This paper proposes a new mathematical model to describe the switching characteristics of thyristor-based HVDC converters when there are low or subsynchronous frequency ripples present in the DC current. The conventional modulation theory assumes valves conduct at equal time intervals, which is not valid when ripples are present. The new model accounts for unequal conduction times due to ripples affecting firing pulses. Simulations of a 7-node hybrid AC-DC system show the new model accurately captures harmonics from switching, unlike the modulation theory. The new model provides a better understanding of harmonics generated under low frequency disturbances in HVDC systems.

1. MODEL FOR THE SWITCHING CHARACTERISTICS OF THYRISTOR
BASED HVDC CONVERTOR
Zhou Li1,2, Xiao-Ping Zhang2, Qiu-lan Wan1
Southeast University1
University of Birmingham2
1
Nanjing, China
Birmingham, UK2
Lizhou1985@163.com, x.p.zhang@bham.ac.uk, qlwan@seu.edu.cn
Abstract – The modulation theory is widely used to describe the operation of the convertor with distortion fed to
it from the DC side. With such a method, all the derivation
of the generated harmonics is based on the assumption
that the valves begin conducting at equal time intervals.
However, in fact, the ripples with low and subsynchronous
frequencies in the DC line may be fed to the HVDC control
system and the valves would begin conducting at unequal
time intervals. Hence, in this paper, a new mathematical
model is proposed to describe the switching characteristics
of the thyristor based HVDC convertor, which takes the
conducting of valves at unequal time intervals in consideration. A seven-node hybrid AC-DC power system, which is
a EMTDC/PSCAD based digital simulation system, with
detailed multi-block generator model and a HVDC transmission link based on the IEEE first benchmark system
model, is employed to re-examine those new generated
harmonics under low and subsynchronous disturbances
due to the switching characteristics of the thyristor based
HVDC convertor.
Keywords: HVDC converter, harmonic, switching
characteristics
cable to the system conditions where the low and subsynchronous frequency ripples are present. Having said
this, it is necessary to propose a new model to describe
the switching characteristics of the thyristor based
HVDC convertor, taking the conducting of valves at
unequal time intervals into consideration. Then the
principle of how the harmonics generated due to the
HVDC converter switching characteristics could be
examined.
This paper is organized as follows. In Section 2, the
applicability of the modulation theory for harmonic
analysis of different frequency ranges is investigated. In
Section 3 a new model of HVDC valves switching characteristics in low and subsynchronous frequency range
is proposed. Then in Section 4, an EMTDC/PSCAD
based seven-node hybrid AC-DC power simulation
system is employed to verify the effectiveness of the
new model, and to investigate the principle of how the
new harmonics generated by the HVDC system, due to
the converter switching characteristics.
2
1 INTRODUCTION
Most research publications have mainly used the
conventional modulation theory to describe the switching characteristics of the thyristor based convertor in
harmonic analysis [1, 3, 4, 6, 7]. In the derivation of the
harmonics generated by the HVDC converter, all of
these publications have assumed that the valves begin
conducting at equal time intervals in order to simply
express the switching function as a function of quasi
square wave with equal width in every cycle, and when
there are ripples in the DC link, the components of the
ripples were added into the modulating function while
the switching function is not affected.
However, when low and subsynchronous frequency
ripples in the DC line are fed to the HVDC control system, they could not be counterbalanced by their positive
and negative part in the HVDC control system as the
high frequency ripples do. Subsequently this would
affect the firing pulses from the control system. As a
result, the valves conducting time interval will not be
constant at equal time intervals as modulation theory
has assumed but change at the frequency of the ripples.
Due to the reason mentioned the switching function
cannot be simply expressed as a function of quasi
square wave with equal width in every cycle.
Hence, the basic assumption of the modulation
theory employed in previous research work is not appli17th Power Systems Computation Conference
THE MODULATION THEORY &
APPLICABILITY
2.1 The Modulation Theory
The modulation theory was firstly introduced by telecommunications engineers, and was then widely employed in electrical power systems to describe the
switching characteristic of the thyristor based convertor
long time ago.
In the modulation theory, the three-phase bridge in
the HVDC convertor was considered as a modulator
[1,3,4,6,7], as shown in Figure 1.
Figure 1: Switching function and modulating function giving modulated AC output.
If the DC ripples are superimposed on the DC current, the alternating current would contain ripples as
Stockholm Sweden - August 22-26, 2011

2. sampled from the direct current, and the output is said to
be 'modulated'.
