Modal analysis of a grid-connected DFIG-based WT considering multi-timescale control interactions

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Modal analysis of a grid-connected DFIG-based WT considering multi-timescale
control interactions
Minhui Wan, Jiabing Hu
State Key Laboratory of Advanced Electromagnetic Engineering and Technology, Huazhong University of Science and
Technology, Wuhan 430074, People’s Republic of China
E-mail: j.hu@hust.edu.cn
Published in The Journal of Engineering; Received on 9th October 2017; Accepted on 2nd November 2017
Abstract: In a grid-connected doubly fed induction generator (DFIG)-based wind turbine system, the multi-timescale control loops of DFIGs
work together to keep the output active/reactive power tracking their references. In weak AC grid condition where the grid impedance cannot
be neglected, interactions among those control loops are very strong. This study presents a modal analysis of the system containing a DFIG.
The DFIG model for analysis considers multi-timescale control and its interactions. The effect of bandwidths of rotor speed control, phase-
locked loop, DC-link voltage control and rotor current control on system stability is presented. Interactions among those control loops are
explored. The analysis results indicate that the interactions among multi-timescale control are strong and will have an effect on the stability
and the dynamics of the DFIG system under weak AC grid condition.
Nomenclature
L
s
,L
r
stator and rotor inductances
L
m
magnetising inductance
R
s,
R
r
stator and rotor resistances
L
RL,
R
RL
choke circuit inductance and resistance
L
TL,
R
TL
transmission line inductance and resistance
C
f
filter capacitance
C
dc
DC-link capacitance
H
t,
H
g
turbine and generator inertias
K
sh,
Dshaft stiffness and damping coefficients
ω
t,
ω
r
turbine and rotor angular frequencies
ω
I
synchronous angular frequency
ω
base
base angular frequency
ω
PLL
angular frequency of PLL-synchronised
reference frame
θ
PLL
PLL output angle in synchronous rotating dq
reference frame
θ
tw
shaft twist angle
T
m,
T
e,
T
sh
mechanical, electromagnetic and shaft torques
P
r,
P
g
rotor and GSC active power
V
dc
DC-link voltage
V
s
terminal voltage magnitude
V
s
,V
r
,V
g
,Vstator, rotor, GSC, and grid voltage vector
I
s
,I
r
,I
g
,Istator, rotor, GSC, and grid current vector
ψ
s
,ψ
r
stator and rotor flux vector
PI
j
=K
Pj
+K
ij
/s transfer function of a generic PI controller
(j=1, 2, …,6)
Superscripts
* reference value
p signals measured in PLL-synchronised reference frame
Subscripts
d, q direct- and quadrature-axis components
s, r stator and rotor components
g, t GSC and turbine components
1 Introduction
With the progress of the human civilisation, the contradiction
between economic development and environmental protection
becomes increasingly acute. An effective solution, which is
adapted worldwide, is to promote the development of new energy
sources. The penetration rate of renewable energy power generation
in the power system, such as wind power generation and photo-
voltaic power generation is increasing rapidly [1]. Among all types
of wind turbines (WTs), DFIGs have more superior characteristics,
which bring it very broad application prospects [2]. The dynamics
of the energy storing element in the system depends on the multi-
timescale control loops of the rotor-side converter (RSC) and the
grid-side converter (GSC) in a DFIG. Commonly, DFIGs are
forced to operate in weak AC grid condition because of long-
distance power transmission [3]. Interactions among those control
loops are very strong in that case [4]. The oscillation modes domi-
nated by one control loop are also affected by other control loops.
Multi-timescale control brings great challenge to the modelling
and analysis of the DFIG system. However, as mentioned in [5], os-
cillation phenomenon may occur in the DFIG system under weak
grid condition. Serious oscillation accidents will cause considerable
damage to the DFIG itself and even the power network. The cause
of oscillation is not clear yet. Therefore, it is a significant work to
figure out the inherent mechanism of DFIGs.
Some studies have already tried to study the stability and the
dynamics of DFIGs. Usually a reduced-order model of DFIG is
adapted in the analysis of the power system containing DFIGs
[6–9]. In [6], the effect of operating point and grid strength on
system stability is explored. In [7], the interaction between a
DFIG and a synchronous machine is studied through modal analysis
of the two machine system. In [8,9], a reduced-order DFIG model
in DC-link voltage control timescale is compared with the detailed
model in terms of eigenvalues. Similarly, the oscillation modes of a
more detailed model are presented in [10]. However, in previous
studies, the effect of multi-timescale control and its interactions
The 6th International Conference on Renewable Power Generation (RPG)
19–20 October 2017
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on system stability is usually overlooked. Therefore, this paper
will present a modal analysis of the grid-connected DFIG-
based WT system considering multi-timescale control and its
interactions.
The rest of this paper is organised as follows: A detailed model of
a grid-connected DFIG-based WT system considering multi-
timescale control schemes is described in mathematical form in
Section 2. Modal analysis results of the detailed model are dis-
played in Section 3 to study the effect of bandwidths of rotor
speed control, phase-locked loop (PLL), DC-link voltage control
(DVC) and rotor current control (RCC) on system stability, and
interactions among those control loops. Finally, conclusions are
drawn in Section 4.
2 Mathematical description of a grid-connected DFIG-based
WT system
2.1 Scheme of a grid-connected DFIG-based WT system
Fig. 1adepicts the topology of the DFIG system [11]. The syn-
chronous machine in the DFIG performs the conversion of mechan-
ical energy into electrical energy. The stator of the generator is
connected directly to the grid, while the rotor through converters
and the choke circuit.
The control scheme of the DFIG system is shown in Fig. 1b
[12,13]. Proportional–integral (PI) controllers are used in both
RSC and GSC to realise two cascaded control. Concretely, rotor
speed control, terminal voltage control (TVC) and RCC are
included in RSC, while DVC and grid-side current control (GCC)
are included in GSC. The d-axis is oriented to the stator voltage
vector in PLL [14].
2.2 Detailed mathematical model of the DFIG system
The direction of current and flux are defined as follows: positive
current flows to the machine and produces positive flux. The
voltage equations and magnetic linkage equations of the generator
can be summarised in the vector form in the synchronous rotating
dq reference frame
Vs=RsIs+d
c
s
dt+j
v
1
c
s
Vr=RrIr+d
c
r
dt+j
v
1−
v
r

