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Abstract—This paper investigates the stability of offshore wind
farms integration through modular multilevel converter-based
high-voltage dc (MMC-HVDC) transmission system. Resonances
or instability phenomena have been reported in between wind
farms and MMC-HVDC systems. They are arguably originated
due to interactions between the MMC and the wind power
inverters. However, the nature of these interactions are neither
well understood nor reported in the literature. In this article, the
impedance-based analytical approach is applied to analyze the
stability and to predict the phase margin of the interconnected
system. For that, analytical impedance models of a three-phase
MMC in compensated modulation case and direct modulation case
are separately derived by using the small-signal frequency domain
method. Moreover, the impedance models of the MMC take the
circulating current control into account. The derived impedance
models are then verified by comparing the frequency responses of
the analytical model with the impedance measured in a nonlinear
time domain simulation model developed in Matlab. The results
show that potential resonances or instability of the interconnected
system can be readily predicted through the Nyquist diagrams. In
addition, the analysis indicates that the circulating current control
of the MMC has a significant impact on the stability of the
interconnected system. Finally, the time domain simulations
validate the theoretical analysis.
Index Terms—Modular multilevel converter (MMC), high-
voltage dc (HVDC), wind farm, stability, subsynchronous
oscillation, impedance.
I. INTRODUCTION
ODULAR multilevel converter-based high-voltage dc
(MMC-HVDC) transmission technology has become a
promising solution for grid integration of large-scale offshore
wind farms, due to its advantages, such as modular design, high
efficiency, low distortion of output voltage, easily scalable in
terms of voltage levels, and so on [1]–[3].There have already
been several MMC-HVDC projects in the world, e.g., TransBay
Cable Project, Shanghai Nanhui MMC-HVDC project,
Guangdong Nan’ao 3-terminal MMC-HVDC project, and
Zhejiang Zhoushan 5-terminal MMC-HVDC project.
However, compared with conventional voltage-source
converters (VSCs), MMC has much more complex internal
Manuscript received May 15, 2015. This work was supported in part by the
National 863 program of China under Grant 2013AA050601 and in part by
Shanghai Science and Technology Committee Fund under Grant 13dz1200202.
Jing Lyu, Xu Cai are with the Wind Power Research Center, Shanghai Jiao
Tong University, Shanghai 200240, China. (e-mail: lvjing@sjtu.edu.cn;
xucai@sjtu.edu.cn).
dynamics [4], i.e., internal circulating current [5], submodule
capacitor voltage balancing [6], etc, which have a large
influence on operation stability of the interconnected system. A
subsynchronous oscillation (SSO) phenomenon in an MMC-
HVDC system with DFIG-based wind farms has been reported
in [7], in which the distribution and propagation mechanisms of
the output SSO current from wind farms in an MMC-HVDC
system were revealed and an SSO current suppression scheme
employed in the rectifier controller of the MMC-HVDC system
was also proposed. However, the mechanism by which these
oscillations are generated in the MMC-HVDC system with
wind farms is still unclear and scarcely understood. Shunt
capacitor banks and high-frequency harmonic filters in wind
power plants can create significant parallel resonance
interaction with the main transformer and any associated load
tap changing apparatus, and harmonic filters were used to
address these resonance concerns in [8]. In [9], the high-
frequency resonance issues in offshore wind farms caused by
the distributed inductance and capacitance of the long-distance
HVAC transmission cable was investigated, and a cascaded
notch-filter-based active damping method was also proposed to
eliminate the resonant peaks. In [10], subsynchronous
resonance (SSR) in DFIG-based wind farms, resulting from the
interaction between the converter control and the series-
compensated line, was studied and impacts of wind speed,
compensation level, and current controller gain on SSR were
then examined. In addition, the interaction between the wind
turbine control system and the wind farm structure was also
investigated in [11]. However, the instability mechanisms
mentioned above, which mainly depend on the physical LC
resonance configuration, could explain common harmonic
resonances whose resonant frequencies are usually above the
fundamental frequency, but wouldn’t provide insight into the
unique SSO phenomenon in an MMC-HVDC system with wind
farms. Voltage stability of offshore wind farms with VSC-
HVDC transmission was studied and the measures for system
stabilization by control modification were then provided in
[12], in which, however, the VSC-HVDC system is based on
two-level converters that lack of internal dynamics so the
instability mechanism would not be able to correctly interpret
the SSO phenomenon in the MMC-HVDC system with wind
Marta Molinas is with the Department of Engineering Cybernetics,
Norwegian University of Science and Technology (NTNU), Trondheim 7491,
Norway. (e-mail: marta.molinas@ntnu.no).
