Mrinal K Pal. Power system stability
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TURBINE-GENERATOR SHAFT TORSIONALS
9-29
q
d
qd
n
n
qd
n
n
e
e
ee
A
Eee
A
E
1
22
o
o
22
o
o
tan
2
,
2
−
−+
−+
==
+
−
=∆+
+
=∆
φφ
ωω
ω
ω
ωω
ω
ω
as derived earlier (see equation (9.34).
Expressions for phases
b and c will be similar but with phases displaced by −120°and +120°,
respectively.
Equation (9.89) shows that the phase voltage is composed of two components -- one at
supersynchronous frequency
ω
o
+
ω
n
, and the other at subsynchronous frequency
ω
o
−
ω
n
. In the
above and subsequent expressions the subscripts
+ and − denote supersynchronous and
subsynchronous quantities, respectively.
The supersynchronous and subsynchronous voltages will drive currents at the corresponding
frequencies. For phase
a the currents are
[]
++
+
+
+
−−+
∆
=∆
θφωω
t
Z
E
i
n
N
a
)(cos
o
(9.90)
[]
−−
−
−
−
−−−
∆
=∆
θφωω
t
Z
E
i
n
N
a
)(cos
o
(9.91)
and
−+
∆
+
∆=∆
aaa
iii (9.92)
where
Z
N+
and Z
N−
are the impedances at supersynchronous and subsynchronous frequencies,
looking into the network from the generator terminal (the driving point impedances).
Z
N+
and Z
N−
can be obtained from a frequency scan of the system network.
θ
+
and
θ
−
are the impedance angles.
Similar expressions can be obtained for phases
b and c.
We now transform back to
d-q components. From equation (9.47) we have
[
]
)120cos()120cos(cos
3
2
oo
++−+=
θθθ
cbad
eeee
Linearizing the above expression
[]
[]
)120cos()120cos(cos
3
2
)120sin()120sin(sin
3
2
oo
oo
+∆+−∆+∆+
∆++−+−=∆
θθθ
θθθθ
cba
cbad
eee
eeee
Using (9.47) and (9.89) the above reduces to
)cos()cos(
−−++
+
∆
+
−
∆=∆−∆
φ
ω
φ
ω
δ
tEtEee
nnqd
(9.93)
TURBINE-GENERATOR SHAFT TORSIONALS
9-30
Similarly,
)cos()cos(
)sin()sin(
−−++
−−++
+∆+−∆−=
+
∆
−
−
∆=
∆
+∆
φωφω
φ
ω
φ
ω
δ
tEjtEj
tEtEee
nn
nndq
(9.94)
The corresponding expressions for currents are
)cos()cos(
−−
−
−
++
+
+
++
∆
+−−
∆
=∆−∆
θφωθφωδ
t
Z
E
t
Z
E
ii
n
N
n
N
qd
(9.95)
and
)cos()cos(
−−
−
−
++
+
+
++
∆
+−−
∆
−=∆+∆
θφωθφωδ
t
Z
E
jt
Z
E
jii
n
N
n
N
dq
(9.96)
In equations (9.93) - (9.96) the voltages and currents are expressed in terms of supersynchronous
and subsynchronous components. Denoting )cos(
++
−
∆
φ
ω
tE
n
by E
+
and )cos(
−−
+
∆
φ
ω
tE
n
by
E
−
, and similarly for currents, it is easy to see that
+
+
+
=
N
Z
E
I
(9.97)
and
∗
−
−
−
=
N
Z
E
I
(9.98)
where
∗
−
N
Z is the complex conjugate of Z
N−.
Equations (9.97) and (9.98) can be written in matrix form as
=
−
+
∗
−
+
−
+
I
I
Z
Z
E
E
N
N
(9.99)
Note that
−+
+
=
∆−∆ EEee
qd
δ
and
−+
+
−
=
∆
+
∆
jEjEee
dq
δ
, from (9.93) and (9.94), and
similarly for currents.
Therefore, equation (9.99) is manipulated as follows:
−
−
−
=
−
−
+
−
∗
−
+
−
+
I
I
jjjj
Z
Z
jj
E
E
jj
N
N
11111111
1
which yields
()()
()()
∆+∆
∆−∆
+−
−
−
+
=
∆+∆
∆−∆
+
∗
−+
∗
−
+
∗
−+
∗
−
δ
δ
δ
δ
dq
qd
NNNN
NNNN
dq
qd
ii
ii
ZZZZ
j
ZZ
j
ZZ
ee
ee
2
1
2
22
1
(9.100)
Note that in the case of the simple system of one generator connected to an infinite bus through a
series compensated transmission line as shown in Figure 9.8,
TURBINE-GENERATOR SHAFT TORSIONALS
9-31
Fig. 9.8 A generator connected to an infinite bus through a series compensated transmission line.
