voltage stability bifurcation analysis for ac/dc systems

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 387167 9 pageshttpdxdoiorg1011552013387167

Research ArticleVoltage Stability Bifurcation Analysis forACDC Systems with VSC-HVDC

Yanfang Wei1 Zheng Zheng1 Yonghui Sun2 Zhinong Wei2 and Guoqiang Sun2

1 School of Electrical Engineering and Automation Henan Polytechnic University Jiaozuo 454000 China2 College of Energy and Electrical Engineering Hohai University Nanjing 210098 China

Correspondence should be addressed to Yanfang Wei weiyanfanghpueducn

Received 24 January 2013 Accepted 1 March 2013

Academic Editor Chuangxia Huang

Copyright copy 2013 Yanfang Wei et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A voltage stability bifurcation analysis approach for modeling ACDC systems with VSC-HVDC is presented The steady powermodel and control modes of VSC-HVDC are briefly presented firstly Based on the steady model of VSC-HVDC a new improvedsequential iterative power flow algorithm is proposed Then by use of continuation power flow algorithm with the new sequentialmethod the voltage stability bifurcation of the system is discussed The trace of the P-V curves and the computation of thesaddle node bifurcation point of the system can be obtained At last the modified IEEE test systems are adopted to illustrate theeffectiveness of the proposed method

1 Introduction

As one of the key technologies of large scale access ofdistributed energy resources theHVDC transmission systemhas great potential for further development [1 2] Thereforein the past decades the problem associated with HVDCconverters connected to weak AC networks has become animportant research field The one of particular interest andhighest in consequences is the AC voltage stability at theHVDC terminals of the ACDC systems [3 4]

Voltage source converter-based HVDC (VSC-HVDC) isa new generation technology of HVDC which overcomessome of the disadvantages of the traditional thyristor-basedHVDC system with a very broad application prospect Com-pared to the conventional HVDC systems the prominentfeatures of the VSC-HVDC system are its potential to beconnected to weak AC systems independent control of activeand reactive power exchange and so on [4ndash6] Due tothose characteristics many researches have been done forthe exploitation of VSC-HVDC to enhance system stabilityof ACDC systems that is the improvement of transientstability [7 8] the power oscillations damping [9 10] theimprovement of stability and power quality for wind farm

based on VSC-HVDC grid-connected [11 12] the stabilityanalysis of multi-infeed DC systems with VSC-HVDC [13]and the keeping voltage stable [3 14ndash17]

In [14 15] the voltage stability analyses of ACDC systemswith VSC-HVDCwere mainly based on simulation softwareand the analysis based on power flow calculation was slightlyinadequate The power flow calculation of ACDC systems isthe premise and foundation of static security analysis tran-sient stability voltage stability small signal stability analysisand so on [18ndash21] At present there are two main types ofpower flow algorithms for ACDC systems sequential itera-tive method [21 22] and integrated iterative method [23 24]The computational practice indicates that the convergence ofintegratedmethod is good but the inheritance of the programis relatively poor and the writing of program code needs hugework The sequential iterative method has better programinheritance for pure AC program but its convergence is notgood In view of these shortcomings a modified sequentialiterative power flow algorithm is proposed in this paper

In [16] the continuation power flow (CPF) algorithmwas presented to solve available transfer capability problembut the saddle node bifurcation point was not discussed In[17] the PV and QV curves were used to investigate voltage

2 Abstract and Applied Analysis

DCnetwork

ACsystem

119894

119877119894 119883119897119894

119883f119894

119868d119894

119880d119894

I119880s119894ang120579s119894

119875s119894 119876s119894

119875c119894 119876c119894

119880c119894ang120579c119894

Figure 1 Simplified circuit diagram of single-phase VSC-HVDC

stability of a weak two area AC networkThoughmanymeth-ods for voltage stability analysis of ACDC systemswithVSC-HVDC have been proposed in different ways few paperscarefully consider the influence induced by different controlmodes ofVSC-HVDCwhich actually plays an important roleon the voltage stability of the ACDC systems Motivated bythe previous discussions our main aim in this paper is toinvestigate the problem of power flow and voltage stabilitybifurcation for ACDC systems with VSC-HVDC

The rest of the paper is organized as follows In Section 2the steady model and control modes of ACDC systemswith VSC-HVDC are presented In Section 3 the improvedsequential iterative algorithm the parameterization powerflow and converter equations of VSC-HVDC and the CPFstrategy are presented In Section 4 the model and methodare applied to the modified IEEE 14- and IEEE 118-bus testsystems with VSC-HVDC Finally in Section 5 the paper iscompleted with a conclusion

2 Steady Model of VSC-HVDC

21 Power Flow Equations of VSC-HVDC Figure 1 showsthe single-line representation of two-terminal VSC-HVDCsystem In Figure 1 119894 is the number of VSC converters I

119894is

the current flowing through transformer Us119894 = 119880s119894ang120579s119894 is theAC side voltage vector of VSC converter Uc119894 = 119880c119894ang120579c119894 is theoutput voltage vector of VSC converter 119877

119894is the equivalent

resistance of internal loss and converter transformer loss ofVSC 119895119883

119897119894is the impedance of converter transformer 119875s119894

and 119876s119894 are the power of AC system injected into convertertransformer 119895119883

119891is the impedance of AC filter Id andUd are

the DC current vector andDC voltage vector respectively119875c119894and119876c119894 are the power flowing through converter bridge119875d119894 isthe DC power From Figure 1 I

119894can be expressed as follows

I119894=Us119894 minus Uc119894119877119894+ 119895119883119897119894

(1)

The complex power 119878s119894 of AC system is given by

119878s119894 = 119875s119894 + 119895119876s119894 = Us119894Ilowast

119894 (2)

To facilitate discussion assume that 120575119894= 120579s119894 minus 120579c119894 |119884119894| =

1radic1198772

119894+ 1198832

119897119894 120572119894= arccot(119883

119897119894119877119894) and substituting (1) into (2)

yields

119875119904119894= minus

10038161003816100381610038161198841198941003816100381610038161003816 119880s119894119880c119894 cos (120575119894 + 120572119894) +

10038161003816100381610038161198841198941003816100381610038161003816 1198802

s119894 cos120572119894

119876119904119894= minus

10038161003816100381610038161198841198941003816100381610038161003816 119880s119894119880c119894 sin (120575119894 + 120572119894) +

10038161003816100381610038161198841198941003816100381610038161003816 1198802

s119894 sin120572119894 +1198802

s119894119883f119894

(3)

Similarly one has

119875c119894 =10038161003816100381610038161198841198941003816100381610038161003816 119880s119894119880c119894 cos (120575119894 minus 120572119894) minus

10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 cos120572119894

119876c119894 = minus10038161003816100381610038161198841198941003816100381610038161003816 119880s119894119880c119894 sin (120575119894 minus 120572119894) minus

10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 sin120572119894(4)

Since the loss of converter bridge-arm is equivalent by 119877119894 119875d119894

is qual to 119875c119894 and thus

119875d119894 = 119880d119894119868d119894 =10038161003816100381610038161198841198941003816100381610038161003816 119880s119894119880c119894 cos (120575119894 minus 120572119894) minus

10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 cos120572119894 (5)

And 119880c119894 can be described as

119880c119894 =radic6119872119894119880d119894

4 (6)

where 119872 (0 lt 119872 lt 1) is defined as the PWMrsquos amplitudemodulation index

The steadymodel of VSC-HVDC is given by (1)ndash(6) in theper-unit system (pu)

22 Steady-State Control Modes of VSC-HVDC Several reg-ular control modes for each VSC converter are chiefly asfollows

(1) constant DC voltage control constant AC reactivepower control

(2) constant DC voltage control constant AC voltagecontrol

(3) constant AC active power control constant AC reac-tive power control

(4) constant AC active power control constant AC volt-age control

Abstract and Applied Analysis 3

3 Voltage Stability Model for ACDC Systemswith VSC-HVDC

31 The Modified Power Flow Algorithm Based on SequentialIteration Method For the previous model of ACDC systemwith VSC-HVDC the simplified power flow equations aregiven by

fac = 0

fac-dc = 0

fdc = 0

(7)

where fac = [Δ119875a1 Δ119876a1 Δ119875a119899AC Δ119876a119899AC]T fac-dc = [Δ119875t1

