Analysis of VOltage and Power Interactions in Multi-Infeed HVDC Systems

816 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 28, NO. 2, APRIL 2013

Analysis of Voltage and Power Interactions inMulti-Infeed HVDC Systems

Denis Lee Hau Aik, Member, IEEE, and Göran Andersson, Fellow, IEEE

Abstract—In recent CIGRÉ literature, an index was proposed toassess the degree of voltage interaction among HVDC convertersin a multi-infeed configuration. However, the index is derived fromempirical information, its usage is somewhat arbitrary or “rule-of-thumb” basis. In this paper, a power-flow model is used as themathematical framework to derive an analytic equivalent of theempirical index, to facilitate rigorous analysis of voltage/power in-teractions in multi-infeed HVDC systems. Moreover, with other al-lied new indices proposed here and in past works, it is shown thata taxanomy of indices can be established with their interrelation-ships derived analytically. Furthermore, it is shown that these an-alytic indices can give theoretical insight into the parametric de-pendence of the power/voltage interactions.

Index Terms—Multi-infeed HVDC systems, nodal indices,power/voltage interactions, quasistatic analytic models.

I. INTRODUCTION

R ECENT concerns on multi-infeed HVDC systems havemostly focused on the power/voltage stability aspects

[1]–[4]. These have been carried over from similar historicalconcerns on single-infeed HVDC systems.However, in recent works by CIGRÉ [5], the issue of voltage

interaction among HVDC converters in a multi-infeed configu-ration was addressed. This had been recognized to affect systemperformance even though the multi-infeed HVDC systemmightbe operating in a stable regime, for example, concurrent com-mutation failure of multiple HVDC converters in close electricalproximity might occur which consequently impaired voltagequality or caused high overvoltage at the converter ac buses,etc. [5]–[8]. An index known as the multi-Infeed interactionfactor (MIIF) was therefore proposed and studied in [5] and [9],to give a quantitative assessment of the degree of voltage in-teraction. However, use of the index has been somewhat on a“rule-of-thumb” basis since it is derived from empirical infor-mation which does not lend exhaustiveness from a theoreticalperspective.In this paper, a power-flow model is used as the mathemat-

ical framework to derive an analytic equivalent of the empiricalindex, to facilitate rigorous treatment of voltage/power interac-tions in multi-infeed HVDC systems. Moreover, with other al-lied new indices proposed here and in past works [1]–[3], it is

Manuscript received March 27, 2012; revised September 17, 2012; acceptedOctober 30, 2012. Date of publication January 03, 2013; date of current versionMarch 21, 2013.Paper no. TPWRD-00321-2012.D. H. A. Lee is with the Sarawak Energy, Berhad, Kuching 93050, Malaysia

(e-mail: [email protected]).G. Andersson is with the ETH Zurich, Zurich 8092, Switzerland (e-mail: an-

[email protected]).Digital Object Identifier 10.1109/TPWRD.2012.2227510

shown that a taxanomy of stability and interaction indices forthe analysis of multi-infeed HVDC systems can be establishedwith their interrelationships derived analytically. Furthermore,it is shown that these analytic indices can give a fundamental in-sight of the parametric dependence of the power/voltage inter-actions, which offer a useful tool to aide planning and operationof practical multi-infeed HVDC systems.It is recognized that the analyses in these works have fo-

cused on line-commutated or current-source converters (CSC),but forced-commutated or voltage-source converters (VSC)have also become a viable HVDC transmission alternativeas evidenced by much research and applications using thistechnology recently. Of course, the HVDC converters of higherrating applications and existing multi-infeed systems are stillmostly CSC, but the situation may change in the future withrapidly advancing VSC technology.In principle, the same analyses in these papers are valid for

multi-infeed systems with hybrid VSC and CSC. In such cases,the matrix elements of the Jacobians will be changed dependingon the control mode of the VSC. This will be further consideredin our future work.

