ' f R X : resistance and reactance of generat-or transformer g g X(p),Y(p) : operational impedance and admittance
ZZ Z : impedances 2' 3 Z : impedance of the bar ac R s,X sZ , : starting resistance, reactance and impedance s rr r : resistances 2' 3 L L : bar and core length b c : instantaneous axis flux 4) d ' 4)(1 141 Tqy : complex axis flux : field flux linkages LII?f A , B : Real and imaginary part of admittance
di , d2 : height of bar, cms.
r : ratio of bar width to slot width
K : ratio of starting to full load torque
4/ W : width of the bar, cms. 2' 3 T : instantaneous torque o T ,T ,T : starting, full-load, and pull-out torque s n m q : number of phases
Ns : synchronous r.p.m.
at and Laplace operator
: frequency
s slip
s , s m g : motor and generator slips : load torque 1 14
S , s : slip at point of pull-out torque and full-load slip
H : inertia constant
J :polar moment of ineria (1) (1) Ho , : hankel functions (1) (1) J l) o J1 : bessel functions • I.V. : integrable variables
N.I.V. non-integrable variables
CR : ratio of computer to real time
C.W.R. : conventionally wound rotor
D.W.R. : divided winding rotor
T.C.R. : time constant regulator
: Laplace transform
6 : load angle
6 : load angle of generator
6 : load angle of motor (= sw t) m o t, t', t 1 : time • ,741 : transient and sub-transient short circuit time constants
TI T" : ' o transient and sub-transient open circuit time constants T : armature time constant a , : switching angle
(') ' wo : actual and synchronous angular frequency : resistivity , 0
13 , /311 : impedance angles R f nt I o ' 'o a : skin effect coefficient
4 15
Subscripts
a , : axis components d , direct-and quadrature axis components
g , m : generator and motor quantities kd, kq direct,quadrature axis components of damper winding : field component
It : transient and sub-transient component / : bar above a symbol indicates a Laplace/ixansiorm
: bar below a symbol indicates a phasor (thick letter) 16
CHAPTER 1
INTRODUCTION
1.1 General
The steady rise in the loading and transmission voltage of power systems, and their capital cost make it essential that the capabilities of such systems are utilized to the maximum extent. Although it has been known for a long time that the induction motors can influence the system during a disturbance, it is only in recent years that engineers have shown interest in including these machines in the system studies.
Large induction motors may now form a significant part of the load on a system. Developments in switchgear design have reduced the operating times while the time constants of the motors have increased with their increased size.
1.2 Problems associated with large induction motors
Increasing use of large induction motor drives make it absolutely necessary that their effect on the system should be studied more carefully for the following reasons. In the event of a 3-phase fault in a system containing large induction motors;
i) the induction motors tend to feed current into the fault.
In modern plants, rapid clearance of system faults is
frequently called for in order to minimize the disturbance
to the system, and therefore faster urotective gear has been
developed. In step with this development, circuit breaker
onerating f,; 11105 have also been greatly reduced, sc) that a
total time to contact separation of five cycles is feasible,
and at this time the contribution from any large induction 17
motor connected to the system may still be appreciable.
ii) a sudden loss or reduction of the voltage at the terminals
of a motor driving a pump or other apparatus is experienced
and therefore the speed of the motors fall. The extent
to which the speed falls depends on how long the under
voltage persists at the motor terminals, the amount of
voltage reduction and the characteristics of the drive.
When the voltage is restored (i.e. the fault cleared) the
motors will take a current in excess of normal, and thus
the voltage will not be restored immediately to the nominal
value, owing to the system voltage drop at this higher
current. The voltage will gradually return to normal if
the system is capable of withstanding the disturbance.
In the past the above problems were studied separately as the first
one is important for accessing the capacity of the circuit breaker whereas
the second one is concerned with the stability of the system as a whole.
As has been mentioned above, when induction motors accelerate after a
disturbance, they cause considerable voltage drop in the system reactance
and thus it should be made as small as possible. However, if the system
reactance is reduced, the fault level of the system rises and thus the
choice of the circuit breaker is affected. Requirements for the two cases
are contradictory. In view of this, it is essential to make a unified
_study of the two problems.
1.3 Review of the investigations
. A number of studies were made in the past cn the fault contribution]-l8 19-25 to the system and the recovery problems associated with large induction
motors.
For quite sometime, the induction motor's contribution to the fault 18
1 evel was regarded as insignificant. The work was mainly for single cage 17 machines having no skin effect in the rotor winding. Vorapamorn- analyzed the deepbar induction motor but he could not establish the validity of his theory for the lack of experimental results. Detailed review of this problem is given in Chapter 4.
The recovery problems of the large motors also received a continuous attention since a long time. It is being recognized that a load containing a large concentration of induction motors, as in an oil refinery, or a chemical plant, cannot be satisfactorily represented by a constant impedance. 19-22 Though the methods have been developed to allow for the special
properties of such motor loads, a satisfactory method of analyzing the induction motor having deepbar winding is still not available. Humpage et al25 tried to tackle this kind of machine by first pre-calculating the variation of secondary impedance with frequency and then adjusting the rotor impedance accordingly in the process of solving the equations for a single cage machine. The disadvantage of such a method is that firstly it is not always possible to get design_ information needed for pre-estimating
the rotcr impedance at various frequencies and secondly the calculated impedance may differ appreciably from the actual values if great care is not taken while making the calculation. The past studies on the recovery
problems have been reviewed in greater details in Chapter 9.
In general, draw back of most of the work done in the past for
studying the above problems is that while analyzing the induction motor, it was treated on its own merits without any reference to the synchronous 7-0 1 ' 16 machines.- '4- - The theory was checked mostly on small machines and
was then extrapolated to cover the large machines. Though this might be
true for a few particular cases, no general conclusions could be drawn. 4 22 Vorapamor n 1 7 and Alf or d however adapted the well established approach to 19
synchronous machines to study the transient behaviour of induction motors, but they had h handicap that a suitable machine, on which • 22,26-28 the theory could be checked, was not available. The micro-alternators are in common use for checking the theory of large synchronous machines.
Thus for the purpose of establishing the theory of the transient performance of.the large induction motor, it was essential to design a suitable model motor.
1.4 The model induction motor
With the above in perspective, a model induction motor was designed to simulate a large 1800 h.p. motor on a per-unit basis. The design of the model is fully described in Chapter 2. The two important operating charateristics of an induction motor are the torque-speed curve and the current locus curve, and the model was designed such that these two curves agreed with those of the large machine. The calculations for the rather unusual shape of the bar (Fig. 2.17) used in the final design required some modifications of the published theories of the deenbar machine.
There was some discrepancy between the measured values for the model and the derived values based on the 1800 h.p. motor and these are discussed in Sect. 2.+ and 2.5 .
The design of the model motor is original. It was manufactured by Messrs. Mawdsley's Limited.
1.5 Method cf anal s' .4 The synchronous machine and the induction motor have much in common, both in their construction and in the internal relations between 20
the currents, m.m.f.s, fluxes and voltages. Both consist of a primary
winding, usually three phase, and a secondary system on the rotor,
comprising several circuits, one of which in the synchronous machines
is excited from a d.c. source. Similar equations apply to the two
types of machines, which are essentially the same devices used in
different ways. Nevertheless the method developed for their study,
analysis and design differ radically, both as presented in text books7'8,12 14-1620 and in the papers dealing with transient performance.
Because of the common features mentioned above, it was decided to
follow the well established transient theory of the synchronous machi4,29,30
for analyzing the induction machine. Based upon the concept of operational
impedance (Eqn. B45), the deepbar secondary winding is represented by two
coils on each axis of the rotor. Although the eddy current effects are
only approximately simulated by these coils, the experimental results show
that the simulation is close enough for practical purposes. The fact,
that the induction motor runs below the synchronous speed with a small
slip s, makes the analysis a bit more complicated. On the otherhand,
because of the machine symmetry, the mathematics can be simplified by
using complex numbers to epxress the variables, as explained in Appendix B.
The equations, assumptions and sign conventions are those of Adkins
1.6 Problems studied and the new conclusions
The investigations carried out in this thesis are broadly divided
in sour sections, viz. Parts I , II III 1 end IV .
- The main object of the work in Part II was to find a suitable method
for calculating the fault contribution of the large induction motors
having pronounced deepbar effect. As with the synchronous machine, the
current dies away rapidly and it is reasonable to assume that the speed remains constant during the period considered. With this assumption, the 21
differential equationt of the machine are linear. Based on comprehensive theory, the following) recommendations' are made for the purpose of 1 calculating the fault contribution of large induction motors. Further details of the method are given in Chapter 5.
1. The short-circuit current of an induction motor can be adequately represented by two exponential decay functions determined by four parameters, viz. transient and sub-transient reactances and time constants. The most improtant quantity is the instantaneous peak current after the short, since this determines the "make" rating of the switch.
To determine the peak with reasonable accuracy the only parameter required is.the sub-transient reactance.
2. When the purpose of evaluating the machine parameters is to predict short-circuit current, it is obviously preferable to measure them from the results of suddent short circuit tests on the machine whenever possible. This principle is adopted in B.S.429631 and in I.E.C.
Publication - 34-432.
3. When sudden short circuit tests are not available, the.required machine parameters may be deduced from other test results, but they are likely to be less accurate for the purpose of predicting fault currents.
In the past, the standstill impedance test has been used, adopting, where necessary, empirical correction factors to allow for saturation and other effects. The implicit error in this procedure is explained in Part II.
By using the impedance test in conjunction with others normally available, it is.possible to obtain a much more accurate estimate of the machine transient parameters, using the "Frequency response characteristic" approach described in Chapter-6.
4. The transient parameters can be calculated from the machine design data. In the pas'c the method has been based on the calculated value 22
of standstill impedance and is subject to errors of considerable magnitude. A more accurate method uses the values of resistance and reactance, expressed as functions of slip, which are required to determine the starting characteristic, and allow for any "deepbar" effect. The "Frequency response characteristic" can then be used to determine the transient parameters.
5. In many cases a switch rating based on the make duty at the instantaneous peak is also satisfactory for a break duty at a reduced current after a period of decay. In a multi-machine system a network calculation, using the sub-transient reactance of each machine, would be made to determine the switch rating. When necessary the manner of decay can be estimated using a mean effective time constant, or, an accurate value can be obtained by a step-by-step transient calculation, if the amount of computation is justified.
In addition to the above, the following problems have also been studied for a large deepbar induction motor for the first time.
1. Electrical torque developed-by the machine during the direct
short circuit.
2. Determination of terminal voltage consequent to an interruption
of the supply.
3. Starting current and torque of the induction motor.
4. Transient current and electrical torque when a machine,
with and without trapped flux in the rotor, and running
at a given slip, is suddenly connected to the fixed supply.
To verify the validity and accuracy of the theory, the model motor described in Part I was used. Quite reasonable agreement was obtained between the computed and the measured currents and voltages. The torque was not measured and thus it was not possible to check its computed value. 23
Ps the derivation of the torque equation is also based on the sane theory as the current, it is expected that the computed torque would also exhibit a reasonable agreement with the actual value.
In Part III of the thesis, various methods currently in use for analyzing the transient stability of a system consisting of both synchronous and induction machines, are discussed. When a fault occurs on a multi-machines system the synchronous machines swing in angle and the induction motors drop in speed. A stability analysis is needed to determine whether normal conditions would be restored after removal of the fault without any motor stalling or any alternator pulling out of step. For synchronous machines, there are established methods of 11,22,25,33-35 analysis. Similarly there are a number of methods of :19-23,25 analyzing the single cage induction motor without any skin effect.
Many large squirrel cage induction motors are installed in the system and are mostly started direct on line. The largestknown motor of this kind is of22000h.p.36 capacity. Stich motors normally have pronounced deepbar effect, which should be suitably accounted while making the stability calculations. No records of the. tests on similar large machines are published and thus it is not possible to verify the theory. The present thesis includes a record of the tests on the model motor (described in Part I) when it is connected to an infinite bus as well as when it forms a part of a multi-machine system wherein the model motor and a micro- 27 alternator are connected to the infinite bus through a simulated transmission line. Induction motor representation is similar to that used for the fault contribution studies (Part II) except that the speed is now allowed to vary. The speed is determined by solving the torque equation along with the other differential equations of the machine in a step-by-step method of solving a set of non-linear differential equations. 24
The type of representations studied for each machine are:
Induction motor:
Accurate No l approximations are made in the basic machine
equations (Sect. 10.2.1)
Approximate Method - A, p4(1 and s terms neglected (Sect. 10.2.2.1)
Method - B, pyl term neglected (Sect. 10.2.2.2)
Method - C, Steady state equivalent circuit (Sect. 10.2.2.3)
These representations of the induction motor for the stability studies have been developed for the first time. ,22,35 The various available methodsli of representing synchronous machines were used for the multi-machine studies. These methods are listed below:
Synchronous machine:
Accurate Ho approximations are made in the basic machine
equations. (Sect. 11.2.2)35
Approximate Method A. pyd, pyq and s terms neglected 22,55 (Sect. 11.2;3.1)
Method B. py d, p\.pq terms neglected (Sect. 11.2.3.2)22
Method C.Damping neglected (Sect. 11.2.3.3)11
It was found necessary to account for the effect of saturation in the . synchronous machines. The method discussed in Ref. 2 for steady state stability studies was extended to the transient stability studies.
Part IV deals with the further studies made by the author in the field of power system stabilityt
A contribution was made towards the improvement of steady state stability limit of synchronous machines using controlled excitation on an additional field winding. Using a field winding on the quadrature axis, 4 it has been 'shown that it is possible to maintain stability at large 25
negative reactive tower, at any output. The results are incorporated 2 in a joint paper published recently.
The quadrature winding machine however is not a feasible manufacturing proposition and a D.W.R. machine, having spit rotor winding, has been proposed38 as a practical alternative. A model D.W.R. machine, designed by the author in collaboration with the manufacturer Messrs. Mawdsley's
Limited, is in use at Imperial College for further studies.
4 26
THE LABORATORY EP,UTETEIT 27
CHAPTER2
1 THP: KODEL INDUCTION EOTOR
- 2.1 Introduction
The increasing size of the induction motors and the faster
opening times of the associated switchgear, have made it increasingly
necessary to predict accurately the performance of the induction motors
in a power system and to study their effect on the other equipment
connected to the same source. For example, if a three phase fault occurs
in a system in which there are large induction motors, the following
effects are observed.
1. The fault is fed from the induction motors in the system
as well as from the generators and thus there is an
increase in the fault level.
2. The induction motors lose speed during the fault period.
After removing the fault, the induction motors draw heavy
currents from the system and cause the line voltage to
drop. If the loss in the voltage is excessive, the
induction motors may not be able to recover and therefore
may stall.
The main difficulty in studying the above problem is in carrying
out full scale tests which may cause instability in the system. On the
otherhand, tests performed on small laboratory sized machines give
Unrealistic results, because their parameters are quite different from
:those of the large machines on a per unit basis, (Table 2.1.).
Experiments performed by .41ford 22 and Vorapamorni7 at Imperial College
have confirmed this. 28
TABLE 2.1.
Per unit=ameters of the 1800 h.p. motor and of the 5 h.p. motor
• Particulars 1800 h.p. 5 h.p. motor motor
Base voltage (volts/ph.) 11000/ 3 400 Base current (amps./ph.) 81.33 4.5 Base impedance (ohms/ph.) 78.0 89.0 Stator resistance 0.0074 0.0546 Stator leakage reactance 0.1485 0.09 Magnetizing reactance 4.32 2.54 Rotor resistance, (starting) 0.0315 0.0484 -- do -- (running) 0.0079 0.0484 Rotor leakage reactance, (starting) 0.082 0.09 -- do -- (running) 0.1276 0.09 Pull out torque 1.941 2.05 Slip at point of pull out torque 0.04 0.27 Full-load slip 0.008 0.06 Starting torque (break away) 1.941 1.2 Starting current (break away) 4.569 5.21
Unless mentioned otherwise, all values in the above table are in per-unit.
• 29
are •Model machineP '26,27 in common use for studying the behaviour of synchronous1 machines and they have proved valuable in investigating the abnormal operating conditions which cannot be studied on an actual system. It was therefore decided to develop a similar dynamic model for a large induction motor. The model was designed as far as possible to simulate all the important operational parameters as well as the torque slip characteristic and the current locus, of a large 1800 h.p. motor.
2.2 Comparison of la and small induction motors Fig. (2.1) shows the dimensions of the iron punchings and of the copper bars of typical squirrel cage rotors of a large 1800 h.p. and a small 5 h.p. induction motor. Table 2.1 gives the per-unit values of some of the parameters and operating quantities of the two machines.
The principal differences are in the lower resistances and in the properties of the deep-bar winding of the large machine. Figs. (2.2) and (2.3) give the torque-slip curves and the current loci of the two machines. It can be seen that the values of slip at full load and at the pull-out point are much greater for the small machine. The starting torque of the small motor is twice that of the large motor and the pull-out torque is somewhat greater. The magnetising current of the small machine is much greater. It was therefore necessary to make a model motor of special design. In the further text, the 1800 h.p. induction motor and the model induction motor will be referred as "large motor" and "model motor" respectively.
The new model motor has the same stator winding as one of the model 28 synchronous machines and thus has a suitably low stator resistance and is of comparable rating. The main special features are in the design of the rotor. The rotor bars have a special shape (Fig 2.17) and the motor 762 aia.\
scale 1: 1 .'300
Soile: 1:2.5 Ca) ( All aimensions in millimeters )
Piqure 2.1 Conrcarative- mansions of the rotors a. 1800 h. p. motor ( /1.-poleo b. 5 h. p, /no-:or ( /4.--polo3) 9.()
0.5
r- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Slip P-u- Figure 2.2 : Torlue - slip characteristics r. 1800 h.p.motor b. 5 h.-n.motor 3 2.5_ 44 .5
2 _
,021.,
03 1
.0075 6 .8
. 0025
0 1 2 3 Imaginary girt n.u. Figure 2.3 : Current loci r.. 1800 h.p.motor SP b. 5 h.p.motor Points marked. represent per-unit slip 33
was shown by calculations to have approximately the same characteristics as the 1800 h.p. motor. (Figs. 2.18 and 2.19).Design details of 1800 h.p. motor are given in Table-2.2 The calculations made and the -test results obtained on the model motor are described in the following pages.
2.3 Theory of deepbarTEIWW** induction motor A precise calculation of the motor bar impedance as a function of frequency is essential in order to estimate accurately the variation of the current and the torque with the speed. In a deepbar induction motor, the secondary resistance is large at standstill but decreases as the speed rises. If the squirrel cage winding consists of narrow deep- bars, embedded in the slots, eddy currents are induced by the oscillating transverse magnetic field which causes a non-uniform distribution of current and an increased resistance at standstill. At the full load running speed, when the slip frequency is low, the bars carry more evenly distributed currents and the resistance is low.
Many researchers40 I41 have produced formulae by reducing the multi- dimensional skin effect nroblem to one-dimension by making the following assumptions.
1. The permeability of the tooth and core material is infinite.
2. The flux lines cross the slot in straight lines.
3. The rotor body is laminatedso that the flux linking the
conductor causes no eddy currents in the core.
4. The resistivity of the conductor is uniform over the whole
cross-section.
2.3.1 Basic design of an induction motor having rectangular dee-obars
An attempt will be made here to study the effect of various factors on the performance of the induction motor by neglecting the following: 34. TABLE 2.2
Design details of a J.800 h.p. souirrel cage induction motor
Specifications Nominal horse power 1800. Capacity, KVA 1550. Line voltage, vats 11000. Phases 3. Poles 4. R.P.M. 1488. Connections star Full-load current, amps. 81.33 Unit current, amps. 81.33 Unit admittance, mho 0.0128 Magnetic circuit Flux per pole, weber 0.1255
cross-sec. flux density length Amp. turns sq.m. wb/sq.m. m
Stator core 0.0888 1.410 0.3050 165.406 Stator teeth 0.1620 0.774 0.1215 20.041 Rotor core 0.0942 1.332 0.1680 45.452 Rotor teeth 0.1838 0.683 0.0557 7.187 Air a. 0.2048 0.612 0.00254 1684.980 Total ampere-turns per pole 1923.07 Carter coefficients 1.0278 Magnetising current, p.u. 0.232 Stator finding Pitch 0.83 Winding factor 0.924 Total slots 72. Effective conductors per slot 20. Conductor section, sq. m.m. 31.5 Mean length of turn, m. 3.38 Turn per phase 240. Resistance, p.u. 0.0074 Rotor winding Winding type Deepbar squirrel cage Total slots 60. Bar material copper Bar section, sq. m.m. 307. Bar length, m. '0.674 End ring material copper End ring section, sq. m.m. 1935. Running resistance, n.u. 0.0079 Starting resistance, p.u. 0.0315 Reactances Stator leakage reactance, p.u. 0.1485 Rotor leakage reactance, p.u. (running) 0.1217 -- do -- (starting) 0.082 ' Magneting reactance, p.u. 4.32 Performance FUll-load slip 0.0081 Starting torque, p.u. 0.587 Starting current, D.U. 4.569 Pull out torque, p.u. 1.941 Slip at point of pull out torque, p.u. 0.04
35
1. The magnetizing current as it is normally quite small in
large induction motors .
2. The stator resistance.
3. The skin effect at full load rurring speed for the large induction motors.
The performance of any proposed induction motor can be, more or less, fully defined by specifying the following,
a. Rill load current, torque and speed
b. Starting current and torque
c. Pull out torque and corresponding speed. With the above information the following can be obtained. D.C. resistance of the secondary winding,
T s n n Ro 2 (2.1) n
Starting resistance of the secondary winding,
T s 2 i2 (2.2) K n
Resistance of secondary winding at the point of pull out torque, V2 . m (2.3) 2m 2 T'm and reactance at the point of pull out torque,
X -v2 - X (2.4) 2m - 2 Tm 1
2.3.1.1 A.C. 1T-Tdance
The resistance and reactance of a rectangular bar, d ems. deep and w ems wide, at any frequency, f are given by 42
36
aci-(Sinh 2txcl ÷: Sin 2a4) R = R (2.5) ac Rob (Cosh atd - Cos ala)
agyinh 21y.d - Sin 2ad) = R (2.6) Xac ob - Cos 20A)
sf 6 where a = 2 Tc‘i 10- (2.7) r P 11:L 1 -6 (2.8) and p = 2 74-pr1.- 0 r = slot width/bar width It is possible to expand Eqns. (2.5) and (2.6) as polynomial in ad. If higher powers of ccd are neglected, it can be shown that 112 , 1. (2.9) Rac = Rob (ad) (2.10) Xac = Xob (112a d)
d Ib (2.11) where Rob = I lid . 2f .d 70-6 (2.12) and xob = 3 -61r_
If the resistance and reactance. due to the end rings as well as all the other components of the secondary reactance, except the slot leakage reactance, are neglected, the following relations hold. 1. Under full load operating condition, (2.13) Ro = Rob Nt = X N o ob t (2.14) where Nt = transformation ratio between the primary and the secondary windings . 2. When the motor is at stand still,
- Rs = R0 p (2.15) X = x 1.,5 s opa (2.16) 37
3. At the poInt of the pull out torque,
R =R_ p d (2.17) am v V-sm ' s (2.18) X-2m X'op d m 2.3.1.2 General procedure for desirming an induction motor havin7 rectanGular deelDbars. There can be a number of possible approaches to obtain a desired design. A simple procedure for designing any kind of deepbar induction motor is outlined below. using Eqn. (2.1). 1. Choose suitable value of Ro of the motor 2. Obtain the required starting resistance, Rs from Eqn. (2.2).
3. From Eqns. (2.2) and (2.15), find the value of d and substitute it in Eqn. (2.11), to get w.
4. Obtain effective resistance and reactance of the secondary winding at the point of pull out torque from Eqns. (2.17) and
(2.18). 5. Values of resistance and reactance calculated in the step 4, must agree with the values obtained from Eqns. (2.3) and
(2.4) respectively. 6. If the resistance R andoes not agree with the value obtained from Eqn. (2.3), then it may be necessary to revise the
requirements of Tm and sm.
• If the reactance X2m does not agree with the value obtained from Eqn. (2.4), then the requirements of Tm and K should be revised as the lattercontrols the value of X.
The above stated procedure leads to a trial and error method in designing the suitable machine. A considerable amount of labour is saved if first the rotor is designed to give required starting and full load
38
1 running resistances and reactances. A suitable stator is, then, designed , by varying its winding details and leakage reactance to meet the require- ments of starting current, null out torque and the slip at which it occurs. 2.3.2 The design of the model induction motor
The problem was to design a rotor for use with the micro-machine stator28, which would simulate the large motor. The first attempt used a rectangular bar 57 m.m. deep and 2.5 m.m. wide. The calculated results are given in Table 2.3 and Figs. (2.4) and (2.5). The results in general meet the requirements. However, the shape of the torque-slip characteristic and the current locus do not exhibit a good agreement with that of the large motor.
2.3.3 Inverted T-shaped bar
For the reasons mentioned in the preceding section, a rectangular bar was discarded. For meeting the requirements of higher starting resistance, it was necessary to keep the upper portion of the bar narrow and to achieve lower running resistance, the area of the lower portion had to be increased. The next proposal was to use T-shaped bar (Fig. 2.6). The 43 impedance of such a bar is given as,
L Coshald, Coshaid2 4. Sinhald, Sinh a id2 = b 2 zacW 1 1j (2.19) 1 Cosha ld1 Sinha ld2 + 172- Sinha id, Cosh aid2
where a 1 = (1 j)ct (2.20)
Since the exnression contained two variables a.tdl and ald2, it was decided to vary each dimension in turn to study its effects on the torque slip characteristic.
To study the effect of depths and widths of the two sections of the
T-shaped bar, computed toraue slip characteristics for bars having same 39
TABLE 2.3
Specifications for the model induction motor
Particulars Required Rectangular values bar design
Base voltage (volts/ph.) 220/ 3 220/ 3 Base current (amps./ph.) 7.87 7.87 Base impedance (ohms/ph.) 16.1 16.1 . Stator resistance 0.007 0.005 Stator leakage reactance 0.1485 0.1485 Magnetizing reactance 4.32 3.8 Rotor resistance, (starting) 0.0315 0.0251 -- do -- .1 (running) 0.0079 0.006 Rotor leakage reactance, (starting) 0.082 0.0615 -- do -- ,(rnnning) 0.1276 0.1086 Pull out torque 1.94 1.82 Slip at point of pull out torque 0.04 0.06 Full load slip 0.008 0.01 Starting torque (breakaway) 0.587 0.53 Starting current (breakaway) 4.369 4.84
Unless mentioned otherwise, all values in the above table are in per-unit.
2.2 r 2.0 - 1.8 r 1.6
N fi 1.2 - g
a, -1 0 r 1 I ' 0.8 . 0 E-1 o.6
0.4 j I! 0.2
I L.... 1 L. 1 I 1 i 0.1 o.2 0.3 0.4. Slip - p, u, Figure 2.4. : Torque - slip Characteristics a. 1800 h.p. motor b. rbael motor with rectanrIular bars 2 , oh_ • b a . 06 .0175 f"..%` 02 .,/,6 e0175 6
se 0 0 015Q Q 015 S. .0125/y /.0125 00 .16 9 .01 7/' .01 /I /1 ,/d 0 0075 L .0075 .6 .6 / o h. / .8 s 1. tJ Xiel3 '? 7 -005
I k ,0025 ;,? •00p..5 J1 if I i I L r 1 1 I I I I t 1 I I I I t 0 1 a 3 /1. 4.8 Imaginary part,p.u.
Figure 2.5 : Current loci a. 1800 h.r.motor b. Model motor with rect-tngullr bars Foints'mrked 0 represent per - unit slip 42
d2
N
al
Figure 2.6 : Inverted. T- bir
I 43
depths widths but different/and vice-versa are shown in Figs. (2.7) and (2.8).
It is observed from 4g. (2.7) that an increment in the depth of the upper portion, reduces the starting as well as pull out torques. This is primarily due to increased leakage reactance. On the otherhaud, computed results in Fig. (2.8) suggest that a reduction in the width of the bottom portion has the effect of reducing the starting torque and increasing the slip in the normal operating region of the induction motor.
It was observed that though it was possible to simulate the torque slip characteristic by keeping the width of the lower portion of the bar around 6.m.m., this would have resulted in a tooth having width at root only 2.6 m.m. which was mechanically weak and unacceptable. This tooth also would have saturated heavily.
2.3.4 Compound bars made of to ered and rectangalar sections
To overcome the difficulties associated with the T shaped bars, it was decided to make the lower portion of the teeth of constant width as fixed by magnetic and mechanical considerations. This resulted in a slot having the lower portion narrowed at the bottom than at the top.
To suit the new slot it was essential to choose a bar that had tapered bottom and rectangular upper sections.
2.3.4.1 Exact solution for estimating the impedance of a tapered bar
In practice tapered bars are much used as they are comparatively easier to machine than the inverted T--bars. However, it is more difficult to estimate the a.c. impedance of such bars. Theoretically accurate solutions for the skin effect in such bars involve the use of Bessel and
Hankel functions.
The impedance of a tapered bar, d ems. deep, W2 ems. wide at top and
W at bottom, is Given as,44'45 1 6 a e) -2 c 2,2 25 30
V (-)
20 / 1 \ \ J_ r-6-1 I/ \\ 1.2,..... i; \ \. a \N....N, -..- 1,2 ..--..._ yi - ----,--
-
; )", I 0.6
0. 21-
0' . 1 0 0.1 0.2 0.3 0.4. 0.5 a 1. Slip , p; u.
Figure 2.7 : Torciue - slip characteristics - 1800 h.p.motor Dar a 'be -4 4-2 2 A —r
20 20
--L
25 1 25
6 4. b
---
1 L 0.1 0.2 0.3 0. 4- 0.5 .9 1. szip , p. u. Figure 2.8 : Tor:jue - slip chiractorist ics 1800 h. p. mot or ----- Trtr —9-- B-tr
46
R (2.20) ac
1 14' X (2.21) ac = to Lb C4 pl
where pi and ql are dependent on Bessel and Haukel functions and are
defined as;
(1) Jo(eTlrj) _1A_ fj) - H0(1)(8 T Irj) J71(T Irj) _ (2.22) p1 Ji(eTf j) 111(1)(T rj) - Hi(1)(e.T fj) J1(1: Iri) W1 where 0 = T. ) ) 2 (2.23) ) T = 0: di(a. — 0) )
The above formulae cannot be used conveniently on a digital computer.
Besides these difficulties, it is also not easy to see the effect of
individual factors on the impedance of the bar. Thus it was essential to
have a method that was easier to use on a digital computer.
2.3.4.2 An approximate solution for the impedance of tatered and composite
bars
In view of various approximations listed in Sect.2.3., there would
not be much error if, the tapered sides of the bar are approximated by a 46 suitable exponential. Based on this additional approximation, Douglas
was able to express the impedance of a tapered bar as;
'1) Lb a ± a Coth( (2.24) ac (- 2 2d) ) 2
where cX -4a21- 2jO (2.25) 2 - 1 t 1N and a -Ed- loge (2.26) 2
The above expression is quite handy for slide-rule and computer
assisted calculations.
47
If the bar consists of differently shaped sections, as is the case with the model, the total impedance of the comnosite bar of Fig. (2.9) 46 is given as; 2 2 2 - a 2ad e A2 (2.27) z12 Z, + Z, e-2ad + 2a/A
are a.c. impedances of lower and upper sections (each where Z1, Z2 calculated assuming the other section to be absent) respectively.
A = W2/(PLO (2.28) As will be noticed later, the above expression has been found very convenient for optimizing the model design. 2.3.4.3 Steplajz.Ithad of calculating, the bar impedance
A very useful numerical method for estimating the impedance of any shape of bar can be obtained from Eqn. (2.27). If it is assumed that the arbitrary shape of bar Fig. (2.10) has been divided in small radial sections such that,
1) The skin effect is negligible in each section
2) Each segment is approximated by a rectangle for which a = o then the 8.!.00 impedance of an arbitrary shape of bar (Fig. 2.10) upto the nth step (including nth step) can be given as, p Lb 2 2 Zn_i Zn + j 2 (--17,--) a
Z Z +Z (2.29) n n-1 n where Z th n-1 = Impedance of the bar upto n-1 step and .6•Z = Impedance of nth n segment neglecting skin effect The impedance. of the whole bar is given as, 1 L 2 2 n=m Z n-1 L..\Zn + j)a(-7- Z = b Z (2.30) n-1n n=1 48
Figure 2.9 : Compound tarered bar
Figure 2,10: A bar of an arbitrary shape • 49
While determining the impedance of any particular bar, the number of steps must be limited and considerable judgement must be exercised in the choice of the steps. As a general rule the steps assume more importance towards the top of the bar and; therefore, the depth of these steps should be chosen progressively smaller.
2.3.4.4 Circuit analysis method of calculating_the bar impedance
Like the step-by-step method, the circuit analysis method is also very flexible. The important feature of this representation is the use of an equivalent transmission line circuit with distributed constants instead of lumped constants to represent those pares of the conductor which have large skin effect. In this way any element of the impedance can be examined in as much details as desired, while the remainder of the circuit is treated in the usual simple way with the lumped constant.
If an arbitrary Shape of bar Fig. (2.10) is assumed to be divided in radial section on the basis of considerations discussed in Sec. 2.3.4.3. it can be shOwn that the combined impedance of segments 1 and 2 is given 47 as
Z CoshV.G X, ± 1 -1 Sinh G X 2 G 2 2 Z _ 2 12 (2.31) Cosh 1,(-67-- Sinh V G 2 s 2 ZI1/ X2 2 X2 2 where Z = impedance of segment 1
= VG X_ CothV G Gi 1 1 1 X1 (2.32)
X1, X2 = Reaceances (neglecting skin effect) of the segments 1 and
2 respectively and is given as, Z = j 2ufL.