Frequency Domain Stability Analysis of MMC-
Based HVDC for Wind Farm Integration
Jing Lyu, Student Member, IEEE, Xu Cai, and Marta Molinas, Member, IEEE
M
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farms. Therefore, this original work will disclose the impact of
the internal MMC dynamics on the stability of the
interconnected system and attempt to find a new instability
mechanism for MMC-based systems.
The impedance-based stability analysis method is a simple
method for the stability analysis of complex power electronics-
based power systems [13]–[17]. The impedance- based method
decomposes the overall system into several subsystems and
models the impedance for each subsystem. Then, impedance-
based Nyquist stability criterion, which can not only determine
whether the interconnected system is stable or not, but also
predict the resonant frequency and the stability margin, is
applied to determine the stability of the overall interconnected
system. Moreover, the Nyquist map is also able to demonstrate
the impact of system parameters on stability, which has a great
significance in designing a power electronics-based system
[17]. Based on these features, the impedance-based analytical
approach is adopted to investigate the stability of the system in
this work.
Impedance modeling is the prerequisite for the impedance-
based stability analysis method. Most research has so far
focused on the impedance modeling of two-level converters
[18]–[26]. Positive- and negative-sequence impedances of two-
level grid-connected converters with current control and phase-
locked loop (PLL) dynamics were derived in [12], [19], and the
stability of wind farm integration via VSC-HVDC was then
analyzed based on the analytical sequence impedances in [12].
D-Q impedances of two-level grid-connected inverters with
feedback control and PLL were derived and measured in [20]–
[22], and an impedance-based stability analysis for three-phase
grid-connected paralleled converters was performed in [22].
The ac- and dc-side d-q impedances of a two-level VSC-HVDC
transmission system with the outer and inner control loops were
derived, respectively, and then the ac- and dc-side resonances
of the VSC-HVDC system were examined by using the
impedance-based Nyquist stability criterion in [23], [24].
Furthermore, some methods for measuring the d-q impedances
via impedance measurement circuits were proposed in [25],
[26]. It is noted that the derivation of d-q impedances of three-
phase converters is relatively easy in the case that current
controllers are implemented in d-q reference frame. However,
the major disadvantage of the d-q impedance-based stability
analysis is that it can only detect whether the system is stable or
not, but cannot determine the stability margin [27]. Therefore,
the sequence impedance models will be preferred for the
impedance-based stability analysis method. By far, very few
authors have reported the impedance modeling of an MMC.
DC-side impedances of an MMC without and with circulating
current control) were derived for dc voltage ripple prediction in
[28], but there is no mention of ac-side equivalent impedances.
This paper focuses on the stability analysis of the
interconnection of a wind farm via an MMC-HVDC system.
The impedance-based analytical approach is implemented to
investigate the stability of the interconnected system. The
analytical sequence impedance models of the three-phase MMC
are first derived with consideration of the internal MMC
dynamics. In addition, two modulation strategies, i.e.,
compensated modulation and direct modulation, have been
taken into consideration for the MMC impedance modeling.
(a)
(b)
(c)
Fig. 1. System configuration. (a) Single-line diagram of an offshore wind farm
with an MMC-HVDC transmission system. (b) Wind power conversion system
for each WTG. And (c) MMC structure for both stations of the MMC-HVDC
system.
The analytical impedance models are then verified by the
measured impedances in a nonlinear time domain simulation
model developed in Matlab. Based on the analytical impedance
models, the stability of the MMC-HVDC system connected
with a wind farm is analyzed by using the impedance-based
Nyquist stability criterion. A detailed time domain simulation
model of the interconnected system is built in Matlab to validate
the theoretical analysis results.
The rest of the paper is organized as follows. Section II
describes the interconnected system configuration. Section III
presents the impedance models derivation and verification for
the MMC. The frequency domain stability of the interconnected
system is then analyzed by the impedance-based Nyquist
stability criterion in Section IV. To validate the theoretical
analysis, the time domain simulations are carried out in Section
V. Section VI concludes the paper.
II. SYSTEM CONFIGURATION
The system configuration under study in this paper is
depicted in Fig. 1. Fig. 1(a) presents the single-line diagram of
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an offshore wind farm with an MMC-based HVDC
transmission system, in which the offshore wind farm consists
Fig. 2. Simplified configuration of the interconnected system.
of wind turbine generators (WTGs) based on two-level full-
power back-to-back converters, and the MMC-HVDC system
comprises converter transformer, submarine dc cable, wind
farm side MMC (WFMMC), and grid side MMC (GSMMC).
