Research Article Small Signal Model for VSC-HVDC ...V dc V ac Pmr Pmi d-q d -q V dc ref V ac ref d ref q ref F : Control block diagram of the grid side converter. ac bus working at

Research ArticleSmall Signal Model for VSC-HVDC Connected DFIG-BasedOffshore Wind Farms

Kai Liao Zheng-you He and Bin Sun

School of Electrical Engineering Southwest Jiaotong University Chengdu 610031 China

Correspondence should be addressed to Kai Liao liaokailkfoxmailcom

Received 26 February 2014 Revised 2 June 2014 Accepted 9 June 2014 Published 8 July 2014

Academic Editor Jin Liang

Copyright copy 2014 Kai Liao et alThis 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

Large-scale offshore wind farms are integrated with onshore ac grids through the voltage source converter based high voltage directcurrent (VSC-HVDC) transmission system The impact on the stability of the ac grids will be significant The small signal modelof a wind farm connected with voltage source converter based dc transmission system is studied in this paper A suitable model forsmall signal stability analysis is presented The control system of wind generator and the HVDC system has also been modeled inthis model for small signal stability analysis The impact of the control parameters on the network stability is investigated

1 Introduction

Due to the advantages in speed control and four-quadrantactive and reactive power regular capabilities the largeoffshore wind farms based on doubly fed induction generator(DFIG) have been planned around worldwide [1 2] Withthe anticipated high level penetration of offshore wind powerincreasing voltage source converter based high voltage directcurrent (VSC-HVDC) connections are expected for windpower integration with the onshore ac grids [3] The VSC-HVDC transmission is emerging as the prospective technol-ogy to address the challenges associated with the integrationof future offshore wind power [4]

VSC-HVDC transmission system was employed in [5]for transmitting offshore wind power equipped with fixedspeed generators to the grid In [6 7] a 6 MW wind farmconnected to the grid through the VSC-HVDC transmissionsystem is studied In the future many large wind farms underdevelopment will employ DFIG-based wind turbines whoseoperation and response to network disturbances are signifi-cantly different from other types of generators The controlstrategy of VSC-HVDC transmission system technology forconnecting large DFIG-based wind farms over long distanceis studied in [8] New control strategies for normal and gridfault conditions are proposed To obtain smooth operationthe wind farm side converter is controlled as an infinitevoltage source that automatically absorbs power generated by

the wind farm and maintains a stable local ac network In[9] line commutated HVDC systems had also been used toconnect a large DFIG-based offshore wind farm into the grid

The dynamic behavior of the DFIG has been investigatedby many researchers [10 11] The majority of these studiesare based on time-domain simulation results to investigatethe impact on power system dynamic behavior The controlmethod made DFIG behave like a synchronous generator[12ndash15] Time-domain studies offer a direct appreciation ofthe dynamic behavior in terms of visual clarity Howeverthe time-domain simulation is hard to observe for all theoscillation modals of the systems [16] In [17] a suitablemodel for small signal stability analysis and control designof multiterminal dc networks is presented A typical test net-work that combines conventional synchronous and offshorefixed speed wind generation connected to onshore via a dcnetwork is used to illustrate the design of enhanced voltagesource converter controllers The impact of VSC controlparameters on network dynamic behavior is discussed

In this paper the DFIG-based wind farm via VSC-HVDCtransmission system connected to the grid is studied Thesingle-machine infinite-bus (SMIB) structure is followedThepaper is organized as follows In Section 2 the mathemat-ical model of the system unit is observed In Section 3modal analysis method and the eigenvalues under employedcontroller parameters are studied Section 4 presents theconclusion

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 725209 6 pageshttpdxdoiorg1011552014725209

2 Journal of Applied Mathematics

DFIG

Grid

Cables

WS-VSC

G

GS-VSC

Figure 1 The study system

2 Modeling of Study System

The study system is shown in Figure 1 where a DFIG-basedwind farm (100MW form aggregation of 2MW units) isconnected to a VSC-HVDC linkThe dc transmission voltageis 300 kV pole-to-pole (plusmn150 kV-bipolar) The length of thedc transmission cables is 50 km The onshore ac power gridwhich is comprised of conventional generation is modeledas a synchronous generator with the ratings of 33 kV and2400MVA

21 Doubly Fed Induction Generator The one-phase equiv-alent electric circuit of the DFIG rotor and stator is shownin Figure 2 The dynamic equations of DFIG are usuallydescribed by transforming the machine ldquo119886119887119888rdquo voltage equa-tions into a synchronously rotating frame referred to as theldquo119889-119902rdquo frame

For stability analysis the generators are modeled as anequivalent voltage source based on transient impedance Thedynamic model of DFIG is shown as

119889119894119902119904

119889119905

= minus1205961198901198971198771

120596119904119871 119904

119894119902119904 + 120596119890119897119894119889119904 +120596119890119897119908119903

1205962119904119871 119904

1198901015840

119902119904minus

120596119890119897

1198791199031205962119904119871 119904

1198901015840

119889119904

minus120596119890119897

120596119904119871 119904

V119902119904 +119870119898119903119903120596119890119897

120596119904119871 119904

V119902119903

119889119894119889119904

119889119905

= minus119908119890119897119894119902119904 minus1199081198901198971198771

119908119904119871 119904

119894119889119904 +119908119890119897119908119903

1199082119904119871 119904

1198901015840

119889119904minus

119908119890119897

1198791199031199082119904119871 119904

1198901015840

119902119904

minus119908119890119897

119908119904119871 119904

V119889119904 +119870119898119903119903119908119890119897

119908119904119871 119904

V119889119903

1198891198901015840

119902119904

119889119905

= 1199081198901198971198772119894119889119904 minus119908119890119897

119879119903119908119904

1198901015840

119902119904+ 119908119890119897 (

119908119904 minus 119908119903

119908119904

) 1198901015840

119889119904

minus 119870119898119903119903119908119890119897V119889119903

1198891198901015840

119889119904

119889119905

= minus1199081198901198971198772119894119889119904 minus119908119890119897

119879119903119908119904

1198901015840

119889119904+ 119908119890119897 (

119908119903 minus 119908119904

119908119904

) 1198901015840

119902119904

+ 119870119898119903119903119908119890119897V119902119903

(1)

where 1198901015840119902119904

and 1198901015840119889119904

are the equivalent internal 119902- and 119889-axisvoltages respectively 119894119889119904 and 119894119902119904 are the stator 119902- and 119889- axiscurrents respectively The mean of the parameters can becalculated according to [16]