G1, G2 = conductances (neglecting skin effect) of the segments 1 and
2 respectively. 4 50
Knowing the cdmbined impedance Z12 of segments 1 and 2, impedance I including 3rd segment can be calculated by substituting Z = Z in 1 12 Eqn. (2.31). This process is repeated until the impedance of the whole
bar is known. In composite bars, if the shape of the lower'portion of the bar is such that its impedance can be represented analytically, then by
substituting this value in place of Z1 in Eqn. (2.31), the impedance of
the whole bar can be estimated by following the above procedure.
2.3.4.5 Formulae for computing the other 'aremeters The arrangement of conductors in the stator slot is as shown in
Fig. (2.11). The skin effect coefficient for the portion of the winding
embedded in the slots is given as, 41
[6112 \ 2
4-1 3 W) 116 (1) b Ks a 3 (1 (2.33)
Sinh 21. + Sin 2P._ where T a = cp ) Cosh 2(P - Cos 27P ) Sinh (P - Sin So ) cos (p ) (2.34 ) (P b = "0(P. Cosh (I) -t-. ) cp = 2 11.1/ 42- Formulae due to Alger were used for predicting the magnetizing reactance and the various other components of leakage reactance, with the
exception of the rotor end leakage reactance48 which were estimated by using an expression due to Iiilischitz—Garils: .All the above mentioned formulae are listed in appendix A.
2.3.5 .C.Smalpon of results corn uteri by various methods
To make .optimum use of the available space on the rotor and for reasons mentioned in Sect, 2.3.3., the bar shape of Fig. (2.12) was finally chosen. By adjusting dimensions di, d2 and W i, e an attempt was made 51
A 12.7
51
4
( All dimensions in m.m.)
Figure 2.11 : Stator slot dimensions 52
17 2
Figure 2.12 : Compound bar con7;isting of tapered nd rect,,,ngul,x sections.
4 53
o simulate the torque-slip characteristic 'a' of Fig. (2.2) and the
Current locus 'a' of Fig. (2.3). II The design claculations were performed on an 1E4-7094 computer. The circuit analysis method, step-by-step method and approximate analytical
solution due to Douglas' (Sect. 2.3.4.2) were used to obtain the charact-
eristics of Figs. (2.13) and (2.14). It is evident from these figures that
the results would not have much error if Douglas' method was used for
.optimizing the design, as the other two methods take comparatively larger
computer time. Using Douglas' method, torque-slip characteristics with three
different bars having the same widths but different depths have been plotted
in Fig. (2.15) against the torque-slip characteristic of the large motor. It can be seen that as the depth of the lower portion is decreased, the
pull out torque as well as the slip in the stable region increases. In
Fig. (2.16) two bars having the same depths for the two sections but different width of the lower portions are used. Again it'is evident that the effect of decreasing the width of the bar in the lower section is to
increase the pull out torque at the expense of greater slip in the stable operating region.
Fig. (2.17) shows the finally accepted bar. The torque-slip characteristics of the large motor and the model motor using the bar
shape of Fig. (2.17) are plotted in Fig. (2.18). Good agreement is
observed on major part of the characteristic except between slip values of
0.04 and..30. Fig. (2.19) shows the comprison of the current loci of the model and the large motors. The impedance of the secondary winding of the model machine is plotted in Fig. (2.20).
The main drawback of the model motor is its higher magnetizing
current which means the lower magnetizing reactance in comparison with 2.0
0.4
0 0 0.2 0.4 0.6 0.8 Slip - p.u. Figure 243 : Torque - slip characteristics of the model Dovgli.si method Step-'ny-stop method --4,--Circuit analysis method 2
5-1
0
O
P-1
0 2 5 11-. 5 Imsinary component of current - p. u.
Figure 2.1/4- : St%tor current loci for the model motor - Step-by-step method - Dou3.1r.s' method —0— Ciroint annlysje method 2.0
110 1.6
1, 4 A C
x 0 O
0.4
0 0 0.1 0,2 0.5 0.4 0.5 0,9 Slip - p.u. Figure 2.15 : Torlue - slip charrIcteris'Acs ---- 1800 h.p. motor - model motor
--x DAr 'C ' t; —2 2
2.0 20 20
1.6 25 25
1=-1- 1.2 A B
0.1
0 0 0.1 0.2 0.3 0.4 0.5 0.9 Slip - r.u. Figure 2.16 : Torque - slip characteristics 1800 h.11. moLor 1:foael motor -her 'A' Ijodel motor 'B' 58
[—+ 1.5 I a5 7: 5 _ = 3
d2 = 20 71 = 2 3
7.875 712 = 7'5 I. -
25
All dimensions in millimeters
4..375
Figure 2.17 : Enlar:.:ed view of model rotor slot 2.11.
2.0
•
O.0
0.1:.
0 1 1 I I i , t I 7 0 0.2 0.11. 0.6 0.8 ? Slip - n.u. Figure 2.18 : Torque - slip characteristics 1800 h.p. motor ----- The moacl motor (cnlcul-Ited) 2 .02
••••• Ms, .04 .015 0- .C2 .06 .06 apt .01
0 c.4 o 1 .5 .5 .7 "sc 1
0 I I 1 1 I t 0 1 9 3 4 14.45 Im9gimry inrt of the current - p.V.
Figure 2.19 : Current loci ---- 1800 1.r. motor ----The model motor (colculn.ted) C) Points rm,rked. -o-- indicate per-unit slip. 0.15
0.023 0.
0.026 0.13 -10.,2
0.022 0.11
0.020 0.10
N 0.018 0.09 nn fJ c.) o.olG 0.08
_1 O. 07 .1 0.031,. Cc: 0.012 o.oG 0
0 -t 0.05 O 0.010 r C) 0.003 0.04. 1,4 o.oc6 -, 0.03
0.002!. 0.02
- 0,01
0.0 1 I --1 0. 0.1 0.2 0.3 0.5 0.6 o.7 0.8 0.9 1.0 0. Slip - t.u. (D 1--- ti Figure 2.20 : Dynamic impedance of the secondary a.Resistnnce b.Ilect:Ince 62
that of the large motor. Other parameters agree reasonably closely.
Roughly it can be said that the magnetizing reactance is inversely proportional to the airgap length which could not be reduced below the present gap length (0.55 mm) for mechanical reasons.
Fig. (2.21) shows the cross section of the rotor body. The rotor bars are brazed to copper end rings. The parameters are listed in Tables
2.4 and 2.5 for the model. Six leads of the stator winding are brought out for ease of making 3-phase, 4-wire connections. Plates 2.1 and 2.2 show the details of the stator and rotor of the model motor. Table 2.4 gives complete design details of the model.
2.3.6 Model induction motor set
For ease of interchangeability of the rotors, the shaft is carried on cartridge type bearings. The model induction motor is coupled, through a torsion free flexibox type coupling, to an amplidyne Which can be used to simulate different kinds of load characteristic. At the non-driving end of the amplidyne, through a torsion-free coupling, B.D. type (B.T.H.) d.c. tachogenerator is mounted to obtain a direct current signal proportional to the speed. It was necessary to use torsion free couplings for transmitting fast changes in the torque and speed during the transient studies. For simulating various inertia constants, a flywheel can be mounted at the non-driving end'of the model motor.
The capacity of the amplidyne was of the same order as the model motor.. This made it difficult to hold the speed of induction motor in the unstable part of the torque speed characteristic. The model was, therefore, recoupled to a d.c. dynamometer of 7.5 KW capacity. To enable the set to run steadily at all speeds, the dynamometer (separately excited) was connected Wavd•-Leonard fashion, to another d.c. machine driven by an induction motor. In the following text, the dynamometer will be referred 63
( Dimensions in m.m.)
Figure 2.21 : Rotor dimension; of the model 64 • TABLE 2.4 Design details of the model induction motor
Sucifications Nominal horse ppwer 3.5 Capacity, K'JA 3.0 Line voltage, Volts 220.0 Phases 3. Poles 4. R.P.M. 1490. Connection star Full-load current, amps. 7.873 Unit current, amps. 7.873 Unit admittance, mho 0.062 Physical. dirensions Airgap.diameter, m 0.2286 Gross core length, m 0.1397 Effective core length, m 0.1327 Stator slot dimensions see figure (2.11) Airgap length,.m 0.00035 Pole pitch at airgap, m 0.18 Rotor slot dimensions see figure (2.17) Shaft bore diameter, m 0.057 Magnetic circuit Flux per pole, weber 0.005633 circuit cross sec. flux density length amp. turns sq.m. w/sq.m. m. Stator core 0.0034 O. 355 0.0931 16.34 Stator teeth 0.0104 0.7332 0.0635 9.77 Rotor core 0.0053 0.5356 0.0357 4.02 Rotor teeth 0.0066 1.1611 0.0461 9.00 Aircal) 0.0251 0. 761_ 0.000- 104.71 Total amper turns per pole 143.84 Carter coefficient 1.1028 Magnetizing current, p.u. 0.2658 Stator winding Skew one slot pitch Winding type DLL Pitch 0.889 Winding factor 0.9405 Total slots 54. Conductors per slot 12. Conductor dimension 2 x .045" x .0451 Conductor section, sq. m.m. 23.2 Mean length of turn, m. 1.034 Turns per phase 108. Resistance, p.u. 0.0114 Rotor winding Winding type deepbar squirrel cage Total Slots 46. Bar material copper Bar section, sq. mom. 190. Bar length, in. 0.1955 Endring material. copper Endring section, sq. m.m. 300. Running resistance, p.u. 0.0059 Starting resistance, p.u. 0.0273
65
TABLE 2.5.
Per unit parameters of the 1800 h.n. motor and of the model
------Model motor Particulars 1800 h.p. motor Calculated Measured ------Base voltage (volts/ph.) 11000/ 3 220/ 3 220/ 3 Base current (amps./ph.) 81.33 7.87 7.87 Base impedance (ohms/ph.) 78 16.1 16.1 Stator resistance 0.0074 0.005 0.009 Stator leakage reactance 0.1485 0.1/07 - Magnetizing reactance - .4.32 3.64 3.41 Rotor resistance, (starting) 0.0315 0.0273 0.033 -- do -- , (running) 0.0079 0.0059 0.0072 Rotor leakage reactances, (starting) 0.082 0.0663 0.058 -- do -- , (running) 0.J.276 0.1147 0.128 Pull out torque 1.941 1,8 1.6 Slip at point of pull out torque 0.04 0.02 0.03 Pull load slip 0.008 0.007 0.009 Starting torque (breakaway) 0.587 0.587 0.77 Starting current (breakaway) 4.369 4.733 4.9
Unless mentioned otherwise, all values in the above table are in p.u. Plate 2.1 : Stator of the moael induction motor Plate 2.2 : Rotor of the model induction motor 68
6 the d.c. machine. 2.4 Measurements for checkinf4 the computed results 2.4.1 No-load test The model motor was run at no-load. The input power and current
were measured for different 3-phase balanced voltages. The watts input
is the sum of the friction and windage, core loss and no-load primary
I2R loss. In Fig. (2.22) the power input minus stator copper loss and
the input current are plotted against the applied voltage V. For the
purpose of separating constant friction and windage losses, the input
power (less stator copper loss) are also plotted against V2.
2.4.2 Locked rotor test
During this test, input power and currents to the model motor,
were measured at different 3-phase balanced voltages and the shaft torque
was measured by a spring balance. The readings were taken quickly to avoid
any significant change in resistance, a close check on which was maintained
by measuring the resistance just before and after the test. As the model
motor is much bigger in size than a commercial 3.5 H.P. motor, it was
possible to perform a locked rotor test at full voltage without seriously
overheating the machine. 42 The starting torque of the induction motor is given as, 2 T = K. q I R/N Newton-meterss (2.35) s 9.55 Where K. is a constant whose value is less than unity and it allows for
non-fundamental secondary losses.
In Fig. (2,23), Is, Ts and K. are plotted against the applied
voltage, V . Plot for I indicates that the leakage as well as the main
flux path starts saturating when the applied voltage exceeds 0.6 p.u.
2.4.3 Torcue slpacd characteristic
The test for measuring the electrical torque was performed in two
69
.18
.17
017
.3 15
.12
.11
r:71 ' .-, ------4- 'an O o03 to to O to H 0 o 07 to rd Ll O 0 H o06- H O 0b0 rd e 1 a 05 11 ' ;-- o (4 03 g .0ri 4 o, 0. " [ .0 K
.001
0 Y t r I 1 I .- I 0 1 0,2 0,11.. 0,6 0.8 1.,0 1p 2 9 V or. V- , p, u,
Figuure 2. 22 : Se-oaration of no--10 c1 losses 1:o-lo^.a current b. 107.sas c. V vs, no-lo-,..d losses 70
laO 5
0.9 1•
0.8
0.7
0 0.6 5 Crl
V) 0VI r-I r31 0.5 ga) o • P1 (1.4 2_
0.3
0.2.
0.1
0 0:1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.9 1.0 1.1 1.2 Priry,ry volt;Igc r.u.
Figure 2.23 : Loctea rotor test .4 flon funa7mental loss f',.ctor lei h. Frimy cy.L-rent , I c. Ycasurea torcue 1s 71
arts
Test 1. The model motor was disconnected from the mains and
was driven at the desired steady speed by the d.c.
machine. The input power to the d.c. machine and
torque on the balance were recorded.
Test 2. The model motor was then connected to the a.c. mains
and by adjusting the applied voltage to the d.c. machine
armature, the set was made to run at the same speed as in test 1 above. input.power to the model motor,
output power of the d.c. machine and the torque on the
scale were recorded.
From the above two tests, electrical torque developed by the induction motor can be calculated by the following three methods.
a. Output power method49 •
This is the most accurate method for measuring the torque slip characteristic of the induction. motor. In this method, electrical torque is defined as the sum of power measured at direct-current machine under tests 1 and 2 above, divided by speed.
b. Torque measurement methoa-50
In principle, this method is the same as method-a above. The electrical torque is given as sum of torques recorded under tests 1 and
2 above. The accuracy of this method is directly related to the accuracy
to which the torque can be read from the balance.
c. Input rower method 9
If the primary copper and core losses are neglected, input power
to the model motor is a direct measure of electrical torque.
Torques measured by the above three methods are plotted in Fig. (2.24).
Difference between the toroue measured by rcythods a and c can b& called • stray load losses. Polar plots of the complex primary current are shown
2.0
1.8 -
1.6 - '' fr•.--,;+,. \ \ 'so \ \\ \\\ 1. 4 - I ' \ •\ \'‘. ,\ \\ !II . 1, .---, 1.2 L \ ..`,. r I- ..... 1. 0 - , \N::: -a
r-i a , q Ls- , b-- x--- . 0.6
0.2
0 . L I. 1 I I L. 1 -I I__.-- i 1 0. 0.1 0.2 0,3 0.4 0.5 0.9 1.0 Slip — p.u.
Figure 2.22. : Toroue slir da.lraeteristics Caleulnted — ac— Output power method Torque measured by dynamometer Input rovier r.lethoa r.2
in Fig. (2.25).
It is observed that the general agreement is not very satisfactory
between the computed and the test results. Some general deductions can be
drawn from Figs. (2.24) and (2.25).
1. The starting torque, Ts is greater than the designed value
whereas the magnitude of the starting current is more or less
the same as computed. This indicates that the resistance of
the secondary winding is larger than the computed value.
However, the total reactance seems to be quite close to the
calculated value.
2. The larger value of full load slip is another indication of higher
secondary resistance.
3. Lower pull out torque of the finished model motor indicates
that the total reactance is too high at this point.
4. The fact that the starting current vector lies well above
the computed vector (Fig. 2.25) is also indication of a larger
secondary resistance.
5. The slip points over the whole input current locus are closer'
together than on the computed locus again indicate a higher
secondary resistance.
6. It can also be deduced from tests 1 and 3 that the secondary leakage reactance is greater than computed at lower values of
slip. At higher slips, when the rotor current is large, the
leakage paths get saturated and cause the reactance to decrease.
2.4;5.1 Calculation of the secondary imnedance
The model motor can be represented by the equivalent circuit shown in Fig. (2.26) rl, rm, r2 and Xm can be determined experimentally. It is difficult to separate X1 from X2 and the calculated value of Xi is used to
2
.033 • 02 "'"
0, 17. 6281 • 015 02
'0 • 06 . 015 //' ir• 08 .11-3 '`)< f13-107• • . oli . 0075 .28 /1 X.0133 . 711.E it /1 .18 • • .005 ' 1. .006 , .0O25 .,0025.0
0 1 2 3 I t 4.5 React ive current, r . u.
I, lgUre• 2.25 : Current loci 1800 h.p.motor The rfloael - calculated. -do- measured.
Points marked inaicate per unit slip. is j X j X (s) 1 1 2 oT-- AN\i\
r2( s
va
C.
Figure 2,26 : Equivalent circuit of the induction motor
76
= r2/s + jX at a given value of slip. determine the value of Z2 2 Based on this assumptlon, the impedance of secondary circuit under 1 any operating condition is given as,. Zm(Zi - Z1) (2.36) Z2 - (Z + Z - Z.) m 1 i
r = Real (Z ) s (2.37) 2 2
X2 = Imag (Z2) (2.38)
X jr'm m where ------°in -(r + jX ) ) m m ) Zi = ri + jX1 ) (2.39) ) and Z. = measurelinput impedance )
and X2, measured by this method are plotted against slip in r2 Fig. (2.27). The salient features observed from Fig. (2.27) are recorded
below. Figure 2.27a, (curve - b) is greater than the computed value 1. Measure r2 (curve -..a) at all values of slip. This confirms the
deductions made in Sect. 2.4.3.
2. r2 measured by running the machine at 1.0 p.u. and 0.75 p.u. voltages agree quite closely. This indicates that the
saturation of the rotor body does not affect the resistance
to a noticeable extent.
Figure 2.27b (curve - b), measured at rated voltage, if compared with 1. X2 the computed values (curve - a), is larger at low values of
slip but X2 decreases rapidly as the slip increases. However, 0;036
0.034. r
0,032
0.030
0.023 1- 0.026 0.024
0,022 2 0,020 R 0.018
0,016 Figure 2.27a : Rotor resistance,r, of the model motor 0. 01h- a. Calculated (ed6,7 current of act neglected in the end-rinc ) 0-To b.Measured at 1 p.u. volt-tc'e c.Measured at 0.75 p.u. volta(s%! d. Calculated. (e&iy current; eff3ct in the 010 end-ring conniaerea)
) 0.006 i ii 0.0C4 I 1 I 1 I I I r 1._ I I 1 ____I I I 1 I 3 0. 0.1 0,2 0.3 0.4 0,5 0.6 0.7 o.s 0.9 „o Slip - p.u. 0.13
0.12
0.11
0,10 `mot c
0.09 L
0.08 r
0.07
0.06 :--
0.05 L.
o.o74 0.03 -
0.02 rI
0.01
• O. O. 0. 2. 0.3 0.4 0.5 0.6 0,7 0.8 0.9 1c. 0 p u Figure 2.27o : Rotor reactance / X2 of the moae 1 motor r.cte oretical b Measured. at 0 p. uo voltage C Measured. at 0,73 .o.u. voltage
79
the. percentage discrepancy is appreicable at lower values
of slip but is less at larger slips.
2. Curve - c gives the measured value of X2 at 0.75 p.u. voltage. Curve - c differs appreciably from curve - a at low values
of slip but agrees more closely as the slip increases. In
general curve - c agrees more closely to curve - a than curve - b.
This was expected as in the theoretical calculations the effect
of saturation is not considered and the degree of saturation at
0.75 p.u. voltage is definitely appreciably less than
with the full voltage.
3. Curves b and c show a large discrepancy above about 0.1 slip but the discrepancy is less above about 0.5 slip. The
variation is probably due to reduced saturation of the leakage
paths, since both the main flux and the current are reduced
at 75% voltage. For investigating the cause of discrepancy between the computed and
the measured secondary resistance, it was decided to include the effect of eddy currents in the end.ring!;.Approximately it can be shown that skin
effect coefficient for endrings can be obtained from Egn. (8.38) of Ref. 41 by substituting p = ci= 1 as,
::%ce 4 -sinh 2Xend sin 2pden 6oc. W Lb K - r --"--Ce 2,1 2 2 1. r, 2) cosh -, (2.40) er R n d - Cos 2p r =1,3,5 (xe Pe e r e r-
where a e = 1. (2ca -1- 1)2 + I (2;:1-1)1+ + a-4 /If 2 ) e d e (2.41) 2 I 47 Pe = -(1de-12) Oa) a k /T.2 e
de = depth of the endring in cms. 80
No attempts were made to simplify the above expression. In
Fig. (2.27) curve 'd' is computed secondary resistance after allowing for skin effect in the endring. The discrepancy between the measured and the computed results is only slightly reduced.
The various factors influencing the behaviour of current in the vicinity of the endring can be explained as follows: 1. As the slip increases, the current in the rotor bar is pushed
to the top and it has to travel across the bar before entering
the endring (Fig. 2.23).
2. The current also tries to crowd along the outer side of the overhanging bar.
3. When current goes from one bar to the next, it does not spread over the whole section of the endring.
4. The brazed joint is another source of additional resistance, which can increase due to heating as the secondary current increases. The following manufacturing tolerances were specified.
a. Tolerance on slot dimensions 0.254 m.m., - 0. b. Tolerance on bar dimensions -1- 0., - 0.051 m.m. c. Clearance between the bar and the slot, .375 m.m. Some. of difficulties encountered while manufacturing the model are listed below:-
a. While punching the slots, the tooth width tends to decrease
because of hardening of the lamination material. b. Slightly staggered laminations produce uneven slot surfaces.
Due to the above irregularities, the bars were found a bit tight in slots.
Table 2.5 shows comparison of theoretical and experimental parameterso of the model motor and that of the large motor.
20
25 Bar Core
4 . r 25 2.5 70 Half core length
All aimensions in milli mot
Figure 2.28 : Connection between the bars and the endring 82
2.4.4 Stray load losses
It is usually quite difficult to obtain this component of losses as it is the difference of two large measured quantities. Referring to
Fig. (2.24), the vertical difference in curves c and a can be said to be equal to the stray load losses. The losses are plotted in Fig. (2.29) against the primary current.
2.5 Conclusions
The general agreement between the calculated and the measured results has been found quite satisfactory. The main drawbacks of the model are:
1. Higher r2 which, it is felt, is due to brazed connections
between the bars and the endring because the absolute value
of resistance of each bar is quite low and a very small
contribution due to the joint can increase the value of r2
significantly.
2. X is larger at low slips but drops as slip increases and
the leakage paths get saturated: Nothing could be done at
this stage to remedy this in the model.
3. The model has 29% magnetizing current whereas the large. machine had 239 only. This could not be helped as magnetizing
current is roughly proportional to the airgap length which
could not be reduced below the present value for mechanical
considerations.
If another model of similar capacity is designed in future, the designer is recommended to pay special attention to the following -points.
1. The bridge over the slots should be made deeper to reduce
saturation due to leakage flux.
2. The absolute value of resistance of each bar is quite small.
Any unwanted additional resistance, such as due to brazed joint • 83
0,4.
003"
64 to 0
0
cd 0.2
ICI)
0.1
0,
1 5 6 Erinnry CUI-r=1.7
Figure 2.29 : St7v le=d losses 84-
between bars and endrings, should be kept within allowable
limits. It would be preferable to braze the whole face of
each bar with endrings.
3. Smallest possible airgap length should be used as the magnetizing current is approximately directly proportional to this length.
Inspite of the above short comings, the parameters of the model are still of the same order as that of the large machines (Tables 7.1 and 7.2).
Moreover the model is primarily intended for the investigations aimed at development of the new theories.
4 85
CHAPTER
°TEM EXPERIMITAL
To justify the use of a theory for representing machines in transient stability studies, it is necessary to carry out practical
experiments to verify the validity of the theory. The synchronous 26 27 micromachines ' are ideal for such verification tests and are used, in conjunction with the model induction motor, for the practical tests which are described in Chapter 11.
3.1 The model motor system
In order to check the theories of Part II and III, the model motor, described in Chapter 2, is tested in a simulated one-machine system as
shown in Fig. (3.1). The model induction motor is connected to an infinite bus of negligible source impedance through a reactance simulating a
transformer and transmission line. The induction motor is loaded by an
amplidyne on which a pump load characteristic (Sect.3.2) is simulated. The speed variation during the transient fault is recorded by the scheme given in Sec t.3.3. The signal proportional to the speed is obtained from
a d.c. tachogenerator coupled to the shaft of the amplidyne. The open-
circuit and the short-circuit switches are operated in conjunction with a sequence timer and external switching control devices which enable the
whole sequence of operations including recording the current and voltage
.oscillograms to be completed by pushsing a button. The point on the a.c. wave is controlled by means of a mag-slip which provides the input signal
to the dekatron-counters in the sequence timer unit. A vacuum switch,
capable of interrupting 200 amps. (a.c.), is used for application and
clearance of the faults. A 30 KVA transformer is used to step down the
laboratory bus voltage from 440v to 220v. To U.V.Recoraer R D.C.Tacho.
Transmission Open-circuit line Suitc1-1- 2 CBI
4.1MAto // Transfomer Induction Motor Reactance • Short -c ircui-t Digital Counter Ehvitch,CB2 Probe •- b
Amp"' idyne
Infinite Bus
Resistive Loaa
Figure 3.1 : Diagram of the Transient Tests Q7
13.2 Induction motor load simulation
Loads on induction motors vary considerably and no average load
torque-speed curve can be assumed in a large power system. If a single
large induction motor is to be included in a transient stability study
its particular mechanical load torque-speed curve can be used for
determining the mechanical load at any speed. However if a number of
machines are to be represented by a single machine, an average or mean
curve will have to be assumed, depending on the types of load on individual
machines. This sort of assumptions introduce errors into any prediction of
the machine performance but it will be difficult to include the mechanical
loads on a large number of machines on a general basis, to any degree of
accuracy, in a multi machine stability study.
The model induction motor used for the stability tests is mechanically
coupled to an amplidyne (Sec.2.3.6), which is driven as a generator and
loaded with a fixed resistance. The specification and the connection
diagram of the amplidyne are shown in Fig. (3.2). A square function torque-
speed characteristic is simulated on the amplidyne. Fig. (3.3) shows 'the
scheme used for the simulation. One half of the main field winding is
supplied from the tachogenerator whereas the second half is excited from
the output voltage of the amplidyne. The windings are connected differentially.
By adjusting the comparative excitations of the two windings, a characteristic
such that the torque is proportional to square of speed, is obtained.
The main purpose of the feedback signal proportional to the output voltage
is to assist in the simulation of the required torque-speed characteristic,
but it also ensures that the amplidyne does not behave erratically. Fig.
(3.4) shows the measured torque-speed characteristic.
3.3 Measurement of motor speed
A special arrangement is designed to record directly the slip of the 88
Terminals:
A Al X Ji- 4 A2 3 1 X2 „ X 2-
Diverter Resistance ,uad. Compensating- feed- Winding back 1,rd
Depagnet- 13/ isilg wgg.
Quad. Resistor
Quad. Wdg. Field Winding
A & B Main Brushes C & D Quadrature Brushed
Connection Diagram for the Amplidyne
Specification_
Killouatts 5,0 Amperes 25.0 Volts 220.0 Poles 2 R.P.M. 2920.0
Make B.T.H., Serial No. 65923F27 Conn. Diag. ZD.23721
•k Figure 3.2 : Connection Diagram & Specification of the Amnlidyne T a chog /ler at ol"
t ft> ifb TS0-63,
Figure 3.3: Simulation of the Square Torque-Speed Curve T °roue = k (Speed)2
90
O - 0.2 o.4 o.6 0.0 1..0 1.2 Speed , D.u.
Ficure : Simulated Toraue-Speed Curve 91
induction motor. The output voltage of. the tachogenerator is approximately 150 volts at 1500 r.p.m. An analogue compute is used for this purpose. The circuit diagram of the scheme is shown in. Fig. (3.5).
the voltage from the tachogenerator, In the operational amplifier A1' while the set is running at 1500 r.p.m., is reduced to half value and is then balanced against a reference d.c. supply available in the computer to make the output of Al zero. This ensures that the output of Al is proportional to the slip-speed of the induction motor. The feedback circuit of the amplifier, Al consists of a parallel R-C circuit which acts as a low pass filter to get rid of the high frequency commutator has a gain of 10 and also acts as a ripples. The second amplifier,A2 low pass filter. One of the galvanometers of the U.V. recorder is used as the mechanical movement. Fig. (3.6) shows the calibration of the device. The relationship between the slip speed and the deflection on the U.V. recorder's screen is reasonably linear. 3.4- Simulation of the transformer and the transmission line impedances The transformer impedance is simulated by a high Q.-reactance which has an iron core with an airgap. However for the simulation of the transmission impedance, use is made of a three phase transmission line simulating network, which consists of resistance, and inductance "units". Each unit consists of a number of fixed "elements" so that a range of values are possible. The units are used in parallel to carry safely the large fault currents during the transient operation. 3.5. LaLchronous machine model 22 27 - One of the micro-alternators ' has been used as a generator together with the model induction motor in the multi-machine studies discussed in Chapter 11. For transient stability studies, it has been necessary to include the Time Constant Regulator (T.C.R.)22'35 in the .00310-F .003/A,F 92 1---- ANAA.t---4 1 1 MA 2 111)... 1 N YA \4\A
Tach0
Galva.
Figure 3.5 : Speed Recording Device
•••
-400 -300 -200 -100 100 Sli-p Speed ,
•Figure 3.6 : Calibrated Speed vs. Deflection on U.V.Recorder To Screen 93
system to obtain field transient time constants comparable with those of large machines.
Standard tests32 were carried out to obtain the micro-alternator parameters and Table 3.1 shows their values. Fig. (3.7) shows the magnetization curve and the saturation factor calculated on the basis 2 described elsewhere . The machine inertia constant was obtained by means of a deceleration test, and adjusted by adding additional flywheels to obtain an inertia constant similar to those of typical large machines. The time constant associated with the field winding of the micro-machine is seen to be considerably lower if a T.C.R. is not used. 94-
Parameters of the model alternator
Stator No. 334818 , Rotor No. 334827 Base stator voltage , line volts. 220 Base stator current , phase amps. 7.87 Base armature power , VA 3000 Base stator impedance, ohms. 16.15 Base field voltage volts. 1703 Base field current amps. 0.85 Base field impedance , ohms. 1954
Mutual reactances: Xmd (unsat.) 2.66 Xmg (unsat.) 2.45 Armature leakage reactance, Xa 0.19 Armature resistance , re. 0.0197 Field leakage reactance X. 0.1489 Field resistance 0.0015 Transient reactance Xt 0.336 Sub-transient reactances Xu 0.232 XII 0.264 q
Time constants Ttdo (unsat.) sec. 6.06 T'd sec. o.63 TT? d sec. 0.008 Tr? sec. 0.01 q Inertia constant, H Ws/kVA 3.5
Saturation characteristic Fig.3.7 95
1,2
0 1.0 -P o 0.8
0. a az* " a.a. .saamai.aaaa,. • ' • • 0 0,k 0.8 1.2 1.6 2.0 Field current - ( a)
1.
0.8:- 0 0.6r
0 .r1 0.21- .p ti) 0.8 1,2 1.6 2.0 Airgap voltage -• p.u.
( i )
Figure 3.7 : Saturation characteristics a.Open circuit characteristics b.Saturation factor 96
PART - II
FAULT STUDIES OF LARGE
INDUCT ION NCTORS 97
CHAPTER4
INTRODUCTION
4.1 Review of investigations The earliest attempts to analyze the transient behaviour of an induction machine were made by Spooner and Barnes3 and flayer4 in 1910.
Their conclusion was that the induction generator ceases to contribute any current to the fault as soon as a short is applied. The conclusions were based on results of small machines having large resistances which caused the transient currents to die away very rapidly. Contrary to the previous findings, Doherty and Williamson5 established in 1921 by testing a 150 h.p. motor that the induction motor does contribute to the fault level. They worked out short circuit current solution based on the concept of constant flux linkages. The contribution of induction motor loads to the fault level was not considered important for the following reasons.
a) The induction motor loads were too small in comparison to
the rest of the system.
b) The circuit breakers were a lot slower. The current used
to decay to an insignificant value by the time the circuit
breaker was able to operate.
Sterley5 developed a method of analysis for the induction machine
based on a set of orthogonal co-ordinate axes which were stationary with respect to the stator, but he did not apply them to solve any particular
:problem. Thus no useful work was done in this field until 1950. when tr Rudenberg7 analyzed the transient current in an induction motor following
a sudden change in terminal voltage. His analysis was based on the 98
fundamental differential equations governing the stator and rotor currents. The subsequent expressions for transient currents were complicated and tedious to evaluate in practice.
Based on the concept of symmetrical components, Lyon8 also worked out the analytical expressions for the transient currents and torques in an induction motor following a sudden short circuit. The formuale were in terms of a characteristic equation formed from the basic equations.
In the same year an A.I.E.E. Committee Report9 on "A new basis for rating power circuit breakers" indicated that the fault current contribution due to induction motors did not decay rapidly in all cases and should 10 not, therefore, be ignored. About the same time Huening proposed a method of calculating the time variation of short circuit currents in a power system including rotating machines. Each of them9,10 recommended that the short circuit current contribution calculations should be based on the standstill impedance of the induction motor.
More recently, Enslin, Kaplan and Davies used Rlidenberg's7 equations to derive expressions for transient torques and fluxes developed by a squirrel cage induction motor. Smith and Sriharan15, 16 used Lyon's equations and obtained analytical as well as nuMerical solutions for the transient currents and torques in an induction motor on connection to a fixed supply. A numerical method was used to predict the behaviour of an induction motor after reswitching of the supply. The expressions were too complicated and it was. not easy to visualize the action inside the
-machine. On the otherhand the numerical solutions have the disadvantage that they conceal many details concerning the significance of machine
parameters and the physical phenomena inside a machine.
