ANALYTICAL INVESTIGATIONS ON SUPER-SYNCHRONOUS OPERATION OF STATIC KRAMER DRIVE
A DISSERTATION submitted in partial fulfilment of the requirements for the award of the degree of MASTER OF ENGINEERING in ELECTRICAL ENGINEERING (With Specialization in Power Apparatus and Electric Drives)
t x 76[I a '• Acc. o, . _ ..._ '+S s Dte? 6:..ó? j By ~ r '~.
BHAWANI SHANKIEi4•.... ''-~
DEPARTMENT OF ELECTRICAL ENGINEERING UNIVERSITY OF ROORKEE ROORKEE-247 667 (INDIA) MARCH, 1996 CANDIDATE'S DECLARATION
I hereby certify that the work that is being presented in the dissertation entitled "ANALYTICAL INVESTIGATIONS ON SUPER -
SYNCHRONOUS OPERATION OF STATIC KRAMER DRIVE", in the partial. fulfilment of the requirements for the award of the degree of Master of Engineering in Electrical Engineering with specialization in
Power Apparatus and Electric Drives, submitted in the Department of
Electrical Engineering, University of Roorkee, Roorkee is an authentic record of my own work carried for a period of about fourteen months, from January, 1995 to February, 1996 under the supervision and guidance of Dr. S.P. Gupta, Professor and Dr. Pramod
Agarwal, Lecturer ' in Department of Electrical Engineering,
University of Roorkee, Roorkee.
The matter embodied in this dissertation has not been submitted by me for the award of any other degree- or diploma of any other
University. 3dtL
(BHAWANI SHANKER)
This is to certify that above declaration made by the candidate is correct to the best of our knowledge and belief.
(Dr. Pramod Agarwal) (Dr. S.P. Gupta) Lecturer Professor Department of Elect. Engg. Department of Elect. Engg. University of Roorkee, University of Roorkee, Roorkee - 246 667 (INDIA) Roorkee - 246 667 (INDIA)
(i) ACKNOWLEDGEMENTS
I take this momentous opportunity to express my profound gratitude to Dr. S.P. Gupta, Professor, and Dr. Pramod Agarwal,
Lecturer, Department of Electrical Engineering, University of
Roorkee, for guiding me throughout this dissertation work.
Inspite of their busy schedules, they always listened to my problems and suggested possible solutions.
I am also thankful to honourable chairman and my senior officers of the Rajasthan State Electricity Board who allowed me leave from time to time to come to Roorkee for completing this work. I am also grateful to Mr. R.M. Sharma, Executive Engineer,
R.S.E. Board for his concrete suggestions and help.
I am grateful to Mr. P.S. Puttaswamy, Research scholar of
Electrical Engineering Department, University of Roorkee for their concrete suggestions and help.
I am also thankful to staff of New Computing Facility (NCF) for their cooperation during the execution of this dissertation work.
Finally, I express my appreciation to all those friends who have directly or indirectly extended their helping hands during this work.
(BHAWANI SHANKER) ABSTRACT
The Principle of slip power recovery has been employed over years in very large power induction motor drives to achieve variable speed operation at high efficiency. A static Kramer drive is essentially a constant torque type of slip power recovery drive in which the motor is of wound rotor type. Slip power, which is available at the rotor slip rings, is made to flow in a slip recovery loop consisting of a three phase bridge converter which is connected to slip rings on its ac side and another three phase bridge converter which is connected to supply mains on its ac side, the, two converters being connected to each other in cascade on their dc sides through suitable filtering inductor.
Both converters are of line commutated type.
If motor side converter is operated with zero firing delay, as a bridge diode recifier, and the motor side converter is operated with firing angle varied in a range 900 to 1800, the drive can be operated at varying speeds in sub-synchronous region.
With the same arrangement of firing, it can also be operated as line excited generator at super-synchronous speeds. On the other hand, super-synchronous motoring and sub-synchronous generating modes are possible to be realized if the motor side converter is operated as a fully ON inverter (a = 1800), and the line side converter is operated as a controlled recifier in a firing angle range of 00 to 900.
In present work, a general mathematical model of the static
Kramer drive has been developed which is suitable for investigating steady-state and transient performance' of the drive in any of the four modes of operation mentioned above. The technique of d-q transformation of variables is employed with associated assumptions. Performance curves of the drive are obtained through digital simulation for all the four modes of operation to investigate the steady state behaviour. It is concluded that for large power applications static Kramer drive offers a robust arrangement which can be operated over a wide speed range in both motoring and regenerating mode.
(iv) Page No.
1.2 Principle of Sub-synchronous and Super- 15
Synchronous Operation with Injection of
Voltage in the Rotor Circuit
1.2.1 Drive configuration for sub- and 20:'
super-synchronous operation
1.2.1.1 Sub-synchronous motoring 22
1.2.1.2 Super-synchronous generating 24
1.2.1.3 Super-synchronous motoring 25
1.2.1.4 Sub-synchronous generating 27
1.3 Literature Survey 28
1.3.1 New topology / Technique for power 30
factor improvement
1.3.2 Efficiency improvement technique 31
1.3.3 Steady state analysis of drive 31
1.3.4 Transient performance of Kramer drive 33
2. MATHEMATICAL MODELLING 35
2.1 Introduction 35
2.2 Development of Mathematical Model of the 36
System
2.3 Conclusion 48
3 STEADY STATE ANALYSIS 49
3.1 Introduction 49
3.2 System Equation Under Steady State 49 Page No.
3.3 Expressions for Input Power, Output Power, 50
Losses and Efficiency
3.4 Computed Performance Results 55
3.4.1 Sub-synchronous motoring 55
3.4.2 Super-synchronous generating 63
3.4.3 Super-synchronous motoring 71
3.4.4 Sub-synchronous generating 79
3.5 Conclusions 88
4. CONCLUSION 89
APPENDIX - I SPECIFICATION OF DRIVE MOTOR AND 94
BASE VALUES -
APPENDIX - II STEADY STATE CALCULATION 96
BIBLIOGRAPHY LIST OF SYMBOLS
V Instantaneous voltage at the motor side converter
DC terminals
Vl Instantaneous voltage at the line side converter DC
terminals
V Peak value of phase voltage on ac side of line side sp converter
V Peak value of fundamental phase input voltage
Vqs, Vds Quadrature and direct component of the stator
voltage
Vqr, Vdr Quadrature and direct component of the rotor
voltage
vas'vbs,vcs Instantaneous value of stator phase voltages Instantaneous value of rotor phase voltages var,vbr,vcr igs,ids Quadrature and direct component of the stator
current
igr,idr Quadrature and direct component of the rotor
current
i , i sp rp Peak value of fundamental stator and rotor current i d dc link current
iar'lbr'icr Instantaneous values of rotor phase current ifb Feadback current
is rms value of supply current
(v) Rss Resistance of stator winding
R Resistance of rotor winding rr XM Magnetizing reactance
X M, stator reactance ss Xis +X X X1r+XM, rotor reactance rr Leakage reactance of stator and rotor Xls' Xlr Rf DC link resistance
Lf DC link inductance
Xf DC link reactance (measured at line frequency)
P Number of poles of induction motor
Te Electromagnetic torque
s Slip, (in p. u. )
p Differential operator, d/dt
a m Firing angle of motor side converter al Firing angle of line side converter
f Supply frequency
Electrical angular speed corresponding to
fundamental component of applied voltage (in
electrical radian/sec)
Or Phase angle between fundamental rotor phase voltage
and fundamental rotor current
13 Angle of advancement of the d-axis w.r.t. rotor
phase a-axis
9r Phase angle between stator and rotor voltage of
phase a at t=O
(vi) W Electrical angular speed of the drive (in
electrical radian/sec.) _ (1-s)w
P Power drawn from supply supply Pst Stator input power
Mechanical power Pmech Slip power Pslip Air gap power Pgap P Stator copper loss cu-st P Rotor copper loss cu-rt P Filter copper loss cu-F CHAPTER - 1
NTRODUCTION
The technology of solid state adjustable speed ac drives was launched in the 1960s. Since then many innovations in devices, circuits control theory, and signal electronics. have made a considerable contribution to this technology. The heritage of past experience, and projection of present trends, convey the belief that ac drives will find wide acceptance in near future.
In Induction motor drives, where the power is controlled in the stator circuit only, converters are to be designed to handle the full power flowing to the machine. Between the two classes of induction machines - squirrel cage and wound rotor, the former is almost always preferred, because the wound rotor machine is more
bulky and expensive and has a fragile slip rings brushes arrangement. However, the wound rotor machine has long been used for inexpensive speed control by mechanically varyiniz the rheostats in rotor circuit. For limited range speed-control applications below synchronous speed, where the slip power is only a fraction of the total power of the machine, the power loss in rheostats is low. This advantage offsets the demerits of the wound rotor machine to some extent, and the drive system becomes viable for large horsepower pump and compressor - type applications, where the speed does not usually deviate more than 50% from the
1 synchronous speed and the higher cost of wound rotor motor is justified as it enables superior starting characteristics too. The slip power can be controlled to flow either out of the rotor or into the rotor, allowing speed to be controlled in both sub-synchronous and super-synchronous regions with motoring and regeneration.
1.1 SLIP POWER RECOVERY DRIVES
Slip power recovey drives employ the principle of recovering slip power of the rotor, which is otherwise wasted in rotor resistance based speed control schemes. Due to recovery of slip power the overall efficiency of the system improves. The slip energy recovery drives may be either constant power type or constant torque type. In the former, the recovered slip energy is returned to shaft of main motor with the help of an auxiliary
motor whereas in the latter, it is returned to supply after
modifying its frequency to be equal to supply frequency. The former owes its introduction to Von Ch. Kramer, who presented it
in 1908 and is popularly known after his name. Before advent of semiconductor devices, the cascade connection of speed control was
made with two rotary machines. Two systems were popular
(1) The Kramer System
(2) The Scherbius System
1.1.1 ORIGINAL KRAMER DRIVE
The original Kramer drive, shown schematically in figure
2 (1.1) had three basic components a wound rotor induction motor, a d.c. motor mechanically coupled to the Induction motor and a converter which converts three-phase slip power to d.c. power.
Slip power is recovered through the converter and fed to the d.c. motor armature which converts it into mechanical power and returns it to the shaft. The shaft speed is controlled by adjusting the excitation and hence the back ernf of the dc motor. As the back emf changes, the current in the d.c. link alters. As a consequence, the torque developed by the induction motor changes and its speed varies until its slip voltage is again in equilibrium with the d.c. motor back emf.
