Mrinal K Pal. Power system stability
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SYNCHRONOUS MACHINES
5-49
Since the machine is connected to a capacitive load,
qced
ixxe )(
−
−= (5.237)
dceq
ixxe )(
−
=
(5.238)
where x
e
is the external reactance and x
c
is the capacitive reactance.
From equations (5.236) - (5.238), we have
ced
q
d
xxx
e
i
−+
′
′
= (5.239)
ceq
d
q
xxx
e
i
−+
′
′
−
=
(5.240)
Substituting the expressions for i
d
and i
q
into equations (5.233) and (5.234) respectively, we have
q
ced
ced
fdqdo
e
xxx
xxx
Ee
dt
d
T
′
−+
′
−
+
−=
′′
(5.241)
and
d
ceq
ceq
dqo
e
xxx
xxx
e
dt
d
T
′
−+
′
−
+
−=
′′
(5.242)
The solutions to the above equations are
t
T
K
qo
t
T
K
fd
q
dodo
e
K
E
e
′
−
′
−
′
+−=
′
11
)1(
1
εε
(5.243)
and
t
T
K
dod
qo
ee
′
−
′
=
′
2
ε
(5.244)
where
ced
ced
xxx
xxx
K
−+
′
−+
=
1
and
ceq
ceq
xxx
xxx
K
−+
′
−
+
=
2
For stability, both K
1
and K
2
must be positive. Self excitation will occur when the system
parameters are such that K
1
and/or K
2
become negative. Conditions for self excitation are
therefore
edced
xxxxx
+
′
≥≥+ (5.245)
eqceq
xxxxx
+
′
≥≥+ (5.246)
It can be seen that as the capacitance is gradually increased starting with a very low value, self
excitation will first occur in the d axis, since the condition for self excitation will be satisfied first
in that axis.
5-50
Automatic voltage regulator action effectively reduces the direct axis synchronous reactance.
Therefore, in the presence of automatic voltage regulators self excitation may start in the q axis,
since the condition for self excitation may be satisfied first in that axis.
In the above analysis, variation of machine speed was neglected. Self excitation is usually
accompanied by considerable increase in machine speed. If the deviation from normal speed is
excessive it should be included in the analysis, since it will have a significant impact on the onset
of self excitation. It can be shown that at off-nominal speed the conditions for self excitation are
approximately given by
)()(
o
o
o
eded
LL
C
LL +
′
≥≥+
ω
ω
ω
ω
ω
ω
(5.247)
for the direct axis, and
)()(
o
o
o
eqeq
LL
C
LL +
′
≥≥+
ω
ω
ω
ω
ω
ω
(5.248)
for the quadrature axis,
where
ω
is the actual speed and
ω
o
is the nominal speed.
Problem
1. Derive the conditions for self excitation including the effect of machine speed variation.
2.
For the system shown in Figure 5.12 investigate the possibility of self excitation following
opening of the breaker at the receiving end. If condition for self excitation is found to exist,
estimate the size of the shunt reactor, to be placed at the high voltage side of the step-up
transformer, that will avoid it. Assume a 5% machine over-speed.
Fig. 5.12 A generator connected to a large system
through a long distance transmission line.
Another manifestation of the self excitation phenomenon caused by transients at subharmonic
frequencies that may arise due to the application of series capacitors is discussed in Chapter 9.
The analysis presented there is quite general and includes the self excitation discussed in this
section.
Derivation of Equations (5.144) - (5.146)
Rewriting the direct axis flux linkage and rotor voltage equations from (5.60) - (5.62),
considering one damper winding on each axis,
ddafdafdddd
ixixix
11
+
+
−
=
ψ
(5.249)
ddffdffddafdfd
ixixix
11
+
+
−
=
ψ
(5.250)
ddfddfddad
ixixix
111111
+
+
−
=
ψ
(5.251)
SYNCHRONOUS MACHINES
5-51
fdfdfdfd
ir
dt
d
e +=
ψ
ω
o
1
(5.252)
ddd
ir
dt
d
111
o
1
0
+=
ψ
ω
(5.253)
Substituting
dfd 1
and
ψ
ψ
from (5.250) and (5.251) into (5.252) and (5.253) respectively, then
eliminating
dfd
ii
1
and from the resulting equations, and using the Laplace operator
dt
d
s for , the
following is obtained.
