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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
4 Journal of Applied Mathematics
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)
Journal of Applied Mathematics 5
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
6 Journal of Applied Mathematics
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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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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
4 Journal of Applied Mathematics
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)
Journal of Applied Mathematics 5
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
6 Journal of Applied Mathematics
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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Differential EquationsInternational Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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
4 Journal of Applied Mathematics
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)
Journal of Applied Mathematics 5
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
6 Journal of Applied Mathematics
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
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
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)
Journal of Applied Mathematics 5
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
6 Journal of Applied Mathematics
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
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
6 Journal of Applied Mathematics
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of