Δ119876t1 Δ119875t119899VSC Δ119876t119899VSC ]T fdc = [Δ119889

11 Δ11988912 Δ11988913 Δ11988914

Δ119889119899VSC1

Δ119889119899VSC2

Δ119889119899VSC3

Δ119889119899VSC4

]T

To expand the Taylor series of (7) the second-orderitem and higher-order terms are omitted and the modifiedequation based on Newton-Raphson is given by

fN = minus JNΔxN (8)

where fN = [fTac fTac-dc f

Tdc]

T ΔxN = [Δ119909Tac1 Δ119909

Tac2

Δ119909Tac-dc Δ119909

Tdc]

T Δxac1 = [Δ119880a1 Δ120579a1 Δ119880a119899AC Δ120579a119899AC]T

Δxac2 = [Δ119880t1 Δ120579t1 Δ119880t119899VSC Δ120579t119899VSC ]T Δxac-dc = [Δ119875t1

Δ119876t1 Δ119875t119899VSC Δ119876t119899VSC ]T Δxdc = [Δ119880d1 Δ119868d1 Δ1205751 Δ1198721

Δ119880d119899VSC Δ119868d119899VSC Δ120575119899VSC Δ119872119899VSC ]T

And JN is given by

JN (xac1 xac2 xac-dc xdc)

=

[[[[[[[[

[

120597fac120597xac1

120597fac120597xac2

120597fac120597xac-dc

120597fac120597xdc

120597fac-dc120597xac1

120597fac-dc120597xac2

120597fac-dc120597xac-dc

120597fac-dc120597xdc

120597fdc120597xac1

120597fdc120597xac2

120597fdc120597xac-dc

120597fdc120597xdc

]]]]]]]]

]

= [

[

Ja-a1 Ja-a2 0 0Jad-a1 Jad-a2 Jad-ad 0Jd-a1 Jd-a2 Jd-ad Jd-d

]

]

(9)

The power flow equation of VSC-HVDC is given as follows

[f1

f2

] = minus [J11

J12

J21

J22

] [Δx1

Δx2

] (10)

where f1= fac f2 = [fTac-dc f

Tdc]

T Δx1= Δxac1 Δx2 =

[ΔxTac2 ΔxTac-dc Δx

Tdc]

T J11

= Ja-a1 J21 = [Jad-a1Jd-a1 ] J12 =

[Ja-a2 0 0] J22 = [Jad-a2 Jad-ad 0Jd-a2 Jd-ad Jd-d ]

The number of the power flow equations for (10) is 2(119899 minus1) + 4119899VSC and the variables number is 2(119899 minus 1) + 6119899VSCThe 2119899VSC variables can be eliminated by control modes ofVSC-HVDC So (10) has solutions The dimensions of f

1are

2(119899AC minus 1) The dimensions of Δx1are 2(119899AC minus 1) And the

inverse matrix of J11is existsThe dimensions of f

2are 6119899VSC

The dimensions of Δx2are 8119899VSC and the 2119899VSC dimensions

of Δx2can be eliminated by control modes of VSC-HVDC

and so the inverse matrix of J22exists

To expand (10)

minus (J11Δx1+ J12Δx2) = f1


(11)

Then the new modified sequential iterative form is givenby

Δx1= minus[J11minus J12(J22)minus1J21]minus1

[f1minus J12(J22)minus1f2]



(12)

It can be seen from the previous derivation process thatthe modified algorithm does not make any hypothesis Themutual influence between the AC system and DC system isfully considered in the iteration solution procedure Bymeansof the previousmethod the problemofAC variables couplingDC variables is solved strictly in the mathematics expressionThematrix J

11and J12can be obtained by the program of pure

AC system

32 Parameter-Dependent Power Flow and Converter Equa-tions of VSC-HVDC According to connected or not con-nected with a converter transformer the buses of ACDCsystems with VSC-HVDC are divided into two kinds DC busand pure AC bus [23] The bus connected to primary side ofa converter transformer is considered as a DC bus The busnot connected to a converter transformer is considered as apure AC bus 119899 is the total bus number of the system 119899VSC isthe number of VSC converters and also is the number of DCbuses So the number of pure AC buses is 119899AC = 119899 minus 119899VSC

Considering the load changes in several areas or aparticular area (at a bus andor at a group of buses) of theACDC systems with VSC-HVDC the power flow equationsfor a pure AC bus are given by

Δ119875a119894 = 119875a119894 minus 119880a119894sum119895isin119894

119880119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)

+ (119875G119894 minus 119875L119894) 120582 = 0


119880119895(119866119894119895sin 120579119894119895minus 119861119894119895cos 120579119894119895)

+ (119876G119894 minus 119876L119894) 120582 = 0

(13)

where the subscript ldquoardquo identifies that the bus is a pure ACbus a = 1 2 119899AC The subscript ldquo119894rdquo is the number ofthe buses 119894 = 1 2 119899 The subscript ldquo119895rdquo identifies that allthe buses connected to the bus ldquo119894rdquo (expressed in the termsof 119895 isin 119894) 119880 and 120579 are the bus voltage amplitude and phaseangle respectively 119866 and 119861 are the real part and imaginarypart of nodal admittancematrix respectively 119875G119894 and119876G119894 arethe power of generator 119875L119894 and 119876L119894 are the loads at the bus 119894120582 isin 119877 are the parameters such realreactive power demand atthe buses and transmission line parameters


For a DC bus the power flow equations are given by

Δ119875t119894 = 119875t119894 minus 119880t119894sum119895isin119894

119880119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895) plusmn 119875t119894

+ (119875G119894 minus 119875L119894) 120582 = 0


119880119895(119866119894119895sin 120579119894119895minus 119861119894119895cos 120579119894119895) plusmn 119876t119894

+ (119876G119894 minus 119876L119894) 120582 = 0

(14)

where the subscript ldquotrdquo identifies the bus as a DC bus ldquoplusmnrdquosigns correspond to the rectifiers and inverters of VSC-HVDC respectively

The basic power flow equations for VSC-HVDC convert-ers are given as follows

Δ1198891198961= 119875t119896 +

radic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 + 120572119896)

minus 1198802

t119896 |119884| cos120572119896 = 0

Δ1198891198962= 119876t119896 +

radic6

4119872119896119880t119896119880d119896 |119884| sin (120575119896 + 120572119896) minus 119880

2

t119896 |119884| sin120572119896

minus1198802

t119896119883f119896

= 0

Δ1198891198963= 119880t119896119868d119896 minus

radic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 minus 120572119896)

+3

8(119872119896119880d119896)2|119884| cos120572119896 = 0

(15)

where the subscript ldquodrdquo identifies that the variable as the DCside of VSC ldquo119896rdquo is the 119896th of VSC connected to DC network119896 = 1 2 119899VSC

For the 119894th VSC converter there are four unknownvariables in (15) that is xd119894 = [119880d119894 119868d119894 120575119894119872119894]

T and onemoreequation is needed to solve (15) that is

Id = GdUd (16)

where Gd is the nodal admittance matrix of DC networkMoreover the DC network equation is given by

Δ1198891198964= plusmn 119868d119896 minus

119899AC

sum

119904=1

119892d119896119904119880d119904 = 0 (17)

where 119892d119896 is the matrix element of nodal admittance matrixof DC network 119904 = 1 2 119899AC

In this paper the concerned parameters are the real andreactive loads changes at the buses that can vary according tothe following equations

119875119894= 1198751198940 (1 + 120582)

119876119894= 1198761198940 (1 + 120582)

(18)

where 1198751198940and 119876

1198940are the initial active and reactive loads

respectively 119875119894and 119876

119894are the active and reactive loads at bus

119894 respectively

For the previously ACDC systems with VSC-HVDC thesimplified power flow equation with parameter 120582-dependentis given by

f (x 120582) = 0 (19)

where f isin 1198772(119899minus1)+4119899VSC+1 x isin 1198772(119899minus1)+4119899VSC+1 f is the balanceequation of power flow x is the system state variable such busvoltage magnitude and phase angles DC system variables 119899

1

and 1198992are the number of PQ and PV AC buses The number

of power flow equations for (19) is 2(119899 minus 1) + 4119899VSC + 1 =21198991+ 1198992+ 4119899VSC + 1

