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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 36, NO. 4, DECEMBER 2021 3449
A Complete Impedance Model of a PMSG-Based
Wind Energy Conversion System and Its Effect On
the Stability Analysis of MMC-HVDC Connected
Offshore Wind Farms
Tao Xue , Student Member, IEEE,JingLyu , Senior Member, IEEE,HanWang, Member, IEEE, and Xu Cai
Abstract—Impedance modeling of a permanent magnet syn-
chronous generator (PMSG)-based wind energy conversion system
(WECS) usually ignores the machine-side dynamics, instead, the
machine-side system is simplified as a constant power source, while
merely keeping the detailed grid-side converter model. This simpli-
fication is mainly based upon the intuition that the dc-bus capacitor
is large enough so that the machine- and grid-side dynamics can be
decoupled. However, the simplification probably does not hold in
practice since there is always a tendency to minimize the capacitor
size to achieve a smaller package for installation. Therefore, the
machine-side system should not be ignored, otherwise maybe lead-
ing to error for stability analysis. To address this issue, this paper
developed a complete ac-side impedance model of the PMSG-based
WECS. Then, impact factors about the coupling characteristics be-
tween machine- and grid-side systems of the PMSG-based WECS
are discussed. Finally, a case study of sub/super synchronous os-
cillation is carried out on a practical offshore wind farm inte-
grated with a modular multilevel converter based high-voltage dc
(MMC-HVDC) transmission system in China. The results show
that ignoring the machine-side system would lead to an optimistic
stability judgement, which confirms the necessity of the complete
model of PMSG-based WECS. Furthermore, the system can be
stabilized by either changing the couplings between machine- and
grid-side systems of PMSG-based WECS or implementing active
damping in the wind farm side MMC.
Index Terms—PMSG-based wind energy conversion system,
machine-side system, impedance modeling, MMC-HVDC, stability
analysis.
I. INTRODUCTION
EXPLOITATION of wind energy has rapid development in
recent years, from onshore to offshore wind farms. The
development process of wind power is closely related to the
Manuscript received December 28, 2020; revised March 12, 2021, April 6,
2021, and April 11, 2021; accepted April 17, 2021. Date of publication April 21,
2021; date of current version November 23, 2021. This work was supported in
part by the National Natural Science Foundation of China under Grant 51907125
and by Power Electronics Science and Education Development Program of Delta
Group under Grant DREG2020010. (Corresponding author: Jing Lyu.)
The authors are with the Key Laboratory of Control of Power Trans-
mission and conversion, Ministry of Education, Shanghai Jiao Tong Univer-
sity, Shanghai 200240, China (e-mail: xuetao@sjtu.edu.cn; lvjing@sjtu.edu.cn;
wanghansjtu@sjtu.edu.cn; xucai@sjtu.edu.cn).
Color versions of one or more figures in this article are available at https:
//doi.org/10.1109/TEC.2021.3074798.
Digital Object Identifier 10.1109/TEC.2021.3074798
type of wind energy conversion systems (WECSs). In the early
stages, squirrel-cage induction generator (SCIG)-based fixed-
speed WECSs were firstly developed [1]. Later, thanks to the
progress of power electronics, doubly-fed induction generator
(DFIG)-based variable speed WECSs have been widely used
in onshore wind farms and in early offshore wind farms, due
to its superior performance and low cost [2]. In recent years,
with rapid development of offshore wind power, multi-megawatt
full-power WECSs, e.g., permanent magnet synchronous gen-
erator (PMSG)-based WECS, have become the mainstream
WECS for large-scale offshore wind farms [3], [4]. Compared
with the DFIG-based WECS, the PMSG-based WECS has such
advantages as more superior grid support ability, higher power
scale, no need for gear box, and so on.
However, interactions between PMSG-based wind farms and
weak ac grids [5], series-compensated lines [6], [7] as well as line
commutated converter (LCC) [8], [9], two-level voltage-source
converter (VSC) [10] and modular multilevel converter (MMC)
[4], [11]–[14] based HVDC transmission systems have caused
serious oscillation issues and threatened the safety operation
of power systems, which have been paid much more attention
in recent years. Since the oscillations occur when the system
configurations are changing moderately, they can be classified
into small-signal stability issues according to the disturbance
type. Two methods, i.e., eigenvalue-based analysis [15] and
impedance-based analysis [13], are commonly used to analyze
the small-signal stability of renewable energy integration sys-
tems. Thanks to its easy-to-use and measurable properties, the
impedance-based analysis method has become the mainstream
approach in analyzing interactive oscillations in complex power
electronic interconnected systems.