II. ANALYTIC FRAMEWORK FOR MULTI-INFEEDHVDC SYSTEMS

A. Taxanomy of Interaction and Stability Indices

In past works, a number of indices were proposed to assessthe voltage/power stability of multi-infeed HVDC systems[1]–[3]. Two such indices were the modal voltage sensitivityfactor (MVSF) and modal power sensitivity factor (MPSF),which are, respectively, the modal ac voltage and dc powersensitivity to incremental variation of modal control variablesin each eigenmode dimension of the power flow Jacobian. Onthe contrary, when variation of the control variables occurs at asingle converter ac bus, nodal indices may be pertinent. In thisrespect, it is possible to derive nodal indices, such as the nodalvoltage sensitivity factor (NVSF) and nodal dc power/currentsensitivity factor (NPSF/NCSF), which can be analyticallyinterrelated with their modal counterparts, as would be shown.The modal and nodal indices described before pertained

to stability or proximity concerns on the multi-infeed HVDCsystem.When the system is operating within a stable regime butvoltage interaction among HVDC converters that affect systemperformance is a more dominant issue, then interaction indicesmay be more pertinent. As mentioned earlier, recent CIGRÉworks [5] had recognized this which motivated the multi-infeedinteraction factor (MIIF) to be proposed. In this respect, it isalso possible to derive nodal indices such as the nodal voltageinteraction factor (NVIF), which is an analytic equivalent of

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AIK AND ANDERSSON: ANALYSIS OF VOLTAGE AND POWER INTERACTIONS IN MULTI-INFEED HVDC SYSTEMS 817

Fig. 1. Taxanomy of indices for the analysis of multi-infeed HVDC systems.

Fig. 2. N-converter multi-infeed HVDC system.

the empirical MIIF, and the nodal dc power/current interactionfactor (NPIF/NCIF). It would also be shown that these nodalinteraction indices can be analytically interrelated with themodal and nodal indices.The foregoing considerations of stability and interaction in-

dices, and their analytic interrelationships, indicate that theyare not unrelated entities but are compatible with one another.This enables a taxanomy of indices, classified into two domainsshown in Fig. 1, to be proposed in these works. It is in the con-text of this taxanomy that the new nodal indices are derived inthese works and they are suitable for a comprehensive analysisof multi-infeed HVDC systems.

B. System Model

In these works, the taxanomy of indices as described earlierwill be applied to the analysis of an -converter multi-infeedHVDC system model shown in Fig. 2. For this model, the con-verters are all assumed to be of the conventional line-commu-tated type.

C. Mathematical Model

In ac/dc power flow, the Jacobian matrix is given by twoforms according to the choice of state variables as

(1)

and

(2)

where , , , and are partial derivatives of theconverter ac bus power-flow equations with respect to thestate variables. , , , , , , , ,and are partial derivatives of the converter ac/dc buspower-flow equations with respect to the state variables. In(1), the converter ac bus voltage magnitude and angle arechosen as state variables, while these and additional dc currentare chosen in (2).If there is no active power change at all with the converter ac

buses, then it may be assumed that 0 and (1) reduces to

(3)

where

(4)

is the reduced ac Jacobian matrix. Similarly, if there is no acactive and reactive power change at all of the converter ac buses,then it may be assumed that and (2) reduce to

(5)

where

(6)

is the reduced ac/dc Jacobian matrix. Note that and aresquare matrices, corresponding to the converter ac/dc

buses and other ac buses eliminated.Equations (3)–(6) constitute the mathematical framework to

derive the nodal indices in these works.

III. DEFINITIONS OF NODAL INDICES

A. Empirical Definition

In [5], an index known as the MIIF was defined as

(7)

where and are the incremental changes in voltage atthe converter ac bus and , respectively, due to an incrementalchange in the ac reactive power at bus . As recommendedin [5], is the incremental ac reactive power required tocause an incremental change in voltage of 1% nominal value(i.e., 0.01 p.u.).