The wind power conversion system of each WTG is shown in
Fig. 1(b). The ac terminal voltage of each WTG is the low
voltage with 0.69 kV. Then the low voltage is stepped up to the
medium voltage with 35 kV via the step-up transformer. Fig.
1(c) depicts the MMC topology for both converter stations of
the MMC-HVDC system. Each phase-leg of the MMC consists
of one upper and one lower arm connected in series between the
dc terminals. Each arm consists of N identical series-connected
submodules (SMs), one arm inductor L and its equivalent series
resistor R. Each SM contains a half-bridge as a switching
element and a dc storage capacitor C. In high-voltage
applications, N may be as high as several hundreds [4]. The
offshore wind farm is connected to the ac terminal of the
WFMMC station, and the ac terminal of the GSMMC station is
connected to the ac power grid.
For simplicity, the interconnected system of the aggregated
wind power inverter connected with the WFMMC of the MMC-
HVDC system is studied in this paper, as shown in Fig. 2. Since
the grid-side VSC of the wind power conversion system can be
decoupled with the generator-side VSC by the dc-link capacitor
[12], it is reasonable to replace the turbine mechanical
(including the generator) and the generator-side converter with
a constant power source, as illustrated in Fig. 2. The aggregated
wind power inverter is connected to the point of common
coupling (PCC) via a step-up transformer with the voltage ratio
of 110/0.69 kV. The WFMMC is integrated to the PCC through
a converter transformer with the voltage ratio of 110/166 kV.
Assuming the dc-side voltage of the WFMMC to be constant, a
dc voltage source can be used instead of the HVDC inverter to
simulate the effect of the dc-side voltage.
III. IMPEDANCE MODELING OF THE INTERCONNECTED
SYSTEM
The analytical impedance models of the interconnected
system as shown in Fig. 2 are derived and verified by a
nonlinear time-domain simulation model in this section. The
sequence impedance models of two-level grid-connected VSCs
have been well investigated in [19], so the results of the paper
have been used for the impedance modeling of the wind power
inverter in this paper. In this section, the sequence impedance
models of the WFMMC will be derived in detail. In addition, it
Fig. 3. Single-phase equivalent circuit of the MMC.
should be noted that the positive- and negative-sequence
impedances of a three-phase balanced converter system under
normal operation can be regarded as the same when a phase-
domain control without PLL is employed [19].
A. Definition and Fundamental Relations of the MMC
The switched SM capacitors make the MMC a converter with
internal dynamics. These internal dynamics comprise all
capacitor voltages and the circulating current. The circulating
current transfers charge between the SM capacitors and thus
plays a very important role.
The single-phase equivalent circuit of a three-phase MMC in
phase-k (k=a,b,c) is presented in Fig. 3, where v
du
and v
dl
are the
positive- and negative-pole dc voltages; i
d
is the dc-side current;
k
u
i
and
k
l
i
are the upper and lower arm currents in phase-k;
k
cu
v
and
k
cl
v
are the voltages produced by the SMs in the upper and
lower arm, respectively;
k
c
i
is the circulating current in phase-k;
k
g
v
and
k
s
i
are the ac-side phase voltage and current of the MMC
in phase-k, respectively. The subscript “u” and “l” represent the
upper and lower arm, respectively.
The circulating current is defined as
2
kk
k
ul
c
ii
i
(1)
And the ac phase current can be expressed as
k k k
s u l
i i i
(2)
From the single-phase equivalent circuit in Fig. 3, one can
obtain
k
k k k
u
g u cu du
di
v L Ri v v
dt
(3)
k
k k k
l
g l cl dl
di
v L Ri v v
dt
(4)
According to the continuous model of the MMC [4], [28],
one can obtain
k k k
cu u cu
k k k
cl l cl
v n v
v n v
(5)
where
k
u
n
and
k
l
n
are the insertion indices of the upper and
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lower arms in phase-k, respectively;
k
cu
v
and
k
cl
v
are the sum
capacitor voltages of the upper and lower arms in phase-k,
respectively.