C
n
n
N
C
n
n
N
XjXjRZ
XjXjRZ
ωω
ω
ω
ωω
ωω
ω
ω
ω
ω
−
+
−
−=
+
−
+
+=
−
+
o
o
o
o
*
o
o
o
o
()
()
22
o
2
o
*
22
o
2
o
o
*
2
2
1
n
C
NN
n
Cn
NN
X
XZZ
j
X
XjRZZ
ωω
ω
ωω
ω
ω
ω
−
−=−
−
++=+
∴
+−
+−
and
∆+∆
∆−∆
−
++
−
−
−
−−
−
++
=
∆+∆
∆−∆
δ
δ
ωω
ω
ω
ω
ωω
ω
ωω
ω
ωω
ω
ω
ω
δ
δ
dq
qd
n
Cn
n
C
n
C
n
Cn
dq
qd
ii
ii
X
XjR
X
X
X
X
X
XjR
ee
ee
22
o
2
o
o
22
o
2
o
22
o
2
o
22
o
2
o
o
(9.101)
Equation (9.101) can be derived directly for the simple one-machine system by writing the
system equations in
d-q components (given earlier) using the Laplace operator s for d/dt,
linearizing the equations, and then substituting
j
ω
n
for s. This is left as an exercise.
We will now develop the necessary equations for the generator. By eliminating the rotor currents
the generator stator flux linkages may be expressed as
fdddd
EsGisx )()(
+
−=
ψ
(9.102)
qqq
isx )(−=
ψ
(9.103)
where
()
(
)
()()
(
)
(
)
()()
sTsT
sTsT
xsx
sTsT
sTsT
xsx
qoqo
qq
qq
dodo
dd
dd
′′
+
′
+
′′
+
′
+
=
′′
+
′
+
′′
+
′
+
=
11
11
)(,
11
11
)(
()()
sTsT
sT
sG
dodo
kd
′′
+
′
+
+
=
11
1
)(
The symbols used in the above expressions have the usual meanings.
x
d
(s) and x
q
(s) are known as
the direct and quadrature axis operational impedances, respectively.
We will also need the stator voltage and air-gap torque equations, which are re-stated below.
TURBINE-GENERATOR SHAFT TORSIONALS
9-32
dqdd
irse −−=
ψ
ω
ω
ψ
ω
oo
1
(9.104)
qdqq
irse −+=
ψ
ω
ω
ψ
ω
oo
1
(9.105)
dqqde
iiT
ψ
ψ
−
= (9.106)
Equations (9.102) - (9.105) can be linearized, assuming fixed field voltage (note that
conventional excitation system will not respond to torsional interaction), and written in matrix
form as
o
o
o
)()(
)()(
ω
ω
ψ
ψ
ω
ω
∆
−
+
∆
∆
+
−+
−=
∆
∆
d
q
q
d
qd
qd
q
d
i
i
sx
s
rsx
sxsx
s
r
e
e
(9.107)
In the steady state we have, from equations (9.102) to (9.105),
qdqdqd
qqqddfdd
ireire
ixixE
−=−−=
−
=
−=
ψψ
ψ
ψ
,
,
Therefore, (9.107) can be rearranged as
[]
()
[]
[]
()
[]
δ
ω
ω
δ
δ
ω
ω
δ
δ
∆
−−+−
−−−−
+
∆+∆
∆−∆
+
−+
−=
∆+∆
∆−∆
dqdfdqdq
dqdfdqdq
dq
qd
qd
qd
dq
qd
isxxE
s
isxx
isxxEisxx
s
ii
ii
sx
s
rsx
sxsx
s
r
ee
ee
)()(
)()(
)()(
)()(
o
o
o
o
(9.108)
Linearizing (9.106), and using (9.102) and (9.103),
[
]
(
)
[
]
qdqdfddqdqe
iisxxEiisxxT
∆
−
−
+
∆
−=∆ )()(
(9.109)
At shaft torsional frequencies x
d
(s) and x
q
(s) are approximately equal and each may be assumed
to be equal to their average value. At oscillation frequency
ω
n
, equations (9.108) and (9.109)
may be written as
+
∆+∆
∆−∆
−=
∆+∆
∆−∆
b
a
ii
ii
jZ
ee
ee
dq
qd
nG
dq
qd
δ
δ
ω
δ
δ
)( (9.110)
where
TURBINE-GENERATOR SHAFT TORSIONALS
9-33
+
−+
=
)()(
)()(
)(
o
o
nd
n
nd
ndnd
n
nG
jxjrjx
jxjxjr
jZ
ω
ω
ω
ω
ωω
ω
ω
ω
[]
()
[]
δωω
ω
ω
∆
−−−−=
dnddfdqndq
n
ijxxEijxx
j
a )()(