The most striking feature of this drive is its high operating efficiency and power factor over the entire operating speed range.
In its original form, however the drive could not become a commercial success. This was due to the fact that the rotating converter needed a tight maintenance schedule besides being a costly equipment in itself. Irrespective of the operating speed, the power output of motor remains more or less constant so it is called the constant power cascade.
1.1.2 MODIFIED KRAMER SYSTEM
With progress in technology, rotary converter was replaced by first mercury arc rectifier and then by solid state rectifier made up of six power diodes which are cheap, maintenance free and can be built almost in any size.
3 Fig. 1:1 Schematic diagram of original Kramer drive. The drive called modified Kramer drive now served as a highly favoured proposition where large amount of power was to be handled and the speed-range requirement was limited, such as in
large pumps, and wind tunnel drives.
1.1.3 MODERN STATIC KRAMER SYSTEM
In this system, the d.c. motor is also replaced by a three-phase SCR bridge as shown in Fig. (1.2). This is constant torque type of slip energy recovery drive. In this arrangement slip power is rectified by three-phase diode bridge whose output
is smoothed by a filter choke L and fed to a line commutated
inverter bridge. The inverter returns the slip power to the.a.c. supply through a suitable transformer. To avoid inrush of current at the time of starting, the machine is started with the aid of a resistor connected in the rotor circuit.
The torque v/s speed characteristics are shown in Fig. (1.3) for different firing angles. As a = 900 the average voltage at
d.c. side of the inverter is zero and hence the drive offers
characteristics of plain induction motor.
A typical starting method with resistance switching is shown
in figure (1.4). The motor is started with switch 1 on and switch
2 &' 3 off. As the speed builds up the resistance R1 and R2 are
shorted sequentially and at the designed S value, switch 1 is max opened and the controller is brought into operation. ]-PHASE SUPPLY
WOUND ROTOR INDUCTION MOTOR
DIODE INVERTER RECTIFIER
Fig • 1.2 Static Kramer drive 2.5
a
Q
I.I
U 0.5 1.0
5coeed foul
Figure L.3 TorquL—sped cur cs at different firing angles of invcrtcr.
7 3-phase ac supply
I.M.
Fig. 1.4 Motor starting method 1.1.3.1 ADVANTAGES
(l) This system is popular in large power pump and compressor type
drive where the range of speed variation is usually limited.
The drive system is not only efficient but the converter power
rating is low because it has to handle only the slip power.
This power rating becomes lower for a more restricted speed
range near the synchronous speed.
(2) The additional advantages are that the drive system has d.c.
machine like characteristics and the control circuit is
simple.
1.1.3.2 DISADVANTAGES
Poor input p.f. due to (i) lagging reactive power drawn by
machine to maintain the air gap flux, and (ii) the inverter also draws reactive power for line commutation. The reactive power drawn by inverter increases as a advances from 1800 to 90o. At a
goo the inverter KVA is almost wholly reactive. Total reactive
power drawn from the supply is the sum of the reactive power to
machine and the inverter. So pf becomes very poor at low speed
when the magnitude of active power is comparatively low. At full
load and rated speed, p.f. of the system is about 0.6 and
decreases as speed decreases.
Power factor of the system can be improved if the reactive
power drawn by the inverter can be low, at the lowest value of the
controllable speed range. At minimum speed Vd will be maximum and f
0 this should be matched with that of the inverter voltage with a =
1800. Hence a suitable transformer is required between inverter and supply line.
1.1.4 ORIGINAL SCHERBIUS CASCADE SYSTEM
The original Scherbius cascade system using rotary machine
is shown in Fig. (1.5). In this system the feedback power is
electrical, whereas in original Kramer system is mechanical.
Instead of coupling the d.c. motor directly to the main motor, it
is mechancially coupled to a synchronous generator which converts
the mechanical power to electrical power and returns it to the
supply line.
1.1.5 STATIC SCHERBIUS DRIVE SYSTEM
In this sytem, all rotary machines in recovery loop have
been replaced by SCR converter and inverter, as shown in
Fig. (1.6). The system is quite similar to static Kramer system.
Only diode bridge is replaced by SCR bridge. So power can flow in
either direction. Motor can be operated in sub-synchronous and
super-synchronous regions.
1.1.5.1 ADVANTAGES
Machine speed can be controlled continuously about 50%
above and below the synchronous speed with a converter rating of
about 50% of machine capacity.
10 N
?nerator t --
Load 'If
Fig.1.5: Original Scherbius Cascade System
11 a0
ED 0
w ~
U o v Cl)
a
12 1.1.5.2 DISADVANTAGES
(1)Line commutation of the machine side converter is difficult
near synchronous speed when ac voltage is very small. This
can be solved by using a force commutated ASCI-type converter.
(2)Speed reversal is not possible, it is possible by installing a
phase sequence reversing con€actor on'stator side. This of
course, requires the drive system to go through zero speed,
where the converters are not designed to operate.
(3)Rotor voltage is high at low speed and low at high speed. So,
the KVA rating of the converter for a given load is much
larger than the actual slip power under high speed condition.
1.1.6 STATIC SCHERBIUS DRIVE USING CYCLOCONVERTER
Dual converter in static Scherbius drive system can be
replaced by a phase controlled line commutated cycloconverter, as shown in Fig. (1.7).
The cycloconverter permits the slip power to flow in either
direction, and therefore the machine speed can be controlled in
both sub-synchronous and super-synchronous ranges, with motoring
and regeneration features.
1.1.6.1 ADVANTAGES
(1) A cycloconverter in rotor circuit can regulate slip power flow
in both direction without a d.c. link.
13 a -1
~+a r5 cn
0.-I 30 U w S. N
r-H 0 wa
04m z M H 14
(2) Current is nearly sinusoidal and hence torque pulsation
effects are minimum.
(3) Difficulty of commutation near the synchronous speed is
completely absent.
1.1.6.2 DISADVANTAGES
(1) System is complex and costly. So applications is restricted
to extremely large motor as driving rolling mill, mine fans,
blowers, centrifugal pumps etc.
1.2. PRINCIPLE OF SUB-SYNCHRONOUS AND SUPER-SYNCHRONOUS OPERATION
WITH INJECTION OF VOLTAGE IN THE ROTOR CIRCUIT
Fig. (1.8(a)) shows the equivalent circuit of a wound rotor
induction motor with an injected voltage of V L~ volts per phase. r r Consider the ideal no-load operation for which I must be zero. r In the absence of injected voltage, I is zero when the motor r speed is equal to the synchronous speed. When V is in phase with
E, Ir is zero when sE/a Ti = Vr
a V
or s= TiE r ... (1.1)
The no load speed is
W mo = [I- aT1VrE lwms . (1.2)
15 RO x V 1, 1 sxr Rr
Ir 1 st V ~r r i aTl
(a)
I19 XS X r Rr/s
m z~ r r_ 14' v X s- m
(b)
Fig. 1.8 Induction motor equivalent circuits with rotor-injected voltage
16 According to equation (1.2) the no-load speed can be changed from synchronous to standstill by varying V from 0 to r (E/aTl ). Further if V is reversed, s will be negative and the
motor no-load speed will be higher than the synchronous speed. The
relative speed between the stator field and the rotor will now be
the opposite of that for speed less than the synchronous speed.
Hence, the phase sequence and direction of the rotor induced
voltages will also be opposite . Thus, for operation above
synchronous speed, both the polarity and the phase sequence of
the injected voltage will have to be changed. Further, as the
speed changes, the frequency of the rotor induced voltage changes.
For the injected voltage to balance the induced voltage, the
frequency of the injected voltage must track the frequency of the
induced voltage.
Let us now consider the operation with a fixed V of positive
polarity and phase sequence. The no-load speed of the motor will
be less than the synchronous speed. An application of a positive
load torque will reduce the motor speed from its no-load speed,
causing the slip and the rotor induced emf to increase. A positive
rotor current will flow and motoring torque will be produced. As
the load torque is increased, the motor speed falls to compensate
for the increase in the machine impedance drop due to the increase
in the rotor current. The higher the load torque, the greater the
drop in speed from no-load speed. This is the motor's
subsynchronous motoring mode of operation. Let P denote the r
17 power absorbed by the source Vr. In this mode of operation, both
Pr and Pm are positive and Vr has a positive polarity and
positive phase sequence. (Fig. 1.9(a)).
Let the load torque be removed. This will restore the
operation to the no-load speed. Now let a negative load torque be
applied. A negative load torque will increase the motor speed.
Consequently, the rotor induced emf will decrease (because the
motors is running at a speed less than the synchronous speed)
and a negative rotor current will flow. Because of the negative
rotor. current, the machine will develop a negative torque and
operate under regenerative braking. The motor will run at a speed
higher than the no-load speed but less than the synchronous
speed. In this mode of operation, known as subsynchronous
regenerative braking, both P and P are negative and V, which
acted as a sink of power for subsynchronous motoring, acts as a
source of power now (Fig. 1.9(b)).
Let us now consider the operation of the drive when the
polarity and phase sequence of V are reversed. The no-load r speed is obtained from equation (1.2). It will be higher than
the synchronous speed. The induced voltage will have negative
polarity and negative phase sequence. Let a positive load torque
be applied. The motor will slow down. The induced voltage
magnitude will decrease and a positive rotor current will flow,
producing motoring torque. The motor will run. at a speed less yr
(a) Subsynchronous (b) Subsynchronous motoring regenerative braking (w1 < w1 )
r
(c) Supersynchronous (d) Supersynchronous regenerative braking motoring . ( wm ➢ mc
Fig. 1.9 Polarities of rotor variables for different modes of drive operation for a constaint magnitude of Vr nud variable load torque.
19 than the no-load speed but greater than the synchronous speed.
The speed will fall with an increase in the load torque. This mode of operation is known as super-synchronous motoring. Here P is m positive and P is negative. The voltage Vr has negative polarity
and negative phase sequence (Fig. 1.9(c)).
Let the load torque be removed. This will restore the
operation to the no-load speed which is higher than the
synchronous speed. Now let a negative load torque be applied.
This will increase the motor speed and the induced voltage
magnitude will increase. A negative rotor current will flow
producing a negative torque and regenerative braking. The higher
the load torque, the greater will the increase be of motor speed
beyond the no-load speed. In this mode of operation, known as
super-synchronous regenerative braking, P is negative and P is m r positive. The voltage V has negative polarity and negative phase r sequence (Fig. 1. 9(d) ). V, which acted as a source of power for r super synchronous motoring now acts as a sink of power.