B. Analytic Definition

1) Nodal AC Voltage Indices: It is seen that the MIIF is apractical definition based on empirical information. To developa theoretical perspective of ac voltage interaction analysis ofmulti-infeed HVDC systems, the mathematical framework of(3)–(4) is first applied to the dual converter and then the -con-verter system model of Fig. 2.a) Dual-converter caseFor a dual-converter multi-infeed HVDC system, in(4) is a 2 2 matrix, whose elements , , ,and are given in the Appendix.If it is further assumed that there is only an incrementalchange in the ac reactive power at the converter ac bus

, then 0 and (3) reduces to

(8)

where

(9)

denotes the determinant of the matrix. From (8), twoindices to be known as the nodal voltage sensitivity factor(NVSF) and nodal voltage interaction factor (NVIF) canbe derived as

(10)

Comparing (7) and (10), it is seen that the MIIF and NVIFhave similar definitions but the latter can be derived an-alytically from the matrix elements. Thus, the NVIFis an analytic equivalent of the empirical MIIF. Note alsothat the NVIF and NVSF are nodal indices since they arederived from the incremental change of control variablesin a single degree of freedom (i.e., the ac reactive poweris incrementally changed at the converter ac buses one ata time).Similarly, where there is only an incremental change inthe ac reactive power at the converter ac bus ,then it can be assumed that . Thus, (3) reducesto

(11)

b) N-converter caseFor the -converter system, we may similarly consideran incremental change in the ac reactive power at the -thconverter ac bus only. Rewriting (3), we obtain

(12)

Thus, for and , , ,, (12) yields

...

...

...

...

(13)

where is the reduced ac Jacobian inverse matrix. Di-viding the th by the th row of (13), the nodal voltage in-teraction factor of the th converter ac bus withrespect to an incremental change in the ac reactive powerat the th bus can hence be derived analytically as

(14)

and from the th row of (13), the nodal voltage sensitivityfactor of the th converter ac bus can be derivedanalytically as

(15)

where is the th row, th column, and isthe th row diagonal element of the reduced ac Jacobianinverse matrix. We note that

...

...

(16)

where is the cofactor and is theminor of the th row, th column element of . (Thein (16) denotes the matrix transpose.) Thus, using the

corresponding elements of (16) in (14), (15) also yields

(17)

(18)

In (18), det can also be written as the cofactor ex-pansion about the th row of . Note also in (17) and(18), .It is seen from (17) and (18) that the nodal voltage interac-tion and sensitivity factors for the -converter multi-in-feed HVDC system can be derived analytically from thematrix elements.


2) Nodal DC Power/Current Indices: It is conceivable thatthe interaction among HVDC converters in a multi-infeed con-figuration is also manifested in the electrical coupling betweendc parameters. Hence, it is pertinent to define nodal interactionand stability indices of dc parameters, analogous to those of theconverter ac bus voltages. To derive the nodal indices of dc pa-rameters for the -converter system model in Fig. 2, the math-ematical framework of (5)–(6) is used.If all of the HVDC links operate in constant dc power-con-

stant control mode and only the dc power order of the thHVDC link is incrementally changed, then it may be assumedthat for all . Thus, from (5), we obtain

...

...

...

...

(19)

where is the reduced ac/dc Jacobian inverse matrix. Similarto the approach used in (13) to derive (14) and (15), dividing theth by the th row of (19), the nodal dc current interaction factor

of the th HVDC link with respect to an incrementalchange in the th HVDC link power order for constant dc powercontrol mode can hence be derived analytically as

(20)

and from the th row of (19), the nodal dc current sensitivityfactor of the th HVDC link for constant dc power con-trol mode can be derived analytically as

(21)

where is the th row, th column, and is theth row diagonal element of the reduced ac/dc Jacobian inversematrix. Expressing in terms of the cofactors and minors of, similar to in (16), and using the corresponding ele-

ments in (20) and (21) yields

(22)

(23)

where is the cofactor andis the minor of the th row, th column element of .

On the other hand, if all of the HVDC links operate in constantdc current-constant control mode and only the dc current orderof the th HVDC link is incrementally changed, then it may beassumed that 0 for all . Thus, (5) becomes

...

...

...

...

...

...

...

...

...

(24)

where is the th row, th column element of . Thus,dividing the th by th row in (24), the nodal dc power inter-action factor of the th HVDC link with respect to anincremental change in the th HVDC link current order for con-stant dc current control mode is derived analytically as

(25)

Further, from the th row in (24), the nodal dc power sensitivityfactor of the th HVDC link for constant DC currentcontrol mode is derived analytically as

(26)

Note that if the system model in Fig. 2 reduces to a singleconverter (i.e., ), then (26) is, in fact, the gradient of themaximum power curve of the single-infeed HVDC system.

