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Chapter 3: Models for Power System Analysis
 STEADY-STATE MODEL OF GENERATOR
 STEADY-STATE MODEL OF TRANSFORMER
 PER-UNIT CALCULATIONS
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Cylindrical-Rotor synchronous generator
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Salient-pole Synchronous Generator
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For the simple models of generators for steady-state balanced operation
generators, like transformers and transmission lines, are represented with lumped
elements on substation busses.
SYNCHRONOUS GENERATORS
Large-scale power is generated by three-phase synchronous generators driven
either by steam turbines, hydroturbines, or gas turbines (prime movers).
The armature windings are placed on the stationary part called stator.
 The armature windings are designed for generation of balanced three-phase
voltages and are arranged to develop the same number of magnetic poles as the
field winding that is on the rotor.
Cross-sectional view of a
two-pole, salient-rotor,
three-phase synchronous
machine
Ref:http://www.ewh.ieee.org/soc/es/Nov1998/08/SYNCMACH.HTM
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 The field which requires a relatively small power (0.2-3 percent of the
machine rating) for its excitation is placed on the rotor.
 The rotor is also equipped with one or more short-circuited windings known
as damper windings.
The rotor is driven by a prime mover at constant speed and its field circuit is
excited by direct current.
 The excitation may be provided through slip rings and brushes by means of
dc generators (referred to as exciters) mounted on the same shaft as the rotor
of the synchronous machine.
 In modern excitation systems usually use ac generators with rotating
rectifiers, and are known as brushless excitation.
 The generator excitation system maintains generator voltage and controls
the reactive power flow.
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The rotor of the synchronous machine may be of cylindrical
or salient construction.
- The cylindrical type of rotor, also called round rotor, has
one distributed winding and a uniform air gap. These
generators are driven by steam turbines and are designed for
high speed 3600 or 1800 rpm (two- and four-pole
machines,respectively) operation.
The rotor of these generators has a relatively large axial
length and small diameter to limit the centrifugal forces. Roughly
70 percent of large synchronous generators are cylindrical rotor
type ranging from about 150 to 1500 MVA.
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The salient type of rotor has concentrated
windings on the poles and nonuniform air
gaps. It has a relatively large number of
poles, short axial length, and large
diameter. The generators in hydroelectric
power stations are driven by hydraulic
turbines, and they have salient-pole rotor
construction.
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 Normally synchronous machines are built as internal-field
machines. Machines with poles 2p = 2 have a round rotor
(cylindrical/turbo-rotor) because of high centrifugal forces,
while those with 2p = 4; 6; 8 and more poles mostly have a
salient-pole rotor.
 The stator carries the three phase winding and must be made
of laminated iron sheets in order to reduce eddy currents. Since
the flux in the rotor is constant with time at a particular place on
the rotor, the rotor can be built from massive steel.
 The excitation winding is generally supplied with DC through
the slip rings. In order to reduce oscillations in case of a network
fault, the machine has a damper winding.
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Damper or amortisseur windings are basically extra bars or coils
added to a synchronous machine rotor to 'damp' speed deviations.
The windings behave in the same fashion as the squirrel cage of an
induction machine. When rotor speed differs from the stator-side
electrical speed, currents are induced in the damper windings. These
currents set up a torque that has the effect of pulling the rotor back
toward synchronous speed. This is true whether the rotor is spinning
above
synchronous
or
below
synchronous
speed.
When the rotor is spinning at synchronous speed (i.e. zero slip), no
currents
are
induced
in
the
damper
windings.
Damper windings are commonly found on large, low-speed, salient
pole machines.
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GENERATOR MODEL
 In an elementary two-pole three-phase generator:
- The stator contains three coils, aa', bb', and cc', displaced
from each other by 120 electrical degrees.
- The concentrated full-pitch coils shown here may be
considered to represent distributed windings producing
sinusoidal mmf waves concentrated on the magnetic
axes of the respective phases.
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 When the rotor is excited to produce an air gap flux of 
per pole and is revolving at constant anguar velocity w, the
flux linkage of the coil varies with the position of the rotor
mmf axis wt, where wt is measured in electrical radians
from coil aa' magnetic axis.
- The flux linkage for an N-turn concentrated coil aa'
will be maximum ,
(a)max =N at wt= 0 and
(a)
=0 at wt=/2.
Assuming distributed winding, the flux linkage a will vary
as the cosine of the angle wt. Thus, the flux linkage with
coil a is
a = N cos wt
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 The voltage induced in coil aa‘ is obtained from Faraday's
law as
ea  
d
 wN  sin wt
dt
 E max sin wt


