Mrinal K Pal. Power system stability
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TURBINE-GENERATOR SHAFT TORSIONALS
9-9
+−
+−
−
=
LL
LLL
3
34o
3423
3
o
3
23o
2
23o
2312
2
o
2
12o
1
12o
1
12o
2
)(
22
2
)(
22
22
H
K
KK
HH
K
H
K
KK
HH
K
H
K
H
K
ωωω
ωωω
ωω
A
[]
′
∆∆∆=
521
θθθ
LLx and
′
∆
=
0
2
000
4
o
H
T
e
ω
T
The eigenvalues of the matrix
A corresponds to the various modes of oscillation of the system
(for a more detailed discussion of modes and mode shapes see Chapter 2). Note that, since the
electrical torque has been represented as an independent forcing function, the electrical mode is
absent in the above model. Since there are only four independent variables corresponding to the
four torsional modes, one of the eigenvalues of the matrix
A is zero.
Using a transformation
x = My, where M is a matrix composed of the eigenvectors of A, called
the modal matrix, equation (9.18) transforms into
FyW
y
+=
2
2
dt
d
(9.19)
where
[
]
2
5
2
2
2
1
diag
ωωω
−−−= LLW
and
TMF
1−
=
L,,
2
2
2
1
ωω
−− are the eigenvalues of A and
L,,
21
ω
ω
are the torsional natural (angular)
frequencies. In the above example there are four non-zero values of
ω
corresponding to the four
torsional natural frequencies.
Equation (9.19) is the modal equation of the shaft system. In this model the variables are
separated from each other and the individual modes can be analyzed independently, as in the
simple two mass system considered earlier. The actual shaft motion is obtained by combining the
separate responses using the transformation
x = My. Note that the initial values of the elements
of
y are obtained from
o
1
o
xMy
−
=
.
Development of modal equivalents
For the five mass system of Figure 9.2, equation (9.19) can be broken down as
5,,2,1
2
)(
4
o
4
12
2
2
L=∆+−=
−
iT
H
y
dt
yd
eiii
i
ω
ω
M (9.20)
where
4
1
)(
i
−
M is the element of the ith row and 4th column of the inverse of the modal matrix
M. For a sinusoidal input, ∆T
e
, of frequency
ω
, the response of the system of equations (9.20) is
given by
TURBINE-GENERATOR SHAFT TORSIONALS
9-10
e
i
i
i
T
H
y ∆
−
=
−
4
o
2
2
4
1
2
)(
ω
ωω
M
(9.21)
The actual response is obtained from
x = My. Note that as the frequency of the sinusoidal input,
ω
, approaches one of the natural frequencies, say
ω
k
, the particular element y
k
of the vector y
approaches infinity, and it will dominate over other elements. Therefore, at an input frequency
close to
ω
k
, the system response can be written as
5,,2,1
2
)(
4
o
22
4
1
L=∆
−
==
−
iT
H
m
ymx
e
k
kik
kik
k
i
ω
ωω
M
(9.22)
The superscript
k denotes the response due to an input of frequency close to the natural angular
frequency
ω
k
.
Equation (9.22) is also the response of the system given by
emimi
mimi
TxK
dt
xdH
∆=+
2
2
o
2
ω
(9.23)
where
2
o
4
1
4
2
,
)(
k
mi
mi
kik
mi
H
K
m
H
H
ω
ω
==
−
M
The system represented by equation (9.23) is the modal equivalent of the
ith mass of the shaft
torsional system. For an excitation
∆T
e
at a frequency close to
ω
k
, the modal equivalent has the
same response as the corresponding mass in the actual shaft torsional system.