22 Controller of DFIG Converters The purpose of the rotorside converter controller is to control the active power so asto track the maxim power 119875ref and to maintain the terminalvoltage 119880119904ref as a constant For the grid-side converter thecontroller has to keep the dc-link voltage as a constant andthe desired sharing of reactive power with stator is achievedUsually the reactive power delivered to the grid is onlyproduced from the stator to minimize the converter capacitySo the rotor side converter works at unity power factorAccording to [18] the model of rotor-side convert and itscontroller is shown in

1198891199091

119889119905

= 119875ref minus 119875119904

119894119902119903ref = 1198961199011 (119875ref minus 119875119904) + 11989611989411199091

119889119909119894119902119903

119889119905

= 119894119902119903ref minus 119894119902119903

1198891199092

119889119905

= 119880ref minus 119880119904

119894119889119903ref = 1198961199013 (119880ref minus 119880119904) + 11989611989431199092

119889119909119894119889119903

119889119905

= 119894119889119903ref minus 119894119889119903

ΔV119902119903 = 1198961199012 (1198961199011 (119875ref minus 119875119904) + 11989611989411199091 minus 119894119902119903) + 1198961198942119909119894119902119903

+ 119904119903120596119904119871119898119894119889119904 + 119904119903120596119904119871119903119903119894119902119903

ΔV119889119903 = 1198961199012 (1198961199013 (119880ref minus 119880119904) + 11989611989431199092 minus 119894119889119903) minus 1198961198942119909119894119889119903

minus 119904119903120596119904119871119898119894119902119904 minus 119904119903120596119904119871119903119903119894119889119903

1198891199093

119889119905

= 119880dcref minus 119880dc

119894119889119892ref = minus1198961199014 (119880dcref minus 119880dc) + 11989611989441199093

Journal of Applied Mathematics 3

ics

ias

iar

ibr

br

cricr

A

B

C

120595br

120595ar

120595cr

Rotor side Stator side

arbs

120595cs

120579(t)

120595bsas

120595ascs

a

b

c

ibs

(a) The circuit of rotor and stator

Is Xr Rr Ir

Vs VsEs Xm Vrs E998400

X998400

+

+

+ +

minus minus minusminus

minus

XsRs Is Rs

(b) Equivalent circuit

Figure 2 The equivalent electric circuit of the DFIG

usausbusc

Idc Idl

Im R Luca

ucbucc

ud

2C

2C

minus

+

Figure 3 Converter circuit diagram

119889119909119894119902119892

119889119905

= 119894119889119892ref minus 119894119889119892

119889119909119894119902119892

119889119905

= 119894119902119892ref minus 119894119902119892

ΔV119889119892 = 1198961199015 (minus1198961199014 (119880dcref minus 119880dc) + 11989611989441199093 minus 119894119889119892) + 1198961198945119909119894119902119892

ΔV119902119892 = 1198961199016 (119894119902119892ref minus 119894119902119892) + 1198961198946119909119894119902119892

(2)

23 VSC-HVDC System Converter The VSC-HVDC systemis employed to transfer wind energy fromoffshore to onshoreAs shown in Figure 1 the key unit of the HVDC system isthe converter The wind farm side converter is to transfer theenergy received fromwind turbineThe grid side converter isto keep the DC voltage as a constant value in order Figure 3illustrated a VSC-HVDC converter circuit diagram in ldquo119886119887119888rdquoframe

As shown in Figure 3 it is easy to show the dynamicmodelof the converter in ldquo119886119887119888rdquo frame

[

[

119906119888119886

119906119888119887

119906119888119888

]

]

= 119871119889

119889119905

[

[

119894119886

119894119887

119894119888

]

]

+ 119877[

[

119894119886

119894119887

119894119888

]

]

+ [

[

119906119904119886

119906119904119887

119906119904119888

]

]

(3)

Voltage measurement

SVPWM

PI contr

120579 120579

M

f

Vwf ref1

wf ref

wf ref

f

Figure 4 Control block diagram of the wind farm side converter

The dynamic model in ldquo119889-119902rdquo frame can be easy obtainedby employing the Park transform as shown in (4) and theldquo119889-119902rdquo frame dynamic model is shown in (5)

119875 =2

3

[sin119908119905 sin (119908119905 minus 120∘) sin (119908119905 + 120∘)cos119908119905 cos (119908119905 minus 120∘) cos (119908119905 + 120∘)] (4)

119889

119889119905

[119894119898119889

119894119898119902

] =1

119871

[V119888119889V119888119902] minus

119877

119871

[119894119898119889

119894119898119902

] minus1

119871

[119906119904119889

119906119904119902

]

minus [0 minus120596

120596 0] [119894119898119889

119894119898119902

]

(5)

where 119888means the converter side parameters and 119904means thegrid side parameters Under PWM control the amplitude ofthe converter output fundamental voltage is controlled by themodulation index119872 as

119881119888 =119872 sdot 119881dc2

(6)