1.2.1 DRIVE CONFIGURATION FOR SUB- AND SUPER-SYNCHRONOUS OPERATION
A static slip energy recovery drive is schematically shown
in Fig. (1.10). The slip recovery loop of this drive consists of
a motor side converter and a line side converter, both being line
commutated type. The dc link current can only flow from the upper
terminal of the motor side converter to the upper terminal of the
line side converter, as shown in the Fig. (1.10). Current flow in
20 4- 4-0 U) 0 L ` '4-O a, C> Lr a .? O 4i O v > 4) 0 0> 0 C L m p~ QU N `O O O 0 4) L V E E a M- - O O qp 2 CC 0 0 ED Q) N w L Q •- of I~ E 0 c U d) D C O DS O 6- a
2]_ the opposite direction is not allowed by the converters. Under ideal no-load conditions the dc link current is zero, resulting in zero developed torque.
1.2.1.1 SUB-SYNCHRONOUS MOTORING
The power flow diagram, ignoring ohmic losses, for sub-synchronous motoring operation is given in Fig. (1.11(a)).
Under this operation slip is positive. Air gap power, P, is partly converted into mechanical power, Pm = (1-s)P and the balance power, called slip power, P = sP is returned to supply s mains. Thus, net power drawn from the supply mains is P. = in (1-s)P and is equal to mechanical power output Pm for all positive values of slip.
In order to implement this mode of operation the converters of the slip power recovery loop are so triggered that motor side converter behaves as an uncontrolled rectifier (a =0) and line m side converter behaves as inverter with adjustable firing angle.
This is schematically shown in Fig. (1.11(b)). At inverter firing angle al equal to 900, average value of voltage Vl is zero. This causes the motor to operate at synchronous speed under no-load, since at synchronous speed slip is zero hence V m will be zero and as a result id will also be zero. When firing angle is adjusted at an increased value between 900 to 180°, the voltage V1
increases. This results in the motor no-load speed to reduce to
allow vm to become equal to Vl. At a given firing angle of the
id I
U, C 0 E a a Ps=sPg N U,
kwon 900 ,, * l <$80°
P mech PS =(I- (b)
(a)
Fig.1.11 Sub-Synchronous motoring mode
id
N Ps= sPg 0C E T a a a P me( N
dm=0° 900 41(1<1800 PS
(a) (b)
Fig. 1.12 Super- Synchronous generating mode
23 inverter, when motor is loaded, its speed falls, slip increases,
V increases and hence id increases to produce required torque.
1.2.2.2 SUPER-SYNCHRONOUS GENERATING
The power flow diagram, ignoring ohmic losses, for super-synchronous generating operation is given in Fig. (1.12(a)).
Under this operation the motor is driven by a prime mover at
super-synchronous speed causing slip to be negative. The input
mechanical power, P m = (1+s)P is partly converted in electrical
power, P, and the balance power, Ps = sP is returned to supply
mains through recovery loop. Thus, net power given to supply
mains is Pin = (1+s)P and is equal to mechanical input power P
for all negative values of slip.
In order to implement this mode of operation the converters
of the slip power recovery loop are so triggered that motor side
converter behaves as an uncontrolled rectifier and line side
converter behaves as inverter with adjustable firing angle. This
is schematically shown in Fig. (1.12(b)). At inverter firing
angle aI equal to 90°, average value of voltage VI is zero. This
causes the motor to operate at synchronous speed under no-load,
since at synchronous speed v shall b'zero and hence id will also
be zero. If the operation is considered at higher than
synchronous speed, al remaining equal to 900, the slip increases
causing v to increase. Current id increase representing an
increase in torque. This implementation is also in accordance
24 with Fig. 1.9(d). The slip power flows out of rotor. However, since the direction of d.c. link current can not be reversed, the polarity of vI and id are both opposite in Fig. 1.12(b) with respect to those of Vr and Ir in I'ig. 1.9(d) to keep the direction of slip power unchanged.
0 For aI > 90 , the no-load speed in generating mode will be higher than synchronous speed.
This mode can also be identified as supersynchronous regenerative braking mode of the drive.
1.2.1.3 SUPER-SYNCHRONOUS MOTORING_
The power flow diagram, ignoring ohmic losses, for super-synchronous motoring operation is given in Fig. (1.13(a)).
Under this operation slip is negative. Air gap power, P is converted into mechanical power and the slip power,Ps = sP drawn from supply mains is also converted into mechanical power. Thus, net power drawn from supply mains is P. = (1+s)P and is equal to in mechanical power output Pm = (1+s)P.
Since the direction of slip power has now reversed, and since direction of id can not change, the polarity of v and vl must change with respect to the previous two cases, as shown in Fig.
(1.13(b)). In order to implement this, the converters of the slip power side converter behaves as a fully ON inverter (a =180°) and m
25
id - Ps= sP9 _ - N C O Vm VI E
11 P meth ~1 + + —fI cn
oCm=I80° 0<&I <900 (a)
Ps (b)
Fig. 1.13 Super synchronous motoring mode
id
Ps=sPg c_ NO
c HVm VI E, I /K a P me4 v~ + + cn
o(m=180° (a)
PS (b)
Fig.1.14 Sub—Synchronous generating mode
26 line side converter behaves as a controlled rectifier with adjustable firing angle. This is schematically shown in Fig:
(1.13(b)). At rectifier firing angle al equal to 90°, average value of voltage vl is zero. This causes the motor to operate at synchronous speed under no-load, since at synchronous speed v m shall be zero and hence id will also be zero. When firing angle, al, set a lower value between 900 and 00 , the voltage vl
increases. This results in the motor no-load speed increasing allowing vm to become equal to v1. At a given firing angle, al,
when motor is loaded, its speed falls, slip reduces, v decreases m and hence id increases to produce required torque.
This implementation is also in accordance with Fig. 1.9(c).
1.2.1.4 SUB-SYNCHRONOUS GENERATING
The power flow diagram, ignoring ohmic losses, for sub-synchronous generating operation is given is Fig. (1.14(a)).
Under this operation slip is positive. Mechanical power P =
(1-s)P is converted into air-gap power, and also slip power, P = s sP drawn from supply mains is added to air-gap power. Thus net
power given to supply mains is P. = (1-s)P. zn .
In order to 'implement this mode of operation the converters
of the slip power recovery loop are so triggered that motor side
converter behaves as a fully ON inverter (a =1800) and line 'side m converter behaves as a controlled rectifier with adjustable firing
27 angle, exactly similar to the case of super-synchronous motoring.
This is schematically shown in Fig. (1.14(b)). At rectifier firing angle al equal to 90°, average value of voltage v1 is zero.
This causes the motor to operate at synchronous speed under no load, since at synchronous speed vm shall be zero and hence id will also be zero. When firing angle al, is set at a reduce value between 90° and 00, the voltage V1 increases. This results in the motor no-load speed to reduce allowing V to become equal to V1.
At a given firing angle of the rectifier, when operation is considered at a speed higher than the no-load speed, slip reduces,
V decreases and hence 1d increases which represents availability of generating torque.
This mode can also be identified as sub-synchoronous regenerative braking mode of the drive. r
The power flow diagram in static slip power recovery drive is shown in Fig. (1.15). This diagram is applicable for all four modes operation of present work.
1.3 LITERATURE SURVEY
Recently, there has been a renewed interest in slip energy recovery induction motor drives. This has been prompted by pressures from industry for more efficient and reliable drive system so that slip energy recovery systems are finding their applications in large power fans andpumps that require a limited Lech :U s)Pgap
Pcu-st
Fig.1.15 Power flow in static slip power recovery drive
29 speed range. Limiting speed range Is not a defect of slip energy recovery drives; because the KVA ratings and hence cost of the rectifier/inverter combination is considerably reduced. Wider speed control range from nearly twice synchronous speed to a few rpm is possible provided the required rectifier/inverter ratings are used.
1.3.1 NEW TOPOLOGY/TECHNIQUE FOR POWER FACTOR IMPROVEMENT
The main draw back of the slip power recovery scheme is that
the power factor of the drive system is poor, particularly in the
low-speed range. Shephard and Khalil [1] investigated the
improvement of the power factor using a capacitor in the rotor
circuit. Lavi Polge [21 presented a method of analysis for
determining power factor, speed-torque characteristics.
Sheshgiri R. Doradla et. al. [3] have proposed a new slip
power recovery scheme with a PWM converter in rotor circuit. The
steady-state performance characteristics such as speed torque
characteristics and the overall power factor of the drive system
are determined using simple DC and AC equivalent circuits. Also,
the performance characteristics are verified experimentally. A
good agreement exists between theoretical and experimental
results. The power factor of the drive system is improved
significantly over the entire range of speed control. The drive
provides speed-torque characteristics which are similar to those
of a separately excited dc motor. Oscillograms of typical
30 waveforms from the experimental set up are illustrated to demonstrate the satisfactory working of the new slip power scheme.
1.3.2 EFFICIENCY IMPROVEMENT TECHNIQUE
W. Shephard and J. Stanway [4] have studied that the low speed efficiency of induction motor can be improved by rectifying slip frequency power, inverting this to line frequency, iri,jecting it back into the supply directly (line feedback) or through auxiliarly stator winding (stator feedback). The low power factor and nonsinusoidal supply current of the line feedback connection are improved by use of stator feedback method but the improvement of efficiency is then much less. Line feedback with a two-phase induction motor eliminates the need for a variable control voltage source.
1.3.3 STEADY STATE ANALYSIS OF DRIVE
V.N. Mittle et. al. [5,6,7] provided an analytical approach
based on synchronously rotating reference frame transformation of
phase variables, through which the steady-state performance of such a drive was predicted and compared with the one obtained
experimentally. Although the analysis neglected saturation,
hysteresis and eddy current effects in the machine, voltage drop
in the rectifier and induction motor rotor current harmonics, good
correlation was observed between analytical and experimental
results. P.C. Krause et al. [8] have pointed out fallacy in this
31 work, which is due to wrong interpretation of the relationship between the reference frame and phase axis.
The paper presented by M.S. Takkher and S.P. Gupta [9] is concerned with steady state analysis of a constant torque type static slip energy recovery drive. The drive is analysed in synchronously rotating d-q reference frame employing a correction in a previously published approach. The steady state perform iic" curves are obtained and discussed.