 E max cos  wt  
2

where
Emax=w N =2f N
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 Therefore, the rms value of the generated voltage is
E=4.44 f N
where f is the frequency in hertz.
In actual ac machine windings, the armature coil of each
phase is distributed in a number of slots. Since the emfs
induced in different slots are not in phase, their phasor
sum
is less than their numerical sum.
Thus, a reduction factor Kw, called the winding factor, must
be applied. For most three-phase windings having
distributed phase winding Kw= 0.85-0.95
Therefore, for a distributed phase winding, the rms value
of the generated voltage is
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 Therefore, for a distributed phase winding, the rms value
of the generated voltage is
E=4.44 Kw, f N
 The magnetic field of the rotor revolving at constant
speed induces three-phase sinusoidal voltages in the
armature, displaced by 2/3 radians.
 The frequency of the induced armature voltages depends
on the speed at which the rotor runs and on the number
of poles for which the machine stator is wound. The
frequency of the armature voltage is given by
f 
P n
2 60
where n is the rotor speed in rpm (synchronous speed.)
n/60 is the speed in revolutions per second.
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 During normal conditions, the generator operates synchronously
with the power grid. This results in three-phase balanced
currents in the armature.
Assume that the phase current ia is lagging the generated emf
ea by an angle , instantaneous armature currents are;
ia  I max sinwt   
2 

ib  I max sin wt   

3 

4 

ic  I max sin wt  

3 

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 Assume there exists a line mn in the space axis
separated from the rotor mmf axis by the same angle .
 Since ia is lagging ea by an angle , when line mn reaches
the axis of coil aa', current in phase a reaches its maximum
value [ia=Imax sin(wt-) and Fa=Kia=Imax sin(wt-) ].
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 At any instant of time, each phase winding produces
a sinusoidally distributed mmf wave with its peak
along the axis of the phase winding. These
sinusoidally distributed fields can be represented by
vectors referred to as space phasors.
ia  fa(); ib  fb();
ic  fc()
The amplitude of the sinusoidally distributed mmf
fa() is represented by the vector Fa along the axis of
phase a.
Similarly, the amplitude of the mmfs fb() and fc()
are shown by vectors Fb and Fc along their respective
axis.
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 At any instant of time, each phase winding produces a
sinusoidally distributed mmf wave with its peak along the
axis of the phase winding.
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 The mmf amplitudes are proportional to the instantaneous
value of the phase current, i.e.,
where K is proportional to the number of armature turns per
phase and is a function of the winding type.
 The resultant armature mmf is the vector sum of the above
mmfs. A suitable method for finding the resultant mmf is to
project these mmfs on line mn and obtain the resultant
in-phase and quadrature-phase components.
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 The resultant in-phase components are
Fa
using the trigonometric identity
sin  cos  =(1/2) sin2,
the above expression becomes
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 The above expression is the sum of three sinusoidal
functions displaced from each other by 2/3 radians,
which adds up to zero, i.e., F1=0.
 The sum of quadrature components results in
The sinusoidal terms of the above expression are displaced
from each other by 2/3 radians and add up to zero, with
F2 
3
Fm
2
Thus, the amplitude of the resultant (quadrature) armature
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mmf or stator mmf becomes
Fs  Fm
2
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Summarizing:
 The resultant armature mmf has a constant amplitude
perpendicular to line mn, and
 Rotates at a constant speed and in synchronism with the
field mmf Fr.
 When the rotor is revolving at synchronous speed and the
armature current is zero,
- the field mmf Fr, produces the no-load generated emf
E in each phase. The no load generated voltage which
is proportional to the field current is known as the
excitation voltage.
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The phasor voltage for phase a, which is lagging Fr by 90o,
is combined with the mmf vector diagram is shown in the
following figure.
mmfs are space
vectors, rotates with
constant speed wmech
whereas
emfs are time phasors
rotates with speed welec