In paper [6], the relationship of the input and output
of the converter is expressed in the following general
form:
uୢୡ = uୟ ∙ S୳ୟ + uୠ ∙ S୳ୠ + uୡ ∙ S୳ୡ
(1)
iୟ = iୢୡ ∙ S୧ୟ
൞ iୠ = iୢୡ ∙ S୧ୠ
(2)
iୡ = iୢୡ ∙ S୧ୡ
In this model, S୳ୟ , S୳ୠ , S୳ୡ are the switching functions of each phase for voltage modulation and
S୧ୟ , S୧ୠ , S୧ୡ are the switching functions for current
modulation. The switching functions in the equations
are related to the conduction states of the thyristors in
each phase. With the modulation theory it has been
assumed that the valves begin conducting at equal time
intervals, and then the Fourier series can be used to
express the switching functions for voltage and current
modulation as follows.
Sal = ∑∞ An cos nwl t
(3-a)
n=1
2π
∞
Sbl = ∑n=1 An cos n ቀwl t- ቁ
(3-b)
Scl = ∑∞ An cos n ቀwl t+ ቁ
n=1
3
2π
3
where
4 1
nπ
Based on the above discussions, the assumption that
the valves begin conducting at equal time intervals and
based on which the modulation theory is no longer applicable in low and subsynchronous frequency range.
Hence, it is necessary to employ a new model to describe the HVDC switching characteristics for the system low and subsynchronous frequency harmonics
analysis.
(3-c)
nπ
An = ∙ ∙sin ∙cos
(4)
π n
2
6
With the assumption that the valves begin conducting
at equal time intervals, the components of the ripples in
the DC line can be added into the modulating function
and this has no effect on the switching function.
2.2 The Applicability of Modulation Theory
In fact, the ripples contained in the direct current can
be sensed by the current measurement of the HVDC
system, and be superimposed onto the control signal.
Then the firing angle output from the control system
will be affected.
Taking the 6-pluse convertor for example, when the
frequency of the harmonics is lower than the AC system
base frequency, inside the HVDC converter, there will
be several firing pulses generated to conduct the valves
during one cycle of the ripples which superimposed on
the direct current, the effect of the ripples cannot counterbalanced in the PI controller of the HVDC control
system under this condition.
For example, if one firing pulse is generated during
the period that covers the positive part of the ripple,
there will be an incremental injection into the control
variable of the HVDC system to increase the firing
angle and the corresponding valve conducting time
interval will be shortened; On the other hand, the firing
pulse generated during the period covers the negative
part of the ripples, will increase the corresponding valve
conducting time interval. Only when the firing pulse is
generated during the period that covers both positive
and negative part equally, the effect will be counterbalanced by the positive and negative part of the ripples,
then the corresponding valve conducting time interval
will be retained at normal value;
17th Power Systems Computation Conference
3
NEW MODEL FOR HVDC CONVERTER
SWITCHING CHARACTERISTICS
The low and subsynchronous frequency ripples can
be sensed by the direct current measurement unit and
fed into the control system. It is found the low and subsynchronous component of the ripples cannot be counterbalanced in the HVDC control system and will retain
in the control variable. As the HVDC rectifier control
system uses the deviation control to provide the firing
pulses to the valves, the ripple component will be superimposed on the value of firing angle output from the
control system and make the conducting time interval of
the valves varying around the normal operating point at
the frequency of the ripples.
Besides, under the effect of the ripples at low or subsynchronous frequency, the valves could be conducted
more than six times during one cycle of the ripples,
which means the cycle of the ripple is much longer than
one conducting time intervals of the valves. As a result,
the change rate of the alternating current amplitude
under the effect of the low or subsynchronous ripple is
so small during each conducting time intervals of the
valves, that the amplitude of the alternating current can
be seemed as a constant.
Figure 2: Alternating current generated from the convertor
in different status. (a) Status at normal operating point. (b)
Status under the effect of low or subsynchronous frequency
ripple.
Furthermore, the low and subsynchronous frequency
ripple components are mainly caused by system faults in
the AC and DC system or the noise in the control system or small disturbances in the electrical power system. Thus the amplitude of the ripples is very small and
hence we can assume that the HVDC system would not
change its operating point. Generally, at the normal
operating point, the conducting time interval of each
valve is 120° (2π/3 in radian measure). Considering this,
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3. the alternating current under the effect of low or subsynchronous frequency ripple illustrated in Figure 2 can
be given as follows:
f (t)=
ቀ (k-1)T1 ≤t≤ T1 + A sin(ω2 t) ቁ
⎧ 0
⎪
⎪ C
k
ቀ T1 + A sin(ω2 t) <t≤ T1 ቁ
6
k
⎨ 0
⎪
⎪
⎩-C
ቀ T1 <t≤
6
k
k
2
2k
(5)
T1 +A sin(ω2 t)ቁ
ቀ T1 +A sin(ω2 t) <t<kT1 ቁ
2
2k
3
3
( k = 1, 2, 3 ⋯ )
where T1 is the cycle of the alternating system,
݂ଶ = ω2 /2π is the frequency of the ripples, the constant
A and C are amplitudes of the ripple and the alternating
current, respectively.
