Research ArticleAn Improved Phase-Locked-Loop Control with AlternativeDamping Factors for VSC Connected to Weak AC System
Bin Yuan Jianzhong Xu Chengyong Zhao and Yijia Yuan
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources North China Electric Power UniversityBeijing 102206 China
Correspondence should be addressed to Bin Yuan yuanbingemini126com
Received 12 October 2015 Revised 1 January 2016 Accepted 4 January 2016
Academic Editor Ahmed M Massoud
Copyright copy 2016 Bin Yuan et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The gains of phase-locked-loop (PLL) have significant impacts on the power transfer limits for the voltage source converter (VSC)connected to weak AC systemTherefore in this paper an improved PLL control respectively with alternative damping factors forrectifier and inverter is proposed First it is proved that the impedance angle of AC system has a great impact on the small-signalstability of the VSC system With the same variation tendency of Thevenin equivalent resistance the limits of power transmissionare changing in opposite trends for rectifier and inverter Second the improved PLL with alternative damping factors is proposedbased on the participation factor analysisThird the optimal damping factors of the improved PLL control for rectifier and inverterare calculated Simulations and calculations validated the following three conclusions (1) in rectifying operation the equivalentsystem resistance has a negative impact on the stability of the system and this is not the case for inverting operation (2) addingthe alternative damping factors to PLL control shows similar results compared with changing the impedance angle of AC system(3) the proposed optimal damping factors of PLL can effectively extend the power transfer limits under both rectifier and invertermodes
1 Introduction
Renewable energy resources are emerging as a future energyvector and the voltage source converters (VSCs) are widelyused to integrate such energies into power system [1ndash4] TheVSC-HVDC link connected to weak AC system with verylow short circuit ratio (SCR lt 2) will emerge quite often inthe future [5ndash7] However the conventional vector-currentcontrol in 119889-119902 frame exhibits poor dynamic performancewhen applied to VSC connected to it This brings a problemthat the transmitted power cannot reach the ideal limitationfor the unstable of small-signal model [8ndash17]
There are three possible approaches to solve this problemThe first approach is shown in [8] which proposed anadvanced vector-current control to decouple the 119889-119902 outer-loop control completely by optimizing the control parame-ters However the provided method is quite complicated andit is not suitable for frequent and rapid power changing
The second approach shown in [9] is adopting powersynchronization control (PSC) as the main control strategy
PSC is similar with power angle controlThis control strategywill not cause stable operating problems in extremely weakAC systems However it behaves in relatively low responsespeed due to the lack of the inner-loop current control andhence it cannot satisfactorily meet the requirement of the ACsystem
The last approach shown in [10] is changing the parame-ters of phase-locked-loop (PLL) especially the proportionalgain in PI controller It has been recognized that the challeng-ing for VSCs operating in weak AC system is caused by thePLL dynamics The response speed and small-signal stabilityare contradictory characteristics of the system With a highproportional gain the system response becomes quickerwhile the power transfer limitations decrease Further [16]reported that PLL has negative impact on the stability ofVSC connected to weak AC system with reduced ordermodel However quite few literatures have attempted tooptimize PLL control system to enhance the stability of VSCsconnected to weak AC system
Hindawi Publishing CorporationJournal of Control Science and EngineeringVolume 2016 Article ID 9537342 13 pageshttpdxdoiorg10115520169537342
2 Journal of Control Science and Engineering
PCC
minus
minus+
+ LgRg
i1abc icabc
Cf
P Q
Lc
abc i2abc cabc
Rc
eabc
Udc
Figure 1 The benchmark test model of VSC connected to weak AC system
This paper aims to propose an improved PLL controlto extend the power transfer limitations In the research ofthis work it has been recognized that the impedance angleof weak AC system can also influence the power transferlimits Meanwhile an important observation is that for VSCsunder different operation modes the equivalent resistance(related to the impedance angle directly) has opposite effectson the stability of VSCs An advanced PLL with dampingfactor is proposed in this paper to enhance the power transferlimitations of grid-connected VSCs
The rest of this paper is organized as follows Section 2presents the fundamental analysis of VSC connected to weakAC system Section 3 studies the influence on power transferlimitations caused by impedance angle Section 4 proposes anadvanced PLL control system to enhance the stability of VSCconnected to weakAC system Section 5 verifies the proposedcontrol by several case studies And Section 6 concludes thispaper
2 Small-Signal Model of VSC Connected toWeak AC System
21 Benchmark Test Model A two-level VSC is adopted inthis paper as the topologyThe testmodel is shown in Figure 1The weak AC system is represented by aThevenin equivalentcircuit and the equivalent impedance is 119877119892 + 119895120596119871119892 TheDC side of the converter is represented by a DC voltagesource With the consideration of the current limitationof transformer a capacitor is shunted at PCC to providereactive power compensation 119871119888 is the leakage inductanceof transformer and 119877119888 is the resistance between PCC andconverter
Vector-current control is selected as system controlstrategy and the control diagram is shown in Figure 2(a)Active power (119875) control and AC voltage (119881) control areadopted Direct and quadrature current reference signals forthe inner-loop are generated from outer-loop control [18]The simplified PLL model [19] is shown in Figure 2(b) WithPI controller the quadrature voltage at PCC point equals zeroand the voltage phase angle can be accurately locked andmeasured
22 System and Control Equations The state-space modelderived in this paper includes AC system and VSC controllershown in Figures 1 and 2 respectively
221 AC System Equations The equations of AC system are
[
119890119889
119890119902
] minus [
V119889V119902] = 119877119892 [
1198941119889
1198941119902
] + 119871119892
119889
119889119905
[
1198941119889
1198941119902
]
+ 120596119871119892 [
minus1198941119902
1198941119889
]
[
V119889V119902] minus [
V119888119889V119888119902] = 119877119888 [
1198942119889
1198942119902
] + 119871119888
119889
119889119905
[
1198942119889
1198942119902
]
+ 120596119871119888 [
minus1198942119902
1198942119889
]
[
1198941119889
1198941119902
] minus [
1198942119889
1198942119902
] = 119862119891
119889
119889119905
[
V119889V119902] + 120596119862119891 [
minusV119902V119889]
(1)
In this case 119890119889 equals 119864119898 cos 120575 and 119890119902 equals minus119864119898 sin 120575and 120575 is the angle to which PCC voltage V leads equivalentAC system voltage E
222 Control System Equations Active power control andAC voltage control are adopted as system control 119875ref and119881ref are the references for active power and AC RMS voltageInner-loop references of direct current 1198942119889ref and quadraturecurrent 1198942119902ref are calculated by outer-loop control
119896pp (119875ref minus 119875) + 119896ip int (119875ref minus 119875) 119889119905 = 1198942119889ref
119896pv (119881ref minus 119881) + 119896iv int (119881ref minus 119881) 119889119905 = 1198942119902ref
(2)
The inner-loop control equations are shown in
1198961199011 (1198942119889ref minus 1198942119889) + 1198961198941 int (1198942119889ref minus 1198942119889) 119889119905
= 119871119888
1198891198942119889
119889119905
+ 1198771198881198942119889
Journal of Control Science and Engineering 3
abc
dq0
Pref
P
minus
minus
minus
minus
minusminus
minus+
+
+
++
++
Vref
V
i2d
i2q
i2dref
i2qref
120596Lc
120596Lc
d
q
120579
ca
cb
cc
kpp +kip
s
kpv +kivs
kp1 +ki1s
kp2 +ki2s
(a) Vector-current control model
abc
dq0
d
q
0minus
++
+
120579
aq
b
c
qref = 0
Δ120596 120596 1
s
1205960
kpll +ckpll
s
(b) The simplified model of PLL
Figure 2 Control diagram of VSC connected to weak AC system
1198961199012 (1198942119902ref minus 1198942119902) + 1198961198942 int(1198942119902ref minus 1198942119902) 119889119905
= 119871119888
1198891198942119902
119889119905
+ 1198771198881198942119902
(3)
The simplified PLL model is given by
120596 = 1205960 + 119896pllV119902 + 119888119896pll int V119902119889119905
120579 = int120596119889119905
(4)
In this paper 1205960 is specified as 100120587 rads
223 State-Space Model The detailed derivation of the state-variable equations is given in the Appendix Consideringthat the references of active power and AC voltage are notsupposed to change for small-signal model both Δ119875ref andΔ119881ref equal zeroThe linearized state-space model is given by
ΔX = AΔX (5)
in which ΔX = [Δ1198941119889 Δ1198941119902 Δ1198942119889 Δ1198942119902 ΔV119889 ΔV119902 Δ1199091 Δ1199092Δ1199093 Δ1199094 Δ120579 Δ1199095]
119879 A is a 12-order matrix (the former 6variables are AC system variables and the others are controlsystem variables) The detailed information of matrix A andthe definitions of 1199091sim1199095 are shown in the Appendix
The operating point of VSC system and the powercontroller parameters are shown in Table 1
The eigenvalues of test model at the operating point of119875ref = 133 and 119875ref = 150 are shown in Table 2
Table 1 Parameters of VSC connected to weak AC system
Parameter symbols Value
Main circuitparameters
Equivalent AC sourcevoltage 119864 10 pu 50Hz
Equivalent impedanceof AC system119877119892+ 119895119883119892
0547 + 1198950048 puimpedance angle 85∘
Equivalent AC systemSCR 183
Rated DC powerrating 10 pu
PCC voltage 10 puConverter impedance
119877119888+ 119895119883119888
0003 + 119895015 pu
Capacitor 119862119891 015 pu
Controllerparameters
Power controllergains (119896pp 119896ip)
(05 50)
AC voltage controllergains (119896pv 119896iv)
(035 30)
Inner 119894119889controller
gains (1198961199011 1198961198941) (1 10)
Inner 119894119902controller
gains (1198961199012 1198961198942) (1 10)
PLL gains (119896pll 119888) (50 10)
With the eigenvalues shown inTable 2 (see the bold italic)the predominant poles [20] ofmatrixA are selected to be 12058289The root-locus of 12058289 with active power changing is shown inFigure 3 And some of the values in Figure 3 are picked up andshown in Table 3
Table 3 The detailed information of predominant poles
119875 12058289
Damping ratio Oscillationfrequency (Hz)
130 pu minus986 plusmn 1198952408 0379 383133 pu minus530 plusmn 1198952310 0224 368137 pu minus280 plusmn 1198952200 0126 350140 pu 022 plusmn j2190 mdash 348143 pu 151 plusmn 1198952171 mdash 346
Figure 3 and Table 3 both show that with the powerrising the small-signal stability of test model is getting worseand system becomes unstable when the active power reachesabout 14 pu
224 Participation Factor Analysis Participation factor canbe utilized to analyze the relationship between predominantpoles and state-variables [21ndash23] The participation factor oftest model is shown in Table 4
FromTable 4 it can be discovered that the outer-loop andPLL control diagrams are likely to have more impacts on thestability of VSC connected to weak AC system Reference [8]has proposed the outer-loop control approach and this paperwill mainly focus on the PLL improvements
3 The Impacts of System Impedance Angle onPower Transfer Limitations
This section will analyze the impact of the system impedanceangle on VSC working in either rectifier or inverter modesregarding the power transfer limitations
The active power transmission at PCC can be calculatedusing
119875 =
119881
1198772119892+ 1198832119892
(minus119864119883119892 sin 120575 + 119864119877119892 cos 120575 minus 119881119877119892) (6)
The power angle curves for different impedance angles(120593) of AC system are shown in Figure 4 It can be found that
minus20
minus10
0
10
20
Imag
inar
y ax
is
minus20 minus10minus30 100Real axis
Figure 3 Root-locus of the predominant poles
Inverting operation
Rectifying operation
minus100minus200 100 2000120575 (deg)
minus3
minus2
minus1
0
1
2Ac
tive p
ower
(pu)
120593 = 90∘
120593 = 87∘
120593 = 83∘
120593 = 80∘
Figure 4 Power angle curves in different impedance angles
120593 has a great influence on power transfer limits The small-signal stability ofVSC systemwith different120593will be analyzedin the subsections below
31 Rectifying Operation With the analysis of state-spacematrix A it is concluded that the equivalent resistance of ACsystem has a negative impact on VSC operating as a rectifierThe root-locus of predominant poles with 120593 changing isshown in Figure 5(a) It shows that for VSC operating inrectifier mode lower resistance of AC system will enhancethe small-signal stability of VSC system
32 Inverting Operation Again with the analysis of state-spacematrixA an opposite conclusion can be drawn that forVSCworking in invertermode lower resistance of AC systemwill weaken the VSC system stability as shown in Figure 5(b)
The power transfer limitations of system with different 120593are expressed in Table 5
Journal of Control Science and Engineering 5
Table 4 The calculated participation factors of test model
Figure 5 Root-locus of predominant poles with 120593 change
Table 5 Calculation result of power transfer limitations withdifferent 120593
120593 Max 119875 (rectifier) Max 119875 (inverter)80∘ 1284 153381∘ 1302 152482∘ 1323 152183∘ 1358 151884∘ 1383 151085∘ 1400 1505
It can be concluded that lower resistance will enlargethe stable margin of VSC working at rectifier mode and willreduce the stable margin for inverter mode
4 Improved PLL Control for VSC Connectedto Weak AC System
In Section 3 a conclusion can be drawn that the dampingcharacteristic of AC network (impedance angle) has a greatinfluence on the system stability With the analysis of partic-ipation factor it is acknowledged that PLL control also has agreat impact on the stability of VSC connected to weak ACsystemTherefore an improved PLL control suitable for VSCconnected to weak AC system is proposed as follows whichis the main contribution of this paper
Considering that the active power is proportional to thedirect-axis current a supplementary damping control withstate-variable 1198942119889 is added in PLL control system (119863 is thedamping factor) And the new PLL equations are shown in(7) in which ldquo+rdquo is for rectifying operation and ldquominusrdquo is forinverting operation Figure 6 shows the improvedPLL controldiagram
120596 = 1205960 + 119896pllV119902 + 119888119896pll int V119902119889119905
120579 = int [120596 plusmn 119863 (1198942119889ref minus 1198942119889)] 119889119905
(7)
The small-signal model is changed and the state-variablematrix A with the improved PLL is shown in the AppendixFigure 7 shows the root-locus of predominant poles of themodified A
From Figure 7 it can be seen that with the rising ofdamping factor119863 in a certain range the small-signal stabilityof VSC system is enhanced After a critical value the stabilityis reduced with the rising of119863 The root-locus of VSC systemwith 119875 rising in different damping factors has also verifiedthe effectiveness of the improved PLL controller Figure 7(c)shows that with a proper damping factor the relationshipbetween PLL and dominant poles is weak It also meansthat the most effective parts are changing from PLL to theouter-loop controller which is the purpose of the improvedPLL control strategy There should be an optimal value for
6 Journal of Control Science and Engineering
D
abc
dq0
d
q
0
Supplementary control
i2dref
+
+
+
+minus
minus
Δ120596
1205960
120579
+ is for rectifying operationminus is for inverting operation
1
s
a
b
c
q
plusmn
i2d
qref = 0
kpll +ckpll
s
Figure 6 Improved PLL control for VSC connected to weak AC system
Rising direction of D
0 20minus40 minus20minus60
Real axis
minus100
minus50
0
50
100
Imag
inar
y ax
is
(a) Root-locus of VSC with 119875 = 14 pu
Rising direction of D
minus100
minus60
minus20
020
60
100
Imag
inar
y ax
is
minus20 0 20 60minus60minus100
Real axis(b) Root-locus of VSC with different transfer power
D ri
sing
dire
ctio
n
0
01
02
03
04
05
06
07
Part
icip
atio
n fa
ctor
2 3 4 5 6 7 8 9 10 11 121State-variables
(c) Participation factor analysis
Figure 7 Root-locus and participation factor of system with advanced PLL control
the damping factor 119863 When impedance angle of AC systemis 85∘ the optimal value of 119863 is approximately 1100 forrectifying operation and 300 for inverting operation
5 Case Studies
51 Validation of the Small-Signal Model Table 6 shows thebase values of the system parameters of benchmark testmodel The accuracy of the small-signal model including
the oscillation frequency and damping ratio will be verifiedin this section
Figure 8 shows the simulation results of the system inwhich the impedance angle is 85∘ The active power PCCvoltage and 119863- and 119876-axis current are shown in Figures8(a)sim8(d) respectively with the active power being changingfrom 360MW to 370MW
It can be seen fromFigure 8 that themaximumerror valuebetween EMTmodel and small-signal model of active power
Journal of Control Science and Engineering 7
EMT modelSmall-signal model
Maximum error value 08
119
12
121
122
123
124
125Ac
tive p
ower
(pu)
72 73 74 75 76 7771Time (s)
(a) Active power
EMT modelSmall-signal model
Maximum error value 015
098
099
10
101
PCC
volta
ge (p
u)
71 72 73 74 75 76 777Time (s)
(b) PCC voltage
D-a
xis c
urre
nt (k
A)
EMT modelSmall-signal model
Maximum error value 027
176
178
18
182
184
186
188
71 72 73 74 75 76 777Time (s)
(c) 119863-axis current
Q-a
xis c
urre
nt (k
A)
EMT modelSmall-signal model
Maximum error value 027
066
067
068
069
07
071
072
073
074
075
71 72 73 74 75 767Time (s)
(d) 119876-axis current
Figure 8 Simulation results of VSC system (impedance angle is 85∘)
Table 6 Parameters of test model
Parameters ValueDc voltage plusmn200 kVRated active power 300MWEquivalent AC source voltage 230 kVPCC voltage 230 kVTransformer 230 kV20412 kV
is 08 The maximum value of PCC voltage is 015 Themaximum error value of 119863-axis current and 119876-axis currentboth is 027 Therefore the correctness of the VSC systemsmall-signal model is satisfactorily validated
52 Validations of Impacts of Impedance Angle on Power Tran-sfer Limits The theory on the impact caused by impedance
angle on stability of VSC connected toweakAC systemwhichis proposed in Section 3 is verified in this section
521 Rectifying Operation Figure 9 shows the maximumpower transmission of system with different impedanceangles
It can be seen from the figure that the maximum activepower is 385MW when 120593 equals 80∘ Meanwhile the maxi-mum active power is 420MWwhen 120593 equals 85∘ Simulationresults show that with the rising of impedance angle the VSCsystem operating in rectifier mode gets larger small-signalstable margin
522 Inverting Operation Figure 10 shows the maximumpower transmission of system with different impedanceangles
However the simulation results shown in Figure 10 areopposite with the system working in rectifying operation
8 Journal of Control Science and Engineering
Reference valueSimulation value
09
1
11
12
13
14
15Ac
tive p
ower
(pu)
4 6 8 102Time (s)
(a) 120593 = 80∘Reference valueSimulation value
09
1
11
12
13
14
15
Activ
e pow
er (p
u)
4 6 8 10 122Time (s)
(b) 120593 = 85∘
Figure 9 Maximum power transmission with different impedance angles (rectifying operation)
Reference valueSimulation value
5 10 15Time (s)
minus15
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(a) 120593 = 85∘Reference valueSimulation value
5 10 15Time (s)
minus1533
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(b) 120593 = 80∘
Figure 10 Maximum power transmission with different impedance angles (inverting operation)
The maximum active power is 460MW when 120593 equals 80∘and this value decreased to 450MWwhen 120593 equals 85∘ Withthe rising of impedance angle the VSC system operatingin inverter mode is getting smaller stable margin and evenbecoming unstable
Table 7 shows the power transfer limitations with differ-ent impedance angles
The conclusion drawn from Section 3 is validated byTable 7 Meanwhile it can be seen that the system operating
at rectifier mode is more severe than inverter mode whichmeans that the stable margin for VSC working in rectifiermode is narrower than that of inverter mode
53 Validations of the Proposed PLL Control Figure 11 showsthe simulation results of maximum power transmission withthe proposed improved PLL control
It can be obtained that with the proposed PLL controlstrategy the limits of power transmission are improved
Journal of Control Science and Engineering 9
5 10 15Time (s)
Reference valueSimulation value
1
12
14
16
18Ac
tive p
ower
(pu)
(a) Rectifying operationReference valueSimulation value
5 10 15 20Time (s)
minus18
minus16
minus14
minus12
minus1
Activ
e pow
er (p
u)(b) Inverting operation
Figure 11 The simulation results of system with proposed PLL control
380
400
420
440
460
Max
imum
activ
e pow
er (M
W)
500 1100 15000Damping factor (D)
(a) Rectifying operation
430
440
450
460
470
480
490
Max
imum
activ
e pow
er (M
W)
100 200 300 400 500 6000Damping factor (D)
(b) Inverting operation
Figure 12 The relationship between119863 and limits of power transmission
The test model adopts 85∘ as the AC system impedanceangle Compared with 420MW the maximum active powerat rectifier is enhanced to 460MW Meanwhile the limits ofpower transmission at inverter are enhanced from450MWto480MW The optimal value of damping factor 119863 calculatedat rectifying operation is 1100 and is 300 at inverting oper-ation Figure 12 shows the relationship between VSC powertransfer limitations and damping factors in the proposed PLLcontrol
It can be seen from Figure 12 that the optimal values ofdamping factors are approximately equal to the calculatedresults (1100 and 300 resp) Figure 13 shows the simulationresults of the relationship between system impedance angles
and power transfer limitations with119863 being equal to zero andthe optimal values (ie 1100 and 300 resp)
Note that the power transfer limitation increments areshown in Figure 13 using arrowsThe validity of the advancedPLL control strategy and the previous impedance angleanalysis results are simultaneously validated in Figure 13
6 Conclusions
In this paper based on small-signal analysis and state-space matrix the power transfer limitations of VSC linkedwith weak AC system are analyzed Besides the dynamicmodel with vector-current control is developed to evaluate
10 Journal of Control Science and Engineering
Table 7 Limits of power transmission with different 120593
the improved PLL control The following conclusions aredrawn
(i) With the rising of active power at PCC the stabilityof VSC system is getting worse and even unstableThe stable margin of VSC for rectifying operation isnarrower than inverting situation With the analysisof participation factor the most sensitive parts tothe stability margin are outer-loop control and PLLcontrol
(ii) The impedance angle of AC system has a great impacton the stable margin of VSC For rectifying operationwith high impedance angle the stability of system
is enhanced and this is not the case for invertingoperation The damping characteristic of AC systemcan change the stable margin of VSC as well
(iii) The proposed PLL control appends the dampingfactor 119863 into the existing PLL diagram to changethe damping characteristic of VSC system Withthe calculated optimal damping factors the stablemargins forVSCworking in either rectifier or invertermodes are significantly enhanced
Appendix
A
=
((((((((((((((((((((((((((((((((((((
(
minus
119877119892
119871119892
120596 0 0 minus
1
119871119892
0 0 0 0 0 minus
119864119898 sin 1205750119871119892
0
minus120596 minus
119877119892
119871119892
0 0 0 minus
1
119871119892
0 0 0 0 minus
119864119898 cos 1205750119871119892
0
0 0
1198961199011 (minus (32) 119896ppV1198890 minus 1) minus 119877119888119871119888
0
minus31198961199011119896pp11989421198890
2119871119888
minus31198961199011119896pp11989421199020
2119871119888
minus119896ip1198961199011
119871119888
0
1198961198941
119871119888
0 0 0
0 0 0
minus1198961199012 minus 119877119888
119871119888
minus
3
2
1198961199012119896pvV11988901198810119871119888
0 0 minus
3
2
1198961199012119896ivV11988901198810119871119888
0
1198961198942
119871119888
0 0
1
119862119891
0 minus
1
119862119891
0 0 120596 0 0 0 0 0 0
0
1
119862119891
0 minus
1
119862119891
minus120596 0 0 0 0 0 0 0
0 0
3
2
V1198890 0
3
2
11989421198890
3
2
11989421199020 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 minus
3
2
119896ppV1198890 minus 1 0 minus
3
2
119896pp11989421198890 minus
3
2
119896pp11989421199020 minus119896ip 0 0 0 0 0
0 0 0 minus1 minus
3
2
119896pvV11988901198810
0 0 minus
3
2
119896ivV11988901198810
0 0 0 0
0 0 0 0 0 119896pll 0 0 0 0 0 119888119896pll
0 0 0 0 0 1 0 0 0 0 0 0
))))))))))))))))))))))))))))))))))))
)
(A1)
The mathematical step in the derivation of the state-variable equations is given in this section
119875 =
3
2
(V1198891198942119889 + V1198891198942119902)
119881 = radic3
2
radicV2119889+ V2119902
(A2)
The small-signal model is shown in (A3) In particularthe quadrature component of the PCC voltage at operatingpoint is 0 (V1199020 = 0)
Figure 13 The power transfer limitations with the advanced PLL control
The relationship between 119890119889 119890119902 and 120579 is shown in (A4)in which1205960119905+1198860 is the real time angle of equivalent ac voltagesource
119890119889 = 119864119898cos [120579 minus (1205960119905 + 1205720)]
119890119902 = minus119864119898sin [120579 minus (1205960119905 + 1205720)] (A4)
The small-signal model is shown in (A5) 120575 is mentionedin nomenclature
Δ119890119889 = minus119864119898 sin 1205750Δ120579
Δ119890119902 = minus119864119898 cos 1205750Δ120579(A5)
The small-signal model of (1)sim(4) is shown below
[
minus (119864119898 sin 1205750) Δ120575minus (119864119898 cos 1205750) Δ120575
] minus [
ΔV119889ΔV119902
]
= 119877119892 [
Δ1198941119889
Δ1198941119902
] + 119871119892
119889
119889119905
[
Δ1198941119889
Δ1198941119902
] + 120596119871119892 [
minusΔ1198941119902
Δ1198941119889
]
[
ΔV119889ΔV119902
] minus [
ΔV119888119889ΔV119888119902
]
= 119877119888 [
Δ1198942119889
Δ1198942119902
] + 119871119888
119889
119889119905
[
Δ1198942119889
Δ1198942119902
] + 120596119871119888 [
minusΔ1198942119902
Δ1198942119889
]
[
Δ1198941119889
Δ1198941119902
] minus [
Δ1198942119889
Δ1198942119902
] = 119862119891
119889
119889119905
[
ΔV119889ΔV119902
] + 120596119862119891 [
minusΔV119902ΔV119889
]
minus 119896ppΔ119875 minus 119896ip intΔ119875119889119905 = Δ1198942119889ref
minus 119896pvΔ119881 minus 119896iv intΔ119881119889119905 = Δ1198942119902ref
1198961199011 (Δ1198942119889ref minus Δ1198942119889) + 1198961198941 int (Δ1198942119889ref minus Δ1198942119889) 119889119905
= 119871119888
119889Δ1198942119889
119889119905
+ 119877119888Δ1198942119889
1198961199012 (Δ1198942119902ref minus Δ1198942119902) + 1198961198942 int (Δ1198942119902ref minus Δ1198942119902) 119889119905
The state-variables Δ1199091 Δ1199092 Δ1199093 Δ1199094 and Δ1199095 areassumed below
Δ1199091 = intΔ119875119889119905
Δ1199092 = intΔV119889119889119905
Δ1199093 = int (Δ1198942119889ref minus Δ1198942119889) 119889119905
Δ1199094 = int (Δ1198942119902ref minus Δ1198942119902) 119889119905
Δ1199095 = intΔV119902119889119905
(A7)
12 Journal of Control Science and Engineering
With the improved PLL control matrix A is changedThe modified elements are shown below ldquo+rdquo is for rectifyingoperation and ldquominusrdquo is for inverting operation
A113 = plusmn119863(
3
2
119896ppV1198890 + 1)
A115 = plusmn3
2
119863119896pp11989421198890
A116 = (plusmn3
2
119863119896pp11989421199020 + 119896pll)
A117 = plusmn119863119896ip
(A8)
The participation factormentioned in Section 2 is definedas the relativity between the 119896th state-variable and theeigenvalue 120582119894 The calculation method is shown in
in which u and k are the left eigenvector and right eigenvectorseparately V119896119894 and 119906119896119894 are the corresponding elements ineigenvectors
Nomenclature
119890119886119887119888 Thevenin equivalent voltage of AC systemV119886119887119888 PCC voltageV119888119886119887119888 Converter voltage1198941119886119887119888 Phase current at the system side1198942119886119887119888 Phase current at the converter side119894119888119886119887119888 Phase current in shunt capacitor119875119881 PCC active power and voltage119875ref 119881ref The reference value of PCC active power
and voltage119890119889 119890119902 119889 and 119902 components of AC source voltageV119889 V119902 119889 and 119902 components of PCC voltageV119888119889 V119888119902 119889 and 119902 components of converter voltage1198941119889 1198941119902 119889 and 119902 components of system current1198942119889 1198942119902 119889 and 119902 components of converter current119894119888119889 119894119888119902 119889 and 119902 components of capacitor current119877119892 119871119892 Equivalent resistance and inductance of ac
system119877119888 119871119888 Resistance and inductance of converter119862119891 Shunt capacitance1198942119889ref 119889 component of converter current
