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Damning. Analvsis of Subsvnchronous
Oscillation Caused
by HVDC
Abstract-The correct analysis of damping characteristics in
subsynchronous frequency range is essential to evaluate the
subsynchronous oscillation
(SSO)
caused by
WDC
but the
damping calculation is handicapped by the modeling of HVDC in
such a frequency range. The complex torque coefficient method
realized by time domain simulation is adopted in this paper to
estimate the
SSO
caused by HVDC. Frequency scanning in the
subsynchronous frequency range is performed to calculate the
subsynchronous damping
of
a
unit. Impacts
of
unit interaction
factor UDF),DC power level, firing angle and parameter settings
of the HVDC controller on the electrical damping are studied.
Research of the damping characteristics under inverter operation
is
also
conducted and it shows that only the units adjacent to the
rectifier have the potential for SSO, the units near the inverter
have
no such risk.
Zndex
Terms--complex torque coefficient method; SS0;HVDC
I. INTRODUCTION
ubsynchronous oscillation resulted from the interaction
S etween the electrical power system and the turbine
generator mechanical system can lead to turbine-generator
shaft failure and electrical instability at oscillation frequency
lower than the normal system frequency. The SSO mainly
occurs in the system with series compensated lines, the first
time that HVDC experienced
SSO
was in 1977 at Square
Butte. Extensive research was conducted but till now few
effective approaches were presented for SSO caused by
HVDC except the Unit Interaction Factor---UIF method
provided by EPRI [l]. The
UIF
establishes
a
relationship
between the HVDC interaction with turbine-generator
torsional vibration and the AC system strength, but there is no
damping information given by the
UIF.
The detailed studies
of
SSO
caused by HVDC were usually realized by HVDC
simulator [2 31, which was based on the electrical damping
characteristics of the units in the subsynchronous frequency
range, and the
SSO
stability was also estimated on the basis of
hangchunZhou, Zheng Xu Member,
IEEE
0-7803-8110-6/03/ 17.00
02003
IEEE 30
Project 50277034 supported by National Natural Science Foundation
of
Project No. G1998020310 supported by National Key Basic Research Special
Fund of China
Changchun
Zhou
and Zheng
Xu are
with the Department
of
Electrical
Engineering, Zhejiang University, Hangzhou, 310027 P. R. China
e-mail:
China.
the damping characteristics, this is the so-called complex
torque coefficient method.
The term of complex torque coefficient is proposed by
1.M.Canay in 1982
[4].
But before that time, the method
based on the concepts of damping torque and synchronous
torque for analysis of the SSO problem had been widely
applied [2,3]. In complex torque coefficient method, the
mechanical and electrical damping coefficient are calculated
respectively and used to evaluate the SSO problem. Since it is
a big challenge to establish an appropriate mathematical
model for systems including HVDC or FACTS devices in the
subsynchronous frequency range, it is difficult to obtain the
analytical solutions of the complex torque coefficient for the
study of the SSO problems caused by
HVDC
and FACTS.
However, the complex torque coefficient method realized by
time domain simulation has unique strong point to deal with
such an issue
[5].
In this paper, a time domain simulation
based on PSCADEMTDC is made to calculate the complex
torque coefficients and a damping analysis of SSO caused by
HVDC is also presented.
II.
MECHANISMS
ESCRIPTION
Investigations have revealed that the SSO problem of
HVDC is due primarily to the effects of the controllers
employed in HVDC systems [11. Turbine-generator rotor
motion causes variations in both magnitude and phase angle
of the commutating voltage. For an equidistant firing angle
control, utilized in modem HVDC systems, a shift in voltage
phase causes an equal shift in the firing angle. The change in
firing angle, as well as variations in the voltage magnitude,
will lead to changes in direct voltage and current, and thereby
dc power transfer. A closed loop control on direct current,
direct voltage, or firing angle applied in the HVDC would
respond to correct for these changes, thereby impacting the
magnitude and phase of variations in dc power transfer. The
ultimate effect of the change in dc power is a change in the
generator electrical torque. If the accumulated phase lags
between the changes in the generator shaft speed and the
ultimate resulting change in electrical torque on the generator
rotor exceed 90, the electrical damping becomes negative
[6]. Whether SSO occurs or not depends on the magnitudes of
the positive mechanical damping and negative electrical
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damping at the corresponding subsynchronous frequencies.