4
SIMULATIONS AND ANALYSIS
4.1 Hybrid AC-DC System
In this paper, a seven-node hybrid AC-DC system
was built in the EMTDC /PSCAD simulation environment where the single-ling diagram of the system is
shown in Figure 3.
500
500
B1
B4
500
B3
S2
S1
'
Y
Y
Y
'
L2
P2
jQ2
Y
220
G
'
are 1000MW and 500kV, respectively. A detailed
HVDC pole control system is built in the system, as
shown in Figure 4.
B2
HVDC
Y
L1
P
1
jQ1
Figure 3: Single-line diagram of a 7-node hybrid AC-DC
test system
The hybrid AC-DC test system, which consists of
double AC transmission lines and a HVDC link, 3 synchronous generators. The test system is referred to
China EPRI 7-node test system, while further details of
the system parameters are given in [9]. The parameters
of the generator G are very similar to the generator in
the IEEE first benchmark system with a rating of 892.4
MVA [2]. Π equivalent circuit model is used to
represent the AC transmission lines in the simulations
while constant power load model is used to represent
the load L1 and L2.
4.2 HVDC System
The HVDC system refers to the CIGRE HVDC
benchmark system. The converter stations are using a
12-pulse configuration with two six-pulse valves in
series where the rated transmission power and voltage
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Figure 4: HVDC pole control system block diagram.
4.3 Description of the Test Cases
In the tests to be presented, the following cases are
carried out:
Case 1: The detailed model of the HVDC converter
valve in the PSCAD is used to simulate the dynamic
performance of the HVDC converter with a 35Hz ripple
in the DC current. This can be considered to be the
benchmark.
Case 2: The new model proposed in this paper is
used to simulate the HVDC converter switching characteristics with a 35Hz ripple in the DC current.
Case 3: The model based on the modulation theory is
used to simulate the HVDC converter switching characteristics with a 35Hz ripple in the DC current.
Case 4: This case is similar to Case 1 except that
there is a ripple of 1Hz in the DC current.
Case 5: This case is similar to Case 2 except that
there is a ripple of 1Hz in the DC current.
Case 6: This case is similar to Case 3 except that
there is a ripple of 1Hz in the DC current.
Case 7: This case is similar to Case 1 except that
there is a ripple of 5Hz in the DC current.
Case 8: This case is similar to Case 2 except that
there is a ripple of 5Hz in the DC current.
Case 9: This case is similar to Case 3 except that
there is a ripple of 5Hz in the DC current.
In each group of simulation cases, Case 1, 2 and 3 for
example, one specific single frequency ripple is added
into the measured current value of the DC line to simulate the disturbance component and we assume that this
can be fed into the HVDC control system. The amplitude of the ripple is 1% of the measured direct current
and the ripple is added at the 20th second and removed
0.5s later. And then the output alternating currents of
Phase A from the HVDC converter valves for each
group of cases are compared.
4.4 HVDC converter switching characteristics under
the effect of 35Hz ripple
For Cases 1, 2 and 3, the 35Hz ripple is added into
the DC current. Fig. 5 (a) shows the output AC current
of Phase A from the valves under the effect of 35Hz
ripple in time domain in Case 2 compares with the AC
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4. current of Phase A without any ripple being added,
while Figure 5 (b) shows the same variables in Case 3.
The comparison of Figure 5 (a) and (b) shows that the
main difference of the new HVDC converter model
proposed in this paper and the model based on modulation theory is the AC current wave form output from the
new model is a quasi-square wave with its width varying at the frequency of the ripple added in the DC side
while only the amplitude of AC current wave output
from the model based on modulation theory is changed.
It can be seen in Figure 6 (a) that in Case 1, employing the detailed HVDC converter valve model contained
in PSCAD component library, with the 35Hz ripple
being added in the system, in the low and subsynchronous frequency band, mainly the ripples around the frequency 15Hz are generated due to the valve switching
characteristics. Besides, the ripples around the frequency 5Hz 10Hz 20Hz 25Hz 30Hz 35Hz 40Hz
and 45Hz with smaller amplitude are generated as well.