reference1198942119902ref 119902 component of converter current
reference120596 Frequency of PCC voltage120579 PLL output angle120575 The angle PCC voltage V leading
equivalent source voltage E120593 Impedance angle of AC system119863 Damping factor in the improved PLL
control119880dc DC voltage
119896pp 119896ip Power loop proportional and integrator gains119896pv 119896iv Voltage loop proportional and integrator
gains1198961199011 1198961198941 Inner-loop control (119863-axis current)
proportional and integrator gains1198961199012 1198961198942 Inner-loop control (119876-axis current)
proportional and integrator gains119896pll 119888119896pll Proportional and integrator gains of PLL119901119896119894 Participation factor119864119898 Maximum instantaneous value of source
voltage
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National HighTechnology Research and Development Program of China(863 Program) (no 2013AA050105) in part by the NationalNatural Science Foundation of China (no 51177042) andin part by the Fundamental Research Funds for the CentralUniversities (no 2015XS23 and no 2015QN04)
References
[1] IEEE ldquoIEEE guide for planning dc links terminating at AClocations having low short-circuit capacitiesrdquo IEEE Standard1204-1997 1997
[2] A Gole V K Sood and L Mootoosamy ldquoValidation andanalysis of a grid control system using D-Q-Z transformationfor static compensator systemsrdquo in Proceedings of the CanadianConference on Electrical and Computer Engineering pp 745ndash758 Montreal Canada September 1989
[3] K Liao Z-Y He and B Sun ldquoSmall signal model for VSC-HVDC connected DFIG-based offshore wind farmsrdquo Journal ofAppliedMathematics vol 2014 Article ID 725209 6 pages 2014
[4] M Cespedes and J Sun ldquoAdaptive control of grid-connectedinverters based on online grid impedancemeasurementsrdquo IEEETransactions on Sustainable Energy vol 5 no 2 pp 516ndash5232014
[5] VK SoodVKhatri andH Jin ldquoPerformance assessment usingEMTP of two gate firing units for HVDC converters operatingwith weak AC systemsrdquo in Proceedings of the InternationalConference on Power Systems Transients (IPST rsquo95) pp 517ndash522Lisbon Portugal September 1995
[6] 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
[7] K M Alawasa and Y A-R I Mohamed ldquoImpedance anddamping characteristics of grid-connected VSCs with powersynchronization control strategyrdquo IEEE Transactions on PowerSystems vol 30 no 2 pp 952ndash961 2015
[8] A Egea-Alvarez S Fekriasl F Hassan andO Gomis-BellmuntldquoAdvanced vector control for voltage source converters con-nected to weak ac gridsrdquo IEEE Transactions on Power Systemsvol 30 no 6 pp 3072ndash3081 2015
Journal of Control Science and Engineering 13
[9] L Zhang L Harnefors and H-P Nee ldquoPower-synchronizationcontrol of grid-connected voltage-source convertersrdquo IEEETransactions on Power Systems vol 25 no 2 pp 809ndash820 2010
[10] J Z Zhou H Ding S Fan Y Zhang and A M Gole ldquoImpactof short-circuit ratio and phase-locked-loop parameters onthe small-signal behavior of a VSC-HVDC converterrdquo IEEETransactions on Power Delivery vol 29 no 5 pp 2287ndash22962014
[11] J A Suul S DrsquoArco P Rodriguez and M Molinas ldquoExtendedstability range of weak grids with Voltage Source Convertersthrough impedance-conditioned grid synchronizationrdquo in Pro-ceedings of the 11th International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February 2015
[12] 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
[13] T Midtsund J A Suul and T Undeland ldquoEvaluation ofcurrent controller performance and stability for voltage sourceconverters connected to a weak gridrdquo in Proceedings of the 2ndInternational Symposium on Power Electronics for DistributedGeneration Systems pp 382ndash388 IEEE Hefei China June 2010
[14] Z Miao L Fan D Osborn and S Yuvarajan ldquoWind farmswith HVdc delivery in inertial response and primary frequencycontrolrdquo IEEE Transactions on Energy Conversion vol 25 no 4pp 1171ndash1178 2010
[15] J Zhu C D Booth G P Adam A J Roscoe and C GBright ldquoInertia emulation control strategy for VSC-HVDCtransmission systemsrdquo IEEETransactions on Power Systems vol28 no 2 pp 1277ndash1287 2013
[16] H Ding S Fan J Z Zhou Y Zhang and A M GoleldquoParametric analysis of the stability of VSC-HVDC convertersrdquoin Proceedings of the 11th IET International Conference on ACand DC Power Transmission pp 1ndash6 IEEE Birmingham UKFebruary 2015
[17] B Wen D Boroyevich R Burgos P Mattavelli and Z ShenldquoAnalysis of D-Q small-signal impedance of grid-tied invertersrdquoIEEE Transactions on Power Electronics vol 31 no 1 pp 675ndash687 2016
[18] A Yazdani and R Iravani Voltage-Sourced Converters in PowerSystems Wiley Hoboken NJ USA 2010
[19] G-C Hsieh and J C Hung ldquoPhase-locked loop techniquesmdashasurveyrdquo IEEE Transactions on Industrial Electronics vol 43 no6 pp 609ndash615 1996
[20] G F Franklin J D Powell and A Emami-Naeini FeedbackControl of Dynamic Systems Prentice Hall New York NY USA2005
[21] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1993
[22] D Dong J Li D Boroyevich P Mattavelli I Cvetkovic andY Xue ldquoFrequency behavior and its stability of grid-interfaceconverter in distributed generation systemsrdquo in Proceedingsof the 27th IEEE Applied Power Electronics Conference andExposition (APEC rsquo12) pp 1887ndash1893 IEEE Orlando Fla USAFebruary 2012
[23] J Beerten S DrsquoArco and J A Suul ldquoIdentification and small-signal analysis of interaction modes in VSC MTDC systemsrdquoIEEE Transactions on Power Delivery 2015
Figure 1 The benchmark test model of VSC connected to weak AC system
This paper aims to propose an improved PLL controlto extend the power transfer limitations In the research ofthis work it has been recognized that the impedance angleof weak AC system can also influence the power transferlimits Meanwhile an important observation is that for VSCsunder different operation modes the equivalent resistance(related to the impedance angle directly) has opposite effectson the stability of VSCs An advanced PLL with dampingfactor is proposed in this paper to enhance the power transferlimitations of grid-connected VSCs
The rest of this paper is organized as follows Section 2presents the fundamental analysis of VSC connected to weakAC system Section 3 studies the influence on power transferlimitations caused by impedance angle Section 4 proposes anadvanced PLL control system to enhance the stability of VSCconnected to weakAC system Section 5 verifies the proposedcontrol by several case studies And Section 6 concludes thispaper
2 Small-Signal Model of VSC Connected toWeak AC System
21 Benchmark Test Model A two-level VSC is adopted inthis paper as the topologyThe testmodel is shown in Figure 1The weak AC system is represented by aThevenin equivalentcircuit and the equivalent impedance is 119877119892 + 119895120596119871119892 TheDC side of the converter is represented by a DC voltagesource With the consideration of the current limitationof transformer a capacitor is shunted at PCC to providereactive power compensation 119871119888 is the leakage inductanceof transformer and 119877119888 is the resistance between PCC andconverter
Vector-current control is selected as system controlstrategy and the control diagram is shown in Figure 2(a)Active power (119875) control and AC voltage (119881) control areadopted Direct and quadrature current reference signals forthe inner-loop are generated from outer-loop control [18]The simplified PLL model [19] is shown in Figure 2(b) WithPI controller the quadrature voltage at PCC point equals zeroand the voltage phase angle can be accurately locked andmeasured
22 System and Control Equations The state-space modelderived in this paper includes AC system and VSC controllershown in Figures 1 and 2 respectively
221 AC System Equations The equations of AC system are
[
119890119889
119890119902
] minus [
V119889V119902] = 119877119892 [
1198941119889
1198941119902
] + 119871119892
119889
119889119905
[
1198941119889
1198941119902
]
+ 120596119871119892 [
minus1198941119902
1198941119889
]
[
V119889V119902] minus [
V119888119889V119888119902] = 119877119888 [
1198942119889
1198942119902
] + 119871119888
119889
119889119905
[
1198942119889
1198942119902
]
+ 120596119871119888 [
minus1198942119902
1198942119889
]
[
1198941119889
1198941119902
] minus [
1198942119889
1198942119902
] = 119862119891
119889
119889119905
[
V119889V119902] + 120596119862119891 [
minusV119902V119889]
(1)
In this case 119890119889 equals 119864119898 cos 120575 and 119890119902 equals minus119864119898 sin 120575and 120575 is the angle to which PCC voltage V leads equivalentAC system voltage E
222 Control System Equations Active power control andAC voltage control are adopted as system control 119875ref and119881ref are the references for active power and AC RMS voltageInner-loop references of direct current 1198942119889ref and quadraturecurrent 1198942119902ref are calculated by outer-loop control
119896pp (119875ref minus 119875) + 119896ip int (119875ref minus 119875) 119889119905 = 1198942119889ref
119896pv (119881ref minus 119881) + 119896iv int (119881ref minus 119881) 119889119905 = 1198942119902ref
(2)
The inner-loop control equations are shown in
1198961199011 (1198942119889ref minus 1198942119889) + 1198961198941 int (1198942119889ref minus 1198942119889) 119889119905
= 119871119888
1198891198942119889
119889119905
+ 1198771198881198942119889
Journal of Control Science and Engineering 3
abc
dq0
Pref
P
minus
minus
minus
minus
minusminus
minus+
+
+
++
++
Vref
V
i2d
i2q
i2dref
i2qref
120596Lc
120596Lc
d
q
120579
ca
cb
cc
kpp +kip
s
kpv +kivs
kp1 +ki1s
kp2 +ki2s
(a) Vector-current control model
abc
dq0
d
q
0minus
++
+
120579
aq
b
c
qref = 0
Δ120596 120596 1
s
1205960
kpll +ckpll
s
(b) The simplified model of PLL
Figure 2 Control diagram of VSC connected to weak AC system
1198961199012 (1198942119902ref minus 1198942119902) + 1198961198942 int(1198942119902ref minus 1198942119902) 119889119905
= 119871119888
1198891198942119902
119889119905
+ 1198771198881198942119902
(3)
The simplified PLL model is given by
120596 = 1205960 + 119896pllV119902 + 119888119896pll int V119902119889119905
120579 = int120596119889119905
(4)
In this paper 1205960 is specified as 100120587 rads
223 State-Space Model The detailed derivation of the state-variable equations is given in the Appendix Consideringthat the references of active power and AC voltage are notsupposed to change for small-signal model both Δ119875ref andΔ119881ref equal zeroThe linearized state-space model is given by
ΔX = AΔX (5)
in which ΔX = [Δ1198941119889 Δ1198941119902 Δ1198942119889 Δ1198942119902 ΔV119889 ΔV119902 Δ1199091 Δ1199092Δ1199093 Δ1199094 Δ120579 Δ1199095]
119879 A is a 12-order matrix (the former 6variables are AC system variables and the others are controlsystem variables) The detailed information of matrix A andthe definitions of 1199091sim1199095 are shown in the Appendix
The operating point of VSC system and the powercontroller parameters are shown in Table 1
The eigenvalues of test model at the operating point of119875ref = 133 and 119875ref = 150 are shown in Table 2
Table 1 Parameters of VSC connected to weak AC system
Parameter symbols Value
Main circuitparameters
Equivalent AC sourcevoltage 119864 10 pu 50Hz
Equivalent impedanceof AC system119877119892+ 119895119883119892
0547 + 1198950048 puimpedance angle 85∘
Equivalent AC systemSCR 183
Rated DC powerrating 10 pu
PCC voltage 10 puConverter impedance
119877119888+ 119895119883119888
0003 + 119895015 pu
Capacitor 119862119891 015 pu
Controllerparameters
Power controllergains (119896pp 119896ip)
(05 50)
AC voltage controllergains (119896pv 119896iv)
(035 30)
Inner 119894119889controller
gains (1198961199011 1198961198941) (1 10)
Inner 119894119902controller
gains (1198961199012 1198961198942) (1 10)
PLL gains (119896pll 119888) (50 10)
With the eigenvalues shown inTable 2 (see the bold italic)the predominant poles [20] ofmatrixA are selected to be 12058289The root-locus of 12058289 with active power changing is shown inFigure 3 And some of the values in Figure 3 are picked up andshown in Table 3
Table 3 The detailed information of predominant poles
119875 12058289
Damping ratio Oscillationfrequency (Hz)
130 pu minus986 plusmn 1198952408 0379 383133 pu minus530 plusmn 1198952310 0224 368137 pu minus280 plusmn 1198952200 0126 350140 pu 022 plusmn j2190 mdash 348143 pu 151 plusmn 1198952171 mdash 346
Figure 3 and Table 3 both show that with the powerrising the small-signal stability of test model is getting worseand system becomes unstable when the active power reachesabout 14 pu
224 Participation Factor Analysis Participation factor canbe utilized to analyze the relationship between predominantpoles and state-variables [21ndash23] The participation factor oftest model is shown in Table 4
FromTable 4 it can be discovered that the outer-loop andPLL control diagrams are likely to have more impacts on thestability of VSC connected to weak AC system Reference [8]has proposed the outer-loop control approach and this paperwill mainly focus on the PLL improvements
3 The Impacts of System Impedance Angle onPower Transfer Limitations
This section will analyze the impact of the system impedanceangle on VSC working in either rectifier or inverter modesregarding the power transfer limitations
The active power transmission at PCC can be calculatedusing
119875 =
119881
1198772119892+ 1198832119892
(minus119864119883119892 sin 120575 + 119864119877119892 cos 120575 minus 119881119877119892) (6)
The power angle curves for different impedance angles(120593) of AC system are shown in Figure 4 It can be found that
minus20
minus10
0
10
20
Imag
inar
y ax
is
minus20 minus10minus30 100Real axis
Figure 3 Root-locus of the predominant poles
Inverting operation
Rectifying operation
minus100minus200 100 2000120575 (deg)
minus3
minus2
minus1
0
1
2Ac
tive p
ower
(pu)
120593 = 90∘
120593 = 87∘
120593 = 83∘
120593 = 80∘
Figure 4 Power angle curves in different impedance angles
120593 has a great influence on power transfer limits The small-signal stability ofVSC systemwith different120593will be analyzedin the subsections below
31 Rectifying Operation With the analysis of state-spacematrix A it is concluded that the equivalent resistance of ACsystem has a negative impact on VSC operating as a rectifierThe root-locus of predominant poles with 120593 changing isshown in Figure 5(a) It shows that for VSC operating inrectifier mode lower resistance of AC system will enhancethe small-signal stability of VSC system
32 Inverting Operation Again with the analysis of state-spacematrixA an opposite conclusion can be drawn that forVSCworking in invertermode lower resistance of AC systemwill weaken the VSC system stability as shown in Figure 5(b)
The power transfer limitations of system with different 120593are expressed in Table 5
Journal of Control Science and Engineering 5
Table 4 The calculated participation factors of test model
Figure 5 Root-locus of predominant poles with 120593 change
Table 5 Calculation result of power transfer limitations withdifferent 120593
120593 Max 119875 (rectifier) Max 119875 (inverter)80∘ 1284 153381∘ 1302 152482∘ 1323 152183∘ 1358 151884∘ 1383 151085∘ 1400 1505
It can be concluded that lower resistance will enlargethe stable margin of VSC working at rectifier mode and willreduce the stable margin for inverter mode
4 Improved PLL Control for VSC Connectedto Weak AC System
In Section 3 a conclusion can be drawn that the dampingcharacteristic of AC network (impedance angle) has a greatinfluence on the system stability With the analysis of partic-ipation factor it is acknowledged that PLL control also has agreat impact on the stability of VSC connected to weak ACsystemTherefore an improved PLL control suitable for VSCconnected to weak AC system is proposed as follows whichis the main contribution of this paper
Considering that the active power is proportional to thedirect-axis current a supplementary damping control withstate-variable 1198942119889 is added in PLL control system (119863 is thedamping factor) And the new PLL equations are shown in(7) in which ldquo+rdquo is for rectifying operation and ldquominusrdquo is forinverting operation Figure 6 shows the improvedPLL controldiagram
120596 = 1205960 + 119896pllV119902 + 119888119896pll int V119902119889119905
120579 = int [120596 plusmn 119863 (1198942119889ref minus 1198942119889)] 119889119905
(7)
The small-signal model is changed and the state-variablematrix A with the improved PLL is shown in the AppendixFigure 7 shows the root-locus of predominant poles of themodified A
From Figure 7 it can be seen that with the rising ofdamping factor119863 in a certain range the small-signal stabilityof VSC system is enhanced After a critical value the stabilityis reduced with the rising of119863 The root-locus of VSC systemwith 119875 rising in different damping factors has also verifiedthe effectiveness of the improved PLL controller Figure 7(c)shows that with a proper damping factor the relationshipbetween PLL and dominant poles is weak It also meansthat the most effective parts are changing from PLL to theouter-loop controller which is the purpose of the improvedPLL control strategy There should be an optimal value for
6 Journal of Control Science and Engineering
D
abc
dq0
d
q
0
Supplementary control
i2dref
+
+
+
+minus
minus
Δ120596
1205960
120579
+ is for rectifying operationminus is for inverting operation
1
s
a
b
c
q
plusmn
i2d
qref = 0
kpll +ckpll
s
Figure 6 Improved PLL control for VSC connected to weak AC system
Rising direction of D
0 20minus40 minus20minus60
Real axis
minus100
minus50
0
50
100
Imag
inar
y ax
is
(a) Root-locus of VSC with 119875 = 14 pu
Rising direction of D
minus100
minus60
minus20
020
60
100
Imag
inar
y ax
is
minus20 0 20 60minus60minus100
Real axis(b) Root-locus of VSC with different transfer power
D ri
sing
dire
ctio
n
0
01
02
03
04
05
06
07
Part
icip
atio
n fa
ctor
2 3 4 5 6 7 8 9 10 11 121State-variables
(c) Participation factor analysis
Figure 7 Root-locus and participation factor of system with advanced PLL control
the damping factor 119863 When impedance angle of AC systemis 85∘ the optimal value of 119863 is approximately 1100 forrectifying operation and 300 for inverting operation
5 Case Studies
51 Validation of the Small-Signal Model Table 6 shows thebase values of the system parameters of benchmark testmodel The accuracy of the small-signal model including
the oscillation frequency and damping ratio will be verifiedin this section
Figure 8 shows the simulation results of the system inwhich the impedance angle is 85∘ The active power PCCvoltage and 119863- and 119876-axis current are shown in Figures8(a)sim8(d) respectively with the active power being changingfrom 360MW to 370MW
It can be seen fromFigure 8 that themaximumerror valuebetween EMTmodel and small-signal model of active power
Journal of Control Science and Engineering 7
EMT modelSmall-signal model
Maximum error value 08
119
12
121
122
123
124
125Ac
tive p
ower
(pu)
72 73 74 75 76 7771Time (s)
(a) Active power
EMT modelSmall-signal model
Maximum error value 015
098
099
10
101
PCC
volta
ge (p
u)
71 72 73 74 75 76 777Time (s)
(b) PCC voltage
D-a
xis c
urre
nt (k
A)
EMT modelSmall-signal model
Maximum error value 027
176
178
18
182
184
186
188
71 72 73 74 75 76 777Time (s)
(c) 119863-axis current
Q-a
xis c
urre
nt (k
A)
EMT modelSmall-signal model
Maximum error value 027
066
067
068
069
07
071
072
073
074
075
71 72 73 74 75 767Time (s)
(d) 119876-axis current
Figure 8 Simulation results of VSC system (impedance angle is 85∘)
Table 6 Parameters of test model
Parameters ValueDc voltage plusmn200 kVRated active power 300MWEquivalent AC source voltage 230 kVPCC voltage 230 kVTransformer 230 kV20412 kV
is 08 The maximum value of PCC voltage is 015 Themaximum error value of 119863-axis current and 119876-axis currentboth is 027 Therefore the correctness of the VSC systemsmall-signal model is satisfactorily validated
52 Validations of Impacts of Impedance Angle on Power Tran-sfer Limits The theory on the impact caused by impedance
angle on stability of VSC connected toweakAC systemwhichis proposed in Section 3 is verified in this section
521 Rectifying Operation Figure 9 shows the maximumpower transmission of system with different impedanceangles
It can be seen from the figure that the maximum activepower is 385MW when 120593 equals 80∘ Meanwhile the maxi-mum active power is 420MWwhen 120593 equals 85∘ Simulationresults show that with the rising of impedance angle the VSCsystem operating in rectifier mode gets larger small-signalstable margin
522 Inverting Operation Figure 10 shows the maximumpower transmission of system with different impedanceangles
However the simulation results shown in Figure 10 areopposite with the system working in rectifying operation
8 Journal of Control Science and Engineering
Reference valueSimulation value
09
1
11
12
13
14
15Ac
tive p
ower
(pu)
4 6 8 102Time (s)
(a) 120593 = 80∘Reference valueSimulation value
09
1
11
12
13
14
15
Activ
e pow
er (p
u)
4 6 8 10 122Time (s)
(b) 120593 = 85∘
Figure 9 Maximum power transmission with different impedance angles (rectifying operation)
Reference valueSimulation value
5 10 15Time (s)
minus15
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(a) 120593 = 85∘Reference valueSimulation value
5 10 15Time (s)
minus1533
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(b) 120593 = 80∘
Figure 10 Maximum power transmission with different impedance angles (inverting operation)
The maximum active power is 460MW when 120593 equals 80∘and this value decreased to 450MWwhen 120593 equals 85∘ Withthe rising of impedance angle the VSC system operatingin inverter mode is getting smaller stable margin and evenbecoming unstable
Table 7 shows the power transfer limitations with differ-ent impedance angles
The conclusion drawn from Section 3 is validated byTable 7 Meanwhile it can be seen that the system operating
at rectifier mode is more severe than inverter mode whichmeans that the stable margin for VSC working in rectifiermode is narrower than that of inverter mode
53 Validations of the Proposed PLL Control Figure 11 showsthe simulation results of maximum power transmission withthe proposed improved PLL control
It can be obtained that with the proposed PLL controlstrategy the limits of power transmission are improved
Journal of Control Science and Engineering 9
5 10 15Time (s)
Reference valueSimulation value
1
12
14
16
18Ac
tive p
ower
(pu)
(a) Rectifying operationReference valueSimulation value
5 10 15 20Time (s)
minus18
minus16
minus14
minus12
minus1
Activ
e pow
er (p
u)(b) Inverting operation
Figure 11 The simulation results of system with proposed PLL control
380
400
420
440
460
Max
imum
activ
e pow
er (M
W)
500 1100 15000Damping factor (D)
(a) Rectifying operation
430
440
450
460
470
480
490
Max
imum
activ
e pow
er (M
W)
100 200 300 400 500 6000Damping factor (D)
(b) Inverting operation
Figure 12 The relationship between119863 and limits of power transmission
The test model adopts 85∘ as the AC system impedanceangle Compared with 420MW the maximum active powerat rectifier is enhanced to 460MW Meanwhile the limits ofpower transmission at inverter are enhanced from450MWto480MW The optimal value of damping factor 119863 calculatedat rectifying operation is 1100 and is 300 at inverting oper-ation Figure 12 shows the relationship between VSC powertransfer limitations and damping factors in the proposed PLLcontrol
It can be seen from Figure 12 that the optimal values ofdamping factors are approximately equal to the calculatedresults (1100 and 300 resp) Figure 13 shows the simulationresults of the relationship between system impedance angles
and power transfer limitations with119863 being equal to zero andthe optimal values (ie 1100 and 300 resp)
Note that the power transfer limitation increments areshown in Figure 13 using arrowsThe validity of the advancedPLL control strategy and the previous impedance angleanalysis results are simultaneously validated in Figure 13
6 Conclusions
In this paper based on small-signal analysis and state-space matrix the power transfer limitations of VSC linkedwith weak AC system are analyzed Besides the dynamicmodel with vector-current control is developed to evaluate
10 Journal of Control Science and Engineering
Table 7 Limits of power transmission with different 120593
the improved PLL control The following conclusions aredrawn
(i) With the rising of active power at PCC the stabilityof VSC system is getting worse and even unstableThe stable margin of VSC for rectifying operation isnarrower than inverting situation With the analysisof participation factor the most sensitive parts tothe stability margin are outer-loop control and PLLcontrol
(ii) The impedance angle of AC system has a great impacton the stable margin of VSC For rectifying operationwith high impedance angle the stability of system
is enhanced and this is not the case for invertingoperation The damping characteristic of AC systemcan change the stable margin of VSC as well
(iii) The proposed PLL control appends the dampingfactor 119863 into the existing PLL diagram to changethe damping characteristic of VSC system Withthe calculated optimal damping factors the stablemargins forVSCworking in either rectifier or invertermodes are significantly enhanced
Appendix
A
=
((((((((((((((((((((((((((((((((((((
(
minus
119877119892
119871119892
120596 0 0 minus
1
119871119892
0 0 0 0 0 minus
119864119898 sin 1205750119871119892
0
minus120596 minus
119877119892
119871119892
0 0 0 minus
1
119871119892
0 0 0 0 minus
119864119898 cos 1205750119871119892
0
0 0
1198961199011 (minus (32) 119896ppV1198890 minus 1) minus 119877119888119871119888
0
minus31198961199011119896pp11989421198890
2119871119888
minus31198961199011119896pp11989421199020
2119871119888
minus119896ip1198961199011
119871119888
0
1198961198941
119871119888
0 0 0
0 0 0
minus1198961199012 minus 119877119888
119871119888
minus
3
2
1198961199012119896pvV11988901198810119871119888
0 0 minus
3
2
1198961199012119896ivV11988901198810119871119888
0
1198961198942
119871119888
0 0
1
119862119891
0 minus
1
119862119891
0 0 120596 0 0 0 0 0 0
0
1
119862119891
0 minus
1
119862119891
minus120596 0 0 0 0 0 0 0
0 0
3
2
V1198890 0
3
2
11989421198890
3
2
11989421199020 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 minus
3
2
119896ppV1198890 minus 1 0 minus
3
2
119896pp11989421198890 minus
3
2
119896pp11989421199020 minus119896ip 0 0 0 0 0
0 0 0 minus1 minus
3
2
119896pvV11988901198810
0 0 minus
3
2
119896ivV11988901198810
0 0 0 0
0 0 0 0 0 119896pll 0 0 0 0 0 119888119896pll
0 0 0 0 0 1 0 0 0 0 0 0
))))))))))))))))))))))))))))))))))))
)
(A1)
The mathematical step in the derivation of the state-variable equations is given in this section
119875 =
3
2
(V1198891198942119889 + V1198891198942119902)
119881 = radic3
2
radicV2119889+ V2119902
(A2)
The small-signal model is shown in (A3) In particularthe quadrature component of the PCC voltage at operatingpoint is 0 (V1199020 = 0)
Figure 13 The power transfer limitations with the advanced PLL control
The relationship between 119890119889 119890119902 and 120579 is shown in (A4)in which1205960119905+1198860 is the real time angle of equivalent ac voltagesource
119890119889 = 119864119898cos [120579 minus (1205960119905 + 1205720)]
119890119902 = minus119864119898sin [120579 minus (1205960119905 + 1205720)] (A4)
The small-signal model is shown in (A5) 120575 is mentionedin nomenclature
Δ119890119889 = minus119864119898 sin 1205750Δ120579
Δ119890119902 = minus119864119898 cos 1205750Δ120579(A5)
The small-signal model of (1)sim(4) is shown below
[
minus (119864119898 sin 1205750) Δ120575minus (119864119898 cos 1205750) Δ120575
] minus [
ΔV119889ΔV119902
]
= 119877119892 [
Δ1198941119889
Δ1198941119902
] + 119871119892
119889
119889119905
[
Δ1198941119889
Δ1198941119902
] + 120596119871119892 [
minusΔ1198941119902
Δ1198941119889
]
[
ΔV119889ΔV119902
] minus [
ΔV119888119889ΔV119888119902
]
= 119877119888 [
Δ1198942119889
Δ1198942119902
] + 119871119888
119889
119889119905
[
Δ1198942119889
Δ1198942119902
] + 120596119871119888 [
minusΔ1198942119902
Δ1198942119889
]
[
Δ1198941119889
Δ1198941119902
] minus [
Δ1198942119889
Δ1198942119902
] = 119862119891
119889
119889119905
[
ΔV119889ΔV119902
] + 120596119862119891 [
minusΔV119902ΔV119889
]
minus 119896ppΔ119875 minus 119896ip intΔ119875119889119905 = Δ1198942119889ref
minus 119896pvΔ119881 minus 119896iv intΔ119881119889119905 = Δ1198942119902ref
1198961199011 (Δ1198942119889ref minus Δ1198942119889) + 1198961198941 int (Δ1198942119889ref minus Δ1198942119889) 119889119905
= 119871119888
119889Δ1198942119889
119889119905
+ 119877119888Δ1198942119889
1198961199012 (Δ1198942119902ref minus Δ1198942119902) + 1198961198942 int (Δ1198942119902ref minus Δ1198942119902) 119889119905
The state-variables Δ1199091 Δ1199092 Δ1199093 Δ1199094 and Δ1199095 areassumed below
Δ1199091 = intΔ119875119889119905
Δ1199092 = intΔV119889119889119905
Δ1199093 = int (Δ1198942119889ref minus Δ1198942119889) 119889119905
Δ1199094 = int (Δ1198942119902ref minus Δ1198942119902) 119889119905
Δ1199095 = intΔV119902119889119905
(A7)
12 Journal of Control Science and Engineering
With the improved PLL control matrix A is changedThe modified elements are shown below ldquo+rdquo is for rectifyingoperation and ldquominusrdquo is for inverting operation
A113 = plusmn119863(
3
2
119896ppV1198890 + 1)
A115 = plusmn3
2
119863119896pp11989421198890
A116 = (plusmn3
2
119863119896pp11989421199020 + 119896pll)
A117 = plusmn119863119896ip
(A8)