The accurate impedance model of the PMSG-based WECS is
needed to ensure the precision of stability assessments. Whereas,
the majority of the existing studies ignored the machine-side
system (including synchronous generator and machine-side con-
verter) of the PMSG-based WECS, merely using the grid-side
converter (GSC) with a dc constant power source to represent
the characteristics of the PMSG-based WECS. For simplicity,
this paper defines the PMSG-based WECS with machine-side
system as complete model and with constant power source rep-
resenting machine-side system as simplified model. A sub/super
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3450 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 36, NO. 4, DECEMBER 2021
synchronous oscillation accident in a PMSG-based wind farm
in Northwest China was reported and analyzed by a generalized
RLC series resonance circuit [2]. This accident is a typical
interaction between PMSG-based WECS and low SCR weak
grid, commonly called sub/super synchronous control interac-
tion (SSCI). Phase-locked loop (PLL) of the grid-side converter
has great impact on the sub/super synchronous oscillations [5].
Oscillations may also occur when it is integrated to a series
compensated SCR weak grid [6], [7]. The researches about
oscillations in wind farms connected with HVDC system are
also under heated discussion.
In early stages, the small-signal stability of wind farms con-
nected to an LCC-HVDC [8], [9] or a two-level VSC-HVDC
[10] were analyzed by eigenvalue-based analysis. In recent
years, new oscillatory phenomena happened in MMC-HVDC
system with wind farms which was first reported in [4]. Subse-
quently, the interaction mechanism between the PMSG-based
wind farm and MMC-HVDC system was revealed and stabi-
lization measures on oscillation mitigation were also proposed
using the impedance-based method [11], [12], [16].
However, only a few researchers have found out that GSC is
sometimes hard to represent all the dynamics of a PMSG-based
WECS [17]–[22]. The whole topology including machine-side
system is discussed and recommended to be used in stability
analysis [17], [18]. Ref. [19] points out that simplified model of
the PMSG-based WECS may lead to error for stability analysis
in some cases, so it derives the impedances of both machine-side
system and grid-side system at dc-side, but the ac-side equiv-
alent impedance of the PMSG-based WECS is not developed.
Sequence impedance models of a PMSG-based WECS were
derived by harmonic linearization in [21]–[23], but the outer con-
trol loops of machine-side converter (MSC) were not considered
in the modeling [23] and the coupling characteristics between
the machine- and grid-side systems were not clearly explained
[21], [22]. A PMSG-based WECS considering machine-side
dynamics is established in [20], and both theoretical analysis and
time-domain simulations verify that the complete model is more
accurate in determination of stability when the PMSG-based
WECS is integrated to a low SCR weak grid. However, all those
above papers lack the influence of the complete and simplified
models on the stability analysis of the PMSG-based wind farm
integrated with an MMC-HVDC, and oscillation suppression
through changing the couplings between the machine- and grid-
side systems of the PMSG-based WECS based on coupling
characteristics analysis, which will be discussed in this paper.
The main contributions of this paper include: 1) Coupling
characteristics between the machine- and grid-side systems of
the PMSG-based WECS and impact factors are analyzed. In
this way, simplifying conditions of machine-side system into a
constant power source is discussed. 2) Stability determination
of PMSG-based wind farms integration through MMC-HVDC
with and without considering machine-side dynamics are com-
pared and advices of using complete model are given. 3) Oscil-
lation suppression through changing the couplings between the
machine- and grid-side systems of the PMSG-based WECS is
provided to stabilize the sub/super-synchronous oscillations in
the Wind Farm-MMC interconnected system.
The rest of this article is structured as follows. Section II illus-
trates a complete impedance modeling of a PMSG-based WECS
considering machine-side dynamics. The impedance model of a
wind farm connected MMC considering the internal dynamics
is also established in section III. Both the impedance models
are verified by frequency-scanning in section IV. Section V dis-
cusses about the impact factors of couplings between machine-
and grid-side systems of the PMSG-based WECS. Then, a case
study about the stability determination, mechanism analysis and
suppression methods of a PMSG-based wind farm integrated via
an MMC-HVDC is conducted in section VI. Finally, section VII
concludes this article.
II. COMPLETE IMPEDANCE MODELING OF A
PMSG-BASED WECS
A. Configuration of a PMSG-Based WECS
The system configuration of the PMSG-based WECS is
shown in Fig. 1 and the main parameters are listed in Table I
in the appendix. At grid-side, the GSC is integrated through
a filter with an inductance of Lfand a resistance of Rf.Vabc
is the output voltage of the GSC, Viabc is the voltage at the
point of common coupling (PCC) and Iabc is the output current
of the GSC. There is an impedance Zgon behalf of the grid
impedance and Vgabc is the voltage of the grid. At dc-side, a
capacitor Cdc with the voltage Vdc represents dc-side dynamics.
The GSC applies vector control with inner-loops and outer-loops
include dc voltage Vdc control and reactive power Qcontrol. PLL
provides the phase angle to realize the integration of the device.
At machine-side, the PMSG is connected to the machine-side
converter (MSC) so that power Pmcan be transmitted to the grid-
side. Imabc is the current flowing from the MSC to the PMSG
and Vmabc is the output voltage of the MSC. On the one hand,
the rotor angular speed ωmis measured and the rotor angle θm
can be calculated through an integrator. The rotor electric angle
θeneeded in Park Transformation is the result of rotor angle
multiplying by a gain npwhich represents number of pole pairs.