dd
dododo
ddd
fd
dododo
kd
d
ix
sTTsTT
sTTsTT
E
sTTsTT
sT
2
1
2
2
2
1
)(1
)(1
)(1
1
′′′
+
′
++
′′′
+
′
++
−
′′′
+
′
++
+
=
ψ
(5.254)
where
fddddodo
ETTTT and ,,,
′′′′′′
are as defined earlier, and
d
d
d
d
kd
r
x
T
r
x
T
1o
11
1
1o
1
,
ωω
==
d
d
ad
d
r
x
x
x
T
1o
2
11
2
ω
−
=
Note that (5.254) can also be approximated as
dd
dodo
dd
fd
dodo
kd
d
ix
sTsT
sTsT
E
sTsT
sT
)1()1(
)1()1(
)1()1(
1
′′
+
′
+
′
′
+
′
+
−
′′
+
′
+
+
=
ψ
(5.255)
(The expression for
ψ
d
in this form will be used in Chapter 9.)
Similarly, for the quadrature axis
qq
qo
q
q
ix
sT
sT
′′
+
′
′
+
−=
1
1
ψ
(5.256)
Neglecting the
ψ
&
terms and speed deviations, the armature voltage equations are
dqd
ire
−
−=
ψ
(5.257)
qdq
ire
−
=
ψ
(5.258)
From (5.256) and (5.117)
)()(
qqqqqqqo
ixixsT
+
−
=
′
′
+
′′
ψ
ψ
Setting
dqqq
eix
′
′
−
=
′
′
+
ψ
(5.259)
dqqqdqo
eixxesT
′
′
−
′
′
−
=
′′′′
)( (5.146)
Similarly, from (5.254), and using (5.114) and (5.115),
[
]
)()()()(
21
2
dddfdfdkddddddoddddodo
ixEETixTTTTsixsTT +−=−
′
++
′
++
′′
+
′′′
ψψψ
SYNCHRONOUS MACHINES
5-52
Setting
qddd
eix
′
′
=
′′
+
ψ
(5.260)
the above can be written as
[
]
qdddfdfdkddddddqdoqdodo
eixxEETixTTixeTTsesTT
′′
−
′′
−−=−
′
++
′′
−
′′′
++
′′′′′
)()()()(
21
2
Using an auxiliary variable
[]
fdkddddddqdo
do
qdoq
ETixTTixeTT
T
esTe −
′
++
′′
−
′′′
+
′
+
′′′′
=
′
)()()(
1
21
it follows that
qdddfdqdo
eixxEesT
′
′
−
′
′
−
−
=
′′
)(
(5.261)
and, using (5.115),
[
]
fdkdddddq
do
qdddqqdo
ETixTixeT
T
eixxeesT −+
′′
−
′′
′
−
′′
−
′′
−
′
−
′
=
′′′′
21
)(
1
)( (5.262)
Equations (5.261) and (5.262) can be written in a slightly modified form after neglecting the less
significant fourth term on the right hand side of equation (5.262) and approximately
compensating its effect by changing
ddqq
xxee
′
′
′
′
′
′
toandto in (5.261) to obtain equations (5.144)
and (5.145). Numerous tests have shown this to be a valid approximation.
From (5.257) – (5.260)
qqddd
ixiree
′
′
−
+
=
′′
(5.263)
ddqqq
ixiree
′
′
+
+
=
′′
(5.264)
References
1. C. Concordia, Synchronous Machines, Theory and Performance, Wiley, N.Y. 1951.
2. E.W. Kimbark, Power System Stability, Vol.3, Wiley, N.Y. 1956.
3. B. Adkins, The General Theory of Electrical Machines, Chapman and Hall, 1957.
4. R.H. Park, "Two-Reaction Theory of Synchronous Machines -- Generalized Method of Analysis," Pt.1, AIEE
Trans., Vol.48, pp.716-30, 1929.