E. Akpinar and P. Pillay [10] have stated that the operati:r: of slip energy recovery induction motor (IM) drives is complicatf-d by the fact that the region of interest is centered on the or and not on the stator as in most other IM drives. A detailed study of the rotor phenomena including the rectifier commutation overlap and inverter harmonics are facilitated by the use of the actual rotor state variable instead of transformed variables that are used in most other induction motor drive systems. The use of
"natural" equations however, has the serious drawbacks of having a 6 x 6 inductance matrix that depends on the rotor angle and which must be inverted at each step of integration. The computer
program developed can be used to examine the transient performance of slip energy recovery drives, for the proper rating of semiconductor devices or to examine the effects of faults on the associated power system on the drive performance. Current. and
32 speed controllers designed using linear techniques can be evaluated using this program as well.
1.3.4 TRANSIENT PERFORMANCE OF KRAMER DRIVE
Rao et. al. [11] have studied the improvement of performance characteristics by using a fully controlled converter with half controlled characteristics. Smith [12] investigated the drive system which provided sub- and super-synchronous speed of* operation, the scheme employed a current source inverter and line commutated converter in the rotor circuit.
J.E. Brown et al. [ 13] have taken account of the different
rotor states which arises from the switching of DC link current
between the rotor phases by the action of diode bridge. They further describe that the performance of a static Kramer drive is substantially affected by DC voltage waveform of the thyristor
recovery bridge and the inductance of the DC link smoothing
reactor. The paper describes a novel method of analysis of the
periodic transient performance of such a drive. In which full
account is taken of these practical aspects of the recovery
system. The analytical model obtained has been verified by
comparison of calculated and measured performance of four
different types of recovery bridge. The computed results have
allowed a more comprehensive examination of drive performance
than is possible from the results of practical tests.
33 Herbert W. Weiss [14) has described in his paper that the application and performance of' a cycloconverter type frequency converter and a doubly fed wound rotor ac motor as applicable to adjustable speed drive systems for pump and compressor loads. The paper covers the subject by discussion of the following: (i) basic theory and operation of a cycloconverter and a doubly fed wound rotor motor; (ii) motor and converter rating to meet the performance requirements of typical pump and compressor applications, (iii) control scheme for starting, running and process control, (iv) efficiency and power factor.
E. Akpina.r, P. Pillay and A. Ersak [151 presented a closed- form expression to estimate the overlap angle in the slip energy recovery system. A closed-form expression is derived using a hybrid model of the induction motor and a dynamic model of the rotor rectifier. The ripple content of the dc link current and the
inverter input, voltage are neglected. The predicted results
obtained using the closed form expression are verified
experimentally.
34 CHAPTER - 2
MATHEMATICAL MODELLING
Generalized method for predicting steady state characteristic of slip energy recovery system is described using reference frame theory. With the help of d-q transformation of variables, the basic instantaneous equations for the idealized induction motor are developed. The model is developed in p.u. system. The model is general and is suitable for studying system performance for any settings of firing angles of motor side and line side converters.
2.1 INTRODUCTION
The suitability of any drive to a given application is characterized by its steady-state and dynamic behaviour. In order to study the complete performance, it is essential to develop a mathematical model of the drive, using reasonable assumption, if any.
The drive configuration considered for development of mathematical model has been presented in the previous chapter in
Fig. (1.10). The procedure followed is to write voltage current equation for the various components of the drive namely the motor, the motor side converter, the dc link filter and the line side
35 converter and combine them to obtain the mo r. In developing the model, the following assumptions have been made.
(i) The induction machine has been considered to be symmetrical.
(ii) Supply voltage and current are considered to be sinusoidal.
(iii) Rotor current harmonics due to the motor side converter are
not taken into account.
(iv) The commutation overlap phenomenon in the converters is not
accounted for.
(v) The do link current is considered to be ripple free and only
average values of voltages are considered on do side of the
converters.
(vi) Iron losses in the machine and effect of magnetic saturation
are neglected.
The model is developed to be suitable for dynamic as well as steady-state performance estimation of the drive.
2.2 DEVELOPMENT OF MATHEMATICAL MODEL OF THE SYSTEM
('i) Induction Machine
The per-unit voltage-current equations of symmetrical
induction machine, as represented in Equation (2.1), in
synchronously rotating reference frame are given by [171
36
s +x—_ -x vd.s (R ss ws p ) Xss wM p lds
v i qs Xss (R+ss Xv ssc p) X M MwW qs
- SXM (Rrr+ Xrr p) - SXrr vdr - p ldr
XM Xrr vqr SXM w p SXrr (Rrr+ W p) I i qr
~ J
(2.1)
The electromagnetic torque in per unit is given by
Te = X ids iqr) ... (2.2) M (i qs ldr ' The equations for motor side and line side converters will
now be written, first in actual units and later in per units.
(ii) Motor Side Bridge Converter
For an ideal three phase thyristor bridge the input/output
relationships of voltages and currents are given by
v = 3~ v Cosa ... (2.3) m it rp m
irp = 2 i d ... (2.4)
where vm, id are the average values of the bridge output voltage
and current and vrp, irp are the peak values of the fundamental
37 phase input voltage and current. The effect of harmonics in rotor current and voltage is ignored, in present work.
In ideal converter, the power input to the bridge is equal to the power output. Thus,
v i 3 rp . rp . Cos 0r= - v id where 0r is phase angle between fundamental rotor phase voltage and fundamental rotor phase current and -ve sign is due to opposite sign conventions adopted for rotor phase current and the do link current,
Substituting for v and id from equations (2.3) and (2.4), it is found that
Cos 0 = -Cos a r m Cos 0r = Cos (am + 1800 ) ... (2.5)
This shows that (A = a + 180° r m
Thus for a =0 i.e uncontrolled rectifier, rotor fundamentals m currents are in antiphase with rotor voltages, whereas for a =1800 m (fully ON inverter) they are in phase.
(iii) Filter
The voltage-current equation of the filter is written as
vj_n (Rt, + Lf p) id- vI X 1 or vm = Rf+ Wf p i d -v1 ... (2.6) [ J where Xf is d.c. link reactance measured at line frequency.
(iv) Line Side Converter
For the phase controlled line side converter, neglecting
commutation delay, the input/output voltages are related as :
vl = 3 vsp Cos al ... (2.7)
where vsp is the peak stator phase voltage.
Combining equations (2.6) and (2.7) and substituting for v m and id from equations (2.3) and (2.4),
X it 3 vrp Cos am = r Rf + fW p 1 i rp - 3i vsp Cos al l J 2~
2 X -It f v Cos am 18 Rf + X p it - vs Cos al p p p
.. (2.8)
The angular relationship of phase axes with the d- and q-axes
is shown in Fig. (2.2).
If d-axis of reference frame is considered to be coincident
with phase a-axes of stator and rotor at t=0, the stator and rotor
terminal voltages can be defined as Stator
v =v . Coswt as sp
vbs = v . Cos (wt - 2n/3) ... (2.9) spP v = v Cos (wt + 2n/3) CS sp
Rotor
var = vrp . Cos ((3 + 9r )
vbr = vrp . Cos ((3 + 8r -2/3) ... (2.10)
v = v . Cos ((3 + 9 + 2/3) cr rp r
where (3 is the angle of advancement of the d-axis with respect to rotor phase a-axis, as shown in Fig. (2.2) and is the phase 9r
angle between stator and rotor voltages of phase a at t=0.
If phase voltages of equations (2.9) and (2.10) are
transformed into d, q variables, following relations are readily
established :
vds = vsp ... (2.11)
vqs =0 . (2.12)
vdr = yr . Cos 0r ... (2.13) P v =v . Sin 0 ... (2.14) qr rp r
Also, since the fundamental rotor phase currents are at an
angle 0r = am + 1800 with respect to rotor phase voltages, they
can be expressed as
iar = - irp . Cos ((3 + Ar + am) ... (2.15) ios 'or
STATOR ROTOR
Fig2.l CONNECTION AND CURRENT CONVENTION FOR 3-PHASE INDUCTION MOTOR
d Axis DIRECTION ROTATION OF bs Axis MAGNETIC FIELD,ROTOR br Axis AND d-q FRAME
gAxis \\ Axis
as Axis
p l = (.J P a.r_ = W r
cs Axis cr Axis
Fig2.2AXES OF 2-POLE,3-PHASE SYMMETRICAL INDUCT ION MACHINE
41
ibr = - irp . Cos ((3 + 6r + am - 2n/3) ... (2.16)
i = -i . Cos ((3 + 0 + a + 2ir/3) ... (2.17) cr rp r m
Using d-q transformation for currents, following relations
are established :
idr - irp . Cos (6r + am) ... (2.18)
i = - i . Sin (6 + a ) ... (2.19) qr rp r m
Equation (2.8) may now be rewritten by substituting for V rp from equation (2.13) as
2 X 1 Cosa v = n R + f p (i Cos6 ) -v 1 Cosh dr 18 f w CoCos am rp r I J 1 sp Cosa r m ... (2.20)
Similarly,
v = TI2 X f 1 Cos al [R rp Sin r) - vsp Sin 6r qr 18 f+ w p ] Cos a (i e in Cosa m .. (2.21)
Using the trigonometric identities,
Cos (A+B) = Cos A . Cos B - Sin A . Sin B
Sin (A+B) = Sin A . Cos B + Cos A . Sin B
in equations (2.18) and (2.19), it can be shown that
irp Cos S = - idr Cos am - iqr Sin am ... (2.22) r
ir,p Sin S = - iqr Cos am m ... (2.23) r + idr Sin a
42
Substituting for i Cos 0 from equation (2.22) in equation p (2.20) n2 Xf 1 m - i Sin am ) vdr 18 Rf + w p , Cos a (-ldr Cos a m Cos al -v Cos 0 sp Cos a m r or, X f 1 it2 vdr Cos am = 18 I Rf + w p ( -idr Cos am - i Sin am} J
- vsp Cos al Cos 0r ... (2.24)
Similarly, from equation (2.21) and (2.23)
v = n2 R + Xf 1 m + idr Sin a qr 18 ( f W p ,Cos a (-iqr Cos a m Cos al -v Sin A sp Cos a r m
2 X vqr Cos a m = 18 ( R + Wf p (-i qrCos a + i Sin a m m
- v Cos al Sin 0 ... (2.25)
Conversion of equations (2.24) and (2.25) in p.u.
Referring rotor quantities to stator side in Equation (2.24),
2 r Xf vd r Cos am = ~8 + Rf W p l (-i'r Cos a m - iq Sin a L m vs Cosa Cos 0r p where primed quantities are rotor quantities referred to stator.