33 Continuation Power Flow Algorithm Based on SequentialIteration Method CPF is a powerful tool to numericallygenerate P-V curve to trace power system stationary behaviordue to a set of power injection variations [25 26] It usespredictor-corrector scheme to find a solution path of a set ofpower flow equations that have been reformulated to includea load parameter (119909

119897 120582119897)T is the initial state of the power

flow solution curve where the subscript ldquo119897rdquo is the iterationnumber of predictor-corrector scheme based on Newton-Raphson algorithm in CPF

331 The Predictor Step The predictor step is a stage in first-order differential form Once a base solution has been found(120582 = 0) a prediction of the next solution can be made bytaking an appropriately sized step in a direction tangent tothe solution path Taking the derivatives of (19) will result inthe following total differential form equation

[1198911015840

1199091198911015840

120582] [119889119909

119889120582

] = 0 (20)

where 1198911015840119909= 120597119891120597119909 is the Jacobian matrix of power flow

equation with 119909 1198911015840120582= 120597119891120597120582 is the partial derivative of power

flow equation with 120582 [119889119909119889120582]T is the tangent vector which is

the computation target of the predictor stepSince the insertion of 120582 in the power flow equation added

an unknown variable one more equation is needed to solvethe previously equation This is satisfied by setting one of thecomponents of the tangent vector to +1 orminus1This componentis referred to as the continuation parameter Equation (20)now becomes [23]

[1198911015840

1199091198911015840

120582

119890119870

] [119889119909

119889120582

] = [0

plusmn1] (21)

where 119890119870

is a row vector with all elements equal to zeroexcept for the 119870th element being equal 1 The dimensionsof 119890119870

are 1 times [2(119899 minus 1) + 6119899VSC + 1] The introductionof the additional equation makes the Jacobian matrix non-singular at the critical operation pointThemodified equationof Newton-Raphson algorithm for (21) is

Δf = minusJΔx (22)

where Δf = [fTac fTac-dc f

Tdc f

T120582]T f120582= plusmn1 Δx = [Δ119909Tac1 Δ119909

Tac2

Δ119909Tacminusdc Δ119909

Tdc Δ119909

T120582]T J is referred to as the Jocobian matrix


Table 1 System parameters of operation and control modes of VSC-HVDC

Operation modes Control parameters of VSC converters (pu)VSC1

VSC2

1 Constant DC voltage 119880refd1 = 20000 Constant AC active power 119875ref

s2 = minus08993

Constant AC reactive power 119876refs1 = 01220 Constant AC reactive power 119876ref

s2 = 01734


s2 = minus08993

Constant AC reactive power 119876refs1 = 01220 Constant AC voltage 119880ref

s2 = 10186


s1 = 09194

Constant AC voltage 119880refs2 = 10186 Constant AC reactive power 119876ref

s1 = 01220


s1 = 09194

Constant AC voltage 119880refs2 = 10186 Constant AC voltage 119880ref

s1 = 10203

Table 2 Initial DC variable parameters (pu) of VSC-HVDC system

119873bus 119877 119883l 119875L 119876L 119877d 119880s 120579s 119875s 119876s 119880d

13 00060 01500 01350 00580 00300 10000 00000 09190 01220 2000014 00060 01500 01490 00500 00300 10000 00000 minus08990 01730 20000

of (22) and the dimensions of J are [2(119899 minus 1) + 4119899VSC + 1] times[2(119899 minus 1) + 6119899VSC + 1]

The initial values of the variables of VSC-HVDC systemfor power flow program iteration are given by

119880(0)

d119896 = 119880refd119896 (119896 isin 119862119881)

119880(0)

d119896 = 119880Nd119896 (119896 notin 119862119881)

119868(0)

d119896 =119875t119896

119880(0)

d119896

120575(0)

119896= arctan(

119875t119896(1198802

t119896119883L119896) + (1198802

t119896119883f119896) minus 119876t119896)

119872(0)

119896=2radic6

3

119875t119896119883L119896

119880t119896119880(0)

d119896 sin 120575(0)

119896

(23)

where 119896 isin 119862119881 identifies that the 119896th VSC is constantDC voltage control mode 119896 notin 119862119881 identifies that the 119896this not constant DC voltage control mode Superscript ldquo0rdquoidentifies the initial value of the 0th iteration Superscriptldquoref rdquo identifies that the variable value is reference valueSuperscript ldquoNrdquo identifies rated value

The 119875t119896 is given in estimation by

119875t119896 = minus

119899VSC

sum

119904=1119904notin119862119881

119875reft119904 (24)

Based on the previous analysis the tangent vector[119889119909119889120582]T is obtainedTheprediction value of the next solution

is given by

[

[

1199091015840

119897+1

1205821015840

119897+1

]

]

= [119909119897

120582119897

] + ℎ [119889119909

119889120582

] (25)

where [1199091015840119897+11205821015840

119897+1]T is prediction value which is an approxi-

mate solution ℎ is the step size of the prediction

332The Corrector Step In the corrector step the predictionvalue of [1199091015840

119897+11205821015840

119897+1]T is substituted into (19) and its iteration

form is reformulated as

[1198911015840

1199091198911015840

120582

0 1] [Δ119909

Δ120582] = minus [

119891 (119909 120582)

0] (26)

The iteration form is now reformulated as

[1198911015840

1199091198911015840

120582

119890119870

0] [Δ119909

Δ120582] = minus[

119891(119909 120582)

0] (27)

4 Case Studies and Validations

Two cases are considered and compared (1) the system withexisting AC transmission line and (2) the system with a newdc transmission line based on VSC-HVDC

The systemparameters of the four different controlmodesand the VSC converters adopted in the case studies areprespecified as listed in Table 1 The initial DC variableparameters of VSC-HVDC system are shown in Table 2 InTable 2 the 119873bus identifies bus number of VSC-HVDC linkconnected to AC systems The 119875L and 119876L are the load powerof the bus VSC connected to The 119877d is the resistance of DCnetwork 119877 119883

119897 119875L and 119876L are given parameters 119875s and 119876s

are calculated by power flow calculation of the original ACsystem which is equal to the branch power of the originalAC system

41 Modified IEEE 14-Bus Text System First the proposedmethod has been applied to the modified IEEE 14-bus systemshown in Figure 2 The AC line parameters of the system arethe same as the IEEE 14-bus system The difference is that atwo-terminal VSC-HVDC transmission line is placed at bus13 and bus 14 to replace the AC transmission line 13-14 thatis the VSC

1and VSC

2are connected to AC line of bus 13 and

bus 14 respectivelyThis paper chooses the commutation bus of the buses 9

12 13 and 14 as the research objects According to the different


1

2 3

45

6

78

91011

12

1314VSC1 VSC2

G

G

G

G

G

Figure 2 The modified IEEE 14-bus system with VSC-HVDC

105

1

095

09

085

08

Volta

ge m

agni

tude

(pu

)

Load margin (pu)1 15 2 25 3 35 4 45 5 55

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

Figure 3 PV curves for IEEE-14 bus system

control modes of VSC-HVDC showed in Table 1 the detailedanalysis of the voltage stability bifurcation for the system canbe divided into three cases

411 The Operation Mode of ldquo1rdquo Figure 3 shows the P-Vcurves and load margins of partial buses (buses 9 12 13and 14) of the original pure IEEE-14 system (ldquoACrdquo for short)Figure 4 shows the P-V curves and load margins of partialbuses (buses 9 12 13 and 14) of the modified IEEE-14 system(ldquoACDCrdquo for short) in which the ACDC system operatesunder the control mode ldquo1rdquo in Table 1 as the load increasesAnd Table 3 shows the power flow data of the VSC-HVDCoperating in mode ldquo1rdquo at initial state and the maximum loadstate of the ACDC system Here the maximum load stateis corresponding to the saddle node bifurcation point of thesystem

In Figure 3 when 120582AC = 52120 pu the saddle nodebifurcation point is acquired and the system is in maximumloading state In Table 3 the ldquoAC busrdquo of the lower half part

Table 3 Power flow data at initial and maximum load states formode ldquo1rdquo

Variable 119880d 119868d 120575 119872 119875d 119876d

VSC1

parameters20000 04551 01370 08108 09153 0122020000 04584 01947 06840 09240 01220

VSC2

parameters19863 minus04551 minus01292 09387 minus08993 0173419862 minus04584 minus03065 05505 minus08993 01734