c
r
⎧
⎪
⎨
⎪
⎩
(1)
c
s=LsIs+LmIr
c
r=LmIs+LrIr
(2)
As for the mechanical parts, the simplified two-mass drive train
model is widely used in system stability analysis [15]. The
turbine, gearbox, shaft, and rotor of the generator are modelled as
two masses, a spring and a damper in (3)–(6)
Tm−Tsh =2Ht
d
v
t
dt(3)
Tsh −Te=2Hg
d
v
r
dt(4)
d
u
tw
dt=
v
t−
v
r

v
base (5)
Tsh =Ksh
u
tw +D
u
tw
dt(6)
where the electromagnetic torque can be written as
Te=
c
sqisd −
c
sdisq (7)
The imbalance of power transmitted by the RSC and GSC leads to
the variation of DC capacitor voltage [16]. The active power in the
synchronous rotating dq reference frame can be written as
Pr=vrdird +vrq irq (8)
Pg=vgdigd +vgd igq (9)
and the DC voltage can be obtained by a differential equation as
Pg−Pr=CdcVdc
dVdc
dt(10)
The dynamics of choke circuit, filter capacitor and grid connected
inductance are all included in this paper. The differential equations
of the inductance and capacitance are given as
Vs−Vg=RRLIg+LRL
dIg
dt+j
v
1LRLIg(11)
V−Vs=RTLI+LTL
dI
dt+j
v
1LTLI(12)
I−Is−Ig=Cf
dVs
dt+j
v
1CfVs(13)
Control of the RSC and the GSC is conducted in PLL-synchronised
reference frame. Control equations of rotor speed control, TVC,
DVC, PLL and rotor- and grid-side AC current control are listed as
T∗
e=Kp1 +Ki1
S

v
r−
v
∗
r
 (14)
Fig. 1 Scheme of a grid-connected DFIG-based WT system
aSystem topology
bControl scheme
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i∗p
rd =T∗
e
Lm/Ls