43
Dividing both sides by v base' v' Cos am n 2 1 X , W p (-idr Cos am - iqr Sin am ) vbase i8 vbase I Rf + v - v sp Cos a1 Cos 6r base
Since, vbase lbase * 2base So,
vdr Cos am ~2 1 X f 1 [R . ( id Cos am - iq Sin a v 18 z f + w P, i - r r m ) base base base v sp - y Cos al Cos 6r base
or n2 Xf (p. u. ) Vdr (p. u.) Cos am = 1g `Ri (p.u.) + , w pJ (-idr(p. U. ) Cos a - i qr (p. u.) Sin am )
- vsp (p.u.) Cos al Cos S
dropping p.u. notation for simplicity,
V Cos am = 1 [R + f pl Cos a - i dr g f W (_idr m Sin am)
- vsp Cos al Cos S r ... (2.26)
Similarly, we can find equation (2.25) in p.u.
n2 Xf vqr Cos am = 18 IRf + w p~ (-i qr Cos am + ldr Sin am )
- v Cos a Sin 9r ... (2.27) sp
44
From equation (2.1), X + rr pl i v M p i - (sX ) i + R dr = w ) ds M qs C rr w J dr - (sX ) i rr qr or X Cos a - M Cos am) iqs vdr, Cos am = L M w m,p ids (sX
X Cosa M l i p - (sX Cos a ) i + CR rrCos am + rr w J dr rr m qr
XM p v cos a = M p i - sX' i + [R/ + X'1i sX' i dr m w ds M qs rr rr w ) dr rr qr
... (2.28)
Cos a, Rr, Rrr Cos am X Cos am where XM = XM m rr X, rr
Combining equations (2.26) and (2.28)
X—p1 - sX' i + `R'+ X' p i i -sX i w ds M qs l rr rr w dr rr qr
~2 X1 m - t(Ir Sin am ) - vsp Cos a1 Sin Ar = 18 IRf + w p, (-idr Cos a or p 2 `{ m] + - vsp Cos al Sin Ar = XM w lds -sM lqs+Rrr + 18 R Cos a
2 p 2 f f Cos aml w ] LsX 18 Sin am [Rf CXrr + 18 X ldr rr +w P]qr J ... (2.29)
2 2 f Cos am f ]Cos am Let. Reg1 = R' + 18 R = ~Rrr+ 18 R
45
2 Xeg1 f Cos am = Xrr + 18 X
2 R = sXrr f rr 18— R Sin am
2 It m Xeg2 1g Xf Sin a
Equation (2.29) becomes p r P- - v Cos a Cos 9 = X' -i - sX' i + R + X i sp 1 r M w ds M qs eq1 eq1 wj dr
p EReg2 + Xeg2 wJ 1 qr (2.30)
- Again, from equation (2.1) x X v + rM pl i qr = sX M ids i qs + (sX ) i + Wr p J rr [Rrr + 1 qrr
XM Cos am v Cos a = (sXM Cos am ) i + w i + ( sX Cos a ) i j p qs rr m dr
Xrr Cos am + [R Cosa + pJ1 i rr m w qr or
X' X v Cos am = sXMs id + Mw p iqs + [RI sXrr ldr + rr + rw p, qr ... (2.31)
Combining equations (2.31) and (2.27)
XM X' sXM ids + ~ p iqs + sX, rr rr ldr + [Rrr + w p,l qr
~2 r Xf } = 81 f + w p (-iqr m) - vsp Cos al Sin 6 1 Cos am + ldr Sin a r
46
p or - v sp Cos a Sin 0 r = sXMs id + XMl — iqs
+ Sin am [R f + IS Xrr 18 wf p,] l dr
p 2X[Rf 1 + (R' + X' -- + Cos a ' +fpi lrr rr w 18 m w J qr or P - vsp Cos a1 Sin 6r = sX l M ds + X M w lqs + LR eg2 + Xeg2. w, i dr
P ... (2.32) + CReq 1 + Xeq 1 I I q r The final system equations in p.u. are obtained by replacing
the last two equations in equation (2.1) by (2.30) and (2.32)
vsp l ds 0 i = z qs -v sp Cos a Cos 0 r -i rp Cos (0+a 1 r m -v sp Cos al Sin 6 r -i rp Sin (0 + am)
(2.33) where Xss X M -X - XM (Rss+ w p) w p
Xss p Xss (Rss+ W p) XM XM_ w
[Zl = XM X eg1 Xeg2 W p - sXM Reg1+ w p) - Reg2+ w p)
sXM wM p (R q2+ X~q2 p) (Reg1+ wglp)
Equation (2.33) along with equation (2.2) completely describe
the static slip power recovery drive system.
47 For the specific case whe c=0, corresponding to uncontrolled m rectifier used as motor side converter, R =R , X =X , eq1 eq eq1 eq Reg2=sXrr, Xeg2=0 and X'=XM. Equations (2.33) then reduce to those obtained in reference [9].
In all further developments, only the above equations expressed in per-unit system have been used. These equations can easily be handled for studying the steady-state performance of the system.
2.3 CONCLUSION
Equations (2.33) have four unknown variables' i ds, i q s , it p and 0 whose values may be determined for given values of supply r voltage vsp, operating slip s and al by simultaneously solving these equations.
The idealized model developed on the concept of coupled circuit approach is well suited for simulating the drive for most of the practical purposes. The parameters involved in the model are such that they can be easily measured at the terminal of the machine. The model is extremely useful for carrying out steady state and transient studies of the static slip energy recovery drive. CHAPTER - 3
STEADY STATE ANALYSIS
This chapter deals with the steady state analysis of slip
power recovery drive. Using the mathematical model, developed in
previous chapter, performance characteristics, like torque, supply current, power factor, losses and efficiency are determined. The effects of firing angle control has been investigated and discussed.
0
3.1 INTRODUCTION
Prediction of steady state performance of a drive system is
usually done before manufacturing it, to enable proper selection
of drive parameters for meeting the requirements of load. The
drive system is simulated on a computer and its performance under
varying operating condition is studied.
3.2 SYSTEM EQUATION UNDER STEADY STATE
The equations describing the behaviour of slip power recovery
drive, developed in Chapter - 2 are given below : Xs s X M v (Rss+ w p) - x —p sp W p - XM
0 Xss p Xss (R+ss w p) X M XM _w x x -vsp Cosa 1 Caser wM p -- SXM (Regl+weq 7 p) - (R eg2+ wq p)
-vsp Cosa l Siner SXM wM p (R +- g2p) (Regl +Xwg1 p)
Ids
i qs x -i Case' rp r -i Sine' rp r ... (3.1)
T = X (i i ... (3.2) e M qs dr dsi lqr
Under steady state, the different voltages and current attain a steady value. When transformed into dq voltages and currents, they appear to be dc quantities. This has been shown for the case of voltages through equations (2. 11) to (2.14). Consequenty dq currents will also become d.c. in similar manner. Hence their time derivative will be equal to zero. To represent the balanced steady state operation with harmonics neglected, the time rate of change of all currents should be equated to zero (i.e. p=d/dt=0).
,, Q
l• . 5 0 ~1 ~ 's
Then final steady-state equations becomes
v lds sp i 0 qs = I Z -vs Cosal Cos9r X -i Cos6' P rp r -vs Cosal Sin8r, -i Sine' P qr r r4
... (3.3)
0 - XM Rss Xss X R XM 0 ss ss where z= 0 - SXM Re gl - R eq2 SX4 0 Req2 Regi
The actual and p. u. value of the parameters of system chosen
for performance computation are given in Appendix-1. For given
V al , am and s. Equations (3.3) contain four variables -i ds' -i , i , 0', which are computed by simultaneously solving the qs rp r four equations as described in Appendix-2. After getting i and rp 0', idr and iqr can be found from following relationships, r -i' Cos 9' ... (3.4) 1dr rp r i = -i' Sin 0' ... (3.5) qr rp r
After finding out the steady state dq component currents, the
torque developed by the drive can be calculated by using Equation
(3.2).
51 3.3 EXPRESSIONS FOR INPUT POWER, OUTPUT POWER, LOSSES AND
EFFICIENCY
The power input to stator is given by
... (3.6) Pst 3 Vs rms is rms Cos 40
Expressing equation (3.6) in per unit, power input becomes 1 Pst (3 Vs rms i s rms Cos )0Pbase
2 Vsp isp Cos Pst
1 2 Vbase 'base J
Pst = vsp (p. u.) i sp (p. u.) Cos ... (3.7)
Since,
ids = isp Cos,
pst Vsp ids. (3.8)
The output can be determined from the torque, found earlier using equation (3.2), as follows :
Gross output Power = Torque Developed x Mechanical Angular
Speed
Te wr (2/P) .. (3.9) Pmech where p is' numberof poles.
Expressing in per-unit, power output becomes
Pmech Pbase [Te Wr P
52 = T 1 X r (Te wr P 1 (3.10) base base L
Te w (1-s) 2 Tbase a
Te (p.u.) (1-s) ... (3.11) Pmech Similarly p.u. expression for various powers can be obtained as
Input power to stator = P I ... (3.12) st vs p ds
Stator copper loss = P = (i2 + i2 ) R ... (3.13) cu-st ds qs ss Rotor copper loss = rr ... (3.14) Pcu-rt Irp R 2 Filter copper loss = Irp 8i R ... (3.15) Pcu-F
Air gap power = (3.16) Pgap Pst Pcu-st ...
Slip power returned to supply,
i t vs Cos aI ... (3.17) Psli P p p
At a given slip, positive or negative, the power flow in the slip-energy recovery drive satisfies the following relationship
-p Pst mech [P cu-st + Pcu-rt + P cu-F) + Pslip .. (3.18)
Net Power drawn from supply
••• (3.19) Psupply Pst P
53 Efficiency of the system, under steady state, is given by ratio of input and output power. From equation (3.19) and (3.11).
Efficiency~)sys ... (3.20) Pmech / Psupply
Neglecting harmonics, the line side convertor current or feedback current on a.c. side may be written as
ifb 2n id L - al ... (3.21) or from equation (2.4)
ifb =i alL-
or irp Cos al - j ir1) Sin al ... (3.22) 'fb
The supply current is phasor sum of stator current and
inverter current
is = i sp +1 fb
is (ids + j iqs) + (II,p Cos al - j irp Sin al )
or (3.23) is = is(real) + j ls(irng) where i Cos al s(real) lds + i qs - i Sill al ls(img) = i
__ /72 ( 2 ls(peak) sreal) + 1s(img)
Therefore rms value of supply current _ 1 is is(peak)
Supp1ly power factor = _s(real) 's(peak)
54 3.4 COMPUTED PERFORMANCE RESULTS
Performance of the drive under steady-state has been computed
for the following modes of operation.