Squirrel cage induction motors are now favoured by the oil industry, 4 because of the absence of sparking devices such as brushgear, for use in 99
the inflammable atmosphere. This has caused a rapid increase in the 36 size of induction motors. Induction motors of 22000 h.p. or more are already in operation. The increasing size of the induction motors has lowered their per-unit resistances and increased inertias. The method9 of calculating the initial peak of the short circuit current of an induction motor from the locked rotor impedance and multiplied by a factor is being used to determine the rupturing capacity of power circuit breakers installed at various junctions in a power system. The method is doubtful as it is not based on a rigorous theory. An empirical method 18 has been proposed by Cooper, MacLean, and Williams , who have introduced non-standard time constants. A comprehensive and theoretically sound method for calculating the short-circuit current of large induction motor is still lacking.
1F.2 The object of the thesis
The methods suggested so far in the literature for estimating the fault currents of single cage machines are generally complicated and are difficult to apply to the practical multi-machine problems. Moreover none of the available methods allows for the deepbar effect usually present in large induction motors. Even in the age of high speed computers, an engineer in practice still prefers a simple and straightforward solution 11,3° that gives results within practical tolerances. A standard method for calculating the fault currents of synchronous machines has been in use for a long time. The main object of this port of the thesis is to investigate if an induction motor can be analyzed using the well established synchronous machine theory. The recommendations listed in Sect. 1.6 are based on the studies made in this Part of the thesis.
4.2.1 Fault currents in a system containing synchronous machine 4 .
The calculation of switch ratings in a power system containing 100
Lynchronous machines is based on the peak instantaneous current flowing after a 3-phase fault at the point in question. The calculation depends on a representation of each machine by its sub-transient reactance and the voltage behind its sub-transient reactance and is carried out by making a network calculation to determine the theoretical value of r.m.s. alternating current immediately after the fault. The first actual peak of current is normally calculated from this value by first multiplying by'[ 2, and then by a further factor 1.8 to allow for the doubling effect of full assymmetry together with an assumed decrement between zero time and the peak occuring half a cycle later52.
There can be much argument about whether the method correctly allows for all the factors determining the switch rating, particularly where the requirements and capabilities for opening and closing are different, and the probability that the switch will open at the peak following a three phase fault is low. Nevertheless the method, which was fully 11 described by Crary in 1947 , has been the accepted method and has been the basis used for synchronous machines at least since that date.
A three phase short-circuit at the terminals of an unloaded machine is a special case of the kind of fault considered and has been adopted as a standard method of measuring the four transient parameters of a synchronous machine, namely; sub-transient and transient reactances and time constants31'32 The initial peak current depeds mainly on the sub- transient reactance and the three other quantities determine the manner of its decay with time. For a multi-machine system the simple network ' calculation described above determines the initial peak current, but• an accurate determination of its decay is more difficult. An approximate curve can be derived by estimating "effective" time constants from a knowledge of the parameters of the individual machines. 101
4.2.2 Fault currents in a system containing induction motors
All the considerations of Sec. 4.2.1 apply equally to the induction motor which like the synchronous machine, has coupled circuits on stator and rotor. The principal differences, none of which affects the application of the method of calculation of the peak current, are as follows:
a) The induction motor normally has no rotor excitation
winding, and the rotor secondary winding corresponds
to the damper winding of a synchronous machine. A
wound rotor motor has a single secondary winding and
can be represented under fault condition by a single
reactance (which determines the sub-transient reactance)
and a single time constant. A large cage induction
motor, on the other hand, usually has a pronounced
"deepbar effect" and requires for a reasonable simulation
at least two reactances and two time constants. Although
the parameters of the induction motor apply to somewhat
different winding arrangement from that of the synchronous
machine, they can still be called "sub-transient and
transient". The important quantity for calculating the
instantaneous peak current is the sub-transient reactance.
b) The induction motor normally operates with a small slip.
It is shown later that this is of little significance
as regards the problem under investigation.
c) An induction motor, when isolated from the supplyl ean not
generate its full voltage as none of the windings is excited
from an external source. Consequently the procedure for
carrying out a sudden short circuit test has to be slig:Itly
different. 102
d) The induction motor is fully symmetrical between the direct
and quadratUre axes. Thus with similar approximations the
accuracy of the calculations should be better than for a
synchronous machine.
4.2.3 Determination of the sub-transient reactance of an induction motor
4.2.3.1 The sudden short-circuit test
A "direct short-circuit test" is one in which the induction motor is still connected to the supply. Since the system is also short-circuited, the method is not always acceptable, depending on the nature of the test plant. Also the current is not zero before the short circuit even when the motor is on no load. These disadvantages can be avoided by adopting the "indirect short-circuit test" which is made by disconnecting the machine from the supply and short circuiting it a few cycles later, before the flux in the machine has decayed appreciably. A record must be made of the voltage at the instant of short circuit. To obtain values of the parameters closer to the rated voltage values the motor can be run above rated voltage before disconnection. The current osoillograms can be analysed to find the sub-transient reactance, as well as the other transient parameters, in the same way as for a synchronous machine.31,32
The only difference is that the motor current decays to zero instead of to a steady value.
A sudden short-circuit test is the preferred method for determining the sub-transient reactance of a machine, since the conditions are then most similar to thoseencountered in service under fault conditions.
4.2.3.2 The standstill impedance toot
The locked rotor test is a standard method of evaluating the machine impedance for determining its starting characteristic (neglecting transient effects), but it does not accurately determine the sub-transient 107
reactance for the following reasons.
a) Even from a'theoretical view point the reactance measured at 50 Hz is only an approximation to the sub-transient
reactance (Sec.6.2.3).
b) Because of the approximations on which the theory is based
the parameters calculated from a steady a.c. test differs
to some extent from those obtained from a sudden short
circuit test. The suddent short-circuit method is more
reliable for fault calculations, because the test has more
in common with the conditions for which the parameters are
required to simulate the machine. This has long been
recognized for synchronous machines, for which the sudden
short-circuit test is specified as the preferred standard
method.
c) Last but not least the parameters depend very much on saturation
of the leakage flux paths. Locked rotor tests are rarely
taken at full voltage and there is considerable discrepancy
if the measurements are made at reduced voltage.
4.2.3.3 Measurement of the frequency response characteristic
The frequency response characteristic, obtained by a.c. measurements, is best expressed by an "admittance locus". At the present time there are three methods of determining the characteristic, given in a draft I.E.C. report,53 intended to supplement I.E.C. 34-432 but as yet no single method is preferred. In fact it is convenient to use results from more than one
method on the same machine, because for each method there are difficulties in obtaining accurate test data at certain regions of the characteristic.
Two methods were used for testing the model machine.
a) Variable frequency impedance test (at standstill)
b) Variable speed impedance test 104-
The other problems studied in this part of the thesis are those listed in. Sect. 1.6. Electrical torques developed by a single cage machine under various transient disturbances had been calculated by 15 different authors, both analytically as well as digitally on the 8 basis of Lyon's equations. However none of them allowed for the eddy current effect normally present in large induction motors. In this thesis, in addition to the transient currents, the transient torques are also calculated for the large machines with deepbar effect.
The analytical expressions both for transient currents and torques are 15 simpler than those given elsewhere for the machine having no eddy current effect bars.
To verify the validity and the accuracy of the proposed theory, the model motor described in Part I was used. 105
CHAPTER5
TRANSIENT THEORY OF A DEEPBAR INDUCTION MOTOR
_5.1 Representation of a deepbar induction motor
The two axis theory of the synchronous machine, as developed by 29 Park , takes the direct axis to be fixed on the pole axis of the field
member. For the induction motor, which has a nnifol-m airgap and
symmetrical windings on the secondary member, an alternative method, in
which axis are fixed to the primary member, can be used. There are only
a few papers giving methods of calculating the short-circuit current of 14 an induction motor. Enslin et al used differential equations derived
from first principles and Huening10 used an approximate method based on
the steady state equivalent circuit. Smith and Sr iharan15'16 used a symmetrical component method of Lyon8. The present work follows Park's method, now widely used for synchronous
machine calculations. Multiple secondary windings are readily dealt with
by using the concept of operational impedance. There is also the great
advantage that the theory of the short circuit of the induction motor
follows very closely that of the synchronous machine and leads to a practical
method almost identical with the well established synchronous machine
method. The equations, assumptions and sign conventions are similar to
those set out in Ref. 30.
Fig. (5.1) shows the diagram of the primitive machine used to
represent a deepbar induction motor. The deepbar secondary winding is represented by two coils on each axis. Although the eddy current effects
are only approximately simulated by these coils, the experimental results
show that the simulation is reasonable for the practical purposes. 106
LI
Figure 5.1 : Two ads representation of a deepbrIr inkluci.ion- motor 107
5.2 Short circuit of an induction motor
The standard test condition for a synchronous machine, on which the calculation of fault currents is mainly based, is a sudden three phase short circuit applied to a machine which is excited and on open circuit. Such a condition is not possible with an induction motor, which has no field winding. For the theoretical analysis, it is therefore necessary to consider a short circuit which occurs while the motor is connected to the supply. Moreover the motor runs below synchronous speed with a small slip 'st. In this respect the theory of induction motor is more complicated. On the other hand, because of the machine symmetry, the mathematics can be simplified using complex numbers to express the variables, as explained in Appendix B. As with the synchronous machine, the current dies away rapidly and it is reasonable to assume that the speed remains constant during the period considered.
With this assutcption the differential equations (Appendix B) are linear,
5.2.1 Current and tague after a direct short circuit
A "direct short circuit" occurs when the induction motor is short circuited while it is connected to the supply. The equations are given in Appendix B and the solution for the short-circuit current in Appendix
C.- In the solution of the equations, approximations are made similar to those made in the derivation of the short circuit current of the synchronous machine3°. As explained in the Appendix the approximations depend on the fact that the resistances are small compared with the reactances.
Before the short circuit the induction motor runs with a constant slip s, and it is assumed that s remains constant after the short circuit at zero time. The slip determines the load currentI (complex axis 10 value) in the armature winding before the short circuit. Using the princi7ple.. of superposition, the charge in the axis current II is obtained from the
108
solution of the equations and is added to Ilo to obtain the total
current IZ. The axis .to phase transformation is used to calculate the (t). phase current ia
i (t) = - V (B Cos X - A sin X)e a m tfra
1 f3' Sin(wt + - pt)e -t/TI - vm (—X' X ) cos
vm ( Y - yr ) cos pu Sin(wt + X - P")e -VT" (5.1)
For a short period after zero time, the alternating component can be -t/T' considered to be the sinusoidal wave obtained by putting e and e -tA" equal to unity in Eqn. (C8) and by using Eqn. (B46).
1 - 1 1 1 i = v Re iqty ej(wt + X) s sw T1 ▪ X" Xt • m 1 j o j SWoT" 1
: = V Re j.)0 e j(wt + X1 m (5.2)
The phasor equation is therefore, using Eon. (B.52) and (B.53): IT10 1 — xif f(istooT
r v v." = -10 - 1.1Cl0x" j = (5.3)
where leis the phasor representing the voltage behind the sub- transient reactance, determined from the phasor diagram of Fig. 5.2.
Hence the machine can be renresetned by y and the short circuit current is obtained by dividing V4 by X". The result is the same as that deduced 30 for the synchronous machine by using the constant flux linkage theorem. An approximation has been introduced because the motor runs with a 109
Figure 5.2 : PhAsor illustr-ybing volt?ze comnonrft.s
110
slip s before the short circuit. If the motor was urloaded the result would be exactly true. .It should also be noted that the freQuency of
. the short circuit current is w and not oo Torque
Electrical torque developed by the motor while shorted can be several times larger than the normal value. This torque can cause a severe strain on the shaft. It is shown in Appendix C, that the torque under short circuit is given as,
V2 2t/T a T = B e 0 2
V2 -t/Ta 1) , (1 Cos Vii' sin (wt pt) e -t/T -
+(err - cos p" Sin (cot - err) e (5.4)
Thus the torque consists of three prominent components.
1. Undirectional torque which decays with time constants ra/2.
Its initial value at zero time is equal to the starting torque
of the motor.
2. Symmetrical torque that oscillates at frequency w and decays
with T and Tt. a 3. Symmetrical terve also oscillating at frequency co but
decays with Ta and T".
It should be noticed that the electrical torque does not depend on the switching angle X.
Generally the first peak of the torque is of interest. Thus the alternating component can be considered to be the sinusoidal wave obtained -t/ VT' tT" by putting e e a and e equal to unity in Eqns. (C17). 4 Using Eqn. (B46), while dropping s in factor (1 - s),
111
2 1 , 1 ) 1 jol e = -0.5 Re V `TrTr ' smoT" e • m iqtr 1 + jsrAo TI
2 1 1 = 17 FiT I ejwt] (5.5) A - XciSW 0
Using Eqns. (B52)and (B53), Eqn. (5.5) can be written in terms of phasors as,
5 Y10 10 e-ja Te = - Re X( jsoo )
0.5Y. 10x! -ja] = Re [- (5.6)
="Re {- '5Y 10,111 -j21 Eqn. (5.6) gives the initial value of the symmetrical torque developed by the machine under short circuit. The value of the first peak torque can either be obtained from Eqn. (5.6) after multiplying with a suitable factor or by evaluating Eqn. (5.4) at half a cycle after application of the fault. 5.2.2 Indirect short circuit An "indirect short circuit" occurs when the induction motor is first disconnected from the supply and shorted a few cycles later. The analysis is in two parts; 1. Determination of terminal voltage after disconnection. 2. Determination of current and torque after the short circuit.
5.2.2.1 Terminal voltaa-e after disconnection
The derivation of the open circuit voltage is given in Appendix D. Because of the closed secondary circuits the flinc linking the armature
only changes initially by a small amount and decays from the initial value in a manner determined by the open circuit time constants. The terminal
voltage is proportional to this flux and has a reduced value when the short circuit is applied a few cycles later. The expression derived in 112
the appendix is: I X" va(t) = m Cos(wt 1,t) e 0 -VT! - I (X - xl) cos pl Sin(wt m .Bf o ) e -tA" ° - m(XI - X") Cos .(3'' Sin(wt Xt - pfl) e (5.7)
The first term represents an impulse of large magnitude lasting for a very small time. The second and third terms are the alternating
components at frequency w, which decay exponentially with time constants T1 and T". o 0 As in Sec. 5.2.1 for a short time after zero time, the terminal voltage
can be considered to be sinusoidal wave obtained by putting e and -t/T' equal to unity in Eqn. (D.11). After dropping the impulse term as well as s,
(x x') (x' x") V1 = 1 jsw To 1 4. jSW -10 0 0 0 T"0 11
= j [X(jswo) - X" ] -10 (5.8) Using Eqn. (D2)
V1 = Y1.0 j X" (5.9) Eqn. (5.9) suggests that when the supply is disconnected the terminal voltage of the machine suddently drops by an amount equal to the voltage drop across the sub-transient reactance. By comparing Eqn. (5.9) with Eqn. (5.3)
V = v " 1 (5.10) and thus the phasor diagram of Fig. (5.2) also holds for Eqn. (5.9). 5.2.2.2 Current and torque after a short circuit . Current
As shown in Appendix E, the short circuit current is, 113
i (t) = Sin X' e a o
1. tT f l 1 -t/T" I [r e + e Sin (wt' + X') (5.11) Vo A. " X.1 0
where V' is the amplitude of the open circuit voltage at the instant of short application, and in practice it can be measured from the recorded envelope. V' is defined by Eqn. (E2). For the initial period, putting e-ti/TI and e-V/T" equal to unity, the symmetrical component of the short circuit current is, V' o ia(t) = X" Sin(wt1 + Xt) (5.12) Torque
It is shown in Appendix E that the electrical torque after an indirect short circuit is given as,
e_ttH 4, e-/11 1 -/T-C T = - 0.5 Vol a Sin Lot XI 13) The above expression for the electrical torque is quite similar (5. to that under direct short circuit (Eqn. 5.1;.). The main difference between the two torque equations (5.4) and (5.13) is that the latter does not have any asymmetrical component. However the two torque equations are independent of switching angles X and X4). --"/T a For the initial period, equating e , e and e to unity in Eqn. (5.13), the torque developed by the machine is,
Te = - 0.5 --0 Sin wt1 (5.14) Thus, as in Eqn. (5.6), the initial value of the torque after an indirect short circuit also depends on X". 5.3 Switching an induction machine to the supply
During steady state operation an induction motor fed from a balanced supply develops a constant torque, while the stator currents pulsate sinusoidally at supply frequency. However, following a switching operation
114
or some other change in the operating conditions, the motor experiences a transient condition during which the instantaneous currents and torques may attain several times their steady state values.
5.3.1 Machine runninr, with trapped flux in the rotor
When an induction motor is disconnected from the supply, and the supply is restored while the rotor is still rotating with trapped flux, transient current and transient negative torque occur. The torque can be large if the machine is re-connected a few cycles after the loss of supply. In practice, rapid reconnection of the supply, or of an alternative supply in an automatic supply change-over scheme can initiate this type of disturbance.
Current
The transient current following such a disturbance is given as,
(Appendix F); + ia, + iaIv (5.15) ia(t) = aI I +aIII where 1a1 = Im Cos (wott + Xt +ya.)
= Steady State Primary Current (5.16) V, = [Vm (B cos (y1+.70 - A sin(yi + %))+ o Sin Xt1 e t a aII X" (5.17) = Asymmetrical component of transient current Vt = [1.!Cos pt Sin(mtt (str - 4 e- t" aIII m Sin(cott+X(;) (5.18)- = Symmetrical component of transient current which alternates at frequency co and decays with Tt ,1 1 11..„ Cos pti Sin(wt' +y p")- t i + X- 0 Sin(mt' %(;) ktl-rie (5,19 Symmetrical component of transient current which alternates
at frequency m and decays with T". 115
However if, during the initial period, only the symmetrical Component of transient current is of interest, it can be obtained from Eqns. (5.18) and (5.19) by equating e-tt/TI and e to unity.
+ y vt 1 j(wt' + 1 o is(t) = V R4- Sin(wt' + h(!)) m " - X(jswo e ]+
j(cott + % +yIP 1 o [y" e •.1- Sin(wt' + = Re X" Xff
aN V' % +y k") + Sin(wt' + (5.20) - X" Cos(wt' + 1 X"
Thus Eqn. (5.20) shows that the initial value of symmetrical component of transient current is controlled by X". Torque As is shown in Appendix F, the electrical torque developed by the machine after it has been re-connected to the supply is given as; T = T + T + T + T + T sw (5.21) e ss dc Iwo w
Vac where T = 0.5 Re [-=-7 = - 0.5 Re (5.22) ss 3/Ajsw0) ] [-kr*10 T=7- 10 I = Steady State Component
V- -2VA:a 2 -2tI/Ta = 0.5 B V e Tdc = 0.5 Re[jrc:Ty e (5.23)
= Asymmetrical component
V2 . --OAa jwott 3 V2m -t/Ta -imoti Two = 0.5 Re [ e e jX(jswo) e Y(717
-t'/T = - 0.5 (B Bs)Cos w t' + (A - A e o s)Sin w o t'i V2m `` (5.24)
116
= Alternating component at frequency, wo
j V2 j V2 " 1/T. 1 -Wr" Tto = 0.5 Re [{ - 4 e-t (3%;11 - yr) e 1 + jswo17 1 X • 1 + jsto -0 V2 -t'/z' V'2 . o o . o • -t'A' j i 1 e j -rot e -Tr e Vol X" X'
-tt/T a jut' I e e
2 1 1 t 1/4' pf Sin(wte - pt) e- = 0.5 [Vm X' X
▪V 2m (1xn 17)Cos p" Sin(wtt p") e V/T" e-tl Aa
2 [e 1 1 --OA" e-t'Aa •0. 5 V'o X"— X' - X" X' Sin ott (5.25)
= Alternating component at frequency, w
112 m (1 j f1 1 Tsw = 0.5 Re e ") ‘X6 -T7 e [fr.17+ iswoTI) - jswoT.
e-jswot
2 = 0.5 V Rf. cos pl sin(swotf (sl) s -t1/4'
1 1 ] (5.26) X"- X' cos pu Sin(swo pu) e
= Alternating component at slip frequency, swo
5.3.2 Machine runninqyithout tralyned flux in the rotor • If the case is such that the trapped flux inside the rotor body is initially zero, then the transient current in the complex for: is given by
117
Eqn. (C5). After changing its sign and making the approximations that
1 I ,w, the transient current is given as T T o
17 10 - Y2.0 -( a + jw)t jX(isw o) 717-jwr
pi" -10 Ya.o 1 1 e-t/TI - (1 - s)(1 isw01.1) (xl x - TY-- s)(1 iscoo,i7
fl 1 -tA" e (5.27)
Dropping s in factor (1 - s) and transforming Eqn. (5.27) by Eqn. (B.18), the instantaneous phase current is,
V egwot + 1.) ia(t) = Re jxost,0 m
-t/T = I Cos(w t X') V (B Cos X - A sin X) e a m o m
1 e-tITI •vm(yr X) cospf Sin(wt I- X - fp)
e-t/T" (5.23) ▪vmX" X1 cos pH sin(wt -
The above equation is identical to Eqn. (5.1) except for the first term which represents the steady state component of the primary current.
Thus the transient current under these conditions can be more than that after a three phase fault by an amount equal to the steady state value of the current.
Similarly it can be shown that the electrical torque developed by the machine under these conditions is that given by Eqn. (5.4) plus a steady torque component. The complete torque equation is, 118
2 -2t/Ta = 0.5 Re V B e Te PI'-10 1-10]+ 0.5
- 0.5 V2 e4/1a. [(1xt- - x4 cos pi Sin(wt - p') e-t/Ti
+ qtil. - -)1g) cos P" sin(cot - WI) e_ t/`11 ] (5.29)
5.3.3 Machine initially_pt standstill The derivation of the transient current, when an inert machine initially at standstill, is suddenly connected to the supply, is given in Appendix F. Both'accurate and approximate expressions are deduced for the transient current and torque. Current The two formulae for the phase current are: Accurate ia(t) = Vm Sin 0a [A0 Cos(wot + "X- 0a) - Bo Sin(w t + X - 8a o )]
Sin 0a cos(?, ea) e-t/Ta - Vm
xt x , - V Sing' , Sin (% - 0') e m (1- T a
1 -25t°" Sin (1. -O - Vm Sin 8" rr) e-t/T" (5.30) (1 - T ) a Approximate Vm -t B Cos(wt %) Sin A e a ia(t) = Vm Ao Sin(wot %) o X /`
1 4T. --) - Vm e ) e Sin? (.5.31) (1 - 'E1) (1 - T T a a
The.above equations consist of two main components.
119
1) A steady state component alternating at rated frequency w0.
It's magnitude is fixed by the value of impedance at
standstill. ii) Asymmetrical transient components, decaying with time
constants T , T1 and T". a For a short period after zero time, the transient component can - t/Ta - be obtained by putting e , e VT' and e-VT" all equal. to unity in Eqn. (5.31). Thus from Eqns. (B46) and (5.31), the instantaneous phase current is, V m i= [A Sin(w Cos(w ?) (5.32) a Vm o ot + + Bo ot + - X"
Torque Accurate and approximate formulae for calculating the starting torque are also worked out in Appendix F, and they are listed below.
Accurate 2 2 -tA- T = 0.5 V B Sin" 0 + Sin 0 -• A)0 Sin w t - B Cos w tfe e m o a a a o o o
fl 1) ‘17 Sin 0 el Sin 6' Ta - a Sin ( - 0a) Sin(wot + 0' - Ode-t/1, (1 -
fl X" Xt -t/Ta cjac-t/T” Sin e Sin eu Sin(6" e - Sin(w t + It - T" a ea) o - Ta (5.33) Approximate
T = 0.5 V2 B + f(1.- - A0) Sin w t -- B Cos w t e-t/Ta e m o o o o JJ
Sin wot e-t/TI
4 (1 1 ‘X" -• X'' -tic" + ri Sin w t e (5.34) (1 - 1._.) o Ta 120
The first term in each of the above two equations (5.33) and '
(5.34) is steady statt) torque. Further it must be noted that as long as T11 ;?, T , the subtr'ansient torque component is in phase with the a transient component. 121
CHAPTER 6
'EXPERINFITIAL PROCEDURE
The model motor was connected as shown in Fig. (3.1) to simulate a single machine system in the laboratory. The theory was checked against the experimental results of the above system. The principal parameters of the large motor and of the model motor are given in Table 2.5 together with measured parameters for the .model motor.
6.1 Determination of transient parameters from the admittance locus
The operational admittance locus determined from the resistance and reactance curves of Fig. (2.27), using the equivalent circuit of Fig. (2.26), is shown in Fig. (6.1), which is the same as the per-unit current locus of
Fig. (2.25) drawn on a different scale. The reason for this similarity is that on a per-unit basis, the admittance and the current loci are practically identical. It is obvious that the admittance locus (Fig. 6.1) is not a single semi-circule, but, on the approximate basis, it can be represented by sum of two semi-circles.
Thus the method of determining the transient parameters is to draw a curve of the type appropriate for a machine with two secondary windings, which approximates as closely as possible to the admittance locus which can be either measured experimentally (Sect. 6.2.3. and 6.2.4) or calculated from design. As discussed in Appendix B, the ideal curve consist of a constant vector and t,•ro semi-circular components, of which the large semi-circle coincides with the low frequency part of the accurate curve. The best choice of the small semi-circle is that which causes the admittance to have the accurate value at normal frequency. * From the component semi-circles of the approximate curve, the transient I •
.02 .--•"rs,9 075 ,tr'''r •; • .015 .01
)-.01 .0075 .2
.005 x • 2 6 .1,•5 r.-4(-!..\-- • .1 •,••• . - • .-- - r • - - 2. 1.0025 •/
1 .
0 1 x 3 5 5.5 r . Pea --sc. )1• - 1 A
Figure 6.1 Aamit'qanco locus of the model accurate curve -HA-- Fitted curve
Points markea indicate per unit slip. 123
reactance• and time constants are readily obtained. The semi-circles
for the calculated admittance locus are shown in Fig. (6.1). It is
evident that the standstill impedance at 50 Hz differs appreciably from the sub-transient reactance. In light of this fact, for accurate
determination of short circuit current, it is necessary to modify the
present calculation methods which are normally directed towards starting at present.
6.2 Model test results and comparison with calculations
As stated in Sect. 4.2.3.1, the recommended test is the indirect short circuit test, which is easier to carry out and simpler to use than a short-circuit applied with the motor connected to the supply. The
test requires two circuit breakers (Fig. 3.1), one in addition to the fault breaker for isolating the motor from the supply. However such a test is not generally used at the present time, since the testing of induction motors is usually directed only at the steady a.c. operation.
The usual tests are a light running test to measure the magnetizing current and a standstill impedance test to measure the total leakage reactance at full frequency. Occasionally a load test is carried out, but more often the full load performance is calculated from the above two tests.
Moreover the impedance test is generally made at reduced voltage.
For a complete determination of the transient parameters the static impedance should be measured at a range of frequencies from the rated value to zero, and should apply to the degree of saturation at full
voltage. An approximate method of deterring them from limited information is described in Sec. 6.2.5. Of particular interest is the sub-transient reactance, which is used for calculating the initial fault current.
The parameters of the model machine, determined by different methods, e
given in Table 6.1. TABLE 6.1
Transient parameters for the model motor
• Calculated Direct short- Indirect Variable frequency Variable speed Approx. method from design circuit short-circuit impedance test impedance test . based on data test test standard test Sect. 6.1 Sect. 6.2.1 Sect. 6.2.2 Sect. 6.2.3 Sect. 6.2.4 Sect. 6.2.5 . X' 0.244 0.300 0.292 0.267 0.263 0.268 X" 0.182 0.165 0.184 0.178 0.18 0.177
T1 0.096 0.118 0.1 0.107 0.0937 0.0955 T" 0.0026 0e0114 0.0142 0.0033 0.0036 0.0037 • ----
All values in the above table are in p.u.
1 125
.2.1 The direct short circuit test
The machine is run at no-load from the fixed voltage supply.
A three phase symmetrical short is then applied, and the short circuit current during the short is recorded. Based on Eqn. (5.1), the current .
wave is analyzed into an asymmetrical component, transient and sub-transient components by the well known synchronous machine method, except that the'
current dies away to zero or to a small value depending on residual
magnetism (Fig. 6.2).
6.2.2 The indirect short-circuit test
The machine is disconnected from the supply while it is running
and is short circuited after a few cycles. Oscillograms is
taken of the voltage during the open period as well as of the three currents after the short circuit. The voltage applicable to the short-circuit is taken from the envelope of the voltage wave at the instant of short circuit.
In order to obtain the full voltage at this instant the original voltage may be raised above the normal value.
The current wave is analyzed using Eqn. (5.11) into an asymmetrical component and transient and sub-transient components (Fig. 6.3).
6.2.3 Variable frequency impedance test
The variable frequency static admittance test is a measure of the complex admittance when a variable.frequeney single phase voltage is applied to two phases connected in series and the rotor is locked at 54 standstill. The operational admittance can then be calculated from:
V r 4_ . 21 1 3 "Ms,10-7 (6.1)
A variable frequency supply Into 10 Hz is obtained from an a.c. commutator ruotor(SchraGe type) used as an induction type frequency changer. 4
The input frequency is 50 Hz and the voltage can be continuously varied by means of a three phase variac. The control of speed, and hence frequency
Time - millisecon:ts 126
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0 .02 004- Ta .06 .08 .10 .12 T5mc Seconds
Ficure 6.2 : Diiscot Short Circuit Test on the Ifoacl 4 a. A.C. ex..relop- e
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127
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Figure 6.3 Inclirec-rG Shore Circuit l'ent on the 1iod.31 a. A. C.Iinvelop c b. Transiel-± com?onont of a. c. envelop e c. Stfo-ia-?..nsient.: con-mom-1-ft; of a. c, enrclop e Asyt-razactric? 1 o0:-.117,or.cnt. of shorL., circuit current 128
is obtained by Ward-Leonard control, the d.c. drive motor being
electrically connected to another motor-generator set. In this
manner, it is possible to get a supply having a fixed frequency. A
supply at frequencies above 10 Hz is obtained from an alternator
driven by a d.c. motor connected in a Ward-Leonard system.
The method of measurement is the 'three voltmeter method' of
measuring complex impedances. Conventional voltmeters usually give
maximum accuracy near 50 Hz and are not suitable for measurements at
very low frequency. A peak reading voltmeter which could measure the
voltage accurately between zero and 100 Hz was therefore used. The
circuit diagram of the voltmeter is shown in Fig. (6.4). It was used
to measure voltages between points 1-2, 2-3 and 3-1, Shown in Fig. (6.5).
In order to obtain good results, the value of the voltage drop across
the non-inductive resistor,R ext' connected in series with the primary should be of the same order as the voltage to be measured.
The measured curve is shown in Fig. (6.6). The values of the transient and sub-transient reactances and time constants of the test
machine can be determined by fitting a curve derived from two semi-circles as discussed in Sect. 6.2. The fitted curve is also shown on Fig. (6.6). 6.2.4 Variable speed impedance test
The variable speed test measures the current locus at the normal
frequency, similar to that shown in Fig. (2.25). The operational
admittances curve is approximately the same and hence the transient
parameters can be deduced by the method of Sect. 6.1.
6.2.5 Approximate determination of operational admittance from standard tests
Without obtaining the complete frequency response locus, parameters can be determined within practical tolerances if only three suitable points k are known on the admittance locus. Such points, A, B and C (shown in
129
n
G=1
311.11. A 3111 Operational amplifier Voltage to be measured
r
32p2
2.125KL / AfAVO:
Meter
Figure 6.4 : Peak reading voltmeLer
3
Figure 6.5 : 1,.e7-surement7 of the operation-11 imped,lnoe :028
^- 02
•
irk
( .006 •
0 2 3 5 5.75 Real [Y( jscod] p.u„ Figure 6.6 Admittance locus of the mead :Measured Fittca
Points markea indicate per unit slip 131
Fig. 6.7) can be obtained as follows.
i) The motor is run at no-load and its input impedance is
measured.. This determines point A and yields X = Xl + X
(the core loss is neglected.) •
ii) When the motor is on-load, its input impedance and slip
determines point B.
iii) When the rotor is at standstill, the input admittance
measured at 50 Hz determines point C.
The centre of the first circle is the point of the intersection of the bi-sector of line A-B and the real axis. Thus the first circle is drawn with K as centre and K-A as its radius.X1 is then fixed by point D, and T1 is obtained from,
¶1 = (tan 0)/60 a (6.2) Having determined T1 , the slip scale can be marked on the first circle by using Eqn. (6.2).
The second circle is drawn such that the resultant frequency response curve passes through point C in Fig. (6.7)0 The point H can be fixed by subtracting the contribution DG of the first circle at s = 1. The centre
L of the second circle is the point of intersection between the bi-sector of line D-H and the real axis. X" is then the inverse of OE. Using the coordinates of point H, the sub-transient time constant r can be obtained.from the following equation.
T" = tan Moo ( 6 . 3) 6.2.6 Comparison of the transient parameters determined by various methods
The short-circuit parametels obtained by the various methods can be compared on the basis of the operational admittance locus. Fig. (6.8) shows that the admittance loci ueasured by the variable frequency and variable speed impedance tests asree better with the curves computed from the desir7,1
2
• •
• i • • .."`;•:,2_, .08:.. . -• '. , \ --.. : . .4 : 1 7---2.--- • • .3.,,, ‘------L.L._...1_1--x- ---.---- •‘.3 ;) ' I ;, . • C ; • : . \ , , ,.., 1 , i , .• f.„-,-:ii• • ' / /// / , , :. - I . 1 • ' ,"- • . ./ 61 '''''-- \ - , ; / • or- h' ! / I/ / .4 \• Vii- , • • I. • ,
1 '2 3 L 5 Real [Vjscoo)
Figure 6.7 Addmittance locus of the model --*-- 'Pitted
Points marked is per unit slip 2L
•••
0 008 tr, • y• q--7) ..28 • r) 1, -L • • Mt ••• - 1 \ • • • `41.6 t rc
47i 1 •
11. 2
0 1 2 3 4 5 5.5 ,
Real [ Y.(jscoo) ] , --)k• 11•
Figure 6.8 A6mittance locii of the model Calculated from design aata Calculated using parameters measured by indirect short-circuit test. Ileasured by variable frequency impedance test Measured by variable speed impedance test Points marked -0- represent mr-unit slip 134
data than the locus determined by the indirect short-circuit test.