As an example, consider the two mass systems shown in Figure 9.1. The linearized equation of
motion in matrix form, neglecting damping, is
e
T
H
H
K
H
K
H
K
H
K
dt
d
∆
+
∆
∆
−
−
=
∆
∆
2
o
2
1
2
o
12
2
o
12
1
o
12
1
o
12
2
1
2
2
2
0
22
22
ω
θ
θ
ωω
ωω
θ
θ
(9.24)
The eigenvalues of the system are
21
21
o1221
2
,0
HH
HH
K
+
−==
ωλλ
The corresponding eigenvectors are
1
1
and
−
2
1
1
H
H
Therefore
−
=
2
1
1
11
H
H
M and
−
+
=
−
22
21
21
1
1
HH
HH
HH
M
Note that the system has one non-zero eigenvalue and therefore one natural frequency, given by
TURBINE-GENERATOR SHAFT TORSIONALS
9-11
21
21
o122
2 HH
HH
K
+
=
ωω
(9.25)
Therefore, the modal equivalent of the turbine mass is
emm
mm
TK
dt
dH
∆=+
11
2
1
2
o
1
2
θ
θ
ω
(9.26)
where
and
21
2
21
12
2
2
o
1
1
21
21
2
2
1
)(
2
)(
HH
HH
K
H
K
HH
HH
H
H
H
m
m
m
+
−==
+−=
+
−
=
ω
ω
The modal equivalent of the generator rotor is
emm
mm
TK
dt
dH
∆=+
22
2
2
2
o
2
2
θ
θ
ω
(9.27)
where
()
21
1
2
21
2
2
1
2
2
HH
H
H
HH
H
H
H
H
H
m
+=
+
−
−
=
and
(
)
2
1
2
21
12
2
2
o
2
2
2
H
HH
K
H
K
m
m
+
==
ω
ω
The responses of the turbine and the generator rotor due to an excitation
∆T
e
, at a frequency
close to
ω
2,
are therefore
()
22
2
21
o
1
2
ωω
ω
θ
−
∆
+
−=
e
T
HH
(9.28)
and
()
22
2
21
o
2
1
2
2
ωω
ω
θ
−
∆
+
=
e
T
HHH
H
(9.29)
Note that the modal parameters for the turbine and hence its response are not affected by
interchanging the values
H
1
and H
2
, whereas those for the generator rotor are affected greatly
depending on the relative magnitudes of
H
1
and H
2
. For example, comparing the case where H
1
=
1,
H
2
= 2 with that where H
1
= 2, H
2
= 1, it is seen that the amplitude of oscillation in the second
case will be larger by a factor of 4. Therefore, the severity of torsional oscillations can be
expected to be less when the generator rotor inertia is relatively high, as in the case of hydro-
generators.
The mode shapes for the two cases are shown in Figure 9.3
TURBINE-GENERATOR SHAFT TORSIONALS
9-12
Fig. 9.3 Mode shapes corresponding to two sets of inertia constants as shown.
From equations (9.28) and (9.29), the relative displacement of the generator rotor with respect to
the turbine is
22
2
2
o
1221
2
ωω
ω
θθθ
−
∆
=−=
e
T
H
and the shaft torque is
22
2
12
2
o
2121
2
ωω
ω
θ
−
∆
=
e
T
K
H
K
Damping
The torsional modes and mode shapes determined on the basis of zero damping will be close to
their actual values, since the overall mechanical damping is small. It is impractical to determine
the damping coefficients of equation (9.17) from either design data or tests. A good estimate of
damping of the individual modal equivalent (equation 9.23) can, however, be obtained from test.
This can be done by exciting the torsional system corresponding to a particular mode (say the kth
mode) and observing the rate of decay of the resulting oscillation.
In the presence of damping, equation (9.23) will be modified to
emimi
mimimimi
TxK
dt
dxD
dt
xdH
∆=++
o
2
2
o
2
ωω
(9.30)
The free response is of the form
)cos(
θβε
α
+=
−
tAx
t
mi
TURBINE-GENERATOR SHAFT TORSIONALS
9-13
where
mi
mi
mi
mi
H
K
H
D
2
,
4
o
ω
βα
==
The free response is illustrated in Figure 9.4
Fig. 9.4 Typical free response of system represented by equation 9.30.
It is customary to express damping in terms of logarithmic decrement which is defined as the
natural logarithm of the ratio of two successive peaks, or, from Figure 9.4,
k
t
t
t
t
n
n
ω
α
ε
ε
ε
ε
α
α
α
α
π2
ln
1
1
lnDec Log
1
2
1
=
−
==
−
−
−
−
Dec Log
π
2
mik
k
mi
H
D
ω
=
∴
(9.31)
Evaluation of Torsional Stress and Fatigue in Turbine-Generator Shafts
Turbine-generators are designed to withstand, without damage, a limited number of terminal
three phase short circuits during their lifetime. The duty associated with major out-of-phase
synchronizations is also recognized and turbine-generators are designed to withstand these
without gross failures, although significant fatigue life may be lost following each incident. Both
of these are, however, relatively rare occurrences. The more commonly encountered disturbances
producing abnormal stresses in the turbine-generator shafts are: single and multiple phase faults
followed by three phase clearing, fast reclosing, line switching, faulty synchronizing and load
rejection. The mechanism of high shaft torque production following any of these incidents has
been explained earlier. The actual magnitudes of the peak torques depend on many factors, some
of which are not predictable.