Many factors may influence the characteristics of the
electrical damping, such as the coupling between HVDC and
turbine-generator, direct power level, magnitude of firing
angle, characteristics of dc controller, parameters of the dc
lines and so on.
HI.
REAJJZATION
OF
COMPLEX
TORQUE
OEFFICIENT METHOD
IN HVDC
In complex torque coefficient method, the increments of
the electromagnetic torque and mechanical torque of a
machine under a
hHz h < f ,
and f, is the base system
frequency) disturbance can be represented as:
A i' e = K , ( h )A ; + D e ( h ) A h
(1)
A T , =
K, ( h )A S+ D ,(h)ACO (2)
In (l),
K,
and De are called as the electrical spring
coefficient and electrical damping coefficient respectively; In
( 2 ) , K , and D , are balled as the mechanical spring
coefficient and mechanical damping coefficient respectively.
When
K , ( h ) + K e ( h ) z O
and
D , ( h ) + D , ( h ) < O ,
the
torsional mode of oscillation at hHz is regarded as unstable.
For torsional modes of turbine-generator oscillation, the value
ofK, s relatively small in comparison to that of
K,
Hence,
the electrical spring coefficient has little effect on rotor
torsional oscillations. However, the inherent damping of the
turbine-generator torsional modes is extremely low, and the
damping contribution of the electrical system can be a
significant factor. Hence, the emphasis of this paper is to
examine the damping contribution of the power system. From
(1).
the electrical damping coefficient can be represented as:
D e
=
R e ( A i ' e / A & ( 3 )
where
A
T e
A h
= the increment of the electrical torque
= the increment of the electrical speed
The prerequisite for the complex torque coefficient method
is that the increment of the electromagnetic torque of a
generator can be represented as a linear function of the
generator's increment power angle and increment angular
frequency
[SI so
the complex torque coefficient method is
only valid for power system with only one generator and some
fixed frequency sources but not valid for multi-machine
power systems. In studying the
SSO
problem of a multi-
machine power system, the generator interested can always be
separated from the system and replace the rest of the units
with an equivalent source. Then the complex torque
coefficient for the unit interested can be calculated in the
equivalent network. The complex torque coefficient of each
unit in the system can be obtained after several times of
equivalence and calculation. A simplified equivalent network
for calculation of complex torque coefficient
is
shown in Fig.1
For the
SSO
study of this kind of network shown in Figl, a
relationship between the magnitude of the
HVDC
interaction
with turbine-generator torsional vibration and the AC system
strength has been established [13. This relationship, as
represented in
4),
with no damping characteristics provided,
Fig.1 E quivalentconfigurationof
A C D C
system
has been a cursory index and often used as a quantitative
screening tool.
where
UIFi
MVAi
sci
sc
=Unit Interaction Factor of the ith unit
=HVDC rating
=Rating of the ith unit
=Short circuit capacity at
HVDC
commutation bus excluding the ith unit
=Short circuit capacity at
HVDC
commutation bus including the ith unit
MVA ,,
According to the related guideline [l], the interaction
between the HVDC and turbine-generator can be ignored in a
network configuration with a UIF less than about 0.1, hence,
there is no potential risk
of
SSO in such a system. Equation 4)
can also be expressed in the form of impedance as shown in
(5).Thereby, supposing that the rating of the generator is not
changed, the
UIF
can be changed by varying the impedance.
where
zs =Equivalent AC system impedance
as seen from the converter excluding
the ith unit.
=Equivalent AC system impedance as
seen from the converter including the
ith unit and
Z ,
=
Zs
/
Z , .
It is convenient to realize the complex torque coefficient
method by time domain simulation. For a certain operating
point, after the system enters steady state, add a series of
small oscillating torques, whose frequencies are in the
subsynchronous range, to the rotor of the turbine-generator
under study, then the electrical damping coefficients can be
calculated according to (3) after getting the resulting changes
of the electromagnetic torque. Reference [SIhas provided the
steps of the time domain simulation, and a noticeable question
is how many small oscillating torques should be added in one
time. Due to the strongly nonlinear characteristics of HVDC
or
FACTS,
the different frequency quantities may interfere
with each other if more than one frequency oscillating torques
Ze q
31
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are added in one time. Thereby, only one frequency
oscillating torque can be added in one time in the simulation
for a system associated with HVDC or FACTS.