The participation factormentioned in Section 2 is definedas the relativity between the 119896th state-variable and theeigenvalue 120582119894 The calculation method is shown in
in which u and k are the left eigenvector and right eigenvectorseparately V119896119894 and 119906119896119894 are the corresponding elements ineigenvectors
Nomenclature
119890119886119887119888 Thevenin equivalent voltage of AC systemV119886119887119888 PCC voltageV119888119886119887119888 Converter voltage1198941119886119887119888 Phase current at the system side1198942119886119887119888 Phase current at the converter side119894119888119886119887119888 Phase current in shunt capacitor119875119881 PCC active power and voltage119875ref 119881ref The reference value of PCC active power
and voltage119890119889 119890119902 119889 and 119902 components of AC source voltageV119889 V119902 119889 and 119902 components of PCC voltageV119888119889 V119888119902 119889 and 119902 components of converter voltage1198941119889 1198941119902 119889 and 119902 components of system current1198942119889 1198942119902 119889 and 119902 components of converter current119894119888119889 119894119888119902 119889 and 119902 components of capacitor current119877119892 119871119892 Equivalent resistance and inductance of ac
system119877119888 119871119888 Resistance and inductance of converter119862119891 Shunt capacitance1198942119889ref 119889 component of converter current
reference1198942119902ref 119902 component of converter current
reference120596 Frequency of PCC voltage120579 PLL output angle120575 The angle PCC voltage V leading
equivalent source voltage E120593 Impedance angle of AC system119863 Damping factor in the improved PLL
control119880dc DC voltage
119896pp 119896ip Power loop proportional and integrator gains119896pv 119896iv Voltage loop proportional and integrator
gains1198961199011 1198961198941 Inner-loop control (119863-axis current)
proportional and integrator gains1198961199012 1198961198942 Inner-loop control (119876-axis current)
proportional and integrator gains119896pll 119888119896pll Proportional and integrator gains of PLL119901119896119894 Participation factor119864119898 Maximum instantaneous value of source
voltage
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National HighTechnology Research and Development Program of China(863 Program) (no 2013AA050105) in part by the NationalNatural Science Foundation of China (no 51177042) andin part by the Fundamental Research Funds for the CentralUniversities (no 2015XS23 and no 2015QN04)
References
[1] IEEE ldquoIEEE guide for planning dc links terminating at AClocations having low short-circuit capacitiesrdquo IEEE Standard1204-1997 1997
[2] A Gole V K Sood and L Mootoosamy ldquoValidation andanalysis of a grid control system using D-Q-Z transformationfor static compensator systemsrdquo in Proceedings of the CanadianConference on Electrical and Computer Engineering pp 745ndash758 Montreal Canada September 1989
[3] K Liao Z-Y He and B Sun ldquoSmall signal model for VSC-HVDC connected DFIG-based offshore wind farmsrdquo Journal ofAppliedMathematics vol 2014 Article ID 725209 6 pages 2014
[4] M Cespedes and J Sun ldquoAdaptive control of grid-connectedinverters based on online grid impedancemeasurementsrdquo IEEETransactions on Sustainable Energy vol 5 no 2 pp 516ndash5232014
[5] VK SoodVKhatri andH Jin ldquoPerformance assessment usingEMTP of two gate firing units for HVDC converters operatingwith weak AC systemsrdquo in Proceedings of the InternationalConference on Power Systems Transients (IPST rsquo95) pp 517ndash522Lisbon Portugal September 1995
[6] 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
[7] K M Alawasa and Y A-R I Mohamed ldquoImpedance anddamping characteristics of grid-connected VSCs with powersynchronization control strategyrdquo IEEE Transactions on PowerSystems vol 30 no 2 pp 952ndash961 2015
[8] A Egea-Alvarez S Fekriasl F Hassan andO Gomis-BellmuntldquoAdvanced vector control for voltage source converters con-nected to weak ac gridsrdquo IEEE Transactions on Power Systemsvol 30 no 6 pp 3072ndash3081 2015
Journal of Control Science and Engineering 13
[9] L Zhang L Harnefors and H-P Nee ldquoPower-synchronizationcontrol of grid-connected voltage-source convertersrdquo IEEETransactions on Power Systems vol 25 no 2 pp 809ndash820 2010
[10] J Z Zhou H Ding S Fan Y Zhang and A M Gole ldquoImpactof short-circuit ratio and phase-locked-loop parameters onthe small-signal behavior of a VSC-HVDC converterrdquo IEEETransactions on Power Delivery vol 29 no 5 pp 2287ndash22962014
[11] J A Suul S DrsquoArco P Rodriguez and M Molinas ldquoExtendedstability range of weak grids with Voltage Source Convertersthrough impedance-conditioned grid synchronizationrdquo in Pro-ceedings of the 11th International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February 2015
[12] 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
[13] T Midtsund J A Suul and T Undeland ldquoEvaluation ofcurrent controller performance and stability for voltage sourceconverters connected to a weak gridrdquo in Proceedings of the 2ndInternational Symposium on Power Electronics for DistributedGeneration Systems pp 382ndash388 IEEE Hefei China June 2010
[14] Z Miao L Fan D Osborn and S Yuvarajan ldquoWind farmswith HVdc delivery in inertial response and primary frequencycontrolrdquo IEEE Transactions on Energy Conversion vol 25 no 4pp 1171ndash1178 2010
[15] J Zhu C D Booth G P Adam A J Roscoe and C GBright ldquoInertia emulation control strategy for VSC-HVDCtransmission systemsrdquo IEEETransactions on Power Systems vol28 no 2 pp 1277ndash1287 2013
[16] H Ding S Fan J Z Zhou Y Zhang and A M GoleldquoParametric analysis of the stability of VSC-HVDC convertersrdquoin Proceedings of the 11th IET International Conference on ACand DC Power Transmission pp 1ndash6 IEEE Birmingham UKFebruary 2015
[17] B Wen D Boroyevich R Burgos P Mattavelli and Z ShenldquoAnalysis of D-Q small-signal impedance of grid-tied invertersrdquoIEEE Transactions on Power Electronics vol 31 no 1 pp 675ndash687 2016
[18] A Yazdani and R Iravani Voltage-Sourced Converters in PowerSystems Wiley Hoboken NJ USA 2010
[19] G-C Hsieh and J C Hung ldquoPhase-locked loop techniquesmdashasurveyrdquo IEEE Transactions on Industrial Electronics vol 43 no6 pp 609ndash615 1996
[20] G F Franklin J D Powell and A Emami-Naeini FeedbackControl of Dynamic Systems Prentice Hall New York NY USA2005
[21] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1993
[22] D Dong J Li D Boroyevich P Mattavelli I Cvetkovic andY Xue ldquoFrequency behavior and its stability of grid-interfaceconverter in distributed generation systemsrdquo in Proceedingsof the 27th IEEE Applied Power Electronics Conference andExposition (APEC rsquo12) pp 1887ndash1893 IEEE Orlando Fla USAFebruary 2012
[23] J Beerten S DrsquoArco and J A Suul ldquoIdentification and small-signal analysis of interaction modes in VSC MTDC systemsrdquoIEEE Transactions on Power Delivery 2015
Figure 2 Control diagram of VSC connected to weak AC system
1198961199012 (1198942119902ref minus 1198942119902) + 1198961198942 int(1198942119902ref minus 1198942119902) 119889119905
= 119871119888
1198891198942119902
119889119905
+ 1198771198881198942119902
(3)
The simplified PLL model is given by
120596 = 1205960 + 119896pllV119902 + 119888119896pll int V119902119889119905
120579 = int120596119889119905
(4)
In this paper 1205960 is specified as 100120587 rads
223 State-Space Model The detailed derivation of the state-variable equations is given in the Appendix Consideringthat the references of active power and AC voltage are notsupposed to change for small-signal model both Δ119875ref andΔ119881ref equal zeroThe linearized state-space model is given by
ΔX = AΔX (5)
in which ΔX = [Δ1198941119889 Δ1198941119902 Δ1198942119889 Δ1198942119902 ΔV119889 ΔV119902 Δ1199091 Δ1199092Δ1199093 Δ1199094 Δ120579 Δ1199095]
119879 A is a 12-order matrix (the former 6variables are AC system variables and the others are controlsystem variables) The detailed information of matrix A andthe definitions of 1199091sim1199095 are shown in the Appendix
The operating point of VSC system and the powercontroller parameters are shown in Table 1
The eigenvalues of test model at the operating point of119875ref = 133 and 119875ref = 150 are shown in Table 2
Table 1 Parameters of VSC connected to weak AC system
Parameter symbols Value
Main circuitparameters
Equivalent AC sourcevoltage 119864 10 pu 50Hz
Equivalent impedanceof AC system119877119892+ 119895119883119892
0547 + 1198950048 puimpedance angle 85∘
Equivalent AC systemSCR 183
Rated DC powerrating 10 pu
PCC voltage 10 puConverter impedance
119877119888+ 119895119883119888
0003 + 119895015 pu
Capacitor 119862119891 015 pu
Controllerparameters
Power controllergains (119896pp 119896ip)
(05 50)
AC voltage controllergains (119896pv 119896iv)
(035 30)
Inner 119894119889controller
gains (1198961199011 1198961198941) (1 10)
Inner 119894119902controller
gains (1198961199012 1198961198942) (1 10)
PLL gains (119896pll 119888) (50 10)
With the eigenvalues shown inTable 2 (see the bold italic)the predominant poles [20] ofmatrixA are selected to be 12058289The root-locus of 12058289 with active power changing is shown inFigure 3 And some of the values in Figure 3 are picked up andshown in Table 3
Table 3 The detailed information of predominant poles
119875 12058289
Damping ratio Oscillationfrequency (Hz)
130 pu minus986 plusmn 1198952408 0379 383133 pu minus530 plusmn 1198952310 0224 368137 pu minus280 plusmn 1198952200 0126 350140 pu 022 plusmn j2190 mdash 348143 pu 151 plusmn 1198952171 mdash 346
Figure 3 and Table 3 both show that with the powerrising the small-signal stability of test model is getting worseand system becomes unstable when the active power reachesabout 14 pu
224 Participation Factor Analysis Participation factor canbe utilized to analyze the relationship between predominantpoles and state-variables [21ndash23] The participation factor oftest model is shown in Table 4
FromTable 4 it can be discovered that the outer-loop andPLL control diagrams are likely to have more impacts on thestability of VSC connected to weak AC system Reference [8]has proposed the outer-loop control approach and this paperwill mainly focus on the PLL improvements
3 The Impacts of System Impedance Angle onPower Transfer Limitations
This section will analyze the impact of the system impedanceangle on VSC working in either rectifier or inverter modesregarding the power transfer limitations
The active power transmission at PCC can be calculatedusing
119875 =
119881
1198772119892+ 1198832119892
(minus119864119883119892 sin 120575 + 119864119877119892 cos 120575 minus 119881119877119892) (6)
The power angle curves for different impedance angles(120593) of AC system are shown in Figure 4 It can be found that
minus20
minus10
0
10
20
Imag
inar
y ax
is
minus20 minus10minus30 100Real axis
Figure 3 Root-locus of the predominant poles
Inverting operation
Rectifying operation
minus100minus200 100 2000120575 (deg)
minus3
minus2
minus1
0
1
2Ac
tive p
ower
(pu)
120593 = 90∘
120593 = 87∘
120593 = 83∘
120593 = 80∘
Figure 4 Power angle curves in different impedance angles
120593 has a great influence on power transfer limits The small-signal stability ofVSC systemwith different120593will be analyzedin the subsections below
31 Rectifying Operation With the analysis of state-spacematrix A it is concluded that the equivalent resistance of ACsystem has a negative impact on VSC operating as a rectifierThe root-locus of predominant poles with 120593 changing isshown in Figure 5(a) It shows that for VSC operating inrectifier mode lower resistance of AC system will enhancethe small-signal stability of VSC system
32 Inverting Operation Again with the analysis of state-spacematrixA an opposite conclusion can be drawn that forVSCworking in invertermode lower resistance of AC systemwill weaken the VSC system stability as shown in Figure 5(b)
The power transfer limitations of system with different 120593are expressed in Table 5
Journal of Control Science and Engineering 5
Table 4 The calculated participation factors of test model
Figure 5 Root-locus of predominant poles with 120593 change
Table 5 Calculation result of power transfer limitations withdifferent 120593
120593 Max 119875 (rectifier) Max 119875 (inverter)80∘ 1284 153381∘ 1302 152482∘ 1323 152183∘ 1358 151884∘ 1383 151085∘ 1400 1505
It can be concluded that lower resistance will enlargethe stable margin of VSC working at rectifier mode and willreduce the stable margin for inverter mode
4 Improved PLL Control for VSC Connectedto Weak AC System
In Section 3 a conclusion can be drawn that the dampingcharacteristic of AC network (impedance angle) has a greatinfluence on the system stability With the analysis of partic-ipation factor it is acknowledged that PLL control also has agreat impact on the stability of VSC connected to weak ACsystemTherefore an improved PLL control suitable for VSCconnected to weak AC system is proposed as follows whichis the main contribution of this paper
Considering that the active power is proportional to thedirect-axis current a supplementary damping control withstate-variable 1198942119889 is added in PLL control system (119863 is thedamping factor) And the new PLL equations are shown in(7) in which ldquo+rdquo is for rectifying operation and ldquominusrdquo is forinverting operation Figure 6 shows the improvedPLL controldiagram
120596 = 1205960 + 119896pllV119902 + 119888119896pll int V119902119889119905
120579 = int [120596 plusmn 119863 (1198942119889ref minus 1198942119889)] 119889119905
(7)
The small-signal model is changed and the state-variablematrix A with the improved PLL is shown in the AppendixFigure 7 shows the root-locus of predominant poles of themodified A
From Figure 7 it can be seen that with the rising ofdamping factor119863 in a certain range the small-signal stabilityof VSC system is enhanced After a critical value the stabilityis reduced with the rising of119863 The root-locus of VSC systemwith 119875 rising in different damping factors has also verifiedthe effectiveness of the improved PLL controller Figure 7(c)shows that with a proper damping factor the relationshipbetween PLL and dominant poles is weak It also meansthat the most effective parts are changing from PLL to theouter-loop controller which is the purpose of the improvedPLL control strategy There should be an optimal value for
6 Journal of Control Science and Engineering
D
abc
dq0
d
q
0
Supplementary control
i2dref
+
+
+
+minus
minus
Δ120596
1205960
120579
+ is for rectifying operationminus is for inverting operation
1
s
a
b
c
q
plusmn
i2d
qref = 0
kpll +ckpll
s
Figure 6 Improved PLL control for VSC connected to weak AC system
Rising direction of D
0 20minus40 minus20minus60
Real axis
minus100
minus50
0
50
100
Imag
inar
y ax
is
(a) Root-locus of VSC with 119875 = 14 pu
Rising direction of D
minus100
minus60
minus20
020
60
100
Imag
inar
y ax
is
minus20 0 20 60minus60minus100
Real axis(b) Root-locus of VSC with different transfer power
D ri
sing
dire
ctio
n
0
01
02
03
04
05
06
07
Part
icip
atio
n fa
ctor
2 3 4 5 6 7 8 9 10 11 121State-variables
(c) Participation factor analysis
Figure 7 Root-locus and participation factor of system with advanced PLL control
the damping factor 119863 When impedance angle of AC systemis 85∘ the optimal value of 119863 is approximately 1100 forrectifying operation and 300 for inverting operation
5 Case Studies
51 Validation of the Small-Signal Model Table 6 shows thebase values of the system parameters of benchmark testmodel The accuracy of the small-signal model including
the oscillation frequency and damping ratio will be verifiedin this section
Figure 8 shows the simulation results of the system inwhich the impedance angle is 85∘ The active power PCCvoltage and 119863- and 119876-axis current are shown in Figures8(a)sim8(d) respectively with the active power being changingfrom 360MW to 370MW
It can be seen fromFigure 8 that themaximumerror valuebetween EMTmodel and small-signal model of active power
Journal of Control Science and Engineering 7
EMT modelSmall-signal model
Maximum error value 08
119
12
121
122
123
124
125Ac
tive p
ower
(pu)
72 73 74 75 76 7771Time (s)
(a) Active power
EMT modelSmall-signal model
Maximum error value 015
098
099
10
101
PCC
volta
ge (p
u)
71 72 73 74 75 76 777Time (s)
(b) PCC voltage
D-a
xis c
urre
nt (k
A)
EMT modelSmall-signal model
Maximum error value 027
176
178
18
182
184
186
188
71 72 73 74 75 76 777Time (s)
(c) 119863-axis current
Q-a
xis c
urre
nt (k
A)
EMT modelSmall-signal model
Maximum error value 027
066
067
068
069
07
071
072
073
074
075
71 72 73 74 75 767Time (s)
(d) 119876-axis current
Figure 8 Simulation results of VSC system (impedance angle is 85∘)
Table 6 Parameters of test model
Parameters ValueDc voltage plusmn200 kVRated active power 300MWEquivalent AC source voltage 230 kVPCC voltage 230 kVTransformer 230 kV20412 kV
is 08 The maximum value of PCC voltage is 015 Themaximum error value of 119863-axis current and 119876-axis currentboth is 027 Therefore the correctness of the VSC systemsmall-signal model is satisfactorily validated
52 Validations of Impacts of Impedance Angle on Power Tran-sfer Limits The theory on the impact caused by impedance
angle on stability of VSC connected toweakAC systemwhichis proposed in Section 3 is verified in this section
521 Rectifying Operation Figure 9 shows the maximumpower transmission of system with different impedanceangles
It can be seen from the figure that the maximum activepower is 385MW when 120593 equals 80∘ Meanwhile the maxi-mum active power is 420MWwhen 120593 equals 85∘ Simulationresults show that with the rising of impedance angle the VSCsystem operating in rectifier mode gets larger small-signalstable margin
522 Inverting Operation Figure 10 shows the maximumpower transmission of system with different impedanceangles
However the simulation results shown in Figure 10 areopposite with the system working in rectifying operation
8 Journal of Control Science and Engineering
Reference valueSimulation value
09
1
11
12
13
14
15Ac
tive p
ower
(pu)
4 6 8 102Time (s)
(a) 120593 = 80∘Reference valueSimulation value
09
1
11
12
13
14
15
Activ
e pow
er (p
u)
4 6 8 10 122Time (s)
(b) 120593 = 85∘
Figure 9 Maximum power transmission with different impedance angles (rectifying operation)
Reference valueSimulation value
5 10 15Time (s)
minus15
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(a) 120593 = 85∘Reference valueSimulation value
5 10 15Time (s)
minus1533
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(b) 120593 = 80∘
Figure 10 Maximum power transmission with different impedance angles (inverting operation)
The maximum active power is 460MW when 120593 equals 80∘and this value decreased to 450MWwhen 120593 equals 85∘ Withthe rising of impedance angle the VSC system operatingin inverter mode is getting smaller stable margin and evenbecoming unstable
Table 7 shows the power transfer limitations with differ-ent impedance angles
The conclusion drawn from Section 3 is validated byTable 7 Meanwhile it can be seen that the system operating
at rectifier mode is more severe than inverter mode whichmeans that the stable margin for VSC working in rectifiermode is narrower than that of inverter mode
53 Validations of the Proposed PLL Control Figure 11 showsthe simulation results of maximum power transmission withthe proposed improved PLL control
It can be obtained that with the proposed PLL controlstrategy the limits of power transmission are improved
Journal of Control Science and Engineering 9
5 10 15Time (s)
Reference valueSimulation value
1
12
14
16
18Ac
tive p
ower
(pu)
(a) Rectifying operationReference valueSimulation value
5 10 15 20Time (s)
minus18
minus16
minus14
minus12
minus1
Activ
e pow
er (p
u)(b) Inverting operation
Figure 11 The simulation results of system with proposed PLL control
380
400
420
440
460
Max
imum
activ
e pow
er (M
W)
500 1100 15000Damping factor (D)
(a) Rectifying operation
430
440
450
460
470
480
490
Max
imum
activ
e pow
er (M
W)
100 200 300 400 500 6000Damping factor (D)
(b) Inverting operation
Figure 12 The relationship between119863 and limits of power transmission
The test model adopts 85∘ as the AC system impedanceangle Compared with 420MW the maximum active powerat rectifier is enhanced to 460MW Meanwhile the limits ofpower transmission at inverter are enhanced from450MWto480MW The optimal value of damping factor 119863 calculatedat rectifying operation is 1100 and is 300 at inverting oper-ation Figure 12 shows the relationship between VSC powertransfer limitations and damping factors in the proposed PLLcontrol
It can be seen from Figure 12 that the optimal values ofdamping factors are approximately equal to the calculatedresults (1100 and 300 resp) Figure 13 shows the simulationresults of the relationship between system impedance angles
and power transfer limitations with119863 being equal to zero andthe optimal values (ie 1100 and 300 resp)
Note that the power transfer limitation increments areshown in Figure 13 using arrowsThe validity of the advancedPLL control strategy and the previous impedance angleanalysis results are simultaneously validated in Figure 13
6 Conclusions
In this paper based on small-signal analysis and state-space matrix the power transfer limitations of VSC linkedwith weak AC system are analyzed Besides the dynamicmodel with vector-current control is developed to evaluate
10 Journal of Control Science and Engineering
Table 7 Limits of power transmission with different 120593
the improved PLL control The following conclusions aredrawn
(i) With the rising of active power at PCC the stabilityof VSC system is getting worse and even unstableThe stable margin of VSC for rectifying operation isnarrower than inverting situation With the analysisof participation factor the most sensitive parts tothe stability margin are outer-loop control and PLLcontrol
(ii) The impedance angle of AC system has a great impacton the stable margin of VSC For rectifying operationwith high impedance angle the stability of system
is enhanced and this is not the case for invertingoperation The damping characteristic of AC systemcan change the stable margin of VSC as well
(iii) The proposed PLL control appends the dampingfactor 119863 into the existing PLL diagram to changethe damping characteristic of VSC system Withthe calculated optimal damping factors the stablemargins forVSCworking in either rectifier or invertermodes are significantly enhanced
Appendix
A
=
((((((((((((((((((((((((((((((((((((
(
minus
119877119892
119871119892
120596 0 0 minus
1
119871119892
0 0 0 0 0 minus
119864119898 sin 1205750119871119892
0
minus120596 minus
119877119892
119871119892
0 0 0 minus
1
119871119892
0 0 0 0 minus
119864119898 cos 1205750119871119892
0
0 0
1198961199011 (minus (32) 119896ppV1198890 minus 1) minus 119877119888119871119888
0
minus31198961199011119896pp11989421198890
2119871119888
minus31198961199011119896pp11989421199020
2119871119888
minus119896ip1198961199011
119871119888
0
1198961198941
119871119888
0 0 0
0 0 0
minus1198961199012 minus 119877119888
119871119888
minus
3
2
1198961199012119896pvV11988901198810119871119888
0 0 minus
3
2
1198961199012119896ivV11988901198810119871119888
0
1198961198942
119871119888
0 0
1
119862119891
0 minus
1
119862119891
0 0 120596 0 0 0 0 0 0
0
1
119862119891
0 minus
1
119862119891
minus120596 0 0 0 0 0 0 0
0 0
3
2
V1198890 0
3
2
11989421198890
3
2
11989421199020 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 minus
3
2
119896ppV1198890 minus 1 0 minus
3
2
119896pp11989421198890 minus
3
2
119896pp11989421199020 minus119896ip 0 0 0 0 0
0 0 0 minus1 minus
3
2
119896pvV11988901198810
0 0 minus
3
2
119896ivV11988901198810
0 0 0 0
0 0 0 0 0 119896pll 0 0 0 0 0 119888119896pll
0 0 0 0 0 1 0 0 0 0 0 0
))))))))))))))))))))))))))))))))))))
)
(A1)
The mathematical step in the derivation of the state-variable equations is given in this section
119875 =
3
2
(V1198891198942119889 + V1198891198942119902)
119881 = radic3
2
radicV2119889+ V2119902
(A2)
The small-signal model is shown in (A3) In particularthe quadrature component of the PCC voltage at operatingpoint is 0 (V1199020 = 0)
Figure 13 The power transfer limitations with the advanced PLL control
The relationship between 119890119889 119890119902 and 120579 is shown in (A4)in which1205960119905+1198860 is the real time angle of equivalent ac voltagesource
119890119889 = 119864119898cos [120579 minus (1205960119905 + 1205720)]
119890119902 = minus119864119898sin [120579 minus (1205960119905 + 1205720)] (A4)
The small-signal model is shown in (A5) 120575 is mentionedin nomenclature
Δ119890119889 = minus119864119898 sin 1205750Δ120579
Δ119890119902 = minus119864119898 cos 1205750Δ120579(A5)
The small-signal model of (1)sim(4) is shown below
[
minus (119864119898 sin 1205750) Δ120575minus (119864119898 cos 1205750) Δ120575
] minus [
ΔV119889ΔV119902
]
= 119877119892 [
Δ1198941119889
Δ1198941119902
] + 119871119892
119889
119889119905
[
Δ1198941119889
Δ1198941119902
] + 120596119871119892 [
minusΔ1198941119902
Δ1198941119889
]
[
ΔV119889ΔV119902
] minus [
ΔV119888119889ΔV119888119902
]
= 119877119888 [
Δ1198942119889
Δ1198942119902
] + 119871119888
119889
119889119905
[
Δ1198942119889
Δ1198942119902
] + 120596119871119888 [
minusΔ1198942119902
Δ1198942119889
]
[
Δ1198941119889
Δ1198941119902
] minus [
Δ1198942119889
Δ1198942119902
] = 119862119891
119889
119889119905
[
ΔV119889ΔV119902
] + 120596119862119891 [
minusΔV119902ΔV119889
]
minus 119896ppΔ119875 minus 119896ip intΔ119875119889119905 = Δ1198942119889ref
minus 119896pvΔ119881 minus 119896iv intΔ119881119889119905 = Δ1198942119902ref
1198961199011 (Δ1198942119889ref minus Δ1198942119889) + 1198961198941 int (Δ1198942119889ref minus Δ1198942119889) 119889119905
= 119871119888
119889Δ1198942119889
119889119905
+ 119877119888Δ1198942119889
1198961199012 (Δ1198942119902ref minus Δ1198942119902) + 1198961198942 int (Δ1198942119902ref minus Δ1198942119902) 119889119905
The state-variables Δ1199091 Δ1199092 Δ1199093 Δ1199094 and Δ1199095 areassumed below
Δ1199091 = intΔ119875119889119905
Δ1199092 = intΔV119889119889119905
Δ1199093 = int (Δ1198942119889ref minus Δ1198942119889) 119889119905
Δ1199094 = int (Δ1198942119902ref minus Δ1198942119902) 119889119905
Δ1199095 = intΔV119902119889119905
(A7)
12 Journal of Control Science and Engineering
With the improved PLL control matrix A is changedThe modified elements are shown below ldquo+rdquo is for rectifyingoperation and ldquominusrdquo is for inverting operation
A113 = plusmn119863(
3
2
119896ppV1198890 + 1)
A115 = plusmn3
2
119863119896pp11989421198890
A116 = (plusmn3
2
119863119896pp11989421199020 + 119896pll)
A117 = plusmn119863119896ip
(A8)
The participation factormentioned in Section 2 is definedas the relativity between the 119896th state-variable and theeigenvalue 120582119894 The calculation method is shown in
in which u and k are the left eigenvector and right eigenvectorseparately V119896119894 and 119906119896119894 are the corresponding elements ineigenvectors
Nomenclature
119890119886119887119888 Thevenin equivalent voltage of AC systemV119886119887119888 PCC voltageV119888119886119887119888 Converter voltage1198941119886119887119888 Phase current at the system side1198942119886119887119888 Phase current at the converter side119894119888119886119887119888 Phase current in shunt capacitor119875119881 PCC active power and voltage119875ref 119881ref The reference value of PCC active power
and voltage119890119889 119890119902 119889 and 119902 components of AC source voltageV119889 V119902 119889 and 119902 components of PCC voltageV119888119889 V119888119902 119889 and 119902 components of converter voltage1198941119889 1198941119902 119889 and 119902 components of system current1198942119889 1198942119902 119889 and 119902 components of converter current119894119888119889 119894119888119902 119889 and 119902 components of capacitor current119877119892 119871119892 Equivalent resistance and inductance of ac
system119877119888 119871119888 Resistance and inductance of converter119862119891 Shunt capacitance1198942119889ref 119889 component of converter current
reference1198942119902ref 119902 component of converter current
reference120596 Frequency of PCC voltage120579 PLL output angle120575 The angle PCC voltage V leading
equivalent source voltage E120593 Impedance angle of AC system119863 Damping factor in the improved PLL
control119880dc DC voltage
119896pp 119896ip Power loop proportional and integrator gains119896pv 119896iv Voltage loop proportional and integrator
gains1198961199011 1198961198941 Inner-loop control (119863-axis current)
proportional and integrator gains1198961199012 1198961198942 Inner-loop control (119876-axis current)
proportional and integrator gains119896pll 119888119896pll Proportional and integrator gains of PLL119901119896119894 Participation factor119864119898 Maximum instantaneous value of source
voltage
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National HighTechnology Research and Development Program of China(863 Program) (no 2013AA050105) in part by the NationalNatural Science Foundation of China (no 51177042) andin part by the Fundamental Research Funds for the CentralUniversities (no 2015XS23 and no 2015QN04)
References
[1] IEEE ldquoIEEE guide for planning dc links terminating at AClocations having low short-circuit capacitiesrdquo IEEE Standard1204-1997 1997