5. A.W. Rankin, “Per Unit Impedances of Synchronous Machines," AIEE Trans., Vol.64, Pt.1 pp.569-72, Pt.2
pp.839-41, 1945.
6. G. Shackshaft, “General Purpose Turbo-Alternator Model,” Proc. IEE, Vol.110, No.4, April, 1963.
7. D.G. Taylor, “Representation of Synchronous Machines in Transient Stability Studies,” Proceedings of the
Conference on Analytical Methods for Power System Design and Operation for Use with Digital Computers,
held at Queen Mary College, University of London, September, 1963.
8. D.G. Taylor, “Analysis of Synchronous Machines Connected to Power System Networks,” Proc. IEE,
Vol.109C, pp.606-610, July 1962.
9. M. K. Pal, "Synchronous Machine Representation in Power System Stability Studies," English Electric Co.
Report NSp 85, 1967.
10. M.K. Pal, "Mathematical Methods in Power System Stability Studies," Ph.D. Thesis, University of Aston in
Birmingham, 1972.
11. C.C. Young, "Equipment and System Modeling for Large-Scale Stability Studies," IEEE Trans., PAS - 91,
pp.99-109, 1972.
12. Power Technologies, Inc. Course Notes, “Electrical Machine Dynamics – I,” 1976.
6-1
CHAPTER 6
EFFECT OF EXCITATION CONTROL ON STABILITY
When a power system is subjected to a large disturbance such as a fault, the terminal voltages of
the generators are depressed significantly and the generators' ability to transfer electrical power
is reduced. Fast acting excitation controls can aid in system stability by quickly boosting the
generators' field, thereby holding the terminal voltages to as high a level as possible. During the
brief period immediately following a fault an excitation system with high speed and high ceiling
voltage will have the most beneficial effect on the system. Following the clearing of the fault, if
the machines have survived the first impact of the disturbance, they will go into oscillations.
Complex interactions among the various machines and excitation and other control equipment
can, under certain conditions, cause the oscillations to grow resulting in system instability. The
requirement on the performance of the excitation control during this period is quite different
from that during the initial period, and these requirements are often conflicting.
Under small-disturbance conditions, such as a load impact, the situation is similar to that existing
beyond the initial period following a large disturbance, after the system has withstood the initial
shock of the disturbance and has entered into a state of oscillation. For satisfactory performance
it is necessary that the system and control parameters are such that sufficient damping is
provided.
Since for small-disturbance studies we are concerned with small excursions of system variables
about their steady state values, linearization of system equations is permissible and we can study
the resulting linear system using the methods of linear system analysis.
Effect of Excitation on Generator Power Limit
In order to illustrate the potential of the excitation system in extending stability limit, let us
consider the system shown in Figure 1, where a generator is supplying power to a large system
represented by an equivalent machine of zero impedance and infinite inertia. The machine and
system parameter values are as shown in the figure.
Fig. 6.1 A one machine-infinite bus system.
The power output of the machine is given by
δδ
sin
2
sin
21
21
21
EE
XX
EE
P =
+
=
(6.1)
where
21
δ
δ
δ
−=
The phasor diagram at a given load at unity terminal voltage and power factor is shown in Figure
6.2. At this operating condition E
1
and E
2
are equal as shown on the phasor diagram. If E
1
and E
2
EFFECT OF EXCITATION ON STABILITY
6-2
are held constant at these values, the maximum power transfer occurs when
δ
= 90° as seen from
equation (6.1).
Fig. 6.2 Phasor diagram for unity terminal voltage and power factor.
Now assume perfect excitation control on both machines such that the terminal voltage, V
t
, and
the power factor are maintained at unity. This will be done by adjusting E
1
and E
2
instantaneously following any change in the load level.
From the phasor diagram of Figure 6.2,
2/
21
δ
δ
δ
=
=
, and therefore
2/cos
1
2/cos
21
δδ
===
t
V
EE
(6.2)
Substituting (6.2) into (6.1), we obtain
2/tansin
2/cos2
1
2
δδ
δ
==P
(6.3)
From equation (6.3) we can see that the power transfer can increase without limit. Of course, the
excitation control as assumed above is not realizable in practice, since there is always a time lag
in the excitation response. Also, the excitation control of the infinite bus voltage is not practical.