Variable 119876s1 119876s2 119876s3 119876s6 119876s8 mdashAC busparameters

minus06999 01404 02347 00010 01045 mdashminus00438 76425 15550 00014 05365 mdash

105

1

095

09

085

08

075

07

065

06

055

Volta

ge m

agni

tude

(pu

)

1 15 2 25 3 35 4 45 5 55 6

1198809 (ACDC)11988012 (ACDC)

11988013 (ACDC)11988014 (ACDC)

Load margin (pu)

Figure 4 PV curves for modified IEEE-14 bus system in operatingmode ldquo1rdquo

includes the four PV buses (bus 2 bus 3 bus 6 and bus 8)and a slack bus (bus 1) of the modified IEEE-14 test systemThe upper row of the twin-row is the initial state data andthe lower row of the twin-row is the maximum load state(corresponding with the 120582AC DC = 56358 pu in Figure 4)data of power flow

It can be seen in Figure 4 and Table 3 in operation modeldquo1rdquo that the voltagemagnitude decreases as the load increasesand the bus 14 (ACDC) has the weakest voltage profile soit is the critical voltage bus needing reactive power supportWhen the load margin at 120582AC DC = 56358 pu the modifiedACDC systems present a collapse or saddle node bifurcationpoint where the system Jacobian matrix becomes singularAnd now the voltage drop of bus 14 (ACDC) is the mostobvious 119880

14= 06451 lt 119880

13= 08381 lt 119880

12= 08816 lt

1198809= 09083 (the parameters are all the ACDC systems)

412 The Operation Modes of ldquo2rdquo and ldquo3rdquo Figures 5 and 6show the P-V curves and load margins of partial buses (buses9 12 13 and 14) of theACDC systemunder the controlmodeof ldquo2rdquo and ldquo3rdquo

As shown in Figures 5 and 6 in operating mode ldquo2rdquo andmode ldquo3rdquo the voltage magnitudes of bus 14 (ACDC) and bus13 (ACDC) remain almost constant as the load increasesrespectively This is because the VSC

2in mode ldquo2rdquo and VSC

1

in mode ldquo3rdquo both are in constant AC voltage control mode


1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

105

1

095

09

085

08

075

07

1 2 3 4 5 6 7

Load margin (pu)

Volta

ge m

agni

tude

(pu

)


Besides the load margins obtained in mode ldquo2rdquo (120582AC DC =

69192 pu) and mode ldquo3rdquo (120582AC DC = 58980 pu) are biggerthan those in mode ldquo1rdquo which demonstrated that the VSC-HVDC system supplies voltage support to theACbus voltagethanks to the benefits of the fast and independent reactivepower output of VSC-HVDC

413 The Operation Mode of ldquo4rdquo Figure 7 shows the P-Vcurves and load margins of partial buses (buses 9 12 13 and14) of the ACDC system under the control mode of ldquo4rdquo inTable 1 with the load varies

As shown in Figure 7 the voltage magnitude of bus 13(ACDC) and bus 14 (ACDC) remains almost constant asthe load increasesThis because to the control modes of VSC

1

and VSC2are constant AC voltage The load margin for this

operation mode is 120582AC DC = 77192 pu which is biggerthan those in other modesTherefore in the case of the VSC-HVDC operation in mode ldquo4rdquo the ACDC system has bettervoltage stabilization But as has been pointed out in [27]when theACDC system is disturbed (such as kinds of faults)in order to maintain AC bus voltage the VSC converter inmode ldquo4rdquo has to provide large amount of reactive power to theACDC system Consequently the overload degree of VSCconverter is more severe under this operation mode thanoverload degree ofVSC converter under other controlmodes

By contrasting the four control modes of VSC convertersthe results show that the requirements of essential reactivepower for AC system can be supplied by VSC-HVDC systemand the certain voltage support capability to AC bus by VSC-HVDC link is validated But it should be pointed out that theappropriate control pattern is the basis to exploit the reactivepower compensation property of the VSC-HVDC system

42 Modified IEEE 118-Bus Text System The modified IEEE118-bus system is analyzed in this section [28] The relevantpart of the network is shown in Figure 8 which shows thelocations of the VSC-HVDC link The VSC-HVDC replacedan existing AC transmission line (75ndash118) as shown in

12

11

1

09

08

07

06

05

1 15 2 25 3 35 4 45 5 55 6

Load margin (pu)

Volta

ge m

agni

tude

(pu

)

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)


104

102

1

098

096

094

092

09

1 2 3 4 5 6 7 8

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

Load margin (pu)

Volta

ge m

agni

tude

(pu

)


Table 4 Performance comparison of the modified IEEE-118 testsystem for four different operating modes

The two cases Operationmodes

Load margin120582 (pu)

Iterationnumbers

CPU timein seconds

ACDC system 1 46188 15 297642ACDC system 2 48750 15 320300ACDC system 3 48625 15 312754ACDC system 4 62750 15 764324

Figure 8 and the VSC1and VSC

2are connected to AC line

of bus 75 and bus 118 respectivelyThe performance comparison of the modified IEEE-118

test system for four different operating modes is shown inTable 4 In Table 4 the CPU time for operation mode of ldquo4rdquois longer than other operation modes The reason is thatthe operation mode of ldquo4rdquo is in constant AC voltage controlmode As the load increases the VSC-HVDC system suppliesvoltage support to the AC bus voltage and as a result theload margin of the ACDC system in mode ldquo4rdquo is bigger


107

VSC1 VSC2

74

75

76

77

78

79

81

82

8384

85

86

87

88 89

9091

92

93

9495

96

97

98

9980

100

101102

103

104 105

106

108

109

110

111

112

118

G

G G G

G

G G G G

G

G

G

G

G G

G G

G

G

Figure 8 Relevant part of the modified IEEE 118-bus system withVSC-HVDC

than other operation modes and the calculation programfor the load margin of the mode ldquo4rdquo needs more iterationso the CPU time is longer Table 4 shows that the modeland algorithm presented in this paper have certain flexibilitywith the increase of the network scale However it is to beremarked that the CPU time is long The reason is that withthe embedded VSC-HVDC transmission line the types andthe number of the system variables have greatly increasedthat is 119875c119894 119876c119894 119880d119894 119868d119894 120575119894119872119894 119880c119894 120579c119894 and so forth and thedimensions of the system equations and the Jacobian matrixof the ACDC system have a higher order than pure ACsystems

5 Conclusions

In this paper a new method has been developed to analyzevoltage stability for ACDC systems with VSC-HVDC Theimpacts of load variations and different VSC-HVDC controlpatterns on P-V curves and saddle node bifurcation pointof the system were numerically analyzed The simulationresults indicate that the constant AC voltage control of VSCconverter is superior to other control modes in voltage sta-bility and also show that VSC-HVDC significantly improvedthe stability of the system compared to a pure AC lineAt last the importance of suitable control mode for theoperating of VSC-HVDCwas discussed and some numericalexamples have been included to demonstrate the validity ofthe obtained results

Acknowledgments

This work is supported in part by the National NaturalScience Foundation of China (Grant nos 61104045 5127705251107032 and 51077125) and in part by the FundamentalResearch Funds for the Central Universities of China (Grantno 2012B03514)

References

[1] S Kouro M Malinowski K Gopakumar et al ldquoRecentadvances and industrial applications of multilevel convertersrdquoIEEE Transactions on Industrial Electronics vol 57 no 8 pp2553ndash2580 2010

[2] N Flourentzou V G Agelidis and G D Demetriades ldquoVSC-based HVDC power transmission systems an overviewrdquo IEEETransactions on Power Electronics vol 24 no 3 pp 592ndash6022009

[3] Z Huang B T Ooi L A Dessaint and F D Galiana ldquoExploit-ing voltage support of voltage-source HVDCrdquo IEE ProceedingsGeneration Transmission and Distribution vol 150 no 2 pp252ndash256 2003

[4] L Zhang L Harnefors and H P Nee ldquoInterconnection oftwo very weak AC systems by VSC-HVDC links using power-synchronization controlrdquo IEEE Transactions on Power Systemsvol 26 no 1 pp 344ndash355 2011