IV SIMULATION
ND
ANALYSIS
A. Studied System
The studied system model is shown in Fig.2. Remain the
turbine-generator G which is connected to the rectifier via a
step-up transformer and reduce the rest of the machines to
fixed frequency sources. Each reduced machine is represented
by its sub-transient reactance in series with a voltage source.
Further, combine the reduced machines and represent them
with a fixed frequency source. In this way, the AC system is
reduced to be a single unit in parallel with a single fix
frequency source system, as the rectifier part shown in Fig.2.
The impedance Z , consists of the sub-transient reactance and
leak reactance of the step-up transformer, Z is only used in
the calculation of UIF n the real simulation the generator
G is modeled by the Park equations with a single mass. The
rating
of
generator
G
is 892MVA. The dc power and voltage
are rated at lOOOMW and 500kV respectively. The inverter
side AC system is also represented with an infinite source
behind an impedance.
Fig.2 Studied system model
The dc system is a mono-pole and
12
pulse system with a
current controller at the rectifier and an extinction angle
controller at the inverter. The block diagram of the current
controller is described in Fig.3, the dc current error is
processed through a PI regulator to produce the firing angle
order abrd.
r
-
Fig.3 Block diagram
of
current regulator
B.
Dam ping characteristics o the rectifier unit
I Impact of UIF on electrical damping
For the studied system in Fig.2, keep the rating of the
generator
G
and the equivalent impedance Z as constants.
Change the AC system strength by varying the impedance Z
to obtain different UIF Corresponding to different UIF , he
curves of electrical damping in the frequency range from
5Hz to 50Hz are depicted in Fig.4. For the situation
of
UIF being 0.1,a positive damping exits almost over the entire
~
32
subsynchronous frequency range. Negative damping only
occurs at some isolated frequency with small magnitude.
Considering the positive contribution of the mechanical
damping, it is deemed that the SSO will not occur in such a
situation. As shown in Fig.4, the magnitudes of the negative
damping are increased with the increase of
UIF ,
he damping
coefficients equal - 0 . 8 ~ ~nd -2.Opu for UIF up to 0.36
and 0.79 respectively. If these negative effects can not be
counteracted by the mechanical damping of the shaft itself,
SSO will definitely occur.
0
20
4
20
4
20 4
Frequency Hz) Frequency
Hz)
Frequency Hz)
Fig.4 Impacts ofUlF on electrica l damplng re ctifieroperation)
The magnitude of
UIF
reflects the coupling between the
HVDC and the unit. In principle, the impact of the
disturbance of the generator terminal voltage on the
commutating voltage is decreased by the parallel
impedance Z s R and the resulting perturbation of dc current is
also shunted by Z , . Thereby the ultimate change of A ; ~ is
smaller than that in the situation without
Z ,
. Supposing that
the ratings of unit and dc power transfer
are
not changed, it
can be seen from (4) and (5) that the UIF is determined by
SC, and SC , or Z sR and Z . UIF can be of small value
only
if
Z,is very large or
Z,is
quite small,
that is
to say,
only on the condition that the electrical distance between the
unit and HVDC is very large, or the parallel equivalent AC
system is fairly strong, the SSO can be avoided effectively.
Another characteristic reflected by Fig.4 is that the
damping for UIF being 0.36 and 0.79 is negative only in the
frequency range about 5-2OHz This phenomenon is
resulted from the effects of the current regulator employed in
the rectifier. HVDC current regulators typically have
bandwidth in the range of
10
to 20
Hz
[7], which include the
first few torsional modes of vibration of turbine-generator
units. Only the disturbances in the range of the bandwidth can
be transferred through the closed loop control and produce
negative damping. Thereby, in studying the
SSO
of HVDC,
special attentions should be paid to the first few torsional
modes which have the same frequency range as the controller
bandwidth.
2 ) Impact
of
dc po wer level
on
electrical damping
From the analysis above, it can be seen that the
perturbation of dc power is the direct cause of
SSO
in HVDC.