There is also a maximum limit to which a machine's internal voltage can be raised.
Now consider a more practical situation where the infinite bus voltage, E
2
, is held constant and
the power factor is allowed to vary. Assuming both the terminal voltage, V
t
, and the infinite bus
voltage, E
2
, held constant at 1.0 pu, we can draw the phasor diagram as shown in Figure 6.3.
Fig. 6.3 Phasor diagram for unity terminal and infinite bus voltage.
From the phasor diagram, since V
t
and E
2
are held constant and equal to 1.0 pu,
EFFECT OF EXCITATION ON STABILITY
6-3
)2/sin(2/
2
δ
t
VI =
or
)2/sin(2
2
δ
=
I (6.4)
E
1
can be obtained by adding the IX
1
drop to the terminal voltage V
t
. Taking V
t
as reference
)2/cos()2/sin(1
)2/sin()2/cos([1
22
2211
δδ
δ
δ
jII
jIIjXE
++=
−
+=
(6.5)
Since V
t
and E
2
are held constant, it can be seen that the maximum power transfer occurs, from
the relationship
2
2
2
sin
δ
X
VE
P
t
=
, when
o
90
2
=
δ
.
Therefore, at maximum power transfer,
2/2/1
1
jIIE ++=
From (6.4)
22/2 ==I
Therefore
o
6.26235.22
1
∠=+= jE
and
ooo
6.1166.2690 =+=
δ
Thus, in this case there is a limit to the power that can be transmitted and the maximum power
transfer occurs at a value of
δ
well over 90°. Again, in this illustration instantaneous control of
the excitation voltage, E
1
, has been assumed.
Effect of Excitation Control on Small-Disturbance Performance of Synchronous Machines
We will now analyze the effect of excitation control on the small-disturbance performance of a
synchronous machine connected to an infinite bus through an external reactance. We will study
the effects of exciter gain and time constants on machine stability. A more rigorous synchronous
machine model than used in the previous elementary discussion will be employed. The
synchronous machine model used is based on the two-axis representation as discussed in Chapter
5. The effects of damper windings will first be neglected. The natural damping caused by the
various damper windings will be evaluated later on.
Neglecting damper windings, the equations for the synchronous machine (model 3) are as
follows:
qdddfd
q
do
eixxE
dt
ed
T
′
−
′
−−=
′
′
)( (6.6)
e
d
m
P
dt
d
K
P
dt
dH
−−=
δ
ω
δ
ω
o
2
2
o
2
(6.7)
qddqqqe
iixxieP )(
′
−
+
′
= (6.8)
qqd
ixe =
(6.9)
EFFECT OF EXCITATION ON STABILITY
6-4
ddqq
ixee
′
−
′
= (6.10)
We also have
δ
sin
bqed
Vixe
+
−
=
(6.11)
δ
cos
bdeq
Vixe
+
= (6.12)
The phasor diagram of the synchronous machine-infinite bus system is shown in Figure 6.4.
Fig. 6.4 Phasor diagram of a synchronous machine connected to infinite bus through a reactance.
The terminal voltage is given by
222
qdt
eeV += (6.13)
From (6.10) and (6.12), we have
de
b
de
q
d
xx
V
xx
e
i
′
+
−
′
+
′
=
δ
cos
(6.14)
Similarly, from (6.9) and (6.11), we have
qe
b
q
xx
V
i
+
=
δ
sin
(6.15)
For small-disturbance analysis the equations can be linearized about the operating point.