IV. RELATIONSHIP BETWEEN INDICES

The underlying relationships of the taxanomy of indices inFig. 1 are derived analytically in this section.

A. Nodal and Modal AC Voltage Sensitivity Factor

It is seen from (18) that has an inverse relationshipwith , given by

(27)

Since the MVSF, which is the minimum eigenvalue of , be-comes identically zero with det as shown in [1], [2], thismeans that tends to infinity when the MVSF becomeszero as seen from (27).

B. Nodal DC Current and Modal Power Sensitivity Factor

Similarly, as seen from (23), has an inverse relation-ship with det , given by

(28)

Since the MPSF, which is the minimum eigenvalue of , be-comes identically zero with det as shown in [1] and [2],


this means that tends to infinity when the MPSF be-comes zero as seen from (28).

C. Nodal AC Voltage Sensitivity and Interaction Factor

The relationship between and can be derivedfrom (13) by rewriting it as

...

...

...

...

(29)

Dividing each side of (29) by and using the definitionsof (14) and (15) yields

...

...

...

...

(30)

It is seen that (30) is the product of the th column vector ofthe NVIF matrix with , and that it is orthogonal to all of theth row of for , that is, . 0.Thus, writing (30) for all of the th column of the NVIF matrix,

, yields its matrix form

(31)

where

. . .

. . .. . .

. . .

...

...

Note that is the identity matrix, and . Thus,from (31)

(32)

Thus, is interrelated with the indirectly through. Note that (31) also implies that can be constructed from

NVSF and NVIF. Thus, if it is impractical to form the elementsof analytically, then simulated or measured values of thenodal indices may be used instead.

D. Nodal DC Current Sensitivity and Interaction Factor

Since (19) is of the same form as (13), similar results as in(29)–(32) can be derived to interrelate and .Thus

(33)

(34)

where

. . .

. . .

...

...

Thus, is interrelated with the indirectly through. Note that (33) also implies that can be constructed from

simulated or measured values of NCSF and NCIF.

E. Nodal DC Power Sensitivity and Interaction Factor

From (25) and (26), the and are interrelatedas

(35)

F. Nodal DC Power and AC Voltage Interaction Factor

The definition of in (25) can be rewritten as

(36)

Thus, if the incremental change in current order becomes infin-itesimal in the limit, then

(37)

where , , and for constant dc current controlis given in [1] as

. Note that if , , , then. , where .

Since it is possible for to be greater than unity, thenthe dc interaction for constant dc current control mode may begreater than the ac interaction in a multi-infeed HVDC system.

G. Summary of Relationships Between Indices

The analytic interrelationships among the indices as derivedin Sections IV-A–F may be summarized by the diagram shownin Fig. 3. In Fig. 3, the solid and dashed arrow lines, respectively,denote a direct and an indirect interrelationship between twoindices. For each interrelationship arrow line, the correspondingequation governing the interrelationship, as derived analyticallyin the preceding sections, is also indicated.


Fig. 3. Relationship between interaction and stability indices.

Fig. 4. Nodal voltage indices versus tie-line.

V. PARAMETRIC ANALYSIS OF INTERACTIONS IN THEDUAL-CONVERTER SYSTEM

The parametric dependence of power/voltage interactions ofthe dual-converter multi-infeed HVDC system inFig. 2 is analyzed in this section using the foregoing analyticderivations. Here also, the effective short-circuit ratiolink of the th ac/dc system is defined in the same way as thatof the single-infeed configuration. Thus

(38)

where PBR is the power base ratio of the dc power rating ofthe th HVDC link and that of the referenced th HVDC link.The parameter symbols and values used to derive the plots inFigs. 4–8 are described in the Appendix. These parameter sym-bols used also correspond to those indicated in Fig. 2. Wheredifferent parameter values are used, they are indicated on theassociated figures.