[2] A Gole V K Sood and L Mootoosamy ldquoValidation andanalysis of a grid control system using D-Q-Z transformationfor static compensator systemsrdquo in Proceedings of the CanadianConference on Electrical and Computer Engineering pp 745ndash758 Montreal Canada September 1989
[3] K Liao Z-Y He and B Sun ldquoSmall signal model for VSC-HVDC connected DFIG-based offshore wind farmsrdquo Journal ofAppliedMathematics vol 2014 Article ID 725209 6 pages 2014
[4] M Cespedes and J Sun ldquoAdaptive control of grid-connectedinverters based on online grid impedancemeasurementsrdquo IEEETransactions on Sustainable Energy vol 5 no 2 pp 516ndash5232014
[5] VK SoodVKhatri andH Jin ldquoPerformance assessment usingEMTP of two gate firing units for HVDC converters operatingwith weak AC systemsrdquo in Proceedings of the InternationalConference on Power Systems Transients (IPST rsquo95) pp 517ndash522Lisbon Portugal September 1995
[6] 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
[7] K M Alawasa and Y A-R I Mohamed ldquoImpedance anddamping characteristics of grid-connected VSCs with powersynchronization control strategyrdquo IEEE Transactions on PowerSystems vol 30 no 2 pp 952ndash961 2015
[8] A Egea-Alvarez S Fekriasl F Hassan andO Gomis-BellmuntldquoAdvanced vector control for voltage source converters con-nected to weak ac gridsrdquo IEEE Transactions on Power Systemsvol 30 no 6 pp 3072ndash3081 2015
Journal of Control Science and Engineering 13
[9] L Zhang L Harnefors and H-P Nee ldquoPower-synchronizationcontrol of grid-connected voltage-source convertersrdquo IEEETransactions on Power Systems vol 25 no 2 pp 809ndash820 2010
[10] J Z Zhou H Ding S Fan Y Zhang and A M Gole ldquoImpactof short-circuit ratio and phase-locked-loop parameters onthe small-signal behavior of a VSC-HVDC converterrdquo IEEETransactions on Power Delivery vol 29 no 5 pp 2287ndash22962014
[11] J A Suul S DrsquoArco P Rodriguez and M Molinas ldquoExtendedstability range of weak grids with Voltage Source Convertersthrough impedance-conditioned grid synchronizationrdquo in Pro-ceedings of the 11th International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February 2015
[12] 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
[13] T Midtsund J A Suul and T Undeland ldquoEvaluation ofcurrent controller performance and stability for voltage sourceconverters connected to a weak gridrdquo in Proceedings of the 2ndInternational Symposium on Power Electronics for DistributedGeneration Systems pp 382ndash388 IEEE Hefei China June 2010
[14] Z Miao L Fan D Osborn and S Yuvarajan ldquoWind farmswith HVdc delivery in inertial response and primary frequencycontrolrdquo IEEE Transactions on Energy Conversion vol 25 no 4pp 1171ndash1178 2010
[15] J Zhu C D Booth G P Adam A J Roscoe and C GBright ldquoInertia emulation control strategy for VSC-HVDCtransmission systemsrdquo IEEETransactions on Power Systems vol28 no 2 pp 1277ndash1287 2013
[16] H Ding S Fan J Z Zhou Y Zhang and A M GoleldquoParametric analysis of the stability of VSC-HVDC convertersrdquoin Proceedings of the 11th IET International Conference on ACand DC Power Transmission pp 1ndash6 IEEE Birmingham UKFebruary 2015
[17] B Wen D Boroyevich R Burgos P Mattavelli and Z ShenldquoAnalysis of D-Q small-signal impedance of grid-tied invertersrdquoIEEE Transactions on Power Electronics vol 31 no 1 pp 675ndash687 2016
[18] A Yazdani and R Iravani Voltage-Sourced Converters in PowerSystems Wiley Hoboken NJ USA 2010
[19] G-C Hsieh and J C Hung ldquoPhase-locked loop techniquesmdashasurveyrdquo IEEE Transactions on Industrial Electronics vol 43 no6 pp 609ndash615 1996
[20] G F Franklin J D Powell and A Emami-Naeini FeedbackControl of Dynamic Systems Prentice Hall New York NY USA2005
[21] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1993
[22] D Dong J Li D Boroyevich P Mattavelli I Cvetkovic andY Xue ldquoFrequency behavior and its stability of grid-interfaceconverter in distributed generation systemsrdquo in Proceedingsof the 27th IEEE Applied Power Electronics Conference andExposition (APEC rsquo12) pp 1887ndash1893 IEEE Orlando Fla USAFebruary 2012
[23] J Beerten S DrsquoArco and J A Suul ldquoIdentification and small-signal analysis of interaction modes in VSC MTDC systemsrdquoIEEE Transactions on Power Delivery 2015
Table 3 The detailed information of predominant poles
119875 12058289
Damping ratio Oscillationfrequency (Hz)
130 pu minus986 plusmn 1198952408 0379 383133 pu minus530 plusmn 1198952310 0224 368137 pu minus280 plusmn 1198952200 0126 350140 pu 022 plusmn j2190 mdash 348143 pu 151 plusmn 1198952171 mdash 346
Figure 3 and Table 3 both show that with the powerrising the small-signal stability of test model is getting worseand system becomes unstable when the active power reachesabout 14 pu
224 Participation Factor Analysis Participation factor canbe utilized to analyze the relationship between predominantpoles and state-variables [21ndash23] The participation factor oftest model is shown in Table 4
FromTable 4 it can be discovered that the outer-loop andPLL control diagrams are likely to have more impacts on thestability of VSC connected to weak AC system Reference [8]has proposed the outer-loop control approach and this paperwill mainly focus on the PLL improvements
3 The Impacts of System Impedance Angle onPower Transfer Limitations
This section will analyze the impact of the system impedanceangle on VSC working in either rectifier or inverter modesregarding the power transfer limitations
The active power transmission at PCC can be calculatedusing
119875 =
119881
1198772119892+ 1198832119892
(minus119864119883119892 sin 120575 + 119864119877119892 cos 120575 minus 119881119877119892) (6)
The power angle curves for different impedance angles(120593) of AC system are shown in Figure 4 It can be found that
minus20
minus10
0
10
20
Imag
inar
y ax
is
minus20 minus10minus30 100Real axis
Figure 3 Root-locus of the predominant poles
Inverting operation
Rectifying operation
minus100minus200 100 2000120575 (deg)
minus3
minus2
minus1
0
1
2Ac
tive p
ower
(pu)
120593 = 90∘
120593 = 87∘
120593 = 83∘
120593 = 80∘
Figure 4 Power angle curves in different impedance angles
120593 has a great influence on power transfer limits The small-signal stability ofVSC systemwith different120593will be analyzedin the subsections below
31 Rectifying Operation With the analysis of state-spacematrix A it is concluded that the equivalent resistance of ACsystem has a negative impact on VSC operating as a rectifierThe root-locus of predominant poles with 120593 changing isshown in Figure 5(a) It shows that for VSC operating inrectifier mode lower resistance of AC system will enhancethe small-signal stability of VSC system
32 Inverting Operation Again with the analysis of state-spacematrixA an opposite conclusion can be drawn that forVSCworking in invertermode lower resistance of AC systemwill weaken the VSC system stability as shown in Figure 5(b)
The power transfer limitations of system with different 120593are expressed in Table 5
Journal of Control Science and Engineering 5
Table 4 The calculated participation factors of test model
Figure 5 Root-locus of predominant poles with 120593 change
Table 5 Calculation result of power transfer limitations withdifferent 120593
120593 Max 119875 (rectifier) Max 119875 (inverter)80∘ 1284 153381∘ 1302 152482∘ 1323 152183∘ 1358 151884∘ 1383 151085∘ 1400 1505
It can be concluded that lower resistance will enlargethe stable margin of VSC working at rectifier mode and willreduce the stable margin for inverter mode
4 Improved PLL Control for VSC Connectedto Weak AC System
In Section 3 a conclusion can be drawn that the dampingcharacteristic of AC network (impedance angle) has a greatinfluence on the system stability With the analysis of partic-ipation factor it is acknowledged that PLL control also has agreat impact on the stability of VSC connected to weak ACsystemTherefore an improved PLL control suitable for VSCconnected to weak AC system is proposed as follows whichis the main contribution of this paper
Considering that the active power is proportional to thedirect-axis current a supplementary damping control withstate-variable 1198942119889 is added in PLL control system (119863 is thedamping factor) And the new PLL equations are shown in(7) in which ldquo+rdquo is for rectifying operation and ldquominusrdquo is forinverting operation Figure 6 shows the improvedPLL controldiagram
120596 = 1205960 + 119896pllV119902 + 119888119896pll int V119902119889119905
120579 = int [120596 plusmn 119863 (1198942119889ref minus 1198942119889)] 119889119905
(7)
The small-signal model is changed and the state-variablematrix A with the improved PLL is shown in the AppendixFigure 7 shows the root-locus of predominant poles of themodified A
From Figure 7 it can be seen that with the rising ofdamping factor119863 in a certain range the small-signal stabilityof VSC system is enhanced After a critical value the stabilityis reduced with the rising of119863 The root-locus of VSC systemwith 119875 rising in different damping factors has also verifiedthe effectiveness of the improved PLL controller Figure 7(c)shows that with a proper damping factor the relationshipbetween PLL and dominant poles is weak It also meansthat the most effective parts are changing from PLL to theouter-loop controller which is the purpose of the improvedPLL control strategy There should be an optimal value for
6 Journal of Control Science and Engineering
D
abc
dq0
d
q
0
Supplementary control
i2dref
+
+
+
+minus
minus
Δ120596
1205960
120579
+ is for rectifying operationminus is for inverting operation
1
s
a
b
c
q
plusmn
i2d
qref = 0
kpll +ckpll
s
Figure 6 Improved PLL control for VSC connected to weak AC system
Rising direction of D
0 20minus40 minus20minus60
Real axis
minus100
minus50
0
50
100
Imag
inar
y ax
is
(a) Root-locus of VSC with 119875 = 14 pu
Rising direction of D
minus100
minus60
minus20
020
60
100
Imag
inar
y ax
is
minus20 0 20 60minus60minus100
Real axis(b) Root-locus of VSC with different transfer power
D ri
sing
dire
ctio
n
0
01
02
03
04
05
06
07
Part
icip
atio
n fa
ctor
2 3 4 5 6 7 8 9 10 11 121State-variables
(c) Participation factor analysis
Figure 7 Root-locus and participation factor of system with advanced PLL control
the damping factor 119863 When impedance angle of AC systemis 85∘ the optimal value of 119863 is approximately 1100 forrectifying operation and 300 for inverting operation
5 Case Studies
51 Validation of the Small-Signal Model Table 6 shows thebase values of the system parameters of benchmark testmodel The accuracy of the small-signal model including
the oscillation frequency and damping ratio will be verifiedin this section
Figure 8 shows the simulation results of the system inwhich the impedance angle is 85∘ The active power PCCvoltage and 119863- and 119876-axis current are shown in Figures8(a)sim8(d) respectively with the active power being changingfrom 360MW to 370MW
It can be seen fromFigure 8 that themaximumerror valuebetween EMTmodel and small-signal model of active power
Journal of Control Science and Engineering 7
EMT modelSmall-signal model
Maximum error value 08
119
12
121
122
123
124
125Ac
tive p
ower
(pu)
72 73 74 75 76 7771Time (s)
(a) Active power
EMT modelSmall-signal model
Maximum error value 015
098
099
10
101
PCC
volta
ge (p
u)
71 72 73 74 75 76 777Time (s)
(b) PCC voltage
D-a
xis c
urre
nt (k
A)
EMT modelSmall-signal model
Maximum error value 027
176
178
18
182
184
186
188
71 72 73 74 75 76 777Time (s)
(c) 119863-axis current
Q-a
xis c
urre
nt (k
A)
EMT modelSmall-signal model
Maximum error value 027
066
067
068
069
07
071
072
073
074
075
71 72 73 74 75 767Time (s)
(d) 119876-axis current
Figure 8 Simulation results of VSC system (impedance angle is 85∘)
Table 6 Parameters of test model
Parameters ValueDc voltage plusmn200 kVRated active power 300MWEquivalent AC source voltage 230 kVPCC voltage 230 kVTransformer 230 kV20412 kV
is 08 The maximum value of PCC voltage is 015 Themaximum error value of 119863-axis current and 119876-axis currentboth is 027 Therefore the correctness of the VSC systemsmall-signal model is satisfactorily validated
52 Validations of Impacts of Impedance Angle on Power Tran-sfer Limits The theory on the impact caused by impedance
angle on stability of VSC connected toweakAC systemwhichis proposed in Section 3 is verified in this section
521 Rectifying Operation Figure 9 shows the maximumpower transmission of system with different impedanceangles
It can be seen from the figure that the maximum activepower is 385MW when 120593 equals 80∘ Meanwhile the maxi-mum active power is 420MWwhen 120593 equals 85∘ Simulationresults show that with the rising of impedance angle the VSCsystem operating in rectifier mode gets larger small-signalstable margin
522 Inverting Operation Figure 10 shows the maximumpower transmission of system with different impedanceangles
However the simulation results shown in Figure 10 areopposite with the system working in rectifying operation
8 Journal of Control Science and Engineering
Reference valueSimulation value
09
1
11
12
13
14
15Ac
tive p
ower
(pu)
4 6 8 102Time (s)
(a) 120593 = 80∘Reference valueSimulation value
09
1
11
12
13
14
15
Activ
e pow
er (p
u)
4 6 8 10 122Time (s)
(b) 120593 = 85∘
Figure 9 Maximum power transmission with different impedance angles (rectifying operation)
Reference valueSimulation value
5 10 15Time (s)
minus15
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(a) 120593 = 85∘Reference valueSimulation value
5 10 15Time (s)
minus1533
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(b) 120593 = 80∘
Figure 10 Maximum power transmission with different impedance angles (inverting operation)
The maximum active power is 460MW when 120593 equals 80∘and this value decreased to 450MWwhen 120593 equals 85∘ Withthe rising of impedance angle the VSC system operatingin inverter mode is getting smaller stable margin and evenbecoming unstable
Table 7 shows the power transfer limitations with differ-ent impedance angles
The conclusion drawn from Section 3 is validated byTable 7 Meanwhile it can be seen that the system operating
at rectifier mode is more severe than inverter mode whichmeans that the stable margin for VSC working in rectifiermode is narrower than that of inverter mode
53 Validations of the Proposed PLL Control Figure 11 showsthe simulation results of maximum power transmission withthe proposed improved PLL control
It can be obtained that with the proposed PLL controlstrategy the limits of power transmission are improved
Journal of Control Science and Engineering 9
5 10 15Time (s)
Reference valueSimulation value
1
12
14
16
18Ac
tive p
ower
(pu)
(a) Rectifying operationReference valueSimulation value
5 10 15 20Time (s)
minus18
minus16
minus14
minus12
minus1
Activ
e pow
er (p
u)(b) Inverting operation
Figure 11 The simulation results of system with proposed PLL control
380
400
420
440
460
Max
imum
activ
e pow
er (M
W)
500 1100 15000Damping factor (D)
(a) Rectifying operation
430
440
450
460
470
480
490
Max
imum
activ
e pow
er (M
W)
100 200 300 400 500 6000Damping factor (D)
(b) Inverting operation
Figure 12 The relationship between119863 and limits of power transmission
The test model adopts 85∘ as the AC system impedanceangle Compared with 420MW the maximum active powerat rectifier is enhanced to 460MW Meanwhile the limits ofpower transmission at inverter are enhanced from450MWto480MW The optimal value of damping factor 119863 calculatedat rectifying operation is 1100 and is 300 at inverting oper-ation Figure 12 shows the relationship between VSC powertransfer limitations and damping factors in the proposed PLLcontrol
It can be seen from Figure 12 that the optimal values ofdamping factors are approximately equal to the calculatedresults (1100 and 300 resp) Figure 13 shows the simulationresults of the relationship between system impedance angles
and power transfer limitations with119863 being equal to zero andthe optimal values (ie 1100 and 300 resp)
Note that the power transfer limitation increments areshown in Figure 13 using arrowsThe validity of the advancedPLL control strategy and the previous impedance angleanalysis results are simultaneously validated in Figure 13
6 Conclusions
In this paper based on small-signal analysis and state-space matrix the power transfer limitations of VSC linkedwith weak AC system are analyzed Besides the dynamicmodel with vector-current control is developed to evaluate
10 Journal of Control Science and Engineering
Table 7 Limits of power transmission with different 120593
the improved PLL control The following conclusions aredrawn
(i) With the rising of active power at PCC the stabilityof VSC system is getting worse and even unstableThe stable margin of VSC for rectifying operation isnarrower than inverting situation With the analysisof participation factor the most sensitive parts tothe stability margin are outer-loop control and PLLcontrol
(ii) The impedance angle of AC system has a great impacton the stable margin of VSC For rectifying operationwith high impedance angle the stability of system
is enhanced and this is not the case for invertingoperation The damping characteristic of AC systemcan change the stable margin of VSC as well
(iii) The proposed PLL control appends the dampingfactor 119863 into the existing PLL diagram to changethe damping characteristic of VSC system Withthe calculated optimal damping factors the stablemargins forVSCworking in either rectifier or invertermodes are significantly enhanced
Appendix
A
=
((((((((((((((((((((((((((((((((((((
(
minus
119877119892
119871119892
120596 0 0 minus
1
119871119892
0 0 0 0 0 minus
119864119898 sin 1205750119871119892
0
minus120596 minus
119877119892
119871119892
0 0 0 minus
1
119871119892
0 0 0 0 minus
119864119898 cos 1205750119871119892
0
0 0
1198961199011 (minus (32) 119896ppV1198890 minus 1) minus 119877119888119871119888
0
minus31198961199011119896pp11989421198890
2119871119888
minus31198961199011119896pp11989421199020
2119871119888
minus119896ip1198961199011
119871119888
0
1198961198941
119871119888
0 0 0
0 0 0
minus1198961199012 minus 119877119888
119871119888
minus
3
2
1198961199012119896pvV11988901198810119871119888
0 0 minus
3
2
1198961199012119896ivV11988901198810119871119888
0
1198961198942
119871119888
0 0
1
119862119891
0 minus
1
119862119891
0 0 120596 0 0 0 0 0 0
0
1
119862119891
0 minus
1
119862119891
minus120596 0 0 0 0 0 0 0
0 0
3
2
V1198890 0
3
2
11989421198890
3
2
11989421199020 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 minus
3
2
119896ppV1198890 minus 1 0 minus
3
2
119896pp11989421198890 minus
3
2
119896pp11989421199020 minus119896ip 0 0 0 0 0
0 0 0 minus1 minus
3
2
119896pvV11988901198810
0 0 minus
3
2
119896ivV11988901198810
0 0 0 0
0 0 0 0 0 119896pll 0 0 0 0 0 119888119896pll
0 0 0 0 0 1 0 0 0 0 0 0
))))))))))))))))))))))))))))))))))))
)
(A1)
The mathematical step in the derivation of the state-variable equations is given in this section
119875 =
3
2
(V1198891198942119889 + V1198891198942119902)
119881 = radic3
2
radicV2119889+ V2119902
(A2)
The small-signal model is shown in (A3) In particularthe quadrature component of the PCC voltage at operatingpoint is 0 (V1199020 = 0)
Figure 13 The power transfer limitations with the advanced PLL control
The relationship between 119890119889 119890119902 and 120579 is shown in (A4)in which1205960119905+1198860 is the real time angle of equivalent ac voltagesource
119890119889 = 119864119898cos [120579 minus (1205960119905 + 1205720)]
119890119902 = minus119864119898sin [120579 minus (1205960119905 + 1205720)] (A4)
The small-signal model is shown in (A5) 120575 is mentionedin nomenclature
Δ119890119889 = minus119864119898 sin 1205750Δ120579
Δ119890119902 = minus119864119898 cos 1205750Δ120579(A5)
The small-signal model of (1)sim(4) is shown below
[
minus (119864119898 sin 1205750) Δ120575minus (119864119898 cos 1205750) Δ120575
] minus [
ΔV119889ΔV119902
]
= 119877119892 [
Δ1198941119889
Δ1198941119902
] + 119871119892
119889
119889119905
[
Δ1198941119889
Δ1198941119902
] + 120596119871119892 [
minusΔ1198941119902
Δ1198941119889
]
[
ΔV119889ΔV119902
] minus [
ΔV119888119889ΔV119888119902
]
= 119877119888 [
Δ1198942119889
Δ1198942119902
] + 119871119888
119889
119889119905
[
Δ1198942119889
Δ1198942119902
] + 120596119871119888 [
minusΔ1198942119902
Δ1198942119889
]
[
Δ1198941119889
Δ1198941119902
] minus [
Δ1198942119889
Δ1198942119902
] = 119862119891
119889
119889119905
[
ΔV119889ΔV119902
] + 120596119862119891 [
minusΔV119902ΔV119889
]
minus 119896ppΔ119875 minus 119896ip intΔ119875119889119905 = Δ1198942119889ref
minus 119896pvΔ119881 minus 119896iv intΔ119881119889119905 = Δ1198942119902ref
1198961199011 (Δ1198942119889ref minus Δ1198942119889) + 1198961198941 int (Δ1198942119889ref minus Δ1198942119889) 119889119905
= 119871119888
119889Δ1198942119889
119889119905
+ 119877119888Δ1198942119889
1198961199012 (Δ1198942119902ref minus Δ1198942119902) + 1198961198942 int (Δ1198942119902ref minus Δ1198942119902) 119889119905
The state-variables Δ1199091 Δ1199092 Δ1199093 Δ1199094 and Δ1199095 areassumed below
Δ1199091 = intΔ119875119889119905
Δ1199092 = intΔV119889119889119905
Δ1199093 = int (Δ1198942119889ref minus Δ1198942119889) 119889119905
Δ1199094 = int (Δ1198942119902ref minus Δ1198942119902) 119889119905
Δ1199095 = intΔV119902119889119905
(A7)
12 Journal of Control Science and Engineering
With the improved PLL control matrix A is changedThe modified elements are shown below ldquo+rdquo is for rectifyingoperation and ldquominusrdquo is for inverting operation
A113 = plusmn119863(
3
2
119896ppV1198890 + 1)
A115 = plusmn3
2
119863119896pp11989421198890
A116 = (plusmn3
2
119863119896pp11989421199020 + 119896pll)
A117 = plusmn119863119896ip
(A8)
The participation factormentioned in Section 2 is definedas the relativity between the 119896th state-variable and theeigenvalue 120582119894 The calculation method is shown in
in which u and k are the left eigenvector and right eigenvectorseparately V119896119894 and 119906119896119894 are the corresponding elements ineigenvectors
Nomenclature
119890119886119887119888 Thevenin equivalent voltage of AC systemV119886119887119888 PCC voltageV119888119886119887119888 Converter voltage1198941119886119887119888 Phase current at the system side1198942119886119887119888 Phase current at the converter side119894119888119886119887119888 Phase current in shunt capacitor119875119881 PCC active power and voltage119875ref 119881ref The reference value of PCC active power
and voltage119890119889 119890119902 119889 and 119902 components of AC source voltageV119889 V119902 119889 and 119902 components of PCC voltageV119888119889 V119888119902 119889 and 119902 components of converter voltage1198941119889 1198941119902 119889 and 119902 components of system current1198942119889 1198942119902 119889 and 119902 components of converter current119894119888119889 119894119888119902 119889 and 119902 components of capacitor current119877119892 119871119892 Equivalent resistance and inductance of ac
system119877119888 119871119888 Resistance and inductance of converter119862119891 Shunt capacitance1198942119889ref 119889 component of converter current
reference1198942119902ref 119902 component of converter current
reference120596 Frequency of PCC voltage120579 PLL output angle120575 The angle PCC voltage V leading
equivalent source voltage E120593 Impedance angle of AC system119863 Damping factor in the improved PLL
control119880dc DC voltage
119896pp 119896ip Power loop proportional and integrator gains119896pv 119896iv Voltage loop proportional and integrator
gains1198961199011 1198961198941 Inner-loop control (119863-axis current)
proportional and integrator gains1198961199012 1198961198942 Inner-loop control (119876-axis current)
proportional and integrator gains119896pll 119888119896pll Proportional and integrator gains of PLL119901119896119894 Participation factor119864119898 Maximum instantaneous value of source
voltage
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National HighTechnology Research and Development Program of China(863 Program) (no 2013AA050105) in part by the NationalNatural Science Foundation of China (no 51177042) andin part by the Fundamental Research Funds for the CentralUniversities (no 2015XS23 and no 2015QN04)
References
[1] IEEE ldquoIEEE guide for planning dc links terminating at AClocations having low short-circuit capacitiesrdquo IEEE Standard1204-1997 1997
[2] A Gole V K Sood and L Mootoosamy ldquoValidation andanalysis of a grid control system using D-Q-Z transformationfor static compensator systemsrdquo in Proceedings of the CanadianConference on Electrical and Computer Engineering pp 745ndash758 Montreal Canada September 1989
[3] K Liao Z-Y He and B Sun ldquoSmall signal model for VSC-HVDC connected DFIG-based offshore wind farmsrdquo Journal ofAppliedMathematics vol 2014 Article ID 725209 6 pages 2014
[4] M Cespedes and J Sun ldquoAdaptive control of grid-connectedinverters based on online grid impedancemeasurementsrdquo IEEETransactions on Sustainable Energy vol 5 no 2 pp 516ndash5232014
[5] VK SoodVKhatri andH Jin ldquoPerformance assessment usingEMTP of two gate firing units for HVDC converters operatingwith weak AC systemsrdquo in Proceedings of the InternationalConference on Power Systems Transients (IPST rsquo95) pp 517ndash522Lisbon Portugal September 1995
[6] 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
[7] K M Alawasa and Y A-R I Mohamed ldquoImpedance anddamping characteristics of grid-connected VSCs with powersynchronization control strategyrdquo IEEE Transactions on PowerSystems vol 30 no 2 pp 952ndash961 2015
[8] A Egea-Alvarez S Fekriasl F Hassan andO Gomis-BellmuntldquoAdvanced vector control for voltage source converters con-nected to weak ac gridsrdquo IEEE Transactions on Power Systemsvol 30 no 6 pp 3072ndash3081 2015
Journal of Control Science and Engineering 13
[9] L Zhang L Harnefors and H-P Nee ldquoPower-synchronizationcontrol of grid-connected voltage-source convertersrdquo IEEETransactions on Power Systems vol 25 no 2 pp 809ndash820 2010
[10] J Z Zhou H Ding S Fan Y Zhang and A M Gole ldquoImpactof short-circuit ratio and phase-locked-loop parameters onthe small-signal behavior of a VSC-HVDC converterrdquo IEEETransactions on Power Delivery vol 29 no 5 pp 2287ndash22962014
[11] J A Suul S DrsquoArco P Rodriguez and M Molinas ldquoExtendedstability range of weak grids with Voltage Source Convertersthrough impedance-conditioned grid synchronizationrdquo in Pro-ceedings of the 11th International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February 2015
[12] 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
[13] T Midtsund J A Suul and T Undeland ldquoEvaluation ofcurrent controller performance and stability for voltage sourceconverters connected to a weak gridrdquo in Proceedings of the 2ndInternational Symposium on Power Electronics for DistributedGeneration Systems pp 382ndash388 IEEE Hefei China June 2010
[14] Z Miao L Fan D Osborn and S Yuvarajan ldquoWind farmswith HVdc delivery in inertial response and primary frequencycontrolrdquo IEEE Transactions on Energy Conversion vol 25 no 4pp 1171ndash1178 2010
[15] J Zhu C D Booth G P Adam A J Roscoe and C GBright ldquoInertia emulation control strategy for VSC-HVDCtransmission systemsrdquo IEEETransactions on Power Systems vol28 no 2 pp 1277ndash1287 2013
[16] H Ding S Fan J Z Zhou Y Zhang and A M GoleldquoParametric analysis of the stability of VSC-HVDC convertersrdquoin Proceedings of the 11th IET International Conference on ACand DC Power Transmission pp 1ndash6 IEEE Birmingham UKFebruary 2015
[17] B Wen D Boroyevich R Burgos P Mattavelli and Z ShenldquoAnalysis of D-Q small-signal impedance of grid-tied invertersrdquoIEEE Transactions on Power Electronics vol 31 no 1 pp 675ndash687 2016
[18] A Yazdani and R Iravani Voltage-Sourced Converters in PowerSystems Wiley Hoboken NJ USA 2010
[19] G-C Hsieh and J C Hung ldquoPhase-locked loop techniquesmdashasurveyrdquo IEEE Transactions on Industrial Electronics vol 43 no6 pp 609ndash615 1996
[20] G F Franklin J D Powell and A Emami-Naeini FeedbackControl of Dynamic Systems Prentice Hall New York NY USA2005
[21] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1993
[22] D Dong J Li D Boroyevich P Mattavelli I Cvetkovic andY Xue ldquoFrequency behavior and its stability of grid-interfaceconverter in distributed generation systemsrdquo in Proceedingsof the 27th IEEE Applied Power Electronics Conference andExposition (APEC rsquo12) pp 1887ndash1893 IEEE Orlando Fla USAFebruary 2012
[23] J Beerten S DrsquoArco and J A Suul ldquoIdentification and small-signal analysis of interaction modes in VSC MTDC systemsrdquoIEEE Transactions on Power Delivery 2015
Figure 5 Root-locus of predominant poles with 120593 change
Table 5 Calculation result of power transfer limitations withdifferent 120593
120593 Max 119875 (rectifier) Max 119875 (inverter)80∘ 1284 153381∘ 1302 152482∘ 1323 152183∘ 1358 151884∘ 1383 151085∘ 1400 1505
It can be concluded that lower resistance will enlargethe stable margin of VSC working at rectifier mode and willreduce the stable margin for inverter mode
4 Improved PLL Control for VSC Connectedto Weak AC System
In Section 3 a conclusion can be drawn that the dampingcharacteristic of AC network (impedance angle) has a greatinfluence on the system stability With the analysis of partic-ipation factor it is acknowledged that PLL control also has agreat impact on the stability of VSC connected to weak ACsystemTherefore an improved PLL control suitable for VSCconnected to weak AC system is proposed as follows whichis the main contribution of this paper