Linearizing equations (9.6) through (6.10) and (6.13) through (6.15), we have
qdddfd
q
do
eixxE
dt
ed
T
′
∆−∆
′
−−∆=
′
∆
′
)( (6.16)
e
d
m
P
dt
d
K
P
dt
dH
∆−
∆
−∆=
∆
δ
ω
δ
ω
o
2
2
o
2
(6.17)
][)(
qdqddqqqqqe
iiiixxieieP
∆
+
∆
′
−
+
∆
′
+
′
∆=
∆
(6.18)
qqd
ixe
∆
=
∆ (6.19)
ddqq
ixee
∆
′
−
′
∆
=
∆ (6.20)
qqddtt
eeeeVV
∆
+
∆
=
∆ (6.21)
EFFECT OF EXCITATION ON STABILITY
6-5
δ
δ
∆
′
+
+
′
∆
′
+
=∆
de
b
q
de
d
xx
V
e
xx
i
sin
1
(6.22)
δ
δ
∆
+
=∆
qe
b
q
xx
V
i
cos
(6.23)
i
d
, i
q
, e
d
, e
q
, etc. are the values at the operating point and the prefix ∆ denotes the small changes
in the variables about the operating point.
Substituting (6.22) and (6.23) into (6.18), we have
[]
∆
′
+
+
′
∆
′
+
′
−+
∆
+
′
−+
′
+
′
∆=∆
δ
δ
δ
δ
de
b
de
qdq
qe
b
ddqqqqe
xx
V
xx
ixx
xx
V
ixxeieP
sin
e
1
)(
cos
)(
q
which can be written as
qe
eKKP
′
∆
+
∆
=∆
21
δ
(6.24)
where
[]
δ
δ
sin
cos
)(
1 bq
de
dq
qe
b
ddqq
Vi
xx
xx
xx
V
ixxeK
′
+
′
−
+
+
′
−+
′
= (6.25)
q
de
qe
i
xx
xx
K
′
+
+
=
2
(6.26)
Substituting (6.22) into (6.16) and using the Laplace variable s for d/dt, we have
δ
∆−
′
∆−∆=
′
∆
′
4
3
1
Ke
K
EesT
qfdqdo
where
de
de
xx
xx
K
+
′
+
=
3
(6.27)
and
δ
sin
4 b
de
dd
V
xx
xx
K
′
+
′
−
= (6.28)
The above equation can be rearranged as
δ
∆
′
+
−∆
′
+
=
′
∆
sTK
KK
E
sTK
K
e
do
fd
do
q
3
43
3
3
11
(6.29)
Substituting (6.22) and (6.23) into (6.20) and (6.19), respectively, and then the results in (6.21),
we obtain
qt
eKKV
′
∆
+
∆
=∆
65
δ
(6.30)
where
EFFECT OF EXCITATION ON STABILITY
6-6
δδ
sincos
5 b
t
q
de
d
b
t
d
qe
q
V
V
e
xx
x
V
V
e
xx
x
K
′
+
′
−
+
=
(6.31)
and
t
q
de
e
V
e
xx
x
K
′
+
=
6
(6.32)
It may be seen that all the constants K
1
through K
6
, except K
3
, are functions of the machine
operating point. Since, as we will see, the stability characteristics are dependent on these
constants, the system performance can be quite different at different operating points for the
same exciter settings. Note that K
5
can become negative under heavily loaded condition (i.e., for
high values of δ).
The initial values of the various variables at an operating point may be calculated from the given
operating condition as follows.
Referring to Figure 6.4, let the power output P, the terminal voltage V
t
, and the infinite bus
voltage V
b
be given. Then
α
sin
e
bt
x
VV
P =
(6.33)
e
btt
x
VVV
Q
α
cos
2
−
=
(6.34)
Therefore
bt
e
VV
xP
1
sin
−
=
α
(6.35)
and
P
Q
1
tan
−
=
φ
(6.36)
Also
IVjQP
t
ˆ
=−
, with V
t
as reference
−+=
−=
∴
tt
qtq
tt
V
Q
j
V
P
jxVE
V
Q
j
V
P
I
ˆ
ˆ
QxV
Px
qt
q
+
=
∴
−
2
1
tan
θ
(6.37)
α
θ
δ
+
= (6.38)
Also
φ
cos
t
V
P
I
= (6.39)
)sin(
φ
θ
+
=
∴
Ii
d
(6.40)