The situation is somewhat different when series capacitors are present in transmission lines.
Although high shaft torques can be produced following system disturbances in the same way as
in systems without series capacitors, the presence of series capacitors constitutes a special
hazard, because of the possibility of torque magnification due to interaction between the
torsional system and the electrical system. This will be discussed in more detail later.
An accurate calculation of shaft torques is possible by simultaneously solving the equations
describing the shaft system, equations (9.11) - (9.15), along with the equations describing the
electrical system comprising the generator and the system network. In these calculations a more
detailed representation of the generator and system network than is usually required for
conventional stability calculation is necessary, in order to capture all the higher frequency
TURBINE-GENERATOR SHAFT TORSIONALS
9-14
electrical transients that affect the shaft torsional oscillations but have little effect on
conventional stability results. The detailed generator and network modeling is described later.
Although a simple network configuration has been used in the illustration, the modeling can be
extended to more complex networks.
In the absence of series capacitors and excitation control incorporating power system stabilizer
employing speed feedback, there is little interaction between the shaft mechanical system and the
electrical system. Therefore the torsional and electrical system equations can be solved
separately. The generator and network equations can be solved corresponding to the disturbances
being investigated, and the resulting air-gap torque can then be used as input to the shaft system
equations. The shaft torques obtained in this way will differ very little from those obtained from
a simultaneous solution of the mechanical and electrical system. The advantage is a reduction in
computational complexity. When the network disturbance and the period over which the
computation is to be extended are such that considerable rotor movement occurs, the
conventional swing equation (9.16) should be included in the electrical system equations.
In situations where significant interaction between mechanical and electrical system exists, the
above computation based on one-way coupling is not valid, and the mechanical and electrical
system equations must be solved simultaneously.
Estimation of fatigue
Once the response torque has been calculated for the various shaft spans, fatigue life expenditure
can be estimated provided that reliable information on fatigue strength characteristics of the
shafts is available. Unfortunately, due to the many complexities and uncertainties involved in
determining shaft fatigue capability under cyclic torsional stresses, an assessment of the loss of
life with a high degree of confidence is difficult, although significant progress has been made in
recent years due to concerns for high speed reclosures and subsynchronous resonance
A valid fatigue model should take into consideration the effects of stress concentration, non-
linear stress-strain characteristics, material behavior to cyclic deformation, geometry and size,
occasional overstrain, processing, etc.
A typical torsional fatigue characteristic is shown in Figure 9.5. Instead of a single curve a band
Fig. 9.5 A typical shaft life expenditure curve.
with an upper and a lower bound curve has been used in order to illustrate the variation in fatigue
strength that is possible due to the effect of surface finish, corrosion, surface damage due to
TURBINE-GENERATOR SHAFT TORSIONALS
9-15
handling and manufacturing process, lack of homogeneity in the shaft material, porosity,
impurities, forging tears etc. It is evident that the predicted value of the number of cycles to
failure can vary considerably depending on whether the upper or the lower curve is used. The
uncertainty is more pronounced in the high cycle fatigue domain. There is considerable
difference in opinion in the definition of shaft failure. Failure can be defined as an outright
failure, or as reaching the point where cracks appear, necessitating inspection, resurfacing or
replacement. The definition of failure accepted by the industry seems to be crack initiation.
100% fatigue implies that crack initiation has been reached.
Any transient event that produces shaft torques exceeding the high cycle fatigue strength will
result in some loss of life. The cumulative effect of the number of cycles exceeding the fatigue
strength will determine the fatigue damage for the whole event. The fatigue damage for each
cycle can be determined using a fatigue life expenditure curve such as the one shown in Figure
9.4. Corresponding to the peak value of the torque during the torque cycle, the number of cycles
to failure is read from the curve. The inverse of this number multiplied by one hundred is the
percent loss of fatigue life for that torque cycle. The cumulative life expenditure for the
completed incident is obtained from the sum total of the life expenditures for the individual
cycles. The procedure is to be repeated for each shaft section.
A representative assortment of disturbances based on historic data could be considered for the
assessment of the shaft torsional stresses and the accompanying loss of fatigue life during the
operating life of a turbine-generator. Due to the many uncertainties involved and the lack of a
thorough understanding of the subject of material fatigue, the life expenditure curves supplied by
the manufacturers can be expected to be conservative. Therefore, if the estimated loss of life is
well below 100%, then, barring extraordinary events, shaft failure can be considered unlikely. In
a marginal situation or where shaft failure is indicated, the case should be studied in more detail
with the manufacturer.