(a)Sub-synchronous Motoring.
(b)Super-synchronous Generating.
(c)Super-synchronous Motoring, and
(d)Sub-synchronous Generating.
As discussed in Chapter 1 the implementation of the above
four modes of operation requires that the firing angles of the two
converters be set as in Table 3.1.
Table 3.1 : Firing Angles For Different Modes
S. No. Mode Firing Angle Settings
Motor side Line side converter converter (am) (range of al )
0 0 1. Sub-synchronous Motoring. 0 90 to 1800
2. Super-synchronous Generating. 00 900 to 180° 0 3. Super-synchronous Motoring, and 1800 90 to 00 0 0 4. Sub-synchronous Generating. 180 90 to 00
Performance curve for each of the four modes of operation
shall now be presented.
3.4.1 Sub-synchronous Motoring
The steady state performance of the drive in this mode of
55 operation has been computed for line side converter fixing angle 0 setting as 90°, 100°, 110°, 120°, 130 and 140°. For a given angle setting the drive's performance is obtained in the full range of speed applicable to the given setting.
The torque/speed characteristics are presented in Fig. (3.1).
At a1=90°, slip power in the recovery loop is zero and normal
induction motor behaviour is obtained with no-load speed of 1.0
p.u. As al is increased characteristics which maintain
parallelism with respect to each other from no load to full load are obtained. The locus of full :Load operating point is marked on
the figure by broken line. Full load condition is identified when
the stator current becomes equal to rated current, the rms value
of which is 0.707 p.u. It is seen that for all firing angles a
full load torque of 0.826 p.u. is obtained.
The nature of dc link current for different setting of aI is
shown in Fig. (3.2). Here again the locus of full load operating
point is seen to correspond to a fixed value of dc link current
equal to 0.833 p.u. From no-load to full load the dc link current
is seen to vary linearly with speed. Since in this region torque
also varies linearly with speed Fig. (3.1), torque is directly
proportional to the dc link current.
X11 N
U o ~
V cc Z_ W 0 I cc - 0 0 I- 0 0 o cc 2 oN. = U Z) o. O 0 Z o W 0 cc LO a 0) o 0 z W o (I) cc o"? 0F-
o a T LL 0
O UO CO It N 0 00 CD I' N N T r^ T ~- O O O O •
57 N r
(I)
0V
I O U) E
0 W VI Z_ I- E 0. 0. ~ 0 W F- oc~ 0 o Q a 2 N. _~C L C II. WO..c W p 0 Z o ^ m 0 v\J cr O Fz LL O3 U r- Z w4- Ci) ~ U--J D N Z_ j U o p N CO N 00 d O (O N 0) It O p Cl) C'_, C N N N r- T O O O ' LL 0
m
E OC c 0 `Q U 0 Co t~
If) 0 I C) 10 O If) 0 CO CO CV N O 0
59 U)
ra V_ 05 H E w I- 0
Z V
J (I) cr a 0 a U IR m cc a) w m cz 0 CL 0 J U. 11 CL 3 0 D a
aa C) 0) c0 LO CC) N O O O O O O LL z c O I- 0 2 0 Z) ci- 0 a z a. 0 oC U 2 z U W_ z U LL U- w 2 W
vJ
LO
O 00 cc It N o o 0 0 LL 61 The stator current/slip characteristics are presented in
Fig. (3.3). Under no-load condition the stator current is equal to the magnetising current which is independent of al setting.
Full load current is marked as 0.707 p.u.
The system power factor/slip characteristics are presented in
Fig. (3.4). As mentioned previously in Chapter-1, this type of drive suffers from poor power factor on account of additional reactive power requirement of the line side converter. However, on account of recovery of slip power, system efficiency is greatly improved in the whole sub-synchronous range, as shown in
Fig. (3.5).
Full load performance of drive is summarized in Table 3.2.
Table 3.2 . Full Load Performance of the Drive in Sub-synchronous
Motoring Mode
(All values in p.u.) Sator Current (rms) = 0.707
S.No. Line side converter 900 1000 1100 1200 1300 1400 Firing angle al
(L) (2) (3) (4) (5) (6) (7) (8)
1. Speed 0.916 0.723 0.535 0.360 0.201 0.064
2. Slip 0.084 0.277 0.465 0.640 0.799 0.936
3. Torque 0.826 0.826 0.826 0.826 0.826 0.826
4. do link current 0.833 0.833 0.833 0.833 0.833 0.833
62 (1) (2) '(3) (4) (5) (6) (7) (8)
5. Rotor current (rms)
= Feedback current 0.919 0.918 0.918 0.918 0.918 0.918
6. Supply current (rms) 1.16 1.097 1.024 0.942 0.854 0.759
7. Supply power 0.885 0.725 0.570 0.425 0.294 0.181
8. Stator power 0.884 0.884 0.884 0.884 0.884 0.884
9. Air gap power 0.826 0.826 0.826 0.826 0.826 0.826
10. Slip power 0.000 0.159 0.314 0.459 0.590 0.703
11. Mechanical power 0.757 0.597 0.442 0.297 0.166 0.053
12. Stator copper losses 0.058 0.058 0.058 0.058 0.058 0.058
13: Rotor copper losses 0.070 0.070 0.070 0.070 0.070 0.070
14. System efficiency 0.855 0.824 0.776 .0.699 0.565 0.293
15. Stator power factor 0.885 0.885 0.885 0.885 0.885 0.885
16. Supply power factor 0.538 0.468 .0.394 0.319 0.224 0.169
3.4.2 Super-synchronous Generating
The steady-state performance of the drive in this mode of operation has been computed for line side converter firing angle settings as 90°, 1000 , 110°, 120°, 130° and 140°. For a given firing angle setting the drive's performance is obtained in the full range of speed (above synchronous speed) applicable to the given setting.
63 The torque-speed characteristics are presented in Fig. (3.6).
At a1 =90°, slip power in recovery loop is zero and system behaves as a normal induction motor with no-,load speed of 1.0 p.u. As al is increased no-load speed is increased and modified torque-speed characteristics maintaining parallelism with respect to each other from no load to full load are obtained.
The locus of full load operating point is marked on the figure by broken line. Full load condition is identified when the stator current becomes equal to rated current, the rms value of
which is 0.707 p.u. It is seen that for all firing angles a full
load torque of -0.907 p.u. is obtained.
The nature of dc link current for different settings of a1 is shown in Fig. (3.7). Here again the locus of full load operating
points is shown by broken line to correspond to a fixed value of
dc link current of 0.823 p.u. From no-load to full load the dc
link current is seen to vary linearly with speed. Since in this
region torque also varies linearly with speed Fig. (3.6), torque
is directly proportional to the dc link current.
The stator current (rms p.u.)/slip characteristics are
presented in Fig. (3.8). Under no-load condition the stator
current is equal to the magnetising current which is independent
of a1 setting. Full load current is marked at 0.707 p.u. The
64 0 N
C) V 0 z 0o cc W cc W r 0.2 z W DCa °a - rn~ 0 co = c c vJ V Z) 0 ~--Q. QW 0c Y2 z 0 J cc (f! _ c V W u z 0 0 co Co r 0U cc 0 J W Q. N (o rt Co a LL-" a
In o In 0 In 0 In 0 O O O N C)
65 0 N
rn U
V Cc z W V .) OC w QN Z < .a_0 w I C C3 U Q ~ C Z) 0 W Qoo O z L 0 1.L 'o cr 5 3 V z L Z cW 0- ' R c r O rr U w C 0 Z) Z. Cl) U J r r U J 0
0 I w i i i i i i t I_1 O Co 00 't O CO N 00 't O tD N 00 It O '^ d U. 0 1
(!) U
0 z LU I- Uto LU Qc z LL o w 0 a a Uro D 0 o2L z V\J cu >1 0 1--io. 0 z 33 wus q- Ø V C z co V 0" co Cc E LU 0 CL 0 D C/) C
0 0 0 0 C') N
67 VJ U I- V / M W H C3 U . z
2 W- U z W CD J 0 Cl) Cr 0 a 0 z I- O a 0 cc 0 cc z W U) 0 C cc W N 0 D Oi
• co O O CO 1p t • V T O O O O O O O O O • LL U)V_ I- . C3 00. 0 z r O cc W I- cc 0 w 0Cl) z ------i------. T . cc w V O V (l) °~N 0 CL J z ----- 0 a ~ cc / Cn I - O z V Z 1•° 7 V LL w cc w w U // °° Q_ _1 O o D V ...... W Co v=, W 0 U) O E a) c U)
---...-.-.-.------J O T7 COL O Co 0 ~t N O O O O O 0 I system power factor or supply power factor/slip characteristics are presented in Fig. (3.9). In super-synchronous generating mode, generator power factor is leading, and line side converter requires lagging reactive power so overall system power factor improves as we increase line side converter firing angle a1. On account of recovery of slip power, system efficiency improves as shown in Fig. (3.10).
Full load performance of drive in super-synchronous generating mode is summarized in Table 3.3.
Table 3.3 : Full Load Performance of the Drive in Super-synchronous
Generating Mode
(All values in p.u.) Sator Current (rms) =0.707
S. No. Line side converter 900 1000 1100 1200 1300 1400 Firing angle al ---3
(1) (2) (3) (4) (5) (6) (7) (h)
1. Speed 1.075 1.249 1.417 1.576 1.719 1,842
2. Slip -0.075 -0.249 -0.417 -0.576 -0.719 -0.842
3. Torque -0.906 -0.906 -0.906 -0.906 -0.906 -0.906
4. dc link current 0.823 0.823 0.823 0.823 0.823 0.823
5. Rotor current (rms)
= Feedback current 0.907 0.907 0.907 0.907 0.907 0.907
6. Supply current (rms) 1.180 1.232 1.274 1.308 1.334 1.345
7. Supply power -0.849 -1.006 -1.158 -1.302 -1.432 -1.544
8. Stator power -0.848 -0.848 -0.848 -0.848 -0.848 -0.848
70 (1) (2) (3) (4) (5) (6) (7) (8)
9. Air gap power -0.906 -0.906 -0.906 -0.906 -0.906 -0.906
10. Slip power 0.000 0.157 0.310 0.453 0.583 0.695
11. Mechanical power -0.975 -1.132 -1.284 -1.428 -1.558 -1.670
12. Stator copper losses 0.058 0.058 0.058 0.058 0.058 0.058
13. Rotor copper losses 0.068 0.068 0.068 0.068 0.068 0.068
14. System efficiency 0.870 0.888 0.902 0.912 0.919 0.924
15. Stator power factor -0.848 -0.848 -0.848 -0.848 -0.848 -0.848
16. Supply power factor -0.508 -0.577 -0.642 -0.703 -0.760 -0.811
3.4.3 Super-synchronous Motoring
The steady state performance of the drive in this mode of operation has been computed for motor side converter firing angle set. at 180° and line side converter firing angle setting as 80°,
70°, 60° and 50°.