Moreover the time constants deduced from the short circuit test are
longer than those deduced either from the design or from the impedance
tests. The difference is also shown by Fig. (6.9), in which the short-
circuit current decays less rapidly than that calculated from the design
or the impedance tests.
f, •
.4 ; 't
• pr 1 12
NI b -p 3L •:-1
IC) -el Jr
0 to
\ . •)Ce ;•".1 *TN 0{e e - c• C'2 .06 .03 .174_ -2 Time - Seconds
Figure 6.9 Calculated short-circuit current of the model &Tit ching angle,X=9 0° ; Speed = 11,.88 r.p.m. Volts = 1. r.u. ; Fully loaded Using Parameters obtained from :- Design data Indirect si-lort-circust test 0 —Variable frequency impedance test X Variable speed impedance test • 136
CHAPTER 7
COMPARISON OF C014PITTED AND TEST RESULTS
7.1 The model motor The tests were performed on a one-machine system shown in Fig. (3.1).
The motor was connected to an infinite bus through appropriate reactance.
The motor was loaded by an amplidyne which was used as a d.c. generator supplying a resistive load. A vacuum circuit breaker, capable of break- ing currents upto 200 amps. (a. c.) and a small switch capable of breaking currents upto 20 amps. (a.c.) were used. For all the tests in this section, switch CB was the vacuum breaker and CB was the small switch. 1 2 As mentioned in Sec. 3.1, both the switches and the recorder were operated through a sequence timer unit. In the direct short circuit tests, the switch CB was set to open about 20 milli-seconds after the short circuit. 1 The section of the transmission line close to the infinite bus was sufficient to limit the short circuit current of the bus to a safe level during the 20 milli-second period. As the machine was not supplied from an individual transformer, the corresponding reactance in Fig. 3.1 was zero. On the other hand a 30 KVA transformer was used to step down the laboratory bus voltage from 440 V to 220 volts and reactance of this transformer is taken into account while calculating the switching transients, (Sect.5.3).
7.1.1 Direct short circuit test
The short circuit current was recorded during the three phase symmetrical short circuit test. The parameters as determined from the direct and indirect short circuit tests were used in Eqn. (5.1) for computing the fault current. Fig. (7.1) shows the measured and calculated curves. The following conclusions can be drawn. Time - seconds
. 02 .0/.;. • C6 .08 .10 .16 • 18 .20
-3
-5
-6 LOAD CONDITION -7 Volts' ,220 (line) Current 4..38 amps. Power 1215 ( input ) U Speed 14.92 r.p.m. -9 Unit Power 3000 VA ' -10 Unit Volts 220/,J Unit Amps. 7.87 0 Switching angle 'X = 290 Three phase symmetrical short-circuit current Measured Calculated (parameters measured by direct short-circuit test. ) Calculated (parameters measured by indirect short-circuit test.) Dotted curve is the envelope. 138
i) The agreement between the measured and the calculated
curve is good for the first 3 to 4 cycles.
ii) The meas4ed and computed envelopes exhibit closer
agreement for much longer period.
iii) The close agreement between the currents computed by
using parameters determined from direct and indirect
short circuit tests confirms the validity of the indirect
short circuit test method of determining the parameters.
Fig. (7.2) shows the upper envelope curves of the short circuit current computed by using various sets of parameters from Table 6.1.
The curve calculated from the approximate parameters (last column of
Table 6.1) exhibits a reasonable agreement. 7.1.2 Electrical torque developed during the short circuit
Fig. (7.3) shows the calculated electrical torque developed by the model motor during a 3-phase symmetrical short circuit. Transient parameters determined by indirect short circuit test were used in
Eqn. (5.4). No attempts were made to measure the torque experimentally.
Since the torque Eqn. (5.4) is based on the same theory as Eqn. (5.1) and as the calculated current exhibits a good correlation with the measured results, it can be expected that the torque computed by Eqn. (5.4) will not be much different from the experimentally determined value.
7.2 Application of the method of calculation to the large machines
Design information as well as indirect short-circuit oscillograms
were available for four large machines of which the main details are given in. Table 7.1. The parameters determined from the indirect short-circuit
test and those determined from the admittance locus derived from design
details are listed in Table 7.2. The stenastill impedance measured at
different values of voltage is given in Table 7.3.
139
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0 .02 .o6 .03 10 .12 . .15
Time - seconds
Figure 7.2 : Direct short-circuit current for the model comDuted with parameters listed in clable-11. Parameters casured from the indirect short circuit test. -- do -- theoretical admittance locus -- do -- variable frecuency impedance Veo
-- Jo -- appro::. admittance locus 4 ,
0 .01 .02 .03 01;. . 05 .c6 .07 .09 .10
Time seconds ,
Figure 7.3: Computed 'Electrical T orrlue during the Short Circuit 141 TABLE 7..i
Details of large induction motors
22000 hope 6040 h.p. 4 700 hop. 2500 h.p. Particulars motor motor motor motor
Unit line voltage, KV 11 11. 6.6 11
Unit current, Amps 1000 ' 270 365 121 Base impedance, ..filph 6.34 23.4 10.4 520 4
Stator resistance 0003 °005 00074 .0074
Stator leakage reactance -1345 *1875 -109 *127
Magnetizing reactance 4.27 5.562 3-62 4.6 Full-load slip % e3 o95 1- o933 Stator winding star star star star
Rotor winding sq. cage sqe cage sqe cage sq. cage Noe of poles 4 4 2 8
Unless mentioned otherwise, all the parameters in the above table are in p.u.
TABLE 7.2
Short-circuit parameters for the large induction motors
22000 h.p. motor 60LI0 hop. motor 4700 h.p. motor 2500 hop. motor
Parti- Admitt Admitt Admitt Admitt culars locus locus locus locus s/c test from s/c test from s/c test from s/c test from design design design design data data data data
X' .216 -236 -294 -316 .2 .218 '378 -42 _.. xr, -182 -191 0242 .234 -16 0156 .283 -302 —...... T, '077 •o89 °114 '129 -0765 00955 -141 . 0112
Tll -0076 -003 -0176 -0032 .0162 -0032 -025 -005
The parameters in above table are in p.u. 142
TABLE ' 7.3
Measured standstill impedance of lar.Te motors at different voltages
22000 hope motor 6040 hope motor 4700 h.p. motor 2500 h.p. motor
Z Z s volts Zst volts t volts Zst volts
0210 -864 -242 -741 -186 -764
0227 .303 -278 -437 -1915 728 -338 1.0
e228 -202 -291 .288 -1935 -189 -387 -338
The parameters in above table are in p.u.
4 143
For the 4700 h.p. motor oscillograms of torque and current during a run-up test were used to deduce the transient parameters. Fig. (7.4) shows the admittance locus deduced from these oscillograms compared with those obtained by other methods.
Oscillograms of the direct short-circuit current were available for 17 the 2500 h.p. motor, for two different load conditions. Fig. (7.5) shows the test curves compared with those deduced from parameters derived alternatively from the design and from an indirect short-circuit test.
Fig. (7.6) is an oscillogram of the short-circuit current of the
22000 h.p. motor when there is nearly the maximum amount of disymmetry.
It is interesting to note that the current does not come down to zero for several cycles after the short-circuit. This possibility, which is associated with the deepbar effect in the rotor winding and is more likely to occur with an induction motor than with a synchronous machine, could result in more difficult conditions in the switch when breaking the circuit. .
7.3 Indirect short circuit test
The following test was performed on the model system of Fig. (3.1) to check the validity of the theoretical arguments in Sect.5.2.2. The machine was loaded while it was supplied from a fixed voltage source.
At a preset point in the a.c. wave, the supply was disconnected by using the vacuum circuit breaker (CB 1) and at a latter instant a three phase symmetrical fault was applied at the machine terminals using the second switch (CB ). 2 Th plot of the open circuit voltage after the interruption of the supply is shown in Fig. (7.7) together with the computed curve using the parameters determined by the indirect short circuit test. A sudden change equal to ;MY in the measured voltage is clearly indicated at zero Points 0 markea represent per-unit slip
QM. all• al.111111.:t J. .ftm
mo, 1 'I do, .02 '1110 •••
O ./-;?""1.02 r-, -,„--7 i.)--- ,---, y i'... Z.,' ^p 0 l029 4F/ , . i i'l! It .27 - •r- ed -.. ,..„, ,,... ,„ •-..-., i .4.1/ -...s1 .30. st. .t„..4 1 ,i 'a-a. Z.....-.-. Nt-J ,e, .." h"".. 9 N. it .01 r...) re Pj tO / 1 - .. " •-• ''''''''''''' ‘ /....4 .*'."."4.a".4 n 1 ,,,,, • ...1.' -..,-, H f /If 4'N,...,_ .6 • 1 ---- -09 i•-• ,.< t0051' / ,...... --iN a 1 I j '\\ 1 I/ ::). 1 1 1 f r ' g.t 7,:,i".„ 2 2-...... _.-ri, 1 - r
1 2 3 5 Real [f(jscoo)j Figure 7./1- : Admittance locii for the 1 700 h.p.motor Cn.lculated. from design --N.-- Derived. from the parameters measured inclirect short-circuit test - Measured by variable speed impedance test
0 145
2.
• j,v1A- 1,‘t 1k " _ 14 • :1'.• - 2 A /
;. \ t 0 _h A ..... • . 1 _ I 0 Load condition P. i.:1...5. P.,------#.2 Voltage 1.072- P.u. tio....." Current 0.233 p.u. QJ .i.'i -8 • ' Loading N.L. Slip - 0.0005p.u. U Switching- angle1 X 85,6°
.02 .04 .06 .08 .10 .12 Time - seconds
( a )
2 _ --A:. ---...--- 0------,...1,•— ,Ii ty 1,1/4. ‘z •,/, '4; 1k, 7' *... . \ • - - 1-: .r. • t %:?. .- -"""----.- Pi _ A r Load condition 0 -6 - Voltage 1.0 p.u. Current 0.807 p.u. —0 — Loading 0.792 p.u. Slip 0.0065 p.u. Switching- r angle, A 220°
r 0 .02 .04 .06 .08 .10 .12 Time - seconds
( b )
Figure 7.5 : Three phase direct short-circuit current of the 2500 h.p. motor Measured curve 0 Calculated with the par-,me::ers derived from the inairect short-circuit test Os Calculated with the parameters obtained from the desgn details c
aoqom • 000F Gqq- JOJ urea 3oTiToso qTnoa-co-q.aucts q.oaa-cpui : 992, eanSTJ
0,491 -
• •. ' •I! 104 ••••• I"- -1-.1. 1.2
1.0
0.8 0.6
0.1,.
0. 2 t t ! 0.0 1 •,01 .02 e. . 03 , 1 0'Y ! lo .05 .06 P.08 .09 4 Cf. v I , i 1 007 . - 0.2 1 - . 1 - , 1 \ Tim . 11 ; -cconds I I I k 1 I 1 I ' r! '1 .-00 6
-0.8
0 0 1 1 -10.2 t jt,
-1 .4, Figure 7.7 : Terminal voltage of the model after disconnection fron the supply 0 I. Switching angle , = 26.5 ljeasured --0-- Calculated 148
time. The general agreement between the measured and the computed voltages is reasonable.'
Fig. (7.8) shows the, curves for the open circuit voltage following the interruption of the supply and the short circuit current when the terminals of the machine are shorted later. The agreement between calculated and measured curves is quite satisfactory. This establishes the validity of assumptions made while deriving the equations. Fig. (7o9) shows the electrical torque (computed from Eqn. 5.13) developed by the machine during the Short circuit.
7.4 Transient current and torque followiu the suddent connection of a machine to the supply
These tests were conducted on the model system of Fig. (3.1) with no reactance between the bus and the machine. Switch CB was the vacuum 1 circuit breaker. A 30 KVA transformer was used to step down the laboratory bus voltage from 440 V to 220 V. The resistance and leakage reactance of the transformer were included in the calculations. Tests were made when the machine was running at slip ist with and without trapped flux in the rotor and when it was initially at standstill.
7.4.1 Machine initially running with trapped flux
The machine was loaded to its rated capacity while it was fed from the fixed supply. At a preset point in the a.c. wave, the switch CB1
(Fig. 3.1) disconnected the supply and after a predetermined period of time, it restored the supply. The voltage and current were recorded throughout the test. Fig. (7.10) shows the calculated and test results.
The voltage during the open period was calculated from Eqn. (5.7).
The measured and calculated values of open circuit voltage showed a reasonable agreement. The primary current after the reconnection was • computed from Eqn. (5.15) by assuming the speed fixed at the value at zero 1 - .1, '1 o I
' t ._ .t .1 .
Poure 7..8 . An Indirect Short Circuit on the 1.Toae1 • (a) Terminal Voltage , p.u. (b) Primary- Current p.u. I:easurea (a) —0 Calculated_
Switching As 1e , X = 109° 5
3 p
1
r.1
P. •3
Time scconas 1
0
Cl ' o -2 Ei
0 .01 0 02 .03 .04. .05 .C6 .07 .08 .09 1.0 Time - seconds
Figure 7.9 : Computed Electrical Torque during the Indirect Short-Circuit •••, ••• 011:-._•27C114; r. To it •
lJ — — N °I N) 0\ co 0 ..„,_ 01 . f 4 0 -*
*.
Fd F-.• 9 Pi 0 0 1 1 I 1 -.-..1 e 1---' I 1 0 0 a t—' a PPPp ,,,,w ci- 1--.1r. 197! • C) 0 p(ca.... II c. & 0 $ 0 p 13' I--I 0 0 p F-1 p fi Ii CQ Pi Li P-1 cl- p vi r Icli c+ 0 -,4, 0 co W 0 I-I P 0 0 11 ct- 0 liI 0 — a o u7 H) tfl En 0 0 ° 11 ;1 0 1-13 H) 0 , 0 H. 1.... 0 0 • o X cr . 0 . N r P. cl• '6 P-(1) , pirCrs''< 0 Fi. 0 •Jt 0 0 p p • 1-4 c+ ci- c+ V I'• 0 P. l- ca (a Ca 4 II) p , 1--i 1---_. F.: 0 0
•• SA." 7 5,1. Or7 WI 0 ••7‹..0 •••
0.11, Ire.01,1• - • _ ;a 0 0. • ,,501' " (78 - 41.4 - .04 — Per-
0 O.- JP, "
•••
4 M
• " iN -= --- 0 ct•
••• ••• •e.
a an/
41.11 .N1
0
•:•
••• • 152
ime. As shown in Fig. (7.10), the computed curve did not agree with the measured curve. In another step-by-step calculation the speed was allowed to change during the open period and the current after the reconnection was computed assuming the speed fixed to the value at the instant of the reconnection. The value of the slip at this instant was o.046 p.u. The first peak of the calculated and measured currents agreed reasonably whereas the subsequent peaks were again out. The main reason for the discrepancy is that after the reconnection, the machine draws a large current and accelerates quickly to make up the lost speed during the open period. Thus the assumption of constant speed gives fallacious results. However the change in the speed during the open period did not affect the calculated open circuit voltage appreciably. If electrical torque is computed from Eqn. (5.21), it will also show a similar discrepancy.
7.4.2 Machine initiallynninr; without trapped flux
While the machine was running at slip 'st without trapped flux, it was suddenly connected to the supply. Fig. (7.11) shows the measured and the computed curves for the transient current following the above switching.
The agreement is reasonably good for the first few cycles after the zero time. Latter the phase angle between the measured and the computed curves starts increasing due to change in the speed.
The computed torque developed by the machine under these conditions is shown in Fig. (7.12). The torque pulsates for a few cycles and then gradually settles to the steady value.
7.4.3 Machine initially at standstill
While the machine was at standstill, it was directly connected to the supply of rated value, and the starting current was recorded. Fig. (7.13) shows the measured and calculated currents. Both accurate and approximate equations (Eqns. 5.30 and 5.31) were used for calculations. The 153.
Applied. %resit ?_ge. 1 p.u.(r.m.s.) Switching Angle,?,.. 37°
Transformer resistance - 0.014 p.u. It reactance - 0.011 p . u. -9 Slip - 0.0093 . • $ 0 .02 .06 , 03 .10 .12 T3rae ; see.
Figure 7.31 T _ns7.Cit currtAt folloviir2,- the sur.-Z.Len conn3::tion of -w innert m%Chine to supply.
0 Ca1cm.1.-.1: CCZ
2 4 .01 .02 .03 .05 .06 .07 .08 Time seconds
Figure 7.12: Transient torcmc fol7o77ing the sudden correction or an inncrt rfnchinc to the supply. 1-1 Slip = 0.0093 p. u. - .1 1 2- •1 •, 9. 0 Applied Volt ise 0.85 p.u. (r.m. s. ) afit chir3L-,r 2.‘ e X 24.6° 8.0 • Transformer imoednnce 7.03 ( 0.071-;. j 0.011) p. u. 1 6.0
3.0
6#
1 0 4;6 t U —le 0 Vi
—2.0
7 -5 .0 • 1 ' • " • r r• kia . I - 0,0 40,0" ' • i ' ••••4 i' - . - , •-
-7 .01 Figure 7.13 : Starting current of the model motor Measured e, Calculated Ecin. 5.30) Njt !I (Eqn. 5.31) 156
impedance of the 30 KVA bus transformer was taken into account while making the calculations. The computed curves agree reasonably well with the measured cure. It is clear that the approximate method
(Eqn. 5.31) gives results within reasonable accuracy.
Fig. (7.14) shows the starting torque computed by accurate and approximate methods (viz. Eqns. 5.33 and 5.34). It must be noted that when the transformer impeOnnce is included in the calculations, the first peak of the starting torque is maximum. However if the machine was started from a source of negligible impedance, the second peak of the torque would have been maximum (Fig. 7.15). Thus a very useful conclusion can be drawn that if the armature time constant T is of a same order as T", the subsequent peaks of the instantaneous torque may be of increasing magnitude until the sub-transient component decays to an insignificant amount. In order to minimize the starting torque Tit peaks, the factor (1 -should be as large as possible. In a machine T a T t without any deepbar effect (1 - should be large. This phenomenon ,a was noted by Smith and Sriharan but due to the complexity of their equations, they could not explain the different starting torque pattern in the large and the.small machines. 157 Ap-olied voltage 4=- 0.85 D.U. (r.ii.s.) Switching angle , ?• - 246° Transio.,-mor Imineaanc (0.01,1_ .1- j 0.011) p.u.
: IP = . \.. iJ- N -Li.- ...‘ - \ - % ' r ' : • :- / 1, N _ ' , "" \ ----/ r , / Air . . • • II ... A,w 4, \ - . ' . - - i - _ _ • _ _ _ ss - -_ - : 1 0 ,--,...,... ,4.- . — "---- i , ______-____7'____..-______•\ - • ./' • - ' ' ,...4 •/ -- •-k . /- ;.-1 0 El
0 .01 .02 03 04 .05 .06 .07 .03 Time - Seconds
Figure 7.11+; Calculated Star(;irk; Torque Accurate ( )t1n.5.33) Aporoximatc (Fqn. 5.34)
0
;-t 0 -1
0. .01 .02 .03 .04 .05 .06 .07 .c8 Time - Seconas
Figure 7.15: Calci).1-_-%ce,. Star-ting Tor:iue Accurate (:.':41n. 5.33) (:an. 5.34) 158
CHAPTER 8
CONCLUSIONS
The main object of this part of the work was to develop a simple.
method of fault current calculations, based on a well established
synchronous machine theory, for the large induction motors having
pronounced deepbar effect. It has been observed that the deepbar 1 effect cannot be ignorea , since there is en appreciable difference
between the impedance at 50 Hz and the true sub-transient reactance
(Fig. 6.1). Another effect noticed in deepbar motors is that the
short-circuit current with maximum asymmetry does not reach zero
until several cycles have elapsed (Fig. 7.6). The good agreement
between the calculated and measured results for the model and large
motors has confirmed that for the practical purposes, the deepbar
effect can be sufficiently simulated by two equivalent windings on the
rotor, and the synchronous machine approach is justified.
Based on detailed investigations, the recommendations listed in
Sect. 1.6 are made. For the model motor, a good agreement has been
observed between the parameters measured by direct- and indirect- short
circuit tests. Thus the indirect short circuit test is suggested as a
preferable method for obtaining the parameters. When it is not convenient
to perform a short circuit test, the parameters can be determined from
the simple design values (Sect. 6.1) or can be derived from the
measured frequency response locus (Sect. 6.2.3 and 6.2.L.). However,
if the complete frequency response locus is not available, the approximate
method described in Sect. 6.2.5 can be used. The errors involved in the
method of calculation are discussed. The method of calculation is directly 159
Lpplicable to any large deepbar cage motor and covers all forms of rotor windings without any need to modify the theory or to use special or non-standard tests. The proposed method, being identical to that used for the synchronous machines, can be adopted without modifying the existing methods of system analysis. It should now be possible to predict more accurately the switchgear rating and will be possible to separate the "making" and "breaking" requirements more precisely. This will also reduce the margins which are allowed in switchgear rating to account for the unknown effects.
The comparison between the calculated and measured terminal voltage of the model machine after interrupting the supply further confirms the validity of the theory. A sudden drop equal to (E10 X") is also noticed in the measured curve of the terminal voltage for the model motor at the instant of disconnection of the supply (Fig. 7.7). Electrical torque has been calculated during the short-circuit and it is observed that the
first peak is negative and can be several times larger than the full-load torque. It has been found that accurate results cannot be obtained by
assuming constant speed, when the machine, running at a fixed speed with
trapped flux, is suddenly connected to the mains. However if a machine, running at a fixed speed without any trapped flux, is connected to the fixed mains, a good agreement between the calculated and measured currents
is observed. The first peak of the developed torque under these conditions
is also negative and can be:many times larger than the normal value.
The computed and measured currents following a suddent connection of a stationary machine to the supply exhibit a reasonable agreement. It has
been found that the initial value of the peak torque can be much larger 4
than the normal value and the subsequent peaks are still larger if the T ratios and 5.1 are very nearly unity. T T a a 160
PART III
TRANSIENT STABILITY OF POWER SYSTEMS CONTAINING BOTH
SYNCHRONOUS AND INDUCTION MACHINES • 161 .
CHAPTER 9
INTRODUCTION
. 9.1 General
In contrast with the synchronous machine, the transient performance
of the induction motor has received relatively little attention in the
past. In the event of a short circuit fault, two main effects mentioned
in Sect. 1.2 are noticed, and this part of the thesis deals with the
latter. A heavy concentration of induction motor loads fed over a long
transmission line can endanger the stability of the whole system. Thus
an accurate determination of the stability margin is of crime economic
importance.
Induction motors often form the major portion of large industrial
loads such as a petroleum refinery or a chemical plant and play an
important part in the behaviour of the system under transient conditions.
Most of the individual plants in such an installation are complex and
expensive, and usually run continuously for years. Often regular
maintenance shut downs are made. In view of the large expense of starting
such a plant and the loss of production during a shut down, it is most
important that under normal operating conditions, vital motors keep on
running. If a shut down becomes unavoidable due to a rower failure, a
considerable damage to the plant can occur. In practice, if a fault of
short duration occurs which results in a reduction in the voltage, the non-
vital machines are stopped by tripping on the under voltage protection,
and the vital machines are allowed to remain connected to the supply for
as long as possible. When the fault is cleared, the vital machines accelerate
and prevent any damage to the plant. The disconnection of the less vital 162
machines allows the remaining motors to carry a larger current sure.
Often a standby altexliator is installed to feed the vital machines in
the event of a disturbance.
9.2 Influence of digital computers on s stem studies
The modern high speed digital computers have greatly influenced
the methods of system analysis and have led to greater accuracy of
system stability studies. The differential equations governing the
operation of generators and motors and their associated equipment are non-
linear and step-by-step methods of integration have to be used for their
solution. Such calculations are best carried out by a digital computer
with which more complicated and accurate machine representations are
possible. The number of machines that can be included in a system study
is limited only by the capacity of the computer.
9.3 The past work 19-25 A good deal of work has been done in the past to improve the
representation of both synchronous and asynchronous machines. Maginniss 19 21 and Schultz and Gabbard and Rowe made theoretical studies based on 51 Stanley's equations, which were solved on a mechanical differential
analyzer. Brereton et x120 studied the induction motor based on
equations written with respect to a set of synchronously rotating reference
axes, and presented results computed by neglecting the stator transients 22 of the motor. Alford , based on Park's equations, studied the behaviour
of an induction motor when it formed part of a composite system
.containing both synchronous and induction machines. In his analysis,
the-effect of stator transients was neglected and the theory was for a 23 single cage machine having no deepbar effect. Cooper also presented
a method (where stator transients are neglected) for studying the 25 synchronous and induction machines. More recently Humpage et al have 163
studied the behaviour of induction motor when it formed a part of a
system containing bc4 kind of machines. The method of representing 1 20 induction motor was that of Brereton et a1 where stator transients
are neglected. However, most of the earlier work, with the exception
of Ref.25, did not account for the eddy current effect which is usually
present in large squirrel cage induction motors.
9.4 The object of the thesis
The main object of the work in Part III of the thesis is to develop
suitable methods for representing a deepbar induction motor in a system
transient study. The procedure adopted for this purpose can be summarized
as follows:
1. Based on machine equations, an accurate method of representation
is developed. (Sect..10.2.1)
2. The validity of the accurate method is checked on a single
model machine system of Fig. (3.1)
3. With varying degree of approximations, three less accurate
_methods of representation are developed. (Sect. 10.2.2)
4. The suitability of the approximate methods is checked against
the accurate method.
Having established the suitable methods of representation for the
induction motor, they are used to study other transient problems. Similar
methods for representing a synchronous machine in a system stability
study are available (Sect. 11.2). The model system of Fig, (11.1) was
tested to verify the validity of the methods of representing both kind of
machines in a multi-machine system.
It was observed that for a single machine system (Fig. 3.1), the
approximate method of relpreseiltation (Sect. 10.2.2) predicts optimistic recovery depending upon the degree of approximations. It is also essential 164
to make a proper allowance for the stray load lOsses.of the induction
motor. Results of the multi-machine stuides have shown that for an
accurate solution, it is essential to rewrite the differential equations
for the primary variables cif each machine. A less accurate method
(Sect. 11.4.1.2) allows the use of the accurate methods of representation
for the two types of machines (Sect. 10.2.1 and 11.2.2) without alterations,
and gives results within reasonable accuracy. However for system
accuracies (error up to 205 alloyed), the method - Z (Sect.. 11.4.3) can be
used. The method predicts optimistic recovery of the motor and pessimistic
performance of the alternator. The main conclusion of the multi-machine
studies is that the machine closer to the fault should be represented
accurately and the distant machines can have less accurate representations
depending upon the desired accuracy. The specially written programme
for the system of Fig. (11.1) has the facility of using any type of
representation for the two machines.
4 165
CHAPTER 10
STUDY OF A' SINGLE INDUCTION MACHINE SYSTEM
10.1 General
Improved methods of representing induction motors in a stability t 20-23,25 study have been sugges ed in the past, but none satisfactorily
accounted for the deepbar effect in the rotor bars of large machines.
For the reasons mentioned in Sect. 1.3, the method suggested in Ref. 25
for considering the deepbar effect is inadequate. Moreover nobody has
used the accurate representation for the large cage motors and compared
calculated and measured results.
The equations derived in Sect. 10.2 are for the deepbar squirrel cage
' induction motor and are based on the theory developed in Part II of the
thesis.
10.2 Mathematical derivations
The basic machine equations, the complex variables and the transformation
are those described in Appendix B.
10.2.1 Accurate representation
The most accurate representation does not make any approximation in
the machine Eqns. (B.25) and (B.31). Since the inclusion of the p011 term
accounts for the stator transient, the voltage drop across the tie line
reactance (Fig. 3.1) should include the inductive volt drop during a
transient. The infinite bus and the machine terminal voltages can be
connected by:
V = V - pI - (r j(1 s)X 1 b we 1 c )1 (10.1) 0 c 1 The following differential equations of the three complex fluxes, 4
namp]..yLk l, 4)2 and ) ., can be obtained from Eqns. (B.25) to (13.30) for
166
any operating condition. B B B PI 1 11 12 13 14 q11 B B B p 12 21 22 23 Li) 2 (10.2) B B B tp3 p'3 31 32 33 V b
The rate of change of primary current is obtained from; (10.3) p11 = all P )1 a12 PY2 a13 13k1)3
The coefficients B11, B12.. and a11, `112". etc. are given in Appendix G. The electrical torque is calculated from Eqn. (B.31). The-
initial value of the primary complex current, II is obtained from Eqn. (B.49)
by substituting jswo for D. The complex voltage V1 is that given by Eqn.
(B.19). As this representation is sensitive to small errors, it will be
necessary to use the accurate expression of X(p) given by Eqn. (B.33). t1i 1 is then calculated from Eqn. (B.32). Similarly by substituting p = jswo in Eqns. (B.26) to (B.30), steady values of 4)2 and T3 can be obtained.
For solving the equations on a digital computer by a step-by-step
method, it is necessary to split those complex Eqns. (10.2) into six real equations, as follows. The bus voltage Vb is also split into components along the d- and q- axes.
d p tPdl 11 wd12 d13 wd14 d15 wd16 d17' vbd t d1 PkVcp. -wd12 dll -wd14 d13 -wd16 d15 dr? .vbq X11 ql d31 0 d33 0 d35 0 0 PY d2 d2 0 d 0 d 0 P q2 42 ill!. d46 0 c2 d 0 d 0 0 0 P d3 51 53 c155 91 d3 0 d 0 d 0 d 0 P (\i cL3 62 64 66 q3
1 (10.4) 167
Who coefficients dii, d12 ... etc. are those given in Appendix G.
The differential equations describing the mechanical motion of
the machine are,
p2e = T - T e 1 (10.5)
and PO = w
In the present investigation T 1 is assumed to be a definite function of w (Fig. 3.4). However it could, if necessary, allow for transient load
characteristics if they are known.
10.2.2 Approximate representation
Reasonable approximations are often made for simplifying the solution
of the equations in order to reduce the computing time without a significant
loss of accuracy in the calculated results.
10.2.2.1 Method - A (pkyi and s terms neglected)
A representation similar to that of- Refs. 20 and 22 is obtained by
making the following approximations'.
a) The speed change during the transient is negligible so far as
its effect on the generated voltage is concerned.
b) The stator voltages induced by the rate of change of flux
linkage are negligible compared with the rotational voltages.
The joint effect of the above two approximations is often referred to as "neglecting stator transient". The differential equations for the secondary complex fluxes are; r 2 [I I- 13412 = (X" XI) -Y 1 T 2 3 d (lo.6)
W04)21 p 3 = (X" - Xi) + X 7 1T 3 2 g X_ X where T = X m d [ 2 ' 73L 471-m)] (13 or2
The terminal voltage and the complex axis current are related by 168
V1 = (ri j X") T1 + V" (lo.8)
where V" = voltage behind the sub-transient reactance y3 = j(X" - o X2 +X 3 (10.9)
Eqn. (10.8) can be re-written with reference to the infinite bus voltage as,
Vb {(ri + r0) + j(Xtt + Xe)] I1 + V" (10ao)
Initial steady values of I and the fluxes y and 1 2 13 are obtained by following the procedure described in Sect. 10.2.1. Alternatively Il can be calculated by using the equivalent circuit of Fig. (B.2). After
splitting the Eqn. (10.6) into real and imaginary parts, they are solved simultaneously along with the equations of mechanical motion (Eqn. 10.5)
by a step-by-step method.
10.2.2.2 Method - B (Pyi term neglected) This representation differs-from the representation of the previous section (Method - A) in the sense that the approximation (a) (Sect. 10.2.2.1)
is not made in Eqn. (B.25). The differntial equations for the secondary
complex fluxes are still Eqns. (10.6) but the equation relating the infinite bus voltage and the primary current becomes,
V (r + r ) + j(1 - s) (X" + X ) I + V" b i c c 1 4'2 3 where V = j(1 - s)(X" Xl) coo (T. + 7z7) (10.12) 2 The solution is performed as before in Sect. 10.2.2.1. 10.2.2.3 Method - C (Stem state equivalent circuit)
In a system stability study, when the steady state equivalent circuit
(Fig. B.2) is used, the induction motor is represetned by a variable
impedance, and its variation depends only on the slip. This method is 169
\)--eliable if the rotor time constants are small. The main disadvantage
of the method is that during a transient, the system impedance is
continuously changing. Thus the use of such a method in a multi-machine
stability study, necessitate the repeated calculation of the system
mutual and transfer impedances at each step, in a step-by-step solution.
10.3 Comparison of computed and test results
To investigate the validity and accuracy of the methods of representing
induction motors in transient stability studies in Sect. 10.2, the test
arrangement shown in Fig. (3.1) was used. The motor was loaded by an
amplidyne, which was used as a d.c. generator supplying a resistive load. 2 A pump load characteristic (T cK w ) was simulated on the amplidyne 1 by using a suitable feedback signal (Sect. 3.2). The vacuum circuit
breaker (Sect. 3.1) was used for applying and clearing the faults (Fig.3.1).
During the period of short circuit fault, the tie line reactance close to
the infinite bus was sufficient to limit the short circuit current from
the bus to a safe level. In the calculations an allowance was made for
the leakage reactance of the 30 KVA transformer used for stepping down the
laboratory voltage from 440 volts to 220 volts. The faults were applied
at the terminals of the machine and the effective value of the tie line
impedance,- including the impedance of the above transformer, was
rc = 0.0142 p.u. = 0.0496 p.u.