On the other hand, the rotor speed ωmis the input of maximum
power point tracking (MPPT), outputting the reference value of
electric torque. The MSC uses a typical control strategy, which
is the zero d-axis current control and electric torque control in
q-axis outer loop. Zero d-axis current control is used so that
the electric torque is proportional to the stator current in q-axis.
Then, the output of electric torque control can be the reference of
the q-axis current control, and it balances mechanical torque and
electric torque so as to control the output power of the generator
[24].
B. Impedance Modeling Process
The whole derivation process is shown in Fig. 2, and the
idea of modularization and multi-port modeling techniques is
used: different parts of the PMSG-based WECS are modeled
separately as modules and assembled together on different ports.
Its circuit Detailed derivation steps of PMSG-based WECS
and grid-tied VSC could be found in [20] and [25]. Circuit
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XUE et al.: COMPLETE IMPEDANCE MODEL OF A PMSG-BASED WIND ENERGY CONVERSION SYSTEM AND ITS EFFECT 3451
Fig. 1. Configuration of a PMSG-based WECS.
Fig. 2. Derivation process of PMSG-based WECS impedance.
explanation is shown in Fig. 3. The detailed modeling process
is properly omitted here due to limited space.
1) Step One: DC-Side Admittance Modeling: Firstly, derive
the dc-side admittance of the machine-side YdcM to represent the
whole machine-side system. The synchro-nous generator equa-
tion and the power balance equation between dc and ac side are
(1) and (2). In the equations below, subscripts d,qare variables
at dq domain, subscript ref means reference value, subscript m
refers to machine-side variable, sis Laplace calculator and ωe
is the rotor electric angular speed.
Vmd
Vmq =Rm+sLm−ωeLm
ωeLmRm+sLm
Z1
dq
Imd
Imq (1)
Fig. 3. Circuit explanation of the modeling process.
Idcm =3
2
VmdImd +Vmq Imq
Vdc
(2)
The outer and inner loop control equations are (3) and (4).
Imd
Imq =⎡
⎢
⎢
⎣
0
Kpt +Kit
s
Ht
(Teref −Te)⎤
⎥
⎥
⎦(3)
Vmd
Vmq =Vdc/2
Vdc0/2Kpim +Kiim
s
Hmc
Imdref −Imd
Imqref −Imq (4)
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3452 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 36, NO. 4, DECEMBER 2021
In the above equation, Vdc/2 is the gain of the power converter,
which is a measured value, and Vdc0/2 is the converter gain of
uniformization before SVPWM, which is a constant value. This
modulation is called direct modulation (DM). If Vdc/2 is the
transfer function of uniformization before SVPWM, which is a
measured value, it will be called compensated modulation (CM).
Rmand Lmare the stator resistance and mutual inductance of the
PMSG. Kpt,Kit and Kpim ,Kiim are the proportional and integral
gains of electric torque and current controller, respectively.
Small-signal linearizing (1)-(4), and then substituting (1) and
(3) into (4), YdcM can be calculated through solving the equation
set of (2) and (4). It is a process of canceling intermediate vari-
ables and finding the relations between dc voltage and current.
Then, YdcM is regarded as a source admittance in paralleled
with dc capacitor as the circuit explanation in Fig. 3 with its
expression in (5). Every matrix used in (5) is explained in the
appendix. In Eq. (5), it can be seen that if the machine-side
system is not considered, the equation of YdcM is only an
admittance of a constant power source (YdcM_sim in Fig. 3).
In addition to this, CM can also cause the simplified form of
YdcM even if the machine-side dynamics are considered, which
will be discussed in the appendix.
YdcM =−Pm
(Vdc)2(Simplif ied)
−Pm
(Vdc)2−YM(Complete)(5)
However, the complete model equation of YdcM (YdcM_com
in Fig. (3) is more complex and the extra items represent the
couplings between the machine- and grid-side systems, which
also have an impact on the ac-side impedance of the PMSG-
based WECS.
2) Step Two: AC-Side Impedance Modeling: The impedance
of the PMSG-based WECS can be calculated through the
impedance modeling method of a grid-tied VSC,
Idref
Iqref =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Kpvdc +Kivdc
s
Hdc
(Vdc −Vdcref )
Kpq +Kiq
s
Hq
(Q−Qref )
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(6)
Vc
d
Vc
q=Vdc/2
Vdc0/2Kpi +Kii
s
Hc
Idref −Ic
d
Iqref −Ic
q(7)
which consists of different control loops. The outer and inner
control loop equations are (6) and (7). In the above equa-
tion, Kpdc ,Kidc ,Kpq ,Kiq, and Kpi ,Kii are the proportional
and integral gains of dc voltage, reactive power and current
controller, respectively. The dc voltage dynamics equation and
power balance equations are (8) and (9).
Cdc
dVdc
dt =−YdcM Vdc −Idc (8)
Idc =3
2
Vs
dIs
d+Vs
qIs
q
Vdc
(9)
Filter dynamics equation is (10).