The torque/speed characteristics are presented in Fig.(3.11).
At al=90°, no load speed in synchronous speed. So, the
torque/speed characteristics is just a point at synchronous speed.
As al is reduced, no-load speed increases and modified
torque-speed characteristics which maintain parallelism with respect to each other from no load to full load are obtained. The
locus of full load operating point is marked on the figure by broken line. Full load operating point is identified when the
71 stator current becomes equal to rated current, the rms value of which is 0.707 p.u.. It is seen that for all firing angles a full load torque of 0.826 p.u. is obtained.
The nature of dc link current for different setting of al is shown in Fig. (3.12).
Here again the locus of full load operating points is seen to correspond to a fix value of dc link current of 0.833 p.u. From no-load to full load the dc link current is seen to vary almost linearly with speed. Since in this region torque also varies linearly with speed Fig. (3.11), torque is directly proportional to the dc link current.
The stator current/slip characteristics are presented in
Fig. (3.13). Under no-load condition the stator current is equal to the magnetising current which is independent of al setting.
Full load current is marked as 0.707 p.u.
The system power factor/slip characteristics are presented in
Fig. (3.14). In this mode, since the line side converter is operating at an angle less than 900, the real power component drawn from supply increases incontrast to the situation when firing angle was more than 900 (sub-synchronous motoring). This leads to include power factor as firing angle is reduced. Due to
72 F-- 0 co SQ Z_ cc W cc F 0 N r Q ~ 0 cc ao 7- Qo,(D c0 = c E r V . D 0 Ci Q a o z LO W O L m 0 LU cc
2 V, zU •-' W 'L 0 0 o U) T QoO cc J W r N D t~ . cj
c.
T LL
D Cr
10 O to T T Q
73 0
O) U T
1 0 Z_ W cc 1 0 U~- I- Q N. 0 o~ c Z Qa° _ U) U` D 0 C 0 n. a Y'° z WLU 0 o 0 cr 2 yJ U ZL 2 Z t
W '~0 N cr
r U W UJ f Z) N T C') Z J U
d N 0 N ( d' 0 (D N 00 d' O T C7 C7 N N I1 r T p O O 0 LL
74 U cn• 0 E zT- Mw O 0 F-0 1.1_0 Z> C E 0 J02 z CJ) D m 0 Z —c U 9 W L O Z o CO lI U)
U cl O a J w co E 0
C U) w CO U co T 0 M co C!) O d! O to 0 LO 0 U) O Q LL MJ M N N T— T-- O O
75 5 z D D 2
D z D I a
I
N O !)
O rn 00 t` CD LO d' C) tV 00000 O 0000
76 C/) U
0 cc z W cC I- 0 0 I- 0 cc 2 I Cl) U Z) 0 0 z 0 cc I U U z z LU V LL cc LL w LU U N C O 2 1 U L LJ
W E U) Co A r m C) 0 CD co P- CO L) 11 M N T T o 0 0 0 0 0 0 0 0 LL 77
recovery of slip power, system efficiency is greatly improved in
the whole super-synchronous range, as shown in Fig. (3.15).
Full load performance of drive is summarized in Table 3.4.
Table 3.4 Full Load Performance of the Drive in Super-synchronous Motoring Mode
(All values in p.u.) Sator Current (rms) = 0.707
S.No. Line side converter 800 700 600 50° Firing angle a1
1. Speed 1.109 1.296 1.471 1.630
2. Slip --0.109 -0.296 -0.471 -0.630
3. Torque 0.826 0.826 0.826 0.826
4. do link current 0.833 0.833 0.833 0.833
5. Rotor current (rms) = Feedback current 0.918 0.918 0.918 0.918
6. Supply current (rms) 1.219 1.266 1.303 1.331
7. Supply power 1,044 1.198 1.343 1.475
8. Stator power 0.884 0.884 0.884 0.884
9. Air gap power 0.826 0.826 0.826 0.826
10." Slip power -0.160 -0.314 -0.459 -0.591
11. Mechanical power 0.916 1.070 1.215 1.347
12. Stator copper losses 0.058 0.058 0.058 0.058
13. Rotor copper losses 0.070 0.070 0.070 0.070
14. System efficiency 0.877 0.893 0.905 0.913
15. Stator power factor 0.884 0.884 0.884 0.884
16. Supply power factor 0.606 0.669 0.729 0.783
iLI 3.4.4 Sub-synchronous Generating
The steady state performance of the drive in this mode of operation has been computed for motor side converter firing angle set at 1800 and line side converter firing angle setting as 80°,
70°, 60 and 50 as shown in Table 3.1.
The torque/speed characteristics are presented in Fig.(3.16).
At a1=900, slip power in the recovery loop is zero, and motor side converter is act as fully ON inverter. So in torque/speed characteristics, we will find a point at synchronous speed. As al is reduced, no-load speed reduces and modified torque-speed characteristics which maintain parallelism with respect to each other are obtained. The locus of full load operating point is marked on the figure by broken line. Full load operating point is identified when the stator current becomes equal to rated current, the rms value of which is 0.707 p.u.. It is seen that for all firing angles a full load torque of -0.907 is obtained.
The nature of dc link current for different setting of aI is shown in Fig. (3.17). Here again the locus of full load operating points is seen to corresponds to a fix value of do link current of
0.823. From no-load to full load the dc link current is seen to vary linearly with speed. Since in this region torque also varies linearly with speed Fig. (3.16), torque is directly proportional to the do link current.
79 Q
U) o F- U) U 00 cc Z o Q Q o a LU cr Z Q C3,Q) W C C CD o U o__ O nY D a W00 0 W .o 00 z O J O 0 Q `" _ C cc = O 2 co 0 o W o z •- rr ~ U Q U) o -' to I- Z) 0 o (1) 0 a) 0 ,__ v U-
0 O LO O O LO 0 1)O 0 N C M M 1 o. (~ T - (I)
cc O W.
F- M U w Z O cc
. •N. cc i W Z a W 0 a J
0 Z o 0 a c J cc Cl) cc O ~ r U U ~N0 Z Co o Z 0 -J.J o V 0 ci c O J c?j
O d .1 O O 0 C N 00 O (0 N co LL 00 •
U 0 O z cc
U~ W z W O I ~J D CIL 0 Q2 z \ Jo.0 0 C/) 1--c cc oC0 z ~~ z ac° a (l) .o0 co J E ti+ W
D/) cc C, O O
0 H L 0 W fo CO o C') O Ln O LO Ln 0' C3 (Y) N N r 0 0 C) LL
82 i
Z_
OC w Z w (-I, 0 (J) D a 0 Z a 0 oc C) R U w Z 0 V) O a.
D LL CL V) a. D ^0 CL C ) 0) c. T `o O CO LO `t C!) N r 0 0 0 0 0 0 o FL C') oO CO E 00 Ui co 1- 0 (D C'J z d cr cc 0 w z a- w —j CO 0 a) 00 a. 0 z LU 0 cc co LL. 00 d Li 0 LO Ui z ------C') Ui I- C') Co > 0 c'J C') d 0 LL OD co LO It c'J 1 - 0 0 06 C' 0* 000d •The stator current/slip characteristics are presented in
Fig. (3.18). Under no-load condition the stator current is equal to the magnetising current which is independent of al setting.
Full load current is marked to 0.707 p.u.
The system power factor or supply power factor/slip characteristics are presented in Fig. (3.19). In this mode of operation the reactive power drawn from supply has two components of opposite nature, whereas the generator reactive power is leading, the line side converter draws a lagging reactive power, whose magnitude reduces as the firing angle is the reduce. This results in overall increase in reactive power as firing angle is reduced. Hence the supply power factor drops at reduced firing angle. So when al reduces power factor of the system poorer. Due to recovery of slip power, system efficiency is greatly improved- in the whole range of speed as shown in Fig. (3.20).
Full load performance of drive is summarized in Table 3.5. Table 3.5 Full Load Performance of the Drive in Sub-synchronous Generating Mode
(All values in p.u.) Sator Current (rms) = 0.707
S.No. Line side converter 800 700 600 500 Firing angle al
1. Speed 0.901 0.733 0.575 0.432
2. Slip 0.099 0.267 0.425 0.568
3. Torque -0.907 -0.907 -0.907 -0.907
4. dc link current 0.823 0.823 0.823 0.823
5. Rotor current (rms)
= Feedback current 0.907 0.907 0.907 0.907
6. Supply current (rms) 1.118 1.048 0.970 0.885
7. Supply power --0,691 -0.538 -0.395 -0.266
8. Stator power -0.849 -0.849 -0.849 -0.849
.9. Air gap power -0.907 -0.907 -0.907 -0.907
10. Slip power -0.158 -0.311 -0.454 -0.583
11. Mechanical power -0.817 -0.664 -0.521 -0.391
12. Stator copper losses 0.058 0.058 0.058 0.058
13.. Rotor copper losses 0.068 0.068 0.068 0.068
14. System efficiency 0.845 0.810 0.758 0.680
15. Stator power factor -0.849 -0.849 -0.849 -0.849
16. Supply power factor -0.437 -0.363 -0.288 -0.212 0 N
O~00 U H
co l~lM w cc z U 0 cc cc N = w r U 0 0 CL W ow w T G) a 0 V! 00 2 0 W cc Z) 0 0 L 0 1
d
Q L
N T- 0 T- (V The complete torque/sped characteristics in four mode is shown by Fig. (3.21). It is readily seen that the drive can
perform in motoring as well as genrating mode over a wide speed
range.