[5] M P Bahrman and B K Johnson ldquoThe ABCs of HVDCtransmission technologiesrdquo IEEE Power and Energy Magazinevol 5 no 2 pp 32ndash44 2007


[6] L Zhang H P Nee and L Harnefors ldquoAnalysis of stabilitylimitations of a VSC-HVDC link using power-synchronizationcontrolrdquo IEEE Transactions on Power Systems vol 26 no 3 pp1326ndash1337 2011

[7] Y Jiang-Hafner M Hyttinen and B Paajarvi ldquoOn the shortcircuit current contribution of HVDC lightrdquo in Proceedingsof IEEE PES Transmission and Distribution Conference andExhibition October 2002

[8] F L ShunM Reza K Srivastava S Cole D vanHertem andRBelmans ldquoInfluence of VSC HVDC on transient stability casestudy of the Belgian gridrdquo in Proceedings of the IEEE Power andEnergy Society General Meeting (PES rsquo10) Minneapolis MinnUSA July 2010

[9] H F Latorre and M Ghandhari ldquoImprovement of powersystem stability by using a VSC-HVdcrdquo International Journal ofElectrical Power and Energy Systems vol 33 no 2 pp 332ndash3392011

[10] F A R Al Jowder and B T Ooi ldquoVSC-HVDC stationwith SSSCcharacteristicsrdquo IEEE Transactions on Power Electronics vol 19no 4 pp 1053ndash1059 2004

[11] J Zhang and G Bao ldquoVoltage stability improvement for largewind farms connection based on VSC-HVDCrdquo in Proceedingsof the Asia-Pacific Power and Energy Engineering Conference(APPEEC rsquo11) Wuhan China March 2011

[12] H Livani J Rouhi and H Karirni-Davijani ldquoVoltage stabi-lization in connection of wind farms to transmission networkusing VSC-HVDCrdquo in Proceedings of the 43rd InternationalUniversities Power Engineering Conference (UPEC rsquo08) PadovaItaly September 2008

[13] C Guo and C Zhao ldquoSupply of an entirely passive AC networkthrough a double-infeed HVDC systemrdquo IEEE Transactions onPower Electronics vol 25 no 11 pp 2835ndash2841 2010

[14] H F Latorre M Ghandhari and L Soder ldquoActive and reactivepower control of a VSC-HVdcrdquo Electric Power Systems Researchvol 78 no 10 pp 1756ndash1763 2008

[15] H F Latorre and M Ghandhari ldquoImprovement of voltagestability by using VSC-HVdcrdquo in Proceedings of IEEE Transmis-sion and Distribution Asia and Pacific Conference Seoul SouthKorea October 2009

[16] Y Gao Z Wang and H Ling ldquoAvailable transfer capabil-ity calculation with large offshore wind farms connected byVSC-HVDCrdquo in Proceedings of IEEE Innovative Smart GridTechnologiesmdashAsia Conference Tianjin China May 2012

[17] L C Azimoh K Folly S P Chowdhury S Chowdhury andA Haddad ldquoInvestigation of voltage and transient stabilityof HVAC network in hybrid with VSC-HVDC and HVDClinkrdquo in Proceedings of the 45th International Universitiesrsquo PowerEngineering Conference (UPEC rsquo10) pp 1ndash6Wales UK August-September 2010

[18] X P Zhang ldquoMultiterminal voltage-sourced converter-basedHVDC models for power flow analysisrdquo IEEE Transactions onPower Systems vol 19 no 4 pp 1877ndash1884 2004

[19] A Pizano-Martinez C R Fuerte-Esquivel H Ambriz-Perezand E Acha ldquoModeling of VSC-based HVDC systems for aNewton-Raphson OPF algorithmrdquo IEEE Transactions on PowerSystems vol 22 no 4 pp 1794ndash1803 2007

[20] C Angeles-Camacho O L Tortelli E Acha and C R Fuerte-Esquivel ldquoInclusion of a high voltage DC-voltage source con-verter model in a Newton-Raphson power flow algorithmrdquo IEEProceedings Generation Transmission andDistribution vol 150no 6 pp 691ndash696 2003

[21] C Liu B Zhang Y Hou F F Wu and Y Liu ldquoAn improvedapproach for AC-DC power flow calculation with multi-infeedDC systemsrdquo IEEE Transactions on Power Systems vol 26 no2 pp 862ndash869 2011

[22] U Arifoglu ldquoThe power flow algorithm for balanced and unbal-anced bipolar multiterminal ac-dc systemsrdquo Electric PowerSystems Research vol 64 no 3 pp 239ndash246 2003

[23] X Wang Y Song and M Irving Modern Power SystemsAnalysis Springer Berlin Germany 2008

[24] T Smed G Andersson G B Sheble and L L Grigsby ldquoA newapproach to ACDC power flowrdquo IEEE Transactions on PowerSystems vol 6 no 3 pp 1238ndash1244 1991

[25] S H Li and H D Chiang ldquoContinuation power flow withnonlinear power injection variations a piecewise linear approx-imationrdquo IEEE Transactions on Power Systems vol 23 no 4 pp1637ndash1643 2008

[26] V Ajjarapu andC Christy ldquoThe continuation power flow a toolfor steady state voltage stability analysisrdquo IEEE Transactions onPower Systems vol 7 no 1 pp 416ndash423 1992

[27] C Zheng X X Zhou and R M Li ldquoDynamic modeling andtransient simulation for voltage source converter basedHVDCrdquoPower System Technology vol 29 no 16 pp 1ndash5 2005

[28] Power Systems Test Case Archive ldquoUniversity of Washingtonrdquohttpwwweewashingtoneduresearchpstca

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014


Stochastic AnalysisInternational Journal of


DCnetwork

ACsystem

119894

119877119894 119883119897119894

119883f119894

119868d119894

119880d119894

I119880s119894ang120579s119894

119875s119894 119876s119894

119875c119894 119876c119894

119880c119894ang120579c119894

Figure 1 Simplified circuit diagram of single-phase VSC-HVDC

stability of a weak two area AC networkThoughmanymeth-ods for voltage stability analysis of ACDC systemswithVSC-HVDC have been proposed in different ways few paperscarefully consider the influence induced by different controlmodes ofVSC-HVDCwhich actually plays an important roleon the voltage stability of the ACDC systems Motivated bythe previous discussions our main aim in this paper is toinvestigate the problem of power flow and voltage stabilitybifurcation for ACDC systems with VSC-HVDC

The rest of the paper is organized as follows In Section 2the steady model and control modes of ACDC systemswith VSC-HVDC are presented In Section 3 the improvedsequential iterative algorithm the parameterization powerflow and converter equations of VSC-HVDC and the CPFstrategy are presented In Section 4 the model and methodare applied to the modified IEEE 14- and IEEE 118-bus testsystems with VSC-HVDC Finally in Section 5 the paper iscompleted with a conclusion

2 Steady Model of VSC-HVDC

21 Power Flow Equations of VSC-HVDC Figure 1 showsthe single-line representation of two-terminal VSC-HVDCsystem In Figure 1 119894 is the number of VSC converters I

119894is

the current flowing through transformer Us119894 = 119880s119894ang120579s119894 is theAC side voltage vector of VSC converter Uc119894 = 119880c119894ang120579c119894 is theoutput voltage vector of VSC converter 119877

119894is the equivalent

resistance of internal loss and converter transformer loss ofVSC 119895119883

119897119894is the impedance of converter transformer 119875s119894

and 119876s119894 are the power of AC system injected into convertertransformer 119895119883

119891is the impedance of AC filter Id andUd are

the DC current vector andDC voltage vector respectively119875c119894and119876c119894 are the power flowing through converter bridge119875d119894 isthe DC power From Figure 1 I

119894can be expressed as follows

I119894=Us119894 minus Uc119894119877119894+ 119895119883119897119894

(1)

The complex power 119878s119894 of AC system is given by

119878s119894 = 119875s119894 + 119895119876s119894 = Us119894Ilowast

119894 (2)

To facilitate discussion assume that 120575119894= 120579s119894 minus 120579c119894 |119884119894| =

1radic1198772

119894+ 1198832

119897119894 120572119894= arccot(119883

119897119894119877119894) and substituting (1) into (2)

yields

119875119904119894= minus

10038161003816100381610038161198841198941003816100381610038161003816 119880s119894119880c119894 cos (120575119894 + 120572119894) +