Vs
d
Vs
q=Vs
id
Vs
iq +Rf+sLf−ωLf
ωLfRf+sLf
Zdqf
Is
d
Is
q(10)
PLL equations are (11) and (12) and every variable with a
tilde represents its small-signal form.
Vc
d
Vc
q=
Vs
d
Vs
q+0Vs
qGpll
0−Vs
dGpll
Vs
id
Vs
iq (11)
Ic
d
Ic
q=
Is
d
Is
q+0Vs
qGpll
0−Vs
dGpll
Vs
id
Vs
iq (12)
In the above equations, Gpll refers to (13), where Kppll ,Kipll
are the proportional and integral gains of PLL.
Gpll =1
Vs
id
Kppll +Kipll/s
s+Kppll +Kipll/s (13)
Small-signal linearizing (6)-(10) first, and then transform
variables under electrical coordinates (superscript s) into control
coordinates (superscript c)by using PLL equation (11) and (12).
Next, substituting (8) into (9), and (6) into (7), we can obtain the
overall ac-side impedance at PCC by solving the equation set of
(7), (9), and (10). It is a process of canceling intermediate vari-
ables and finding the relations between ac voltage and current
as shown in Fig. 2. The ac-side impedance expression is shown
in (14) and every matrix used in it is explained in the appendix
(Iis identity matrix).
ZacW =A−1B−C(sCdc +YdcM +H)−1G−1
(14)
In Eq. (14), YdcM has influences on ac-side impedance ZacW
to different extents determined by both electrical and control
systems, which will be discussed precisely in section V.
III. IMPEDANCE MODELING OF WIND FARM SIDE MMC
Impedance modeling of a wind farm side MMC is well
established in [13] based on harmonic state-space (HSS), so
the derivation process is highly omitted due to limited space.
The system configuration of the wind farm side MMC is shown
in Fig. 4 and the main parameters are listed in Table II in the
appendix. The common-mode current icom , sum voltages of the
upper and lower arm submodule ucpΣand ucnΣas well as ac-side
current isare selected as state variables. The state-space equation
of the MMC is expressed as (15), where a variable with a dot
over represents its derivative.
˙
X(t)=A(t)X(t)+B(t)U(t)(15)
Harmonic values of state variables in perturbation at different
frequencies can be calculated by the combination of HSS/HTF
as well as small-signal perturbation and linearization. The state-
space equation changes its form after small-signal perturbation
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XUE et al.: COMPLETE IMPEDANCE MODEL OF A PMSG-BASED WIND ENERGY CONVERSION SYSTEM AND ITS EFFECT 3453
Fig. 4. System configuration of wind farm-side MMC converter.
and linearization as follows (matrix Bis constant):
˜
˙
X(t)=A˜
X(t)= ˜
A(t)X+B˜
U(t)(16)
By applying the HSS modeling procedure to (16), the time-
domain equations are transferred into frequency domain and
the harmonic values of perturbed state variables at different
frequencies can be calculated by (17), where Trefers to a
function to get the Toeplitz form of the matrix.
˜
X(ω)=−T
A(ω)−N(ω)−1T[B(ω)]
U(ω)(17)
The positive- and negative-sequence impedances can be ob-
tained through dividing harmonic currents of +1 and -1 order
by the perturbation voltage, where subscripts ‘pos’ and ‘neg’
represent positive and negative sequences.
Zpos =Vs+1(ωp)
Is+1(ωp)Zneg =Vs−1(ωn)
Is−1(ωn)(18)
IV. VERIFICATION OF IMPEDANCE MODELS
A. Transformation Among Different Impedance Models
The derived PMSG-based WECS impedance above is dq
impedance, which is a multi-input multi-output (MIMO)
impedance. In this section, the MIMO impedance model in dq
domain will be transformed into the single-input single-output
(SISO) impedance model in sequence domain based on the
method illustrated in [26]. In this way, the traditional impedance
stability criterion can be used instead of the complex Generalized
Nyquist Criterion (GNC). As Fig. 5 shows, this process involves
a linear transformation and a model reduction.
B. Verification of Impedance Models of PMSG-Based WECS
and Wind Farm Side MMC
In order to validate the derived impedance models above, the
impedance measurements in the simulation have been carried out
to compare with those analytical impedance characteristics. In
Fig. 5. Transformation among different impedance models.
the simulation, the dc-side admittance of machine-side system
is measured by means of injecting a series of small perturba-
tion signals at different frequencies with the interval of 1 Hz
at dc-side, while the ac-side impedances of the PMSG-based
WECS and the wind farm side MMC are both measured through
injecting perturbations at PCC, respectively. Then by measuring
the resulting perturbation currents, the small-signal impedances
can be calculated for each frequency. The original simulation
parameters of the PMSG-based WECS and the wind farm side
MMC are listed in Table I.
