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Research ArticleThe Small-Signal Stability of Offshore Wind
Power TransmissionInspired by Particle Swarm Optimization
Jiening Li,1 Hanqi Huang,1 Xiaoning Chen,2 Lingxi Peng ,1,3
Liang Wang ,4
and Ping Luo 5
1School of Mechanical and Electrical Engineering, Guangzhou
University, Guangzhou 510006, China2School of Mathematics and
Information Science, Guangzhou University, Guangzhou 510006,
China3Data Recovery Key Laboratory of Sichuan Province, Neijiang
Normal University, Sichuan 641100, China4School of Public
Administration, Guangzhou University, Guangzhou 510006,
China5School of Economics and Statistics, Guangzhou University,
Guangzhou 510006, China
Correspondence should be addressed to Lingxi Peng;
[email protected], Liang Wang; [email protected], and Ping
Luo;[email protected]
Received 6 May 2020; Accepted 22 May 2020; Published 15 July
2020
Academic Editor: Shuping He
Copyright © 2020 Jiening Li et al. 3is is an open access article
distributed under the Creative Commons Attribution License,
whichpermits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Voltage source converter-high-voltage direct current (VSC-HVDC)
is the mainstream technology of the offshore wind
powertransmission, which has been rapidly developed in recent
years. 3e small-signal stability problem is closely related to
offshore windpower grid-connected safety, but the present study is
relatively small. 3is paper established a mathematical model of the
doubly fedinduction generator (DFIG) integrated into the IEEE9
system via VSC-HVDC in detail, and small-signal stability analysis
of offshorewind farm (OWF) grid connection is specially studied
under different positions and capacities. By selecting two load
nodes and twogenerator nodes in the system for experiments, the
optimal location and capacity of offshore wind power connection are
obtained bycomparing the four schemes. In order to improve the weak
damping of the power system, this paper presents amethod to
determine theparameters of the power system stabilizer (PSS) based
on the particle swarm optimization (PSO) algorithm combined with
differentinertia weight functions. 3e optimal position of the
controller connected to the grid is obtained from the analysis of
modal controltheory. 3e results show that, after joining the PSS
control, the system damping ratio significantly increases. Finally,
the proposedmeasures are verified by MATLAB/Simulink simulation. 3e
results show that the system oscillation can be significantly
reduced byadding PSS, and the small-signal stability of offshore
wind power grid connection can be improved.
1. Introduction
Offshore wind power, as a clean and sustainable technology,has
been developed rapidly in recent years [1]. At present,the total
installed capacity in Europe is increasing every year,among which
the UK and Germany dominate the offshorewind power industry [2].
According to Wind Europe’s HighScenario, it is estimated that the
offshore wind energy ca-pacity in Europe will reach 99GW by 2030
[3]. Nowadays,the mainstream transmission technology is
VSC-HVDC[4–6]. Compared to other traditional transmission modes,
ithas the advantages of independent control of the active
andreactive power output, smaller power loss, and lower voltagedrop
[7–10].
Small-signal stability analysis is to study the dynamicresponse
characteristics of the power system after smalldisturbances
(including random fluctuations in powergeneration or consumption)
and to evaluate their ability tosuppress oscillations [11]. It is
of great significance topromote the development of wind power and
improve thesafe stability of the system, which needs urgent
attention[12]. However, there are few studies on the
small-signalstability analysis of the offshore wind power grid
connectedby VSC-HVDC transmission at present. Based on the
small-signal stability problem of the offshore wind power
systemtransmitted by VSC-HVDC, the damping control of thesystem is
increased by introducing PID regulation at theconverter station. 3e
modeling method does not consider
HindawiComplexityVolume 2020, Article ID 9438285, 13
pageshttps://doi.org/10.1155/2020/9438285
mailto:[email protected]:[email protected]:[email protected]://orcid.org/0000-0002-7376-2925https://orcid.org/0000-0002-1017-0500https://orcid.org/0000-0002-1998-8253https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/9438285
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the aerodynamic model, and the selection of PID parametersbased
on eigenvalues has limitations [13]. Small-signalstability analysis
method from the perspective of voltage andfrequency is discussed in
[14, 15]. In [14], the variability ofwind power was considered as
the interference source of thesystem and used Prony analysis and
swing-based frequencyresponse metric to study the influence of the
small signal onthe large offshore wind power system. However, the
wholesystemmodeling process has not been described in detail butis
focused on analysis. In [15], the model of the offshore ACnetwork
for point-to-point VSC-HVDC transmission isdeduced in detail, and
the droop gain boundary under thestable operation is determined by
eigenvalue analysis. A low-pass filter was introduced to improve
the system damping,but the effect of different grid-connected
positions andcapacity on the small signal of the system was not
consid-ered. At present, neural networks [16, 17], deep learning,
andmachine learning methods have been widely used in variousfields,
and remarkable results have been achieved [18, 19].