Considering that the active power is proportional to thedirect-axis current a supplementary damping control withstate-variable 1198942119889 is added in PLL control system (119863 is thedamping factor) And the new PLL equations are shown in(7) in which ldquo+rdquo is for rectifying operation and ldquominusrdquo is forinverting operation Figure 6 shows the improvedPLL controldiagram
120596 = 1205960 + 119896pllV119902 + 119888119896pll int V119902119889119905
120579 = int [120596 plusmn 119863 (1198942119889ref minus 1198942119889)] 119889119905
(7)
The small-signal model is changed and the state-variablematrix A with the improved PLL is shown in the AppendixFigure 7 shows the root-locus of predominant poles of themodified A
From Figure 7 it can be seen that with the rising ofdamping factor119863 in a certain range the small-signal stabilityof VSC system is enhanced After a critical value the stabilityis reduced with the rising of119863 The root-locus of VSC systemwith 119875 rising in different damping factors has also verifiedthe effectiveness of the improved PLL controller Figure 7(c)shows that with a proper damping factor the relationshipbetween PLL and dominant poles is weak It also meansthat the most effective parts are changing from PLL to theouter-loop controller which is the purpose of the improvedPLL control strategy There should be an optimal value for
6 Journal of Control Science and Engineering
D
abc
dq0
d
q
0
Supplementary control
i2dref
+
+
+
+minus
minus
Δ120596
1205960
120579
+ is for rectifying operationminus is for inverting operation
1
s
a
b
c
q
plusmn
i2d
qref = 0
kpll +ckpll
s
Figure 6 Improved PLL control for VSC connected to weak AC system
Rising direction of D
0 20minus40 minus20minus60
Real axis
minus100
minus50
0
50
100
Imag
inar
y ax
is
(a) Root-locus of VSC with 119875 = 14 pu
Rising direction of D
minus100
minus60
minus20
020
60
100
Imag
inar
y ax
is
minus20 0 20 60minus60minus100
Real axis(b) Root-locus of VSC with different transfer power
D ri
sing
dire
ctio
n
0
01
02
03
04
05
06
07
Part
icip
atio
n fa
ctor
2 3 4 5 6 7 8 9 10 11 121State-variables
(c) Participation factor analysis
Figure 7 Root-locus and participation factor of system with advanced PLL control
the damping factor 119863 When impedance angle of AC systemis 85∘ the optimal value of 119863 is approximately 1100 forrectifying operation and 300 for inverting operation
5 Case Studies
51 Validation of the Small-Signal Model Table 6 shows thebase values of the system parameters of benchmark testmodel The accuracy of the small-signal model including
the oscillation frequency and damping ratio will be verifiedin this section
Figure 8 shows the simulation results of the system inwhich the impedance angle is 85∘ The active power PCCvoltage and 119863- and 119876-axis current are shown in Figures8(a)sim8(d) respectively with the active power being changingfrom 360MW to 370MW
It can be seen fromFigure 8 that themaximumerror valuebetween EMTmodel and small-signal model of active power
Journal of Control Science and Engineering 7
EMT modelSmall-signal model
Maximum error value 08
119
12
121
122
123
124
125Ac
tive p
ower
(pu)
72 73 74 75 76 7771Time (s)
(a) Active power
EMT modelSmall-signal model
Maximum error value 015
098
099
10
101
PCC
volta
ge (p
u)
71 72 73 74 75 76 777Time (s)
(b) PCC voltage
D-a
xis c
urre
nt (k
A)
EMT modelSmall-signal model
Maximum error value 027
176
178
18
182
184
186
188
71 72 73 74 75 76 777Time (s)
(c) 119863-axis current
Q-a
xis c
urre
nt (k
A)
EMT modelSmall-signal model
Maximum error value 027
066
067
068
069
07
071
072
073
074
075
71 72 73 74 75 767Time (s)
(d) 119876-axis current
Figure 8 Simulation results of VSC system (impedance angle is 85∘)
Table 6 Parameters of test model
Parameters ValueDc voltage plusmn200 kVRated active power 300MWEquivalent AC source voltage 230 kVPCC voltage 230 kVTransformer 230 kV20412 kV
is 08 The maximum value of PCC voltage is 015 Themaximum error value of 119863-axis current and 119876-axis currentboth is 027 Therefore the correctness of the VSC systemsmall-signal model is satisfactorily validated
52 Validations of Impacts of Impedance Angle on Power Tran-sfer Limits The theory on the impact caused by impedance
angle on stability of VSC connected toweakAC systemwhichis proposed in Section 3 is verified in this section
521 Rectifying Operation Figure 9 shows the maximumpower transmission of system with different impedanceangles
It can be seen from the figure that the maximum activepower is 385MW when 120593 equals 80∘ Meanwhile the maxi-mum active power is 420MWwhen 120593 equals 85∘ Simulationresults show that with the rising of impedance angle the VSCsystem operating in rectifier mode gets larger small-signalstable margin
522 Inverting Operation Figure 10 shows the maximumpower transmission of system with different impedanceangles
However the simulation results shown in Figure 10 areopposite with the system working in rectifying operation
8 Journal of Control Science and Engineering
Reference valueSimulation value
09
1
11
12
13
14
15Ac
tive p
ower
(pu)
4 6 8 102Time (s)
(a) 120593 = 80∘Reference valueSimulation value
09
1
11
12
13
14
15
Activ
e pow
er (p
u)
4 6 8 10 122Time (s)
(b) 120593 = 85∘
Figure 9 Maximum power transmission with different impedance angles (rectifying operation)
Reference valueSimulation value
5 10 15Time (s)
minus15
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(a) 120593 = 85∘Reference valueSimulation value
5 10 15Time (s)
minus1533
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(b) 120593 = 80∘
Figure 10 Maximum power transmission with different impedance angles (inverting operation)
The maximum active power is 460MW when 120593 equals 80∘and this value decreased to 450MWwhen 120593 equals 85∘ Withthe rising of impedance angle the VSC system operatingin inverter mode is getting smaller stable margin and evenbecoming unstable
Table 7 shows the power transfer limitations with differ-ent impedance angles
The conclusion drawn from Section 3 is validated byTable 7 Meanwhile it can be seen that the system operating
at rectifier mode is more severe than inverter mode whichmeans that the stable margin for VSC working in rectifiermode is narrower than that of inverter mode
53 Validations of the Proposed PLL Control Figure 11 showsthe simulation results of maximum power transmission withthe proposed improved PLL control
It can be obtained that with the proposed PLL controlstrategy the limits of power transmission are improved
Journal of Control Science and Engineering 9
5 10 15Time (s)
Reference valueSimulation value
1
12
14
16
18Ac
tive p
ower
(pu)
(a) Rectifying operationReference valueSimulation value
5 10 15 20Time (s)
minus18
minus16
minus14
minus12
minus1
Activ
e pow
er (p
u)(b) Inverting operation
Figure 11 The simulation results of system with proposed PLL control
380
400
420
440
460
Max
imum
activ
e pow
er (M
W)
500 1100 15000Damping factor (D)
(a) Rectifying operation
430
440
450
460
470
480
490
Max
imum
activ
e pow
er (M
W)
100 200 300 400 500 6000Damping factor (D)
(b) Inverting operation
Figure 12 The relationship between119863 and limits of power transmission
The test model adopts 85∘ as the AC system impedanceangle Compared with 420MW the maximum active powerat rectifier is enhanced to 460MW Meanwhile the limits ofpower transmission at inverter are enhanced from450MWto480MW The optimal value of damping factor 119863 calculatedat rectifying operation is 1100 and is 300 at inverting oper-ation Figure 12 shows the relationship between VSC powertransfer limitations and damping factors in the proposed PLLcontrol
It can be seen from Figure 12 that the optimal values ofdamping factors are approximately equal to the calculatedresults (1100 and 300 resp) Figure 13 shows the simulationresults of the relationship between system impedance angles
and power transfer limitations with119863 being equal to zero andthe optimal values (ie 1100 and 300 resp)
Note that the power transfer limitation increments areshown in Figure 13 using arrowsThe validity of the advancedPLL control strategy and the previous impedance angleanalysis results are simultaneously validated in Figure 13
6 Conclusions
In this paper based on small-signal analysis and state-space matrix the power transfer limitations of VSC linkedwith weak AC system are analyzed Besides the dynamicmodel with vector-current control is developed to evaluate
10 Journal of Control Science and Engineering
Table 7 Limits of power transmission with different 120593
the improved PLL control The following conclusions aredrawn
(i) With the rising of active power at PCC the stabilityof VSC system is getting worse and even unstableThe stable margin of VSC for rectifying operation isnarrower than inverting situation With the analysisof participation factor the most sensitive parts tothe stability margin are outer-loop control and PLLcontrol
(ii) The impedance angle of AC system has a great impacton the stable margin of VSC For rectifying operationwith high impedance angle the stability of system
is enhanced and this is not the case for invertingoperation The damping characteristic of AC systemcan change the stable margin of VSC as well
(iii) The proposed PLL control appends the dampingfactor 119863 into the existing PLL diagram to changethe damping characteristic of VSC system Withthe calculated optimal damping factors the stablemargins forVSCworking in either rectifier or invertermodes are significantly enhanced
Appendix
A
=
((((((((((((((((((((((((((((((((((((
(
minus
119877119892
119871119892
120596 0 0 minus
1
119871119892
0 0 0 0 0 minus
119864119898 sin 1205750119871119892
0
minus120596 minus
119877119892
119871119892
0 0 0 minus
1
119871119892
0 0 0 0 minus
119864119898 cos 1205750119871119892
0
0 0
1198961199011 (minus (32) 119896ppV1198890 minus 1) minus 119877119888119871119888
0
minus31198961199011119896pp11989421198890
2119871119888
minus31198961199011119896pp11989421199020
2119871119888
minus119896ip1198961199011
119871119888
0
1198961198941
119871119888
0 0 0
0 0 0
minus1198961199012 minus 119877119888
119871119888
minus
3
2
1198961199012119896pvV11988901198810119871119888
0 0 minus
3
2
1198961199012119896ivV11988901198810119871119888
0
1198961198942
119871119888
0 0
1
119862119891
0 minus
1
119862119891
0 0 120596 0 0 0 0 0 0
0
1
119862119891
0 minus
1
119862119891
minus120596 0 0 0 0 0 0 0
0 0
3
2
V1198890 0
3
2
11989421198890
3
2
11989421199020 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 minus
3
2
119896ppV1198890 minus 1 0 minus
3
2
119896pp11989421198890 minus
3
2
119896pp11989421199020 minus119896ip 0 0 0 0 0
0 0 0 minus1 minus
3
2
119896pvV11988901198810
0 0 minus
3
2
119896ivV11988901198810
0 0 0 0
0 0 0 0 0 119896pll 0 0 0 0 0 119888119896pll
0 0 0 0 0 1 0 0 0 0 0 0
))))))))))))))))))))))))))))))))))))
)
(A1)
The mathematical step in the derivation of the state-variable equations is given in this section
119875 =
3
2
(V1198891198942119889 + V1198891198942119902)
119881 = radic3
2
radicV2119889+ V2119902
(A2)
The small-signal model is shown in (A3) In particularthe quadrature component of the PCC voltage at operatingpoint is 0 (V1199020 = 0)
Figure 13 The power transfer limitations with the advanced PLL control
The relationship between 119890119889 119890119902 and 120579 is shown in (A4)in which1205960119905+1198860 is the real time angle of equivalent ac voltagesource
119890119889 = 119864119898cos [120579 minus (1205960119905 + 1205720)]
119890119902 = minus119864119898sin [120579 minus (1205960119905 + 1205720)] (A4)
The small-signal model is shown in (A5) 120575 is mentionedin nomenclature
Δ119890119889 = minus119864119898 sin 1205750Δ120579
Δ119890119902 = minus119864119898 cos 1205750Δ120579(A5)
The small-signal model of (1)sim(4) is shown below
[
minus (119864119898 sin 1205750) Δ120575minus (119864119898 cos 1205750) Δ120575
] minus [
ΔV119889ΔV119902
]
= 119877119892 [
Δ1198941119889
Δ1198941119902
] + 119871119892
119889
119889119905
[
Δ1198941119889
Δ1198941119902
] + 120596119871119892 [
minusΔ1198941119902
Δ1198941119889
]
[
ΔV119889ΔV119902
] minus [
ΔV119888119889ΔV119888119902
]
= 119877119888 [
Δ1198942119889
Δ1198942119902
] + 119871119888
119889
119889119905
[
Δ1198942119889
Δ1198942119902
] + 120596119871119888 [
minusΔ1198942119902
Δ1198942119889
]
[
Δ1198941119889
Δ1198941119902
] minus [
Δ1198942119889
Δ1198942119902
] = 119862119891
119889
119889119905
[
ΔV119889ΔV119902
] + 120596119862119891 [
minusΔV119902ΔV119889
]
minus 119896ppΔ119875 minus 119896ip intΔ119875119889119905 = Δ1198942119889ref
minus 119896pvΔ119881 minus 119896iv intΔ119881119889119905 = Δ1198942119902ref
1198961199011 (Δ1198942119889ref minus Δ1198942119889) + 1198961198941 int (Δ1198942119889ref minus Δ1198942119889) 119889119905
= 119871119888
119889Δ1198942119889
119889119905
+ 119877119888Δ1198942119889
1198961199012 (Δ1198942119902ref minus Δ1198942119902) + 1198961198942 int (Δ1198942119902ref minus Δ1198942119902) 119889119905
The state-variables Δ1199091 Δ1199092 Δ1199093 Δ1199094 and Δ1199095 areassumed below
Δ1199091 = intΔ119875119889119905
Δ1199092 = intΔV119889119889119905
Δ1199093 = int (Δ1198942119889ref minus Δ1198942119889) 119889119905
Δ1199094 = int (Δ1198942119902ref minus Δ1198942119902) 119889119905
Δ1199095 = intΔV119902119889119905
(A7)
12 Journal of Control Science and Engineering
With the improved PLL control matrix A is changedThe modified elements are shown below ldquo+rdquo is for rectifyingoperation and ldquominusrdquo is for inverting operation
A113 = plusmn119863(
3
2
119896ppV1198890 + 1)
A115 = plusmn3
2
119863119896pp11989421198890
A116 = (plusmn3
2
119863119896pp11989421199020 + 119896pll)
A117 = plusmn119863119896ip
(A8)
The participation factormentioned in Section 2 is definedas the relativity between the 119896th state-variable and theeigenvalue 120582119894 The calculation method is shown in
in which u and k are the left eigenvector and right eigenvectorseparately V119896119894 and 119906119896119894 are the corresponding elements ineigenvectors
Nomenclature
119890119886119887119888 Thevenin equivalent voltage of AC systemV119886119887119888 PCC voltageV119888119886119887119888 Converter voltage1198941119886119887119888 Phase current at the system side1198942119886119887119888 Phase current at the converter side119894119888119886119887119888 Phase current in shunt capacitor119875119881 PCC active power and voltage119875ref 119881ref The reference value of PCC active power
and voltage119890119889 119890119902 119889 and 119902 components of AC source voltageV119889 V119902 119889 and 119902 components of PCC voltageV119888119889 V119888119902 119889 and 119902 components of converter voltage1198941119889 1198941119902 119889 and 119902 components of system current1198942119889 1198942119902 119889 and 119902 components of converter current119894119888119889 119894119888119902 119889 and 119902 components of capacitor current119877119892 119871119892 Equivalent resistance and inductance of ac
system119877119888 119871119888 Resistance and inductance of converter119862119891 Shunt capacitance1198942119889ref 119889 component of converter current
reference1198942119902ref 119902 component of converter current
reference120596 Frequency of PCC voltage120579 PLL output angle120575 The angle PCC voltage V leading
equivalent source voltage E120593 Impedance angle of AC system119863 Damping factor in the improved PLL
control119880dc DC voltage
119896pp 119896ip Power loop proportional and integrator gains119896pv 119896iv Voltage loop proportional and integrator
gains1198961199011 1198961198941 Inner-loop control (119863-axis current)
proportional and integrator gains1198961199012 1198961198942 Inner-loop control (119876-axis current)
proportional and integrator gains119896pll 119888119896pll Proportional and integrator gains of PLL119901119896119894 Participation factor119864119898 Maximum instantaneous value of source
voltage
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National HighTechnology Research and Development Program of China(863 Program) (no 2013AA050105) in part by the NationalNatural Science Foundation of China (no 51177042) andin part by the Fundamental Research Funds for the CentralUniversities (no 2015XS23 and no 2015QN04)
References
[1] IEEE ldquoIEEE guide for planning dc links terminating at AClocations having low short-circuit capacitiesrdquo IEEE Standard1204-1997 1997
[2] A Gole V K Sood and L Mootoosamy ldquoValidation andanalysis of a grid control system using D-Q-Z transformationfor static compensator systemsrdquo in Proceedings of the CanadianConference on Electrical and Computer Engineering pp 745ndash758 Montreal Canada September 1989
[3] K Liao Z-Y He and B Sun ldquoSmall signal model for VSC-HVDC connected DFIG-based offshore wind farmsrdquo Journal ofAppliedMathematics vol 2014 Article ID 725209 6 pages 2014
[4] M Cespedes and J Sun ldquoAdaptive control of grid-connectedinverters based on online grid impedancemeasurementsrdquo IEEETransactions on Sustainable Energy vol 5 no 2 pp 516ndash5232014
[5] VK SoodVKhatri andH Jin ldquoPerformance assessment usingEMTP of two gate firing units for HVDC converters operatingwith weak AC systemsrdquo in Proceedings of the InternationalConference on Power Systems Transients (IPST rsquo95) pp 517ndash522Lisbon Portugal September 1995
[6] 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
[7] K M Alawasa and Y A-R I Mohamed ldquoImpedance anddamping characteristics of grid-connected VSCs with powersynchronization control strategyrdquo IEEE Transactions on PowerSystems vol 30 no 2 pp 952ndash961 2015
[8] A Egea-Alvarez S Fekriasl F Hassan andO Gomis-BellmuntldquoAdvanced vector control for voltage source converters con-nected to weak ac gridsrdquo IEEE Transactions on Power Systemsvol 30 no 6 pp 3072ndash3081 2015
Journal of Control Science and Engineering 13
[9] L Zhang L Harnefors and H-P Nee ldquoPower-synchronizationcontrol of grid-connected voltage-source convertersrdquo IEEETransactions on Power Systems vol 25 no 2 pp 809ndash820 2010
[10] J Z Zhou H Ding S Fan Y Zhang and A M Gole ldquoImpactof short-circuit ratio and phase-locked-loop parameters onthe small-signal behavior of a VSC-HVDC converterrdquo IEEETransactions on Power Delivery vol 29 no 5 pp 2287ndash22962014
[11] J A Suul S DrsquoArco P Rodriguez and M Molinas ldquoExtendedstability range of weak grids with Voltage Source Convertersthrough impedance-conditioned grid synchronizationrdquo in Pro-ceedings of the 11th International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February 2015
[12] 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
[13] T Midtsund J A Suul and T Undeland ldquoEvaluation ofcurrent controller performance and stability for voltage sourceconverters connected to a weak gridrdquo in Proceedings of the 2ndInternational Symposium on Power Electronics for DistributedGeneration Systems pp 382ndash388 IEEE Hefei China June 2010
[14] Z Miao L Fan D Osborn and S Yuvarajan ldquoWind farmswith HVdc delivery in inertial response and primary frequencycontrolrdquo IEEE Transactions on Energy Conversion vol 25 no 4pp 1171ndash1178 2010
[15] J Zhu C D Booth G P Adam A J Roscoe and C GBright ldquoInertia emulation control strategy for VSC-HVDCtransmission systemsrdquo IEEETransactions on Power Systems vol28 no 2 pp 1277ndash1287 2013
[16] H Ding S Fan J Z Zhou Y Zhang and A M GoleldquoParametric analysis of the stability of VSC-HVDC convertersrdquoin Proceedings of the 11th IET International Conference on ACand DC Power Transmission pp 1ndash6 IEEE Birmingham UKFebruary 2015
[17] B Wen D Boroyevich R Burgos P Mattavelli and Z ShenldquoAnalysis of D-Q small-signal impedance of grid-tied invertersrdquoIEEE Transactions on Power Electronics vol 31 no 1 pp 675ndash687 2016
[18] A Yazdani and R Iravani Voltage-Sourced Converters in PowerSystems Wiley Hoboken NJ USA 2010
[19] G-C Hsieh and J C Hung ldquoPhase-locked loop techniquesmdashasurveyrdquo IEEE Transactions on Industrial Electronics vol 43 no6 pp 609ndash615 1996
[20] G F Franklin J D Powell and A Emami-Naeini FeedbackControl of Dynamic Systems Prentice Hall New York NY USA2005
[21] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1993
[22] D Dong J Li D Boroyevich P Mattavelli I Cvetkovic andY Xue ldquoFrequency behavior and its stability of grid-interfaceconverter in distributed generation systemsrdquo in Proceedingsof the 27th IEEE Applied Power Electronics Conference andExposition (APEC rsquo12) pp 1887ndash1893 IEEE Orlando Fla USAFebruary 2012
[23] J Beerten S DrsquoArco and J A Suul ldquoIdentification and small-signal analysis of interaction modes in VSC MTDC systemsrdquoIEEE Transactions on Power Delivery 2015
+ is for rectifying operationminus is for inverting operation
1
s
a
b
c
q
plusmn
i2d
qref = 0
kpll +ckpll
s
Figure 6 Improved PLL control for VSC connected to weak AC system
Rising direction of D
0 20minus40 minus20minus60
Real axis
minus100
minus50
0
50
100
Imag
inar
y ax
is
(a) Root-locus of VSC with 119875 = 14 pu
Rising direction of D
minus100
minus60
minus20
020
60
100
Imag
inar
y ax
is
minus20 0 20 60minus60minus100
Real axis(b) Root-locus of VSC with different transfer power
D ri
sing
dire
ctio
n
0
01
02
03
04
05
06
07
Part
icip
atio
n fa
ctor
2 3 4 5 6 7 8 9 10 11 121State-variables
(c) Participation factor analysis
Figure 7 Root-locus and participation factor of system with advanced PLL control
the damping factor 119863 When impedance angle of AC systemis 85∘ the optimal value of 119863 is approximately 1100 forrectifying operation and 300 for inverting operation
5 Case Studies
51 Validation of the Small-Signal Model Table 6 shows thebase values of the system parameters of benchmark testmodel The accuracy of the small-signal model including
the oscillation frequency and damping ratio will be verifiedin this section
Figure 8 shows the simulation results of the system inwhich the impedance angle is 85∘ The active power PCCvoltage and 119863- and 119876-axis current are shown in Figures8(a)sim8(d) respectively with the active power being changingfrom 360MW to 370MW
It can be seen fromFigure 8 that themaximumerror valuebetween EMTmodel and small-signal model of active power
Journal of Control Science and Engineering 7
EMT modelSmall-signal model
Maximum error value 08
119
12
121
122
123
124
125Ac
tive p
ower
(pu)
72 73 74 75 76 7771Time (s)
(a) Active power
EMT modelSmall-signal model
Maximum error value 015
098
099
10
101
PCC
volta
ge (p
u)
71 72 73 74 75 76 777Time (s)
(b) PCC voltage
D-a
xis c
urre
nt (k
A)
EMT modelSmall-signal model
Maximum error value 027
176
178
18
182
184
186
188
71 72 73 74 75 76 777Time (s)
(c) 119863-axis current
Q-a
xis c
urre
nt (k
A)
EMT modelSmall-signal model
Maximum error value 027
066
067
068
069
07
071
072
073
074
075
71 72 73 74 75 767Time (s)
(d) 119876-axis current
Figure 8 Simulation results of VSC system (impedance angle is 85∘)
Table 6 Parameters of test model
Parameters ValueDc voltage plusmn200 kVRated active power 300MWEquivalent AC source voltage 230 kVPCC voltage 230 kVTransformer 230 kV20412 kV
is 08 The maximum value of PCC voltage is 015 Themaximum error value of 119863-axis current and 119876-axis currentboth is 027 Therefore the correctness of the VSC systemsmall-signal model is satisfactorily validated
52 Validations of Impacts of Impedance Angle on Power Tran-sfer Limits The theory on the impact caused by impedance
angle on stability of VSC connected toweakAC systemwhichis proposed in Section 3 is verified in this section
521 Rectifying Operation Figure 9 shows the maximumpower transmission of system with different impedanceangles
It can be seen from the figure that the maximum activepower is 385MW when 120593 equals 80∘ Meanwhile the maxi-mum active power is 420MWwhen 120593 equals 85∘ Simulationresults show that with the rising of impedance angle the VSCsystem operating in rectifier mode gets larger small-signalstable margin
522 Inverting Operation Figure 10 shows the maximumpower transmission of system with different impedanceangles
However the simulation results shown in Figure 10 areopposite with the system working in rectifying operation
8 Journal of Control Science and Engineering
Reference valueSimulation value
09
1
11
12
13
14
15Ac
tive p
ower
(pu)
4 6 8 102Time (s)
(a) 120593 = 80∘Reference valueSimulation value
09
1
11
12
13
14
15
Activ
e pow
er (p
u)
4 6 8 10 122Time (s)
(b) 120593 = 85∘
Figure 9 Maximum power transmission with different impedance angles (rectifying operation)
Reference valueSimulation value
5 10 15Time (s)
minus15
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(a) 120593 = 85∘Reference valueSimulation value
5 10 15Time (s)
minus1533
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(b) 120593 = 80∘
Figure 10 Maximum power transmission with different impedance angles (inverting operation)
The maximum active power is 460MW when 120593 equals 80∘and this value decreased to 450MWwhen 120593 equals 85∘ Withthe rising of impedance angle the VSC system operatingin inverter mode is getting smaller stable margin and evenbecoming unstable
Table 7 shows the power transfer limitations with differ-ent impedance angles
The conclusion drawn from Section 3 is validated byTable 7 Meanwhile it can be seen that the system operating
at rectifier mode is more severe than inverter mode whichmeans that the stable margin for VSC working in rectifiermode is narrower than that of inverter mode
53 Validations of the Proposed PLL Control Figure 11 showsthe simulation results of maximum power transmission withthe proposed improved PLL control
It can be obtained that with the proposed PLL controlstrategy the limits of power transmission are improved
Journal of Control Science and Engineering 9
5 10 15Time (s)
Reference valueSimulation value
1
12
14
16
18Ac
tive p
ower
(pu)
(a) Rectifying operationReference valueSimulation value
5 10 15 20Time (s)
minus18
minus16
minus14
minus12
minus1
Activ
e pow
er (p
u)(b) Inverting operation
Figure 11 The simulation results of system with proposed PLL control
380
400
420
440
460
Max
imum
activ
e pow
er (M
W)
500 1100 15000Damping factor (D)
(a) Rectifying operation
430
440
450
460
470
480
490
Max
imum
activ
e pow
er (M
W)
100 200 300 400 500 6000Damping factor (D)
(b) Inverting operation
Figure 12 The relationship between119863 and limits of power transmission
The test model adopts 85∘ as the AC system impedanceangle Compared with 420MW the maximum active powerat rectifier is enhanced to 460MW Meanwhile the limits ofpower transmission at inverter are enhanced from450MWto480MW The optimal value of damping factor 119863 calculatedat rectifying operation is 1100 and is 300 at inverting oper-ation Figure 12 shows the relationship between VSC powertransfer limitations and damping factors in the proposed PLLcontrol
It can be seen from Figure 12 that the optimal values ofdamping factors are approximately equal to the calculatedresults (1100 and 300 resp) Figure 13 shows the simulationresults of the relationship between system impedance angles
and power transfer limitations with119863 being equal to zero andthe optimal values (ie 1100 and 300 resp)
Note that the power transfer limitation increments areshown in Figure 13 using arrowsThe validity of the advancedPLL control strategy and the previous impedance angleanalysis results are simultaneously validated in Figure 13
6 Conclusions
In this paper based on small-signal analysis and state-space matrix the power transfer limitations of VSC linkedwith weak AC system are analyzed Besides the dynamicmodel with vector-current control is developed to evaluate
10 Journal of Control Science and Engineering
Table 7 Limits of power transmission with different 120593
the improved PLL control The following conclusions aredrawn
(i) With the rising of active power at PCC the stabilityof VSC system is getting worse and even unstableThe stable margin of VSC for rectifying operation isnarrower than inverting situation With the analysisof participation factor the most sensitive parts tothe stability margin are outer-loop control and PLLcontrol
(ii) The impedance angle of AC system has a great impacton the stable margin of VSC For rectifying operationwith high impedance angle the stability of system
is enhanced and this is not the case for invertingoperation The damping characteristic of AC systemcan change the stable margin of VSC as well
(iii) The proposed PLL control appends the dampingfactor 119863 into the existing PLL diagram to changethe damping characteristic of VSC system Withthe calculated optimal damping factors the stablemargins forVSCworking in either rectifier or invertermodes are significantly enhanced
Appendix
A
=
((((((((((((((((((((((((((((((((((((
(
minus
119877119892
119871119892
120596 0 0 minus
1
119871119892
0 0 0 0 0 minus
119864119898 sin 1205750119871119892
0
minus120596 minus
119877119892
119871119892
0 0 0 minus
1
119871119892
0 0 0 0 minus
119864119898 cos 1205750119871119892
0
0 0
1198961199011 (minus (32) 119896ppV1198890 minus 1) minus 119877119888119871119888
0
minus31198961199011119896pp11989421198890
2119871119888
minus31198961199011119896pp11989421199020
2119871119888
minus119896ip1198961199011
119871119888
0
1198961198941
119871119888
0 0 0
0 0 0
minus1198961199012 minus 119877119888
119871119888
minus
3
2
1198961199012119896pvV11988901198810119871119888
0 0 minus
3
2
1198961199012119896ivV11988901198810119871119888
0
1198961198942
119871119888
0 0
1
119862119891
0 minus
1
119862119891
0 0 120596 0 0 0 0 0 0
0
1
119862119891
0 minus
1
119862119891
minus120596 0 0 0 0 0 0 0
0 0
3
2
V1198890 0
3
2
11989421198890
3
2
11989421199020 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 minus
3
2