Fig. 6(a) shows the analytical and measured dc-side ad-
mittances of the machine-side system when the complete and
simplified models are used. ‘Com’ and ‘Sim’ in legends re-
fer to complete and simplified models, respectively. By com-
paring the curves of different models, it is observed that the
machine-side system could not be simplified into a constant
power source because the impedance models differ from each
other in the frequency range higher than 10 Hz. Fig. 6(b) and
(c) shows the analytical and measured ac-side impedances of
the complete and simplified PMSG-based WECS, respectively,
where ‘P’ and ‘N’ represent the positive- and negative-sequence
impedances, respectively. Besides, the positive- and negative-
sequence impedances of the wind farm side MMC are verified
in Fig. 6(d). It can be seen that the analytical impedances
match well with the measured ones, which validates the derived
analytical impedance models.
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3454 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 36, NO. 4, DECEMBER 2021
Fig. 6. Verification of impedance models. (a) dc-side admittances of PMSG-
based WECS, (b) ac-side impedances of PMSG-based WECS with complete
model, (c) ac-side impedances of PMSG-based WECS with simplified model,
(d) ac-side impedances of the MMC.
Fig. 7. Impacts of the machine-side system on the ac-side impedances of the
PMSG-based WECS.
Fig. 8. Impacts of the dc-bus capacitance. (a) 50% value of the original
capacitance, (b) 200% value of the original capacitance.
V. C OUPLING CHARACTERISTICS AND ITS IMPACT FACTOR S
ANALYSIS BETWEEN MACHINE-AND GRID-SIDE SYSTEMS OF
THE PMSG-BASED WECS
Dynamics of the machine-side system are represented by an
admittance and coupled with the gird-side system. According
to the dc-side admittance equation (5), the main impact factors
are the dc-bus capacitance, current controller and the electric
torque controller of the MSC. Therefore, the following parts
of this section will discuss the couplings of the machine- and
grid-side under the same operating point.
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XUE et al.: COMPLETE IMPEDANCE MODEL OF A PMSG-BASED WIND ENERGY CONVERSION SYSTEM AND ITS EFFECT 3455
Fig. 9. Impacts of the current controller of the MSC. (a) controller bandwidth
200 Hz, (b) controller bandwidth 300 Hz.
A. Impacts of the Whole Machine-Side System
In Fig. 7, impacts of the whole machine-side system
on the positive- and negative-sequence impedances of the
PMSG- based WECS are compared. For positive-sequence
impedance (blue and red curves), differences exist in sub/super-
synchronous range and frequency range of several hundred Hz.
For negative-sequence impedance (black and green curves),
differences can be neglected. Therefore, stability determination
of sub/super-synchronous and medium frequency oscillations
may have wrong results if the simplified model is used.
B. Impacts of the DC-Bus Capacitance
The DC link capacitance is selected based on two aspects:
1) the output following capability and 2) the anti-disturbance
capability, which is in line with the current industry practice.
Impacts of the dc-bus capacitance are shown in Fig. 8, (a)
and (b) display the results when the dc-bus capacitance de-
creases to half and increases to 2 times. It is obvious that
the discrepancies of both the positive- and negative-sequence
impedances between the complete and simplified model become
larger and discrepancies start to appear in mid frequency range
of negative-sequence impedance when the dc-bus capacitance
becomes smaller. Therefore, only when the dc-bus capacitance
is large enough can the machine-side system be simplified into
a constant power source.
C. Impacts of the Current Controller of the MSC
The original current controller bandwidth of the MSC is 100
Hz. When the bandwidth increases 100 and 200 Hz respectively,
Fig. 10. Impacts of the electric torque controller of the MSC. (a) controller
bandwidth 15 Hz, (b) controller bandwidth 25 Hz.
the results are shown in Fig. 9. It can be observed that as
the bandwidth of the current controller of the MSC increases,
the discrepancies between the complete and simplified models
become smaller, especially in sub/super synchronous range.
D. Impacts of the Electric Torque Controller of the MSC
The electric torque controller usually operates at a lower
bandwidth and the original value is 5 Hz. Fig. 10(a) and (b) show
the cases that the electric torque controller bandwidth increases
to 15 Hz and increases to 25 Hz. Similar to the current controller,
the discrepancies become smaller when the bandwidth becomes
larger. But the impacts of torque controller are less obvious than
the current controller.
VI. A CASE STUDY ABOUT THE STABILITY DETERMINATION IN
APMSG-BASED WIND FARM INTEGRATION VIA
MMC-HVDC SYSTEM
In this section, a case study based on the practical Rudong
offshore wind farms with MMC-HVDC in China is implemented
to show how the different models of the PMSG-based WECS
affect the stability of the interconnected system. The system
structure diagram is presented in Fig. 11. The offshore wind
farm consists of 267 PMSG-based WECSs, which is aggregated
into one PMSG-based WECS. The output active and reactive
power of the wind farm is 35% and 0% of the rated power.
The main parameters of the PMSG-based wind farm and MMC
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3456 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 36, NO. 4, DECEMBER 2021
Fig. 11. A PMSG-based Wind Farm integration via an MMC-HVDC system.