In ideal condition the dynamic of the dc side transmis-sion system is shown in

119889119881dc119889119905

=1

119862

sdot 119868dc minus3

4119862

(119872119889 sdot 119894119898119889 +119872119902 sdot 119894119898119902) (7)

where the 119894119898119889 and 119894119898119902 are the 119889- and 119902- axis convertercurrents respectively119872 is the modulation index 119881dc is theDC voltage and 119868dc is the DC current

24 Controller of Wind Farm Side Converter The wind farmside converter (WFVSC) should keep the wind farm side


md

md

PWMconverter

Currentmeasurement

Voltage controller Current controller

I

Ii

IdIr

IIq

PI

PI PI

PI

PLL transftransf

P

P

+minus+minus

+minus+minus

cos(ref)

sin(ref)

Vdc

Vac

Pmr

Pmi

d-qd-q

Vdc ref

Vac ref

d ref

q ref

Figure 5 Control block diagram of the grid side converter

ac bus working at constant frequency and voltage So thecontroller of wind farm side converter should maintain theac bus frequency and voltage as a constant After that theprimary requirement of the WFVSC is to transfer energycollected from thewind farmTheoutput power formDFIG iscontrolled by power electronic converters and MPPT systemand the network frequency variations when little influenceis on the power generation The main task for the WFVSCis to collect energy from the wind farm and to control theac voltage and frequency of the local wind farm networkTherefore the control strategy adopted on the WFVSC is tocontrol the wind farm side converter as an infinite voltagesource with constant frequency voltage amplitude and phaseangle The control block diagram for theWFVSC is shown inFigure 4

Consider the voltage measurement devise delay themeasurement unit can be modeled as a first-order process

119881mea = 119881119908119891 sdot1

1 + 119879119904

(8)

as

119889119881mea119889119905

=

119881119908119891 minus 119881mea

119879

(9)

The controller of the converter is respected to the bus voltageas shown in Figure 4 The dynamic model of the wind farmside converter controller is shown as

119889119872

119889119905

= minus1198701199017 (

119881119908119891 minus 119881mea

119879

) + 1198701198947119881119908119891 ref1 minus 1198701198947119881mea (10)

Thus the control strategy makes the wind farm beconnected to an infinite ac system the power generated by thewind farm is automatically absorbed by the source resembledby the wind farm side converters and then transmitted to thegrid via the DC lines

25 Controller of Grid Side HVDC Converter The WSVSCcollects energy from wind farm and then transmits it to thepower grid via the dc transmission and grid side converter(GSVSC) (Figure 5) For a dc transmission system weexpect the dc voltage to maintain a constant value under allcondition to keep the system work stability A constant dcvoltage indicates balanced active power flow between the two

Vdcref

VGref

kp10 +ki10s

kp11 +ki11s

VG

kp8 +ki8s

kp9 +ki9s

Vdc

ilowastsd

isd

ilowastsq

isq

uq

ud

minus

minusminus

minus

Figure 6 The control system of the grid side converter

sides Unstable dc link voltage can cause the system to tripand disrupt its normal operation Therefore the grid sideconverter is assigned to control the dc voltage as the expectedvalue The PI control system of the grid side converter isshown in Figure 6The control method can ensure the energycollected by the wind farm is transmitted to the grid networkalmost completely The reference angle for 119886119887119888-119889119902 transfer isobtained from the PLL loop

The current control loop is designed as

119906119889 =119889119894119904119889

119889119905

= 1198961199018 (119894lowast

119904119889minus 119894119904119889) + 1198961198948 int (119894

lowast

119904119889minus 119894119904119889) 119889119905

119906119902 =

119889119894119904119902

119889119905

= 1198961199019 (119894lowast

119904119902minus 119894119904119902) + 1198961198949 int(119894

lowast

119904119902minus 119894119904119902) 119889119905

(11)

where 119894lowast119904119889

is the dc voltage control loop and 119894lowast

119904119902is the

connection bus voltage control loop

119894lowast

119904119889= 11989611990110 (119881

lowast

dc minus 119881dc) + 11989611989410 int (119881lowast

dc minus 119881dc) 119889119905

119894lowast

119904119902= 11989611990111 (119881

lowast

119866minus 119881119866) + 11989611989411 int (119881

lowast

119866minus 119881119866) 119889119905

(12)

The VSC control modulation index119872 in 119889119902 frame is shownin

119872119889 =2119871

119881dc(119906119889 +

119877

119871

119894119904119889 minus 120596119904119894119904119902 +1

119871

V119904119889)

119872119902 =2119871

119881dc(119906119902 +

119877

119871

119894119904119902 + 120596119904119894119904119889 +1

119871

V119904119902) (13)


Linearized model of dc transmission system

Wind farm side VSC controller

Linearized model

of DFIG generator and its controller

Linearized model of wind farm side

converter

Linearized model

converter

ac grid side VSC controller

Linearized model ΔisdΔisq Δdc Δdc

Δdc

ΔgdΔgq

ΔG

ΔsdΔsqΔd

Δq

ΔigdΔigqΔd

Δq

Δidc

Δwf

Δidc of ac gridof grid side

Figure 7 The combined small signal model for whole system

Table 1 Controller parameters

1198961199011

1198961199012

1198961199013

1198961199014

1198961199015

1198961199016

1198961199017

1198961199018

1198961199019

11989611990110

11989611990111

045 05 0025 0005 001 025 15 1 1 10 0011198961198941

1198961198942

1198961198943

1198961198944

1198961198945

1198961198946

1198961198947

1198961198948

1198961198949

11989611989410

11989611989411

025 045 02 015 025 015 20 8 8 200 300

For small signal model four middle variables named119909Vdc 119909119894119904119889 119909V119892 and 119909119894119904119902 are employed 119909Vdc = int(119881

lowast

dc minus 119881dc)

119909119894119904119889 = int(119894lowast

119904119889minus 119894119904119889) 119909V119892 = int(119881

lowast

119866minus 119881119866) and 119909119894119904119902 = int(119894

lowast

119904119902minus 119894119904119902)