In Figure 6 (b), the new HVDC converter model proposed in this paper is employed to simulate the effect of
the HVDC converter switching characteristics on the
output alternating current in Case 2. With the 35Hz
ripple being added, the very similar AC current spectrum in frequency domain compared with Case 1 can be
obtained.
However, in Figure 6 (c), when the HVDC converter
model based on the modulation theory is employed in
Case 3, it can be found that no ripples of other frequencies but only the 35Hz ripple itself contained in the
alternating current, and this obviously does not match
the phenomenon simulated in Case 1 and 2.
4.5 More Cases study
Figure 5: The AC current of phase A output from the valves
under the effect of 35Hz ripple in time domain in different
simulation cases
Then the alternating current outputs of phase A from
the HVDC converter valves under the effect of 35Hz
ripple are sampled for Case 1, 2 and 3 respectively, then
the FFT is used to analyze the alternating currents and
the comparison is shown in Figure 6.
Figure 7: Comparison among the alternating current of
phase A in frequency domain under ripples of different
frequencies in different cases.
Figure 6: The AC current of phase A output from the HVDC
converter valves under the effect of 35 in frequency domain
in different simulation cases
17th Power Systems Computation Conference
Furthermore, Figure 7 compares the simulation results in more cases: Figure 7 (a) compares the alternating current outputs of phase A from the HVDC converter valves in frequency domain under the effect of 1Hz
ripple in Case 4, 5 and 6; while Figure 7 (b) compares
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5. the same variable under the effect of 5Hz ripple in Case
7, 8 and 9.
In Case 1, 2, 4, 5, 7 and 8, it can be seen from Figure
6 and 7 that if one single frequency ripple was added
into the DC current, new harmonics of other frequencies
could be generated due to the HVDC converter switching characteristics and be injected into the AC system
connects to the HVDC system. Especially the amplitude
of harmonics whose frequencies is around the complementary frequency (the fundamental frequency of the
AC system minus the frequency of the ripple being
added in) is much larger than the new generated harmonics of other frequencies. For example, under the
effect of 35Hz ripple, the amplitude of the 15Hz harmonic is the largest among all the new generated harmonics. In a similar way, under the effect of 1Hz and
5Hz ripple, the 49Hz and 45Hz harmonics are the largest harmonic components respectively. Such simulation
results in Case 1 and 2 are similar, as well as in Case 4
and 5, and in Case 7 and 8.
However in Case 3, 6 and 9 showed in Figure 6 and
7, due to the original purpose of employing the modulation theory is for high frequency harmonics analysis and
the basic assumption for employing the modulation
theory is not close to real under the low and subsynchronous harmonics. The simulation results employing
the model based on the modulation theory are different
from those in PSCAD.
Hence it has been confirmed again that under low
and subsynchronous ripples, compared with the model
based on the widely used modulation theory, employing
the new model proposed in this paper to simulate the
valve switching characteristics can get very similar AC
current to that simulated by the detailed PSCAD model.
5 CONCLUSIONS
When there are disturbances in the DC line of the
HVDC transmission system or noise superimposed in
the control signal of the HVDC control system, some
harmonics contained in these disturbances may be fed
into the HVDC control system. However for low and
subsynchronous frequency harmonics, unlike the high
frequency harmonics, they cannot be counterbalanced
by their positive and negative part in the HVDC control
system but superpose upon the control variable consistently. Subsequently, they could affect the generation of
firing angles of the valves and make the conducting
time interval of the valves changes at the frequency of
the harmonics. The basic assumption of the modulation
theory that the valves begin conducting at equal time
intervals is no longer true. Hence the model based on
the modulation theory is not applicable in this situation.
In this paper, a new model based on the principle of
the valve firing angle generation and considering the
change of the valve conducting time intervals has been
proposed to describe the valve switching characteristics.
It can be seen in the comparison that, in low and subsynchronous frequency band, rather than the modulation
theory, the new model proposed in this paper can obtain
results closer to the detailed simulations using EMTDC.
17th Power Systems Computation Conference
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The simulation result based on the new model indicates that if low and subsynchronous frequency disturbances be fed in the HVDC control system, harmonics
of other frequencies in low and subsynchronous frequency band would be generated and be injected into
the AC system due to the HVDC valve switching characteristics. Especially the harmonics around the complementary frequency of the disturbances have significantly larger amplitudes than the other new generated
harmonics. Such phenomenon has not been reported
before when the modulation theory has been employed.
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