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This combined phasor/vector diagram leads to a circuit
model for the synchronous machine:
- When the armature is carrying balanced three-phase
currents, Fs is produced perpendicular to line mn.
-The interaction of armature mmf, Fs and the field mmf, Fr
known as armature reaction, gives rise to the resultant
air gap mmf Fsr or,
-The resultant mmf Fsr is the vector sum of the field mmf,
Fr and the armature mmf, Fs.
-The resultant mmf, Fsr is responsible for the resultant air
gap flux sr, that induces the generated emf on-load, Esr.
-The armature mmf Fs induces the emf Ear, known as the
armature reaction voltage, which is perpendicular to Fs.
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 The voltage Ear, leads Ia by 90o and thus can be represented by a
voltage drop across a reactance Xar, due to the current Ia. Xar is known
as the reactance of the armature reaction.
 The phasor sum of E and Ear, is shown by Esr perpendicular to Fsr,
which represents the on-load generated emf.
E=Esr + j XarIa
The terminal voltage V is less than Esr by the amount of resistive
voltage drop RaIa and leakage reactance voltage drop Xl Ia. Thus;
E=V + [Ra+ j (Xl +Xar )]Ia
or
E=V + [Ra+ j Xs ]Ia
Xs=Xl +Xar is known as the
synchronous reactance. 25
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A simple per-phase model for a cylindrical rotor generator
based on the equation
E=V + [Ra+ j Xs ]Ia
is as shown in the following figure
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 The armature resistance is generally much smaller
than the synchronous reactance and is often neglected.
The equivalent circuit connected to an infinite bus
becomes that shown in figure reduces to
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 The phasor diagrams of the generator with terminal voltage
as reference for excitations corresponding to lagging, unity,
and leading power factors.
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 The voltage regulation of an alternator is used for
comparison with other machines. It gives an indication of the
change in field current required to maintain system voltage
when going from no-load to rated load at some specific
power factor.
The no-load voltage Vnl for a specific power factor may be
determined by operating the machine at rated load
conditions and then removing the load and observing the
no load voltage.
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POWER FACTOR CONTROL
Cylindrical Rotor
 Most synchronous machines are connected to large
interconnected electric power networks.
These networks have the important characteristic that
the system voltage at the point of connection is constant
in magnitude, phase angle, and frequency.
Such a point in a power system is referred to as an infinite
bus.That is, the voltage at the generator bus will not be
altered by changes in the generator's operating condition.
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The ability to vary the rotor excitation is an important
feature of the synchronous machine,
The effect of rotor excitation a variation
When the machine operates as a generator with constant
mechanical input power. neglecting the armature
resistance, the output power is equal to the power
developed, which is assumed to remain constant given by


P3   3 V I a  3 V I a cos 
where V is the phase-to-neutral terminal voltage assumed
to remain constant. Here, for constant developed power at
a fixed terminal voltage V Ia cos  must be constant.
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 Thus, the tip of the armature current phasor must fall on a
vertical line as the power factor is varied by varying the
field current as shown in the figure.
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The variation in the magnitude of armature current as the
excitation voltage is varied is best shown by a curve.
 Keeping the field current as the abscissa the curve of the
armature current as the function of the field current
resembles the letter V and is often referred to as the V
curve of synchronous machines.
These curves constitute one of the generator's most
important characteristics.
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POWER ANGLE CHARACTERISTICS
The three-phase complex power at the generator terminal is
S 3  3V I a
Expressing the phasor voltages in polar form, the
armature current is
Ia 
E V0
Zs
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
Substituting for I a results in
S3  3
EV
V2
(   )  3 
Zs
Zs
Thus, the real power P3 and reactive power Q3 are
P3  3
EV
V2
cos(   )  3 cos
Zs
Zs
Q3  3
EV
V2
sin(   )  3 sin
Zs
Zs
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If Ra is neglected, then Zs=jXs and =90o then these equations
can be written as
EV
P3  3
sin 
Zs
V
Q3  3
E cos   V 
Zs
If E and V are held fixed and the power angle  is
changed by varying the mechanical driving torque, the
power transfer varies sinusoidally with the angle . The
theoretical maximum power occurs when =90o
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The limit beyond which the excitation cannot be reduced.
when  = 90o.
Any reduction in excitation below the stability limit for a
particular load will cause the rotor to pull out of synchronism.
P3  3
EV
sin 
Zs
Pmax
P
Ia
E