A. Tie-Line Power Flow

The tie-line power flow is varied between 0 and 1 p.u.,while the power injected into the ac/dc system 1 is maintainedconstant (i.e., 1 p.u.).As seen from Fig. 4, the NVIF’s and VSF’s of the converter

ac buses decrease slightly as increases, with a greater de-crease for the converter ac bus 1. The diminishing as

Fig. 5. Nodal voltage indices versus ESCR1.

increases implies a progressively lesser incremental voltagechange at this bus due to the same incremental reactivepower change at the converter ac bus .On the other hand, the consistent implies a consistent

incremental voltage change at the converter ac bus dueto this change . As a result, the ratio of and ,which is , decreases as increases.It is also seen that is greater than for all .

This is because ac/dc system 1 remained electrically strongerthan system 2 , suchthat an incremental reactive power change in the strongersystem 1 causes greater voltage interaction of theweaker system 2 than that of the stronger system 1 caused bythe same change in the weaker system 2 .

B. ESCR

The ESCR of ac/dc system 1 (ESCR1) is varied, while thatof ac/dc system 2 (ESCR2) is maintained constant.AC/DC system 1 is electrically stronger than system 2 as

seen from the NVSF’s in Fig. 5 , resultingin greater than as explained earlier. How-ever, and converge as the electrical strengthof ac/dc system 1 decreases. Before this occurs, the system be-comes unstable as determined by the det or NVSF be-coming zero or infinite, respectively, at .It is also noted that themagnitude of NVIF can be greater than

unity, as seen from Fig. 5 for at .Thus, this can be the case for a multi-infeed configuration withconstituent ac/dc systems that are electrically weak.

C. Power Base Ratio

The power rating of HVDC link 2 is varied while thatof HVDC link 1 is maintained constant. Thus, the PBR isvaried.As seen from Fig. 6, the NVIF of converter ac bus 2 in-

creases with PBR, which is expected since the electrical strengthof ac/dc system 2 decreases correspondingly. This is also seenfrom the becoming increasingly greater than .For example, as PBR increased from 0.5 to 2, ESCR2 decreased


Fig. 6. Nodal voltage indices power base ratio.

Fig. 7. Nodal voltage indices versus .

from 6.17 to 1.54 as computed using (37) with 0.3,0.24, 0.54, 1.08.It is noted that is equal to at ,

meaning that both converter ac buses have the same voltageinteraction propensity. This is due to the equal electrical strengthof both ac/dc systems at this PBR value as also indicated bythe equal and (i.e., ESCR1 and ESCR2 equalto 2.8), as computed using (37) with and otherparameters as before.

D. Tie-Line Impedance

The tie-line impedance is varied while the electricalstrength of both ac/dc systems (ESCR1, ESCR2) are main-tained constant.As seen from Fig. 7, the NVIF’s of both converter ac buses

tend to unity and zero, as tends to zero and a large value, re-spectively, which correspondingly relates to the electrical amal-gamation and decoupling of the ac/dc systems. Furthermore,the NVIF’s are less than unity for all , which is the casewhen at least one of the constituent ac/dc systems is electricallystrong (dashed line plots, , ).However, the NVIF’s can be greater than unity when all con-stituent ac/dc systems are electrically weak and in close elec-trical proximity (solid line plot, 0.4 p.u.and , ), as observed earlier.

Fig. 8. Nodal voltage indices versus ESCR for different control mode.

It is also seen that the NVSF’s of both converter ac busesdecrease as increases, when both constituent ac/dc systemsare electrically weak (solid line NVSF plots). This implies thatthe weak constituent ac/dc systems in close electrical proximityadversely affect instead of lending voltage support to each othermutually. On the other hand, when one of the constituent ac/dcsystems is electrically stronger, increasing increases theNVSF’s of the converter ac bus in the weaker ac/dc system 2(dashed line NVSF2 plot). This implies that the stronger ac/dcsystem is lending positive voltage support to the weaker ac/dcsystem, which is progressively diminished as increases.

E. HVDC Link Control Mode

The control mode of HVDC link 2 is changed to constantcurrent control. In the previous sections, HVDC links 1 and 2operate in constant power control. Similar to Section V-B, theESCR of the ac/dc system 1 (ESCR1) is varied, while that ofac/dc system 2 (ESCR2) is maintained constant.As seen from Fig. 8, the NVIF’s and NVSF’s of converter

ac bus 2 are less than those of converter ac bus 1, even thoughthe electrical strength of ac/dc system 2 is relatively lower

. This is opposite ofthe voltage interaction and sensitivity characteristics in Fig. 5,when both HVDC links operate in constant power control andthe electrical strength of ac/dc system 2 is relatively higher( 1.54, ). This implies the in-herent higher electrical strength of ac/dc system 2 when HVDClink 2 operates in constant current compared to constant powercontrol. The electrical strength of ac/dc system 1 overtakes thatof ac/dc system 2 only when the ESCR1 becomes exceedinglyhigher .