To improve the stability of the system under the smallsignal and
large signal, the PSO algorithm is used to optimizethe parameters
of the DFIG controller. However, the PSOalgorithm will fall into
local optimum, and more controllerequations increase the dimension
of the state matrix, whichwill slow down the fast operation of the
system [20]. In thesestudies, although scholars have studied the
small-signalstability of offshore wind power grid connection in
differentfields, it is still in the exploratory stage to change the
grid-connected position and capacity of offshore wind farms
forsmall-signal stability analysis and to apply damping
control.
Aiming at the problem of small-signal stability of theoffshore
wind power grid-connected system, this paperfirstly establishes a
complete VSC-HVDC transmissionoffshore wind power grid-connected
mathematical modeland then analyzes the impact of OWFs on the
small-signalstability of the system under different positions and
ca-pacities. Secondly, PSS is introduced for damping control.3e
optimal position of the PSS connected to the power gridis
determined by modal control theory, and the parametersof the
controller are determined by the PSO algorithmcombined with
different inertia weight functions. Finally,the correctness of the
establishedmodel and the effectivenessof the proposed control
measures are verified by MATLAB/Simulink simulation.
2. Modeling of Offshore Wind PowerTransmitted by VSC-HVDC
3is paper uses the typical offshore wind power grid con-nection
system [21] for reference, and the designed systemtopology is shown
in Figure 1. 3e OWFs are composed of10 2-MW DFIGs, and they are
equivalent to a DFIG rep-resentation according to the aggregated
model [22]. 3ewind turbine (WT) runs on the low-speed (LS) shaft,
thegenerator runs on the high-speed (HS) shaft, and thegearbox
connects the two to act as the booster. Each 690VDFIG is connected
to 10 kV bus through the boost trans-former XT0 and then sent to
110 kV bus through the boosttransformer XT1 and the transmission
line XT1. VSC1 and
VSC2 are converter stations, which play the roles of
therectifier and the inverter, respectively. 3e HVDC plays therole
of power transmission. In addition, there are
resistance-capacitance (RC) filters and phase reactors, whose
influenceon the system is ignored in this paper. 3e VSC-HVDCoutlet
us2 is connected to the 230 kV 3-machine, 9-bus testsystem [23] via
the boost transformer XT2 and the trans-mission line XL2, which is
represented by IEEE9.
2.1. AerodynamicModel. WTis driven by the wind, which
isconverted into mechanical energy through three blades.
3eequations of the aerodynamic model [24] are given by
CP � 0.22116λi
− 0.4β − 5 e− 12.5/λi ,
1λi
�1
λ + 0.08β2−0.035β3 + 1
,
Tt �Pt
ωt�ρπr2CPv3
2ωt,
(1)
where Cp is the coefficient of wind energy utilization; β is
thepitch angle; λ is the tip speed ratio (λ�ωtr/]); ωt is the
me-chanical angular velocity of the WT; r is the radius of the
windwheel; ] is the wind speed; Pt is the mechanical power output
bythe WT; Tt is the mechanical torque; and ρ is the air
density.
2.2. Shafting Model. In order to ensure the accuracy of
cal-culation, this paper selects the two-mass block shafting
model.3emathematical model of the shaftingmodel [25] is as
follows:
2Htdωtdt
� Tt − Kθ − D ωt − (1 − s)ωs( ,
dθdt
� ωb ωt − (1 − s)ωs( ,
− 2Hgωsdsdt
� Kθ + D ωt − (1 − s)ωs( − Te,
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(2)
where Ht and Hg are the inherent inertial time constants ofthe
WT and the generator; Tt and Te are the mechanicaltorque of the WT
and the electromagnetic torque of thegenerator; K andD are the
stiffness and damping coefficientsof the shaft; θ is the shaft
torsional angle; ωt and ωr are thespeeds of the rotor of the WT and
the generator; ωs is thestator speed of the generator; and ωb is
the base value ofrotational speed.
2.3. Pitch Angle ControlModel. To ensure the smooth poweroutput
of the OWFs, the pitch angle [26] needs to becontrolled. 3e
equations are given by
dβdt
�1
Tββref − β( ,
βref � Kp0dωtdt
+ K10Δωt,
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(3)
2 Complexity
-
where Tβ is the inertial time constant of the pitchangle control
model; βref and ωt_ref are the referencevalues of the pitch angle
and the WTspeed; and Kp0 and KI0are the proportional and integral
coefficients of thecontroller.