c
s
(15)
i∗p
rq =Ki2
SVs−V∗
s
 (16)
vp
rd =Kp3+Ki3
S

i∗p
rd −i∗p
rd
 (17)
vp
rq =Kp3+Ki3
S

i∗p
rq −i∗p
rq
 (18)
i∗p
gd =Kp4+Ki4
S

V∗
dc −Vdc
 (19)
vp
gd =− Kp5+Ki5
S

i∗p
gd −i∗p
gd
 (20)
vp
gq =− Kp5+Ki5
S

i∗p
gq −i∗p
gq
 (21)
v
PLL =Kp6+Ki6
s

vp
rq (22)
u
PLL =
v
PLL
S(23)
2.3 Linearisation of the detailed DFIG model
According to the equations in Section 2.2, a block diagram of the
detailed DFIG model can be drawn as Fig. 2. The non-linearity
of this detailed model is mainly caused by the coordinate transform-
ation and calculation of power, terminal voltage, and DC voltage.
The detailed DFIG model can be expressed in a set of differen-
tial–algebraic equations (DAEs)
dx/dt=fx,z,u()
0=gx,z,u()
(24)
where x,z, and u, respectively, represent state, algebraic and input
variables; fand gare a set of differential and algebraic equations.
The equilibrium point (x
0
,z
0
,u
0
) is calculated by solving all the
DAEs with dx/dt= 0. The non-linear DFIG system is linearised
locally at the calculated equilibrium point.
3 Modal analysis of the linearised DFIG model
3.1 Base case
The dynamics of the linearised DFIG model is studied through
modal analysis. In the base case, short-circuit ratio (SCR) is set
to be 2, which means the AC grid is weak [17]. The mechanical
torque, as the input of the system, is considered unchanged in the
study. The bandwidths of rotor speed control, PLL, DVC, RCC,
and GCC are assigned 1, 6, 9, 193, and 263 Hz, respectively. The
effect of the variation of bandwidths will be studied later.
Eigenvalues and their response modes of the base case are dis-
played in Table 1. The response modes are identified by participa-
tion factor analysis. The pole diagram is drawn in Fig. 3. Three pairs
of poles are far away from the imaginary axis, so they are not indi-
cated on the diagram. It is easy to judge that the system is stable
because all the poles are located on the left of the imaginary axis.
3.2 Effect of control bandwidths on system stability
The effect of control bandwidths on system stability is explored
through eigenvalue locus analysis in Figs. 4–7. The bandwidths
of rotor speed control, PLL, DVC, and RCC are altered in a
range, respectively, while the damping ratio remains unchanged.
Fig. 2 Block diagram of the DFIG model
Table 1 Eigenvalue analysis of the linearised model when SCR = 2
Mode Eigenvalues Response mode
λ
1,2
−0.59 ± 2.24irotor speed control
λ
3,4
−4.18 ± 36.57ishafting
λ
5,6
−10.05 ± 25.5iPLL
λ
7
−11.3 TVC
λ
8,9
−14.82 ± 305.4istator flux
λ
10,11
−24.57 ± 0.32iGCC
λ
12,13
−67.47 ± 83.11iDVC
λ
14,15
−105.49 ± 148.99iRCC
λ
16,17
−194.09 ± 131.66iRCC
λ
18,19
−7671.5 ± 313.11iGCC
Fig. 3 Pole diagram of the linearised model when SCR = 2
Fig. 4 Eigenvalue locus of the linearised model as rotor speed control
bandwidth varies from 0.1 to 40 Hz
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Fig. 4depicts the eigenvalue locus with varying bandwidth of
rotor speed control. The damping of stator mode decreases gradual-
ly as the rotor speed control bandwidth increases. The system
turns unstable when the bandwidth reaches a specific value. The
damping of the rotor speed control mode first increases and then
decreases in the process, while the damping of the shaft mode
keeps increasing.
Fig. 5depicts the eigenvalue locus with varying bandwidth of
PLL. As in the above case, the damping of stator mode decreases
gradually as the PLL bandwidth increases, and finally the system
turns unstable. In DVC timescale, the damping of PLL mode first
increases and then decreases, while the damping of DVC mode is
just the opposite.
Fig. 6depicts the eigenvalue locus with varying bandwidth of
DVC. The damping of DVC mode increases but the grid mode
turns unstable as the bandwidth increases.
Fig. 7depicts the eigenvalue locus with varying bandwidth of
RCC. It is worth noting that the PLL mode turns unstable when
the bandwidth of RCC decreases to a specific value.
According to the analysis result, the DFIG system will become
unstable when the control bandwidths are not appropriate. PLL