Fig. 12. Theoretical analysis of the case study.
converter station used in the practical Rudong project are listed
in Appendix Table I, II and III.
A. Part one: An Unstable Case
Fig. 12 presents the theoretical analysis of the small-signal
stability of the PMSG-based wind farm connected with MMC-
HVDC. As can be seen, the magnitude-frequency curve (red
line) of the wind farm with complete model of PMSG-based
WECS will intersect with that (blue line) of the wind farm side
MMC at 84 Hz, where the phase margin of the interconnected
system is less than zero, indicating that the system is unstable.
However, no intersections occur if the simplified model of the
PMSG-based WECS is used for the wind farm (black line),
which indicates that the interconnected system is stable.
The analytical prediction is verified by the time-domain sim-
ulation results in Fig. 13 in which the three-phase voltages,
currents as well as active and reactive power at PCC are pre-
sented. At 0.2s, the wind farm starts to transmit power to the
MMC-HVDC system, where the output active power is 35% of
the rated power. It can be seen that the system is unstable under
the same conditions used for Fig. 12, which is consistent with
the predicted result using the complete model of PMSG-based
WECS. The frequency analysis in Fig. 13 (d) shows that the
oscillatory frequency in the PCC current is about 83 Hz, which
confirms the theoretical analysis. In addition, it is noted that the
frequency component of 17 Hz is the result of the mirror fre-
quency coupling [26]. Therefore, the effectiveness and necessity
of the complete impedance model of a PMSG-based WECS in
stability analysis has been confirmed.
B. Part two: Stabilization through Changing the Couplings
As shown in Fig. 12, the magnitude-frequency curve of the
wind farm impedance with complete model will intersect with
that of the wind farm side MMC impedance, but the wind farm
impedance with simplified model does not. Consequently, we
can weaken the couplings between the machine- and grid-side
by changing certain parameters to make the complete model
closer to the simplified model so as to avoid the intersections.
Fig. 14 shows that the impedance-based analysis and simu-
lation results of the wind farm-MMC interconnected system by
changing the bandwidth of the current controller of the MSC
from 100 Hz to 300 Hz. In Fig. 14 (a), the green line represents
the wind farm impedance with the MSC current controller
bandwidth of 300 Hz. As can be seen, the intersections between
the impedance magnitudes of the wind farm and the wind farm
side MMC are avoided by changing the controller parameters
of the MSC so as to make the interconnected system stable.
The simulation results in Fig. 14(b)–(d) have also validated the
theoretical analysis. The simulation from 0 to 2s is the same as
Fig. 13. At 2s, the bandwidth of the current controller of the
MSC changes from 100 Hz to 300 Hz so that the oscillations
gradually converges and the system becomes stable again.
C. Part three: Oscillation Mechanism and Suppression
The system under study is a wind farm-MMC interconnected
system, which is a typical two-terminal power electronic inter-
connected system, shown in Fig. 11. In this system, the wind
farm side MMC controls the ac voltage at the PCC of the wind
farm and the wind farm side MMC to provide the grid for
the wind farm. Therefore, the unstable modes are influenced
by both wind farm and MMC sides. According to Fig. 12, the
interaction instability is mainly caused by the resonance peak of
the impedance magnitude of the wind farm side MMC at about
84 Hz. Besides, it is noted that the resonance peak at about 16
Hz could cause the interaction instability as well. The above
two resonance peaks of the impedance magnitude of the wind
farm side MMC are originated from the internal dynamics of the
MMC [13].
A suppression method called ‘virtual arm resistance (VAR)’
to stabilize the sub/super-synchronous oscillations caused by
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XUE et al.: COMPLETE IMPEDANCE MODEL OF A PMSG-BASED WIND ENERGY CONVERSION SYSTEM AND ITS EFFECT 3457
Fig. 13. Time-domain simulation results of the case study. (a) Three-phase voltages at PCC, (b) Three-phase currents at PCC, (c) Active and reactive power at
PCC, and (d) Frequency analysis of the PCC current.
Fig. 14. Theoretical and simulation results of stabilizing the oscillation by weakening the couplings: (a) Theoretical analysis, (b) Three-phase voltages at PCC,
(c) Three-phase currents at PCC, and (d) Active and reactive power at PCC.
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3458 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 36, NO. 4, DECEMBER 2021
Fig. 15. Control diagram of virtual arm resistance method.
the MMC internal dynamics was proposed [11], whose control
diagram is shown in Fig. 15. In the diagram, Icoma bc and Icom abc
∗
are the common-mode currents and their references, Rvis the
proportional gain that can be regarded as a virtual arm resistance,
and the output Vdiff˙extra is added to the original modulation
voltages. In this way, a virtual resistance is connected in series to
each arm of the MMC, and circulating currents will be restrained
so that the resonance peaks on the impedance curves will be
suppressed.
