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; scu.peng@gmail.com, Liang Wang; wl_1998@gzhu.edu.cn, and Ping Luo;514923134@qq.com

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:scu.peng@gmail.commailto:wl_1998@gzhu.edu.cnmailto:514923134@qq.comhttps://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

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