VI. NUMERICAL VERIFICATION

The interrelationships between the various indices, derivedanalytically in Section IV, are numerically verified here. For thispurpose, power flows using PSS/E are simulated on the dual-converter multi-infeed HVDC system in Fig. 2, with the systemparameters and conditions as given in the Appendix.For the simulations to derive the ac voltage indices, a

2-MVar capacitor bank is separately switched off at eachconverter ac bus, and the required variables are computed. For


Fig. 9. Power-flow simulation instants to compute indices.

TABLE ISIMULATED RESPONSE OF AC VOLTAGE, DC POWER, AND CURRENT

TABLE IIEMPIRICALLY AND ANALYTICALLY COMPUTED INDICES

TABLE IIICOMPUTATION OF INTERRELATIONSHIP EQUATIONS

the simulations to derive the dc power and current indices,the power/current order of each HVDC link is separatelyincreased approximately 0.5 1%, and the required variablesare computed. The incremental change of these variables is thedifference between their values at and ,where is the time instant of change (i.e., switching thecapacitor bank or raising the power/current order). A diagramdepicting the power-flow simulation instants is shown in Fig. 9,and the simulated incremental changes of the required variablesare given in Table I. From these, the various indices as definedin Section III are empirically and analytically computed asgiven in Table II. These empirical indices are then substitutedinto the equations governing their relationships in Section IV,and compared with their analytic counterparts as shown inTable III.It is seen from Tables II and III that the indices and their in-

terrelationships, as computed analytically and empirically, aresimilar. The small but noticeable differences between them asindicated in the tables are contributed by the low ESCR2 value

(1.54) and, therefore, a weak ac/dc system 2, in the case study.This caused high-voltage sensitivity at the converter ac bus 2which consequently made it practically difficult to estimate thevoltage magnitudes consistently from the power-flow simula-tions. For higher values of ESCR2 , smaller differenceswere obtained. Another factor possibly contributing to the dif-ferences is the reduced Jacobian matrix versus a full representa-tion in the analytic and PSS/E power-flow model, respectively.Despite these small differences, it is seen that the analyti-

cally and empirically computed indices are sufficiently closelymatched. Thus, the theoretical assertions as presented in theseworks are deemed to be validated. Hence, the analytic indicescan alternatively be computed with those empirically computedfrom power-flow simulations.

VII. CONCLUSIONS

In these works, ac and ac/dc power-flow models were usedas the mathematical framework to derive various voltage/powerindices for multi-infeed HVDC systems. The main motivationof this was to derive an analytic equivalent of the empiricalindex proposed in [5], but also other allied new indices, in orderto facilitate a comprehensive and rigorous analysis of voltage/power interactions in multi-infeed HVDC systems. Moreover,it was shown that the indices proposed in this and in past workscan be organized into a taxanomy through establishing theirunderlying interrelationships analytically. Furthermore, it wasshown that these indices can be used for parametric studies togain a fundamental insight of voltage/power interactions, whichwould otherwise not be so apparent from empirical calculations.The theoretical assertions of these indices were also verifiednumerically using power-flow simulations on a dual-convertermulti-infeed HVDC system, and it was shown that the indicesand their interrelationships, computed from the analytic equa-tions and empirical power-flow simulations, closely matched.

APPENDIX

Expressions for the elements of the reduced ac and ac/dc Ja-cobian matrices ( and ) for the dual-converter multi-in-feed HVDC system (i.e., in Fig. 2) are given as follows.

Reduced AC Jacobian Matrix Elements:


where

Reduced AC/DC Jacobian Matrix Elements:

where

and the subscript denotes rated values. is the convertertransformer short-circuit impedance.