After implementing this method, the theoretical impedance
analysis and simulation results are shown in Fig. 16. As can be
seen from Fig. 16 (a), the two resonance peaks of the MMC
at 16 Hz and 84 Hz are effectively reduced by applying the
virtual arm resistance method, thus avoiding the intersections
between the two impedance magnitude curves so as to make
the interconnected system stable. The time-domain simulation
results in Fig. 16 (b)-(d) have validated the theoretical analysis.
The simulation from 0 to 2s is the same as Fig. 13. At 2s, the
VAR scheme is implemented so that the oscillations converge
and the system becomes stable again.
VII. CONCLUSION
This paper developed the complete ac-side impedance model
of a PMSG-based WECS considering the machine-side sys-
tem. The analytical impedance models are first validated by
frequency scanning. Then, the coupling characteristics between
the machine- and grid-side systems are analyzed by representing
the machine-side system as a dc-bus paralleled admittance, and
the impact factors about the couplings between machine- and
grid-side systems of the PMSG-based WECS are discussed,
which can serve as a useful reference for the simplification of
a PMSG-based WECS. In the end, a case study of sub/super-
synchronous oscillation is carried out on a practical MMC-
HVDC connected PMSG-based offshore wind farm in China
to show the necessity of the complete model of PMSG-based
WECS in stability determination. The detailed findings and
conclusions of this paper are listed below:
1) The ac-side impedance characteristics of the PMSG-based
WECS considering machine-side dynamics differ from the
simplified model without consideration of the machine-
side dynamics in sub/super-synchronous and medium fre-
quency range (several hundred Hz).
2) A larger dc-bus capacitance, a faster bandwidth of MSC
current and electric torque controller can weaken the cou-
plings between the machine- and grid-side systems, so the
machine-side system could be simplified into a constant
power source without losing much accuracy.
3) The simplified model would lead to an optimistic stability
judgement in PMSG-based wind farm integration. It is
suggested that the complete impedance model of a PMSG-
based WECS considering the machine-side system is used
in stability assessment of PMSG-based wind farms.
4) Sub/super-synchronous oscillations in the Wind Farm-
MMC interconnected system could be suppressed by prop-
erly changing the couplings between the machine- and
grid-side systems of the PMSG-based WECS.
APPENDIX
A. Matrices in Detail
Matrices in equation (5):
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
c=−Hmc 0
0HmcHt3
2npψr+Hmc
Z2
dq
−1Vmd
Vdc
Vmq
Vdc T
d=3Imd Imq −Vmd Vmq (Z2
dq)−1
2Vdc
e=Z1
dq −1+Z2
dq −1−1
YM=3Vmd Vmq
2Vdc
c+dec
Matrices in equation (14):
⎧
⎪
⎪
⎨
⎪
⎪
⎩
Vdcq =3
2Hq00
Vs
iq −Vs
id Idcq =3
2Hq00
−Is
qIs
d
GIpll =0Is
qGpll
0−Is
dGpll GUpll =0Vs
qGpll
0−Vs
dGpll
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
A=HcI+Zdqf −HcVdcq
B=I+HcGIpll +GUpll −HcIdcq
C=−Hc!Hdc 0"T−!Vs
d/Vdc Vs
q/Vdc "T
D=3!Is
dIs
q"/(2Vdc)F=−P/(Vdc)2
E=3#!Vs
dVs
q"+!Is
dIs
q"Zdqf $/(2Vdc)
G=D−EA−1BH=F−EA−1C
Matrices in equation (17):
Harmonic version of modulation indices:
Mp=⎡
⎣
Mpa
Mpb
Mpc ⎤
⎦Mn=⎡
⎣
Mna
Mnb
Mnc ⎤
⎦
where Mpa and Mna represent the Toeplitz matrices of phase A
modulation indices of the upper and lower arm of the MMC.
Harmonic versions of control systems are listed below, where
Kpc ,Kic,Kpv ,Kiv, and Kpcc sc ,Kiccsc are the proportional and
integral gains of current, voltage and CCSC controller, respec-
tively. ‘diag’ means forming a diagonal matrix and ‘h’ represents
-4 to +4 harmonic orders.
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
Hc=diagKpc +Kic
s−jhω1 9×9
Hv=diagKpv +Kiv
s−jhω1 9×9
Gd=diag!e−(s−jhω1)Td"9×9
Hccsc=diagKpccsc +Kiccsc
s−jhω1 9×9
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XUE et al.: COMPLETE IMPEDANCE MODEL OF A PMSG-BASED WIND ENERGY CONVERSION SYSTEM AND ITS EFFECT 3459
Fig. 16. Theoretical and simulation results of stabilizing the oscillation by implementing virtual arm resistance: (a) Theoretical analysis, (b) Three-phase voltages
at PCC, (c) Three-phase currents at PCC, and (d) Active and reactive power at PCC.