For the same studied system shown in Fig.2 with UIF being
0.79, Fig.5 gives the curves of damping characteristic versus
different dc power level. For the dc power Pdc s
1 .Opu
, he
damping is of large negative values in the frequency range
from
5
to 20Hzand is the same as the one shown in Fig.3
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with UIF =0.79 Less destabilization exits as the dc power is
reduced, as illustrated by Fig.5. With the dc power dropping
to
0.2
pu
positive damping exits almost over the entire
subsynchronous frequency range except the frequencies near
5 Hz It is well known that the first natural torsional
frequency of a turbine-generator shaft is generally larger than
lOHz,
hence the small negative damping is not capable of
causing
SSO
at the frequency about
5
z
-0
10
20 30 40 50
Frequency
Hz)
Fig .5 Impact of dc power level on electrical damping
Being a kind of oscillation of active power, the SSO caused
by HVDC is heavily dependent on the dc power transfer and
the output of generator. UIF is an index of interaction in the
rated operation, which indicates the coupling of the HVDC
and the unit in rated operation mode. From the view of
coupling, the relationship between the unit and HVDC is
weakened as the dc power transfer drops down. Hence the
interaction between them and the destabilization are also
decreased.
3
Impact offiring angle on electrical damping
The disturbance of generator rotor can lead to the changes
in dc voltage, dc current and hence dc power transfer. The dc
control would respond to these changes. All the control
actions at the rectifier are done by changing the firing angle.
Changes in the fiing angle have a significant impact on the
interaction of HVDC and unit. This influence arises from the
inherent cosine relationship between firing angle and dc
voltage. Fig.6 illustrates the different damping characteristics
for varied firing angle. As shown in Fig.6, destabilization
increases as the firing angle is increased. There will be more
risk in the condition of high firing angle.
I I
10 20 30 40
50
Frequency Hz)
Fig .6 Impactof firing angle on electrical damping
To ensure the operation of valves, firing angle should be
larger than its minimum limit, which is about
3 -
5 In real
HVDC operation, the firng angle is usually set in the range of
10'
-
15 .
In general, high firing angle at the rectifier exist
only for transient situation of at most a few hundred
milliseconds, during which time torsional damping is not of
primary concern. However, there may be situations where it is
operated at reduced voltage in steady-state. Such a case will
have more potential risk of SSO than HVDC system operating
near 15 iring angle.
4
Impact
of
controller settings
on
electrical damping
Since the negative electrical damping is produced by the
HVDC closed loop current controller, the characteristics of
damping
are
heavily dependent upon the parameter settings of
them. The structure of current controller has been shown in
Fig.3. A PI regulator is adopted to produce firing angle order
for HVDC system. The curves of damping corresponding to
different integral time constants
i
and to different
proportional gains K , are depicted in Fig.7 and Fig.8
respectively.
2
1 - - - r - - -
=OiOlS I
10 20 30 4
50
Frequency Hz)
Fig.7 Impact
of
Ti on eiectrical damping
0 10 20 30 40 50
Frequency
Hz)
Fig .8 Impact ofKp on electrical damping
The characteristics of damping have a similar tendency of
change for the three different settings of
T.
,
which is negative
in a low frequency range and becomes positive in a high
frequency range. Define the frequency at which the damping
turns from negative to positive as f,
.
It is clear that
when Ti
>Ti*
T 3 , he relationship of
f,
can be expressed
as f,,
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shown in Fig.8, for different proportional gains
K ,
, he values
of
f,
are almost the same, moreover, the characteristics of
damping are almost the same in the f < f frequency range.
When frequency
f
increases and is higher than f , the
damping characteristics behave obviously different for
different values of. K,
Because of the complexity
of
the HVDC system, it is
difficult to establish a mathematical model for the closed loop
current control including the whole dc system. In normal
operation, different parameters have different impacts on the
dc controller bandwidth and comparing with the proportional
gain
K ,
, the integral time constant is a more significant
factor. The larger the c i s , the lower the bandwidth and f
would be. The unit will benefit from higher time constant to
avoid
SSO,
but the large time constant can surely slow down
the respond speed of the dc controller. Proportional gain
behaves as an amplifier to the dc disturbance, from Fig.8 we
can see, high setting of K , will bring more risks of SSO.
C.
Dam ping characteristics
of
inverter unit
Former researches have led to such a conclusion that
turbine-generators in the vicinity of an inverter station have
no potential risk of SSO
[3].
It is of interest to validate it by
means of the complex torque coefficient method realized in
time domain. Suppose that the unit in parallel with equivalent
AC system is connected to the inverter, or reverse the power
flow and make the rectifier work as an inverter, the studied
system in Fig.2 can be used to examine the SSO
characteristics of the units adjacent to inverter. Fig.9 gives the
electrical damping of the inverter unit for different UZF .