Combine (11) with (13) and the middle variables thedynamic model for GSVSC can be obtained as shown in

119889119909Vdc119889119905

= 119881lowast

dc minus 119881dc

119889119909119894119904119889

119889119905

= 11989611990110 (119881lowast

dc minus 119881dc) + 11989611989410119909Vdc minus 119894119904119889

119889119909V119892

119889119905

= 119881lowast

119866minus 119881119866

119889119909119894119904119902

119889119905

= 11989611990111 (119881lowast

119866minus 119881119866) + 11989611989411119909V119892 minus 119894119904119902

119881119892119888119889 = minus119896119901811989611990110 (119881lowast

dc minus 119881dc) + 119896119901811989611989410119909Vdc

minus 1198961199018119894119904119889 + 1198961198948119909119894119904119889

119881119892119888119902 = 119896119901911989611990111 (119881lowast

119866minus 119881119866) + 119896119901911989611989411119909V119892

minus 1198961199019119894119904119902 + 11989611989411119909119894119904119902

(14)

3 Modal Analysis

The complete state-space representation of the test system isobtained by combining all the linearized models As shownin Figure 7 the small signal model of the whole system iscombined

The small signal stability can be observed by eigenvalueanalysis of the whole system linearized model In this casethe ldquosmall signalrdquo disturbances are considered sufficientlysmall to permit the equations representing the system to belinearized and expressed in state-space form The model of apower system can be expressed as a set of DAEThe linearized

Table 2 Eigenvalues

120582 Value 119891 (Hz) Damping ratio12058212 minus1428 plusmn 1198957654 12187 018312058234 minus316 plusmn 1198951468 2158 02112058256 minus1489 plusmn 119895283 4506 005312058278 minus56 plusmn 119895212 3376 0026120582910 minus451 plusmn 119895521 8296 00861205821112 minus78 plusmn 119895853 1358 00911205821314 minus283 plusmn 1198952196 3497 01281205821516 minus94 plusmn 119895821 131 07531205821718 minus18 plusmn 119895568 09 03021205821920 minus0004 plusmn 119895435 069 0009

model of the test system can be expressed in state-space formas in

Δx = ΑΔx + BΔu (15)

where Δx is the state vector with 26 orders u is the inputvector A is the state matrix and B is the input or controlmatrix The eigenvalue of the state matrix A provides thenecessary information about the small signal stability of thesystem

The purpose to model the small signal model of thesystem is to study the small signal stability and for thefurther study to investigate the controller parameters designto enhance the small signal stability The control parametersemployed in this study are shown in Table 1

The eigenvalue analysis of the small signal model is showin Table 2 Eigenvalues shown here are all the oscillationmode It is known that the whole real part of the eigenvaluesis negative and the oscillation mode is stable under thecontroller parameters in Table 1


4 Conclusion

A linearized mathematical model for small signal stabilityanalysis of VSC-HVDC transmission system collected with aDFIG-based wind farm has been presented in this paperThelinearizedmodel is based on the state-spacemodelsThe statematrix is employed to investigate the small signal stabilityperformance of the studied system through the eigenvalueanalysis It was validated that using the small signal stabilitymodel it was possible to design improved controllers forthe VSCs of the dc network which ensure stable networkoperation and enhanced dynamic performance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National High TechnologyResearch and Development Program (no 2012AA050208)

References

[1] R Pena J CClare andGMAsher ldquoDouble fed induction gen-erator using back-to-back PWM converter and its applicationto variable speed wind energy generationrdquo IEE ProceedingsmdashElectric Power Applications vol 143 no 3 pp 231ndash241 1996

[2] T Ackermann ldquoTransmission systems for offshore wind farmsrdquoIEEE Power Engineering Review vol 22 no 12 pp 23ndash27 2002

[3] D C Kong and X- Zhang ldquoModelling and control of offshorewind farm with VSC-HVDC transmission systemrdquo in Proceed-ings of the 9th IET International Conference onACandDCPowerTransmission (ACDC rsquo10) pp 1ndash6 London UK October 2010

[4] W Lu and B Ooi ldquoOptimal acquisition and aggregation ofoffshore wind power by multiterminal voltage-source HVDCrdquoIEEE Transactions on Power Delivery vol 18 no 1 pp 201ndash2062003

[5] D Xiang L Ran J R Bumby P J Tavner and S Yang ldquoCoor-dinated control of an HVDC link and doubly fed inductiongenerators in a large offshore wind farmrdquo IEEE Transactions onPower Delivery vol 21 no 1 pp 463ndash471 2006

[6] K H Sobrink P L Sorensen P Christensen N Sandersen KEriksson and PHolmberg ldquoDC feeder for connection of awindfarmrdquo in Proceedings of the Cigre Symposium Kuala LumpurMalaysia September 1999

[7] X I Koutiva T D Vrionis N A Vovos and G B Gian-nakopoulos ldquoOptimal integration of an offshore wind farm toa weak AC gridrdquo IEEE Transactions on Power Delivery vol 21no 2 pp 987ndash994 2006

[8] L Xu L Yao and C Sasse ldquoGrid integration of large DFIG-based wind farms using VSC transmissionrdquo IEEE Transactionson Power Systems vol 22 no 3 pp 976ndash984 2007

[9] S V Bozhko R V Blasco-Gimenez R Li J C Clare and GM Asher ldquoControl of offshore DFIG-based wind farm gridwith line-commutated HVDC connectionrdquo IEEE Transactionson Energy Conversion vol 22 no 1 pp 71ndash78 2007

[10] J G Slootweg Wind power modelling and impact on powersystems dynamics [PhD dissertation] Delft University of Tech-nology Delft The Netherlands 2003

[11] V Akhmatov Analysis of dynamic behaviour of electric powersystems with large amount of wind power [PhD dissertation]Technical University of Denmark Lyngby Denmark 2003