Parameter Values For Parametric Analysis in Section V:Converter extinction angle: .Converter transformer short-circuit impedance:

0.04 p.u.Converter ac bus shunt compensation: 0.54 p.u.,

1.08 p.u.Coupling impedance: 0.5 p.u.DC-link rated power: 1 p.u., 2 p.u.

Power base ratio: .ESCR: 2.8, 1.54.Power flow into the ac/dc system 1: 0.5 p.u.Converter ac bus voltage 1 p.u.System Parameters and Conditions For Numerical Verifi-

cation in Section VI: 0.04, 130MVar,317 MVar, , , 0.3, 0.15,0.25, 300,

300 MW, 600 MW, 400 kV,275 kV, 1.035, 1.045, 80 MW,

REFERENCES

[1] L. Denis, “Voltage and power stability of HVDC systems,” Ph.D. dis-sertation, Dept. Elect. Power Syst., Royal Inst. Technol., TRITA-EES-9801, Stockholm, Sweden, 1998.

[2] L. Denis and G. Andersson, “Voltage stability analysis of multi-infeedHVDC systems,” IEEE Trans. Power Del., vol. 12, no. 1, pp. 547–557,Jan. 1997.

[3] L. Denis and G. Andersson, “Power stability analysis of multi-infeedHVDC systems,” IEEE Trans. Power Del., vol. 13, no. 1, pp. 923–931,Jan. 1998.

[4] CIGRE, “On voltage and power stability in ac/dc systems,” CIGRÉBrochure 222, WG14.05. Paris, France, 2002.

[5] CIGRE, “System with multiple dc infeed,” CIGRÉ WG B4–41 Guide,2008.

[6] E. Rahimi, “Voltage interactions and commutation failure phenomemainmulti-infeed hvdc systems,” Ph.D. dissertation, Dept. Elect. Comput.Eng., Univ. Manitoba, Winnipeg, MB, Canada, 2011.

[7] E. Rahimi, A. M. Gole, J. B. Davies, I. T. Fernando, and K. L. Kent,“Commutation failure analysis in multi-infeed HVDC systems,” IEEETrans. Power Del., vol. 26, no. 1, pp. 378–384, Jan. 2011.

[8] E. Rahimi, A. M. Gole, J. B. Davies, I. T. Fernando, and K. L. Kent,“Commutation failure in single- and multi-infeed HVDC systems,” inProc. 8th Inst. Elect. Eng. Int. Conf. AC—DC Power Transm., London,U.K., p. 182.

[9] I. T. Fernando, K. L. Kent, J. B. Davies, E. Rahimi, and A. M. Gole,“Parameters for planning and evaluation of multi-infeed HVDCschemes,” presented at the CIGRÉ Symp. Syst. Develop. AssetManage. under Restructuring, Osaka, Japan, Nov. 2007.

Denis Lee Hau Aik (M’88) was born in Kuching, Sarawak, East Malaysia. Hereceived the B.Eng. degree in electrical engineering from the National Univer-sity of Singapore in 1984 and the Ph.D. degree in power systems from the RoyalInstitute of Technology, Stockholm, Sweden, in 1998.Since 1984, he has been with the Sarawak Electricity Supply Corporation,

Malaysia, which is wholly owned by its parent company Sarawak EnergyBerhad. His work experiences include power system planning, power systemdesign, and project engineering. His professional and research interests includepower system dynamics, control, stability, and interactions between ac andHVDC systems.

Göran Andersson (M’86–SM’91–F’97) was born in Malmö, Sweden. He re-ceived the M.S. and Ph.D. degrees in engineering physics from the Universityof Lund, Lund, Sweden, in 1975 and 1980, respectively.In 1980, he joined ASESA’s HVDC division and in 1986, he was appointed

Professor in Electric Power Systems, Royal Institute of Technology, Stockholm,Sweden. In 2000, he was appointed to Professor in the Power Systems Group,Swiss Federal Institute of Technology, ETH, Zürich, Switzerland. His researchinterests are in power system analysis and control, particularly issues involvingHVDC and other power-electronics-based equipment.Dr. Andersson is a member of the Royal Swedish Academy of Engineering

Sciences and Royal Swedish Academy of Sciences.

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Analysis of VOltage and Power Interactions in Multi-Infeed HVDC Systems