Faults upto 0.142 Seconds duration were applied at the machine terminals
in-the present investigations to study the theories of representation, rather than to simulate exactly the conditions in a power system. The load torque speed characteristic determined under steady state (Fig. 3.4)
was used for calculating the transient performance of the machine. 4 170
10.3.1 Effect of rotor trapped flux
Under normal operating conditions a flux is set up in the rotor of an induction motor which cannot fall instantaneously to zero if the
machine eithe2/suffers a short-circuit fault or is disconnected from the
power source. During a fault, the rotor trapped flux, which rotates relative to the stator, induces an e.m.f. that produces current in any
closed circuit on the stator. The rotor flux and the induced currents
then decay at a rate dependent on the time constants of the circuit.
As the currents flow through circuits of finite resistance, the power is
consumed in the form of heat. Thus for a short period of time after
the inception of the short-circuit, the machine acts as a generator.
However, under open circuit conditions a comparatively high resistance
path exist in the iron of the stator.
Both under short -circuit and open circuit fault conditions, the motor
possesses only kinetic energy as there is no input power from the supply.
By interaction of the currents in the circuits on the stator and the rotor trapped flux, electrical torque is generated which retards the
machine. The friction and windage losses as well as the copper losses in
the circuits on the stator, consume the kinetic energy. A greater speed reduction is observed under a 3-phase short as then the current generated
in the primary winding and the electrical torque developed are both larger.
Fig. (10.1) shows the calculated speed curves for both open and short circuit
faults. In the Part II of the thesis, formulae have been worked out to
calculate the short circuit current based on constant sneed assumition.
10.3.2 Three phase short-circuit at full-load
Three phase symmetrical faults were applied to the single machine system
of Fig. (3.1) at the machine terminals. Fig. (10.2) shows the recorded *
slip curve as well as the curves computed with various methods discussed - -• t t • ,
.,;
1 1
Pt r-t 2
Prefault operating conditions Voltage = 1.05 p.u. r 4 -1 Current = 1.32 p.u. Slip = 0.01 p.u. Tie line resistance = 0.0142p. u. Tie line reactance = 0.04961).u.
Figure 10.1: Induction motor transient stability
n Speed recovery comparison for short- and open-circuit faults.
lt o (Calculated by the accurate method) --- Short-circuit fault. Fau MO AM al.10 WEI Open circuit fault
- ,-; 77.7 7717 -1-.1-- ••!- ; •• :•''' :T:yne!.. (seconds) 11 1 . .1 • , , - ; • 0.6.: t•
•
, s • ' "i • •1 - •,- .- _ i 43 _ , __ .. " • ' , , • 'I • • '';, ! • , 1 • .-i, °' 1 ' ; ' r : 1, .••• 4 • •A`'' ;. • 1 i s": ,,, ,: ... , , e ,• • ; ' • • ; • • r - - • 1- 1: -•; -••• - '1, T.A. •; :',.':;, 0' i!..7' _ ' O 1`..*‘ .e• i".P. Pre-fault oper-ting conditions .r.1. • ▪• 00431/ 0 Voltage 1.05 p;u:
• cH Current = 1.32 p.v. cH I_ I! • 7 0 I .,! Slip = 0.01 Phu. 1- Tie-line Resistance .120:t 1)::
p•--4 ! Tic-lino Renctqnce
,•
Fig. 10.2 i Speed, recovery of the -model motor after a 3--hasc symetvicrl fult.
MIN01101111W1M1 t on 7.:ensured1 l Comnutea by :-
ION
Fau Accu2:-..te- method Ap-nro:cimate 1.',c;thod-A
•••• at! 11' Liethod-B
Ale Ole Methos.1-C 173
in Sect. 10.2. No allowance was made for the saturation of the leakage or the main magnetic circuit inside the motor.
The stray load losses were measured under steady state and are
plotted against primary current in Fig. (2.29). All the computed curves shown in Fig. (10.2) allow for the stray load losses. It should be noted that when approximate methods are used, the discrepancy is larger. The approximate method A shows better agreement than the method B.
The reason for the discrepancy is that in the method A, term (pk,1 - jswolly is neglected wherein the components pyl and jswcy l have opposite sense.
Thus the error due to the approximations is small in relation to the case
(method B) where only the Op, term is neglected. As expected, the results computed with approximate method C (steady state equivalent circuit) show much greater discrepancy, and may not, therefore, be suitable for studying the transient response of large induction motors.
Fig. (10.3) shows the electrical torque computed by using the accurate method and the approximate method A. It should be noted that the electrical torque oscillates with frequency w during the fault, and for a short period of time after the fault is cleared.
10.3.:5 Effect of stray load losses
It is abosolutely necessary to account for the stray load losses in the machine while calculating both steady and transient performances.
These losses may be as high as 0.% under full load running condition and will increase sharply as the primary current increases. In fact at any-instant, the theoretical torque developed by the machine is higher than the actual torque because of the stray losses. The stray load loss curve
(Fig. 2.29) was fed into the computer by means of 12 points on the curve.
Linear interpolation was used to evaluate the losses between points. Once these losses are known, they are subtracted from the computed electrical •
1 1 % a. n t't r 2 z d1 ♦ `• s • # 0 a # # r - • r,, Sys ; rs'.0= ,•:•---*1 • 0 a • • .11.J " 8 ' 9 Is r*". (U 1 1 0
;• C 0 EA H 0 I
-P 0 (I) rl --11,
-5
0 .16 .20 214. .28 • .32 .36 .11.0
Time - Seconds n t o l u Figure 10.3 : Electrical torque developed by the model motor Fa during the trnnsient operation. ObQ 0 fa Calculated by accurate method
0111MIPIMMO Calculated by aimroximatc method-A l. 175
torque to obtain the mechanical torque. Fig. (10.4) shows the slip curves computed with the accurate method by including and excluding the stray losses. It is clear, when the stray load losses are neglected, the computed recovery performance scan be optimistic.
10.3.4 Open circuit fault at full load
The open circuit faults were applied to the single machine system of Fig. (3.1) by interrupting the supply at the machine terminals with the vacuum circuit breaker. Fig. (10.5) shows the measured slip curve as well.as the curves computed with the accurate and approximate methods.
The agreement seems to be reasonable between the measured curve and the curve
computed by the accurate method. An allowance was made in the calculations
for the stray load losses. For the reasons explained in Sect. 10.3.2, the curves calculated by the less accurate methods do not exhibit good agreement
with the measured curve.
Fig. (10.6) shows the torque computed with the accurate and the approximate method A. The electrical torque computed by the accurate
method oscillates at nearly fundamental frequency for a short period after the reconnection of the supply. The various torque components under these conditions are those given by Eqns. (5.21). 10.4 Numerical integration techniques
The transient behaviour of an induction machine is described by the
non-linear differential equations which can be solved by using numerical integration methods. The time taken to solve a set of equations is related
to their complexity, the nature of the integration method and the speed of
the computer. In this section accuracy of the results computed with
various integration methods and the comparative computing time taken by
each method, is discussed.
The values of the "integrable-variables" (I.V.) at time (t A t) arc •- ; Time seconds '1.0
-3
• -5 Prcfault olierriting conditions
-6 Voltage = 1.05 p.u. Current = 1.32 p. u. Slip = 0.01 r.u. 0 -p Tie line resistance = 0.01l.2 p.u. Tic line ro-;ctance = 0.C4-96 p. u.
•I
, • 1 11 , " 1
Figure 10.1.: Induct ion Motor Transient Stability Calculr,ted using the accurate representation
lt on ---- Stray-load losses neglecUed u
Fa Stray-lold losses included 177
U) 0 cv
0
Fault on Prefault loaa, csns:- Voltage = 1.05 p.u. Current = 1.29, D.11. Slip = 0.009 p. u. Tie-line resistance = 0.012 n. u. Tie-line reactance = 0.01 1D.u,
Figure 10.5 :3peea recoveiT of the model motor after a 5-1-Fhase open-circuit faun.
.111.11111041111.••• Feasurcd Comn.uted by: method
2.1)prote method1-3 4 •,• Frefault load conditions:- Voltage = 1.C5 p.u. Current = 1.29 p.u. Slip = 0.009 p.u. Tie-line resistance = 0.012 p.u.
. Tie-line reactance = 0.01 p.u. 0 IN • r;{ 3 IN,‘; , N- 1 a I s,:I 4 4
.16 .' .20 Time - seconds
n t o
l Figure 10.6 : Electrical torque developed by the model motor after clearing the open-circuit fault of .1k.2 sec. duration Fau Calculated by the accurate method Calculated by the approximate method-A 11 179
found by integrating the set of differential equations written in terms
of IVs at time t. The "non-integrable-variables", (N.I.V.) are the algebraic functions of the I.V.s and can, therefore, be calculated at any instant. The algebraic and the differential equations used in
various methods of representation of the induction motor are summarized in Table (10.1) below:
TABLE 10.1
Method Algebraic Differential Equations Eauations
Accurate 10.1, 10.3 10.4, 10.5 A 10.8, 10.9, 10.10 10.5, 10.6 B 10.11, 10.12. 10.5, 10.6
C Fig. B.2 10.5
The simplest method of solving the non-linear differential equations is Eulers'' "trapezoidal method" which consists of adding
the product of A t and x' (th slope of x at td to xn in order to find
xn 1. It was found that this method takes much longer computer time in relation to other integration methods due to Runge-Kutta, Kutta-Mersion, Hamming etc.
It is possible to "arrange" the equations and the sequence of their
solution in several ways. The term "arrangement" implies the grouping of
certain equations iii a form that is most suitable for a particular method of integration. The arrangement shown in Fig. (10.7) was used for
' calculating the transient behaviour of the machine. The names SYSTILM,INTEG I PLANT etc. appearing in the above said figure are the sub-programmes 180
Read input data and conipute machine constants
ComPute initial values of the variables and the pre-fault admittances.
Compute post- Compute fault fault admittanc C admittance .
Sub-PLANT Sub-INTEg One of the integratior Differential Eqns. methods ETV Eqns.
Write tne regd. information
Is No co ,'.t1,-,cnea
Y03 Fault Fault Removed Applied
STOP
Figure 10.7 : Computer programme flow chart 181
(or sub-routines), and they perform the following functions. SYSTEM: A sub-programme, which finds the solution of the differential
equations using a given time step for a given time of
solution. INTEL: Integration sub-programme is desinged to incorporate a particular method of integration, viz., Runge-Kutta, Runge-Kutta
-Gill, Kutta-Mersion, Hamming etc.
PLANT: This sub-programme contains essentially the main set of
differential equations. The algebraic equations (N.I.V.$) can be solved in the two places in the arrangement of Fig. (10.7) viz.
(a) In the SYSTEM sub-routine (b) In the PLANT sub-routine Some of the differential equations in PLANT are dependent on N.I.V.s which are calculated at the end of each step in the method (a). This implies that th the values of N.I.V.s at the end of the (n) step remain constant during th the (n 1) step while the differential equations are being solved in the "PLANT". Thus the values of "N.I.V.s" are always a step behind that of I.V.s. In order to maintain accuracy, a relatively small time step (A t) should be used. During the process of integration, the equations in the "PLANT" are evaluated four times for each Lt if the Runge-Kutta method of integration is used. Since the major portion of computer time is consumed by the
"PLANT" going through the "INTEL", a computation with the same time interval would generally increase the computer - real time - ratio
(CR - ratio) if the number of equations in "PLANT" are increased.
It was observed that for the same time stew (At)? the accuracy of results computed by arrangement (a) was better than that of (b). Runge-Kutta- 182
Gill method which is an improvement on the Runge-Kutta method has the facility of evaluating the error which can be kept under control by suitably modifying theiSt during the process of integration.
To reduce the CR - ratio, Hamming's Corrector-Predictor method is often used which needs only two evaluations of the equations in the "PLANT" for each A.t. However, the main disadvantage of :the method is that first it is not self starting and secondly, it tends to be unstable at sharp discontinuities. During such conditions of discontinuities, Kutta-Mersion's fifth order method can be used. These methods have also the facility of continuously evaluating the error.
- . For a single machine system, on the CDC-6600 computer, the CR ratio for Hamming's Corrector-Predictor method was 1.12 for &-t = .002, and for the Kutta-Mersion method was 0.383 for tl.t = .01 sec. Thus it is possible to perform the calculations in real time with the Kutta-Mersion method of integration.
10.5 Conclusions
Based on a comprehensive theory, four alternative methods of representing an induction motor having deepbar cage winding, for the transient studies are developed. These methods are used to predict the behaviour of the model induction motor under transient conditions of operation. Of the various curves calculated with the different methods, the curve calculated with the accurate method of representation exhibits a better correlation with the measured curve. Both approximate methods A and B show a bigger discrepancy.
However,the curve computed by method C shows the largest discrepancy and is not recommended for use in stability studies. However if it is used in a mula- machine study, it will be necessary to recalculate the system impedances at each step in the step-by-step integration method. 4 It has been found that in all methods of representations, an allowance 183
for the stray load losses should be made for improving the accuracy of the calculated results. "In the past, the tendency was to ignore these effects. The model induction motor used for the transient tests has a full load stray loss of 2.5%. However in the large machines, these losses are of the order 0.6% at full-load. If the stray load-losses are neglected in the large machine, their effect on recovery may not be as much as was found in the studies on the model. Nevertheless it is still important to allow for these losses for an accurate prediction of the transient performance.
The electrical torque computed by the accurate method during the period of short-circuit is of oscillatory nature and its pattern is generally in agreement with that calculated in Part II (Fig. 7.3) of the thesis on the basis of constant speed assumption.
It was observed that the accurate method of representation takes comparatively longer computing time. With the modern high speed digital computer, it is now possible to solve the machine equations in real time.
In the approximate methods, the axis flux linkages and current vary sloWly in relation to a.c. cycle while the accurate method shows that these quantities contain fundamental frequency components during the fault and for a short period immediately after the fault is cleared.
Various well known integration methods were tried. It was found that the Kutta-Version's fifth order method was most economical as it works with time steps upto .01 sec, and therefore a greater reduction in the CR-ratio is achieved. On the otherhand, if the object is to display the results where it is essential to use very small At, Hamming's Corrector-Predictor method should be used. This method is unstable for time steps greater than .002 sec.
4 184-
CHAPTER 11
MULTIMACHINE SYSTEi•i STUDIES
The analysis of a multi-machine system involves the solution of,
1.The differential equations of the machines
2.The algebraic equations of the machines, and 3.The equations of the network
11.1 System under investigation
A model system consisting of a synchronous machine, an induction
motor and an infinite bus was chosen for checking the validity of the representation of the machines. The layout of the experimental system is shown in Fig. (11.1). The synchronous alternator and the induction motor are connected via simulated three phase transmission line to the laboratory bus. Three phase symmetrical short-circuit faults of predetermined time period are applied at a selected point, F (Fig. 11.1). The system shown in Fig. (11.1) was adopted because of its similarity to a practical case of a petroleum refinery or another similar installation.
In the system of Fig. (11.1), the point C is the common link between the generator, the motor and the infinite bus. Usually the
equations describing each machine are written with respect to its own reference axes. The induction motor representations are those discussed in Sect. 10.2, whereas the various methods of representing the synchronous
machine are given in Sect. 11.2. As the reference frame of each machine
differs from any other, it becomes essential to transform quantities of
each machine to a common reference frame fixed to the infinite bus. The axis transformation is given in Sect. 11.3. 4 185
Infinite bus
Generator Mot or
Figure 11.1: Circuit Diagram for System Stability Studies.
.4
186
11.2 Synchronous machine representation
The most commonly used method of representing synchronous machines 29 in transient stability studies is based on Park's equations . Detailed
analysis of synchronous machines has been done by various authors.22,33-35
11.2.1 Alternator equations
The basic single machine system consists of an alternator connected
through a transformer and a transmission line (single or twin) to an
infinite busbar as shown in Fig. (11.2). The effect of saturation, eddy
currents and hysteresis are neglected. The sign convention of Adkins3° is followed. 30 According to the two axis theory, the general equations which
describe the voltages, flux linkages and the torque of an alternator
vd = p a (Avg ra id (11.1) v q = y a p q r q a i (11.2)
= rkd ikd kd (11.3) o - rkq ikq PTkq (11.4) of = rf if f (11.5) The flux linkages associated with each winding are, d L i d d + LMdkd Lmd if Li 4-L" Wq = qq mq ikq
kd = Lma ld + (Lkd Lmdikd + Lmd if i (Lkci Lmq)ikci q? kq = Lmq q L L (L, L )i f = mdd mdd md f The electrical torque is, w T i e = 0 (4)d. q L1Pqd)
4 transformer Infinite alternator tie-line bus R* X - V 1 r6176) R j X2 2
L
R j X
Figure 11.2 : The basic system 188
11.2.2 Accurate re-presentation
Like the induction machine representation of Sect. 10.2.1 in the accurate representation of a synchronous machine, no approximations are made in the machine equations, while deriving the differential equations for the variables. It has been shown35 that the differential equations for i are those given by Eqn. (11.12). It is d' 1q' 4)kd' and also possible to replace pi z_nd pi qby the differential equations for d' The two methods are equivalent. 111 d and 11Pq. . n 1- wag a3 all wa a6 vbd a7 id PId al s Pig w b b wb cub b b 0 1 2 3 4 5 b bq q c 0 0 0 lqf 1 c3 c4 1 f (11.12) di 0 d d 0 0 0 Ptd 3 4 Wkd 0 e 0 0 e5 0 0 PlYkq 2 Pkq v - -
The coefficients al, b1, cl etc. are given in appendix H. Since the inclusion of the pq d and p 4) qterms allow for the stator transients, the voltage equations relating the bus and machine terminals voltages must also account for the transmission line and transformer inductive volt drop during a transient (Fig. 11.2). Thus
X v X td = vbd w p i - R i - T w iq o d d } X . X = v - pi -RI + —0) tq bq w q w o o d
The equation for mechanical motion is,
2 (m P 6 = H Te Tl) (11.14) 4
189
11.2.3. Approximate representations Appropriate approximations are normally made in the machine equations
such that a reduction in computing time is achieved without significant
loss of accuracy.
11.2.3.1 Method-A (py d, pk:V ci and s terms neglected) In normal synchronous machine calculations, it is usual to reduce
the Eqns. (11.1) and (11.2) to22: = woo) + r i vd q a d vq -')01) d + ra iq
When the speed deviates from the synchronous speed and if the slip
is large, pti'd and py q terms are not themselves small but the values of
factors (pyd - swoy q) and (p y) q swoY d) are negligible. The above approximations remove the asymmetrical component from the variables. The
method allows for damping, system resistances, saliency and speed variation. 22 It has been shown that the differential equations for the variables andtOkci. are still those given by Eqns. (11.12) . It can also be (Al f tl' shown that, xn a 4)f + kd ) d = d (X"d - -1) X w o f (11.16) XII ) kg } i) q = cocl (x" - x1) 4---- c Xk i 0
Substituting Eqns. (11.16) in Eans. (11.15)
v = r i 4. X" i -i- v" d a d q q d . v = r i - X" i + v" q a ci d d q
4
190
\VIca where v" = w (X"o q- X1 ). 27
qjf ‘A)14..d. .01 = -mo(Xici - Xi) (c- ka)
Thus an alternator can be represented by a voltage behind its
sub-transient reactance. After dropping pi and pi terms in Eqns (11.13), the equations d q relating the machine and the bus voltages, are:
-Xiq vtd = vbd -Rid } v, = vbq -Ri +Xi The generator terminal voltage is then,
2 2 (11.20) vt = ilvtd vtq The mechanical motion of the machine is still governed by the 0 under steady Eqn. (11.14). By setting Af - P‘Ykd PlYkq state, the field voltage is obtained as,
(Xicl - Xi) Xma ( • (XA )1
v i 7 - 1 qo (11.21) fo = [Yfo do 1D 775 (5o 1 cd 11.2.3.2 Method-B (pYd and pLYcl terms neglected) When only the 1:AVd and pyq terms are neglected in Eqns. (11.1) and (11.2), it can be shown that Eqns. (11.17) become v = r i. + (1 - s) X" i + v" d a a q d } (11.22) v • = r - (1 - s) X" i + v" q aq d d q 4)k where v" = w(X" - X__) -•=11 d q -1 X ha (11.23) -P - kd } v" = -w(X" - Xi) (--=.. , + .7-21-) d f ..3. kci 191
To conform with Eons. (11.22), Eqns. (11.19) can be rewritten as, bd d vtd = ir i - (1 - s) X ia, } (11.24) v = v - R i + (1 - s) X tq bq -d f The differential equations for the variables , kd and t'll)kci are still those given by Eqns. (11.12) 11.2.3.3.Method-C (Daispin, neglected)
In this method the effect of the damper winding is neglected while deriving the equations but an allowance is made for the electrical torque.
The set of equations describing the behaviour of the synchronous machine 22 are;
= r i + Xt d a d q q } (11.25) vq = r iaq - Xt ddqi + vt
d where vt = -(X Xpid and, 1f x } 1 v md o and pvt 77" 'q - r f - (Xd - Xi)i 11 do f d d Te = vcti
The damping torque is allowed for by the following imperical equation given by Crary.11 2 2 v (Xt - X")s Sin 172(X - X")sa Cos% T b d d b damp = -KT+ 4--yd occi 40c4 :77c1 (11.28)
where ad direct axis damper decrement factor (Xt + X) 1 (X" + Tao
a = quadrature axis damper decrement factor q (Xa + X) = q T"qo 4 192
11.3 System network representation
11.3.1 Transformation.,from the machine axis(d,a) variables to the system (Mi 0 axis) variables Equations representing the individual machine are with respect to its own reference frame which in general is different from that of any other machine as well as from the common reference frame ( a, p axes) rotating at synchronous speed, wo. To relate any machine to the other machines in the system, it is necessary to transform it's axis quantities to the a , p axis. Because of the dissymmetry of the rotors of synchronous machines, it is possible but complicated to transform the secondary equations to obtain a relationship entirely in terms of system phasors. Thus only the primary quantities are transformed to. the common reference frame.
Synchronous machine
Eqn. (B.15) defines a phasor corresponding:to the system voltage given by Eqn. (B.14).
With Park's transformation equation, the instantaneous phase-A voltage is related to the axis voltages by,
va = vd Cos e v q Sin 0 (11.29) where 0 is the angular displacement of the axis of phase-A from the d-axis. The machine load angle, S is defined in relation to the reference voltage of fixed magnitude and frequency, wo given by Eqn. (B.14). Thus
0 = wot - 6 (11.30) Substituting Eqn. (11.30) in Eqn. (11.29),
j(w" 4 _ - 6)1 v Re I v )e a d q (11.31)* ot - - Re ej(w 6) ga 193
where V = v - j v (11.32) ga d i q = alternator complex axis voltage
Comparing Eqns. (B.14) and (11.31),
V = V e j(6 4. %) ga m vd = V- Cos(b + X) (11.33) or m , v = _V J q m Sin(6 -:- X) Eqns. (11.33) relates the axis voltage to the system voltage. Also substituting Eqn. (B.15) in Eqn. (11.33), (e gating Vg = 17-10) ' iT = V eib (11.34) ga g Thus the above equation relates the complex alternator voltage Vga with the complex system voltage, V on a, p axes which is defined as g V •(11.35) g =Vg cc -jVgP Induction motor
Eqn. (11.34) is also applicable to the case of an induction motor. The only variation is that, 6 = swot (11.36) A relation similar to Eqns. (11.34) also holds for the currents in the two kind of machines.
214,3.2 Accurate representation of the network
In Fig. (11.1), the voltage (Vb) at the bus and the current (Ib) in the common tie line are the complex quantities defined in the common reference frame ( a, p axes) as, V =V b a -jV (11.37) Ib = Ia j I
The complex voltage Vc of point C (Fig. 11.1) is also defined in the similar manner. The voltages Vb and Vc are related by a complex equation similar to Eqn. (10.1) as, 194 •
X b (11.38) Vb = Vc + w pIb + + j X) ' where I I + I b g m -jr5; Ig = Iga e 6 -ism I Im ma e pIb = p Is + p Im (11.39)
pI = e 6g pI j s w I g ga g o g -j pIm =e6m pima -jscoIm o m Eqn. (11.38) in the accurate method for accounting the voltage drops in the network impedances. 11.3.3 1L222.-oximate representation of the network
In this case, the pIb term in Eqn. (11.38) is neglected, which implies that the effect of system damping is neglected. Thus equation relating Vb and \To is, Vb = Vo + (R + jX)lb (11.40) 11.4 The method of analysis for the system under investigation There are four methods of representing each synchronous and induction
machine. To check the validity of the first two methods, they were
applied to the system of Fig. (11.1). Three methods, formed by combining the first two representation of each king of machine, were used and they are listed in Table 11.1.
.4 195
TABLE: 11.1
Representations Method Induction motor Synchronous machine Network
XI Accurate Accurate Accurate Sect. 10.2.1. Sect. 11.2.2 Sect. 11.3.2 X X - do - - do - Approximate 2 Sect. 11.3.3
- do - Y1 - do - Approximate Sect. 11.2.3.1 Y Y Approximate Accurate - do - 2 Sect. 10.2.2.1 Sect. 11.2.2
Z - do - Approximate - do - Sect. 11.2.3.1 _
From the knowledge of the computed results of these combinations, it is possible to estimate the calculated performance with the other combinations. 11.4.1 Method-X
The method uses the most accurate representation for each kind of machine for studying the operation of the system of Fig. (11.1) under transient conditions. Two possible variations of this method exist, depending on whether pIb voltage drop in Eqn. (11.38) is considered or neglected. The two methods are classified as and X2 (Table 11.1). 1 11.4.1.1 Method-X 1 (accurate)
This is the most accurate method of representation for the model system under investigation. In this case, it becomes necessary to formulate the equations of each machine with respect to the infinite bus voltage, Vb. 4
The analysis is given below.
196
Substituting.Eqns. (11.39) in Eqn. (11.38), the voltage Vc of the
point C (Fig. 11.1) is obtained as, L -j6 X .= V - w e m pI — pI c b o ma wo ga
- R [Im Ig] - jX 1 - sm) Im (1 - s ) I (11.41) g g Now V c is the applied voltage for each machine. After transforming V by Eqn. (11.34) to the reference frame(d-q-axes) of each machine, its axis components are treated as the applied voltages in Eqns. (11.12) for
the synchronous machine and Eqns. (10.4) for the induction motor. It
should be noted that in both machines, only the differential equations
for the primary flux (or current) arealtered. After carrying out the above exercise, the differential equations for the primary currents of the two machines are:-
pIag A pI -1 B qg mo [G] (11.42) pIdm C pI qm D where °)(3 .A.X+ + — X Cos 6 X Sin 6 a6 gm gm CO • + -X Sin 6 X Cos 6 6 gm gm
[G] X Cos 6 -X Sin 5 X 4. X" (11.43) gm gm
X Sin 6 X Cos 6 X -1- X" gm gm
'6 = 5 - 6 gm g m (11.44) The quantities A, B, C and D are given in Appendix I. Eqns. (11.42) .. are the differential equations in terms of the axis variables of the two machines. 197
The differential equations for the primary currents (Eqns. 11.42), 1 the secondary variables of the two machines (Eqns. 10.4 and 11.12), and their
motion equations can be solved simultaneously on a digital computer as
explained below. Steady state condition
In a multimachine system study, various methods of calculating the•
steady current, load angle (or slip), terminal voltage and power flow
of each machine have been derived. However for the present study, a load flow technique was used to find the steady condition. For this purpose
the induction motor is represented by its equivalent circuit, Fig. (B.2). The system of Fig. (11.1) is treated as a 3 machine system wherein
the infinite bus is represented as a source of fixed voltage and frequency. The terminal conditions of each machine so obtained, are then transformed
through Eqns. (11.34) to get the variables referred to the machines_own
axes. For the synchronous machine, under the steady state, the left handside of Egns. (11.12) is zero. By applying this condition, the initial
values of all the variables is obtained. However for the induction motor,
jscoo is substituted for p in Eqns. (10.2) which are then solved for the steady values of t)1, tp2, and q)3. Transient condition
During a transient, Eqns. (10.4), (10.5), (11.12), (11.14) and
(11.42) are solved simultanesouly following the method explained in
Sect. 10.4. The major difference conformed with a single machine study
. is that it is necessary to evaluate the inverse of. matrix [G1(Eqn.11.43) at each step during the process of integration. The three phase symmetrical fault is applied at point F in Fig. (11.1)
for a predetermined period of time. The three phase short circuit fault
is simulated by equating the 210 X.b and Vb to zero and is cleared when .4
Xb and Vb are equated to their original prefault values. 198
The initial conditions of the system are listed in Table 11.2.
The calculated and measured swing curve for the generator and the speed
curve for the induction motor and the current and voltage curves for the two machines are shown in Fig. (11.3). The agreement throughout is reasonable. Due to the presence of the short lived asymmetrical component, the initial portion of the current curves is shown by an envelope.
The swing in the load angle of the synchronous machine is very small.
The system under investigation was stiff for the following reasons:
a.The induction motor is rated at 220v which is also the voltage of point C. To make the synchronous machine less rigid, it is necessary to increase the reactance, X . This raises the terminal voltage of the generator when it is supplying lagging reactive power. Under these conditions, the generator voltage can be more than 1.1 p.u., -which saturates it to a large extent (Fig. 3.7). Thus to keep the terminal voltage of the generator low, it is necessary to keep Xg as small as possible. b. The other method of making the system less stiff is to increase
X in Fig. (11.1). When this is donel then for maintaining 1.0 p.u.
voltage at the motor terminals, the bus voltage increases. Under the
present case it is 240v (= 1.09 p.u.). The reactance units which make X, are designed for operation at more than 230 volts and 13 amps., under
steady state. It is not, therefore, advisable to operate these reactances at larger voltages. Two reactance units were used in parallel. Further a larger value of X slows the recovery of the voltage of point C after a fault and hence that of the motor.
c.These difficulties can be overcome by reducing the operating
voltage of the induction motor but this will make its characteristics
quite different from that of a large motor. ..• 199
TABLE 11.2
Initial conditions of the system under investiation
Particulars values
0:0065 Network impeannce Rb . X.10 0.093 0.0045 Rf 0.031 Xf Rg 0.0062
0.2
Bus voltage 1.0226
Bus current 0.78
Motor Terminal voltage 0.96
Current 1.36
Slip 0.0107 Power 1.14
Generator Terminal voltage 1.03 Current 0.58 Load angle (bus) 29.6°
.4 Power 0.49
Vars - 0.33
Unless mentioned otherwise, all values in the above table are in per-unit
200 0 ul r—I 0 130. 32.5 , / N.,,bs' • *
rd 30 0 0 27.5
.r--3• 1.2 F-t • ^ --a • 0 .8 O 4-, o Ho 0 . 0 0-.-
3 F-i — • -P 2_ -Pcr.; pit • rl • . o 11-1 Pi . .._.. . •-•-•N,... p.:-..-.,-0--- ,-::-.- - 0 IA....,•. • -, - -. -----,-0-.._ „-v,--,_ C3 - 0
. 4
1.0 0 0.8 f-t -P 0 H 43 0 0.4 0 0
3 -
0 .---- • . • •
0 0
t--.— i.... t___..--...... I..— -a..--- t..- ...J. t..— --a 0 A 0 . 0 . 4 0.8 1.2 - 1.6 2.0 2.4 „..,f ,z) ,- , Time — sec.onds ..--: Hp, 1.;- $.---1 5 cl f-1 ;121 H 0 cz,'
.4 Figure 11.3: System transient characteristics lleasurea Calculated. with accurate methoa-X 1 . 201
The effect of saturation of the main flux path can be taken into account by the method explained in Sect. 3.5. In the present case, the saturated values of Xmd and Xmq determined under steady state from Fig. (3.7) were used. The.effect of modifying them during the transient was insignificant. However if the swing in the load angle- is large, it may be necessary to modify these reactances during the transient.
Inspite of all the above limitations, the system is quite suitable for checking the validity of the method of calculation.
11.4.1.2 Method-X2(approximate) '
In this case, during a transient, 1)III term in Eqn. (11.38) is neglected. The representation is then similar to that of Refs. 34 and 55. The steady state terminal conditions are obtained as before (Sect. 11.4.1.1).
However, during the transient operation, the machine equations and the network equations are solved alternatively. During each step, while solving the equations of each machine in turn, its terminal voltage is assumed to remain constant and the new value of the current is calculated. Eqns.
(10.3), (10.4) and (10.5), and Eqns. (11.12) and (11.14) give the rate of change of the variables with respect to time for the induction motor and the synchronous machine respectively. These equations are used to find the values of the variables after a time,At.
For a short period of time, immediately after the application of the fault and its clearance, the primary current and flux contain asymmetrical components. It is difficutl to separate the symmetrical and asymmetrical components from the instantaneous transient current.
Hence for the purpose of calculating the voltage drop in R and X,
(Fig. 11.1) an assumption is made that the instantaneous current obtained by summation of the currents of the individual machines, is an equivalent 202
low changing current with respect to the 50 Hz wave.
The step-by-step process of solution can be summarized as follows:
1.Using a load flow technioue, calculate current and voltage
at terminals of each machine under steady state.
2.During the transient, assuming that the terminal voltage of
each machine remains constant for the interval At, find the
new values of the variables.
3.Find the new value of the current in the common tie line
I,), which is the algebraic sum of the Ib'(= Im currents of the two machines transformed to the a axes. .4. The new value of the voltage V at point C is, c Ve = Vb - (R jX) Ib
Subtracting from Ve, the voltage drop in each machine's transformer
impedance, the terminal voltage of each machine is obtained and
it is transformed to machine's own axes using Eqns. (11.34). 5. Return to step-2 and repeat until such time as a new steady state condition is reached.
In general the flow diagram of Fig. (10.7) is also valid for the
present case.
The computed and measured curves for the two machines are shown
in Fig. (11.4). The agreement is not as good as was obtained with the
method-X1, particularly during a short period immediately after application
of the fault. Due to the approximations in the calculation of the voltage
. drop in R, X (Fig. 11.1), perhaps the effect of the initial kick is lost
to some extent.