o
[]
()
[]
δω
ω
ω
ω
∆
−−+−=
dnddfd
n
qndq
ijxxE
j
ijxxb )()(
o
and
[
]
(
)
[
]
qdnddfddqndqe
iijxxEiijxxT
∆
−
−
+
∆
−=∆ )()(
ω
ω
(9.111)
Note that
()()
()()
+−
−−+
=
+−+−
+−+−
GGGG
GGGG
nG
ZZZZ
j
ZZ
j
ZZ
jZ
**
**
2
1
2
22
1
)(
ω
(9.112)
where
)(
)(
o
o
o
o
nd
n
G
nd
n
G
jxjrZ
jxjrZ
ω
ω
ωω
ω
ω
ω
ω
−
−
+=
+
+=
−
+
and
)()(
*
ndnd
jxjx
ωω
−=
From (9.100) and (9.110) we have
()()
()()
=
∆+∆
∆−∆
+−
−−+
+−+−
+−+−
b
a
ii
ii
ZZZZ
j
ZZ
j
ZZ
dq
qd
δ
δ
**
**
2
1
2
22
1
where
−−−+++
+
=
+=
GNGN
ZZZZZZ ,
which yields
+
−−
−
+
=
∆+∆
∆−∆
−
+
−
+
−
+
−
+
b
a
Z
Z
Z
Z
j
Z
Z
j
Z
Z
ii
ii
dq
qd
**
**
11
2
111
2
11
2
11
2
1
δ
δ
(9.113)
Substituting the expressions for ∆i
d
and ∆i
q
from (9.113) into (9.111), the air-gap torque can be
expressed as
TURBINE-GENERATOR SHAFT TORSIONALS
9-34
δ
ω
∆
=∆ )(
ne
jKT (9.114)
The imaginary part of the above expression will be indicative of electrical damping. Note that the
damping will depend not only on the system parameters but also on the operating point. The
computation of electrical damping from equations (9.111) and (9.113) is quite straightforward
once a frequency scan of the system network is obtained. Note that it is not necessary to set x
d
(s)
= x
q
(s), although this allows some simplification in that the generator and network impedances
can be combined.
As a specific example, consider the generator under no-load. At no-load,
qe
n
fd
qd
iT
jb
a
E
ii
∆=∆
∆=
∆−=
∴
≈
=
=
δ
ω
ω
δ
o
0.1
0
From (9.113)
()
δ
ω
ω
δ
∆
++∆−
−−=∆
−
+
−
+
o
**
11
2
111
2
n
q
j
Z
Z
Z
Z
j
i
Therefore, the electrical damping is obtained as
+
−
+
+=
−
−
−
−
+
+
+
+
2
o
22
o
2
o
2
1
2
1
Z
R
Z
R
Z
R
Z
R
D
nn
n
e
ω
ω
ω
ω
ω
ω
or
2
o
2
o
2
1
2
1
−
−
+
+
−
−
+
=
Z
R
Z
R
D
n
n
n
n
e
ω
ω
ω
ω
ω
ω
(9.115)
The second term is usually more dominant and therefore the electrical damping due to torsional
interaction tends to be negative. Also, when the subsynchronous frequency
ω
o
−
ω
n
(the
complement of the torsional natural frequency
ω
n
) approaches the network resonant frequency,
X
−
= 0. Under these conditions the negative damping can reach a high magnitude, since the
resistance is usually small.
Note that if the magnitude of the negative electrical damping exceeds the modal mechanical
damping corresponding to that particular frequency, the combined mechanical-electrical system
will be unstable.
The method described above can also be used in the analysis of the phenomenon of hunting (the
oscillation of the combined turbine-generator system with respect to the power system), which is
usually in the .05 to 2 Hz range, with or without series capacitor in the system. However, since
ω
n
is small, the approximation x
d
(j
ω
n
) = x
q
(j
ω
n
) cannot be used.