2.4. DFIG Model. Currently, DFIG is the most widely usedWT.
Ignoring the electromagnetic transient process of statorwinding,
the mathematical model of the DFIG [27] is asfollows:
ded′dt
� −1
T0′ed′ + sωseq′ −
Xs − Xs′( T0′
iqs −ωsLm
Lrruqr,
deq′
dt− sωsed′ −
1T0′
eq′ +Xs − Xs′(
T0′ids +
ωsLmLrr
udr,
udr � Rridr + Tb′didrdt
− sωsTb′iqr,
uqr � Rriqr + Tb′diqrdt
+ sωs Ta′uqs + Tb′idr ,
ids �1
Lssuqs −
Lm
Lssidr,
iqs � −Lm
Lssiqr,
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(4)
where T0′= Lrr/Rr; Ta′= Lm/Lss; Tb′= Lrr − Lm/Lss; Xs =ωsLss;Lss
= Ls + Lm; and Lrr = Lr + Lm are the sum of self-inductanceof the
stator and the rotor; Rs and Rr are the resistances of thestator
and the rotor; Lm is the mutual inductance betweenthe stator and
the rotor; Xs is the reactance of the stator;ed′= − (ωsLm/Lrr)ψqr;
eq′= (ωsLm/Lrr)ψdr; Xs′=ωs/Lrr(LssLrr− Lm); ed′ and eq′ are the d-
and q-axis components ofthe transient voltages; Xs′ is the
transient resistance of thestator; and ψdr and ψqr are the d- and
q-axis components ofthe rotor flux, respectively. ids, iqs, idr,
and iqr are the d- and q-axis components of the stator and rotor
currents, respec-tively. Let the stator flux ψs always reunite with
the d-axis;then, uds = 0 and uqs =ψs. uds, uqs, udr, and uqr are
thevoltages of the stator and the rotor.
3e active and the reactive power output of OWFs are
Pe �32
Rr P2s + Q
2s(
T′2a u2s
+2RrQsT′2a Lss
+ (1 − sω)Ps +Rru
2s
L2m ,
Qe �32
− sωTb′P2s + Q
2s
T′2a u2s
+2Qs
T′2a Lss+
u2sL2m
+ 1 − sωs( Qs −sωsu2s
Lss .
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(5)
2.5. VSC-HVDC Model. According to the relationship be-tween the
three-phase stationary coordinate system and thed and q synchronous
rotating coordinate system, after thePark transformation, the
7-order mathematical model[13, 21] can be obtained as follows:
L1di1ddt
� − R1i1d − ω1L1i1q + usd1 − K1ud1 cos δ1,
L1di1qdt
� − R1i1q + ω1L1i1d + usq1 − K1ud1 sin δ1,
L2di2ddt
� − R2i2d − ω2L2i2q + usd2 − K2ud2 cos δ2,
L2di2ddt
� − R2i2q − ω2L2i2d + usq2 − K2ud2 sin δ2,
Cdud1dt
�3K12
i1d cos δ1 + i1q sin δ1 − id,
Cdud2dt
�3K22
i2q cos δ2 + i2q sin δ2 − id,
Lddidt
� ud1 − ud2 − Rdid,
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(6)
where the physical quantities corresponding to the rectifierand
the inverter station are denoted by subscripts “1” and“2,”
respectively. i1d, i1q, i2d, and i2q represent the
currentcomponents of the AC network; usd1, usq1, usd, and
usq2represent the voltage components of the AC network; ud1,uq1,
ud2, and uq2 represent the voltage components betweenbuses of the
DC system; and K1 and K2 are the voltageutilization coefficients of
the DC system. δ1 and δ2 are thedeviation angles between the
voltage of the converter station
110kV 110kV
10111213
IEEE9
9
DFIG
230kV10kV690V
VSC1 VSC2
RC filter1 RC filter2Equivalent offshorewind farm
LS HSGearbox
G
HtTt Hg
Tevwind
K
D
Xd
Xd
X2C is2
XT2 XL2
us2 u
HVDC
X1
is1
isXT1XT0 XL1
us1us
ωrωt
Figure 1: 3e topology of the whole system.
Complexity 3
-
and the bus voltage of the AC system; ω1 and ω2 are
thefundamental angular frequencies of the AC system. id is theDC
current transmitted by the high-voltage transmissionline.
2.6. System InterfaceModel. Position the stator voltage us onthe
d-axis, and the voltage vector between the DFIG and therectifier
station is shown in Figure 2.
Its mathematical relationship is
usd1
usq1
⎡⎣ ⎤⎦ �uds
uqs
⎡⎣ ⎤⎦ +0 − XTL1
XTL1 0
ids
iqs
⎡⎣ ⎤⎦, (7)
where XTL1 is the total impedance of line 1; XT0 is
theequivalent transformer impedance of OWFs; and XT1 andXL1 are the
transformer and the circuit impedance of line 1.
Position the terminal voltage u of the power system onthe
x-axis, and the vector between the inverter station andthe power
system is shown in Figure 3.
Its mathematical relationship is
usd2
usq2
⎡⎣ ⎤⎦ �cosφ
− sinφ u −
0 − XTL2XTL2 0
i2d
i2q
⎡⎣ ⎤⎦. (8)
2.7. Complete Model. All the equations of DFIG and VSC-HVDC are
combined and linearized near the stable value:
dΔx10dt
0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦�
A B
C D
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
Δx10
Δy10
⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦, (9)
where x10 � [ωt θ s β ed′ eq′ i1d i1q i2d i2q u1d u2d id]T
and
y10 � ids iqs T.