10038161003816100381610038161198841198941003816100381610038161003816 1198802

s119894 cos120572119894

119876119904119894= minus

10038161003816100381610038161198841198941003816100381610038161003816 119880s119894119880c119894 sin (120575119894 + 120572119894) +

10038161003816100381610038161198841198941003816100381610038161003816 1198802

s119894 sin120572119894 +1198802

s119894119883f119894

(3)

Similarly one has

119875c119894 =10038161003816100381610038161198841198941003816100381610038161003816 119880s119894119880c119894 cos (120575119894 minus 120572119894) minus

10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 cos120572119894

119876c119894 = minus10038161003816100381610038161198841198941003816100381610038161003816 119880s119894119880c119894 sin (120575119894 minus 120572119894) minus

10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 sin120572119894(4)

Since the loss of converter bridge-arm is equivalent by 119877119894 119875d119894

is qual to 119875c119894 and thus

119875d119894 = 119880d119894119868d119894 =10038161003816100381610038161198841198941003816100381610038161003816 119880s119894119880c119894 cos (120575119894 minus 120572119894) minus

10038161003816100381610038161198841198941003816100381610038161003816 1198802

c119894 cos120572119894 (5)

And 119880c119894 can be described as

119880c119894 =radic6119872119894119880d119894

4 (6)

where 119872 (0 lt 119872 lt 1) is defined as the PWMrsquos amplitudemodulation index

The steadymodel of VSC-HVDC is given by (1)ndash(6) in theper-unit system (pu)

22 Steady-State Control Modes of VSC-HVDC Several reg-ular control modes for each VSC converter are chiefly asfollows

(1) constant DC voltage control constant AC reactivepower control

(2) constant DC voltage control constant AC voltagecontrol

(3) constant AC active power control constant AC reac-tive power control

(4) constant AC active power control constant AC volt-age control




fac = 0

fac-dc = 0

fdc = 0

(7)



11 Δ11988912 Δ11988913 Δ11988914

Δ119889119899VSC1

Δ119889119899VSC2

Δ119889119899VSC3

Δ119889119899VSC4

]T




Tdc]

T ΔxN = [Δ119909Tac1 Δ119909

Tac2

Δ119909Tac-dc Δ119909

Tdc]





And JN is given by


=

[[[[[[[[

[

120597fac120597xac1

120597fac120597xac2

120597fac120597xac-dc

120597fac120597xdc

120597fac-dc120597xac1

120597fac-dc120597xac2


120597fac-dc120597xdc

120597fdc120597xac1

120597fdc120597xac2

120597fdc120597xac-dc

120597fdc120597xdc

]]]]]]]]

]

= [

[


]

]

(9)


[f1

f2

] = minus [J11

J12

J21

J22

] [Δx1

Δx2

] (10)


Tdc]



Tdc]

T J11

= Ja-a1 J21 = [Jad-a1Jd-a1 ] J12 =



1are



2are 6119899VSC




To expand (10)



(11)






(12)



AC system




119880119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)

+ (119875G119894 minus 119875L119894) 120582 = 0


119880119895(119866119894119895sin 120579119894119895minus 119861119894119895cos 120579119894119895)

+ (119876G119894 minus 119876L119894) 120582 = 0

(13)





119880119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895) plusmn 119875t119894

+ (119875G119894 minus 119875L119894) 120582 = 0



+ (119876G119894 minus 119876L119894) 120582 = 0

(14)



Δ1198891198961= 119875t119896 +

radic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 + 120572119896)

minus 1198802

t119896 |119884| cos120572119896 = 0

Δ1198891198962= 119876t119896 +

radic6

4119872119896119880t119896119880d119896 |119884| sin (120575119896 + 120572119896) minus 119880

2

t119896 |119884| sin120572119896

minus1198802

t119896119883f119896

= 0

Δ1198891198963= 119880t119896119868d119896 minus

radic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 minus 120572119896)

+3

8(119872119896119880d119896)2|119884| cos120572119896 = 0

(15)




Id = GdUd (16)


Δ1198891198964= plusmn 119868d119896 minus

119899AC

sum

119904=1

119892d119896119904119880d119904 = 0 (17)



119875119894= 1198751198940 (1 + 120582)

119876119894= 1198761198940 (1 + 120582)

(18)

where 1198751198940and 119876




119894 respectively


f (x 120582) = 0 (19)


1







[1198911015840

1199091198911015840

120582] [119889119909

119889120582

] = 0 (20)






[1198911015840

1199091198911015840

120582

119890119870

] [119889119909

119889120582

] = [0

plusmn1] (21)

where 119890119870





Tdc f


Tac2


Tdc Δ119909





VSC2


s2 = minus08993


s2 = 01734


s2 = minus08993


s2 = 10186


s1 = 09194


s1 = 01220


s1 = 09194


s1 = 10203


119873bus 119877 119883l 119875L 119876L 119877d 119880s 120579s 119875s 119876s 119880d

13 00060 01500 01350 00580 00300 10000 00000 09190 01220 2000014 00060 01500 01490 00500 00300 10000 00000 minus08990 01730 20000



119880(0)

d119896 = 119880refd119896 (119896 isin 119862119881)

119880(0)

d119896 = 119880Nd119896 (119896 notin 119862119881)

119868(0)

d119896 =119875t119896

119880(0)

d119896

120575(0)

119896= arctan(

119875t119896(1198802

t119896119883L119896) + (1198802

t119896119883f119896) minus 119876t119896)

119872(0)

119896=2radic6

3

119875t119896119883L119896

119880t119896119880(0)

d119896 sin 120575(0)

119896

(23)



119875t119896 = minus

119899VSC

sum

119904=1119904notin119862119881

119875reft119904 (24)


is given by

[

[

1199091015840

119897+1

1205821015840

119897+1

]

]

= [119909119897

120582119897

] + ℎ [119889119909

119889120582

] (25)

where [1199091015840119897+11205821015840




119897+11205821015840



[1198911015840

1199091198911015840

120582

0 1] [Δ119909

Δ120582] = minus [

119891 (119909 120582)

0] (26)


[1198911015840

1199091198911015840

120582

119890119870

0] [Δ119909

Δ120582] = minus[

119891(119909 120582)

0] (27)







1and VSC





1

2 3

45

6

78

91011

12

1314VSC1 VSC2

G

G

G

G

G


105

1

095

09

085

08

Volta

ge m

agni

tude

(pu

)

Load margin (pu)1 15 2 25 3 35 4 45 5 55

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)






Variable 119880d 119868d 120575 119872 119875d 119876d

VSC1

parameters20000 04551 01370 08108 09153 0122020000 04584 01947 06840 09240 01220

VSC2




105

1

095

09

085

08

075

07

065

06

055

Volta

ge m

agni

tude

(pu

)

1 15 2 25 3 35 4 45 5 55 6

1198809 (ACDC)11988012 (ACDC)

11988013 (ACDC)11988014 (ACDC)

Load margin (pu)




14= 06451 lt 119880

13= 08381 lt 119880

12= 08816 lt





1



1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

105

1

095

09

085

08

075

07

1 2 3 4 5 6 7

Load margin (pu)

Volta

ge m

agni

tude

(pu

)






1





12

11

1

09

08

07

06

05

1 15 2 25 3 35 4 45 5 55 6

Load margin (pu)

Volta

ge m

agni

tude

(pu

)

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)


104

102

1

098

096

094

092

09

1 2 3 4 5 6 7 8

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

Load margin (pu)

Volta

ge m

agni

tude

(pu

)





Iterationnumbers

CPU timein seconds







107

VSC1 VSC2

74

75

76

77

78

79

81

82

8384

85

86

87

88 89

9091

92

93

9495

96

97

98

9980

100

101102

103

104 105

106

108

109

110

111

112

118

G

G G G

G

G G G G

G

G

G

G

G G

G G

G

G



5 Conclusions


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






fac = 0

fac-dc = 0

fdc = 0

(7)



11 Δ11988912 Δ11988913 Δ11988914

Δ119889119899VSC1

Δ119889119899VSC2

Δ119889119899VSC3

Δ119889119899VSC4

]T




Tdc]

T ΔxN = [Δ119909Tac1 Δ119909

Tac2

Δ119909Tac-dc Δ119909

Tdc]