[12] M V A Nunes J A P Lopes H H Zurn U H Bezerra and RG Almeida ldquoInfluence of the variable-speed wind generatorsin transient stability margin of the conventional generatorsintegrated in electrical gridsrdquo IEEE Transactions on EnergyConversion vol 19 no 4 pp 692ndash701 2004

[13] F M Hughes O Anaya-Lara N Jenkins and G StrbacldquoControl of DFIG-based wind generation for power networksupportrdquo IEEE Transactions on Power Systems vol 20 no 4 pp1958ndash1966 2005

[14] D Alvira J Arevalo C Bermudez et al ldquoFeasibility studiesof the HVDC submarine interconnection between the Spanishpeninsula and the Balearic island of Mallorcardquo in Proceedings ofthe 41st International Conference on Large High Voltage ElectricSystems (CIGRE rsquo06) pp 4ndash104 Paris France September 2006

[15] R S Pena Vector control strategies for a doubly-fed inductiongenerator driven by a wind turbine [PhD dissertation] Univer-sity of Nottingham Nottingham UK 1996

[16] F Mei and B Pal ldquoModal analysis of grid-connected doubly fedinduction generatorsrdquo IEEE Transactions on Energy Conversionvol 22 no 3 pp 728ndash736 2007

[17] G O Kalcon G P Adam O Anaya-Lara S Lo and K UhlenldquoSmall-signal stability analysis of multi-terminal VSC-basedDC transmission systemsrdquo IEEETransactions on Power Systemsvol 27 no 4 pp 1818ndash1830 2012

[18] F Wu X-P Zhang K Godfrey and P Ju ldquoSmall signal stabilityanalysis and optimal control of a wind turbine with doublyfed induction generatorrdquo IET Generation Transmission andDistribution vol 1 no 5 pp 751ndash760 2007

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DFIG

Grid

Cables

WS-VSC

G

GS-VSC

Figure 1 The study system

2 Modeling of Study System

The study system is shown in Figure 1 where a DFIG-basedwind farm (100MW form aggregation of 2MW units) isconnected to a VSC-HVDC linkThe dc transmission voltageis 300 kV pole-to-pole (plusmn150 kV-bipolar) The length of thedc transmission cables is 50 km The onshore ac power gridwhich is comprised of conventional generation is modeledas a synchronous generator with the ratings of 33 kV and2400MVA

21 Doubly Fed Induction Generator The one-phase equiv-alent electric circuit of the DFIG rotor and stator is shownin Figure 2 The dynamic equations of DFIG are usuallydescribed by transforming the machine ldquo119886119887119888rdquo voltage equa-tions into a synchronously rotating frame referred to as theldquo119889-119902rdquo frame

For stability analysis the generators are modeled as anequivalent voltage source based on transient impedance Thedynamic model of DFIG is shown as

119889119894119902119904

119889119905

= minus1205961198901198971198771

120596119904119871 119904

119894119902119904 + 120596119890119897119894119889119904 +120596119890119897119908119903

1205962119904119871 119904

1198901015840

119902119904minus

120596119890119897

1198791199031205962119904119871 119904

1198901015840

119889119904

minus120596119890119897

120596119904119871 119904

V119902119904 +119870119898119903119903120596119890119897

120596119904119871 119904

V119902119903

119889119894119889119904

119889119905

= minus119908119890119897119894119902119904 minus1199081198901198971198771

119908119904119871 119904

119894119889119904 +119908119890119897119908119903

1199082119904119871 119904

1198901015840

119889119904minus

119908119890119897

1198791199031199082119904119871 119904

1198901015840

119902119904

minus119908119890119897

119908119904119871 119904

V119889119904 +119870119898119903119903119908119890119897

119908119904119871 119904

V119889119903

1198891198901015840

119902119904

119889119905

= 1199081198901198971198772119894119889119904 minus119908119890119897

119879119903119908119904

1198901015840

119902119904+ 119908119890119897 (

119908119904 minus 119908119903

119908119904

) 1198901015840

119889119904

minus 119870119898119903119903119908119890119897V119889119903

1198891198901015840

119889119904

119889119905

= minus1199081198901198971198772119894119889119904 minus119908119890119897

119879119903119908119904

1198901015840

119889119904+ 119908119890119897 (

119908119903 minus 119908119904

119908119904

) 1198901015840

119902119904

+ 119870119898119903119903119908119890119897V119902119903

(1)

where 1198901015840119902119904

and 1198901015840119889119904

are the equivalent internal 119902- and 119889-axisvoltages respectively 119894119889119904 and 119894119902119904 are the stator 119902- and 119889- axiscurrents respectively The mean of the parameters can becalculated according to [16]

22 Controller of DFIG Converters The purpose of the rotorside converter controller is to control the active power so asto track the maxim power 119875ref and to maintain the terminalvoltage 119880119904ref as a constant For the grid-side converter thecontroller has to keep the dc-link voltage as a constant andthe desired sharing of reactive power with stator is achievedUsually the reactive power delivered to the grid is onlyproduced from the stator to minimize the converter capacitySo the rotor side converter works at unity power factorAccording to [18] the model of rotor-side convert and itscontroller is shown in

1198891199091

119889119905

= 119875ref minus 119875119904

119894119902119903ref = 1198961199011 (119875ref minus 119875119904) + 11989611989411199091

119889119909119894119902119903

119889119905

= 119894119902119903ref minus 119894119902119903

1198891199092

119889119905

= 119880ref minus 119880119904

119894119889119903ref = 1198961199013 (119880ref minus 119880119904) + 11989611989431199092

119889119909119894119889119903

119889119905

= 119894119889119903ref minus 119894119889119903

ΔV119902119903 = 1198961199012 (1198961199011 (119875ref minus 119875119904) + 11989611989411199091 minus 119894119902119903) + 1198961198942119909119894119902119903