3.5 CONCLUSIONS
In this Chapter computed steady state performance of the
static slip power recovery drive has been presented in the four
identified modes of operation. In each mode curves for torque, dc
link current, stator current, supply power factor and system
efficiency are presented for different settings of the firing
angles of the line and motor side converters. The performance is
found to be fairly satisfactory. CHAPTER - 4
CONCLUSION
The main contribution of present work is investigation of performance of static Kramer drive in (a) super-synchronous motoring mode and (b) regenerating mode both under sub- and super-synchronous speed range.
A general model of the drive has been developed which allows performance investigation for any combination of the firing angles settings of motor and line side converters. The model ignores saturation effect and effect of rotor current harmonics. Only copper losses in the machine and dc link filter are considered.
The effect of commutation overlap in the converters has also been ignored. These assumptions are commonly made in the analysis of the static Kramer drive [8].
For investigating steady-state performance through digital simulation, A 5 H.P. wound rotor induction motor has been considered. The simulation results for torque/speed characteristics under sub-synchronous motoring mode are found to exactly match with those published earlier for the same motoring by Krause, Wasynczuk and Hilderbrandity [8]. Performance of the drive is computed under sub- and super-synchronous speed range for both motoring as well as generating mode. Curves depicting variations of torque, dc link current, stator current, supply power factor and system efficiency are systematically presented and discussed for each case. The
performance under full load condition is tabulated in detail for each case. The full load condition is identified when the stator
is carrying rated current whose rms value is 0.707 p.u. The full
load performance is summarized in Fig. (4.1) for motoring mode operation in both sub- and super-synchronous speed range. The top curve presents variation in supply current over a speed range 0 -
1.75 p.u. This current varies from about 0.75 p.u. at zero speed
to about 1.2 p.u. at 1.7 p.u. speed. Since the stator current is always 0.707 p.u. (full load value), the additional component is
the feedback current present in the slip power recovery loop. The
middle curve represents system efficiency which is seen to be more
than 0.75 p.u. in good part of the considered speed range.
Efficiency drops significantly as speed approaches zero. The
bot-tom curve represents system power factor. The static Kramer drive is known to suffer from poor power factor. It is noted that
power factor steadily improves from about 0.2 t 0.6 as speed is
increased from 0 to 1.7 p. u.
The full load performance under generating mode is summarized
in Fig. (4.2). The supply current is found to vary in a manner U C N 0 U I W r W- E cc N D GO C) CO W N Ia T U II z Q 0
ii 0 LL O CC Z W a. CC o 0
J 0 J J 0 C L U r IC) N O 0. LL_
N O 00 CD N r r r do Q Q
91 O O
U
ti W r W
0 W V Z W LO Q O 1 0 2 O O o cw
a a. °Q fw w O z 0 TI J C'3W J 0 J 0 a) LL U N
LO Q CV a 0 ~ V
i • N O 00 CO I' CV r r r Q Q Q d •
92 almost similar to the motoring mode, discussed above. The system efficiency remains high in sub-synchronous and super-synchronous mode. The power factor variation is similar as in the motoring mode.
It is thus noted the static Kramer drive is capable of operating in a wide speed range with reasonably good efficiency both under motoring and regenerating modes.
FURTHER SCOPE OF WORK
The mathematical model developed in present work can be further improved to incorporate
1. Effect of magnetic saturation.
2. Effect of rotor current harmonics.
3. Effect of commutation overlap.
93 APPENDIX -
SPECIFICATION OF DRIVE MOTOR AND BASE VALUES
A1.1 INDUCTION MOTOR PARAMETERS
Volts 230/400 V; Ph 3;
CY 50 Hz; Pole 4;
KW 3.75; HP 5;
Amp 7.5; Rotor Volts 140 V;
Rotor Amps 22 A;
Rotor Connection Y;
Class of Insulation A
The per unit parameters of the drive are,
Xss = 3.0 Xrr = 3.0 XM = 2.9
R = 0.058 R = 0.072 ss rr Rf = 0.02 Xf = 1.0
A1.2 BASE VALUES FOR VARIOUS QUANTITIES
Unit current = Ibase - Peak value of rated phase current of
induction motor in amps (I p = 7.5 % = 10.6 Amps.
Unit Voltage = Vbase = Peak value of rated phase voltage of
induction motor in volts (V sp 400V
= 326.6 volts
Unit Impedance = Z = V / I S1 base base base = 326.6 / 10.6 Q
= 30.81 0
Jnit power = Rated Apparent Power = Pbase (3/2) Vsp sp I = (3/2) x 326.6 x 10.6
= 5. 193 KW
Unit Mechanical Speed = v = 2w = 2 x 2 x it x f base p p = 50 it rad./sec.
Unit torque = T P / v base base base
= 5.193 x 103 / (50 x it)
= 16.53 Nw-Metre
Full load stator current (rms value)
rated rms current (in amps) base current
= 7.5 = 0.707 p. u. x 7.5
95
Second equation becomes
ss lqs + XM (-irp Cos (0r+am)) ... (A2.3) 0 = X ss Ids + R
Let -vsp Cos ai = A2
Third equation becomes
A2 Cos er = (-sX') iqs + Reg1 (-irp Cos (er+am))
M- Reg2 (-ir,p Sin (9r+am)) ... (A2.4)
Fourth equation becomes
A2 Sin 0r = (sX') ids + Reg2 (-irp Cos (er+am))
M+ Re (-i Sin (8r +am )) (A2.5) q1 r P
Let e' = 0 + a (A2. 5a) r r m Combining equation (A2.2) and (A2.3), (A2.2) + j (A2.3) gives as
v R sp ss (ids + j i qs ) + j X ss (ids + j i qs ) + j XM (-irp). (Cos 0' + j Sin 0') or
vsp . LO = (R x sp - j XM irp Z8P (A2.6) ss + j ss ) i
Similarly, (A2.4) + j (A2.5) gives
A (Cos 8r + j Sin er) = j sX' r (ids + i i q s ) + Req 1 (-i p). [Cos 0' + j Sin 0P) + j Re q2 (-irp).[Cos 8r + j Sin 0'] r
A2 L@r = is X' isp + (Reg1 + j Reg2) (-irp) L8' (A2.7)
97
From (A2.7), 1 isp =[A L6r + (Reg1 + j Reg2) irp - r] 2 jsXM
substituting for isp in equation (A2.6) . v . LO = (R Xss) [A Ler + (Reg1 + j R ) ss + j eg2 1 irp z9] xM s ir. Zer js XM
,js X;, vsp 'I' _ [A2 L8r + (Reg1 + j Reg2) .irp L6r ] (R +jX ) ss ss js XM - (j XM lr Ler) X p IKSS TJ ss sp is XM v L-o = A LA -8' + (R + jR ) i (Rs s + jXss ) r 2 r r e~q1 eq2 rp
J2 XM XM s irp (R + j X ss ss S. XM XM = A /er-er+ (Re q1 + `)Re ~i 2) itP + irt p (Rss+`~Xss )
or Y'1 Le - 8'r = Y'2 + j Y'3 .. (A2.8) where Y - v sp s XM 1 R2 + X2 ss ss
• 6 = - tan-1 Xss 2n Rss
s.X X' R M M ss it Y2 = A2 2 Cos am + R lr + e q1 P (R +x2 ) P ss ss s.XM Y3 = - A Sin a XM Xss + Req 2 rpi r (R 2 + X2 I rp
ss ss or Y' =A' +y' --_ —.2
where A2 = A Cos am
s. XM X' R s s. Y7 = R e q 1 + (R2 + X2 ) = X2 Rss Re q + ( R l Cos a 2 + X2 )J m ss ss ss ss) and Y3 = - A Sin am + y8 lr P where M _ s.X XM Y8 = Req 2 Xss s + X2 ) ss ss 2 s . XM XM . X = sX' - ~8 Rf Sin am - s (R2 +X2 ss ss )
Comparing magnitudes on both sides in (A2.8) y /2 = y22 + Y1 2
2 (A2 + Y7 irp) + (-A Sin am + Yg irp)2
,2 = A' 2 + i2P + 2 A' Y' i rp + A2i S n2am
+ Y82 i2- 2 A Y8 S rp i n am rp
[Y'2 + y.12 ] i2 + [2 A Y' - 2 A 7 8 rp 2 7 2 Y'8 Sin am l irp
+ [A'2 - Y12 + A Sin2aml = 0 or [Y72 + Y,2] irp + [2 A2 Y7 - 2 A2 Y8 Sin am I it P
+ 1A - Y 12 ] = 0 or .2p + Y, irp + Y5 = 0 ... (A2.9)
(2 A2 Y -2 A2 Y8 Sin a) Y4 =______[ Y72+Y82______]
2 A2 (Cos am Y' - Yg Sin am) Y72 + Yg2
[A - Y' 3 Y5 = [Y12 + Y82]
- Y4 ± /Y42 - 4 Y~ So, i ... (A2.10) rp - 2
Comparing the angles in (A2.8)
0 - 0r = tan-1 (Y' / YZ) or 0' = 0 - tan-1 (Y3 / Y2) ... (A2.11) r
If Y' and Y2 are both negative, then
o - 0' = n + tan-1 (Y' / Y2 ) r 3 0' = e - n - tan 1 (Y3 / Y') ... (A2.12) r Thus, both 0' and i are found. r rp Now, From equation (A2.5a),
o = 9' - am, So 0r is known. r r Finding i sp From equation (A2.6)
sp . 10 + j XM i L@r
sp (R ss + J Xss)
100
vsp . LO + j XM irp (Cos 9r + j Sin 0'
(R ss + j Xss) (vsp - XM Sin 0' irk) + j (x irP Cos 0') r
SE; SS
Y9 + j Yi0 (Y9 + j Y10)(Rss j Xss) (A2.13) 2 2 Rss + j Xss (Rss + Xss)
where Y9 = vs - X Sin 0' i P rp Y10 XM it Cos 0' P
(Y' R + Y X) + j (Y ' ss 10' Rss Y9 Xss) isp z
• • • (A2. 14 ) Y11 + j Y12 where
Y11 = (Y9 Rss + Y' Xss) / Z __ , Y12 Y10 Rss - Y9 X)ss / Z
Here, 2 = (R2 + X2) ss ss
,2 ,2 SP I = Y11 + Y12 (A2.15)
= tan-1(Y12 / Y11) ... (A2.16)
If Y~ 1 and Y12 are both negative, then
= n + tan-1(Y12 / Y11 )
and ids = i Cos 0 = Y1 1 ... (A2.17) Sp iqs = 1 Sin 0 = Y12 ... (A2.18)
101 BIBLIOGRAPHY
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