3e offshore wind power transmitted by VSC-HVDC isintegrated into
the IEEE9 system, as shown in Figure 4. 3esmall-signal equation of
the system is
dΔxdt
� AΔx, (10)
where x � x1 x2 x3 x10 T; x1 � δ1 ω1
T; x2 �δ2 ω2 eq2′ ed2′ efq2 vR2 vM2
T; and x3 �
δ3 ω3 eq3′ ed3′ efq3 vR3 vM3 T.
After deducing the complete small-signal stability modelof
offshore wind power transmitted by VSC-HVDC, it isnecessary to do
further research on the influence of small-signal stability on the
system of offshore wind power indifferent positions and
capacities.
3. Small-Signal Stability Analyses
3is section will combine with the concrete example of
theoffshore wind power grid-connected position and capacityfor
small-signal stability analysis. Firstly, the parameters ofeach
subsystem are initialized, which are detailed in
Appendix and then the state of OWFs is found before
in-corporation into the system. Finally, two load nodes and
twogenerator nodes are chosen, and the influence of the
offshorewind power grid connection is analyzed in detail.
3.1. EigenvalueAnalysis. When the OWFs are added at node6 and
the active power output is 0.5 pu, the eigenvalues of thesystem are
shown in Table 1. It can be seen from Table 1 thatthe real part of
the eigenvalues is all negative, which are onthe left side of the
imaginary axis, indicating that the originalpower system is running
in a stable state.
λ7,8, λ10,11, λ12,13, and λ16,17 are related to the
generatorrotational speed (Δω1, Δω2, and Δω3) and the
rotationalspeed difference Δs of wind generator 10, respectively,
be-longing to the electromechanical oscillation mode.
O
q
y
x
XTL1is
us
us1 jXTL1is
d
δφ
Figure 2: 3e voltage vector between the DFIG and the
rectifierstation.
xuθ
O
q
y
us2
–jXTL2i2
d
φ
XTL2i2
Figure 3: 3e voltage vector between the inverter station and
thepower system.
2 3
11
2 3
4
5 6
7 8 9
18kV 230kV 13.8kV
16.5kV
110kV
10
VSC-HVDC DFIG
Equivalent offshore wind farm
Figure 4: Schematic diagram of the equivalent offshore wind
farmconnected to the IEEE9 system by VSC-HVDC.
4 Complexity
-
3.2. Damping Ratio Analysis. Damping ratio [28] can reflectthe
speed and the characteristics of oscillation attenuation.3e
expression is given by
ζ � −σ
������σ2 + ω2
√ , (11)
where σ and ω are the real and the imaginary part of
theeigenvalue, respectively.
In the actual power system, it is generally required thatthe
damping ratio of the electromechanical oscillation modeshould be
above 0.05, and then the operating state of thesystem could be
accepted. However, this principle is not
unchanged. If the mode of fluctuation is not large when
thesystem operation mode changes, it is acceptable to have sucha
low damping ratio (for example, 0.03) [28].
By changing different active power outputs and grid-connected
positions of OWFs, the damping ratio curve invarious cases can be
obtained as shown in Figure 5.
4. Control Measures
In order to improve the weak damping instability of thesystem,
it is necessary to add a controller to the system forauxiliary
regulation. Firstly, PSS needs to be modeled, andthe speed
difference is used as the feedback signal to beadded to the system
voltage equation. Secondly, the optimalposition of the PSS is
determined by the detailed analysis ofmodal control theory.
Finally, the PSO algorithm is introduced, and the ob-jective
function is established according to the researchcontent of this
paper. 3e parameters of PSS controllers aredetermined with
different inertia weight functions.
4.1. PSS Model. 3e main function of the PSS is to
increasedamping or suppress low-frequency oscillation of the
powersystem [29]. 3e mathematical expression of the PSS is
dΔV1dt
�Kgain
T6Δω −
1T6ΔV1,
dΔV2dt
�Kgain
T6Δω −
1T6ΔV1 −
1T5ΔV2,
dΔV3dt
�KgainT1
T2T6Δω −
T1
T2T6ΔV1 −
T1 − T5
T2T5ΔV2 − ΔV3,
dΔVSdt
�KgainT1T3
T2T4T6Δω −
T1T3T2T4T6
ΔV1 −T3 T1 − T5(
T2T4T5ΔV2 −
T3 − T2T2T4ΔV3 −
1T4ΔVS,
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(12)
where the generator speed difference VIS �ω-ωt is selected asthe
input signal. VS is the output signal, which serves as theauxiliary
input signal of the stator voltage equation Δuqi(i� 1,2,3) of the
IEEE9 system. Kgain is the gain of theamplification link; T6 is the
time constant of the measure-ment link. T5 is the time constant of
the filtering link. T1, T3,T2, and T4 are the lead and lag time
constants of two phasecompensation links, respectively.
If the PSS is added to IEEE9 system generator 2, thevoltage
equation needs to be
Δud2 � Δed′ − Ra2Δid2 + Xq2′ Δiq2,
Δuq2 � Δeq′ − Ra2Δiq2 + Xd2′ Δid2 + ΔVS.
⎧⎨
⎩ (13)
4.2. PSS Grid-Connected Position Selection. When OWFsintegrate
into load node 6 and in normal operation, thechanges of the
electromechanical oscillation mode areshown in Table 2.