+ 119904119903120596119904119871119898119894119889119904 + 119904119903120596119904119871119903119903119894119902119903

ΔV119889119903 = 1198961199012 (1198961199013 (119880ref minus 119880119904) + 11989611989431199092 minus 119894119889119903) minus 1198961198942119909119894119889119903

minus 119904119903120596119904119871119898119894119902119904 minus 119904119903120596119904119871119903119903119894119889119903

1198891199093

119889119905

= 119880dcref minus 119880dc

119894119889119892ref = minus1198961199014 (119880dcref minus 119880dc) + 11989611989441199093


ics

ias

iar

ibr

br

cricr

A

B

C

120595br

120595ar

120595cr


arbs

120595cs

120579(t)

120595bsas

120595ascs

a

b

c

ibs


Is Xr Rr Ir


X998400

+

+

+ +


minus

XsRs Is Rs



usausbusc

Idc Idl

Im R Luca

ucbucc

ud

2C

2C

minus

+


119889119909119894119902119892

119889119905

= 119894119889119892ref minus 119894119889119892

119889119909119894119902119892

119889119905

= 119894119902119892ref minus 119894119902119892


ΔV119902119892 = 1198961199016 (119894119902119892ref minus 119894119902119892) + 1198961198946119909119894119902119892

(2)



[

[

119906119888119886

119906119888119887

119906119888119888

]

]

= 119871119889

119889119905

[

[

119894119886

119894119887

119894119888

]

]

+ 119877[

[

119894119886

119894119887

119894119888

]

]

+ [

[

119906119904119886

119906119904119887

119906119904119888

]

]

(3)

Voltage measurement

SVPWM

PI contr

120579 120579

M

f

Vwf ref1

wf ref

wf ref

f



119875 =2

3


119889

119889119905

[119894119898119889

119894119898119902

] =1

119871

[V119888119889V119888119902] minus

119877

119871

[119894119898119889

119894119898119902

] minus1

119871

[119906119904119889

119906119904119902

]


120596 0] [119894119898119889

119894119898119902

]

(5)


119881119888 =119872 sdot 119881dc2

(6)


119889119881dc119889119905

=1

119862


4119862

(119872119889 sdot 119894119898119889 +119872119902 sdot 119894119898119902) (7)




md

md

PWMconverter

Currentmeasurement


I

Ii

IdIr

IIq

PI

PI PI

PI

PLL transftransf

P

P

+minus+minus

+minus+minus

cos(ref)

sin(ref)

Vdc

Vac

Pmr

Pmi

d-qd-q

Vdc ref

Vac ref

d ref

q ref




119881mea = 119881119908119891 sdot1

1 + 119879119904

(8)

as

119889119881mea119889119905

=

119881119908119891 minus 119881mea

119879

(9)


119889119872

119889119905

= minus1198701199017 (

119881119908119891 minus 119881mea

119879

) + 1198701198947119881119908119891 ref1 minus 1198701198947119881mea (10)



Vdcref

VGref

kp10 +ki10s

kp11 +ki11s

VG

kp8 +ki8s

kp9 +ki9s

Vdc

ilowastsd

isd

ilowastsq

isq

uq

ud

minus

minusminus

minus




119906119889 =119889119894119904119889

119889119905

= 1198961199018 (119894lowast

119904119889minus 119894119904119889) + 1198961198948 int (119894

lowast

119904119889minus 119894119904119889) 119889119905

119906119902 =

119889119894119904119902

119889119905

= 1198961199019 (119894lowast

119904119902minus 119894119904119902) + 1198961198949 int(119894

lowast

119904119902minus 119894119904119902) 119889119905

(11)

where 119894lowast119904119889


119904119902is the


119894lowast

119904119889= 11989611990110 (119881

lowast


dc minus 119881dc) 119889119905

119894lowast

119904119902= 11989611990111 (119881

lowast

119866minus 119881119866) + 11989611989411 int (119881

lowast

119866minus 119881119866) 119889119905

(12)


119872119889 =2119871

119881dc(119906119889 +

119877

119871

119894119904119889 minus 120596119904119894119904119902 +1

119871

V119904119889)

119872119902 =2119871

119881dc(119906119902 +

119877

119871

119894119904119902 + 120596119904119894119904119889 +1

119871

V119904119902) (13)




Linearized model



converter

Linearized model

converter



Δdc

ΔgdΔgq

ΔG

ΔsdΔsqΔd

Δq

ΔigdΔigqΔd

Δq

Δidc

Δwf




1198961199011

1198961199012

1198961199013

1198961199014

1198961199015

1198961199016

1198961199017

1198961199018

1198961199019

11989611990110

11989611990111

045 05 0025 0005 001 025 15 1 1 10 0011198961198941

1198961198942

1198961198943

1198961198944

1198961198945

1198961198946

1198961198947

1198961198948

1198961198949

11989611989410

11989611989411

025 045 02 015 025 015 20 8 8 200 300


lowast

dc minus 119881dc)

119909119894119904119889 = int(119894lowast

119904119889minus 119894119904119889) 119909V119892 = int(119881

lowast

119866minus 119881119866) and 119909119894119904119902 = int(119894

lowast

119904119902minus 119894119904119902)


119889119909Vdc119889119905

= 119881lowast

dc minus 119881dc

119889119909119894119904119889

119889119905

= 11989611990110 (119881lowast


119889119909V119892

119889119905

= 119881lowast

119866minus 119881119866

119889119909119894119904119902

119889119905

= 11989611990111 (119881lowast

119866minus 119881119866) + 11989611989411119909V119892 minus 119894119904119902

119881119892119888119889 = minus119896119901811989611990110 (119881lowast

dc minus 119881dc) + 119896119901811989611989410119909Vdc

minus 1198961199018119894119904119889 + 1198961198948119909119894119904119889

119881119892119888119902 = 119896119901911989611990111 (119881lowast

119866minus 119881119866) + 119896119901911989611989411119909V119892

minus 1198961199019119894119904119902 + 11989611989411119909119894119904119902

(14)