Fig. (11.5) shows the comparison between the results calculated with
methods Xi and X2. The swing curve calculated with method-X2 has higher
,amplitude and frequency of oscillation than that calculated by the 35 • • -0 203
• e- - . / 3 I • ,•-• - \ N-, --- ;.''''.-. 0 0 27.5-
1 . 2
FA Pi • 0 Irc • -P0 0 . 8 cJ •o 4-) 1H 0 . O 0 0 I> 0
• F-1 O 4) 'JP RI 2 c.;r: r-4 • • pi 1 0 0 o - o
P 1 .ri 2 H • , •
Pec•-I 1.0 • • O. 8
•O r 0 . 4_
0 - •
• , 3
•O 2 2
P • - • - P Pi o 1
0 0 • . 0 . 6 1. 2 1 . 6 2.0 2 , 4 rz; C.)
Figure 11. Syst em transient charc.,ct eristics Meacurea - -o - Calculated vzith appro:Kimate method-a 2 A
„a 4, .4,, r.,1! 35 , Od• / H 1) /-)t•-• 0-> rd Kr • • c>•.. o •• • 4-, a) H 30 tO c rd 01) - 4/ 8 0
20 -21
0
I IL •o 2
6 ii 0 0.4 0.8 1.2 1:6 , 2.0 2.2 , _.,2.1}. Time - seconds.
Figure 11.5: System transient characteristic Measured Calculated with method - X -0-- Accurate Approximate . 205
accurate method. The' approximate method predicts a faster recovery of the induction motor. !Though the descrepancy between the results calculated by the two, methods for the system under investigation is less than 5% it can be more in a less rigid system.
It is recommended in Ref. 34 that in addition to the pIb term in
Eqn. (11.38) the changes in the electrical angle between the terminal voltage of each machine and the common reference axis (in the present case it is the infinite bus) should be neglected during a transient.
Fig. (11.6) shows the instantaneous angle between the bus and the terminal voltage of the generator calculated by the methods X1 and X2. The degree of variation suggests that the above assumption is not justified. The variation in the angle can be greater if the system reactances are large. However the method-X2 has an advantage over the method-X1 that for the former the CR ratio is roughly 30% less.
11.4.2 Method-Y 'In this method, the induction motor and the synchronous machine have accurate and approximate methods of representation alternatively.
The two methods (Y and Y ) are defined in Table 11.1.. The voltage drop 1 2 due to pIb in X is neglected. The method described in Sect. 11.4.1.2 is used for the solution.
Fig. (11.7) shows the comparison between the measured and calculated results with method-Y wherein the synchronous and induction machines 1, are represented approximately (method-A) and accourately (Sect. 10.2.1) respectively. For the induction motor, except the current curve, the speed and voltage curves show reasonable agreement. The calculated current
.of the induction motor, for a short period after the removal of the fault, is more than the measured value. On the other hand, the calculated .! swing curve of the generator has higher amplitude and frequency of
- 80 , —' 60 11
. -10 , •i -10 ji ; 0,3 1111 r-11 7 20 1:-.1.!, t 1. • ea ti 1,-. r---.... b 0 Itl i \ y , g 20 '1 . 01?. • 0.2 0.3 Time - sec. 4-0
P. A
" " •• • " " " • o "
-2 L
L • ' 0 0.2 0.4. 0.6 0.8 1.0 1.6 1.8 2,0 2.2 Time - seconds Figure 11.6 Phase angle between the generator terminal votage and the bus voltage. - - - (.31 Calculated with accurate & approximate method -X
207- • ' 3 5 - s • / , o o ,, 0: --. . H (.9 /• .„./'''-*.---''''•s. S - , ss N • b tU ri2. \ .• , 4 \ I ...... "-...'.. • 5 , % ;. Vf'f. ,„,. 'r,v• 'b. . rd n 30 ...,..••• I 0 rt.; s o 0 -27. 5-
• 1 2 Pi Pi ...••"—...... 0 /... a: 42) 0.8 i':20 A•1 1-1 0 o (). o 0
-a-
1.0 • 0.8
0 . [
0 -
3 - - • 2 p, a -poo O 0 - • 0 o.1I. 0.8 1.2 1.6 2.0 2.- 1
e • Time - secortias rl c P. C;
Figure 11,7:Sys t em transient characteristics ----- Measured -- 0-- Calculated with method-7-1 208
oscillation. The swing of load angle is bigger because when p (pa and
py terms are neglected in a synchronous machine, the effect of additional
damping due to these terms is lost. Once disturbed, the rotor takes longer to settle to a steady value, because the damping effects are reduced when stator transients are neglected. Another reason for the
better agreement for the motor quantities is that for accelerating, the
motor draws most of the power from the infinite bus and very little from the generator.
The measured and computed results with method-Y2 are shown in Fig (11.8). In this method, the synchronous machine is represented accurately (Sect. 11.2.2) whereas the induction motor is represented approldmately by method-A. The swing curve of the generator shows a better agreement, but it is not as good as was obtained with method-X. This is
probably due to the fact that the induction motor is much larger than the synchronous machine and during a transient, it affects the behaviour of the latter to a greater degree. The induction motor speed and current curves show reasonable but not very satisfactory agreement. A quicker recovery is predicted than that measured or calculated using accurate representation (Sect. 10.2.1). The disagreement in the current curves is due to the earlier recovery of the motor, as after having recovered,
demand of the motor on the system for abnormal currents is reduced.
11.4.3. Method-Z
This time both synchronous and induction machines are represented
by- approximate methods of Sects. 11.2.3.1 and 10.2.2.1 respectively. The voltage drop due to pIlo in Xis neglected. The calculated and measured results are shown in Fig. (11.9). The calculated performance of the synchronous machine is similar to that calculated with method-Y-1 (Fig. 11.7)* and the induction motor behaves as in method-Y 2 (Fig. 11.8). Thus when the
209 - • -eN 0 • • - H . / to 0 32 . 5 t OJ rd 30 0 1'1 27.5-
1.2 -- tz, ofo -P 0 0.8 P d a) -p g H 0.4 0 0 o
3.
—
_ •rl 2 4 •
..cop0) 0.8
OH 0 • LI. -P 0
41) 0
3
O
0 Lr, • 0 .\ 0.4 0.8 1.2 1.6 2.0 2.4 A,„; 0 0fJ Time — seconds
4
Figure 11.8: System transient characteristics ----- Measured Calculateawith method T _ _
-et 35 - • . - 210 _ t ...e.,...... S...„, 0 VI - I %, ...,..\ o • i .„.••fe • .t.0 — • ...1 ' I . o P l' • \ A ...__ P C-1 1 \ .d...... ',.. If ..l. O" \ N.- ‘ - 0 F:1 27. 5- S_ I - 9 1 • 1 2 P Cs • - • 4-) 0 . 8 • fa, c-i r--1 0 . 4 O 0 0
3. IN* P O -P pi • 2 c.; G pi P f-4 • • f--t Pi 1 . .,b\711- • 0 - •O 0 • - .
0 --• . . - 2 Ce.
cn • 4
0.8
OH -P o 0 :--=• &!, 0
,,: . - . - 3 .. - 11 -....,1„, ....,. .•...... _.„ • 0 • ,D -- 0 , 0 r-, " •--.. f-t • ' rt:I;?( • __ • r-t f-1 Pt W.' • .•••••••• • 0 •1 / _ -P 0 .t. .O ,-.... . L__..,.._ . • ....1 —_t---. - , __1___ , 0 1 A . 0 t. 0.4 0. 8 1.2 1.6 2.0 2.4 A _. ...,0 --f.0 c'ir...-- sec alias ;fi r Pi c; •;-1 g ..: 4-1 Pi Figure 11.9:' System transient characteristics
Measured Calculated with methoa-Z 211
calculations are done with this method, the induction motor indicates a quicker recovery whereas the synchronous machine shows bigger oscillations and takes longer'to settle to a steady value. Thus the calculated performances of the two machines are contradictory. Recovery of the motor is optimistic whereas that of synchronous machine is pessimistic.
However for the system accuracies, the present method can be used.
11.5 Application of the method of calculation to a general n-machine system
It is possible but rather complicated to write the differential equations in a form similar to Eqns. (11.42) for a general n-machine system.
However the method foicalculations, described in Sect. 11.4.1.2, can be extended to multi-machines system. It is usually advantageous to include the infinite bus as one of the machines in the system studies. As the angular position of the rotor of a machine is different from any other, before manipulating the quantities of different machines, it is necessary to transform them to the common a, p axes fixed to the infinite bus.
This set of axes rotates at synchronous speed. The inclusion of the infinite bus in the-multi-machine studies has the following advantages.
a. Each machine swings with respect to the infinite bus.
b. The network frequency is fixed.
c. The lumped parameters of the network remain constant
except when they are modified to represent the initiation
of a disturbance or a switching action.
Thus the algebraic equations of the network can be written as,
.4.
212
R21 Xil R12 X12 Rln Xln Ial X3:11 R11 -X17 R12 -Xln Rln 'p1 RXRX R X 21 21 22 22 2n 2n I -X21 R21 -X22 R22 X2n R2n p2 (11.45)
• • • • • •
. • • •
• • • •
R X R X an n1 nl n2 n2 Rnn Xnn V -X R -X R I pn nl nl -Xn2n2 n2 nn nn pn
A load flow study gives the steady state values of the network voltages
and the power flows. From these, the initial values of the machine variables
are calculated.
As explained in Sect. 11.4.1.2, the machine equations and the
network equatiOns (Eqns. 11.45) are solved alternatively during the
transient solution. During a step, while the equations of each machine
are solved, its terminal voltage is assumed to constant and the
termirn1 current is calculated. The currents so calculated, are first transformed to the common reference frame ( a, p axes) and are then substituted in Eqns. (11.45) to obtain the new value of the machine terminal
voltages.
.If the accurate method and the approximate methods A and B are used
for representing either kind of machines, it becomes essential that the
machine and network equations are solved as explained above. However
when the sub-transient saliency is neglected in the synchronous machines
Xlci) and the approximate methods A and B of representations are
used for machines, it is possible to simplify the solution considerably
by using the complex phasors. Eqns. (11.17) for the synchronous machine 4 and Eqns. (10.8) for the induction motor can be converted to the system
213
phasors by Eqn. (11.34). For a system of n-machine with terminal voltages
V and currents In, the relationship between voltages and currents may be n written as;. V z Z 1 Z11 z12 - 13 ln I1 V Z Z I 2 21 22 23 2n 2 V Z Z Z Z 1 3 31 32 33 3n 3 (11.46)
• •
• •
• • •
V Z I n n1 n2 n3 nn n In the above equation, ... and etc., are the quantities referred to tilt common m,p a'es. When each machine is represented by the sub-transient reactance and the
voltage behind the sub-transient reactance, then
V1 =(ri j VV (11.47)
Substituting for the machine terminal voltage V1 from Eqn. (11.47) in Eqns. (11.46)
lar ropl Z12 . Z Z Z11-(r11-i 13 ln 1 Via Z Z22 -(r -1-jX") Z 1 2 21 1 d 2 Z23 Z2n 2 V" Z Z32 Z.33-(r14-jX1d3 Z3n 1 (11.48) 31 3
•
Z Z -(r q-jX") I m n1 n2 n) nn 1 d n n Inverting the above matrix, 214
Y Y V" Y11 Y12 13 ln 1 Y 12 Y2121. Y22 23 Yen 2 I Y I V" 1 Y 32 3 (11.49) 3 31 33. 3n . . . . • . . , . . • . . . . • I I vu n nl n2 n3 Ynn n Vb.
In Eqns. (11.49) the system admittance I now includes the primary
and sub-transient reactance X" of each machine. This resistance r1 equation defines the system perforamcne, in terms of the system phasors of current and voltage behind the sub-transient reactance, of all the machines in the system. All the parameters remain fixed unless changed due to a switching:action.
However if approximate method-C is used for representing either kind of machines, the terminal conditions of both types of machine cannot be defined by a single equation of the form of Eqn. (1147). When method
C is adopted for the synchronous machines, the terminal conditions are given by Eqns. (11.25) and no correspondence.with the induction motor is possible. If the method-C is used for representing the induction motor, the machine is reduced to a variable impedance which is included as part of the system impedance. One of the main advantage of using the general equation representation for both types of machine is that each machine, whether it be synchronous or asynchronous, is linked to the system impedance matrix by the single Eqns. (11.47).
11.6 Conclusions
One accurate and three approximate methods are proposed in Chapter 10 for representing the deepbar squirrel cage induction motor. Similar 11,22,35 methods of representations for a synchronous machine are available elsewhere
It has been shown that when the machines are represented accurately, for an 215
accurate prediction of the system performance, it is necessary to rewrite the differential equations of the system with respect to the infinite source in such a way that they could be solved on a digital computer.
In an alternative method, the voltage drop due to pIb in the network is neglected. Thus it is possible to solve the equations of the machines and the network in the manner described in Sects. 11.4.1.2 and 11.5.
For the system under investigation, it was observed that the method X 2 predicts bigger swing in the load angle, whereas the recovery of the induction motor is not so much affected. From the results of
Figs. (11.7) and (11.8), it is evident that the two machines are not influencing each other to a noticeable extent, and results of Fig. (11.9) confirm this deduction. The reason for this type of behaviour are given in Sect. 11.4.2. However for system accuracies, it is sufficient to represent each kind of machine approximately as in the method-Z. With this method, an optimistic recovery is calculated for the induction motor and a pessimistic one for the generator. Thus to be safe, it is essential that the induction motors closer to the fault are represented accurately.
As discussed in Sect. 11.4.1.2i-the electrical phase angle between the terminal voltage of the generator and the infinite bus voltage should not be held constant at its value under steady state.
If approximate method-A of representing both synchronous and induction machines is used and the transient saliency in the synchronous machines is neglected, it is possible to link each machine by a single equation, whether it be synchronous or induction, to the system impedance matrix equation, and the primary resistance and sub-transient reactance of each machine, are included as part of the system impedance. 216
Comparison between the test and calculated results has shown the importance and necessity of making an allowance for the stray losses in induction machines and saturation in the synchronous machines. 217
CHAPTER 12
SUGGESTIONS FOR THE FUTURE WORK
The first part of the thesis deals with the design of the model induction motor and explains the reasons for its drawbacks. If another model is designed in future, special notice of the recommendations made in Sect. 2.5 should be taken.
The proposed theory of representing the deepbar cage motor by two fictitious coils on the rotor can be extended to study unbalanced faults both on the constant and variable speed assumptions. Analytical expressions for the electrical torques developed by the machine under different three phase symmetrical faults are developed and by measuring the torque, the validity of these expressions could be checked.
The pump load characteristic was chosen for the present studies but there is no reason why any other kind of load characteristic cannot be used. Further studies can also include the effect of the load transients if they are known.
Another study can be made to investigate if the recovery of the induction motor is assisted by introducing a reactance suitably controlled by feedback signals. One obvious scheme could be to use a saturable reactor whose reactance is automatically varied by a signal proportional to the change in speed during a transient.
The three machine system studied in the latter part of the thesis was a little stiff and thus the degree of accuracy of the method-X2 is not fully established. The effect of saturation of the generator is insignificant in the present case, but it can be more if the generator load angle is allowed to swing to a greater degree. In future studies, 218
the effect of voltage regulators and the turbine characteristic can be included. The one way of making the system less stiff is to make the generator of comparable size to that of the motor.
In future multi-machine studies, the infinite bus may not be included as one of the machines. With little modifications, the programmes written for the multi-machines studies of the thesis, can be extended to include only the synchronous machines. Thus a OWR and DWR machines can be connected together to the bus to study the interaction of one machine on the other. Some interesting results should be obtained as it is quite difficult to visualize the transient performance of such a system otherwise. 219
PART IV
OTHER STUDIES ON THE SYNCHRONOUS MACHINES 220
CHAPTER 13
STUDIES ON THE ALTERNATORS
13.1 quadrature axis excitation studies 2 It was shown by Kapoor , that a quadrature field winding, controlled
from the bus load angle (0 feedback, is an effective way of extending
the range of stable operation of a generator to cover the whole region
of negative reactive .power, at any active power up to the heating limit
of the machine. In practice, it is often difficult to obtain a signal
proportional to the bus angle, and therefore it becomes essential to
•use the machine terminal load angle (at) feedback. The author
collaborated with Sarma39 to extend Kapoor's work to include the effect
of feedback, and the conditions were found a little less favourable t with this arrangement. However the results are still sufficiently good
for practical purposes. A new method was developed to account for the
saturation of the main flux path. Good agreement was obtained with the
corresponding computation for the both kinds of feedback signals. The 2 results of the above studies are incorporated in the following paper . 221 Improvement of alternator stability by controlled quadrature excitation
S. C. Kapoor, M.Tech., Ph.D., C.Eng., M.I.E.E., S. S. Kalsi, B.Tech., M.Sc., and B. Adkins, M.A., D.Sc.(Eng.), C.Eng., F.I.E.E.
Abstract Operation of a synchronous generator at leading power factor has been severely limited in the past because of stability considerations. A regulator, acting on a normal direct-axis field winding, can only extend the range of stability when the generator is loaded, and has no effect under unloaded conditions. An additional winding on the quadrature axis, provided with a suitable control, can, however, ensure stable operation at any leading-power-factor load, within the heating limit of the generator. The most effective control uses a closed loop actuated by a signal derived from the load angle. The theoretical treatment in the paper consists of two parts. First, some general results are deduced from simplified equations, particularly relating to the limitations of a direct-axis regulator and the benefit of using an angle signal with the quadrature regulator. More complete computations are then made to obtain stability-limit curves for many alternative schemes. The work is concerned with the steady-state stability of a 1-machine system, in which a generator is connected to an infinite bus through a reactance. Experiments to confirm the theoretical results were carried out on the micromachine equipment at Imperial College. The alternatives studied included simple proportionate regulators and more elaborate schemes using first- and second- derivative elements, and the angle signal was taken alternatively from the infinite bus and the generator terminals. Good agreement was obtained with the corresponding computations.
List of principal symbols Rq max, Rqmia = maximum and minimum quadrature-axis vd, vc, = direct- and quadrature-axis voltages regulator gains id, iq = direct- and quadrature-axis currents L(p), Lq(p) = direct- and quadrature-axis open-loop trans- yid , vfq = direct- and quadrature-axis field voltages fer functions ifd, ifq = direct- and quadrature-axis field currents o = subscript to denote steady-state value v, = machine-terminal voltage
Capital letters V and I, with the same suffixes, are used to 1 Introduction denote phasor components of voltages and currents Recent developments in electrical-supply systems have V= infinite-bus voltage tended to bring about a change in the conditions under which V j = voltage behind leakage reactance the generators operate.' At times of low power consumption, Va = voltage behind synchronous reactance the generator, because of increased charging currents in the ra armature resistance high-voltage transmission network, often has a leading power rfd, ri, = direct- and quadrature-axis field resistances factor, and may need to operate beyond the normal stability Xa = machine-leakage reactance (includes line limit. Moreover, the higher reactances of modern generators reactance) reduce the normal range of stable operation at leading X,, transmission-line reactance and admittance currents. The paper describes an investigation of a new method of control, by means of which the range can be extended. Xmdt Xmq = direct- and quadrature-axis magnetising reactances The normal range of stable operation of a generator with Km = saturation factor for magnetising reactances fixed excitation is severely limited at leading current if a KmA saturation factor for incremental mag- reasonable margin is allowed, but it is well known that the netising reactances range can be extended somewhat, under loaded conditions, by Xd(P), Yd(P) = direct-axis operational impedance and ad- means of a continuously acting voltage regulator acting on a mittance direct-axis field winding.2 At low power, this method is much ;(P), Yq(P) = quadrature-axis operational impedance and less effective. The present paper shows how, by using an admittance excitation regulator acting on a quadrature-axis field winding, 1 it is possible to maintain stability at large negative reactive direct-axis synchronous, transient and sub- power, at any output. Yd,Xd, Xd,Xd transient reactances and admittances Fig. 1 shows typical stability-limit curves on a diagram of Xa, x4, quadrature-axis synchronous, transient and active power P (watts) against reactive power Q (reactive subtransient reactances and admittances volt amperes). The scales are on a per-unit basis. In curve a, for an unregulated machine with a small degree of saliency, Td direct- and quadrature-axis short-circuit, the limit at no load is at the point A, where Q = — V 2Y Tqv T"q f — q• transient and subtransient time constants The region of stability can be extended by means of a direct- T:jal direct- and quadrature-axis open-circuit, axis regulator, as indicated by curve b.2• 3'4 The feedback Tq"a f = transient and subtransient time constants signal for such a regulator may be derived from the terminal • voltage, the current or the load angle, but the curve always rotor angle with respect to infinite bus and 8, St = passes through the point A. In general terms,, the reason is machine terminals that the direct-axis regulator has the effect of reducing Xd , active and reactive power at infinite bus P, Q = but does not affect Xq. A more rigorous theoretical proof of moment of inertia J = this result is given in Section 2.1. w = angular frequency Alternatively, a regulator acting on a quadrature-axis field Rq(p) = quadrature-axis regulator transfer function winding can modify Xq and thereby extend the stability limit to curve c. It is found that, of the various possible signals, the Paper 5773 P, first received 4th September and in revised form 14th most effective is one depending on the load angle. The present December 1968 Dr, Kapoor was formerly, and Dr. Adkins and Mr. Kalsi are, with investigation' is mainly concerned with the system of Fig. 2, the Department of Electrical Engineering, Imperial College, London in which an alternator, connected to an infinite supply SW7, England. Dr. Kapoor is with the Department of Electrical Engineering, Indian Institute of Technology, New Delhi, India through a reactance, is controlled by feeding back an angle PROC. lEE, Vol. 116, No. 5, MAY 1969 771 772 as ameansofbuildingupquadratureexcitationduring transient disturbance. recently beendemonstratedbyadigitalstudy.?Amorerecent tion, itsexcitationistakentobeadjustablebutunregulated. own voltageregulator,but,forthepurposesofinvestiga- direct-axis fieldwinding,which,inpractice,wouldhaveits Schematic ofquadrature-axisangleregulator Fig. 2 Powerlreactive-power chartshowingstability-limitcurves Fig. 1 signal toaquadrature-axisfieldwinding.Theangleusedmay the alternatorterminalvoltage.Thealsohasa be 8,asshownonFig.2,oralternatively8„derivedfroth 222 The useofaquadrature-fieldwindingwasfirstproposed
uadratu re - ax 1 on full loadpower reactive power,VAr Fig. 3 Block diagramfor direct-axis-excitationcontrol 6 Thefeasibilityofsuchasystemhas ea -1 AV
f b d
R(p) V 2 Yq 0 C to tiVfd A HE3 t o 47 0 a. ,
Bi(P) B2(0 3 (01
1 . 1 _,_,H q
2.1 intermediate apparatushavingatransferfunction study wasmadeofasysteminwhichsignaldependenton feedback, andmanyalternativetypesofregulatorwere 2 considered. Subsequently,atheoreticalandexperimental is state stabilityofageneratorwithregulatorusingvoltage the rateofchangefieldcurrentwascombinedwith voltage signal. illustrated bytheblockdiagramofFig.3.Thefeedbacksignal on thequadratureaxis. had beenreplacedbyadistributedslip-ringwinding,acting the testshadsalientpoles,anddifferedfromnormal micromachine rotorsinthatthesquirrel-cagedamperbars machine equipmentatImperialCollege. system theory,twoparticularmethodshavebeenused: (b) (a) are derived,andcombinedwiththetransferfunctionsof ways ofestablishingstabilitycriteriadevelopedfromcontrol- generator. Equations relatingsmalloscillationsofthealternatorvariables the componentswhichconstitutefeedback.Ofmany system showninFig.2.Manyoftheconclusionsarerelevant based onthegeneralequationsofsynchronousmachine. section, whichperformsasimilarfunctiontothequadrature practical methodofachievingthedesiredresultonalarge to thedivided-windingscheme,whichprovidesaneffective is dividedintotwosections,onedisplacedfromtheother.One scheme, developedbytheCEGB, derived fromthegeneratorloadangle. which determinetherangeofsteady-statestability winding ofthepresentpaper,isactuatedbyafeedbacksignal generator hasa'dividedwinding',thatis,therotorwinding greatly improvesthetransientstability.Inthisscheme, the extensionofrangesteady-statestability,butitalso
The experimentalworkwascarriedoutonthemicro- M(p)Ai The theoreticalpartofthework,explainedinSection2,is The purposeofthepresentpaperistoanalyseconditions some generalpropositionshavebeenobtained,e.g.the feedback circuits.Thestability-limitcurveshavebeen computed byusingtheNyquistcriterion,whichcandeal with complicatedsystemswithoutunduedifficulty. limitation ofthedirect-axisregulator,alreadymentioned. simplified form.BytheuseofRouth'scriteria,proofs For detailedcalculations,equationsgivingamorecom- representation ofthealternatorandregulatingdevices nator resistance,andfordelaysderivativesinthe To makesomegeneraldeductions,anapproximate plete representationhavebeenused,allowingforalter- has 'beenused,sothatthecharacteristicequationtakesa — The investigationofReference2dealtwiththesteady- Limitations ofdirect-axis-excitationcontrol Theory ofquadrature-excitationcontrol 1 401-L A A'Ail—r i 3 , (p) being derivedfromthefieldcurrent 5 Thescheme,firstsuggestedbyKron, e. .
PROC. IEE, Vol.116, No. 5,MAY1969 M(p)A if 1 has,asitsfirstobjective, utput 9 Therotorusedfor M(p). i f , through 19 This is 8
223 arrangement was found to have the same limitation as the Now Air(p) is the ratio of two polynomials in p. Let it be normal voltage regulator, namely that the reactive absorption at no load cannot exceed V2Yg. d,,Pn a2P2 + alp a„ Alt(P) bnipm The following theory shows that, no matter what signal is b2 p2 biP b0 fed back from the alternator to the direct-axis field winding, The characteristic equation of the closed-loop system, the limitation still holds. The method follows that developed obtained by equating the numerator of [1 + L(p)] to zero, is in Reference 2, with the assumption that alternator armature resistance and damping can be neglected. The external-line D(p)(bn,pm . . . bi p bo ) reactance is combined with that of the alternator. • 3'd(Q0 v2yq jp2) The equations for small oscillations of the alternator variables are, (W' al p -I- ao) =0 (8) [G(p)Avfl By Routh's criterion, the system is unstable if the term without p is negative, i.e. if 0 = (b, aoYd )(Q 0 0 112Y,i ) < 0 • (9) This shows that the system is unstable if Q, is negative and vdo ra Lid — Xd(P) greater than V2 Yq, regardless of the regulator used. Many —4vd0 + raid, —(Qo±Jp2) roi„][6,8 experiments, covering a wide range of operating conditions, [ —ro V qo —X a(p) Ai confirmed the correctness of this deduction.