TURBINE-GENERATOR SHAFT TORSIONALS
9-35
Self-Excitation
Whenever there is a perturbation in a series compensated system, transient subsynchronous
currents of both positive and negative sequence will flow in the network. The frequency of the
subsynchronous currents will be given by
L
C
e
X
X
o
ωω
=
where X
L
and X
C
are the total equivalent series inductive and capacitive reactances, respectively.
Considering the positive sequence subsynchronous current, since the generator rotor is rotating at
or close to synchronous speed, the machine will act as an induction generator feeding energy into
the system. Whether or not the subsynchronous current will grow can be determined from the
following consideration.
Since )()(
ω
ω
jxjx
qd
≈ for values of
ω
under consideration, the generator impedance to the
positive sequence subsynchronous current, excluding armature resistance, is given by
)(
o
ω
ω
ω
jxjZ
d
e
G
= (9.116)
where
o
ω
ω
ω
−
=
e
From the expression for the generator operational impedance given earlier,
(
)
(
)
()()
dodo
dd
dd
TjTj
TjTj
xjx
′′
+
′
+
′
′
+
′
+
=
ωω
ω
ω
ω
11
11
)(
(9.117)
Since )(
o
ω
ω
ω
−=
e
is negative, and
ddoddo
TTTT
′
′
>
′
′
′
>
′
, , it is evident that Z
G
will have a
negative real part the value of which can be readily calculated from equations (9.116) and
(9.117). If the magnitude of this negative resistance exceeds the combined resistance of the
generator armature and the network, the subsynchronous current will grow, since the decrement
factor (R/2L) will be negative.
It is evident from equations (9.116) and (9.117) that, the lower the magnitude of
ω
(i.e., higher
ω
e
) the larger the value of the negative real part of Z
G
. Therefore, the possibility of self-
excitation increases at higher series compensation. Similarly, larger values of the time constants
will reduce the magnitude of the negative real part and hence lower the possibility of self-
excitation. Larger values of
dod
TT
′
′
′′
and may be obtained by using low resistance pole-face
damper windings.
Following the above procedure it can be seen that the real part of Z
G
, corresponding to the
negative sequence subsynchronous current, is positive and therefore this component will always
damp out.
Note that self-excitation of purely electrical origin is possible only in the presence of series
compensation. For self-excitation to occur the effective system resistance at system natural
(subsynchronous) frequency must be negative. Positive resistance, on the other hand, is
TURBINE-GENERATOR SHAFT TORSIONALS
9-36
responsible for the negative electrical damping due to torsional interaction (see equation 9.115),
which can occur in systems with or without series compensation.
In the state-space approach described earlier both the torsional interaction effect and the self-
excitation effect are accounted for simultaneously.
Control of Subsynchronous Oscillation Problems
A number of control measures are available for mitigating the subsynchronous oscillation
problems arising from series capacitor compensation of transmission lines. None of these control
measures, applied singly, can however be considered to be a complete solution to the problems.
Some of these are effective only in specific applications. More than one measure may be needed
in any given situation.
From the analysis presented earlier it is evident that the most direct way of eliminating most of
the shaft torsional problems would be to provide for large mechanical damping of the shaft
system. However, there does not appear to be any practical means of achieving this at the present
time.
The theory and principle of operation of some of the control measures will now be described [26
- 30]. Although recent control measures employing power electronics are not included here, the
principles of operation are similar.
Static blocking filter
This is a multi-element blocking filter placed in series with the high-voltage winding of the
generator step-up transformer at the neutral end. Each element is a high Q parallel LC circuit
tuned to block electrical current at a frequency which coincides with the complement of a
torsional natural frequency, with negligible increase in impedance to 60 Hz current. The
combination of the filter impedance and the transmission impedance introduces parallel
resonances at the torsional complementary frequencies and shifts system series resonance points
to frequencies that cannot damage the machine.
The static filter provides control of transient torque and torsional interaction effects. Since each
filter is tuned to protect an individual unit, the effect of system changes is minimal. The filters’
effectiveness is reduced when detuned due to normal temperature variations, capacitor failure,
and changes in system frequency during a disturbance. This may be counteracted by augmenting
the natural shaft damping by a supplementary excitation control. This device provides damping
by injecting into the voltage regulator of a high initial response excitation system a properly
phased sinusoidal signal derived from rotor motion. Also, it may be necessary to increase the
basic insulation level of the generator step-up transformer.
Field tests have demonstrated the effectiveness of the static filter.
Line filter
The flow of subsynchronous current at a specific frequency can be blocked by connecting an
appropriately sized reactor in parallel with an existing series capacitor. The filter thus formed
will have a higher net capacitive reactance at 60 Hz than the original series capacitor and a lower
allowable line current. The original capacitive reactance can be restored by connecting additional
capacitors in parallel.