119896ppV1198890 minus 1 0 minus
3
2
119896pp11989421198890 minus
3
2
119896pp11989421199020 minus119896ip 0 0 0 0 0
0 0 0 minus1 minus
3
2
119896pvV11988901198810
0 0 minus
3
2
119896ivV11988901198810
0 0 0 0
0 0 0 0 0 119896pll 0 0 0 0 0 119888119896pll
0 0 0 0 0 1 0 0 0 0 0 0
))))))))))))))))))))))))))))))))))))
)
(A1)
The mathematical step in the derivation of the state-variable equations is given in this section
119875 =
3
2
(V1198891198942119889 + V1198891198942119902)
119881 = radic3
2
radicV2119889+ V2119902
(A2)
The small-signal model is shown in (A3) In particularthe quadrature component of the PCC voltage at operatingpoint is 0 (V1199020 = 0)
Figure 13 The power transfer limitations with the advanced PLL control
The relationship between 119890119889 119890119902 and 120579 is shown in (A4)in which1205960119905+1198860 is the real time angle of equivalent ac voltagesource
119890119889 = 119864119898cos [120579 minus (1205960119905 + 1205720)]
119890119902 = minus119864119898sin [120579 minus (1205960119905 + 1205720)] (A4)
The small-signal model is shown in (A5) 120575 is mentionedin nomenclature
Δ119890119889 = minus119864119898 sin 1205750Δ120579
Δ119890119902 = minus119864119898 cos 1205750Δ120579(A5)
The small-signal model of (1)sim(4) is shown below
[
minus (119864119898 sin 1205750) Δ120575minus (119864119898 cos 1205750) Δ120575
] minus [
ΔV119889ΔV119902
]
= 119877119892 [
Δ1198941119889
Δ1198941119902
] + 119871119892
119889
119889119905
[
Δ1198941119889
Δ1198941119902
] + 120596119871119892 [
minusΔ1198941119902
Δ1198941119889
]
[
ΔV119889ΔV119902
] minus [
ΔV119888119889ΔV119888119902
]
= 119877119888 [
Δ1198942119889
Δ1198942119902
] + 119871119888
119889
119889119905
[
Δ1198942119889
Δ1198942119902
] + 120596119871119888 [
minusΔ1198942119902
Δ1198942119889
]
[
Δ1198941119889
Δ1198941119902
] minus [
Δ1198942119889
Δ1198942119902
] = 119862119891
119889
119889119905
[
ΔV119889ΔV119902
] + 120596119862119891 [
minusΔV119902ΔV119889
]
minus 119896ppΔ119875 minus 119896ip intΔ119875119889119905 = Δ1198942119889ref
minus 119896pvΔ119881 minus 119896iv intΔ119881119889119905 = Δ1198942119902ref
1198961199011 (Δ1198942119889ref minus Δ1198942119889) + 1198961198941 int (Δ1198942119889ref minus Δ1198942119889) 119889119905
= 119871119888
119889Δ1198942119889
119889119905
+ 119877119888Δ1198942119889
1198961199012 (Δ1198942119902ref minus Δ1198942119902) + 1198961198942 int (Δ1198942119902ref minus Δ1198942119902) 119889119905
The state-variables Δ1199091 Δ1199092 Δ1199093 Δ1199094 and Δ1199095 areassumed below
Δ1199091 = intΔ119875119889119905
Δ1199092 = intΔV119889119889119905
Δ1199093 = int (Δ1198942119889ref minus Δ1198942119889) 119889119905
Δ1199094 = int (Δ1198942119902ref minus Δ1198942119902) 119889119905
Δ1199095 = intΔV119902119889119905
(A7)
12 Journal of Control Science and Engineering
With the improved PLL control matrix A is changedThe modified elements are shown below ldquo+rdquo is for rectifyingoperation and ldquominusrdquo is for inverting operation
A113 = plusmn119863(
3
2
119896ppV1198890 + 1)
A115 = plusmn3
2
119863119896pp11989421198890
A116 = (plusmn3
2
119863119896pp11989421199020 + 119896pll)
A117 = plusmn119863119896ip
(A8)
The participation factormentioned in Section 2 is definedas the relativity between the 119896th state-variable and theeigenvalue 120582119894 The calculation method is shown in
in which u and k are the left eigenvector and right eigenvectorseparately V119896119894 and 119906119896119894 are the corresponding elements ineigenvectors
Nomenclature
119890119886119887119888 Thevenin equivalent voltage of AC systemV119886119887119888 PCC voltageV119888119886119887119888 Converter voltage1198941119886119887119888 Phase current at the system side1198942119886119887119888 Phase current at the converter side119894119888119886119887119888 Phase current in shunt capacitor119875119881 PCC active power and voltage119875ref 119881ref The reference value of PCC active power
and voltage119890119889 119890119902 119889 and 119902 components of AC source voltageV119889 V119902 119889 and 119902 components of PCC voltageV119888119889 V119888119902 119889 and 119902 components of converter voltage1198941119889 1198941119902 119889 and 119902 components of system current1198942119889 1198942119902 119889 and 119902 components of converter current119894119888119889 119894119888119902 119889 and 119902 components of capacitor current119877119892 119871119892 Equivalent resistance and inductance of ac
system119877119888 119871119888 Resistance and inductance of converter119862119891 Shunt capacitance1198942119889ref 119889 component of converter current
reference1198942119902ref 119902 component of converter current
reference120596 Frequency of PCC voltage120579 PLL output angle120575 The angle PCC voltage V leading
equivalent source voltage E120593 Impedance angle of AC system119863 Damping factor in the improved PLL
control119880dc DC voltage
119896pp 119896ip Power loop proportional and integrator gains119896pv 119896iv Voltage loop proportional and integrator
gains1198961199011 1198961198941 Inner-loop control (119863-axis current)
proportional and integrator gains1198961199012 1198961198942 Inner-loop control (119876-axis current)
proportional and integrator gains119896pll 119888119896pll Proportional and integrator gains of PLL119901119896119894 Participation factor119864119898 Maximum instantaneous value of source
voltage
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National HighTechnology Research and Development Program of China(863 Program) (no 2013AA050105) in part by the NationalNatural Science Foundation of China (no 51177042) andin part by the Fundamental Research Funds for the CentralUniversities (no 2015XS23 and no 2015QN04)
References
[1] IEEE ldquoIEEE guide for planning dc links terminating at AClocations having low short-circuit capacitiesrdquo IEEE Standard1204-1997 1997
[2] A Gole V K Sood and L Mootoosamy ldquoValidation andanalysis of a grid control system using D-Q-Z transformationfor static compensator systemsrdquo in Proceedings of the CanadianConference on Electrical and Computer Engineering pp 745ndash758 Montreal Canada September 1989
[3] K Liao Z-Y He and B Sun ldquoSmall signal model for VSC-HVDC connected DFIG-based offshore wind farmsrdquo Journal ofAppliedMathematics vol 2014 Article ID 725209 6 pages 2014
[4] M Cespedes and J Sun ldquoAdaptive control of grid-connectedinverters based on online grid impedancemeasurementsrdquo IEEETransactions on Sustainable Energy vol 5 no 2 pp 516ndash5232014
[5] VK SoodVKhatri andH Jin ldquoPerformance assessment usingEMTP of two gate firing units for HVDC converters operatingwith weak AC systemsrdquo in Proceedings of the InternationalConference on Power Systems Transients (IPST rsquo95) pp 517ndash522Lisbon Portugal September 1995
[6] 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
[7] K M Alawasa and Y A-R I Mohamed ldquoImpedance anddamping characteristics of grid-connected VSCs with powersynchronization control strategyrdquo IEEE Transactions on PowerSystems vol 30 no 2 pp 952ndash961 2015
[8] A Egea-Alvarez S Fekriasl F Hassan andO Gomis-BellmuntldquoAdvanced vector control for voltage source converters con-nected to weak ac gridsrdquo IEEE Transactions on Power Systemsvol 30 no 6 pp 3072ndash3081 2015
Journal of Control Science and Engineering 13
[9] L Zhang L Harnefors and H-P Nee ldquoPower-synchronizationcontrol of grid-connected voltage-source convertersrdquo IEEETransactions on Power Systems vol 25 no 2 pp 809ndash820 2010
[10] J Z Zhou H Ding S Fan Y Zhang and A M Gole ldquoImpactof short-circuit ratio and phase-locked-loop parameters onthe small-signal behavior of a VSC-HVDC converterrdquo IEEETransactions on Power Delivery vol 29 no 5 pp 2287ndash22962014
[11] J A Suul S DrsquoArco P Rodriguez and M Molinas ldquoExtendedstability range of weak grids with Voltage Source Convertersthrough impedance-conditioned grid synchronizationrdquo in Pro-ceedings of the 11th International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February 2015
[12] 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
[13] T Midtsund J A Suul and T Undeland ldquoEvaluation ofcurrent controller performance and stability for voltage sourceconverters connected to a weak gridrdquo in Proceedings of the 2ndInternational Symposium on Power Electronics for DistributedGeneration Systems pp 382ndash388 IEEE Hefei China June 2010
[14] Z Miao L Fan D Osborn and S Yuvarajan ldquoWind farmswith HVdc delivery in inertial response and primary frequencycontrolrdquo IEEE Transactions on Energy Conversion vol 25 no 4pp 1171ndash1178 2010
[15] J Zhu C D Booth G P Adam A J Roscoe and C GBright ldquoInertia emulation control strategy for VSC-HVDCtransmission systemsrdquo IEEETransactions on Power Systems vol28 no 2 pp 1277ndash1287 2013
[16] H Ding S Fan J Z Zhou Y Zhang and A M GoleldquoParametric analysis of the stability of VSC-HVDC convertersrdquoin Proceedings of the 11th IET International Conference on ACand DC Power Transmission pp 1ndash6 IEEE Birmingham UKFebruary 2015
[17] B Wen D Boroyevich R Burgos P Mattavelli and Z ShenldquoAnalysis of D-Q small-signal impedance of grid-tied invertersrdquoIEEE Transactions on Power Electronics vol 31 no 1 pp 675ndash687 2016
[18] A Yazdani and R Iravani Voltage-Sourced Converters in PowerSystems Wiley Hoboken NJ USA 2010
[19] G-C Hsieh and J C Hung ldquoPhase-locked loop techniquesmdashasurveyrdquo IEEE Transactions on Industrial Electronics vol 43 no6 pp 609ndash615 1996
[20] G F Franklin J D Powell and A Emami-Naeini FeedbackControl of Dynamic Systems Prentice Hall New York NY USA2005
[21] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1993
[22] D Dong J Li D Boroyevich P Mattavelli I Cvetkovic andY Xue ldquoFrequency behavior and its stability of grid-interfaceconverter in distributed generation systemsrdquo in Proceedingsof the 27th IEEE Applied Power Electronics Conference andExposition (APEC rsquo12) pp 1887ndash1893 IEEE Orlando Fla USAFebruary 2012
[23] J Beerten S DrsquoArco and J A Suul ldquoIdentification and small-signal analysis of interaction modes in VSC MTDC systemsrdquoIEEE Transactions on Power Delivery 2015
Figure 8 Simulation results of VSC system (impedance angle is 85∘)
Table 6 Parameters of test model
Parameters ValueDc voltage plusmn200 kVRated active power 300MWEquivalent AC source voltage 230 kVPCC voltage 230 kVTransformer 230 kV20412 kV
is 08 The maximum value of PCC voltage is 015 Themaximum error value of 119863-axis current and 119876-axis currentboth is 027 Therefore the correctness of the VSC systemsmall-signal model is satisfactorily validated
52 Validations of Impacts of Impedance Angle on Power Tran-sfer Limits The theory on the impact caused by impedance
angle on stability of VSC connected toweakAC systemwhichis proposed in Section 3 is verified in this section
521 Rectifying Operation Figure 9 shows the maximumpower transmission of system with different impedanceangles
It can be seen from the figure that the maximum activepower is 385MW when 120593 equals 80∘ Meanwhile the maxi-mum active power is 420MWwhen 120593 equals 85∘ Simulationresults show that with the rising of impedance angle the VSCsystem operating in rectifier mode gets larger small-signalstable margin
522 Inverting Operation Figure 10 shows the maximumpower transmission of system with different impedanceangles
However the simulation results shown in Figure 10 areopposite with the system working in rectifying operation
8 Journal of Control Science and Engineering
Reference valueSimulation value
09
1
11
12
13
14
15Ac
tive p
ower
(pu)
4 6 8 102Time (s)
(a) 120593 = 80∘Reference valueSimulation value
09
1
11
12
13
14
15
Activ
e pow
er (p
u)
4 6 8 10 122Time (s)
(b) 120593 = 85∘
Figure 9 Maximum power transmission with different impedance angles (rectifying operation)
Reference valueSimulation value
5 10 15Time (s)
minus15
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(a) 120593 = 85∘Reference valueSimulation value
5 10 15Time (s)
minus1533
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(b) 120593 = 80∘
Figure 10 Maximum power transmission with different impedance angles (inverting operation)
The maximum active power is 460MW when 120593 equals 80∘and this value decreased to 450MWwhen 120593 equals 85∘ Withthe rising of impedance angle the VSC system operatingin inverter mode is getting smaller stable margin and evenbecoming unstable
Table 7 shows the power transfer limitations with differ-ent impedance angles
The conclusion drawn from Section 3 is validated byTable 7 Meanwhile it can be seen that the system operating
at rectifier mode is more severe than inverter mode whichmeans that the stable margin for VSC working in rectifiermode is narrower than that of inverter mode
53 Validations of the Proposed PLL Control Figure 11 showsthe simulation results of maximum power transmission withthe proposed improved PLL control
It can be obtained that with the proposed PLL controlstrategy the limits of power transmission are improved
Journal of Control Science and Engineering 9
5 10 15Time (s)
Reference valueSimulation value
1
12
14
16
18Ac
tive p
ower
(pu)
(a) Rectifying operationReference valueSimulation value
5 10 15 20Time (s)
minus18
minus16
minus14
minus12
minus1
Activ
e pow
er (p
u)(b) Inverting operation
Figure 11 The simulation results of system with proposed PLL control
380
400
420
440
460
Max
imum
activ
e pow
er (M
W)
500 1100 15000Damping factor (D)
(a) Rectifying operation
430
440
450
460
470
480
490
Max
imum
activ
e pow
er (M
W)
100 200 300 400 500 6000Damping factor (D)
(b) Inverting operation
Figure 12 The relationship between119863 and limits of power transmission
The test model adopts 85∘ as the AC system impedanceangle Compared with 420MW the maximum active powerat rectifier is enhanced to 460MW Meanwhile the limits ofpower transmission at inverter are enhanced from450MWto480MW The optimal value of damping factor 119863 calculatedat rectifying operation is 1100 and is 300 at inverting oper-ation Figure 12 shows the relationship between VSC powertransfer limitations and damping factors in the proposed PLLcontrol
It can be seen from Figure 12 that the optimal values ofdamping factors are approximately equal to the calculatedresults (1100 and 300 resp) Figure 13 shows the simulationresults of the relationship between system impedance angles
and power transfer limitations with119863 being equal to zero andthe optimal values (ie 1100 and 300 resp)
Note that the power transfer limitation increments areshown in Figure 13 using arrowsThe validity of the advancedPLL control strategy and the previous impedance angleanalysis results are simultaneously validated in Figure 13
6 Conclusions
In this paper based on small-signal analysis and state-space matrix the power transfer limitations of VSC linkedwith weak AC system are analyzed Besides the dynamicmodel with vector-current control is developed to evaluate
10 Journal of Control Science and Engineering
Table 7 Limits of power transmission with different 120593
the improved PLL control The following conclusions aredrawn
(i) With the rising of active power at PCC the stabilityof VSC system is getting worse and even unstableThe stable margin of VSC for rectifying operation isnarrower than inverting situation With the analysisof participation factor the most sensitive parts tothe stability margin are outer-loop control and PLLcontrol
(ii) The impedance angle of AC system has a great impacton the stable margin of VSC For rectifying operationwith high impedance angle the stability of system
is enhanced and this is not the case for invertingoperation The damping characteristic of AC systemcan change the stable margin of VSC as well
(iii) The proposed PLL control appends the dampingfactor 119863 into the existing PLL diagram to changethe damping characteristic of VSC system Withthe calculated optimal damping factors the stablemargins forVSCworking in either rectifier or invertermodes are significantly enhanced
Appendix
A
=
((((((((((((((((((((((((((((((((((((
(
minus
119877119892
119871119892
120596 0 0 minus
1
119871119892
0 0 0 0 0 minus
119864119898 sin 1205750119871119892
0
minus120596 minus
119877119892
119871119892
0 0 0 minus
1
119871119892
0 0 0 0 minus
119864119898 cos 1205750119871119892
0
0 0
1198961199011 (minus (32) 119896ppV1198890 minus 1) minus 119877119888119871119888
0
minus31198961199011119896pp11989421198890
2119871119888
minus31198961199011119896pp11989421199020
2119871119888
minus119896ip1198961199011
119871119888
0
1198961198941
119871119888
0 0 0
0 0 0
minus1198961199012 minus 119877119888
119871119888
minus
3
2
1198961199012119896pvV11988901198810119871119888
0 0 minus
3
2
1198961199012119896ivV11988901198810119871119888
0
1198961198942
119871119888
0 0
1
119862119891
0 minus
1
119862119891
0 0 120596 0 0 0 0 0 0
0
1
119862119891
0 minus
1
119862119891
minus120596 0 0 0 0 0 0 0
0 0
3
2
V1198890 0
3
2
11989421198890
3
2
11989421199020 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 minus
3
2
119896ppV1198890 minus 1 0 minus
3
2
119896pp11989421198890 minus
3
2
119896pp11989421199020 minus119896ip 0 0 0 0 0
0 0 0 minus1 minus
3
2
119896pvV11988901198810
0 0 minus
3
2
119896ivV11988901198810
0 0 0 0
0 0 0 0 0 119896pll 0 0 0 0 0 119888119896pll
0 0 0 0 0 1 0 0 0 0 0 0
))))))))))))))))))))))))))))))))))))
)
(A1)
The mathematical step in the derivation of the state-variable equations is given in this section
119875 =
3
2
(V1198891198942119889 + V1198891198942119902)
119881 = radic3
2
radicV2119889+ V2119902
(A2)
The small-signal model is shown in (A3) In particularthe quadrature component of the PCC voltage at operatingpoint is 0 (V1199020 = 0)
Figure 13 The power transfer limitations with the advanced PLL control
The relationship between 119890119889 119890119902 and 120579 is shown in (A4)in which1205960119905+1198860 is the real time angle of equivalent ac voltagesource
119890119889 = 119864119898cos [120579 minus (1205960119905 + 1205720)]
119890119902 = minus119864119898sin [120579 minus (1205960119905 + 1205720)] (A4)
The small-signal model is shown in (A5) 120575 is mentionedin nomenclature
Δ119890119889 = minus119864119898 sin 1205750Δ120579
Δ119890119902 = minus119864119898 cos 1205750Δ120579(A5)
The small-signal model of (1)sim(4) is shown below
[
minus (119864119898 sin 1205750) Δ120575minus (119864119898 cos 1205750) Δ120575
] minus [
ΔV119889ΔV119902
]
= 119877119892 [
Δ1198941119889
Δ1198941119902
] + 119871119892
119889
119889119905
[
Δ1198941119889
Δ1198941119902
] + 120596119871119892 [
minusΔ1198941119902
Δ1198941119889
]
[
ΔV119889ΔV119902
] minus [
ΔV119888119889ΔV119888119902
]
= 119877119888 [
Δ1198942119889
Δ1198942119902
] + 119871119888
119889
119889119905
[
Δ1198942119889
Δ1198942119902
] + 120596119871119888 [
minusΔ1198942119902
Δ1198942119889
]
[
Δ1198941119889
Δ1198941119902
] minus [
Δ1198942119889
Δ1198942119902
] = 119862119891
119889
119889119905
[
ΔV119889ΔV119902
] + 120596119862119891 [
minusΔV119902ΔV119889
]
minus 119896ppΔ119875 minus 119896ip intΔ119875119889119905 = Δ1198942119889ref
minus 119896pvΔ119881 minus 119896iv intΔ119881119889119905 = Δ1198942119902ref
1198961199011 (Δ1198942119889ref minus Δ1198942119889) + 1198961198941 int (Δ1198942119889ref minus Δ1198942119889) 119889119905
= 119871119888
119889Δ1198942119889
119889119905
+ 119877119888Δ1198942119889
1198961199012 (Δ1198942119902ref minus Δ1198942119902) + 1198961198942 int (Δ1198942119902ref minus Δ1198942119902) 119889119905
The state-variables Δ1199091 Δ1199092 Δ1199093 Δ1199094 and Δ1199095 areassumed below
Δ1199091 = intΔ119875119889119905
Δ1199092 = intΔV119889119889119905
Δ1199093 = int (Δ1198942119889ref minus Δ1198942119889) 119889119905
Δ1199094 = int (Δ1198942119902ref minus Δ1198942119902) 119889119905
Δ1199095 = intΔV119902119889119905
(A7)
12 Journal of Control Science and Engineering
With the improved PLL control matrix A is changedThe modified elements are shown below ldquo+rdquo is for rectifyingoperation and ldquominusrdquo is for inverting operation
A113 = plusmn119863(
3
2
119896ppV1198890 + 1)
A115 = plusmn3
2
119863119896pp11989421198890
A116 = (plusmn3
2
119863119896pp11989421199020 + 119896pll)
A117 = plusmn119863119896ip
(A8)
The participation factormentioned in Section 2 is definedas the relativity between the 119896th state-variable and theeigenvalue 120582119894 The calculation method is shown in
in which u and k are the left eigenvector and right eigenvectorseparately V119896119894 and 119906119896119894 are the corresponding elements ineigenvectors
Nomenclature
119890119886119887119888 Thevenin equivalent voltage of AC systemV119886119887119888 PCC voltageV119888119886119887119888 Converter voltage1198941119886119887119888 Phase current at the system side1198942119886119887119888 Phase current at the converter side119894119888119886119887119888 Phase current in shunt capacitor119875119881 PCC active power and voltage119875ref 119881ref The reference value of PCC active power
and voltage119890119889 119890119902 119889 and 119902 components of AC source voltageV119889 V119902 119889 and 119902 components of PCC voltageV119888119889 V119888119902 119889 and 119902 components of converter voltage1198941119889 1198941119902 119889 and 119902 components of system current1198942119889 1198942119902 119889 and 119902 components of converter current119894119888119889 119894119888119902 119889 and 119902 components of capacitor current119877119892 119871119892 Equivalent resistance and inductance of ac
system119877119888 119871119888 Resistance and inductance of converter119862119891 Shunt capacitance1198942119889ref 119889 component of converter current
reference1198942119902ref 119902 component of converter current
reference120596 Frequency of PCC voltage120579 PLL output angle120575 The angle PCC voltage V leading
equivalent source voltage E120593 Impedance angle of AC system119863 Damping factor in the improved PLL
control119880dc DC voltage
119896pp 119896ip Power loop proportional and integrator gains119896pv 119896iv Voltage loop proportional and integrator
gains1198961199011 1198961198941 Inner-loop control (119863-axis current)
proportional and integrator gains1198961199012 1198961198942 Inner-loop control (119876-axis current)
proportional and integrator gains119896pll 119888119896pll Proportional and integrator gains of PLL119901119896119894 Participation factor119864119898 Maximum instantaneous value of source
voltage
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National HighTechnology Research and Development Program of China(863 Program) (no 2013AA050105) in part by the NationalNatural Science Foundation of China (no 51177042) andin part by the Fundamental Research Funds for the CentralUniversities (no 2015XS23 and no 2015QN04)
References
[1] IEEE ldquoIEEE guide for planning dc links terminating at AClocations having low short-circuit capacitiesrdquo IEEE Standard1204-1997 1997
[2] A Gole V K Sood and L Mootoosamy ldquoValidation andanalysis of a grid control system using D-Q-Z transformationfor static compensator systemsrdquo in Proceedings of the CanadianConference on Electrical and Computer Engineering pp 745ndash758 Montreal Canada September 1989
[3] K Liao Z-Y He and B Sun ldquoSmall signal model for VSC-HVDC connected DFIG-based offshore wind farmsrdquo Journal ofAppliedMathematics vol 2014 Article ID 725209 6 pages 2014
[4] M Cespedes and J Sun ldquoAdaptive control of grid-connectedinverters based on online grid impedancemeasurementsrdquo IEEETransactions on Sustainable Energy vol 5 no 2 pp 516ndash5232014
[5] VK SoodVKhatri andH Jin ldquoPerformance assessment usingEMTP of two gate firing units for HVDC converters operatingwith weak AC systemsrdquo in Proceedings of the InternationalConference on Power Systems Transients (IPST rsquo95) pp 517ndash522Lisbon Portugal September 1995
[6] 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
[7] K M Alawasa and Y A-R I Mohamed ldquoImpedance anddamping characteristics of grid-connected VSCs with powersynchronization control strategyrdquo IEEE Transactions on PowerSystems vol 30 no 2 pp 952ndash961 2015
[8] A Egea-Alvarez S Fekriasl F Hassan andO Gomis-BellmuntldquoAdvanced vector control for voltage source converters con-nected to weak ac gridsrdquo IEEE Transactions on Power Systemsvol 30 no 6 pp 3072ndash3081 2015
Journal of Control Science and Engineering 13
[9] L Zhang L Harnefors and H-P Nee ldquoPower-synchronizationcontrol of grid-connected voltage-source convertersrdquo IEEETransactions on Power Systems vol 25 no 2 pp 809ndash820 2010
[10] J Z Zhou H Ding S Fan Y Zhang and A M Gole ldquoImpactof short-circuit ratio and phase-locked-loop parameters onthe small-signal behavior of a VSC-HVDC converterrdquo IEEETransactions on Power Delivery vol 29 no 5 pp 2287ndash22962014
[11] J A Suul S DrsquoArco P Rodriguez and M Molinas ldquoExtendedstability range of weak grids with Voltage Source Convertersthrough impedance-conditioned grid synchronizationrdquo in Pro-ceedings of the 11th International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February 2015
[12] 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
[13] T Midtsund J A Suul and T Undeland ldquoEvaluation ofcurrent controller performance and stability for voltage sourceconverters connected to a weak gridrdquo in Proceedings of the 2ndInternational Symposium on Power Electronics for DistributedGeneration Systems pp 382ndash388 IEEE Hefei China June 2010
[14] Z Miao L Fan D Osborn and S Yuvarajan ldquoWind farmswith HVdc delivery in inertial response and primary frequencycontrolrdquo IEEE Transactions on Energy Conversion vol 25 no 4pp 1171ndash1178 2010
[15] J Zhu C D Booth G P Adam A J Roscoe and C GBright ldquoInertia emulation control strategy for VSC-HVDCtransmission systemsrdquo IEEETransactions on Power Systems vol28 no 2 pp 1277ndash1287 2013
[16] H Ding S Fan J Z Zhou Y Zhang and A M GoleldquoParametric analysis of the stability of VSC-HVDC convertersrdquoin Proceedings of the 11th IET International Conference on ACand DC Power Transmission pp 1ndash6 IEEE Birmingham UKFebruary 2015
[17] B Wen D Boroyevich R Burgos P Mattavelli and Z ShenldquoAnalysis of D-Q small-signal impedance of grid-tied invertersrdquoIEEE Transactions on Power Electronics vol 31 no 1 pp 675ndash687 2016
[18] A Yazdani and R Iravani Voltage-Sourced Converters in PowerSystems Wiley Hoboken NJ USA 2010
[19] G-C Hsieh and J C Hung ldquoPhase-locked loop techniquesmdashasurveyrdquo IEEE Transactions on Industrial Electronics vol 43 no6 pp 609ndash615 1996
[20] G F Franklin J D Powell and A Emami-Naeini FeedbackControl of Dynamic Systems Prentice Hall New York NY USA2005
[21] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1993
[22] D Dong J Li D Boroyevich P Mattavelli I Cvetkovic andY Xue ldquoFrequency behavior and its stability of grid-interfaceconverter in distributed generation systemsrdquo in Proceedingsof the 27th IEEE Applied Power Electronics Conference andExposition (APEC rsquo12) pp 1887ndash1893 IEEE Orlando Fla USAFebruary 2012
[23] J Beerten S DrsquoArco and J A Suul ldquoIdentification and small-signal analysis of interaction modes in VSC MTDC systemsrdquoIEEE Transactions on Power Delivery 2015
Figure 9 Maximum power transmission with different impedance angles (rectifying operation)
Reference valueSimulation value
5 10 15Time (s)
minus15
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(a) 120593 = 85∘Reference valueSimulation value
5 10 15Time (s)
minus1533
minus16
minus15
minus14
minus13
minus12
minus11
minus1
minus09
Activ
e pow
er (p
u)
(b) 120593 = 80∘
Figure 10 Maximum power transmission with different impedance angles (inverting operation)
The maximum active power is 460MW when 120593 equals 80∘and this value decreased to 450MWwhen 120593 equals 85∘ Withthe rising of impedance angle the VSC system operatingin inverter mode is getting smaller stable margin and evenbecoming unstable
Table 7 shows the power transfer limitations with differ-ent impedance angles
The conclusion drawn from Section 3 is validated byTable 7 Meanwhile it can be seen that the system operating
at rectifier mode is more severe than inverter mode whichmeans that the stable margin for VSC working in rectifiermode is narrower than that of inverter mode
53 Validations of the Proposed PLL Control Figure 11 showsthe simulation results of maximum power transmission withthe proposed improved PLL control
It can be obtained that with the proposed PLL controlstrategy the limits of power transmission are improved
Journal of Control Science and Engineering 9
5 10 15Time (s)
Reference valueSimulation value
1
12
14
16
18Ac
tive p
ower
(pu)
(a) Rectifying operationReference valueSimulation value
5 10 15 20Time (s)
minus18
minus16
minus14
minus12
minus1
Activ
e pow
er (p
u)(b) Inverting operation
Figure 11 The simulation results of system with proposed PLL control
380
400
420
440
460
Max
imum
activ
e pow
er (M
W)
500 1100 15000Damping factor (D)
(a) Rectifying operation
430
440
450
460
470
480
490
Max
imum
activ
e pow
er (M
W)
100 200 300 400 500 6000Damping factor (D)
(b) Inverting operation
Figure 12 The relationship between119863 and limits of power transmission
The test model adopts 85∘ as the AC system impedanceangle Compared with 420MW the maximum active powerat rectifier is enhanced to 460MW Meanwhile the limits ofpower transmission at inverter are enhanced from450MWto480MW The optimal value of damping factor 119863 calculatedat rectifying operation is 1100 and is 300 at inverting oper-ation Figure 12 shows the relationship between VSC powertransfer limitations and damping factors in the proposed PLLcontrol