• • (1) 2.2 Machine and regulator equations for quadratureraxis control The suffix o denotes the steady-state conditions, and A block diagram of the system, using any type of feed- Q0 = i(vgoido — V doi qo) = Vdot qo — V gold° • . (2) back, is shown in Fig. 4. The equations for small oscillations A • AV qi BOP) Aqi(P) ttVfb... AVb. AVfq A6 Rq(p) eq2(o) output A Di Bq3(P) .1Aq3(p)I- 7
Fig. 4 Block diagram for quadrature-axis-excitation control
Inverting the matrix, the transfer functions of the alternator of the alternator are similar to those in eqn. 1, but are modified (see Fig. 3) are obtained as because of the presence of the quadrature winding, Bi(p) = Aid /Avfd 0 B2(p) = A811Xvid (3) r 0 = B3(p) AiglAvfd [Gq(p)Avfq
For zero power, neglecting armature resistance and damping, [ _xd(p) Vdo ra the steady-state quantities vdo, i„, and So are all zero, and 11 —ivdo - f- raja° —(Q0 + JP2) rai,„ V qo = — V2 V. The transfer functions are then as follows: —ra V qo —X,(p) d BA P) = + V2 37 7, + Jp2) [Ai/ (4) A 8 (10) BAP) = 0 Aig B2(p) = 0 For the present investigation, the direct-axis field voltage is where assumed constant, so that Ovid = 0. Y4(p) is modified, and the function G4(p) is introduced, because of the presence of D(p) = T,'1.1p3 V 2Yq Jp2 )P the quadrature-field winding. Transfer functions are obtained + (Qo + v2Yq) . (5) as before, by inverting the matrix, giving and Q0 is the reactive power at the point of operation. Bo(p) = AidlAvf,} Whatever feedbacks or regulator transfer functions are used, B1,2(p) = A 8/Avfq (11) the overall transfer functions can be calculated to be Bo(p) = Aig/Avfq = AvblAid } For any feedback signal and any regulating equipment, A2,(p) = Avb/A S (6) transfer functions can be obtained using the symbols indi- A3,(p) = Avbb6d, cated in Fig. 4: In the example of Fig. 3, two separate feedbacks are shown, Ao(p) = Avfbobodd but in eqns. 6, which are more general, Ai,(p), etc. are the A q2(p) = 1vfb,2/6.8 (12) transfer functions of the complete regulating system, relating the resultant feedback signal vb to the three output quantities. Aq3(p) = Avfb,30i, Since B2(p) and B3(p) are zero, the complete open-loop If the loop is opened at A, the overall open-loop transfer transfer is function is L(p) = AvdAvfd L(p) = Avbq/Avf, = Rq(P){,40(P)Bo(P) = Alt(P)Bt(P) (7) + Ao(P))3,72(P) + Ao(p);3(p)} (13) PROC. IEE, Vol. 116, No. 5, MAY 1969 773 224 2.3 Choice of the feedback signal the rotor angle 8 must be such that the diagram can close, The main purpose of the quadrature excitation is to and 8 therefore varies as the current phasor changes. With increase the reactive absorption at no load, and it is necessary fixed excitation, the system becomes unstable if 8 exceeds. to determine the feedback signal which does this most effec- tively. At no load, and neglecting armature resistance and damping as before, the alternator transfer functions, from eqns. 10 and 11, are Bo(p) = 0 VY Bq2(P) A/2Dq(p) (14)
(Q0 + JP2)Yq Bo(P., Dq(P) where Da(p) = 4Ip3 Jp2 7'4 (Q. + 1/2174)P Q,+ v2Yq (15) Consider first a terminal-voltage signal. The values obtained in Reference 2 for this signal show that, at no load, both Aq2(p) and Aq3(p) are zero. Hence L(p) = 0, and there is effectively no regulator. Tests on the micromachine equip- ment, using several alternative arrangements of this type of a regulating system, confirmed that it was not possible to obtain any increase of reactive absorption at no load.5 The effect of using the three output quantities individually, as feedback signals, can be examined by the same method. Clearly, a signal depending on id is ineffective, because Bo(p) = 0. For a signal iq, the open-loop transfer function is L(p) = Rq(p)Aq3(p)B03(p). Hence, using eqn. 14, the characteristic equation is 1310(p) Rq(p)A03(p)(Q0 Jp2)Y0 = 0 . . (16) Voq Thus the characteristic equation is modified, but the stabilising action is unsatisfactory, because the term Q,Yq changes sign with Q0. The signal could possibly be mixed with other feed- backs, but is not useful in itself. There remains the angle 8, for which L(p) = R0(p)A,2(p) B02(p). Using eqn. 14, the characteristic equation is
Dq(p) Ra(p)A0(p)(-VYA2--2q) --- 0 . . (17)
Eqn. 17 shows that the angle signal is the most effective feed= back, because VYq is always positive. With different regulator transfer functions Rq(p), the coefficients of the characteristic equation can be modified as desired. In a practical system, it is more convenient to use the angle 8„ derived from the alternator-terminal voltage, as the - K feedback signal. The following equation for small changes lq must then be used: 1 0 iv AS, — o Ye + /a o)A id KdIfd ( V„ c — /do)2{ (Tid (V2 Y, Q0)Y48 Fig. 5 (V00Y, — Ido)A10 ) (18) Lagging-current condition with bus-angle control (6 = 0) a Axis diagram of normal machine As shown later, this method is also very effective in increasing b Axis diagram of machine with quadrature control the reactive compensation. c Armature-current phasor diagram about 90°, but phasor diagrams can still be drawn for larger 2.4 The equilibrium diagrams angles, and stability can often be maintained by using an For any possible operating condition, that is, any appropriate regulator. point on the power/reactive-power chart (Fig. 1), a phasor Fig. 5b shows the diagram for the same operating condition, diagram, referred to here as an 'equilibrium diagram', can when the machine has a quadrature-winding control which be drawn, but the system may or may not be stable. The holds 8 at zero. V is now vertical. Vo makes the same angle distinction between equilibrium and stability is an important with V as in Fig. 5a, but it now consists of two components, one in any control system. The phasor diagram tells nothing namely Voq produced by the direct-axis field current Ird about the stability, which must be determined by one of the and Vod produced by the quadrature-axis field current /A. stability criteria of control-system theory. If, however, the The axis currents /,'/ and 14 (Fig. 5b) now differ from /d and /q system is in fact stable, the equilibrium diagram is useful in (Fig. 5a), but the voltage diagram can be completed by the indicating relations between the variables as the operating drops Xd/:, and Xq4. Note that Vod and Voq differ from condition changes. In Figs. 5 and 6, V is the fixed supply Vdo and Vqo, the components of the steady-state bus voltage. voltage, and the line reactance X, is included with the alter- Fig. 5c shows the resultant armature current I, the power nator reactances. component 4, and the reactive component Io, relative to the Fig. 5a is the conventional phasor diagram for a generator voltage V. Comparison of Figs. 5b and 5c shows that Ip = Iq operating at a lagging power factor, with excitation on the and = Id. Thus, when 8 is held at zero by a quadrature- direct axis. The axis currents are Id and I. For equilibrium, winding control, the power component in the phasor diagram 774 PROC. ZEE, Vol. 116, No. 5, MAY 1969
225
equals the quadrature-axis current, and the reactive component The power components P and Q (Fig. 1) are obtained by equals the direct-axis current. multiplying the current components /j, and 4 by V. The important result is thus obtained; that the reactive power Q is controlled by the direct-axis field which may therefore be called a 'reactive winding'. The active power P is controlled by the quadrature-axis field, which may be called a 'torque winding'. It should be re-emphasised that this result is only obtained because there is an independent control holding the angle 8 at zero. Xd Fig. 6 shows diagrams corresponding to those of Fig. 5, for a condition at leading power factor. The shapes of the diagrams change considerably, but the relations derived above still hold. Assuming that stability is maintained by the closed- loop control, the direct-axis field current is negative when the reactive compensation exceeds V2Yd. It is well known that a salient-pole synchronous machine, with fixed excitation, remains stable at no load with a limited amount of negative excitation. The field current changes sign when Q = — V2Yd , but the machine is stable a up to Q = — V 2Y q• The conditions are somewhat modified when the generator angle Sr is used as the feedback signal,. since 8 is not then held at zero. Fig. 7a is the same as Fig. 6a, but with Xj, V,
Vod b
Vod b ip= Kg IN Fig. 7 Leading-current condition with terminal angle control (6, = 0) N a Axis diagram of normal machine b Axis diagram of machine with quadrature control
and Sr added. Fig. 7b shows how Fig. 6b is modified when c Fig. 6 8, is held at zero, instead of 8. It is still approximately true Leading-current condition with bus-angle control (6 = 0) that the quadrature-field controls the active power, and the a Axis diagram of normal machine direct-axis field controls the reactive power. b Axis diagram of machine with quadrature control c Armature-current phasor diagram 2.5 Calculation of the stability limit curves Hence, from Fig. 5b, 2.5.1 The general method Vod = Xo1:7 = Xoto At any given steady operating condition defined by P (19) V° , = V + = V + Xd1q } and Q, the equations for small oscillations can be used to determine whether the system is stable or not. Because the Vod and Vo, depend on the field currents as follows: system is nonlinear, the coefficients.in the equations depend 1 jr on the steady quantities, and hence there are stable and V = d A -d unstable regions, separated by the stability curve. The open- °qV2 °I (20) loop transfer function L(p) contains the numerical gain factor 1 Rq of the regulator, and can be written Vod = 72: Kmq i fq L(p) = Rall(p) (23) Hence Ip = K QIfg Using any of the well known stability criteria, the function (21) H(p) can be used to determine the Rq which causes the con- = — VYd + Mid} dition PIQ to be at the stability limit. The Routh method was used to make some general deduc- Xmd where Kd = - tions for simplified conditions, as explained in Sections 2.1 V2 Kd (22) and 2.3. Other calculations were made using the root-locus 1 Kmd method.5 Both of these methods, however, rapidly become K g = \72 more complicated when it is necessary to allow for derivatives and delays in the regulator and angle device, or for resistance If M is a point on Fig. 5c, at a distance VYd from 0 in a and damping in the alternator, since these cause the equation nbgative direction, the direct-axis field current ifd is propor- to be of a high order. The method of domain separation tional to MN = 4 + VYd. The quadrature-axis field current would be useful for synthetising a design, but would not help If, is proportional to in the analytical process required for the present purpose. The PROC. IEE, Vol. 116, No. 5, MAY 1969 775
226 Nyquist method, on the other hand, presents no difficulty and, the limiting vio yq is still approximately V2 Yq, as indicated moreover, shows more clearly the contributions made by on Fig 1. different features. Rqmax depends on the point (B on Fig. 8) at which the The method used was to calculate a Nyquist frequency- response locus for each operating condition, and to find the a2 limiting Rq for stability. Curves relating the reactive power IMM1111111111 Qo at the stability limit to the regulator gain 12„ were then k,5 ME111111111 mum obtained, for a given active power Po. mmomnii /am.. For any control scheme in which a signal derived from 6 12 MIIIEN1111111W.MIMEN is fed back to a quadrature winding, the open-loop transfer MIIMI1111111111.21•11111111 function is o . 11111MENIMMIEEIRMI 1111,111 L(p) = Rq(p)A „2(p)B q2(p) (24) .6 04 midaSCIEMIIIIIMIIII where /30(p) is obtained by solving eqns. 10, and Aq2(p) is MEI the transfer function of the angle device. If no approximations MENEM IN MU 02 03 05 07 1 2 3 4 5 6 are made, the expressions become complicated,5 but there is regulator ga n Rq no difficulty in calculating the required curves with a digital Fig. 9 computer. The calculation can readily allow for alternator Steady-state stability-limit curves for proportionate regulator resistance and damping, and for delays or derivative elements a With ideal angle device in the angle device and the regulator. Saturation can be allowed b With practical angle device for, as explained in Section 2.5.4. When the angle signal is 81, eqn. 18 must be used to calculate L(p). This introduces a Nyquist loci cross the axis. If the frequency is con, eqn. 25 further complication, but the computation presents no shows that difficulty. nJ( jcvn)3 T4(Qo V 2YO(!taa ) = 0 2.5.2 Simplified system with a proportionate regulator and hence Consider, as a simple example, the case of a propor- 1 V2V(Y4 — Yq) tionate regulator using 8 as the feedback signal, and assume Rqmax — . (27) 11(icon) Yq that the angle device is ideal. The combined transfer function Rq(p)Aq2(p) of the regulator and the angle device is then the Thus B on Fig. 8 is a fixed point, and the BC of Fig. 9 is constant-gain factor Rq, and, from eqns. 23 and 24, the a vertical line. For higher Rq, stable operation is not possible. function H(p) is equal to Bo(p). If alternator damping and The maximum reactive compensation —Q,, is given by resistance are neglected, from eqns. 10, Rq „,in = R",„" whence V„Y q/V2 = V2Y4 (28) H(P)— TWP3 +Jp2 (Q 0 + V g20Y OP±Q0+ Vi0 17 q The possible reactive compensation at no load is therefore . . . (25) increased, because of the quadrature-field control, to a mag- Eqn. 25 does not contain the active power Po, and hence nitude dependent on the transient reactance X instead of on the same result is obtained at any power level. Fig. 8 shows the synchronous reactance Xq. the Nyquist loci for four values of Q0, and Fig. 9 gives the It can be deduced from the above analysis that, for gains curve relating Qo at the stability limit to the gain Rq. The less than ;max, instability is of the drifting type, because it upper part of the curve, for the simplified system, is the is associated with zero frequency; whereas at higher gain, vertical line BC, marked (a). For each Q0, there are two gains, above Rg max, instability is oscillatory, with a natural fre- Rvnin and ;max, corresponding to the two regions AB and quency con. This is confirmed by experiment. BC of Fig. 9. When 8, is used as the angle signal, it is no longer true ;min depends on the zero-frequency points on Fig. 8, and, that H(p), and hence the limiting gain, is independent of Po, consequently, Qo increases with 12, up to about Qo =1.6. when a proportionate regulator is used. At no load, the function is (V2Ya Qo Jp2)VY„IV2 11(P)— (29 ) (V,2Y Q0.){7Vp3 + 1p2 + 7',(V 2Y Q0 )p (V 2Yq + Qo)}
Rgmin is therefore given by The expression is similar to that in eqn. 25, but contains , „ is given by eqn. 26, and hence the V 2Yq ) additional factors. Rq01 R _ V2(Q0 . . (26) part AB of Fig. 9 still applies. Rqmax is different, and now qmin H(0) YY depends on Qo, since Thus, between the prescribed limits, the gain is proportional Yq to the amount by which the reactive compensation —Qo (V2)(ri — )(VYc + Qo) (30) exceeds 110Yq. With a perfect control, such that 8 = 0, Rninax Yq( Ye — Y,) Vqo = V. In practice, 8 may depart slightly from zero, but Hence the upper portion of the curve corresponding to 0-8 08 01 1 Fig. 9 is no longer a vertical line. Moreover, it changes for different Po (Fig. 19). 03 dee ...... -- sP.7=-18 0 The maximum reactive power for Pa = 0, obtained by 07 00= -16 equating ;min = Rq „, "is still given by eqn. 28. 01 However, there is a afurther limitation determined by the factor (V2Y0 Qo), namely that the system is unstable if —\‘...... \0-1 0 =-16 A058 :A. (): 2 the reactive compensation —Qo exceeds V2K, with any 02 Q. =-0 4 regulator. For 'the system used in the experiments, the factor 07 03 is 3.12p.u.
04 2.5.3 Practical systems 03 The curves are modified by the complications of a -0.8 -0.6 -0.2 0 practical system. For example, a calculation which allowed Fig. 8 for the delays in the practical angle device (Fig. 11) showed Nyquist plot for proportionate regulator that the part AB of Fig. 9 was unchanged, but that the part —s— points marked with frequency in Hz BC was modified, as shown by curve b. 776 PROC. IEE, Vol. 116, No. 5, MAY 1969 227 Results obtained with more complicated practical systems condition, V1 is obtained by adding (X, + X„)/a to the bus are discussed in Section 4. The following terminology is voltage, and is proportional to the air-gap flux. At leading adopted: power factors, Vi is numerically smaller than V. The dotted (a) Two types of feedback are used: line in Fig. 10A is an extension of the initial straight portion, and its slope is Xmdo, the unsaturated synchronous reactance. (i) 'bus-angle feedback', depending on S For any steady operating condition, the reactance is Xmd - (ii) 'generator-angle feedback', depending on 8, kmXmdo = Villp km is plotted against Vi in Fig. 10B. (b) Three types of regulator are distinguished: If X„,,,„ is the unsaturated quadrature reactance, it is (i) 'proportionate regulator', for which the transfer func- assumed that the true reactance is Xmq = tion is a constant For the stability calculation, using the equations for small (ii) 'first-derivative regulator', in which a component oscillations, incremental reactances Xmd = kmAXmdc, and dependent on the first derivative of the angle is added Xmq = kmA X,,,, must be used. Xmdc, is obtained from the to the proportional signal slope of the tangent to the curve of Fig. 10A at voltage V1. (iii) 'second-derivative regulator', having first- and km6, is plotted against V1 in Fig. 10s. second-derivative components added to the pro- portionate signal. 3 The model equipment 2.5.4 Modified method allowing for saturation 3.1 The experimental system The computed stability-limit curve for the second- The experiments were carried out on the micromachine derivative regulator differed appreciably from the measured equipment, arranged as in Fig. 2. The microalternator is one (Fig. 18). The method of calculation was therefore modi- connected to the infinite supply through a series reactance X. fied to allow for saturation, by using factors km and kmA, The machine has no damper winding, since the slot space derived from the open-circuit magnetisation curve, to deter- normally occupied by the damper bars is used for the mine the modified Xmd and Xmq. quadrature-axis field winding. In this diagram, the quadrature The following assumptions were made: field is controlled by a feedback regulator, to which is fed a (a) that the same factors can be used for both direct and signal derived from the angle between the machine rotor and quadrature axes the infinite supply. The direct-axis field is supplied from a (b) that all leakage reactances are constant. constant, but adjustable, direct voltage. Fig. 10A shows the magnetisation curve relating V1 and If on open circuit, using per-unit scales. Under any operating 3.2 The alternator and its time constant regulator (t.c.r.) The microalternator is a small machine which is specially designed to simulate a large synchronous generator. The main per-unit parameters correspond well to those of a typical large machine, except that the field-winding resistances need to be reduced by means of auxiliary electronic apparatus, referred to as a 'time-constant regulator'.11 In the experi- mental equipment, the direct-axis field is used without a time-constant regulator, so that its resistance is higher than a that of a typical large machine, but a t.c.r. is used with the cirj; 1-0—•—.-- quadrature field, which is performing the main regulating .9 G9- function. However, the normal time-constant regulator could 9 not be used directly, because it was designed for use with a direct-axis winding provided with an auxiliary shadow coil. a 07,- Since the quadrature winding has no shadow coil, the feedback circuit in the time-constant regulator had to be modified. The G5- new arrangement worked well, but was somewhat sensitive to temperature variation and brush-contact voltage drop in the quadrature-field circuit, particularly at low currents.5 0.3- The machine parameters, allowing for the effect of the time- constant regulator, and including the external reactance with that of the alternator, are given in Table 1. The synchronous 0.1— •• Table 1 0 02 04 06 0.8 10 field current, p.u. MACHINE PARAMETERS Fig. 10A Open-circuit magnetosation curve Machine rating 2KVA Unit voltampere 1825 VA Unit voltage . . 186V (line-line) E Unit current . . . . • 5.66A Unit quadrature-field voltage • 646V F Unit quadrature-field impedance 41852 E 0 Xd 2.471 p.u. Xq 1.93p.u. 9 Xa 0.431 p.u. u 0.6 Xc 0.321p.u. ra 0.0384p.u. 8 Y./4 0.00482p.u. .04 X:1 0.91p.u. Xy 0.615 p.u. 0.2 Tdo 1.2s T‘,0 1. 1 s 0.442s T'd 0.4 0 6 0.8 1.2 1.4 0.35s open-circuit voltage, p.0 J 0.0318s Fig. 10B Main and incremental saturation factors All the machine parameters include Xc PROC. IEE, Vol. 116, No. 5, MAY 1969 777 45 P15
228 reactances Xd and Xq were obtained from steady-state tests, amplification factor. If the tachogenerator voltage is using the equilibrium diagrams. There was some variation 2E sin (wt — S), the d.c. component of the curve is with the load, and the values given are for P = 0.2 p.u. The transient reactances and time constants were obtained from 2o.).1f standstill-impedance tests, for both direct and quadrature — E sin (cot + 8)dt Tr o axes, taken over a range of frequencies. Since the machine has no damper winding, there are no subtransient quantities. 2 E cos (WI — 8)dt} = — E cos 8 o 7r 3.3 The angle device Fig. 11 shows the diagram of the angle device and its To obtain a signal proportional to the rotor angle 8, the filtering unit. An a.c. tachogenerator supplies a voltage of tachogenerator is displaced by 90°, and the curve is then
Rat Rb2 R3
R2 't C3 Fig. 11 Cl 134 2C3 V, The'angle device 101(0 R3 = 2501(11 — fir/Rai/Ea) = 200k0 R4 = 5.3kfll Rat Al Rbl Rattail/42) = 50kf2 Ct =0.31µF •_ T I Rao = 47k0 C2 = 0.10 R2 = 10k0 C3 = 0.311F first section--•-lh_secon--dEsectio ----- phase detecting unit filtering unit
constant amplitude and variable phase to the input terminals 1 0 and 2, and a reference voltage, derived from one phase of the infinite supply, is applied at the point marked 'ref', which is 6 the junction of the base resistances Rba of the transistors Tr1
and Tr2. Tr, is a p-n-p transistor, and Tr2 is an n-p-n V transistor. e, 12
Fig. 12 shows how the potential at B is determined by the ltag action of the transistors. The filter unit ensures that the output I vo 8 at Vo is the mean of the curve in Fig. 12e, multiplied by an t u tp u
I
o 4
0 30 20 1 0 1 0 2 0 30 4 0 backwards forwards r o to r angle, deg. - 4 E sin(Wt-ö) 8
12 -E sin (wt-E)
16 gradient 570 mWdeg.
-20 c Fig. 13 Angle-device output characteristic
iiiiiMEMII MIIIIII IIIIa d 1. ow 11111111103•11111111 1114c 0 111111.111iiii=101111111 1111 3 sum maromili nrc .:3• 0 Emuwino silscl LO phase ang e C2): -BO minuElm loonounk . curve ii!.1 0. I'll e slim ampl rude-- 911.0 o0, ■111111 1111110111.011111,1°8, 0 2 IIIIIII .1111111101111101111 0 11. 211 1111111IIIIIIIIIKNISCI22 0 mum Inn Iiiii:!! : Fig. 12 el 1 et q rtiR 7 10 an an frequency, Hz Angle-device voltage waveforms a Reference waveform Fig. 14 b Tacho waveforms at 1 and 2w.r.t. earth Frequency response of the angle device Tacho input (1 — 2): 2E sin (tot — 8) c Potential At 0 experimental points d Potential A2 computed amplitude e Potential AtAt ---- computed phase angle 778 PROC. IEE, Vol. 116, No. 5, MAY 1969 229 very close to a straight line, as verified experimentally by the at which this occurred, particularly because of erratic varia- curve of Fig. 13. Fig. 14 shows the frequency-response tions of the a.c. and d.c. supplies. The matter is discussed in characteristics of the combined angle device and filter. more detail in Reference 2. Instability was most clearly indicated by the rotor-angle variation relative to the equili- 3.4 The quadrature-axis regulator brium condition, and it was decided to consider the condition For a proportionate regulator, only an amplifier is to be unstable if the angle required. In order to add derivative terms, it is necessary (a) drifted by 2° from equilibrium, and subsequently did not not only to introduce differentiating circuits, but also addi- settle back within two minutes tional filtering. Moreover, there must be some delay in the (b) built up an oscillation of 2° about the mean, either as a system, to prevent the function from becoming infinite when limit cycle or as an increasing oscillation.
p•F I—gain. K —I 1_ J. 0111F WO output fronL . angle device n comp
Hi 11/S11.1F limiter pot 0—v1AA--• Li0.211F 1Mg
0.1 MS? Fig. 15 Quadrature-axis derivative regulator
p tends to infinity. Fig. 15 shows the circuit used to obtain 4.2 Experiments and computations with bus-angle first- and second-derivative components. The transfer function feedback of the circuit used for the stability tests with the second Fig. 17 shows curves obtained with a proportionate derivative regulator, was regulator. The computed curve ADC, which allows for delays in the angle device, is the same at all power levels, as 0.1p • explained in connection with Fig. 9. However, there is some R(p) Rq {1 + (1 + (1•01 p)(1 ± 0.01P) variation with power between the three experimental curves shown on Fig. 17. 0.02p2 + (1 + 0.01p) (1 0 .01p) (1 ± 0.02p)(1 0.01p)} . . . . (31) zo `3) The calculated and measured frequency-response loci are & M11111•111111111111111•1111=11119111 shown in Fig. 16. 1 6 5 14 1-2 X 1 0INIIEN1111111111111WKW 1) 08 •• got is O q 0 241:111111:111111 V2 y ...MEMBIIIIIIMINIIIIii1111 0-1 02 04 06 1 4 5 6 regulator gain Rq
Fig. 17 Steady-state stability-limit curves for proportionate regulator theoretical experimental at 0.2p.u. power —x— experimental at 0.5p.u. power —A— experimental at 0.8p.u. power
4 4 th 4 0 Q36 B 3.2 8_2 3 a t.) 2 4 Fig. 16 X 20 Frequency-response curve for the derivative regulator E 16 Frequencies marked in Hz X 12 —•-- computed curve 0 0 A 0 experimental points Eri0 4 c 0 03 06 3 5 10 C 20 40 60 4 Comparison of measurements and regulator gain computations Fig. 18 4.1 Stability-limit tests Steady-state stability-limit curves for second-derivative regulator at 0- 2 p.u. power The tests were made to determine the reactive power Q a •—•—• theoretical, neglecting resistance and saturation at which stability was lost, for a given active power P and a b x — x—x theoretical, allowing for saturation only c theoretical, allowing for saturation and resistance given regulator gain. It was not easy to find a precise point —5— experimental PROC. /EE, Vol. 116, No. 5, MAY 1969 779 230 The computations showed that the use of a first derivative of no saturation, gave curve a of Fig. 18 and confirmed the effectively moved the point D on Fig. 17 to D', but left A and increase in reactive compensation and regulator gain, but did C unchanged. Experiments confirmed that there was little not agree at all well with the experimental curve, particularly benefit in using a first derivative. in the region DB, where the stability is mainly dependent on Fig. 18 shows curves obtained for a second-derivative the second derivative. It was clear that saturation must have regulator, for a power level Po = 0. 2 p.u. The experimental an important effect at such high leading currents. Curve b curve shows a marked increase both in the reactive com- shows the computed curve, when saturation is allowed for pensation obtainable, and in the permissible range of regu- by the method explained in Section 2.5.4. Curve c allows for lator gain. It was possible to operate the machine at any the alternator resistance, as well as saturation. The agreement power level, with reactive currents up to three times higher is better with these corrections, but it is thought that the than normal. The micr6machine could be tested at these very discrepancy at very high currents is due to saturation of the high, negative reactive powers because the losses are very low, leakage paths, which is not taken into account. but, in a practical large generator, currents greater than 1 p.u. The stability-limit curve, with the second-derivative regu- would not be permissible. The wider range of regulator gain lator, is in three sections, depending on three separate criteria. brings the advantage of greater accuracy of control of the The part AD does not depend on the derivatives, and is angle. associated with drifting instability. In the section DB, the A large margin is also desirable, because transient changes instability is of the oscillatory type. Section BC, which sets a may cause instability, even though the operating condition is fairly definite limit to the regulator gain, is also associated inside the steady-state limit. Some simple tests were made by with oscillatory instability. applying large sudden changes of direct-axis excitation. The recovery was rapid, and there was no tendency to pull out of 4.3 Experiments and computations with synchronism, so long as the steady-state operation was stable. generator-angle feedback A much more extensive study of transient performance, using The conditions are somewhat less favourable when the an analogue computer, is recorded in Reference 1. The load feedback angle is taken from the generator terminals, but angle 8, which introduces a second-derivative inertia term in results would be sufficiently good for practical purposes. The the equations, is an important factor affecting the stability of computed curves vary as the power level changes. Fig. 19a a synchronous machine. It is thought that the great improve- shows curves at power levels of 0.2 and O. 85p.u. power, ment of stability obtained with quadrature-field control results for a generator with a proportionate regulator, and Figs. 19b from the fact that the angle is held approximately constant. and 19c show similar curves for first- and second-derivative The first computation, made as before with the assumption regulators. The agreement between measurement and computation is less good than before. However, the general deduction can be t2.7 2 4 2.2 made that the proportionate regulator can extend the stable k 2 0 region to reactive compensations appreciably greater than 18 1 p.u., but that there is much less benefit from the use of (g 1.6 derivative components than when bus-angle feedback is used. -0 I 4 0 1 2 E 10 A ta 08 5 Conclusions 06 1 $_10 4,-- \. The use of a quadrature-field winding, controlled from E 0 2 a load-angle feedback, is an effective way of extending the 0 01 0 2 0 1 gain 2R, 4 5 10 20 30 range of stable operation of a generator to cover the whole regu5lator region of negative reactive power, at any active power up to a the heating limit of the machine. Although the widest range is obtained by using a derivative regulator actuated by an o- 24 angle signal derived from the bus voltage, it appears to be e 22 MENIIIIIIIM111111111111111•11M1 adequate to use the generator-angle feedback without deriva- 3 2 0 • 111111111111111=1111111111111111MINIIM tive elements in the regulator. &1 8IMMENIIIIIMMT411111MIIM tu ,s IM11111111111111111111=1112,44111111111111111M 0 1 6 Acknowledgments F 11•1111MIIIIINVEMINItollitligNM1111 0 The authors wish to acknowledge assistance received > 1.61•1111111115SHIPT2Orid1111111MIIAMM from the South-Eastern Region of the CEGB and from the 0 • NIIIMECH1112111•111111111111i• Science Research Council in carrying out the investigation. 4c3"10 211111111•11111M1111=11M1111111111111111•111111111 0IM1111111111111111111111MININ111111111111111111MII They also wish to thank A. N. Sarma for assistance in 01 02 04 2 3 5 10 20 30 carrying out some of the tests and computations. regulator gain Ra b 7 References 24 1 SOPER, .1. A., and FAGG, A. R.: 'Divided-winding-synchronous generator', Proc. IEE, 1969, 116, (1), pp. 113-126 I2 0 ".% 2 JACOVIDES, L. J., and ADKINS, B.: 'Effect of excitation regulation on synchronous-machine stability', ibid., 1966, 113, (6), pp. 1021- 16 / 1034 • 3 CONCORDIA, c.: 'Steady-state stability of synchronous machines as V, 1 2 affected by voltage-regulatar characteristics', Trans. Amer. Inst. 4 Elect. Engrs., 1944, 63, pp. 215-220 E 4 VENIKOV, V. A., and LITKENS, 1. v.: 'Experimental and analytical 08 investigation of power-system stability with automatically regulated generator excitation', CIGRE, Paper 324, 1956 004 5 KAPOOR, S. c.: 'The steady-state stability of synchronous machines as affected by direct- and quadrature-axis excitation regulators', F 0 Ph.D. thesis, University of London, 1967 01 02 04 2 3 5 10 30 'Process for increasing the transient-stability regulator ga n ft, 6 HAMDI-SEPEN, C.: power limits on a.c. transmission systems', CIGRE, Paper 305, 1962 7 NICHOLSON, H.: 'Integrated control of nonlinear turboalternator Fig. 19 model under fault conditions', Proc. IEE, 1967, 114, (6), pp. 834-844 8 ADKINS, a.: 'The general theory of electrical machines' (Chapman & Steady-state stability-limit curves with generator angle control Hall, 1957) a Proportionate regulator 9 ADKINS, B.: 'Micro-machine studies at imperial college', Elect. b First-derivative regulator Times, 7th July 1960, 138, pp. 3-8 c Second-derivative regulator 10 KRON, G.: 'A super regulator', Matrix Tensor Quart., 1955, (5), p. 71 theoretical at 0.85 p.u. power 11 ROBERT, R.: `Micro-machine and micro-research study of problems —•— experimental at 0.85p.u. power of transient stability by use of models similar electromechanically theoretical at 0.20p.u. power -- • experimental at 0.20p.u. power to existing machines and systems', CIGRE, Paper 338, 1950 780 PROC. IEE, Vol. 116, No. 5, MAY 1969 231
13.2 Design of a D.W.R. micro-alternator
As mentioned in Sect. 1.6, it was decided to obtain a D.W.R. micro-alternator to pursue further studies to establish the theoretical basis for the analysis of the D.W.R. machines. The micro-alternator was designed partly by the author at the sametime as the model induction motor. Both machines have identical Mawdleyts28 stators. Fig. (13.1) shows the winding diagram for the rotor. The angle between the axes of two-•halves is fixed at 67.5°. The value of the angle was arrived at by making a compromise between the mechanical strength of the rotor teeth and the relative degree of saturation of the direct - and quadrature axis flux paths in the rotor body. It was ensured that the armature reaction m.m.f. does not saturate the quadrature axis severely under leading power factor loads. The only drawback of the micro-alternator is its higher field winding resistance in comparison with that of a large machine. In order to reduce the field winding resistance artificially 2235 by a time constant regulator (T.C.R.) , shadow windings are placed in the same slots as the two halves of the main field winding. It was thought appropriate to have cage damper winding in the top of the rotor slots,instead of Mawdsleyts standard wound damper windings which require additional T.C.R.s for reducing their resistance.
At the design stage, no parameters were available for a large D.W.R. alternator. Thus for calculating the parameters of the model, the two halves of the rotor winding were assumed to be connected in series
(cumulatively). This made the field winding exactly identical to a C.W.R. alternator. On this basis, the design parameters• were compared with those . of a typical large turbo-generator.
The details of the D.W.R. are shown in Plate 13.1. Table (13.1) and
Fig. (13.2) show respectively the parameters and the magnetization curve of the model alternator. This alternator was used37 for steady state and transient stability studies. "`•
°R1 T4 oT '1 R2c T2 0 T3
41.10.1, --- X x x x 0 0 0 x X X 0 0 0 'I #Th r- -• rJ / X Y. X x 0 0 ,••• x x x X X X 0 0
I. • r
H i I 1 r
•••r ,=••• ••••1 d--axis q-axis r-axis t-axis
‘..)4 Figure 13.1: Divided rotor winding of the micro-alternator IN) r. 233
TABLE 13.1
MICRb DWR ALTERNATOR PARAMETERS
Machine rating 3 KVA Base stator voltage 220, r.m.s. line Base current 7.87 amps Base stator impedance 16.14 ohms Short circuit ratio 0.5 Base field voltage 714 volts Base field current 2.1 ohms Base field power 1.5 KVA Base field impedance 341. ohms Mutual reactance, Xmd 2.3 p.u. Stator leakage reactance, X a 0.106 p.u. Armature resistance, ra 0.005 p.u. Field leakage reactance, Xf 0.08 p.u. Field resistance, rf 0.026 p.u. Transient reactance, Xd 0.18 p.u. Sub-transient reactance, Xd 0.15 p.u.
Time constants, TdoI 107 sec. TI 0.08 sec. T" 0.008 sec.
4 aoplua:04:[1:-oao-;w our jo ao4o,z-2u-ouTmp-ppTATa : 235
1.8
1.
1.2
1.0
0.8
0.6
0.4
0.2
0 0.h. 0.8 1.2 1.6 2.0
Figure 13.2: Magnetisation curve for the Micro-alternator. 4 236
APPENDICES 237
APPENDIX A.
Formulae for calculating the parameters of the model motor
In addition to the main list of symbols, the following list consists of symbols being used in Appendix A only.
a Primary tooth width/slot pitch 1 a Secondary tooth width/slot pitch 2 Depth of slot
Airgap diameter of the machine
D Stator bore diameter 1 D = Rotor diameter 2 G = Effective airgap of the machine eff K Pitch factor of primary winding p K Distribution factor of primary winding d N = No. of turns per phase in primary 1 Pole pairs
Number of phases
S Number of slots on stator 1 S Number of slots on rotor 2 Angle of skew
Pole pitch at airgap
All dimensions are in centimeters
Magnetizing reactance
2 S q f N2 K2 K2 1 D L X - 1 p d m 2 0 phase (A.1) K. G p 10-8
4
238
Stator slot leakage reactance, I 2 3.16 f q '11. Psi 10-7 S Q/phase (A.2) XlS 1 : d 61 2 (A.3) where P (d --- (1 K ) (Ks sl = w1 3 3 12w1 s - 4w d and w refer to stator slot dimensions. Zigzag leakage reactance, 2 n Xi(6a1 - 1) r (6a - 1) Ks X sc 2 Q/phase (A.4) z - 12 5 4 5 S2 K2 K2 2 P
In squireel cage machines having skewed slots, Fsc can be taken unity. Leakage reactance due to akew,
4 2 0/2 1 (1 - Sin ' 0/phase (A.5) Xakew = Xm -02 End leakage reactance of stator winding, 184 q f N1 D1 .8D P [tan (Du - Sin 17g) Xel - Tc il -(4) 106 P2 1 1.4 D . 93 K2 {log 1 7,-;D log 0.54Df- )) p1 sl '1
0/phase (A.6)
Leakage reactance of vacant part of rotor slot,
-7 PS2 X2r = 3.16 f q L10Ni -7 2 K) 0/phase (A•7) 52 48 End leakage reactance of rotor winding,
2 S2 f -6 e2 = 0.47c "--p—ci 3 (L b - L) Ke T ] 10 P/bar (A.8)
.4 239
APPEIDIX B
Short circuit current of an induction motor with two secondary windings
B.1 The two-axis equations The equations relating the voltages, flux linkages and currents in the six coils of Fig. (5.1) are Eqns. (B.1) to (B.12), and Eqn. (B.13) gives the electrical torque. The assumptions are the same as those usually made in the theory of the double squirrel cage induction motor, except that the core loss is neglected and that the leakage flux, which links the two secondary windings but not the primary, is negligible. Thus in Eqn. (B.7) to (B.12) there is a common mutual inductance, Lm and three individual leakage reactances Li, L2, L3. These additional assumptions are commonly made in synchronous machine theory.
For convenience the Heaviside notation, in which p is written for dt' is used at this stage, although Laplace transforms are used for the later solution. A symbol with a bar above it indicates a Laplace transform, and a symbol with a bar below indicates a phasor (thick letter).
va = P'1 a + w Tql "1 ion (B.1) ir = q1 -4°T dl P lj ql r1 iql (B.2) o = r2 id2 (B.3) 0 = r 2 ici2 Pq) q2 (B.4) 0 = r (B.5) 3 id3 d3 0 r i 4. ptil (B.6) 3 q3 The flux linkages associated with each winding are;
(L L )i + L + L i (B.7) 11 a = m 1 dl m d2 m d3 T ql (Lm + + Lm ig2 + Lm io (B.8) idi + `Yd2 m m + L2)id2 + Lm id3 (B.9) LI/ q2 = L + + + L i mql m L2)iq2 m q3 (B.10) 240
L id, + L i + (L + i .(B.11) d3 = m m d2 m 3 L i + (L + L ) i (B.12) q3 = m ql + Lm q2 m 3 q3
Electrical torque is given as,
W T o=(i ) e 2 dl ql L.11 ql idl - (B.13)
Figure (B.1) is the operational equivalent circuit in which the variables of either axis satisfy Eqn. (B.3) to (B.12). It is similar to those of the synchronous machine, for which two circuits are needed because of the dissymmetry of the axes. For the present purpose the speed is assumed to remain constant, so that the voltage equations are linear differential equations with constant coefficients, and can be solved independently of the torque.
In obtaining the solution, the instant of disturbance is taken as zero time, and the voltage at the terminal of phase A is,
va = Vm Cos(wot + (B.14) = Re [V. egwe't ÷
The phasor representing this voltage alternating at frequency wo, is
V10= Vm ej% (B.15)
The angle % defines the instant of switching in the a.c. cycle. The actual position of the rotor at zero time is different for different values of %, but because of the symmetry of the magnetic circuit and because the rotor windings are shorted, the axes can be located in any angular position. In order to simplify the mathematics, the direct axis is located to coincide with the position occupied by the axis of armature phase A at
zero time. Then •
8 = wt = (1 s) wot (B.16) 241
Figure B.1: Operational equivalent circuit of an induction motor
242
11.2 Relation between axis quantities and stator ,Vilasors during ; • steady operation
To determine the current during steady operation at slip s, when
the stator is supplied with a balanced three phase voltage, the conventional induction motor equations or equivalent circuit for a
double squirrel cage motor can be used. The standard text book
treatment for steady operation of the induction motor is quite different
from that of a synchronous machine. For the present purpose of determin- ing the transient behaviour after a sudden disturbance; the steady current
can be determined by the following method, based on the above equations.