If compensated modulation is assumed, the resulting
insertion indices
k
u
n
and
k
l
n
are therefore calculated by
refk
k
du s
u
k
cu
refk
k
dl s
l
k
cl
vv
n
v
vv
n
v
(6)
where
refk
s
v
is the modulation voltage in phase-k generated by
the ac voltage control.
If however, the sum capacitor voltages of the upper and lower
arms
k
cu
v
and
k
cl
v
are approximated by a constant value (i.e.,
the dc bus voltage v
d
), the direct modulation is assumed in this
case. Then, the resulting insertion indices can be written as
refk
k
du s
u
d
refk
k
dl s
l
d
vv
n
v
vv
n
v
(7)
where v
d
is the pole-to-pole dc bus voltage,
d du dl
v v v
.
The MMC can be successfully controlled either by using (6)
or (7). However, if the MMC is assumed to be controlled by (6),
i.e., using compensated modulation, then the internal dynamics
of the MMC can be disregarded. As a result, the arm voltages
k
cu
v
and
k
cl
v
produced by the SMs are identical to their
corresponding control references, as the capacitor voltage
dynamics have been compensated. Nevertheless, it is worth
noticing that the compensated modulation is difficult to be
implemented in practice as voltage measurement delays will be
introduced by the processing speed limitations of the real
system implementation. By contrast, the direct modulation is
easy to be carried out in practice but the internal dynamics of
the MMC need to be considered in this case, e.g., the circulating
current. In this paper, the compensated modulation is only for
theoretically comparing the impedance characteristics with
those obtained by using the direct modulation.
B. Control of WFMMC
For wind farm integration through MMC-HVDC system, the
WFMMC must provide an ac power supply for the wind farm.
Hence, ac voltage control strategy is adopted in the WFMMC.
In this subsection, with ac voltage closed-loop control, the ac-
side impedance models of the WFMMC without and with
circulating current control are derived, respectively. Fig. 4
depicts the block diagram of the ac voltage closed-loop control
in abc frame employed in the WFMMC, where H
v
(s) is an ac
voltage control compensator, a proportional-resonant (PR)
controller is used to realize the zero steady-state error for a
sinusoidal quantity; k
f
is a feedforward gain for improving the
dynamic response of the control system;
refk
g
v
is the
fundamental-frequency voltage reference in phase-k.
The ac voltage control compensator H
v
(s) is expressed as
Fig. 4. Block diagram of the ac voltage closed-loop control in abc frame.
22
1
vr
v vp
Ks
H s K
s
(8)
where K
vp
and K
vr
are the proportional coefficient and the
resonant coefficient of PR controller, respectively; ω
1
is the
angular frequency of the control signal, that is the fundamental
angular frequency in this paper.
Assuming that the delay time of sampling and computing is
ignored, according to Fig. 4, the modulation voltage
refk
s
v
in
frequency-domain can be written as
refk refk k k
s v g g f g
v s H s v s v s k v s
(9)
C. Impedance Modeling of WFMMC in Compensated
Modulation Case
Subtracting (3) and (4) yields
( ) ( ) ( )
k k k k k k
u l u l cu cl d
d
L i i R i i v v v
dt
(10)
Substituting (1), (5), and (6) into (10) leads to
0
k
k
c
c
di
L Ri
dt
(11)
Equation (11) implies that the circulating current naturally
decays to zero in the steady-state when the compensated
modulation is used. As a result, the circulating current
suppression controller is not required in the compensated
modulation case.
Adding (3) and (4) yields
2 ( ) ( ) ( )
k k k k k k k
g u l u l cu cl d
d
v L i i R i i v v v
dt
(12)
where
d
v
is the imbalance dc-bus voltage, i.e.,
d du dl
v v v
,
for a three-phase MMC,
0
d
v
[4].
Substituting (2), (5), and (6) into (12) yields
22
k
k refk k
s
s s g
di
LR
i v v
dt
(13)
Equation (13) indicates that the dynamic behavior of the
MMC in compensated modulation case is identical to that of a
controlled two-level VSC. In other words, the ac-side
equivalent impedance of the MMC in compensated modulation
case is not related to the internal MMC dynamics.
Substituting (9) into (13), and then applying linearization in
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frequency-domain will result in
1
1
2
kk
s f v g
Ls R i s k H s v s
(14)
Hence, the ac-side equivalent impedance of the WFMMC in
compensated modulation case is obtained
21
k
g
MMC
k
s
fv
vs
Ls R
Zs
is
k H s
(15)
It can be observed from (15) that the ac-side equivalent
impedance of the MMC in compensated modulation case,
depending on the arm inductance, parasitic resistance, and
converter controller, is not related to the internal dynamics of
the MMC, which is essentially a controlled two-level VSC
impedance equivalent.