It can be seen from Figure 12 that the optimal values ofdamping factors are approximately equal to the calculatedresults (1100 and 300 resp) Figure 13 shows the simulationresults of the relationship between system impedance angles
and power transfer limitations with119863 being equal to zero andthe optimal values (ie 1100 and 300 resp)
Note that the power transfer limitation increments areshown in Figure 13 using arrowsThe validity of the advancedPLL control strategy and the previous impedance angleanalysis results are simultaneously validated in Figure 13
6 Conclusions
In this paper based on small-signal analysis and state-space matrix the power transfer limitations of VSC linkedwith weak AC system are analyzed Besides the dynamicmodel with vector-current control is developed to evaluate
10 Journal of Control Science and Engineering
Table 7 Limits of power transmission with different 120593
the improved PLL control The following conclusions aredrawn
(i) With the rising of active power at PCC the stabilityof VSC system is getting worse and even unstableThe stable margin of VSC for rectifying operation isnarrower than inverting situation With the analysisof participation factor the most sensitive parts tothe stability margin are outer-loop control and PLLcontrol
(ii) The impedance angle of AC system has a great impacton the stable margin of VSC For rectifying operationwith high impedance angle the stability of system
is enhanced and this is not the case for invertingoperation The damping characteristic of AC systemcan change the stable margin of VSC as well
(iii) The proposed PLL control appends the dampingfactor 119863 into the existing PLL diagram to changethe damping characteristic of VSC system Withthe calculated optimal damping factors the stablemargins forVSCworking in either rectifier or invertermodes are significantly enhanced
Appendix
A
=
((((((((((((((((((((((((((((((((((((
(
minus
119877119892
119871119892
120596 0 0 minus
1
119871119892
0 0 0 0 0 minus
119864119898 sin 1205750119871119892
0
minus120596 minus
119877119892
119871119892
0 0 0 minus
1
119871119892
0 0 0 0 minus
119864119898 cos 1205750119871119892
0
0 0
1198961199011 (minus (32) 119896ppV1198890 minus 1) minus 119877119888119871119888
0
minus31198961199011119896pp11989421198890
2119871119888
minus31198961199011119896pp11989421199020
2119871119888
minus119896ip1198961199011
119871119888
0
1198961198941
119871119888
0 0 0
0 0 0
minus1198961199012 minus 119877119888
119871119888
minus
3
2
1198961199012119896pvV11988901198810119871119888
0 0 minus
3
2
1198961199012119896ivV11988901198810119871119888
0
1198961198942
119871119888
0 0
1
119862119891
0 minus
1
119862119891
0 0 120596 0 0 0 0 0 0
0
1
119862119891
0 minus
1
119862119891
minus120596 0 0 0 0 0 0 0
0 0
3
2
V1198890 0
3
2
11989421198890
3
2
11989421199020 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 minus
3
2
119896ppV1198890 minus 1 0 minus
3
2
119896pp11989421198890 minus
3
2
119896pp11989421199020 minus119896ip 0 0 0 0 0
0 0 0 minus1 minus
3
2
119896pvV11988901198810
0 0 minus
3
2
119896ivV11988901198810
0 0 0 0
0 0 0 0 0 119896pll 0 0 0 0 0 119888119896pll
0 0 0 0 0 1 0 0 0 0 0 0
))))))))))))))))))))))))))))))))))))
)
(A1)
The mathematical step in the derivation of the state-variable equations is given in this section
119875 =
3
2
(V1198891198942119889 + V1198891198942119902)
119881 = radic3
2
radicV2119889+ V2119902
(A2)
The small-signal model is shown in (A3) In particularthe quadrature component of the PCC voltage at operatingpoint is 0 (V1199020 = 0)
Figure 13 The power transfer limitations with the advanced PLL control
The relationship between 119890119889 119890119902 and 120579 is shown in (A4)in which1205960119905+1198860 is the real time angle of equivalent ac voltagesource
119890119889 = 119864119898cos [120579 minus (1205960119905 + 1205720)]
119890119902 = minus119864119898sin [120579 minus (1205960119905 + 1205720)] (A4)
The small-signal model is shown in (A5) 120575 is mentionedin nomenclature
Δ119890119889 = minus119864119898 sin 1205750Δ120579
Δ119890119902 = minus119864119898 cos 1205750Δ120579(A5)
The small-signal model of (1)sim(4) is shown below
[
minus (119864119898 sin 1205750) Δ120575minus (119864119898 cos 1205750) Δ120575
] minus [
ΔV119889ΔV119902
]
= 119877119892 [
Δ1198941119889
Δ1198941119902
] + 119871119892
119889
119889119905
[
Δ1198941119889
Δ1198941119902
] + 120596119871119892 [
minusΔ1198941119902
Δ1198941119889
]
[
ΔV119889ΔV119902
] minus [
ΔV119888119889ΔV119888119902
]
= 119877119888 [
Δ1198942119889
Δ1198942119902
] + 119871119888
119889
119889119905
[
Δ1198942119889
Δ1198942119902
] + 120596119871119888 [
minusΔ1198942119902
Δ1198942119889
]
[
Δ1198941119889
Δ1198941119902
] minus [
Δ1198942119889
Δ1198942119902
] = 119862119891
119889
119889119905
[
ΔV119889ΔV119902
] + 120596119862119891 [
minusΔV119902ΔV119889
]
minus 119896ppΔ119875 minus 119896ip intΔ119875119889119905 = Δ1198942119889ref
minus 119896pvΔ119881 minus 119896iv intΔ119881119889119905 = Δ1198942119902ref
1198961199011 (Δ1198942119889ref minus Δ1198942119889) + 1198961198941 int (Δ1198942119889ref minus Δ1198942119889) 119889119905
= 119871119888
119889Δ1198942119889
119889119905
+ 119877119888Δ1198942119889
1198961199012 (Δ1198942119902ref minus Δ1198942119902) + 1198961198942 int (Δ1198942119902ref minus Δ1198942119902) 119889119905
The state-variables Δ1199091 Δ1199092 Δ1199093 Δ1199094 and Δ1199095 areassumed below
Δ1199091 = intΔ119875119889119905
Δ1199092 = intΔV119889119889119905
Δ1199093 = int (Δ1198942119889ref minus Δ1198942119889) 119889119905
Δ1199094 = int (Δ1198942119902ref minus Δ1198942119902) 119889119905
Δ1199095 = intΔV119902119889119905
(A7)
12 Journal of Control Science and Engineering
With the improved PLL control matrix A is changedThe modified elements are shown below ldquo+rdquo is for rectifyingoperation and ldquominusrdquo is for inverting operation
A113 = plusmn119863(
3
2
119896ppV1198890 + 1)
A115 = plusmn3
2
119863119896pp11989421198890
A116 = (plusmn3
2
119863119896pp11989421199020 + 119896pll)
A117 = plusmn119863119896ip
(A8)
The participation factormentioned in Section 2 is definedas the relativity between the 119896th state-variable and theeigenvalue 120582119894 The calculation method is shown in
in which u and k are the left eigenvector and right eigenvectorseparately V119896119894 and 119906119896119894 are the corresponding elements ineigenvectors
Nomenclature
119890119886119887119888 Thevenin equivalent voltage of AC systemV119886119887119888 PCC voltageV119888119886119887119888 Converter voltage1198941119886119887119888 Phase current at the system side1198942119886119887119888 Phase current at the converter side119894119888119886119887119888 Phase current in shunt capacitor119875119881 PCC active power and voltage119875ref 119881ref The reference value of PCC active power
and voltage119890119889 119890119902 119889 and 119902 components of AC source voltageV119889 V119902 119889 and 119902 components of PCC voltageV119888119889 V119888119902 119889 and 119902 components of converter voltage1198941119889 1198941119902 119889 and 119902 components of system current1198942119889 1198942119902 119889 and 119902 components of converter current119894119888119889 119894119888119902 119889 and 119902 components of capacitor current119877119892 119871119892 Equivalent resistance and inductance of ac
system119877119888 119871119888 Resistance and inductance of converter119862119891 Shunt capacitance1198942119889ref 119889 component of converter current
reference1198942119902ref 119902 component of converter current
reference120596 Frequency of PCC voltage120579 PLL output angle120575 The angle PCC voltage V leading
equivalent source voltage E120593 Impedance angle of AC system119863 Damping factor in the improved PLL
control119880dc DC voltage
119896pp 119896ip Power loop proportional and integrator gains119896pv 119896iv Voltage loop proportional and integrator
gains1198961199011 1198961198941 Inner-loop control (119863-axis current)
proportional and integrator gains1198961199012 1198961198942 Inner-loop control (119876-axis current)
proportional and integrator gains119896pll 119888119896pll Proportional and integrator gains of PLL119901119896119894 Participation factor119864119898 Maximum instantaneous value of source
voltage
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National HighTechnology Research and Development Program of China(863 Program) (no 2013AA050105) in part by the NationalNatural Science Foundation of China (no 51177042) andin part by the Fundamental Research Funds for the CentralUniversities (no 2015XS23 and no 2015QN04)
References
[1] IEEE ldquoIEEE guide for planning dc links terminating at AClocations having low short-circuit capacitiesrdquo IEEE Standard1204-1997 1997
[2] A Gole V K Sood and L Mootoosamy ldquoValidation andanalysis of a grid control system using D-Q-Z transformationfor static compensator systemsrdquo in Proceedings of the CanadianConference on Electrical and Computer Engineering pp 745ndash758 Montreal Canada September 1989
[3] K Liao Z-Y He and B Sun ldquoSmall signal model for VSC-HVDC connected DFIG-based offshore wind farmsrdquo Journal ofAppliedMathematics vol 2014 Article ID 725209 6 pages 2014
[4] M Cespedes and J Sun ldquoAdaptive control of grid-connectedinverters based on online grid impedancemeasurementsrdquo IEEETransactions on Sustainable Energy vol 5 no 2 pp 516ndash5232014
[5] VK SoodVKhatri andH Jin ldquoPerformance assessment usingEMTP of two gate firing units for HVDC converters operatingwith weak AC systemsrdquo in Proceedings of the InternationalConference on Power Systems Transients (IPST rsquo95) pp 517ndash522Lisbon Portugal September 1995
[6] 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
[7] K M Alawasa and Y A-R I Mohamed ldquoImpedance anddamping characteristics of grid-connected VSCs with powersynchronization control strategyrdquo IEEE Transactions on PowerSystems vol 30 no 2 pp 952ndash961 2015
[8] A Egea-Alvarez S Fekriasl F Hassan andO Gomis-BellmuntldquoAdvanced vector control for voltage source converters con-nected to weak ac gridsrdquo IEEE Transactions on Power Systemsvol 30 no 6 pp 3072ndash3081 2015
Journal of Control Science and Engineering 13
[9] L Zhang L Harnefors and H-P Nee ldquoPower-synchronizationcontrol of grid-connected voltage-source convertersrdquo IEEETransactions on Power Systems vol 25 no 2 pp 809ndash820 2010
[10] J Z Zhou H Ding S Fan Y Zhang and A M Gole ldquoImpactof short-circuit ratio and phase-locked-loop parameters onthe small-signal behavior of a VSC-HVDC converterrdquo IEEETransactions on Power Delivery vol 29 no 5 pp 2287ndash22962014
[11] J A Suul S DrsquoArco P Rodriguez and M Molinas ldquoExtendedstability range of weak grids with Voltage Source Convertersthrough impedance-conditioned grid synchronizationrdquo in Pro-ceedings of the 11th International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February 2015
[12] 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
[13] T Midtsund J A Suul and T Undeland ldquoEvaluation ofcurrent controller performance and stability for voltage sourceconverters connected to a weak gridrdquo in Proceedings of the 2ndInternational Symposium on Power Electronics for DistributedGeneration Systems pp 382ndash388 IEEE Hefei China June 2010
[14] Z Miao L Fan D Osborn and S Yuvarajan ldquoWind farmswith HVdc delivery in inertial response and primary frequencycontrolrdquo IEEE Transactions on Energy Conversion vol 25 no 4pp 1171ndash1178 2010
[15] J Zhu C D Booth G P Adam A J Roscoe and C GBright ldquoInertia emulation control strategy for VSC-HVDCtransmission systemsrdquo IEEETransactions on Power Systems vol28 no 2 pp 1277ndash1287 2013
[16] H Ding S Fan J Z Zhou Y Zhang and A M GoleldquoParametric analysis of the stability of VSC-HVDC convertersrdquoin Proceedings of the 11th IET International Conference on ACand DC Power Transmission pp 1ndash6 IEEE Birmingham UKFebruary 2015
[17] B Wen D Boroyevich R Burgos P Mattavelli and Z ShenldquoAnalysis of D-Q small-signal impedance of grid-tied invertersrdquoIEEE Transactions on Power Electronics vol 31 no 1 pp 675ndash687 2016
[18] A Yazdani and R Iravani Voltage-Sourced Converters in PowerSystems Wiley Hoboken NJ USA 2010
[19] G-C Hsieh and J C Hung ldquoPhase-locked loop techniquesmdashasurveyrdquo IEEE Transactions on Industrial Electronics vol 43 no6 pp 609ndash615 1996
[20] G F Franklin J D Powell and A Emami-Naeini FeedbackControl of Dynamic Systems Prentice Hall New York NY USA2005
[21] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1993
[22] D Dong J Li D Boroyevich P Mattavelli I Cvetkovic andY Xue ldquoFrequency behavior and its stability of grid-interfaceconverter in distributed generation systemsrdquo in Proceedingsof the 27th IEEE Applied Power Electronics Conference andExposition (APEC rsquo12) pp 1887ndash1893 IEEE Orlando Fla USAFebruary 2012
[23] J Beerten S DrsquoArco and J A Suul ldquoIdentification and small-signal analysis of interaction modes in VSC MTDC systemsrdquoIEEE Transactions on Power Delivery 2015
(a) Rectifying operationReference valueSimulation value
5 10 15 20Time (s)
minus18
minus16
minus14
minus12
minus1
Activ
e pow
er (p
u)(b) Inverting operation
Figure 11 The simulation results of system with proposed PLL control
380
400
420
440
460
Max
imum
activ
e pow
er (M
W)
500 1100 15000Damping factor (D)
(a) Rectifying operation
430
440
450
460
470
480
490
Max
imum
activ
e pow
er (M
W)
100 200 300 400 500 6000Damping factor (D)
(b) Inverting operation
Figure 12 The relationship between119863 and limits of power transmission
The test model adopts 85∘ as the AC system impedanceangle Compared with 420MW the maximum active powerat rectifier is enhanced to 460MW Meanwhile the limits ofpower transmission at inverter are enhanced from450MWto480MW The optimal value of damping factor 119863 calculatedat rectifying operation is 1100 and is 300 at inverting oper-ation Figure 12 shows the relationship between VSC powertransfer limitations and damping factors in the proposed PLLcontrol
It can be seen from Figure 12 that the optimal values ofdamping factors are approximately equal to the calculatedresults (1100 and 300 resp) Figure 13 shows the simulationresults of the relationship between system impedance angles
and power transfer limitations with119863 being equal to zero andthe optimal values (ie 1100 and 300 resp)
Note that the power transfer limitation increments areshown in Figure 13 using arrowsThe validity of the advancedPLL control strategy and the previous impedance angleanalysis results are simultaneously validated in Figure 13
6 Conclusions
In this paper based on small-signal analysis and state-space matrix the power transfer limitations of VSC linkedwith weak AC system are analyzed Besides the dynamicmodel with vector-current control is developed to evaluate
10 Journal of Control Science and Engineering
Table 7 Limits of power transmission with different 120593
the improved PLL control The following conclusions aredrawn
(i) With the rising of active power at PCC the stabilityof VSC system is getting worse and even unstableThe stable margin of VSC for rectifying operation isnarrower than inverting situation With the analysisof participation factor the most sensitive parts tothe stability margin are outer-loop control and PLLcontrol
(ii) The impedance angle of AC system has a great impacton the stable margin of VSC For rectifying operationwith high impedance angle the stability of system
is enhanced and this is not the case for invertingoperation The damping characteristic of AC systemcan change the stable margin of VSC as well
(iii) The proposed PLL control appends the dampingfactor 119863 into the existing PLL diagram to changethe damping characteristic of VSC system Withthe calculated optimal damping factors the stablemargins forVSCworking in either rectifier or invertermodes are significantly enhanced
Appendix
A
=
((((((((((((((((((((((((((((((((((((
(
minus
119877119892
119871119892
120596 0 0 minus
1
119871119892
0 0 0 0 0 minus
119864119898 sin 1205750119871119892
0
minus120596 minus
119877119892
119871119892
0 0 0 minus
1
119871119892
0 0 0 0 minus
119864119898 cos 1205750119871119892
0
0 0
1198961199011 (minus (32) 119896ppV1198890 minus 1) minus 119877119888119871119888
0
minus31198961199011119896pp11989421198890
2119871119888
minus31198961199011119896pp11989421199020
2119871119888
minus119896ip1198961199011
119871119888
0
1198961198941
119871119888
0 0 0
0 0 0
minus1198961199012 minus 119877119888
119871119888
minus
3
2
1198961199012119896pvV11988901198810119871119888
0 0 minus
3
2
1198961199012119896ivV11988901198810119871119888
0
1198961198942
119871119888
0 0
1
119862119891
0 minus
1
119862119891
0 0 120596 0 0 0 0 0 0
0
1
119862119891
0 minus
1
119862119891
minus120596 0 0 0 0 0 0 0
0 0
3
2
V1198890 0
3
2
11989421198890
3
2
11989421199020 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 minus
3
2
119896ppV1198890 minus 1 0 minus
3
2
119896pp11989421198890 minus
3
2
119896pp11989421199020 minus119896ip 0 0 0 0 0
0 0 0 minus1 minus
3
2
119896pvV11988901198810
0 0 minus
3
2
119896ivV11988901198810
0 0 0 0
0 0 0 0 0 119896pll 0 0 0 0 0 119888119896pll
0 0 0 0 0 1 0 0 0 0 0 0
))))))))))))))))))))))))))))))))))))
)
(A1)
The mathematical step in the derivation of the state-variable equations is given in this section
119875 =
3
2
(V1198891198942119889 + V1198891198942119902)
119881 = radic3
2
radicV2119889+ V2119902
(A2)
The small-signal model is shown in (A3) In particularthe quadrature component of the PCC voltage at operatingpoint is 0 (V1199020 = 0)
Figure 13 The power transfer limitations with the advanced PLL control
The relationship between 119890119889 119890119902 and 120579 is shown in (A4)in which1205960119905+1198860 is the real time angle of equivalent ac voltagesource
119890119889 = 119864119898cos [120579 minus (1205960119905 + 1205720)]
119890119902 = minus119864119898sin [120579 minus (1205960119905 + 1205720)] (A4)
The small-signal model is shown in (A5) 120575 is mentionedin nomenclature
Δ119890119889 = minus119864119898 sin 1205750Δ120579
Δ119890119902 = minus119864119898 cos 1205750Δ120579(A5)
The small-signal model of (1)sim(4) is shown below
[
minus (119864119898 sin 1205750) Δ120575minus (119864119898 cos 1205750) Δ120575
] minus [
ΔV119889ΔV119902
]
= 119877119892 [
Δ1198941119889
Δ1198941119902
] + 119871119892
119889
119889119905
[
Δ1198941119889
Δ1198941119902
] + 120596119871119892 [
minusΔ1198941119902
Δ1198941119889
]
[
ΔV119889ΔV119902
] minus [
ΔV119888119889ΔV119888119902
]
= 119877119888 [
Δ1198942119889
Δ1198942119902
] + 119871119888
119889
119889119905
[
Δ1198942119889
Δ1198942119902
] + 120596119871119888 [
minusΔ1198942119902
Δ1198942119889
]
[
Δ1198941119889
Δ1198941119902
] minus [
Δ1198942119889
Δ1198942119902
] = 119862119891
119889
119889119905
[
ΔV119889ΔV119902
] + 120596119862119891 [
minusΔV119902ΔV119889
]
minus 119896ppΔ119875 minus 119896ip intΔ119875119889119905 = Δ1198942119889ref
minus 119896pvΔ119881 minus 119896iv intΔ119881119889119905 = Δ1198942119902ref
1198961199011 (Δ1198942119889ref minus Δ1198942119889) + 1198961198941 int (Δ1198942119889ref minus Δ1198942119889) 119889119905
= 119871119888
119889Δ1198942119889
119889119905
+ 119877119888Δ1198942119889
1198961199012 (Δ1198942119902ref minus Δ1198942119902) + 1198961198942 int (Δ1198942119902ref minus Δ1198942119902) 119889119905
The state-variables Δ1199091 Δ1199092 Δ1199093 Δ1199094 and Δ1199095 areassumed below
Δ1199091 = intΔ119875119889119905
Δ1199092 = intΔV119889119889119905
Δ1199093 = int (Δ1198942119889ref minus Δ1198942119889) 119889119905
Δ1199094 = int (Δ1198942119902ref minus Δ1198942119902) 119889119905
Δ1199095 = intΔV119902119889119905
(A7)
12 Journal of Control Science and Engineering
With the improved PLL control matrix A is changedThe modified elements are shown below ldquo+rdquo is for rectifyingoperation and ldquominusrdquo is for inverting operation
A113 = plusmn119863(
3
2
119896ppV1198890 + 1)
A115 = plusmn3
2
119863119896pp11989421198890
A116 = (plusmn3
2
119863119896pp11989421199020 + 119896pll)
A117 = plusmn119863119896ip
(A8)
The participation factormentioned in Section 2 is definedas the relativity between the 119896th state-variable and theeigenvalue 120582119894 The calculation method is shown in
in which u and k are the left eigenvector and right eigenvectorseparately V119896119894 and 119906119896119894 are the corresponding elements ineigenvectors
Nomenclature
119890119886119887119888 Thevenin equivalent voltage of AC systemV119886119887119888 PCC voltageV119888119886119887119888 Converter voltage1198941119886119887119888 Phase current at the system side1198942119886119887119888 Phase current at the converter side119894119888119886119887119888 Phase current in shunt capacitor119875119881 PCC active power and voltage119875ref 119881ref The reference value of PCC active power
and voltage119890119889 119890119902 119889 and 119902 components of AC source voltageV119889 V119902 119889 and 119902 components of PCC voltageV119888119889 V119888119902 119889 and 119902 components of converter voltage1198941119889 1198941119902 119889 and 119902 components of system current1198942119889 1198942119902 119889 and 119902 components of converter current119894119888119889 119894119888119902 119889 and 119902 components of capacitor current119877119892 119871119892 Equivalent resistance and inductance of ac
system119877119888 119871119888 Resistance and inductance of converter119862119891 Shunt capacitance1198942119889ref 119889 component of converter current
reference1198942119902ref 119902 component of converter current
reference120596 Frequency of PCC voltage120579 PLL output angle120575 The angle PCC voltage V leading
equivalent source voltage E120593 Impedance angle of AC system119863 Damping factor in the improved PLL
control119880dc DC voltage
119896pp 119896ip Power loop proportional and integrator gains119896pv 119896iv Voltage loop proportional and integrator
gains1198961199011 1198961198941 Inner-loop control (119863-axis current)
proportional and integrator gains1198961199012 1198961198942 Inner-loop control (119876-axis current)
proportional and integrator gains119896pll 119888119896pll Proportional and integrator gains of PLL119901119896119894 Participation factor119864119898 Maximum instantaneous value of source
voltage
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National HighTechnology Research and Development Program of China(863 Program) (no 2013AA050105) in part by the NationalNatural Science Foundation of China (no 51177042) andin part by the Fundamental Research Funds for the CentralUniversities (no 2015XS23 and no 2015QN04)
References
[1] IEEE ldquoIEEE guide for planning dc links terminating at AClocations having low short-circuit capacitiesrdquo IEEE Standard1204-1997 1997
[2] A Gole V K Sood and L Mootoosamy ldquoValidation andanalysis of a grid control system using D-Q-Z transformationfor static compensator systemsrdquo in Proceedings of the CanadianConference on Electrical and Computer Engineering pp 745ndash758 Montreal Canada September 1989
[3] K Liao Z-Y He and B Sun ldquoSmall signal model for VSC-HVDC connected DFIG-based offshore wind farmsrdquo Journal ofAppliedMathematics vol 2014 Article ID 725209 6 pages 2014
[4] M Cespedes and J Sun ldquoAdaptive control of grid-connectedinverters based on online grid impedancemeasurementsrdquo IEEETransactions on Sustainable Energy vol 5 no 2 pp 516ndash5232014
[5] VK SoodVKhatri andH Jin ldquoPerformance assessment usingEMTP of two gate firing units for HVDC converters operatingwith weak AC systemsrdquo in Proceedings of the InternationalConference on Power Systems Transients (IPST rsquo95) pp 517ndash522Lisbon Portugal September 1995
[6] 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
[7] K M Alawasa and Y A-R I Mohamed ldquoImpedance anddamping characteristics of grid-connected VSCs with powersynchronization control strategyrdquo IEEE Transactions on PowerSystems vol 30 no 2 pp 952ndash961 2015
[8] A Egea-Alvarez S Fekriasl F Hassan andO Gomis-BellmuntldquoAdvanced vector control for voltage source converters con-nected to weak ac gridsrdquo IEEE Transactions on Power Systemsvol 30 no 6 pp 3072ndash3081 2015
Journal of Control Science and Engineering 13
[9] L Zhang L Harnefors and H-P Nee ldquoPower-synchronizationcontrol of grid-connected voltage-source convertersrdquo IEEETransactions on Power Systems vol 25 no 2 pp 809ndash820 2010
[10] J Z Zhou H Ding S Fan Y Zhang and A M Gole ldquoImpactof short-circuit ratio and phase-locked-loop parameters onthe small-signal behavior of a VSC-HVDC converterrdquo IEEETransactions on Power Delivery vol 29 no 5 pp 2287ndash22962014
[11] J A Suul S DrsquoArco P Rodriguez and M Molinas ldquoExtendedstability range of weak grids with Voltage Source Convertersthrough impedance-conditioned grid synchronizationrdquo in Pro-ceedings of the 11th International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February 2015
[12] 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
[13] T Midtsund J A Suul and T Undeland ldquoEvaluation ofcurrent controller performance and stability for voltage sourceconverters connected to a weak gridrdquo in Proceedings of the 2ndInternational Symposium on Power Electronics for DistributedGeneration Systems pp 382ndash388 IEEE Hefei China June 2010
[14] Z Miao L Fan D Osborn and S Yuvarajan ldquoWind farmswith HVdc delivery in inertial response and primary frequencycontrolrdquo IEEE Transactions on Energy Conversion vol 25 no 4pp 1171ndash1178 2010
[15] J Zhu C D Booth G P Adam A J Roscoe and C GBright ldquoInertia emulation control strategy for VSC-HVDCtransmission systemsrdquo IEEETransactions on Power Systems vol28 no 2 pp 1277ndash1287 2013
[16] H Ding S Fan J Z Zhou Y Zhang and A M GoleldquoParametric analysis of the stability of VSC-HVDC convertersrdquoin Proceedings of the 11th IET International Conference on ACand DC Power Transmission pp 1ndash6 IEEE Birmingham UKFebruary 2015
[17] B Wen D Boroyevich R Burgos P Mattavelli and Z ShenldquoAnalysis of D-Q small-signal impedance of grid-tied invertersrdquoIEEE Transactions on Power Electronics vol 31 no 1 pp 675ndash687 2016
[18] A Yazdani and R Iravani Voltage-Sourced Converters in PowerSystems Wiley Hoboken NJ USA 2010
[19] G-C Hsieh and J C Hung ldquoPhase-locked loop techniquesmdashasurveyrdquo IEEE Transactions on Industrial Electronics vol 43 no6 pp 609ndash615 1996
[20] G F Franklin J D Powell and A Emami-Naeini FeedbackControl of Dynamic Systems Prentice Hall New York NY USA2005
[21] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1993
[22] D Dong J Li D Boroyevich P Mattavelli I Cvetkovic andY Xue ldquoFrequency behavior and its stability of grid-interfaceconverter in distributed generation systemsrdquo in Proceedingsof the 27th IEEE Applied Power Electronics Conference andExposition (APEC rsquo12) pp 1887ndash1893 IEEE Orlando Fla USAFebruary 2012
[23] J Beerten S DrsquoArco and J A Suul ldquoIdentification and small-signal analysis of interaction modes in VSC MTDC systemsrdquoIEEE Transactions on Power Delivery 2015
the improved PLL control The following conclusions aredrawn
(i) With the rising of active power at PCC the stabilityof VSC system is getting worse and even unstableThe stable margin of VSC for rectifying operation isnarrower than inverting situation With the analysisof participation factor the most sensitive parts tothe stability margin are outer-loop control and PLLcontrol
(ii) The impedance angle of AC system has a great impacton the stable margin of VSC For rectifying operationwith high impedance angle the stability of system
is enhanced and this is not the case for invertingoperation The damping characteristic of AC systemcan change the stable margin of VSC as well
(iii) The proposed PLL control appends the dampingfactor 119863 into the existing PLL diagram to changethe damping characteristic of VSC system Withthe calculated optimal damping factors the stablemargins forVSCworking in either rectifier or invertermodes are significantly enhanced
Appendix
A
=
((((((((((((((((((((((((((((((((((((
(
minus
119877119892
119871119892
120596 0 0 minus
1
119871119892
0 0 0 0 0 minus
119864119898 sin 1205750119871119892
0
minus120596 minus
119877119892
119871119892
0 0 0 minus
1
119871119892
0 0 0 0 minus
119864119898 cos 1205750119871119892
0
0 0
1198961199011 (minus (32) 119896ppV1198890 minus 1) minus 119877119888119871119888
0
minus31198961199011119896pp11989421198890
2119871119888
minus31198961199011119896pp11989421199020
2119871119888
minus119896ip1198961199011
119871119888
0
1198961198941
119871119888
0 0 0
0 0 0
minus1198961199012 minus 119877119888
119871119888
minus
3
2
1198961199012119896pvV11988901198810119871119888
0 0 minus
3
2
1198961199012119896ivV11988901198810119871119888
0
1198961198942
119871119888
0 0
1
119862119891
0 minus
1
119862119891
0 0 120596 0 0 0 0 0 0
0
1
119862119891
0 minus
1
119862119891
minus120596 0 0 0 0 0 0 0
0 0
3
2
V1198890 0
3
2
11989421198890
3
2
11989421199020 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 minus
3
2
119896ppV1198890 minus 1 0 minus
3
2
119896pp11989421198890 minus
3
2
119896pp11989421199020 minus119896ip 0 0 0 0 0
0 0 0 minus1 minus
3
2
119896pvV11988901198810
0 0 minus
3
2
119896ivV11988901198810
0 0 0 0
0 0 0 0 0 119896pll 0 0 0 0 0 119888119896pll
0 0 0 0 0 1 0 0 0 0 0 0
))))))))))))))))))))))))))))))))))))
)
(A1)
The mathematical step in the derivation of the state-variable equations is given in this section