V
0
90o
180o
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Control of the reactive power;
V
Q3  3 E cos   V 
Zs
for small , cos is nearly unity and the reactive power can
be approximated to
Q3  3
V
(E V )
xs
-When E>V the generator delivers reactive power to the bus,
and the generator is said to be overexcited.
-When E<V, the reactive power delivered to the bus is
negative; that is, the bus is supplying positive reactive power
to the generator.
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 Generators are normally operated in the overexcited mode
since the generators are the main source of reactive power
for inductive load throughout the system.
 The flow of reactive power is governed mainly by the
difference in the excitation voltage E and the bus bar
voltage V.
The adjustment in the excitation voltage E for the control of
reactive power is achieved by the generator excitation
system.
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SALIENT-POLE SYNCHRONOUS GENERATORS
The salient-pole rotor results in nonuniformity of the
magnetic reluctance of the air gap.
The reluctance along the polar axis  the rotor direct axis
is less than that along the interpolar axis  the quadrature
axis.
Therefore, the reactance has a high value Xd along the
direct axis, and a low value Xq along the quadrature axis
Xd>Xq
These reactances produce voltage drop in the armature
and can be taken into account by resolving the armature
current Ia into two components Iq, in phase, Id in time
quadrature, with the excitation voltage.
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The phasor diagram with the armature resistance neglected
is
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It is no longer possible to represent the machine by a
simple equivalent circuit. The excitation voltage magnitude
is
E  V cos   X d I d
The three-phase real power at the generator terminal is
P  3 V I a cos 
The power component of the armature current can be
expressed in terms of Id and Iq as follows:
Ia cos  = ab + de
= Iq cos  + Id sin 
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or the real power can be rewritten as
P  3 V ( I q cos   I d sin  )
V sin  = Xq Iq
V sin 
Xq
or
Iq 
from
E  V cos  X d I d
Id 
Id is given by
E  V cos
Xd
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Substituting for Id and Iq into
P  3 V ( I q cos   I d sin  )
the real power with armature current neglected becomes
P3  3
Xd  Xq
EV
sin   3 V 2
sin 2
Xd
2X d X q
The real power equation contains an additional term known
as the reluctance power.
For short circuit analysis, assuming a high X/R ratio, the
power factor approaches zero, and the quadrature
component of current can often be neglected. In such a
case, Xd merely replaces the Xq used for the cylindrical
rotor machine. Generators are thus modeled by their direct
axis reactance in series with a constant-voltage power
source.
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POWER TRANSFORMER
Power transformers are essential in power systems.
They are used to increase voltage level for transmission.
They are used to decrease voltage level for distribution and consumer use.
In modern utility systems there are five or more voltage transformations.
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A single voltage level is obtained by Referring
Referring is done either primary or secondary side
This simplifies analysis of systems with transformers
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EFFICIENCY and VOLTAGE REGULATION of POWER TRANSFORMER
efficiency
95% - 99% in real transformers
No referring
Referred to primary side
Referred to secondary side
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A typical 50 MVA three-phase power transformer
Ref: http://www.energy.siemens.com/hq/en/power-transmission/transformers/power-transformers/#content=Power%20Transformer%2050%20MVA
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THREE-PHASE TRANSFORMER CONNECTIONS
WYE-WYE
DELTA-DELTA
No phase-shift between
HV side and LV side
WYE-DELTA
DELTA-WYE
30-degrees phase-shift
between HV side and LV
side: HV side is leading
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COMMON CONNECTION CONFIGURATIONS
WYE-DELTA
DELTA-WYE
for step-down
for step-up
Advantages:
 High voltage side is grounded so the insulation requirements for
the high-voltage transformer windings are reduced
 One advantage of the Δ winding is that the undesirable third harmonic magnetizing current, caused by
the nonlinear core B-H characteristics, remains trapped inside the Δ winding.
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VOLTAGE CONTROL OF TRANSFORMERS
Voltage control is required for
o To compensate voltage drops
o To control reactive power flow over transmission line
TAP CHANGING TRANSFORMERS
 Off-load tap changing transformers
 Requires disconnection of transformer
 infrequent change in voltage ratio due to load growth or seasonal change
 Typically 4-taps each has 2.5 %, a total regulation of ±5 % of the nominal voltage
 TAP CHANGING UNDER LOAD (TCUL) TRANSFORMERS
 No requirement of disconnection of transformer
 frequent change in voltage ratio
 HV side: Typically 4-taps each has 2.5 %, a total regulation of ±5 % of the nominal voltage
 LV side: Typically 32-incremental step of 5/8 each, giving an automatic range of ±10 % of the
nominal voltage
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TAP SETTING EQUATION:
Transmission Line
Tap setting (in pu)
for sending-end side
Tap setting (in pu)
for receiving-end side
P: real power flow per phase
Q: reactive power flow per phase
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PER-UNIT (PU) SYSTEM
Advantages of PU system:
o Different voltage levels are disappeared to reduce a single level,
so the analysis of power system becomes easy.
o Physical quantities of the power system (voltage,power,current,impedance) are represented
as percentage or decimal fraction of base quantites.
Actual value
Base value
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PER-UNIT (PU) SYSTEM
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CHANGE OF BASE:
 required to match different base values on a common base value
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2-machine 6-bus system
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End of Chapter-3
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