TURBINE-GENERATOR SHAFT TORSIONALS
9-37
If X
C
and X
Cf
are the 60 Hz capacitive reactances of the original series capacitor and the filter
capacitor, respectively, and X
L
is the 60 Hz inductive reactance of the filter reactor, then
−
=
−
=
1
60
,1
60
22
f
XX
f
XX
CLCfC
where f is the tuned frequency of the filter
The 60 Hz MVAR ratings of the filter capacitor and reactor, Q
Cf
and Q
L
, are given by
1
60
2
−
=
f
Q
Q
C
fC
, and
1
60
2
−
=
f
Q
Q
C
L
where Q
C
is the 60 Hz MVAR rating of the original series capacitor bank.
The above expressions show that the MVAR requirements of the filter elements go up rapidly as
the tuned frequency increases. The filter would have the best application at low electrical
frequencies.
The line filter could be effective in controlling transient torque and torsional interaction
problems when a single frequency is involved and the source of the problem is a single line.
Bypass damping filter
The filter consists of a resistor in series with a parallel combination of a reactor and capacitor
tuned to the system frequency (60 Hz). When connected across the series capacitor, the filter
provides an inductive/resistive bypass path for the flow of subsynchronus currents while
blocking the normal frequency current. By introducing positive resistances in the circuit for
subsynchronous currents the filter can be effective in controlling electrical self-excitation (the
induction generator effect).
When a single line and a single torsional frequency is involved, the bypass damping filter can be
used in conjunction with the line filter to counter torsional interaction as well as self-excitation.
Pole-face damper windings
Pole-face damper windings can be effective in controlling electrical self-excitation (the induction
generator effect).
Self-excitation results from a negative net system resistance at series resonance. Since the only
source of negative resistance is the generator, an effective solution would be to reduce the
negative resistance contribution of the generator at subsynchronous frequencies. This can be
achieved by adding pole-face damper windings to the generators.
In general, it is not practical to install pole-face damper windings on existing machines.
However, it is relatively inexpensive to install them on new machines.
Note that pole-face damper windings would not be effective in controlling torsional interaction
and transient torque problems.
TURBINE-GENERATOR SHAFT TORSIONALS
9-38
Dynamic filter
The dynamic filter is an active device placed in series with the generator to cancel the
subsynchronous voltage generated by torsional oscillations. A signal derived from rotor motion
is used to produce a voltage in phase opposition and of sufficient magnitude to counter the
subsynchronous voltage generated by rotor oscillation.
The dynamic filter would be effective in controlling torsional interaction problems. However, it
would do little to alleviate transient torque problems because of the inherent long time constants
of the control system.
Dynamic stabilizer
The dynamic stabilizer is a thyristor modulated shunt inductive load connected to the terminals
of the generator. The modulation of the reactance is determined by control signals derived from
rotor speed deviation.
The dynamic stabilizer can be visualized as a high-impedance inductive load where the inductive
susceptance is being sinusoidally modulated at the rotor oscillation frequency
ω
n
. For an applied
voltage at system frequency
ω
o
, the modulation will produce currents through the reactor at
frequencies
n
ω
ω
±
o
. The reactor currents will be divided between the generator and the
transmission network.
If ∆i is the current through the reactor at the subsynchronous frequency, the portion of this
current through the generator, considering an equivalent generator transmission system, is
i
XXXjRR
XXjR
i
CNGNG
CNN
G
∆
−+++
−
+
=∆
)(
)(
where the subscripts G and N refer to the generator and network quantities respectively, and X
C
is the series capacitor reactance.
At electrical resonance
i
RR
jX
i
RR
jXR
i
XXX
NG
G
NG
GN
G
CNG
∆
+
−
≈∆
+
−
=∆
∴
=
−
+
0
Thus the generator current is amplified in inverse proportion to the net system resistance. This is
the same amplification as that for the current resulting from the voltage produced by rotor
oscillation. Therefore, with proper gain and phase control, the two currents will cancel resulting
in no net oscillating torque.
The supersynchronous component of the reactor current will not be amplified by electrical
resonance and therefore is not a primary consideration in determining the control strategy.
Since the stabilizer responds to the electrical network in the same manner as the torsional
interaction, the stabilizer control will be independent of network configuration.
The dynamic stabilizer does not provide protection against self-excitation and transient torque
problems although it can provide damping for rotor oscillations.