119875 =
3
2
(V1198891198942119889 + V1198891198942119902)
119881 = radic3
2
radicV2119889+ V2119902
(A2)
The small-signal model is shown in (A3) In particularthe quadrature component of the PCC voltage at operatingpoint is 0 (V1199020 = 0)
Figure 13 The power transfer limitations with the advanced PLL control
The relationship between 119890119889 119890119902 and 120579 is shown in (A4)in which1205960119905+1198860 is the real time angle of equivalent ac voltagesource
119890119889 = 119864119898cos [120579 minus (1205960119905 + 1205720)]
119890119902 = minus119864119898sin [120579 minus (1205960119905 + 1205720)] (A4)
The small-signal model is shown in (A5) 120575 is mentionedin nomenclature
Δ119890119889 = minus119864119898 sin 1205750Δ120579
Δ119890119902 = minus119864119898 cos 1205750Δ120579(A5)
The small-signal model of (1)sim(4) is shown below
[
minus (119864119898 sin 1205750) Δ120575minus (119864119898 cos 1205750) Δ120575
] minus [
ΔV119889ΔV119902
]
= 119877119892 [
Δ1198941119889
Δ1198941119902
] + 119871119892
119889
119889119905
[
Δ1198941119889
Δ1198941119902
] + 120596119871119892 [
minusΔ1198941119902
Δ1198941119889
]
[
ΔV119889ΔV119902
] minus [
ΔV119888119889ΔV119888119902
]
= 119877119888 [
Δ1198942119889
Δ1198942119902
] + 119871119888
119889
119889119905
[
Δ1198942119889
Δ1198942119902
] + 120596119871119888 [
minusΔ1198942119902
Δ1198942119889
]
[
Δ1198941119889
Δ1198941119902
] minus [
Δ1198942119889
Δ1198942119902
] = 119862119891
119889
119889119905
[
ΔV119889ΔV119902
] + 120596119862119891 [
minusΔV119902ΔV119889
]
minus 119896ppΔ119875 minus 119896ip intΔ119875119889119905 = Δ1198942119889ref
minus 119896pvΔ119881 minus 119896iv intΔ119881119889119905 = Δ1198942119902ref
1198961199011 (Δ1198942119889ref minus Δ1198942119889) + 1198961198941 int (Δ1198942119889ref minus Δ1198942119889) 119889119905
= 119871119888
119889Δ1198942119889
119889119905
+ 119877119888Δ1198942119889
1198961199012 (Δ1198942119902ref minus Δ1198942119902) + 1198961198942 int (Δ1198942119902ref minus Δ1198942119902) 119889119905
The state-variables Δ1199091 Δ1199092 Δ1199093 Δ1199094 and Δ1199095 areassumed below
Δ1199091 = intΔ119875119889119905
Δ1199092 = intΔV119889119889119905
Δ1199093 = int (Δ1198942119889ref minus Δ1198942119889) 119889119905
Δ1199094 = int (Δ1198942119902ref minus Δ1198942119902) 119889119905
Δ1199095 = intΔV119902119889119905
(A7)
12 Journal of Control Science and Engineering
With the improved PLL control matrix A is changedThe modified elements are shown below ldquo+rdquo is for rectifyingoperation and ldquominusrdquo is for inverting operation
A113 = plusmn119863(
3
2
119896ppV1198890 + 1)
A115 = plusmn3
2
119863119896pp11989421198890
A116 = (plusmn3
2
119863119896pp11989421199020 + 119896pll)
A117 = plusmn119863119896ip
(A8)
The participation factormentioned in Section 2 is definedas the relativity between the 119896th state-variable and theeigenvalue 120582119894 The calculation method is shown in
in which u and k are the left eigenvector and right eigenvectorseparately V119896119894 and 119906119896119894 are the corresponding elements ineigenvectors
Nomenclature
119890119886119887119888 Thevenin equivalent voltage of AC systemV119886119887119888 PCC voltageV119888119886119887119888 Converter voltage1198941119886119887119888 Phase current at the system side1198942119886119887119888 Phase current at the converter side119894119888119886119887119888 Phase current in shunt capacitor119875119881 PCC active power and voltage119875ref 119881ref The reference value of PCC active power
and voltage119890119889 119890119902 119889 and 119902 components of AC source voltageV119889 V119902 119889 and 119902 components of PCC voltageV119888119889 V119888119902 119889 and 119902 components of converter voltage1198941119889 1198941119902 119889 and 119902 components of system current1198942119889 1198942119902 119889 and 119902 components of converter current119894119888119889 119894119888119902 119889 and 119902 components of capacitor current119877119892 119871119892 Equivalent resistance and inductance of ac
system119877119888 119871119888 Resistance and inductance of converter119862119891 Shunt capacitance1198942119889ref 119889 component of converter current
reference1198942119902ref 119902 component of converter current
reference120596 Frequency of PCC voltage120579 PLL output angle120575 The angle PCC voltage V leading
equivalent source voltage E120593 Impedance angle of AC system119863 Damping factor in the improved PLL
control119880dc DC voltage
119896pp 119896ip Power loop proportional and integrator gains119896pv 119896iv Voltage loop proportional and integrator
gains1198961199011 1198961198941 Inner-loop control (119863-axis current)
proportional and integrator gains1198961199012 1198961198942 Inner-loop control (119876-axis current)
proportional and integrator gains119896pll 119888119896pll Proportional and integrator gains of PLL119901119896119894 Participation factor119864119898 Maximum instantaneous value of source
voltage
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National HighTechnology Research and Development Program of China(863 Program) (no 2013AA050105) in part by the NationalNatural Science Foundation of China (no 51177042) andin part by the Fundamental Research Funds for the CentralUniversities (no 2015XS23 and no 2015QN04)
References
[1] IEEE ldquoIEEE guide for planning dc links terminating at AClocations having low short-circuit capacitiesrdquo IEEE Standard1204-1997 1997
[2] A Gole V K Sood and L Mootoosamy ldquoValidation andanalysis of a grid control system using D-Q-Z transformationfor static compensator systemsrdquo in Proceedings of the CanadianConference on Electrical and Computer Engineering pp 745ndash758 Montreal Canada September 1989
[3] K Liao Z-Y He and B Sun ldquoSmall signal model for VSC-HVDC connected DFIG-based offshore wind farmsrdquo Journal ofAppliedMathematics vol 2014 Article ID 725209 6 pages 2014
[4] M Cespedes and J Sun ldquoAdaptive control of grid-connectedinverters based on online grid impedancemeasurementsrdquo IEEETransactions on Sustainable Energy vol 5 no 2 pp 516ndash5232014
[5] VK SoodVKhatri andH Jin ldquoPerformance assessment usingEMTP of two gate firing units for HVDC converters operatingwith weak AC systemsrdquo in Proceedings of the InternationalConference on Power Systems Transients (IPST rsquo95) pp 517ndash522Lisbon Portugal September 1995
[6] 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
[7] K M Alawasa and Y A-R I Mohamed ldquoImpedance anddamping characteristics of grid-connected VSCs with powersynchronization control strategyrdquo IEEE Transactions on PowerSystems vol 30 no 2 pp 952ndash961 2015
[8] A Egea-Alvarez S Fekriasl F Hassan andO Gomis-BellmuntldquoAdvanced vector control for voltage source converters con-nected to weak ac gridsrdquo IEEE Transactions on Power Systemsvol 30 no 6 pp 3072ndash3081 2015
Journal of Control Science and Engineering 13
[9] L Zhang L Harnefors and H-P Nee ldquoPower-synchronizationcontrol of grid-connected voltage-source convertersrdquo IEEETransactions on Power Systems vol 25 no 2 pp 809ndash820 2010
[10] J Z Zhou H Ding S Fan Y Zhang and A M Gole ldquoImpactof short-circuit ratio and phase-locked-loop parameters onthe small-signal behavior of a VSC-HVDC converterrdquo IEEETransactions on Power Delivery vol 29 no 5 pp 2287ndash22962014
[11] J A Suul S DrsquoArco P Rodriguez and M Molinas ldquoExtendedstability range of weak grids with Voltage Source Convertersthrough impedance-conditioned grid synchronizationrdquo in Pro-ceedings of the 11th International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February 2015
[12] 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
[13] T Midtsund J A Suul and T Undeland ldquoEvaluation ofcurrent controller performance and stability for voltage sourceconverters connected to a weak gridrdquo in Proceedings of the 2ndInternational Symposium on Power Electronics for DistributedGeneration Systems pp 382ndash388 IEEE Hefei China June 2010
[14] Z Miao L Fan D Osborn and S Yuvarajan ldquoWind farmswith HVdc delivery in inertial response and primary frequencycontrolrdquo IEEE Transactions on Energy Conversion vol 25 no 4pp 1171ndash1178 2010
[15] J Zhu C D Booth G P Adam A J Roscoe and C GBright ldquoInertia emulation control strategy for VSC-HVDCtransmission systemsrdquo IEEETransactions on Power Systems vol28 no 2 pp 1277ndash1287 2013
[16] H Ding S Fan J Z Zhou Y Zhang and A M GoleldquoParametric analysis of the stability of VSC-HVDC convertersrdquoin Proceedings of the 11th IET International Conference on ACand DC Power Transmission pp 1ndash6 IEEE Birmingham UKFebruary 2015
[17] B Wen D Boroyevich R Burgos P Mattavelli and Z ShenldquoAnalysis of D-Q small-signal impedance of grid-tied invertersrdquoIEEE Transactions on Power Electronics vol 31 no 1 pp 675ndash687 2016
[18] A Yazdani and R Iravani Voltage-Sourced Converters in PowerSystems Wiley Hoboken NJ USA 2010
[19] G-C Hsieh and J C Hung ldquoPhase-locked loop techniquesmdashasurveyrdquo IEEE Transactions on Industrial Electronics vol 43 no6 pp 609ndash615 1996
[20] G F Franklin J D Powell and A Emami-Naeini FeedbackControl of Dynamic Systems Prentice Hall New York NY USA2005
[21] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1993
[22] D Dong J Li D Boroyevich P Mattavelli I Cvetkovic andY Xue ldquoFrequency behavior and its stability of grid-interfaceconverter in distributed generation systemsrdquo in Proceedingsof the 27th IEEE Applied Power Electronics Conference andExposition (APEC rsquo12) pp 1887ndash1893 IEEE Orlando Fla USAFebruary 2012
[23] J Beerten S DrsquoArco and J A Suul ldquoIdentification and small-signal analysis of interaction modes in VSC MTDC systemsrdquoIEEE Transactions on Power Delivery 2015
Figure 13 The power transfer limitations with the advanced PLL control
The relationship between 119890119889 119890119902 and 120579 is shown in (A4)in which1205960119905+1198860 is the real time angle of equivalent ac voltagesource
119890119889 = 119864119898cos [120579 minus (1205960119905 + 1205720)]
119890119902 = minus119864119898sin [120579 minus (1205960119905 + 1205720)] (A4)
The small-signal model is shown in (A5) 120575 is mentionedin nomenclature
Δ119890119889 = minus119864119898 sin 1205750Δ120579
Δ119890119902 = minus119864119898 cos 1205750Δ120579(A5)
The small-signal model of (1)sim(4) is shown below
[
minus (119864119898 sin 1205750) Δ120575minus (119864119898 cos 1205750) Δ120575
] minus [
ΔV119889ΔV119902
]
= 119877119892 [
Δ1198941119889
Δ1198941119902
] + 119871119892
119889
119889119905
[
Δ1198941119889
Δ1198941119902
] + 120596119871119892 [
minusΔ1198941119902
Δ1198941119889
]
[
ΔV119889ΔV119902
] minus [
ΔV119888119889ΔV119888119902
]
= 119877119888 [
Δ1198942119889
Δ1198942119902
] + 119871119888
119889
119889119905
[
Δ1198942119889
Δ1198942119902
] + 120596119871119888 [
minusΔ1198942119902
Δ1198942119889
]
[
Δ1198941119889
Δ1198941119902
] minus [
Δ1198942119889
Δ1198942119902
] = 119862119891
119889
119889119905
[
ΔV119889ΔV119902
] + 120596119862119891 [
minusΔV119902ΔV119889
]
minus 119896ppΔ119875 minus 119896ip intΔ119875119889119905 = Δ1198942119889ref
minus 119896pvΔ119881 minus 119896iv intΔ119881119889119905 = Δ1198942119902ref
1198961199011 (Δ1198942119889ref minus Δ1198942119889) + 1198961198941 int (Δ1198942119889ref minus Δ1198942119889) 119889119905
= 119871119888
119889Δ1198942119889
119889119905
+ 119877119888Δ1198942119889
1198961199012 (Δ1198942119902ref minus Δ1198942119902) + 1198961198942 int (Δ1198942119902ref minus Δ1198942119902) 119889119905
The state-variables Δ1199091 Δ1199092 Δ1199093 Δ1199094 and Δ1199095 areassumed below
Δ1199091 = intΔ119875119889119905
Δ1199092 = intΔV119889119889119905
Δ1199093 = int (Δ1198942119889ref minus Δ1198942119889) 119889119905
Δ1199094 = int (Δ1198942119902ref minus Δ1198942119902) 119889119905
Δ1199095 = intΔV119902119889119905
(A7)
12 Journal of Control Science and Engineering
With the improved PLL control matrix A is changedThe modified elements are shown below ldquo+rdquo is for rectifyingoperation and ldquominusrdquo is for inverting operation
A113 = plusmn119863(
3
2
119896ppV1198890 + 1)
A115 = plusmn3
2
119863119896pp11989421198890
A116 = (plusmn3
2
119863119896pp11989421199020 + 119896pll)
A117 = plusmn119863119896ip
(A8)
The participation factormentioned in Section 2 is definedas the relativity between the 119896th state-variable and theeigenvalue 120582119894 The calculation method is shown in
in which u and k are the left eigenvector and right eigenvectorseparately V119896119894 and 119906119896119894 are the corresponding elements ineigenvectors
Nomenclature
119890119886119887119888 Thevenin equivalent voltage of AC systemV119886119887119888 PCC voltageV119888119886119887119888 Converter voltage1198941119886119887119888 Phase current at the system side1198942119886119887119888 Phase current at the converter side119894119888119886119887119888 Phase current in shunt capacitor119875119881 PCC active power and voltage119875ref 119881ref The reference value of PCC active power
and voltage119890119889 119890119902 119889 and 119902 components of AC source voltageV119889 V119902 119889 and 119902 components of PCC voltageV119888119889 V119888119902 119889 and 119902 components of converter voltage1198941119889 1198941119902 119889 and 119902 components of system current1198942119889 1198942119902 119889 and 119902 components of converter current119894119888119889 119894119888119902 119889 and 119902 components of capacitor current119877119892 119871119892 Equivalent resistance and inductance of ac
system119877119888 119871119888 Resistance and inductance of converter119862119891 Shunt capacitance1198942119889ref 119889 component of converter current
reference1198942119902ref 119902 component of converter current
reference120596 Frequency of PCC voltage120579 PLL output angle120575 The angle PCC voltage V leading
equivalent source voltage E120593 Impedance angle of AC system119863 Damping factor in the improved PLL
control119880dc DC voltage
119896pp 119896ip Power loop proportional and integrator gains119896pv 119896iv Voltage loop proportional and integrator
gains1198961199011 1198961198941 Inner-loop control (119863-axis current)
proportional and integrator gains1198961199012 1198961198942 Inner-loop control (119876-axis current)
proportional and integrator gains119896pll 119888119896pll Proportional and integrator gains of PLL119901119896119894 Participation factor119864119898 Maximum instantaneous value of source
voltage
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National HighTechnology Research and Development Program of China(863 Program) (no 2013AA050105) in part by the NationalNatural Science Foundation of China (no 51177042) andin part by the Fundamental Research Funds for the CentralUniversities (no 2015XS23 and no 2015QN04)
References
[1] IEEE ldquoIEEE guide for planning dc links terminating at AClocations having low short-circuit capacitiesrdquo IEEE Standard1204-1997 1997
[2] A Gole V K Sood and L Mootoosamy ldquoValidation andanalysis of a grid control system using D-Q-Z transformationfor static compensator systemsrdquo in Proceedings of the CanadianConference on Electrical and Computer Engineering pp 745ndash758 Montreal Canada September 1989
[3] K Liao Z-Y He and B Sun ldquoSmall signal model for VSC-HVDC connected DFIG-based offshore wind farmsrdquo Journal ofAppliedMathematics vol 2014 Article ID 725209 6 pages 2014
[4] M Cespedes and J Sun ldquoAdaptive control of grid-connectedinverters based on online grid impedancemeasurementsrdquo IEEETransactions on Sustainable Energy vol 5 no 2 pp 516ndash5232014
[5] VK SoodVKhatri andH Jin ldquoPerformance assessment usingEMTP of two gate firing units for HVDC converters operatingwith weak AC systemsrdquo in Proceedings of the InternationalConference on Power Systems Transients (IPST rsquo95) pp 517ndash522Lisbon Portugal September 1995
[6] 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
[7] K M Alawasa and Y A-R I Mohamed ldquoImpedance anddamping characteristics of grid-connected VSCs with powersynchronization control strategyrdquo IEEE Transactions on PowerSystems vol 30 no 2 pp 952ndash961 2015
[8] A Egea-Alvarez S Fekriasl F Hassan andO Gomis-BellmuntldquoAdvanced vector control for voltage source converters con-nected to weak ac gridsrdquo IEEE Transactions on Power Systemsvol 30 no 6 pp 3072ndash3081 2015
Journal of Control Science and Engineering 13
[9] L Zhang L Harnefors and H-P Nee ldquoPower-synchronizationcontrol of grid-connected voltage-source convertersrdquo IEEETransactions on Power Systems vol 25 no 2 pp 809ndash820 2010
[10] J Z Zhou H Ding S Fan Y Zhang and A M Gole ldquoImpactof short-circuit ratio and phase-locked-loop parameters onthe small-signal behavior of a VSC-HVDC converterrdquo IEEETransactions on Power Delivery vol 29 no 5 pp 2287ndash22962014
[11] J A Suul S DrsquoArco P Rodriguez and M Molinas ldquoExtendedstability range of weak grids with Voltage Source Convertersthrough impedance-conditioned grid synchronizationrdquo in Pro-ceedings of the 11th International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February 2015
[12] 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
[13] T Midtsund J A Suul and T Undeland ldquoEvaluation ofcurrent controller performance and stability for voltage sourceconverters connected to a weak gridrdquo in Proceedings of the 2ndInternational Symposium on Power Electronics for DistributedGeneration Systems pp 382ndash388 IEEE Hefei China June 2010
[14] Z Miao L Fan D Osborn and S Yuvarajan ldquoWind farmswith HVdc delivery in inertial response and primary frequencycontrolrdquo IEEE Transactions on Energy Conversion vol 25 no 4pp 1171ndash1178 2010
[15] J Zhu C D Booth G P Adam A J Roscoe and C GBright ldquoInertia emulation control strategy for VSC-HVDCtransmission systemsrdquo IEEETransactions on Power Systems vol28 no 2 pp 1277ndash1287 2013
[16] H Ding S Fan J Z Zhou Y Zhang and A M GoleldquoParametric analysis of the stability of VSC-HVDC convertersrdquoin Proceedings of the 11th IET International Conference on ACand DC Power Transmission pp 1ndash6 IEEE Birmingham UKFebruary 2015
[17] B Wen D Boroyevich R Burgos P Mattavelli and Z ShenldquoAnalysis of D-Q small-signal impedance of grid-tied invertersrdquoIEEE Transactions on Power Electronics vol 31 no 1 pp 675ndash687 2016
[18] A Yazdani and R Iravani Voltage-Sourced Converters in PowerSystems Wiley Hoboken NJ USA 2010
[19] G-C Hsieh and J C Hung ldquoPhase-locked loop techniquesmdashasurveyrdquo IEEE Transactions on Industrial Electronics vol 43 no6 pp 609ndash615 1996
[20] G F Franklin J D Powell and A Emami-Naeini FeedbackControl of Dynamic Systems Prentice Hall New York NY USA2005
[21] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1993
[22] D Dong J Li D Boroyevich P Mattavelli I Cvetkovic andY Xue ldquoFrequency behavior and its stability of grid-interfaceconverter in distributed generation systemsrdquo in Proceedingsof the 27th IEEE Applied Power Electronics Conference andExposition (APEC rsquo12) pp 1887ndash1893 IEEE Orlando Fla USAFebruary 2012
[23] J Beerten S DrsquoArco and J A Suul ldquoIdentification and small-signal analysis of interaction modes in VSC MTDC systemsrdquoIEEE Transactions on Power Delivery 2015
With the improved PLL control matrix A is changedThe modified elements are shown below ldquo+rdquo is for rectifyingoperation and ldquominusrdquo is for inverting operation
A113 = plusmn119863(
3
2
119896ppV1198890 + 1)
A115 = plusmn3
2
119863119896pp11989421198890
A116 = (plusmn3
2
119863119896pp11989421199020 + 119896pll)
A117 = plusmn119863119896ip
(A8)
The participation factormentioned in Section 2 is definedas the relativity between the 119896th state-variable and theeigenvalue 120582119894 The calculation method is shown in
in which u and k are the left eigenvector and right eigenvectorseparately V119896119894 and 119906119896119894 are the corresponding elements ineigenvectors
Nomenclature
119890119886119887119888 Thevenin equivalent voltage of AC systemV119886119887119888 PCC voltageV119888119886119887119888 Converter voltage1198941119886119887119888 Phase current at the system side1198942119886119887119888 Phase current at the converter side119894119888119886119887119888 Phase current in shunt capacitor119875119881 PCC active power and voltage119875ref 119881ref The reference value of PCC active power
and voltage119890119889 119890119902 119889 and 119902 components of AC source voltageV119889 V119902 119889 and 119902 components of PCC voltageV119888119889 V119888119902 119889 and 119902 components of converter voltage1198941119889 1198941119902 119889 and 119902 components of system current1198942119889 1198942119902 119889 and 119902 components of converter current119894119888119889 119894119888119902 119889 and 119902 components of capacitor current119877119892 119871119892 Equivalent resistance and inductance of ac
system119877119888 119871119888 Resistance and inductance of converter119862119891 Shunt capacitance1198942119889ref 119889 component of converter current
reference1198942119902ref 119902 component of converter current
reference120596 Frequency of PCC voltage120579 PLL output angle120575 The angle PCC voltage V leading
equivalent source voltage E120593 Impedance angle of AC system119863 Damping factor in the improved PLL
control119880dc DC voltage
119896pp 119896ip Power loop proportional and integrator gains119896pv 119896iv Voltage loop proportional and integrator
gains1198961199011 1198961198941 Inner-loop control (119863-axis current)
proportional and integrator gains1198961199012 1198961198942 Inner-loop control (119876-axis current)
proportional and integrator gains119896pll 119888119896pll Proportional and integrator gains of PLL119901119896119894 Participation factor119864119898 Maximum instantaneous value of source
voltage
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported in part by the National HighTechnology Research and Development Program of China(863 Program) (no 2013AA050105) in part by the NationalNatural Science Foundation of China (no 51177042) andin part by the Fundamental Research Funds for the CentralUniversities (no 2015XS23 and no 2015QN04)
References
[1] IEEE ldquoIEEE guide for planning dc links terminating at AClocations having low short-circuit capacitiesrdquo IEEE Standard1204-1997 1997
[2] A Gole V K Sood and L Mootoosamy ldquoValidation andanalysis of a grid control system using D-Q-Z transformationfor static compensator systemsrdquo in Proceedings of the CanadianConference on Electrical and Computer Engineering pp 745ndash758 Montreal Canada September 1989
[3] K Liao Z-Y He and B Sun ldquoSmall signal model for VSC-HVDC connected DFIG-based offshore wind farmsrdquo Journal ofAppliedMathematics vol 2014 Article ID 725209 6 pages 2014
[4] M Cespedes and J Sun ldquoAdaptive control of grid-connectedinverters based on online grid impedancemeasurementsrdquo IEEETransactions on Sustainable Energy vol 5 no 2 pp 516ndash5232014
[5] VK SoodVKhatri andH Jin ldquoPerformance assessment usingEMTP of two gate firing units for HVDC converters operatingwith weak AC systemsrdquo in Proceedings of the InternationalConference on Power Systems Transients (IPST rsquo95) pp 517ndash522Lisbon Portugal September 1995
[6] 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
[7] K M Alawasa and Y A-R I Mohamed ldquoImpedance anddamping characteristics of grid-connected VSCs with powersynchronization control strategyrdquo IEEE Transactions on PowerSystems vol 30 no 2 pp 952ndash961 2015
[8] A Egea-Alvarez S Fekriasl F Hassan andO Gomis-BellmuntldquoAdvanced vector control for voltage source converters con-nected to weak ac gridsrdquo IEEE Transactions on Power Systemsvol 30 no 6 pp 3072ndash3081 2015
Journal of Control Science and Engineering 13
[9] L Zhang L Harnefors and H-P Nee ldquoPower-synchronizationcontrol of grid-connected voltage-source convertersrdquo IEEETransactions on Power Systems vol 25 no 2 pp 809ndash820 2010
[10] J Z Zhou H Ding S Fan Y Zhang and A M Gole ldquoImpactof short-circuit ratio and phase-locked-loop parameters onthe small-signal behavior of a VSC-HVDC converterrdquo IEEETransactions on Power Delivery vol 29 no 5 pp 2287ndash22962014
[11] J A Suul S DrsquoArco P Rodriguez and M Molinas ldquoExtendedstability range of weak grids with Voltage Source Convertersthrough impedance-conditioned grid synchronizationrdquo in Pro-ceedings of the 11th International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February 2015
[12] 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
[13] T Midtsund J A Suul and T Undeland ldquoEvaluation ofcurrent controller performance and stability for voltage sourceconverters connected to a weak gridrdquo in Proceedings of the 2ndInternational Symposium on Power Electronics for DistributedGeneration Systems pp 382ndash388 IEEE Hefei China June 2010
[14] Z Miao L Fan D Osborn and S Yuvarajan ldquoWind farmswith HVdc delivery in inertial response and primary frequencycontrolrdquo IEEE Transactions on Energy Conversion vol 25 no 4pp 1171ndash1178 2010
[15] J Zhu C D Booth G P Adam A J Roscoe and C GBright ldquoInertia emulation control strategy for VSC-HVDCtransmission systemsrdquo IEEETransactions on Power Systems vol28 no 2 pp 1277ndash1287 2013
[16] H Ding S Fan J Z Zhou Y Zhang and A M GoleldquoParametric analysis of the stability of VSC-HVDC convertersrdquoin Proceedings of the 11th IET International Conference on ACand DC Power Transmission pp 1ndash6 IEEE Birmingham UKFebruary 2015
[17] B Wen D Boroyevich R Burgos P Mattavelli and Z ShenldquoAnalysis of D-Q small-signal impedance of grid-tied invertersrdquoIEEE Transactions on Power Electronics vol 31 no 1 pp 675ndash687 2016
[18] A Yazdani and R Iravani Voltage-Sourced Converters in PowerSystems Wiley Hoboken NJ USA 2010
[19] G-C Hsieh and J C Hung ldquoPhase-locked loop techniquesmdashasurveyrdquo IEEE Transactions on Industrial Electronics vol 43 no6 pp 609ndash615 1996
[20] G F Franklin J D Powell and A Emami-Naeini FeedbackControl of Dynamic Systems Prentice Hall New York NY USA2005
[21] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1993
[22] D Dong J Li D Boroyevich P Mattavelli I Cvetkovic andY Xue ldquoFrequency behavior and its stability of grid-interfaceconverter in distributed generation systemsrdquo in Proceedingsof the 27th IEEE Applied Power Electronics Conference andExposition (APEC rsquo12) pp 1887ndash1893 IEEE Orlando Fla USAFebruary 2012
[23] J Beerten S DrsquoArco and J A Suul ldquoIdentification and small-signal analysis of interaction modes in VSC MTDC systemsrdquoIEEE Transactions on Power Delivery 2015
[9] L Zhang L Harnefors and H-P Nee ldquoPower-synchronizationcontrol of grid-connected voltage-source convertersrdquo IEEETransactions on Power Systems vol 25 no 2 pp 809ndash820 2010
[10] J Z Zhou H Ding S Fan Y Zhang and A M Gole ldquoImpactof short-circuit ratio and phase-locked-loop parameters onthe small-signal behavior of a VSC-HVDC converterrdquo IEEETransactions on Power Delivery vol 29 no 5 pp 2287ndash22962014
[11] J A Suul S DrsquoArco P Rodriguez and M Molinas ldquoExtendedstability range of weak grids with Voltage Source Convertersthrough impedance-conditioned grid synchronizationrdquo in Pro-ceedings of the 11th International Conference on AC and DCPower Transmission pp 1ndash10 Birmingham UK February 2015
[12] 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
[13] T Midtsund J A Suul and T Undeland ldquoEvaluation ofcurrent controller performance and stability for voltage sourceconverters connected to a weak gridrdquo in Proceedings of the 2ndInternational Symposium on Power Electronics for DistributedGeneration Systems pp 382ndash388 IEEE Hefei China June 2010
[14] Z Miao L Fan D Osborn and S Yuvarajan ldquoWind farmswith HVdc delivery in inertial response and primary frequencycontrolrdquo IEEE Transactions on Energy Conversion vol 25 no 4pp 1171ndash1178 2010
[15] J Zhu C D Booth G P Adam A J Roscoe and C GBright ldquoInertia emulation control strategy for VSC-HVDCtransmission systemsrdquo IEEETransactions on Power Systems vol28 no 2 pp 1277ndash1287 2013
[16] H Ding S Fan J Z Zhou Y Zhang and A M GoleldquoParametric analysis of the stability of VSC-HVDC convertersrdquoin Proceedings of the 11th IET International Conference on ACand DC Power Transmission pp 1ndash6 IEEE Birmingham UKFebruary 2015
[17] B Wen D Boroyevich R Burgos P Mattavelli and Z ShenldquoAnalysis of D-Q small-signal impedance of grid-tied invertersrdquoIEEE Transactions on Power Electronics vol 31 no 1 pp 675ndash687 2016
[18] A Yazdani and R Iravani Voltage-Sourced Converters in PowerSystems Wiley Hoboken NJ USA 2010
[19] G-C Hsieh and J C Hung ldquoPhase-locked loop techniquesmdashasurveyrdquo IEEE Transactions on Industrial Electronics vol 43 no6 pp 609ndash615 1996
[20] G F Franklin J D Powell and A Emami-Naeini FeedbackControl of Dynamic Systems Prentice Hall New York NY USA2005
[21] P Kundur Power System Stability and Control McGraw-HillNew York NY USA 1993
[22] D Dong J Li D Boroyevich P Mattavelli I Cvetkovic andY Xue ldquoFrequency behavior and its stability of grid-interfaceconverter in distributed generation systemsrdquo in Proceedingsof the 27th IEEE Applied Power Electronics Conference andExposition (APEC rsquo12) pp 1887ndash1893 IEEE Orlando Fla USAFebruary 2012
[23] J Beerten S DrsquoArco and J A Suul ldquoIdentification and small-signal analysis of interaction modes in VSC MTDC systemsrdquoIEEE Transactions on Power Delivery 2015