Effect of damping
The amount of damping present in the torsional oscillations determine the number of fatigue
cycles experienced by each turbine-generator shaft section following a transient event. It has
been recognized that the decay rates of torsional oscillations at natural frequencies are very
small. Due to the complexities of the damping mechanisms, it is difficult to reliably predict the
damping values from design data. Different damping values have been observed in tests
conducted on nominally identical units under similar operating conditions. Tests conducted on a
number of turbine-generators have shown the decay time constants of the various oscillation
modes to be in the range from 5 to 15 seconds. Tests have also shown the damping values to be
functions of generator power output and network configuration -- in general, damping tends to
go down as the power output is reduced.
Since the oscillation frequencies are high and the damping is low, the initial peaks of the
torsional oscillations are not affected by damping. However, the amount of damping present
determines the number of cycles the shafts experience before the amplitudes fall below the high
cycle fatigue levels. Estimates of shaft fatigue life expenditure will therefore be critically
dependent on the damping assumptions. Since the damping values cannot be precisely
determined at the design stage, manufacturers recommend that station tests be performed to
accurately measure the damping values, if torsional response evaluations show significant fatigue
duty for disturbances that occur relatively frequently on the system.
TURBINE-GENERATOR SHAFT TORSIONALS
9-16
Series Compensation and Subsynchronous Instability
The presence of series capacitors in transmission systems creates oscillatory electrical circuits
that have natural frequencies in the subsynchronous (i.e., below fundamental frequency) range.
Consider a synchronous generator connected through a series compensated transmission line to a
large power system. If
X
C
is the per unit capacitive reactance, and X
L
is the per unit inductive
reactance, the electrical natural frequency is given by
L
C
e
X
X
ff
o
=
(9.32)
where
f
o
is the rated synchronous frequency (60 Hz). For practical levels of compensation
1<
L
C
X
X
, and therefore f
e
< f
o
.
The subsynchronous oscillations arising from series compensation can, in certain situations, give
rise to phenomena leading to instability and/or equipment damage. Before detailed mathematical
analyses are undertaken, an introductory discussion of these phenomena will be in order.
Transient torque amplification
In a series compensated system, a large system disturbance, such as a system fault that does not
spark over the capacitor protective gap, will cause transient current at electrical natural
frequencies (subsynchronous) to flow. The frequency of the currents during the fault will depend
on the location of the fault and could have any value within a fairly wide range. It is easy to
verify that the transient subsynchronous currents in the three phases are unbalanced and therefore
will have both positive and negative sequence components. The positive sequence current
flowing in the generator armature will interact with the air-gap flux rotating at synchronous
speed and produce an electrical air-gap torque at the complementary frequency (
f
o
- f
e
).
Assuming that the per unit air-gap flux is close to unity, the transient torque will be proportional
to the per unit subsynchronous current. Therefore, the magnitude of the torque can be
considerable. If the frequency of the torque is close to one of the shaft natural frequencies, great
amplification of this torque may result. As illustrated in equation (9.10), shaft torques under
resonant conditions can reach high values very quickly.
In the absence of resonance, or if capacitors are bypassed during the fault, the shaft system will
still undergo high amplitude oscillations due to the large electrical torques generated by the fault.
During the fault the series capacitors in the unfaulted lines will be charged to high voltages. If
these voltages are below the capacitor protective gap sparkover level, a large amount of energy
will be stored in the
L-C circuit. After the fault is cleared, the excess energy will discharge,
driving considerable transient subsynchronous currents through the generator. The frequencies of
these subsynchronous currents are usually predictable since the post-fault network configuration
is known. Torque amplification will result if the frequency of the resulting air-gap torque
matches one of the torsional natural frequencies. Note that since the shaft system is already in a
state of oscillation initiated by the fault, torque amplification after fault clearing will be greater
than that during the fault.
The transient subsynchronous torque will decay more slowly than would be calculated from
normal circuit inductance and resistance due to the contribution of a negative resistance by the
TURBINE-GENERATOR SHAFT TORSIONALS
9-17
generator at subsynchronous frequencies. In some situations, as will be seen later, it may even
grow.