And JN is given by


=

[[[[[[[[

[

120597fac120597xac1

120597fac120597xac2

120597fac120597xac-dc

120597fac120597xdc

120597fac-dc120597xac1

120597fac-dc120597xac2


120597fac-dc120597xdc

120597fdc120597xac1

120597fdc120597xac2

120597fdc120597xac-dc

120597fdc120597xdc

]]]]]]]]

]

= [

[


]

]

(9)


[f1

f2

] = minus [J11

J12

J21

J22

] [Δx1

Δx2

] (10)


Tdc]



Tdc]

T J11

= Ja-a1 J21 = [Jad-a1Jd-a1 ] J12 =



1are



2are 6119899VSC




To expand (10)



(11)






(12)



AC system




119880119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895)

+ (119875G119894 minus 119875L119894) 120582 = 0


119880119895(119866119894119895sin 120579119894119895minus 119861119894119895cos 120579119894119895)

+ (119876G119894 minus 119876L119894) 120582 = 0

(13)





119880119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895) plusmn 119875t119894

+ (119875G119894 minus 119875L119894) 120582 = 0



+ (119876G119894 minus 119876L119894) 120582 = 0

(14)



Δ1198891198961= 119875t119896 +

radic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 + 120572119896)

minus 1198802

t119896 |119884| cos120572119896 = 0

Δ1198891198962= 119876t119896 +

radic6

4119872119896119880t119896119880d119896 |119884| sin (120575119896 + 120572119896) minus 119880

2

t119896 |119884| sin120572119896

minus1198802

t119896119883f119896

= 0

Δ1198891198963= 119880t119896119868d119896 minus

radic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 minus 120572119896)

+3

8(119872119896119880d119896)2|119884| cos120572119896 = 0

(15)




Id = GdUd (16)


Δ1198891198964= plusmn 119868d119896 minus

119899AC

sum

119904=1

119892d119896119904119880d119904 = 0 (17)



119875119894= 1198751198940 (1 + 120582)

119876119894= 1198761198940 (1 + 120582)

(18)

where 1198751198940and 119876




119894 respectively


f (x 120582) = 0 (19)


1







[1198911015840

1199091198911015840

120582] [119889119909

119889120582

] = 0 (20)






[1198911015840

1199091198911015840

120582

119890119870

] [119889119909

119889120582

] = [0

plusmn1] (21)

where 119890119870





Tdc f


Tac2


Tdc Δ119909





VSC2


s2 = minus08993


s2 = 01734


s2 = minus08993


s2 = 10186


s1 = 09194


s1 = 01220


s1 = 09194


s1 = 10203


119873bus 119877 119883l 119875L 119876L 119877d 119880s 120579s 119875s 119876s 119880d

13 00060 01500 01350 00580 00300 10000 00000 09190 01220 2000014 00060 01500 01490 00500 00300 10000 00000 minus08990 01730 20000



119880(0)

d119896 = 119880refd119896 (119896 isin 119862119881)

119880(0)

d119896 = 119880Nd119896 (119896 notin 119862119881)

119868(0)

d119896 =119875t119896

119880(0)

d119896

120575(0)

119896= arctan(

119875t119896(1198802

t119896119883L119896) + (1198802

t119896119883f119896) minus 119876t119896)

119872(0)

119896=2radic6

3

119875t119896119883L119896

119880t119896119880(0)

d119896 sin 120575(0)

119896

(23)



119875t119896 = minus

119899VSC

sum

119904=1119904notin119862119881

119875reft119904 (24)


is given by

[

[

1199091015840

119897+1

1205821015840

119897+1

]

]

= [119909119897

120582119897

] + ℎ [119889119909

119889120582

] (25)

where [1199091015840119897+11205821015840




119897+11205821015840



[1198911015840

1199091198911015840

120582

0 1] [Δ119909

Δ120582] = minus [

119891 (119909 120582)

0] (26)


[1198911015840

1199091198911015840

120582

119890119870

0] [Δ119909

Δ120582] = minus[

119891(119909 120582)

0] (27)







1and VSC





1

2 3

45

6

78

91011

12

1314VSC1 VSC2

G

G

G

G

G


105

1

095

09

085

08

Volta

ge m

agni

tude

(pu

)

Load margin (pu)1 15 2 25 3 35 4 45 5 55

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)






Variable 119880d 119868d 120575 119872 119875d 119876d

VSC1

parameters20000 04551 01370 08108 09153 0122020000 04584 01947 06840 09240 01220

VSC2




105

1

095

09

085

08

075

07

065

06

055

Volta

ge m

agni

tude

(pu

)

1 15 2 25 3 35 4 45 5 55 6

1198809 (ACDC)11988012 (ACDC)

11988013 (ACDC)11988014 (ACDC)

Load margin (pu)




14= 06451 lt 119880

13= 08381 lt 119880

12= 08816 lt





1



1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

105

1

095

09

085

08

075

07

1 2 3 4 5 6 7

Load margin (pu)

Volta

ge m

agni

tude

(pu

)






1





12

11

1

09

08

07

06

05

1 15 2 25 3 35 4 45 5 55 6

Load margin (pu)

Volta

ge m

agni

tude

(pu

)

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)


104

102

1

098

096

094

092

09

1 2 3 4 5 6 7 8

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

Load margin (pu)

Volta

ge m

agni

tude

(pu

)





Iterationnumbers

CPU timein seconds







107

VSC1 VSC2

74

75

76

77

78

79

81

82

8384

85

86

87

88 89

9091

92

93

9495

96

97

98

9980

100

101102

103

104 105

106

108

109

110

111

112

118

G

G G G

G

G G G G

G

G

G

G

G G

G G

G

G



5 Conclusions


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






119880119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895) plusmn 119875t119894

+ (119875G119894 minus 119875L119894) 120582 = 0



+ (119876G119894 minus 119876L119894) 120582 = 0

(14)



Δ1198891198961= 119875t119896 +

radic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 + 120572119896)

minus 1198802

t119896 |119884| cos120572119896 = 0

Δ1198891198962= 119876t119896 +

radic6

4119872119896119880t119896119880d119896 |119884| sin (120575119896 + 120572119896) minus 119880

2

t119896 |119884| sin120572119896

minus1198802

t119896119883f119896

= 0

Δ1198891198963= 119880t119896119868d119896 minus

radic6

4119872119896119880t119896119880d119896 |119884| cos (120575119896 minus 120572119896)

+3

8(119872119896119880d119896)2|119884| cos120572119896 = 0

(15)




Id = GdUd (16)


Δ1198891198964= plusmn 119868d119896 minus

119899AC

sum

119904=1

119892d119896119904119880d119904 = 0 (17)



119875119894= 1198751198940 (1 + 120582)

119876119894= 1198761198940 (1 + 120582)

(18)

where 1198751198940and 119876




119894 respectively


f (x 120582) = 0 (19)


1







[1198911015840

1199091198911015840

120582] [119889119909

119889120582

] = 0 (20)






[1198911015840

1199091198911015840

120582

119890119870

] [119889119909

119889120582

] = [0

plusmn1] (21)

where 119890119870





Tdc f


Tac2


Tdc Δ119909





VSC2


s2 = minus08993


s2 = 01734


s2 = minus08993


s2 = 10186


s1 = 09194


s1 = 01220


s1 = 09194


s1 = 10203


119873bus 119877 119883l 119875L 119876L 119877d 119880s 120579s 119875s 119876s 119880d

13 00060 01500 01350 00580 00300 10000 00000 09190 01220 2000014 00060 01500 01490 00500 00300 10000 00000 minus08990 01730 20000



119880(0)

d119896 = 119880refd119896 (119896 isin 119862119881)

119880(0)

d119896 = 119880Nd119896 (119896 notin 119862119881)

119868(0)

d119896 =119875t119896

119880(0)

d119896

120575(0)

119896= arctan(

119875t119896(1198802

t119896119883L119896) + (1198802

t119896119883f119896) minus 119876t119896)

119872(0)

119896=2radic6

3

119875t119896119883L119896

119880t119896119880(0)

d119896 sin 120575(0)

119896

(23)



119875t119896 = minus

119899VSC

sum

119904=1119904notin119862119881

119875reft119904 (24)


is given by

[

[

1199091015840

119897+1

1205821015840

119897+1

]

]