As can be seen from Table 2, in the eigenvectors of thefirst and
fourthmodes, the modulus value of Δs is the largest,but the
directions of each component are basically the same,indicating that
the influence of each generator on the modeis similar. In the
secondmode, themodulus value ofΔω3 andΔs is larger and opposite to
the direction of other compo-nents (the average argument is about −
154.57°), indicatingthat the oscillation mainly exists in
generators 3, 10 andgenerators 1, 2. 3e oscillation frequency f�
8.65/(2π)�1.3767Hz, belonging to the local oscillation mode. In
the
Table 1: Eigenvalues of the system during stable state.
Number Eigenvalue Number Eigenvalueλ1 − 99.9360 λ2 − 74.4198λ3 −
52.7251 λ4 − 51.3813λ5 − 31.1510 λ6 − 28.8327λ7,8 − 2.6655±
18.2668i λ9 − 12.9409λ10,11 − 0.6959± 12.8937i λ12,13 − 0.1431±
8.6461iλ14,15 − 4.7160± 8.0754i λ16,17 − 0.1222± 4.0611iλ18,19 −
4.4655± 3.1380i λ20,21 − 5.4943λ21 − 4.1027 λ22 − 0.4476λ23,24 −
0.8287± 0.9010i λ25,26 − 0.3560± 0.5849iλ27 − 0.0290 λ28,29 −
0.0291± 0.0502i
Complexity 5
-
third mode, the modulus values of Δω2 and Δs are larger andare
basically opposite to Δω1 in the direction (the argumentis −
108.95°), indicating that the oscillation mainly exists
ingenerators 2, 10 and generator 1. 3e oscillation frequency
f� 12.89/(2π)� 2.0515Hz, belonging to the local
oscillationmode.
In the participation vector, if the generator speedcomponent is
positive and the value is large, it indicates that
0.25
0.2
0.15
0.1
0.050.03
00 0.2 0.4 0.6 0.8 1
Wind turbine 10 active output (pu)
Dam
ping
ratio
at n
ode 6
ω1ω2
ω3s
(a)
0.25
0.2
0.15
0.1
0.050.03
00 0.2 0.4 0.6 0.8 1
Wind turbine 10 active output (pu)
Dam
ping
ratio
at n
ode 7
ω1ω2
ω3s
(b)
0.25
0.2
0.15
0.1
0.050.03
00 0.2 0.4 0.6 0.8 1
Wind turbine 10 active output (pu)
Dam
ping
ratio
at n
ode 8
ω1ω2
ω3s
(c)
0.25
0.2
0.15
0.1
0.050.03
00 0.2 0.4 0.6 0.8 1
Wind turbine 10 active output (pu)
Dam
ping
ratio
at n
ode 9
ω1ω2
ω3s
(d)
Figure 5: 3e damping ratio of OWF grid connection.
Table 2: Corresponding components of generator speed in the
right eigenvector.
Generatorλ7,8 λ10,11 λ12,13 λ16,17
Modulus value(pu)
Argument(°)
Modulus value(pu)
Argument(°)
Modulus value(pu)
Argument(°)
Modulus value(pu)
Argument(°)
1 0.0000 − 65.65 0.0002 − 158.19 0.0009 − 108.95 0.0007 − 73.932
0.0001 − 50.39 0.0012 − 150.94 0.0024 69.35 0.0006 − 81.423 0.0003
− 57.68 0.0040 26.21 0.0013 72.60 0.0005 − 81.4910 0.0115 − 42.98
0.0046 23.87 0.0025 72.36 0.0021 − 78.95
6 Complexity
-
the generator is the best candidate position for installing
thePSS, which can significantly increase the damping of themode.
For increasing system damping, it is better to applycontrol on a
larger capacity generator [30].
3e local oscillation modes λ10,11 and λ12,13 are specif-ically
analyzed. 3e components of the participation vectorcorresponding to
the rotation speed of each generator areshown in Table 3.
As can be seen from Table 3, for the participation vectorof
local mode λ10,11, the value of generator 3 is the largest andthat
of generator 10 is the smallest, indicating that thecontrol is
mainly applied in generator 3, and the damping ofthe system can be
significantly increased. Similarly, for theparticipation vector of
local mode λ12,13, the value of gen-erator 2 is the largest and
that of generator 10 is the smallest,indicating that fine effects
can be achieved by applyingcontrol on generator 2.
Furthermore, compared with the capacity of generators 2and 3,
the active power of generator 2 (1.63 pu) is larger thanthat of
generator 3 (0.85 pu), so the damping control ofgenerator 2 can
achieve better suppression of the oscillationeffect.