3 Modal Analysis



Table 2 Eigenvalues








4 Conclusion




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ics

ias

iar

ibr

br

cricr

A

B

C

120595br

120595ar

120595cr


arbs

120595cs

120579(t)

120595bsas

120595ascs

a

b

c

ibs


Is Xr Rr Ir


X998400

+

+

+ +


minus

XsRs Is Rs



usausbusc

Idc Idl

Im R Luca

ucbucc

ud

2C

2C

minus

+


119889119909119894119902119892

119889119905

= 119894119889119892ref minus 119894119889119892

119889119909119894119902119892

119889119905

= 119894119902119892ref minus 119894119902119892


ΔV119902119892 = 1198961199016 (119894119902119892ref minus 119894119902119892) + 1198961198946119909119894119902119892

(2)



[

[

119906119888119886

119906119888119887

119906119888119888

]

]

= 119871119889

119889119905

[

[

119894119886

119894119887

119894119888

]

]

+ 119877[

[

119894119886

119894119887

119894119888

]

]

+ [

[

119906119904119886

119906119904119887

119906119904119888

]

]

(3)

Voltage measurement

SVPWM

PI contr

120579 120579

M

f

Vwf ref1

wf ref

wf ref

f



119875 =2

3


119889

119889119905

[119894119898119889

119894119898119902

] =1

119871

[V119888119889V119888119902] minus

119877

119871

[119894119898119889

119894119898119902

] minus1

119871

[119906119904119889

119906119904119902

]


120596 0] [119894119898119889

119894119898119902

]

(5)


119881119888 =119872 sdot 119881dc2

(6)


119889119881dc119889119905

=1

119862


4119862

(119872119889 sdot 119894119898119889 +119872119902 sdot 119894119898119902) (7)




md

md

PWMconverter

Currentmeasurement


I

Ii

IdIr

IIq

PI

PI PI

PI

PLL transftransf

P

P

+minus+minus

+minus+minus

cos(ref)

sin(ref)

Vdc

Vac

Pmr

Pmi

d-qd-q

Vdc ref

Vac ref

d ref

q ref




119881mea = 119881119908119891 sdot1

1 + 119879119904

(8)

as

119889119881mea119889119905

=

119881119908119891 minus 119881mea

119879

(9)


119889119872

119889119905

= minus1198701199017 (

119881119908119891 minus 119881mea

119879

) + 1198701198947119881119908119891 ref1 minus 1198701198947119881mea (10)



Vdcref

VGref

kp10 +ki10s

kp11 +ki11s

VG

kp8 +ki8s

kp9 +ki9s

Vdc

ilowastsd

isd

ilowastsq

isq

uq

ud

minus

minusminus

minus




119906119889 =119889119894119904119889

119889119905

= 1198961199018 (119894lowast

119904119889minus 119894119904119889) + 1198961198948 int (119894

lowast

119904119889minus 119894119904119889) 119889119905

119906119902 =

119889119894119904119902

119889119905

= 1198961199019 (119894lowast

119904119902minus 119894119904119902) + 1198961198949 int(119894

lowast

119904119902minus 119894119904119902) 119889119905

(11)

where 119894lowast119904119889


119904119902is the


119894lowast

119904119889= 11989611990110 (119881

lowast


dc minus 119881dc) 119889119905

119894lowast

119904119902= 11989611990111 (119881

lowast

119866minus 119881119866) + 11989611989411 int (119881

lowast

119866minus 119881119866) 119889119905

(12)


119872119889 =2119871

119881dc(119906119889 +

119877

119871

119894119904119889 minus 120596119904119894119904119902 +1

119871

V119904119889)

119872119902 =2119871

119881dc(119906119902 +

119877

119871

119894119904119902 + 120596119904119894119904119889 +1

119871

V119904119902) (13)




Linearized model



converter

Linearized model

converter



Δdc

ΔgdΔgq

ΔG

ΔsdΔsqΔd

Δq

ΔigdΔigqΔd

Δq

Δidc

Δwf




1198961199011

1198961199012

1198961199013

1198961199014

1198961199015

1198961199016

1198961199017

1198961199018

1198961199019

11989611990110

11989611990111

045 05 0025 0005 001 025 15 1 1 10 0011198961198941

1198961198942

1198961198943

1198961198944

1198961198945

1198961198946

1198961198947

1198961198948

1198961198949

11989611989410

11989611989411

025 045 02 015 025 015 20 8 8 200 300


lowast

dc minus 119881dc)

119909119894119904119889 = int(119894lowast

119904119889minus 119894119904119889) 119909V119892 = int(119881

lowast

119866minus 119881119866) and 119909119894119904119902 = int(119894

lowast

119904119902minus 119894119904119902)


119889119909Vdc119889119905

= 119881lowast

dc minus 119881dc

119889119909119894119904119889

119889119905

= 11989611990110 (119881lowast


119889119909V119892

119889119905

= 119881lowast

119866minus 119881119866

119889119909119894119904119902

119889119905

= 11989611990111 (119881lowast

119866minus 119881119866) + 11989611989411119909V119892 minus 119894119904119902

119881119892119888119889 = minus119896119901811989611990110 (119881lowast

dc minus 119881dc) + 119896119901811989611989410119909Vdc

minus 1198961199018119894119904119889 + 1198961198948119909119894119904119889

119881119892119888119902 = 119896119901911989611990111 (119881lowast

119866minus 119881119866) + 119896119901911989611989411119909V119892

minus 1198961199019119894119904119902 + 11989611989411119909119894119904119902

(14)