D. Impedance Modeling of WFMMC in Direct Modulation
Case
In this case, the internal dynamics of the MMC have been
taken into consideration. The insertion indices in (7) can be
modified as (16) by introducing the circulating current control.
refk refk
k
du s c
u
d
refk refk
k
dl s c
l
d
v v v
n
v
v v v
n
v
(16)
where
refk
c
v
is the reference voltage generated by the circulating
current control.
Combining (1), (2), (3), (4), (5), and (16), the internal
dynamics of the MMC can be described as a third-order system
[4] in the state variables
k
c
v
,
k
c
v
, and
k
c
i
:
2
(1 )
k refk k refk
k
c s s c
c
dd
dv v i v
C
i
N dt v v
(17)
22
(1 )
2
k refk k refk
k
c c s s
c
dd
dv v i v
C
i
N dt v v
(18)
1
()
22
k k refk k refk k
k
c c c c s c
dc
dd
di v v v v v
L v Ri
dt v v
(19)
where
k
c
v
is the total capacitor voltage in phase-k, i.e.,
k k k
c cu cl
v v v
;
k
c
v
is the unbalance capacitor voltage in
phase k, i.e.,
k k k
c cu cl
v v v
.
The system’s equilibrium point of the MMC described by
(17)–(19) is as follows. It is noted that the steady-state ripples
are disregarded in (20) in order to linearize the system easily.
However, this assumption has less influence on the impedance
model response, which will be further discussed next.
0 0 0
2 , 0,
3
k k k
c d c c
d
P
v v v i
v
(20)
where P is the transferred active power.
Substituting (2), (5), and (16) into (12) yields
1
20
2
k refk refk
k k k k k
s c s
g s c c c
dd
di v v
v L Ri v v v
dt v v
(21)
In the following derivation of the ac-side equivalent
impedances of the WFMMC, two cases (without and with
circulating current control) are considered separately.
1) Without Circulating Current Control: If the circulating
current control is not used, i.e.,
0
refk
c
v
, (21) can thus be
written as
1
20
2
k refk
k k k k
ss
g s c c
d
di v
v L Ri v v
dt v
(22)
When ac voltage open-loop control is used, that is, the
modulation voltage
refk
s
v
is given directly, (22) is a linear
equation in a manner. Therefore, the steady-state ripples in the
sum capacitor voltage and unbalance capacitor voltage have no
effect on the impedance model response. When ac voltage
closed-loop control is used, (22) is a nonlinear equation and the
impact of the steady-state ripples on the model response is
reflected in the fifth term on the left-hand side of (22). However,
since the steady-state ripples are generally very small, which
are also multiplied by a small perturbation and divided by a high
dc voltage, the effects of ignoring these steady-state ripples on
the model response can thus be disregarded.
Combining (9), (17), (18), and (22), then applying
linearization in frequency-domain, and rearranging, we can
obtain
1 1 1
0
k k k
v g i s c c
A v s A i s A i s
(23)
where
10
0
1
2
1
1
2
2
2
44
i
j
ref
f v c f v s
v
dd
fv
i
c
N k H s i Nm k H s I e
A
Cv s Cv s
k H s
N Nm
A sL R
Cs Cs
Nm
A
Cs
(24)
in which m is the modulation index; I
s0
and
10i
are the steady-
state components of amplitude and initial angle of
k
s
i
,
respectively.
In general, the circulating current contains a series of even
harmonic components, in which the dc and second-order
harmonic currents are the dominant components in the
circulating current. Since the effect of the ac-side perturbation
voltage and current on the active power transferred by the MMC
is very small, the resulting perturbation dc current can thus be
ignored. Therefore, the perturbation circulating current can then
be given by
2
kk
c
i s i s
(25)
where
2
k
i
is the second-order harmonic circulating current in
phase-k.
It is assumed that only the fundamental-frequency
component in the pulse pattern is considered, the capacitor
voltages are balanced at all times, and direct modulation is used.
When the dc-link of an MMC is modeled as a constant voltage
source, the second-order harmonic circulating current can be
approximated as a function of ac-side fundamental-frequency
current [29]. Hence, the corresponding perturbation of the