4.3. PSO Algorithm Based on Different Inertia WeightFunctions.
3e updating formulas of particle velocity andposition are as
follows:
vk+1i D � ωv
ki D + c1r1 p
ki D − x
ki D + c2r2 p
kg D − x
kg D ,
xk+1i D � x
ki D + v
k+1i D ,
(14)
where k represents the number of iterations; vki D and xki
Drepresent the velocity and position of the i-th particle in
theD-dimensional space, and the value range is []lb, ]ub] and[xlb,
xub]; and c1 and c2 refer to the individual learningfactors and the
group learning factors, and c1 � c2 � 2 istaken in this paper. r1
and r2 are random numbers between0 and 1; pki D and p
kg D represent the individual optimal
position and the global optimal position of the i-th and
g-thparticle in the D-dimensional space; and ω represents
theinertia weight.
3e inertia weight ω indicates the ability to retainexisting
velocity. 3e larger the value is, the stronger theglobal
optimization is. On the contrary, the smaller the valueis, the
better the local optimization is. In order to weigh theglobal
optimization and local optimization, Shi. Y sum-marized a new
method of calculating weights, linear de-creasing inertia weights
(LDIW) [31], which is specificallyexpressed as
ω1(k) � ωstart − ωstart − ωend( k
Tmax, (15)
where ωstart is the inertia weight in the initial state; ωend
isthe inertia weight at the end of the iteration; k is thenumber of
iterations; and Tmax is the maximum number ofiterations. It is
generally believed that ωstart = 0.9 andωend = 0.4 in which the
algorithm performs best. In ad-dition to LDIW, the common inertia
weight functions [32]are as follows:
ω2(k) � ωstart − ωstart − ωend( k
Tmax
2
,
ω3(k) � ωstart + ωstart − ωend( 2k
Tmax−
k
Tmax
2⎡⎣ ⎤⎦,
ω4(k) � ωendωstartωend
1/ 1+k/Tmax( )
.
(16)
4.4. Determination of Optimization Objective Function. Inorder
to improve the weak damping of the system, it is veryimportant to
determine the optimal target. As can be seenfrom Figure 5, when
OWFs first join the system, thedamping ratio is relatively large.
With the increase of theactive power output, most damping ratios
show a downwardtrend, and some electromechanical oscillation modes
evenhave weakly damped unstable states. In order to improve
thelow-frequency oscillation, when generator 10 active outputis 1.0
pu, the minimum damping ratio of each generator isselected as the
optimal target, and then the maximum valueof this target in the
iterative process can be obtained by thePSO algorithm. If the
optimization target is larger than 0.03,the variation curve of
damping ratio can be ensured to be ina stable running state. 3e
mathematical expressions offitness function are as follows:
Dn � min ζni( ,
fitness � max Dn( ,(17)
where ξni (i� 1,2,3,4) is the damping ratio of the
componentrelated to the rotation speed of generators 1, 2, 3, and
10during the n-th iteration process.
According to the research of this topic, the steps of thePSO
algorithm are as follows:
(1) Initialize particle swarm size, velocity, and
position.Particle specifically refers to PSS parameters.
(2) Take the small-signal stability program as a sub-function,
and then optimize the designed objectivefunction to obtain the
individual and the globaloptimal fitness value of the particle.
(3) Compare the fitness value of each generation par-ticle. If
the current fitness value is better, update thepreviously recorded
optimal fitness value with thecurrent fitness value, and update the
previouslyrecorded optimal position with the current position.If
not, leave the value unchanged.
Table 3: Components of the participation vector corresponding
tothe rotation speed of each generator.
Generator 1 2 3 10λ10,11 0.0099 0.1914 1.0000 0.0264λ12,13
0.3700 1.0000 0.1220 0.0143
Complexity 7
-
0.034
0.032
0.03
0.028
0.026
0.024
0.022
0.02
0.0180 10 20 30 40 50
Number of iterations
Fitn
ess
ω1 optimal fitness :0.033867ω2 optimal fitness :0.034232ω3
optimal fitness :0.033999ω4 optimal fitness :0.033543
Figure 6: Fitness curves under different inertia weights.
0.25
0.2
0.15
0.1
0.050.03
00 0.2 0.4 0.6 0.8 1
Wind turbine 10 active output (pu)
Dam
ping
ratio
at n
ode 6
ω1ω2
ω3s
(a)
0.25
0.2
0.15
0.1
0.050.03
00 0.2 0.4 0.6 0.8 1
Wind turbine 10 active output (pu)
Dam
ping
ratio
at n
ode 7
ω1ω2
ω3s
(b)
Figure 7: Continued.
8 Complexity
-
(4) Update the position and velocity of each particle.(5)
Determine whether the current fitness value reaches
the given standard or reaches the upper limit of thenumber of
iterations. If yes, the program ends. If not,return to (2) and
continue to execute the loop body.
4.5. Effect of Applying PSSControl. According to the analysisof
Section 4.3, PSS installed in generator 2 works well.
Underdifferent inertial weights, the fitness curve can be obtained
bysolving the function for many times and taking the meanvalue, as
shown in Figure 6.