mode’s losing stability with the variation of RCC bandwidth indi-
cates that strong interactions exist between the control loops.
3.3 Interactions among multi-timescale control
With varying control bandwidths, the participation factors of differ-
ent control in oscillation modes are changing. Participation factors
of PLL, DVC, and RCC in three oscillation modes are plotted in
Figs. 8–10. These oscillation modes are considered as PLL mode,
DVC mode and RCC mode in the base case.
Fig. 7 Eigenvalue locus of the linearized model as RCC bandwidth varies
from 12 to 380 Hz
Fig. 6 Eigenvalue locus of the linearised model as DVC bandwidth varies
from 0.6 to 90 Hz
Fig. 5 Eigenvalue locus of the linearised model as PLL bandwidth varies
from 0.75 to 72 Hz
Fig. 8 Participation factors of PLL, DVC and RCC in
aλ
5,6
bλ
12,13
cλ
16,17
with varying PLL bandwidth
Fig. 9 Participation factors of PLL, DVC and RCC in
aλ5,6
bλ12,13
cλ16,17 with varying DVC bandwidth
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Fig. 8depicts participation factors of PLL, DVC and RCC in
three oscillation modes with varying PLL bandwidth. As PLL band-
width increases, the participation levels of DVC and RCC in PLL
mode increase. Meanwhile the participation level of PLL in RCC
mode increases as well.
Fig. 9depicts participation factors of PLL, DVC and RCC in
three oscillation modes with varying DVC bandwidth. The partici-
pation levels of DVC in RCC mode and RCC in DVC mode are
both increasing rapidly with DVC bandwidth increasing. PLL
mode is not affected by DVC and RCC in the process. Fig. 10
depicts participation factors of PLL, DVC and RCC in three oscil-
lation modes with varying RCC bandwidth. When RCC bandwidth
decreases, the participation levels of RCC in PLL mode, PLL in
DVC mode, and DVC in RCC mode increase. This exactly explains
why PLL mode loses stability when RCC bandwidth decreases.
The analysis results show that the participation levels of different
control in the oscillation modes are related to the control band-
widths. The dominant response mode may lose its supremacy in
the process. It is inferred that the interactions among multi-
timescale control are very strong under weak AC grid condition
and will have influence on the system stability. The dynamics of
the system are affected by multi-timescale control and its
interactions.
4 Conclusion
This paper describes a detailed model of a grid-connected
DFIG-based WT system considering multi-timescale control and
its interactions. Eigenvalues and their response modes of the
DFIG system are studied in a base case. The effect of control band-
widths on system stability is explored and the interaction behaviour
among multi-timescale control is studied. Finally, conclusions can
be drawn that the inappropriate control bandwidths will cause in-
stability of the DFIG system under weak AC grid condition. The
interactions among multi-timescale control are strong and will
have an effect on the stability and the dynamics of the DFIG
system. Therefore, interactions among multi-timescale control
should be considered in the modelling and analysis of a DFIG con-
nected to weak AC grid.
5 Acknowledgment
This work was supported in part by the National Key Research and
Development Program under Grant 2016YFB0900104.
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Fig. 10 Participation factors of PLL, DVC and RCC in
aλ5,6
bλ12,13
cλ16,17 with varying RCC bandwidth
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8 Appendix
1.5-MW DFIG parameters
Sbase =1.5MW Vbase =575 V(phase to phase rms value)
fbase =50 Hz
v
base =2
p
fbase Vdcref =1150 V Cdc =0.01 F
LRL =0.3p.u.RRL =0.003 p.u.LTL =0.5p.u.RTL =0.03 p.u.
Cf=0.08 p.u.LS=3.08 p.u.Lr=3.06 p.u.Lm=2.9p.u.
Rs=0.023 p.u.Rr=0.01 6p.u.Hg=0.05 s Ht=3s
Ksh =10 p.u/rad D=0.01 p.u.∗s/rad
v
∗
r=1.2V∗
dc =1V∗
s=1Tm=1/1.2i∗p
gq =0
Controller parameters in the base case (p.u.)
Rotor speed control Kpl =8.5Ki1 =38
TVC Ki2 =20
RCC Kp3 =0.6Ki3 =90
DVC Kp4 =1.5Ki4 =150
GCC Kp5 =8Ki5 =200
PLL Kp6 =53 Ki6 =1400
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J Eng 2017
doi: 10.1049/joe.2017.0503