It is similar to that used to calculate the current of a synchronous • machine. From Park's transformation equation, the instantaneous phase A
voltage is related to the axis voltage by,
Cos 8 + v Sin 8 a vdl ql (v . ) jwt Re dl 3 vql e (B.17)
jwt Similarly is = Re Pcu - •i 'cp.) e (B.18) Since Eqn. (B.14) and (B.17) are identical for all values of t,
j(swot + X) val j vql = V1 = Vm e (B.19) or vdl = Vm Cos(swot X) (B.20) vql = Vm Sin(swot X) Thus II = 1 Vm ejX (B.21) is the phasor representing the direct axis voltage vim, which alternates at frequency swo.
Similar relations are obtained between the currents. Thus the stator
phasor I 1 is the phasor representing the axis current id. .4
= e X - I m ej(X g° (B.22)
243
During steady operation at slip 1st, all the voltages, fluxes and currents in Eqn. (B.1) to (B.12) alternate at slip frequency and the
stator -current can be calculated from the equations by putting p = jswo. 3.3 Complex form of the machine equations
Because of the symmetry of the magnetic system, Eqn. (B.1) to (B.12)
can be replaced by half the number of complex equations if new complex variables V1, Il, 22,.13, 1, Q)2, q?3 are used. These are obtained by combining pairs of variables, for example,
V v j v (B.23) 1 di ql B and I1 id, j ql The minus sign is chosen so that, during steady state conditions, the
expression agrees with the left hand side of Eqn. (B.19). The equations
become;
V1 1 (B.25) = p q?i + i441 + r Il 0 = r I + -o - 2 2 - ,§) 2 (B.26) o = r3 13 + p T. 3 (B.27)
q" l = (1,131 + L1)11 -I- Lin 12 + Lm 13 (3.28) '1' 2 = Lm Il + (Lm + L2)I2 + Lm I, (B.29) , = L 1 + L 1 + (L + L )i T3 m 1 m 2 m 3 3 (B.30) to
Te 70 Re(ju2 i (B.31)
.B.4 The operational impedance of the induction motor
By substituting 12 and qr!3- in Eqn. (B.26) and (B.27) and eliminating I and 1 from Eqn. (B.26) to (B.28) in a manner similar to that used for • 2 3 the synchronous machine, lilis determined as a function of w o1 = X(p) (B.32)
X(p) is the operational impedance of the induction motor. It has 4 the same value for both axes, and for the machine with two secondary
21 4
windings, has the form,
(1 DT') (1 DT") + (T — T")p .X(p) 5 X (1 p.0)(1 (3.33) pr")0 (7.2 - T7-00-13
where X = w (L o m Li) (B.34)
X is the "synchronous reactance", equal to the complete magnetizing reactance (including leakage) of the induction motor.
By making approximations depending upon the fact that the .r23' operational impedance becomes,
(1 = In") X (B.35) X(p) CL pV)(r in")
In addition to the six time constants appearing in Eqn. (B.33), the armature time constant, Ta which is used later, is also included in the following list.
T1 = Transient short circuit time constant
[x (1 5n ] (B.36) = 2 /6 r 14. m X) o 2
TI = Transient open circuit time constant
X2 + m w 1' (B.37) o 2 T" = Sub-transient short circuit time constant X1XXXmm [ X..X 4. X X /co r 3 (B.38) 2 2 m Xm,, o
T"o = Sub-transient open circuit time constant [ + X2 Xm = X, X ----y- i /0) r _ (B.39) ' 2 4. m 0 ) , T [_X o 2 = , + Xm 1 /co0 r _ , • (B.40)
X X T = [X + m 1Xri /co r (B.41) 5 3 o 3
2 4 5
T = Armature time constant a = XVW r o 1 (B.42) Transient and sub-transient reactances can also be defined in
relation to Eqn. (B.35) in the same way as for the synchronous machine. X Tt X Xt = •-•-••• X = Xi m 2 (B.38) TI X:11,1 -I- X 2
X X X m 2 3 X" = 7.1 ¶" (B.39) T 1 T" X = X3. + X X + X X3. T x 0 0 m 2 2 3 m where the reactances are c times the corresponding inductance. 1 In the Solutions given later -5c-m- and X(p) are required as slim
of partial fractions. If T", T" ‘Tt , T t from Eqn. (B.35) the expressions 17 are approximately,
11 r1 DT" TCpT = X +-1".1 1+/T' 21-XI1 1 -1- yr" (B.45) or ..... 1 1 11 11 r-r- _ +1 .J1 1 1 Xkp) X" - - X p±' L3 - X' 1 pr" (3.46)
- x') (X' - XJ!) X(p) = X" -1- 1 + pTI 71 + yr" (B.47)
When the second cage is ineffective, the operational impednlice function becomes,
(X - X') (B.48) X(p) = X' 1 pTI
Eqn.,(B.45) shows that the frequency response locus of the admittance function can be obtained by summation of two semi-circles. Conversely
'if a measured admittance locus is approximated by a curve derived from
the two semi-circles, values of X', X", T1 and 7." can be deduced from it.
Combining Eqns. (B.25) and (B.32)a relation between V an& I is 1 1 found; V I = o 1 1 (B.49) X(p) p+ jw ri 6;j X p)
246
For the solution of the current following a disturbance, X(p) may
be replaced by X". The approximation, in which the term in r1 is
simplified by neglecting the resistances in the factors of X(p), is 30 again similar to that made for the synchronous machine. Eqn. (B.49)
simplifies to;
(D o (B.50) I1 X(p)( p + a+ jw),
rl wo 1 where oc - (See Eqn. B.42) - X" ti a
B.5 Steady current precedinf; the disturbance
•During steady operation at slip s the axis currents alternate with . The current phasor X.'10 is found by putting angular frequency swo in Eqn. (B.49). The manipulation introduces the imaginary unit p = jswo j-in two different ways, but it is valid because j has the same
significance in both cases.
3: 11 10 ( B . 51) —1.0 — r1 + j X(jswo)
where10 = Vm ej?'" (B.52) 11 Combining this result with the equivalent circuit of Fig. (B.1),
the conventional equivalent circuit of Fig. (B.2) is obtained.
If r is neglected, 1 -10 E10 - j x(iswo) (13.53)
. -The instantaneous value of the complex axis current is,
V ej(swt X) m (B.510 -1.0 — j x (j s (;)
.4 By -substituting Eqn. (B.53) in (B.32) the primary flux phasor under 247
Figure B.2 : Equivalent circuit for a double cage induction motor.
.4 248
steady state can be obtained. 1
wA/10 = j 110 (B.55) and e jsw°t c°o110 = j L10 (B.56)
249
APPENDIX C
Solution of the equations to determine
the short circuit current and torque
Current: Using the principle of super-position the solution is obtained
by adding the original current Ilo (Eqn. B.51) to the superimposed
obtained when a change of voltage -V is applied to an current I'1, 1 initially dead system. From Eqn. (B.50),
wo V1 It = (c.i) 1 17(37- .( p +a + jca.) To solve by Laplace transforms, Ii and V1 must be replaced by the • transformed functions{ and V1, where,
= 0 , [v egswot V1 m :;):11110 ejsw°t]
110 (C.2) (p - iscoo)
Hence Eqn. (C.i) becomes, - v 0 -10 ( 3 ) -(p jswo)(p + a + jw) X(p)
where X(p) is given by Eqn. (B.45).
Hence using the standard transform for a linear function with denominator f(p) qr(p - pn),
g (-0 g(D) • -n I 1 ePnt (c.4) f Cp) (nn-)
250
1 1 jswot I' = - V 1 0.)o -10 [X(jsw) (a + jwc7
1 1 _ — e-( Of, + jw)t X -( cc. + jw) ( + jwo)
oc + Jo) — 2Ti—• TI Jsco0 )
1 (1 -t/T" - (0.5) 1 1 ‘X" X11‘1 w - 1T" / e . ( + j w ) ( 7,77 - jswo)
Now CC , 1 1— are all negligible compared with w and w, although Oc T 1 , T" o must not be neglected in the exponential factor. The first term is then equal to - 110 (Eqn. B.54).
I1(t)' = I10 + It1 v j e ( + jw)t 1 1 -t/T Xt, X e x w ) (1 - s)(1 + jswo
(1 -t/T" + (1 - 6)(1 + jswoT") X" XI' e (c.6)
Till this point no approximations depending upon the slip being small have been made. However (1 - s) can be replaced by unity, since s is less than 1% for large motors. The instantaneous value of the phase
current is is derived from Eqns. (B.18).and (B.24)
(t) (0.7) ia(t) Re [Il
-at = - V m e Re Pr‘jw,e-41 j(wt + X) + V -t/T t Reki e m (XI - 1 e X 1 + jswo TrY j(wt + X) # 1 1 t /T" + V ( Re e (c.8) M X" + jswo T") 251
tet 1 ICTE7c747 - AS j Bs
yurly = A 4- jB (0.9)
pf SWo TI = tan
T" = tan pu SWo Then Re Cos X-A Sin 7, 1, Re = cos pf Sin (wt X - pf) (cox))
Re = cos pu Sin (wt + X - pu)
Eqn. (5.1) in Sect. 5.2.1 is obtained by substituting Eqns. (C.10) in Eqn. (0.8). Torque: The elctrical torque, Te developed by an induction motor under transient operation is given by Eqn. (B.31). Thus for determing Te, knowledge of 91 and Li is essential during the short circuit. Il is given by Eqn. (C.6) and 1 can be obtained as explained below. Similar to the case of short circuit current, the resultant primary flux under short circuit is obtained by adding the original steady state
(Eqn. B.56) to the superimposed flux 17 I obtained by applying flux 'Pio a change of voltage - V1 to an initially dead machine. From Eqns. (C.1)
'and (B.32) - V 1 1 1 - (i) + C + jw) (cal)
Taking the Laplace transform on Eqn. (0.11) and substituting for
V1 from Eqn. (0.2)
252
10 t 7 (C.12) 1 TT7737Ao)(p + a + jwY
Using Eqn. (C.4), inverse transform of Eqn. (C.12) is. - -10 [ jsco t oc jco)t e e-( (c.13) 11) (0c+ two
Since 463o' ( a .1- jw) t w _ v r. imot - e (C.14) 0 1 - e j
4) (Eqn. The first term is then equal to - wo 10 B.56)
co + co + wo 1-4 o 10 0 1
= jw)t (C.15) j e( c't The complex conjugate of Eqn. (0.15) is,
(c.16) a)o XTZ *1 j 1110* e-( CC - jw)t
From Eqns. (B.31), (C.6) and (0.16), the electrical torque is,
v —( a + jco)t T 0.5 Re [ j2 y.31') e-( a — ico)t — e e —10 x(— ico)
i 1) e-VT' + -r--ki - s) (1 + jsw0¶') (X' 1 - X
1 1 N -t/T" + - (C.17) (1 - S)(1 + jSW T") % X" '''.11 e o
Using the relations of Eqns. (C.9) and (C.10) and ropping s from the factor (1 - s) for the reasons explained earlier, Eqn. (C.17) reduces to
253
V2 ----m -2t/ra T = 2 B e • 2 Ii Vm e-t/Ta.i 1 1,72 cos pl Sin(wt - plY e-t/Ti - 2 1 GrA7 - 3 fi fi 1 , T kyrr - yr) cos f3" sin(wt (3u) e-t/ " (c.18)
Since the motor normally operates at small value of slip, it can
be assumed that wt..tw, o Thus B = Imag (1(1.7) (C.19) o'
Let the standstill impedance of the induction motor at normal
frequency be Zs . Then if ra is neglected,
Z = jx(jwo) s (C.20) or (R + jXs ) _ --a---A jB s
Comparing both sides of Eqn. (C.20) 'R .B _ s (c.21) R2 4. X2
where R s , and Xs are the effective resistance and reactance of induction motor at standstill. Eqn. (5.4) in Sect. 5.2.1. is obtained when Eqn. (C.19) is substituted in Eqn. (C.18). 254
APPENDIX D
Calculation of the voltapie after open circuiting the induction motor
When the supply to an induction motor is interrupted, the flux linking the inductive winding does not allow the current to reduce instantaneously to zero, but an arc is drawn in the switch. The current
continues to flow until the zero point is reached. Instantaneous interruption of the current would generate a voltage of infinite magnitude.
An exact calculation for the three phase system would be very complicated,
and the method used here assumes that the decay of each current is specified
by multiplying each current by a factor e 6t. The limiting condition
when the current is reduced to zero instantaneously is obtained by putting
C = co •
Before the switch opens at zero time, let the complex axis current
be,
10 -jswot- -110 e (D.1)
where m ixt "10Y =
The corresponding voltage is obtained from Eqn. (B.53).
1T10 = j X(jewo) Z10 (D.2) and V 10 = 1r10 eje%t (D.3)
After the switch opens the current is assumed to be j_ = t I 10 (D.4)
The change in current is therefore, 4
II = - I e -Ct 1 10 4. I10 (D.5)
255
Substituting for Ilo from Eqn. (D.1)
It (D.6) /10 [ei6w°t e-( - jw(3)t
The Laplace transform of Eqn. (D.6) is,
-10 (D.7) I1 - (p - jswo) (p jswo C )
The transform of the voltage change is, from Eqn. (B.50)
7' = (p + a jw) ?iP-1 111 0
X(p) (D.8) -10 wo (p jsw )(p jswo -1777 Transforming Eqn. (D.8) with the help of Eqn. (C.4)
ejswt VI -1.10 ( ajwo) X(jswo) 1 wo a + jwo- 0. ) e-( a - jswo).t x(..cr jswo) • ( wo (a+jw - 7r)1 0 . e o ▪(x - xt) I10 w o (1 jsw0 0)(a - jsw0 - TI T 0
. 1 (a + JD) T" --t/To 0 e ▪ - x10 ---a (D.9) 10 w 4. T.19( - 1 - o (1 _jsw jsw T ) o o 0
Since a is very large (infinite for instantaneous switching) and 1 1 at T• are all small in comparison with w and wo, Eqn. (D.9) becomes, 0 0 V1 = j x(jswo) R 10 ej wot t - CLX„ 0 0 .1
256
(1 - e-tAl ) ▪j(X - xl) X10 (1 + jswo T('))
e-t/To (1 - s) (D.10) ▪g x' x") Ilo (1 + jswo TO)
The net open circuit voltage is,
V1(t) = V10 -L.' V' Substituting V10 from Eqns. (D.2) and (D.3),
V1(t) = - a X" I e- (I t wo 10 -t/T1) j(X X1) (1 - e T10 (1+ jswo o t/T8 j(XI - X") X (1 - s) 10 (1 jswo Tg) (D.11)
In Eqn. (D.11) ifci--* co, the first term represents an impulse of infinite magnitude for an infinitesmal time. The second and third terms represent transient and sub-transient components of the open circuit voltage. SinCes is small in relation to unity, the instantaneous open circuit phase voltage is obtained from Eqns. (D.11) and (B.17) as,
va(t) = Re [Vi(t) ejciA I X" = ----11 cos(mt xi) e-crt wo 4 ej(Wt WI ) 1 • I (X - X1). e-t/T; Re ' m (1 jswo TO -I . ej(wt XI)] + I (X' - X") e-t/Tg Re (D.12) m 1/41 jswooT")
Let sw T1 = tan 13' 0 00 (D.13) and sw T" = tan pu 0 0 0 4
257
Len j(wt + hi)] Re j e Cos Sin(wt + ) (1 + jswo A'0 ) ) ) (D.14) j egut + 70)] ) Re - Cos Ait Sin(cot %I - So) ) (1 + jswo Tg) o )
Eqn. (5.7) is obtained by substituting Eqn. (D.14) in (D.12).
258
APPEflDIX E
Current and torque after an indirect short circuit
Current:
Since the short is applied a few cycles after disconnecting the
machine from the supply, it can be assumed (Fig. E.l) that at the
instant of short application, the value of the component associated
with T" is negligible in comparison to that associated with T1. Following the approach of Sect. 5.2.2.1, after omitting the impulse
term as well as the term associated with T" in Eqn. (D.11),
x') I: e-t/T,1) (E.l) = (1 4. j„„w o To -lo
Using Eqn. (B.48)
(X ') T 3 Tf. -7 jsuao r(c. r-.10 = {X(jsc°0) -
= 1110 - X' Elo
= IV'l ejki (E.2)
is the phasor for the voltage behind transient reactance. Thus Eqn. (E.l) becomes,
V = V' e (E.3)
Let the time interval from disconnection to short-circuit be t 1 seconds. If a new time t' is defined such as t - t = - 1 0, the voltage with respect to the new origin at time t' becomes;
V ={\r' e-q/Tii] e-ti/TL 1 (E.4) Leti -t1/14o =_. vt 1Y 1 e o (E.5) and lc' 4. X 4. cat1 . 70o 259
Time t
Figure E.1: Open-circuit voltage after disconnection
• 260
X6 -Ws.; Then V = V' ej (E.6) 1 The Laplace transformiof Eqn. (E.6) is, lf I V' ej"° (E.7) V - - 1 P 0
Substituting V1 and X(p) from Eqn. (B.35) in Eqn. (B.50).
W VI e-AO ( - , o o p .1 7.jir o (E.8) X"(p 4 247)(p + Tor) (p 4 a + jw)
Using the transform of Eqn. (C.4)
1 -( Cc + jw) t' (.7171 - a - jw) e wo VI o eAO I1 X" 1 1 - - jW)(-77 a - jw)
(1 1•--OA' fi -t'/T" TIT e T" T" 0 h 0 (E,9) "a 1 1 )(1 ) ( a + - - ( a + jw - -
1 T" T . a, Tr, 7 are all negligible compared with w and T", 0 0i 0 0 Using these approximations Eqn. (E.9) simplifies to:
. V' 0 a+ jo.)tt 1 -WTI A t 1 e-( 4 e 11(t) = j(1 s) e ° X"
,p,1 -)1yr -tl/T"e (E.10)
Neglecting s in the factor (1 s) and using Eqns. (B0.8) and (B.24),
the.Instantaneous phase.current is given by Eqn. (5.11) in Sect. 5.2.2.2.
Torque: The torque developed by the induction motor under indirect short • circuit is.obtained by following the approach similar to that of the direct
short-circuit (Appendix C). The open circuit voltage with respect to the 261
iew origin (time zero) is given by Eqn. (E.6). The equation relating the primary voltage and the flux is obtained from Eqns. (B.32) and (B.50) as; V 1 (E.11) 1 (5. a jr.75:
Taking the Laplace transform of Eqn. (E.11) and substituting V1 from Eqn. (E.7): VI ei?'"; (E.12) (p (D + a + jw) '0 Taking the inverse transform of Eqn. (E.12),
41,f VI e4"0 o [-t'/To (a + jw)te (E.13) - e e- jth -)
The first term will cancel with the flux just before the fault. The approximations which can be made in Eqn. (E.13) are, 1 a 2 and wzr: wo o Thus Eqn. (E.15) becomes, -(a+e jw)t' wo+ 1 = J V ,) ei4 (E.14)
The complex conjugate of Eqn. (E.14) is
*41 I Wo T = j V;t e-'1"° e-(a- jw) t' (E.15)
After dropping s in Eqn. (E.10), from Eqns. (B.31), (E.10) and (B.15), the electrical torque is,
1 j(1) -t1/0 T = 0.5 Vol Re [- 1- 3"- 7-, yr e e e X" e •
1 1 " tt -eV (TT - 770 eJw e-tIA" e (E.16) l • Further simplications in Eqn. (E.16) yields Eqn. (5.15) in Sect. 5.2.2.2. 262
APPENDIX F
Switchin transients
F.1. Reconnection after a disconnection
It is possible in a system that the supply is lost for a few seconds.
This section deals with the analysis of this kind of disturbances. The condition at the instant of reconnection can be represented by Fig. (F.1).
As discussed in Appendix E, let the new time tl defined such as t t1 = t', then the open circuit voltage, V1 of the motor at any time t' is that given by Eqn. (E.6) thus
= V' i4 e-t'/7L V1 o e (F.1)
However, Vio is the instantaneous supply voltage with respect to the new time, and is given as, V' - )710 ej"otl jsco-0 10 - — e (F.2) Current:
The solution for the transient current is obtained by super imposing the current obtained after applying a step function Vio to that obtained by applying another step function - V1 at the terminals of the machine at time t' = 0, (Fig. F.1).
The transient current due to a step voltage Vio is that given by Eqn. (C.5) but with opposite sign. After making the approximations suggested in Appendix C and dropping s in the factor (1 - s), the transient current is,
4 263 •
L p
Figure F.1 : Transient conditions at the instant of reconnection.
264
I' V ejmot1 . 1 jswotl . 1 -(c4+ jw)t1 1 -10. - 3 X(jswo)e X(- jw) e
1 .7 - e-t7/Tt -7177-JE071:771)- 4
, (1 1 ) e- tfAil I (F.3) - + jswo T") 'X" - X11
The transient current When step Voltage - V1 is applied is given by Eqn. (E.10). After dropping s in the factor (1 - s).
= j 1/1 Xrr e-( cc+ jw)tT e
+ _ e-tV0 %xli xi' (F.4)
The net transient current is the sum of Eqns. (F.5) and (F.4) jscootl I j Y-10 e iswot‘ 1 X(jswo) e
IT(t) i j V10 eisw°t1 ei4 1 -( cc + jw)-0 X(- jw) X" e
ejswoti V 1 1 ITO ei4 10 - - e (1 jawo TT) (Xi ) -X'
ij 110 eisw°t1 - j (F.5) (1 jswoT") 17) e-ttA"
Using Eqns. (B.18) and (13.24), instantaneous phase current is obtained from Eqn. (F.5) as,
= ejwt1 ] ia(t) Re [I1 , -10 j 17.(!) ej;\"0 = Re •1210 ejsw0t1 ejw°t1 Re _ i+ 1.1.J X(.-jw) X" 4 e-ttA a
265
jswoti .x I V' 04"C) - Re (1 - j 0 e-t'/T' ejwt [E (!!,j (:)jswe o T ) X' X - XI 4 j ejmoti _ j ej74 (1 o-tl/T" (F.6) - Re fla—nswo T") " ' ) e
Let yl "otl (F.7) Eqn. (5.15) in Sect. 5.3.1 is obtained.by substituting Eqns. (B.22), (B.52), (C.9), (C.10) and (F.7) in Eqn. (F.6). Torque: Analysis for obtaining the electrical torque follows the approach of Appendix C. Similar to the case of transient current, the transient torque is also obtained by superimposing the torque obtained after applying
a step function Vio to that obtained by applying another step function - V1 at the terminals of the machine. The transient torque due to a step voltage Vio is obtained from Eqns. (C.5) and (C.14) by following the procedure of Appendix C. The primary flux due to step function V10 is given by Eqn. (C.14) but has opposite signs. The complex conjugate of the flux is,
* * [ -( cc - jw)t1 -jswole -jswoti o!-P 1 - j -Y3.0 e - e (F.8)
Electrical torque due to step function Vio is obtained by substituting Eqn. (F.3) and (F.8) in Eqn. (B.31)
= 1 Re -(a- jw)t' - e-iscoet t 1 jsw tl el 2 io (e { jX(jswo) e 0 -(a. Jw)tl
fi 1, Xl "kJ e kJ' jSWo • %
, 1 -tt/T" Tit) %xi! xli e (F.9) k 1 sw 13
266
The transient torque when step voltage - V1 is applied, is given*by Eqn. (E.13). .
The net transient torque is the sum of Eqns. (F.9) and (E.13).
1 T 0.5 Re [Vm2 j x(jecoo) e ejwatT 7 j e2. —2ttira e jw)
(1. 1) e-t t e-tt/Ta ejwt1 - - (1 jswo TO `X' X
1 1-t'/T" -t' jat' I. - (1 e e e jsw0-7 77 X" X'
- V2 i 1 -swo m j X(jswoi X(-jw e t'
(2. 1) e- jscoot i e-t 1/4' - (1jsc07+ - TIT X'- X'
777= T") 45. e-jswc3t1 e-tiA"
1 + j Irl2_. e-ttfra e...j,,tt + y r e_t,A, 0 f _ xi'
-tl/T" ▪kl.Tr e e-ti/TZ
4 In2 / 1 -41/ra jelyt, -VAt o e 2"X' e e
eiwti e-t'An ) e-tiAa ▪% PI xt (F.1o) Eqn. (5.21) is obtained by further simplifications of Eqn. (F.10).
F.2 Starting current and toroue of an induction motor When an inert machine initially at standstill is switched on to the supply, a large current is drawn and the transient torque may be
several times the normal value. The analysis is given below. *
267
Current:
The complex stating current is given by Eqn. (C.5) but with an opposite sign as the response of the machine is desired when a step function V 10 is applied. Substituting w = 0 and s = 1 in Eqn. (C.5), the complex starting current is;
1 ejwot 1 1 - at wo 110 [573170-) ( a jwo) 3T=5T (cc jwo) e
1 — 1)(_ 2.1,) e—t/TI ( a:1TI )(- !tt jwo) • 1 1 1 e -t/T" 1 1 • ( X" - Y1)(. " l—) (pal) Tiljk T" - j(1)0)
Accourate Solution
Let tan e W a =)o Ta )' .tan Of = w o Tt ) ) (F.12) tan eft = (Do Til
and x(-a) =
Thus Eqn. (F.11) becomes,
. 0 Sin a Sin Oa _io gwot - Oa) _ rczoTT e a -t/Ta 1 =—10 V X(jwo) e e fl 1\ - Sin At e- ‘5'5- e-t/TI j°1 (1- --,z ) a
) 40,] Sin x" - xl - T" e (P.13) T a
Transforming Eqn. (F.13) by Eqns. (B.18) and (B.24), instaneous phase A current is, 268
ia(t) = Re j I1 eiwt 1
= Vm Sin 0e.{Ao - Cos(wot 8a) Bo Sin(wot - a)
Sin 8a - V ----- Cos (X. - 8a) e-t/Ta m ti , v.W7 V Sin el Cos (X - el) e m (1 - -14-) a ,1 1 , v--X" - X'--- ) -tA" -V Sin 8" Cos (X - e") e (Fak) m ) (1 - IIIT a
*where Ao = Real [1/X(j(00)] (F.15) Bo = Imag Li/X(jw0))
Torque: The elctrical torque developed by the machine under these conditions can be obtained by following the approach of Appendix C. The electrical torque Te is given by &in. (B.31). The complex transient current is that given by Eqn. (F.13), and the primary flux is 'obtained from Eqn. (C.13) after substituting w = 0 and s = 1. Thus
-t/T, = - y.3.0 Sin 0 rej(wot - ea) - e e j0a (F.16) woi1 a L The complex conjugate of Eqn. (F.16) is,
[e - i(mot Oa) coo 1) 1* -li-aop Sin ea e jea (F.17)
From Eqns. (B.31),(F.13)and (F.17), electrical torque is obtained as; T = 0.5 Re e [J4)• * 1
2 6 9
2 = 0.5 v B Sin e + Sin2 0 {(z.- - Ao) o a a X [I a, t/T Sin wot - Bo Cos wot } e_ a
Sin 0a Sin 01 fSin (0' - 6a)
t/ta e Sin(wot + 0' - Oa) / e
e Sin a Sin 0" Sin (en - Oa)
11 e_t/Ta Sin(cot + 0" - 0a) e-t/T (F.18)
Approximate solutions:
Current: An approximate solution for the starting current can be obtained from 1 1 Eqn. (F.14) by making the approximations that m , To Til 0a = et = on = n/2 (F.19) Thus the primary current is, ia(t) = Vm LA0 Sin(wot + %) + B0 Cos(wot + 1.)] Sin % -t/Ta - v m- X e (1 57) - V SinX e-VT' m (1 - a (1 1 ) 'X" - X' -t/T" - V Sin 1. e (F .20) m (1 - a .4 270 Torque,: The approximatelsolution for the electrical torque is obtained , 8' and 0" from Eqn. (F.19). Thus from Eqn. (F.13) by substituting 0a -t/Ta 2 1 ) Sin to t - B Cos w t e T = 0.5 Vm [BoX - Ao o o o e c. t/T' Sin wot e Sin wo t (F.21) 271 APPENDIX G ACCURATE REPRESENTATION OF AN INDUCTION MOTOR By eliminating the three complex currents, Il, 12 and 1 from 3 Eqns. (B.25) to (B.30), differential &Ins. (10.2) for the three complex fluxes are obtained. The coefficients B 11' B12 etc. are given as, - B11 = P2Lcal2a21 •r3Lca13a31 (r1 •r c)all jw(Lcall + 1)1 /(1 + B12 = [r2Lca12a22 •r 3Lca13a32 (r1 •r c)a12 jw Lca12 /(1 allijc) B13 = [r2Lca12a23 + r3Lca13a33 (ri + rdal3 jw Lcal33 /(1 + ailLc) B14 ... 1/(1 ailLc) B = 21 -r2 a21 B22 = -r2 a22 B = -r a 23 2 23 (G.1) B24 = 0 B 31 =-r_ a31 B32 = -r3 a32 B33 = -r3 a33 B34 = 0 where L + L L a11 =(1,2L3 L2 m 3 m )/D a = a 12 21 = LmL3/D a13 = a31 = - Lm L2/D (G,2) a22 = (L11,3 + L1Lm + L3V/D a23 = a32 = Lm Ll/D a33 = (1,11,2 + L1Lm + L2Lm)/D D = LiL2L3 + L1L3Lm + L1L2Lm + L (G.3) 2L3Lm 272 However when the fluxes are represented as real quantities , coefficients. d2 .... etc. are;" dil, all = [r2Lcal2a21 + r3Leal3a31 (ri + rdan] /(1 + Lean) 0112= - d1.3 - [r2Leal2a22 + r3Lea13a32 (r1 + rda32 ]/(1 + Lean) = - a12Le/(1 + Lean) 0115 = [r2Lea a23 + r3Lea 3a33 - (r1 + re)al3 1/(1 + Lean) 12 1 - a Le/(1 + Lean) d16 = 13 d 1/(1 Lean) 17 = d d = r 31 = 42 (G.4) d = r a d33 -- 44 2 22 r2a23 d35 = d46 = d d = 51 = 62 r3a31 d, = r3a32 d5353 = 04 d d = r3a33 55 = 66 273 APPENDIX II ACCURATE REPRESENTATION OF A SYNCHRONOUS MACHINE The various coefficients appearing in the differential Eqns. (11.12) are: 2 X m w g3 o a = - Er + R + md '1 61 1 l g w oX I 03 (X + X") a 3 f T do oXkd Tdo " a = I X + Xq 2 + X" 2 y w 1 61 .".md. 6 ] o a - g + - 163 3 [X L3 X2 (X + X") f f xka 'Ao Xkd Xf ado d 2 g w g1 1 Xmd o (X + X") ak - r ricio Xkd - g3 Xkd Xf 1.o ] d g w 2 o a5 = -X775C: + Xp- -w a6 = • (X + X") g w 1 o a7 = Xf(X + X") where g1 = xa - g2 = X l - Xi, g3 =Xd - + Xa) b - I (X + X") g Xmg. w b - r +R+ 2 o 2 = a w X. Tit ] (X 4- X") [ • o ir.q qo q w g b o 1 3 -.X (X + X")- I' 03 c," b 0 `'- 1 Xkd(X + X") 274- co 0 = b 775-TX 5 Xk- q qo • q Coo b6 -(X b = 0 7 gl and C ., - g .1_ O 3 do Xmd 1 C3 =- Xkd Xf AL 6 Xm 1 d •4 =Xkd TC1o g7 =0 and C = 1 G2 =• G 5 = 06 g3 -• w Tit •1 o do g- d --2- TIT 3 • X f - do d _ 1 4 - rItdo d2 = - d6 = d =0 - 6 7 X_Lna_ e = co V? 2 O qo • 1. e5 - ¶ fl qo - el = e3 - ek = e6 = e7 = 0 4 275 APPENDIX I COEFFICIENTS FOR REPRESENTING THE TWO MACHINES ACCURATELY IN THE SYSTEM UNDER INVESTIGATION al a. . V A = Cos 5g + Vp Sin bg + (--- - R) idg + f--17-. w - (1 -S ) X}i a a6 a6 g g qg - (R Cos 6 - (1 - s ) X Sin bgjidm - {11 Sin 6 + (1 - Sm)X Cos 6 / i gm m gm gm -qm 4. a5 .1. alt tv kd .1. a5 to .h 4. a,„ ' v a6 f a6 a6 g 9' kci a6 f bi f la, B = - Va Sin bg + 1/(3 Cos 6 + t(1 - s g)X + --- w )' i + I--= - RI i g b6 g dg kb qg +{R Sin 5gm + (1 - sm)X Cos bon ) idm + {- R Cos 5gm + (1 - sm)X Sin 5gmliqm b b b 3 5 iii + b w L) + -- co 0? + — 1 kq 6 g f b6 g kd b6 C= V Cos 6 + Sin b+ A i + A. i + B i + B. i a m vp m r dm a. qm r dg qg 412 + Ci LI) q2 wm + Dr L d3+ D . 0.) IGO m D = - V Sin 5 +V Cos 5 +A i -A +B -B, a mp.m r qmdm; ir qg 1 dgi + C r Yq2 Ci. Yd2 wm ▪D r tleq3 -Di 4)c1.3 wm - - X" - X, 2 X" - X, 2 A where r - {11 r + ( ) r 1 .X2 j-) r2 ( k3 3 A.1 - (1 - sm)(X X") B R Cos bgm - (1 s ) X Sin 5 r 0 B. = R Sin 5 - s ) X Cos b 1 gm 0 gm (X" -X1) X" - X1 2 r3 Cr - X T ( 2 d 3 "2 0 276 X" — X, C = ) X2 (X" — X2.) X" — X.i. 2 r D r = 2 co X_ T" — ( X ) X o 3 2 3 X" - X,. 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