= [119909119897

120582119897

] + ℎ [119889119909

119889120582

] (25)

where [1199091015840119897+11205821015840




119897+11205821015840



[1198911015840

1199091198911015840

120582

0 1] [Δ119909

Δ120582] = minus [

119891 (119909 120582)

0] (26)


[1198911015840

1199091198911015840

120582

119890119870

0] [Δ119909

Δ120582] = minus[

119891(119909 120582)

0] (27)







1and VSC





1

2 3

45

6

78

91011

12

1314VSC1 VSC2

G

G

G

G

G


105

1

095

09

085

08

Volta

ge m

agni

tude

(pu

)

Load margin (pu)1 15 2 25 3 35 4 45 5 55

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)






Variable 119880d 119868d 120575 119872 119875d 119876d

VSC1

parameters20000 04551 01370 08108 09153 0122020000 04584 01947 06840 09240 01220

VSC2




105

1

095

09

085

08

075

07

065

06

055

Volta

ge m

agni

tude

(pu

)

1 15 2 25 3 35 4 45 5 55 6

1198809 (ACDC)11988012 (ACDC)

11988013 (ACDC)11988014 (ACDC)

Load margin (pu)




14= 06451 lt 119880

13= 08381 lt 119880

12= 08816 lt





1



1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

105

1

095

09

085

08

075

07

1 2 3 4 5 6 7

Load margin (pu)

Volta

ge m

agni

tude

(pu

)






1





12

11

1

09

08

07

06

05

1 15 2 25 3 35 4 45 5 55 6

Load margin (pu)

Volta

ge m

agni

tude

(pu

)

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)


104

102

1

098

096

094

092

09

1 2 3 4 5 6 7 8

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

Load margin (pu)

Volta

ge m

agni

tude

(pu

)





Iterationnumbers

CPU timein seconds







107

VSC1 VSC2

74

75

76

77

78

79

81

82

8384

85

86

87

88 89

9091

92

93

9495

96

97

98

9980

100

101102

103

104 105

106

108

109

110

111

112

118

G

G G G

G

G G G G

G

G

G

G

G G

G G

G

G



5 Conclusions


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






VSC2


s2 = minus08993


s2 = 01734


s2 = minus08993


s2 = 10186


s1 = 09194


s1 = 01220


s1 = 09194


s1 = 10203


119873bus 119877 119883l 119875L 119876L 119877d 119880s 120579s 119875s 119876s 119880d

13 00060 01500 01350 00580 00300 10000 00000 09190 01220 2000014 00060 01500 01490 00500 00300 10000 00000 minus08990 01730 20000



119880(0)

d119896 = 119880refd119896 (119896 isin 119862119881)

119880(0)

d119896 = 119880Nd119896 (119896 notin 119862119881)

119868(0)

d119896 =119875t119896

119880(0)

d119896

120575(0)

119896= arctan(

119875t119896(1198802

t119896119883L119896) + (1198802

t119896119883f119896) minus 119876t119896)

119872(0)

119896=2radic6

3

119875t119896119883L119896

119880t119896119880(0)

d119896 sin 120575(0)

119896

(23)



119875t119896 = minus

119899VSC

sum

119904=1119904notin119862119881

119875reft119904 (24)


is given by

[

[

1199091015840

119897+1

1205821015840

119897+1

]

]

= [119909119897

120582119897

] + ℎ [119889119909

119889120582

] (25)

where [1199091015840119897+11205821015840




119897+11205821015840



[1198911015840

1199091198911015840

120582

0 1] [Δ119909

Δ120582] = minus [

119891 (119909 120582)

0] (26)


[1198911015840

1199091198911015840

120582

119890119870

0] [Δ119909

Δ120582] = minus[

119891(119909 120582)

0] (27)







1and VSC





1

2 3

45

6

78

91011

12

1314VSC1 VSC2

G

G

G

G

G


105

1

095

09

085

08

Volta

ge m

agni

tude

(pu

)

Load margin (pu)1 15 2 25 3 35 4 45 5 55

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)






Variable 119880d 119868d 120575 119872 119875d 119876d

VSC1

parameters20000 04551 01370 08108 09153 0122020000 04584 01947 06840 09240 01220

VSC2




105

1

095

09

085

08

075

07

065

06

055

Volta

ge m

agni

tude

(pu

)

1 15 2 25 3 35 4 45 5 55 6

1198809 (ACDC)11988012 (ACDC)

11988013 (ACDC)11988014 (ACDC)

Load margin (pu)




14= 06451 lt 119880

13= 08381 lt 119880

12= 08816 lt





1



1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

105

1

095

09

085

08

075

07

1 2 3 4 5 6 7

Load margin (pu)

Volta

ge m

agni

tude

(pu

)






1





12

11

1

09

08

07

06

05

1 15 2 25 3 35 4 45 5 55 6

Load margin (pu)

Volta

ge m

agni

tude

(pu

)

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)


104

102

1

098

096

094

092

09

1 2 3 4 5 6 7 8

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

Load margin (pu)

Volta

ge m

agni

tude

(pu

)





Iterationnumbers

CPU timein seconds







107

VSC1 VSC2

74

75

76

77

78

79

81

82

8384

85

86

87

88 89

9091

92

93

9495

96

97

98

9980

100

101102

103

104 105

106

108

109

110

111

112

118

G

G G G

G

G G G G

G

G

G

G

G G

G G

G

G



5 Conclusions


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




1

2 3

45

6

78

91011

12

1314VSC1 VSC2

G

G

G

G

G


105

1

095

09

085

08

Volta

ge m

agni

tude

(pu

)

Load margin (pu)1 15 2 25 3 35 4 45 5 55

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)






Variable 119880d 119868d 120575 119872 119875d 119876d

VSC1

parameters20000 04551 01370 08108 09153 0122020000 04584 01947 06840 09240 01220

VSC2




105

1

095

09

085

08

075

07

065

06

055

Volta

ge m

agni

tude

(pu

)

1 15 2 25 3 35 4 45 5 55 6

1198809 (ACDC)11988012 (ACDC)

11988013 (ACDC)11988014 (ACDC)

Load margin (pu)




14= 06451 lt 119880

13= 08381 lt 119880

12= 08816 lt





1



1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

105

1

095

09

085

08

075

07

1 2 3 4 5 6 7

Load margin (pu)

Volta

ge m

agni

tude

(pu

)






1





12

11

1

09

08

07

06

05

1 15 2 25 3 35 4 45 5 55 6

Load margin (pu)

Volta

ge m

agni

tude

(pu

)

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)


104

102

1

098

096

094

092

09

1 2 3 4 5 6 7 8

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

Load margin (pu)

Volta

ge m

agni

tude

(pu

)





Iterationnumbers

CPU timein seconds







107

VSC1 VSC2

74

75

76

77

78

79

81

82

8384

85

86

87

88 89

9091

92

93

9495

96

97

98

9980

100

101102

103

104 105

106

108

109

110

111

112

118

G

G G G

G

G G G G

G

G

G

G

G G

G G

G

G



5 Conclusions


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

105

1

095

09

085

08

075

07

1 2 3 4 5 6 7

Load margin (pu)

Volta

ge m

agni

tude

(pu

)






1





12

11

1

09

08

07

06

05

1 15 2 25 3 35 4 45 5 55 6

Load margin (pu)

Volta

ge m

agni

tude

(pu

)

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)


104

102

1

098

096

094

092

09

1 2 3 4 5 6 7 8

1198809 (AC)11988012 (AC)

11988013 (AC)11988014 (AC)

Load margin (pu)

Volta

ge m

agni

tude

(pu

)





Iterationnumbers

CPU timein seconds







107

VSC1 VSC2

74

75

76

77

78

79

81

82

8384

85

86

87

88 89

9091

92

93

9495

96

97

98

9980

100

101102

103

104 105

106

108

109

110

111

112

118

G

G G G

G

G G G G

G

G

G

G

G G

G G

G

G



5 Conclusions


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




107

VSC1 VSC2

74

75

76

77

78

79

81

82

8384

85

86

87

88 89

9091

92

93

9495

96

97

98

9980

100

101102

103

104 105

106

108

109

110

111

112

118

G

G G G

G

G G G G

G

G

G

G

G G

G G

G

G



5 Conclusions


Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



voltage stability bifurcation analysis for ac/dc systems

Documents