Unlike the damping of unit near rectifier, the negative
damping with small magnitude is achieved only for range of
high frequency. And the frequency range covered by negative
damping is decreased as the
UIF
increase, as can be seen
from Fig.9,
the
positive damping exists over the entire
subsynchronous frequency range for UIF being 0.36. As no
loads are included in the simulation, the results shown in Fig.9
are very conservative.
B
0
20
40
1.5
1
0.5
0
-0.5
: :
0
2 4
2.5
2
1.5
1
0.5
0
20
4
Frequency Hz) Fresuency (W Frequency Hz)
Fig.9 Impact of UIF on electrical damping (inverter operation)
Power system loads have positive frequency-dependent
characteristic, which will contribute positive damping to
power oscillations with any frequency, obviously including
the oscillation in the subsynchronous frequency range. Being
rigid load, the rectifier station is independent of network
frequency and usually has negative contribution to power
oscillations.
If
the rated power
of
HVDC and
of
the unit are at
the same level, i.e. the UZF is high, there will be more
possibility of SSO occurrence. Inverter is something like an
AC source, and the adjacent units do not supply any power to
the HVDC. Operating parallel with the inverter, the units
supply conventional, frequency-dependent loads. In addition,
an inverter, at least operating with dc voltage regulations, is
similar to the conventional loads----each increase in voltage
lead to an increased reactive power consumption and vice
versa. Thereby, the turbine-generators adjacent to an inverter
are not endangered by SSO problems.
V
CONCLUSION
The complex torque coefficient method is adopted in this
paper to study the SSO caused by HVDC. The method is
realized by a time domain simulation. Frequency scanning in
the subsynchronous range has been performed and the
characteristics of electrical damping are calculated. This
method adapts to not only the SSO caused by HVDC, but also
the SSO problems caused by FACTS
or
other power
electronic devices.
Current control at rectifier would produce negative
damping, and the frequency range of negative damping is
determined by the controller bandwidth. Therefore, properly
settings of the controller parameters will decrease the risk of
SSO,
but can not eliminate the potential danger. Reduce the
coupling between the HVDC and unit, or reduce the dc power
transfer and firing angle can mitigate the SSO stress caused by
HVDC. Units adjacent to inverter have no SSO risk.
W
REFERENCES
Electric Power Research Institute, HVDC System Control
for
Damping of
Subsynchronous Oscillations, EPRl EL-2708,
Final
Report,Ocyober
1982.
SvenssonS, Mortensen K. Damping of subsynchronous oscillations by an
HVDC link, an HVDC simulator study, IEEE Trans on Power Apparatus
and System, vol. 3, pp.1431-1437, 1981.
Ymg Jiang-Haher,Hugo Duchen, Kerstin Linden and Mats Hyttinen,
Improvement of subsynchronous torsional damping using VSC HVDC,
Proceedings of PowerCon2002, vol. 2, pp.998-1003,2002.
Canay I.M.,
A
new approach to the torque inyrtaction and electrical
damping of the synchronous machine, part I and part 11, I EEETrans on
Power Apparatus and Systems, vol. 10, pp.3630-3647, 1982.
Xu Zheng, Feng Zhouyan, A novel unified approach for analysis of
small-
signal stability
of
power system, Proceedings
of
IEEE/PES Winter
Meeting, vol. 2, pp.963-967,2000.
The Institute
of
Electrical and Electronics Engineers, IEEE guide for
planning DC
l i
erminating at
AC
locations having low short-circuit
capacities,I EEEStd 1204, 1997.
P. Kundur, Power System Stability and Control, New YorkMcGraw-Hill ,
1994, p.1026
VIII.
BIOGRAPHIES
Changchun Zhou
was
bom
in Shandong, China, in January 1976. He received
B.S from SouthEast University, Nanjing, China in 1998. He received M.S from
Shandong University, Jinan,
China
in 2000. He is now a Ph.D. student in the E.E.
Department of Zhejiang University. His main field of interest includes power
system stability and control of HVDC.
Zheng Xu was bom in Zhejiang, China, in September 1962. He received the BS,
MS and Ph.D. degrees from Zhejiang University, China in 1983,1986 and 1993
respectively, all in Electrical Engineering. He has been with the Electrical
Engineering Department of Zhejiang University since 1986. Since 1998 he is a
professor of Zhejiang University.
His
research area includes
HVDC, FACTS,
power harmonics and power quality.
34