It can be seen from Figure 6 that the deviation of theoptimal
fitness obtained under the four inertia weights issmall. 3e optimal
fitness of the inertia weight ω2 is0.034232, the effect of ω1 and
ω3 is similar, and the worst isω4. Based on ω2, the control
parameters are deeply opti-mized, and the optimized parameters of
each controller areKgain � 10, T1 � 0.66, T2 � 0.25, T3 � 0.45, T4
� 0.15, T5 � 4.5,and T6 �1.25. By substituting the PSS parameters
into theoriginal state matrix and changing the position and
capacityof the offshore wind power into the power system, the
effectcan be obtained, as shown in Figure 7.
As can be seen from Figure 7, after adding the PSS togenerator
2, each damping ratio is improved correspondingly,and the curves
are all above 0.03, indicating that the systemworks stably. 3e
variation trend of each damping ratio isbasically the same as that
without the PSS, and the dampingratio of generator 2 rotation speed
is significantly increased.Furthermore, it can be verified that the
theoretical analysis isreasonable, and generator 2 is the best
place to add the PSS.
5. Simulink Simulation
3rough theoretical analysis and programming experiments,the
control measures of offshore wind power grid
connection have been obtained. However, how the actualpower
system operates and whether the controller PSSachieves the desired
effect need to be verified by simulation.3e circuit in Figure 1 was
built on MATLAB/Simulink. Byapplying the small signal (wind speed
disturbance andsystem fault), the proposed control measures are
verified andanalyzed.
5.1. Wind Speed Disturbance. Assuming the initial windspeed of
the OWFs is 10m/s, a step signal generator disturbsthe wind speed
to 12m/s at 1.8 s. 3e dynamic responseeffect of the system before
and after adding PSS control togenerator 2 is shown in Figure
8.
From the PSS2 control in Figures 8(a) and 8(c), it can beclearly
seen that the power of the system fluctuates within asmall range
after applying the wind speed disturbance, andthe system runs
stably again at 3.3 s. 3e process has gonethrough 1.5 s; it can be
seen from Figure 8(a) that, after theaddition of PSS2, the
electromagnetic torque of synchronousgenerator 1 is stabilized more
quickly, and the oscillationamplitude is effectively suppressed.
Figure 8(b) is the voltagecurve of synchronous generator 1, and the
effect of addingPSS2 is small. It can be seen from Figure 8(c) that
theamplitude of the synchronous generator 1 active power
issignificantly reduced after the addition of PSS2. Figure
8(d)shows the DC curve of HVDC. By comparison, the am-plitude of
the oscillation is reduced when PSS2 is applied.3e amplitude of the
voltage stability will drop slightlywithout PSS2 control, which is
also the adverse effect of windspeed disturbance. Figures 8(e) and
8(f ) show the active andreactive power of VSC1. It can be seen
that, after the additionof the damping controller, the oscillation
of the system isbetter improved. In summary, when the OWFs are
disturbedby wind speed, adding PSS2 will improve the
small-signalstability of the whole system.
0.25
0.2
0.15
0.1
0.050.03
00 0.2 0.4 0.6 0.8 1
Wind turbine 10 active output (pu)
Dam
ping
ratio
at n
ode 8
ω1ω2
ω3s
(c)
0.25
0.2
0.15
0.1
0.050.03
00 0.2 0.4 0.6 0.8 1
Wind turbine 10 active output (pu)
Dam
ping
ratio
at n
ode 9
ω1ω2
ω3s
(d)
Figure 7: Curve of damping ratio change after the PSS is added
into generator 2.
Complexity 9
-
5.2. System Fault. In order to simulate the actual systemfault,
a three-phase short circuit was applied to the trans-mission line
incorporated into node 8 at 1.5 s, and the faultduration is 0.12 s.
3e dynamic response of the systemcontrolled by PSS2 is shown in
Figure 9.
It can be seen from Figures 9(a) and 9(f) that, afterapplying a
three-phase short-circuit fault, the state of thesystem changes
obviously. 3e system recovers at 2.8 s, and
the adjustment process goes through 1.3 s; Figure 9(a) is
theelectromagnetic torque curve of synchronous generator 1,and the
oscillation of the system is significantly reduced afterthe
addition of PSS2. Figure 9(b) shows the voltage curve ofsynchronous
generator 1, and the effect of adding or notadding PSS2 is small.
It can be seen from Figure 9(c) that theamplitude of the
synchronous generator 1 active power isremarkably suppressed after
the addition of PSS2.
1.5
1
0.5
0
t (s)
Pel (
pu)
0 1 2 3 4 5
Without PSS2 controlWith PSS2 control
(a)
1.5
1
0.5
0
V1 (p
u)
t (s)0 1 2 3 4 5
Without PSS2 controlWith PSS2 control
(b)
2
1.5
1
0.5
0
t (s)0 1 2 3 4 5
P1 (p
u)
Without PSS2 controlWith PSS2 control
(c)
1.5
1
0.5
0
t (s)0 1 2 3 4 5
Udc
(pu)
Without PSS2 controlWith PSS2 control
(d)
t (s)0 1 2 3 4 5
P (p
u)
2
–1
1
0
Without PSS2 controlWith PSS2 control
(e)
t (s)0 1 2 3 4 5
Q (p
u)
2
–1
1
0
Without PSS2 controlWith PSS2 control
(f )
Figure 8: Dynamic response of the system before and after adding
PSS2 in the case of wind speed disturbance. (a) 3e
electromagnetictorque of synchronous generator 1. (b) 3e voltage of
synchronous generator 1. (c) 3e active power of synchronous
generator 1. (d) 3edirect current voltage of HVDC. (e) 3e active
power of VSC1. (f ) 3e reactive power of VSC1.