The negative sequence subsynchronous currents flowing through the generator armature will
produce air-gap torque at a supersynchronous frequency (
f
o
+ f
e
). Torsional natural frequencies
calculated from the lumped mass model representation of the turbine-generator shaft system as
shown in Figure 9.2 are all subsynchronous. The simplified model is adequate if the objective is
to assess the possibility of shaft damage, since the model accurately determines the torsional
modes in which maximum shaft stresses occur. The simplified model neglects the torsional
modes involving deformations of the rotor bodies as well as the additional modes due to the
many flexible members, e.g., turbine bladings attached to the rotor bodies. Calculations based on
a more detailed model in which the rotors and shafts are represented by a large number of
elements, as well as field tests on completely assembled units, show the presence of numerous
natural frequencies extending to 150 Hz. Supersynchronous electrical torques may therefore
produce high responses in some turbine-generator components. Turbine-generators are designed
to keep natural frequencies well away from 60 Hz and 120 Hz, so as to avoid resonance and
consequent torque amplification due to the 60 Hz and 120 Hz electrical torques that result from
balanced and unbalanced faults normally occurring in systems without series compensation. The
presence of series compensation may require consideration of all the natural frequencies in the
supersynchronous range.
Self excitation
A power system is never in a absolutely steady state. In a series compensated system any
perturbation will produce transient subsynchronous currents of both positive and negative
sequence. The positive sequence subsynchronous current flowing in the generator armature will
set up a magnetic field which will rotate in the same direction as the generator rotor but at a
speed corresponding to the subsynchronous frequency. The generator will therefore act like an
induction generator, feeding energy into the system.
From induction machine theory, the generator impedance to the positive sequence
subsynchronous current will have a negative resistance component
R
r
/s, where R
r
is the rotor
resistance referred to the armature, and
s is the per unit negative slip given by
o
o
ω
ω
ω
−
=
e
s (9.33)
If the magnitude of the negative resistance is greater than the net positive resistance of the
generator armature and the transmission system, the circuit will become self excited and the
subsynchronous current will grow.
Since there is no negative resistance associated with the negative sequence current, it will always
damp out.
Torsional interaction
Any perturbation, whether mechanical or electrical, will excite torsional natural frequency
oscillations of the generator shaft system. An oscillatory rotor gives rise to voltages and currents
in the armature circuits at frequencies which are complements of the rotor oscillating frequency.
This can be seen from the following analysis.
TURBINE-GENERATOR SHAFT TORSIONALS
9-18
Consider a generator whose rotor has a small amplitude oscillation superimposed on the steady
synchronous speed.
t
A
tA
n
n
n
ω
ω
θ
ω
ω
cos
sin
−=∆
=
∆
From the synchronous machine equations given in Chapter 5 and repeated later in this chapter,
the changes in the
d-q components of the terminal voltage due to speed change only (i.e.,
retaining only the speed voltage terms in the voltage equations) are given by
ω
ω
ψ
ω
ω
ψ
∆=∆∆−=∆
oo
,
d
q
q
d
ee
The phase a voltage is
θθθθθθ
θ
θ
∆−∆−∆−∆=∆
∴
−
=
cossinsincos
sincos
qqdda
qda
eeeee
eee
Noting that
θ
=
ω
o
t, where
ω
o
is the rated angular frequency of the system, we have
−−
+−=∆
tt
A
ettA
tt
A
ettAe
n
n
qn
d
n
n
dn
q
a
oo
o
oo
o
coscossinsin
sincoscossin
ωω
ω
ωω
ω
ψ
ωω
ω
ωω
ω
ψ
In the steady state
dqqd
ee
ψ
ψ
≈
−≈ ,
[] []
φωω
ωω
ω
ω
φωω
ωω
ω
ω
+−
−
+++
+
=∆
∴
tV
A
tV
A
e
nt
n
n
nt
n
n
a
)(cos
2
)(cos
2
o
o
o
o
o
o
(9.34)
where
q
d
qdt
e
e
eeV
1
22
tan,
−
−=+=
φ
Equation (9.34) shows that the armature voltage induced due to the rotor oscillating at frequency
ω
n
will have components at frequencies (
ω
o
+
ω
n
) (supersynchronous), and (
ω
o
-
ω
n
)
(subsynchronous). These voltages will drive currents through the generator armature and the
transmission system at the corresponding frequencies. The subsynchronous frequency current
can be considerable even for a small induced voltage if the frequency is close to the
(subsynchronous) natural frequency of the electrical system. The armature currents will produce
pulsating air-gap torques of frequency
ω
n
. Under certain condition, as shown later by detailed
mathematical analysis, the torque produced by the subsynchronous current can reinforce the
rotor oscillations so that the oscillations will grow.
Note that torsional interaction and shaft torque magnification are also possible in the presence of
certain excitation and governor control systems, with or without series capacitors in the system.
For example, in the application of power system stabilizer employing speed feedback, depending