3 Modal Analysis



Table 2 Eigenvalues








4 Conclusion




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










md

md

PWMconverter

Currentmeasurement


I

Ii

IdIr

IIq

PI

PI PI

PI

PLL transftransf

P

P

+minus+minus

+minus+minus

cos(ref)

sin(ref)

Vdc

Vac

Pmr

Pmi

d-qd-q

Vdc ref

Vac ref

d ref

q ref




119881mea = 119881119908119891 sdot1

1 + 119879119904

(8)

as

119889119881mea119889119905

=

119881119908119891 minus 119881mea

119879

(9)


119889119872

119889119905

= minus1198701199017 (

119881119908119891 minus 119881mea

119879

) + 1198701198947119881119908119891 ref1 minus 1198701198947119881mea (10)



Vdcref

VGref

kp10 +ki10s

kp11 +ki11s

VG

kp8 +ki8s

kp9 +ki9s

Vdc

ilowastsd

isd

ilowastsq

isq

uq

ud

minus

minusminus

minus




119906119889 =119889119894119904119889

119889119905

= 1198961199018 (119894lowast

119904119889minus 119894119904119889) + 1198961198948 int (119894

lowast

119904119889minus 119894119904119889) 119889119905

119906119902 =

119889119894119904119902

119889119905

= 1198961199019 (119894lowast

119904119902minus 119894119904119902) + 1198961198949 int(119894

lowast

119904119902minus 119894119904119902) 119889119905

(11)

where 119894lowast119904119889


119904119902is the


119894lowast

119904119889= 11989611990110 (119881

lowast


dc minus 119881dc) 119889119905

119894lowast

119904119902= 11989611990111 (119881

lowast

119866minus 119881119866) + 11989611989411 int (119881

lowast

119866minus 119881119866) 119889119905

(12)


119872119889 =2119871

119881dc(119906119889 +

119877

119871

119894119904119889 minus 120596119904119894119904119902 +1

119871

V119904119889)

119872119902 =2119871

119881dc(119906119902 +

119877

119871

119894119904119902 + 120596119904119894119904119889 +1

119871

V119904119902) (13)




Linearized model



converter

Linearized model

converter



Δdc

ΔgdΔgq

ΔG

ΔsdΔsqΔd

Δq

ΔigdΔigqΔd

Δq

Δidc

Δwf




1198961199011

1198961199012

1198961199013

1198961199014

1198961199015

1198961199016

1198961199017

1198961199018

1198961199019

11989611990110

11989611990111

045 05 0025 0005 001 025 15 1 1 10 0011198961198941

1198961198942

1198961198943

1198961198944

1198961198945

1198961198946

1198961198947

1198961198948

1198961198949

11989611989410

11989611989411

025 045 02 015 025 015 20 8 8 200 300


lowast

dc minus 119881dc)

119909119894119904119889 = int(119894lowast

119904119889minus 119894119904119889) 119909V119892 = int(119881

lowast

119866minus 119881119866) and 119909119894119904119902 = int(119894

lowast

119904119902minus 119894119904119902)


119889119909Vdc119889119905

= 119881lowast

dc minus 119881dc

119889119909119894119904119889

119889119905

= 11989611990110 (119881lowast


119889119909V119892

119889119905

= 119881lowast

119866minus 119881119866

119889119909119894119904119902

119889119905

= 11989611990111 (119881lowast

119866minus 119881119866) + 11989611989411119909V119892 minus 119894119904119902

119881119892119888119889 = minus119896119901811989611990110 (119881lowast

dc minus 119881dc) + 119896119901811989611989410119909Vdc

minus 1198961199018119894119904119889 + 1198961198948119909119894119904119889

119881119892119888119902 = 119896119901911989611990111 (119881lowast

119866minus 119881119866) + 119896119901911989611989411119909V119892

minus 1198961199019119894119904119902 + 11989611989411119909119894119904119902

(14)

3 Modal Analysis



Table 2 Eigenvalues








4 Conclusion




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Linearized model



converter

Linearized model

converter



Δdc

ΔgdΔgq

ΔG

ΔsdΔsqΔd

Δq

ΔigdΔigqΔd

Δq

Δidc

Δwf




1198961199011

1198961199012

1198961199013

1198961199014

1198961199015

1198961199016

1198961199017

1198961199018

1198961199019

11989611990110

11989611990111

045 05 0025 0005 001 025 15 1 1 10 0011198961198941

1198961198942

1198961198943

1198961198944

1198961198945

1198961198946

1198961198947

1198961198948

1198961198949

11989611989410

11989611989411

025 045 02 015 025 015 20 8 8 200 300


lowast

dc minus 119881dc)

119909119894119904119889 = int(119894lowast

119904119889minus 119894119904119889) 119909V119892 = int(119881

lowast

119866minus 119881119866) and 119909119894119904119902 = int(119894

lowast

119904119902minus 119894119904119902)


119889119909Vdc119889119905

= 119881lowast

dc minus 119881dc

119889119909119894119904119889

119889119905

= 11989611990110 (119881lowast


119889119909V119892

119889119905

= 119881lowast

119866minus 119881119866

119889119909119894119904119902

119889119905

= 11989611990111 (119881lowast

119866minus 119881119866) + 11989611989411119909V119892 minus 119894119904119902

119881119892119888119889 = minus119896119901811989611990110 (119881lowast

dc minus 119881dc) + 119896119901811989611989410119909Vdc

minus 1198961199018119894119904119889 + 1198961198948119909119894119904119889

119881119892119888119902 = 119896119901911989611990111 (119881lowast

119866minus 119881119866) + 119896119901911989611989411119909V119892

minus 1198961199019119894119904119902 + 11989611989411119909119894119904119902

(14)

3 Modal Analysis



Table 2 Eigenvalues








4 Conclusion




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4 Conclusion




Acknowledgment


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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