10 Complexity
-
Figure 9(d)shows the DC curve of HVDC. It can be seenthat the DC
voltage fluctuation amplitude is reduced, andthe system reaches
stability more quickly. Figures 9(e) and9(f ) show the active and
reactive power of VSC1. It can beseen that, after the damping
controller is added, the os-cillation of the system is better
improved. In summary,when the onshore power system is disturbed by
the fault,adding PSS2 improves the small-signal stability of
thewhole system.
6. Conclusions
Based on the establishment of a complete mathematicalmodel, the
influence of offshore wind power on the systemat different
positions and capacities is analyzed in detailaccording to the
damping characteristics. 3e experimentalresults show that most of
the damping ratios are decreasing,and some of the electromechanical
oscillation modes willshow weak damping instability. By comparing
the
4
2
0
–20 1 2 3 4 5
t (s)
Pel (
pu)
Without PSS2 controlWith PSS2 control
(a)
0 1 2 3 4 5t (s)
Without PSS2 controlWith PSS2 control
1.5
1
0.5
0
V1 (p
u)
(b)
0 1 2 3 4 5t (s)
4
2
0
–2
P1 (p
u)
Without PSS2 controlWith PSS2 control
(c)
0 1 2 3 4 5t (s)
Without PSS2 controlWith PSS2 control
1.5
1
0.5
0
Udc
(pu)
(d)
0 1 2 3 4 5t (s)
P (p
u)
Without PSS2 controlWith PSS2 control
2
1
0
–1
(e)
0 1 2 3 4 5t (s)
Without PSS2 controlWith PSS2 control
Q (p
u)
2
1
0
–1
(f )
Figure 9: Dynamic response of the system before and after adding
PSS2 in the case of the three-phase short-circuit fault. (a)
3eelectromagnetic torque of synchronous generator 1. (b) 3e voltage
of synchronous generator 1. (c) 3e active power of
synchronousgenerator 1. (d) 3e direct current voltage of HVDC. (e)
3e active power of VSC1. (f ) 3e reactive power of VSC1.
Complexity 11
-
experimental results of two load nodes and two generatornodes,
the optimal location and capacity of offshore windfarms are
obtained. In order to improve the weak dampingof the power system,
the optimal positions of the PSS in thepower grid are obtained
through the modal control. PSOalgorithm combining with different
inertia weight func-tions is presented to determine the parameters
of the PSS.Finally, the whole system model is built on
MATLAB/Simulink platform, and the influence of PSS control on
thesystem is compared and analyzed by applying wind
speeddisturbance and system fault. 3e simulation results showthat
the dynamic response of the system is obviously im-proved, the
oscillation amplitude is significantly reduced,and the accuracy of
the proposed control strategy isverified.
Appendix
DFIG parameters of offshore wind farms:
r� 40m; ρ� 1.242 kgm3; v0 � 10.45m s− 1; Rr �
0.0073 pu; Rs � 0.0076 pu; Lr � 0.0884 pu; Ls � 0.1248 pu;Lm �
1.8365 pu; ωt0 �1.1055 rad s− 1; s0 � − 0.003;β0 � 0 rad; Kp0 � 0.1
pu; KI0 � 20 s− 1; Tβ �10 s; Ht � 3 s;Hg � 0.05 s; K� 10 pu.rad− 1;
and D� 0.5 pu.rad− 1.
VSC-HVDC parameters:
R1 �R2 � 0.075 pu; L1 � L2 � 0.016 pu; ω1 �ω2 �1 pu;δ1 � δ2
�10°; K1 �K2 � 0.1; Rd � 1.0425 pu;Ld � 0.0119 pu; and C� 0.1733
pu.
System interface parameters:
XT0 � 0.015 pu; XT1 � 0.016 pu; XL1 � 0.02 pu;XT2 � 0.061 pu;
and XL2 � 0.014 pu.
Data Availability
All data generated or analyzed during this study are includedin
this article.
Conflicts of Interest
3e authors declare no conflicts of interest.
Acknowledgments
3is paper was carefully guided by Professor Ru Yang ofGuangzhou
University and was supported by the NationalNatural Science
Foundation of China under Grant nos.61772147 and 61100150, the
University Innovation TeamConstruction Project of Guangdong
Province under Grantno. 2015KCXTD014, Guangdong Province Philosophy
andSocial Science Foundation under Grant no. GD19CSH03,the key
Project of Science and Technology Plan ofGuangdong Province under
Grant No. 2020b1010010014and Open Research Fund Program of Data
Recovery KeyLaboratory of Sichuan Province. Here, the authors
expresstheir sincere gratitude.
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Complexity 13