TFv
Δ
=−Tinv Gd
GdHc
HcHv
HvTp
TFccsc
Δ
=−Tinv2n Gd
GdHccsc
Hccsc Tp2n
TFvc
Δ
=−Tinv Gd
GdHc
HcTp
−Tinv Gd
GdHc
HcHv
HvTpZL
Harmonic state matrices:
[X(ω)]108×1=⎡
⎢
⎢
⎣
icomabc27×1
ucpSabc 27×1
ucnSabc 27×1
isabc27×1
⎤
⎥
⎥
⎦
where icom a bc ,ucpΣabc,ucnΣabc ,isabc represent the harmonic
matrices (-4 to +4 order) of four state variables in three phases.
Upa =⎡
⎢
⎢
⎢
⎢
⎣
0
···
Vp
···
0
⎤
⎥
⎥
⎥
⎥
⎦9×1
Upb =⎡
⎢
⎢
⎢
⎢
⎣
0
···
Vpe−j2
3π
···
0
⎤
⎥
⎥
⎥
⎥
⎦9×1
Upc
=⎡
⎢
⎢
⎢
⎢
⎣
0
···
Vpe+j2
3π
···
0
⎤
⎥
⎥
⎥
⎥
⎦9×1
⎧
⎨
⎩
U(ω)=!081×1Upa Upb Upc "T
N(ω)=diag!Wp··· Wp"108×108
Wp=diag[(s−jhω1)]9×9
B. Influence of CM on the Couplings
If direct modulation has been implemented like Eq. (4),
small-signal linearization will be conducted as follows. In this
equation, voltages and currents of the machine-side system and
dc voltage are coupled.
−Z2
dq Imd
Imq =Vmd
Vmq +1
Vdc0Vmd
Vmq Vdc
But if compensated modulation has been implemented, we
can get:
Vmd
Vmq =Vdc/2
Vdc/2Hmc Imdref −Imd
Imqref −Imq =Hmc Imdref −Imd
Imqref −Imq
Then small-signal linearization will be conducted as follows.
In the following equation, voltages and currents of the machine-
side system and dc voltage are not coupled with each other. In
this way, machine-side system can be simplified into a constant
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3460 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 36, NO. 4, DECEMBER 2021
power source.
−Z2
dq Imd
Imq =Vmd
Vmq
C. Parameters Used in This Paper
TAB L E I
PARAMETERS OF THE PMSG-BASED WECS
TAB L E I I
PARAMETERS OF THE WIND FARM SIDE MMC
TABLE III
PARAMETERS OF THE WIND FARM
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Tao Xue (Student Member, IEEE) received the
B.Eng. degree in electrical engineering and its au-
tomation from North China Electric Power Univer-
sity, Beijing, China, in 2018 and the M.Eng. degree
in electrical engineering from Shanghai Jiao Tong
University, Shanghai, China, in 2021. His current re-
search interests include modeling of power electronic
devices, wind farm integration, MMC-HVDC, and
stability issues in future power systems.
Jing Lyu (Senior Member,IEEE) received the B.Eng.
degree in electrical engineering and automation from
the China University of Mining and Technology,
Xuzhou, China, in 2009, the M.Eng. and Ph.D. de-
grees in electrical engineering from Shanghai Jiao
Tong University, Shanghai, China, in 2011 and 2016,
respectively. From 2016 to 2017, he was a Post-
doctoral Research Fellow with the Department of
Engineering Cybernetics, Norwegian University of
Science and Technology, Trondheim, Norway. Since
2018, he has been a tenure-track Assistant Professor
with the Department of Electrical Engineering, Shanghai Jiao Tong University.
His current research interests include dynamic stability of MMC-based HVDC
connected offshore wind farms and application of artificial intelligence in power
electronic systems.
Han Wang (Member, IEEE) received the B.Sc. de-
gree in electrical engineering from the China Univer-
sity of Mining and Technology, Xuzhou, China, in
2005, the M.Sc. and Ph.D. degrees in electrical engi-
neering from Shanghai Jiao Tong University, Shang-
hai, China, in 2008 and 2013, respectively. During
2014-2019, he was an Engineer with Shanghai Mit-
subishi Elevator Company, Shanghai Electric Group.
Since 2019, he has been a Postdoctoral Research Fel-
low with the Department of Electrical Engineering,
Shanghai Jiao Tong University. His research inter-
ests include high power converters, renewable energy generation, and grid
integration.
Xu Cai received the B.Eng. degree in electrical engi-
neering from Southeast University, Nanjing, China, in
1983, and the M.Eng. and Ph.D. degrees in electrical
engineering from the China University of Mining
and Technology, Xuzhou, China, in 1988 and 2000,
respectively. From 1989 to 2001, he was with the De-
partment of Electrical Engineering, China University
of Mining and Technology, as an Associate Professor.
From 2010 to 2013, he was the Vice Director of
the State Energy Smart Grid R&D Center, Shanghai,
China. Since 2002, he has been with Shanghai Jiao
Tong University, Shanghai, as a Professor, where he has also been the Director
of the Wind Power Research Center since 2008. His current research interests
include power electronics and renewable energy exploitation and utilization,
including wind power converters, wind turbine control system, large power
battery storage systems, clustering of wind farms and its control system, and
grid integration.
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