Methods for Voltage and Power Stabilty Analisys of Emerging HVDC System Confugurations

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Methods for Voltage and Power Stability Analysis ofEmerging HVDC System Configurations

Denis Lee HauAikSarawak ElectricitySupply Corporation

P.O.Box 149 Kuching93700Sarawak, Malaysia

Abstract - Voltage and power stability in HVDC systems havetraditionally been studied for sim plified system models usinganalytical methods based on quasi-static assumptions. Howeverasusage of direct current transmission increases, new systemconfigurations, the corresponding need for new analytical tools,and new concems about adequacy of system modeling haveemerged. In this paper these emerging issuesand needs arediscussed and new analytical m ethodologies that could potentiallymeet those needs are presented ina single framework.

and m ethodologies for the voltage and power stability analysisof these system configurations. These emerging issues andneeds e ssentially prompted th e recent work s in [7]-[9], 1131.

To date these analytical concepts and methodologies,irrespective for single or multi-infeed HVDC systems, havebeen based solely on quasi-static assumptions. The works in[7]-[9], [13], showed that the propo sed analytica l meth ods formulti-infeed HVD C systems also correctly capture and model-

Keywords: Emerging HVDC system co nfigu ration s, voltage and the most important aspects Of voltage/power stability?akin tothe single-infeed case. Nevertheless, there has also been keeninterest to assess the impact of system dynam ics on the power

ower stability, new analytical methodologies

1. INTRODUCTION

An issue of great concern in the planning and operation ofHVDC links is their voltage and power stubil iv. In manyinstances the limit for operating the HV DC links arose fromweak AC system condit ions, i.e. w hen the AC systemconditions at the HV DC link termination location varied to anextent that the AC short-circuit capacity became low relativeto the power rating of the HV DC link. Historically this issuemainly concerned single point-to-point HVDC links,or single-infeed HVDC systems as they are commonly known, sincethese were the most common occurrence of ACDCinterconnections. However,as usage of HVDC transmissionincreases, new situations are also expected to emerge. Theseare where multiple HVD C links terminate electrically close toeach other in an AC system area. Already such system

configurations, which are generally knownas muhi-infeedHVDC systems, are taking shape in the Scandinavian andNorth European power system interconnections, or canpossibly emerge in the ASEAN (Association of South EastAsian N ations) region. Since multi-infeed HV DC systems arean outgrowth of single-infeed HVDC systems, voltage andpower stability problems under weak AC system conditionsconceivably also affect such em erging system configurations.Until recently, however, the system models, concep ts andmethodologies for voltage and power stability analysis ofHVDC systems have mainly been applicable to single-infeedconfigurations. With the emergence of multi-infeed HVDCsystems, there is thus a need for new system models, concepts

stability limit derived under quasi-static assumptions[lo].Also recently, the issue of voltage instability in HVDCsystems due to nonlinear dynamics was addressed in[ll],particularly in situations where nonlinear loads and longHVDC cables are increasingly becoming more common.Investigating these em erging issues conceming the effects ofsystem dynamics require new HVDC system models andanalytical approaches that incorporate the system dynamicsconcerned which hitherto have been neglected under quasi-static assumptions. These issues and needs thus motivated theworks in [lo], [ l l ] , [13].

This paper thus collates the recent works done by the author(s)as describe d above and in [7]-[13], and presents here themethodologies proposed in those w orksas a concerted tool foranalyzing the voltage and power stability of these emergingsystem HVDC configurations, also addressing the emergingissues pertaining to the need for appropriate models andadequacy of HVDC system m odeling.

2. SYSTEM MODELS

2.1 Modeling Emerging System Configurations

Single-infeed HVD C systems are typically represented by aquasi-static system model for voltage and power stabilityanalysis, as shown in Figure 1. Though simplified, it can

0-7803-6338-8/00/ 10.00(~)2000 EEE

Figure I: Classical simpl9ed model of a single-infeed H V Kconfguration

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capture many of the important system phenomena in physicalACDC interconnections. Thus until recently, this systemmodel had been predominantly used in the literature forvoltage and power stability analysis of HV DC systems.

However, recently emerging multi-infeed HVDC systemsworldwide as mentioned earlier have motivated newer systemmodels to be proposed for voltage and power stability analysis

[7]-[9], [13], [4]-[6]. Ma ny types of mu lti-infeed HVDCsystem configurations are possible, for example the chain-type(see Figure 5) or ring-type (see [13]). In general,a multi-infeed HVD C system configurat ion can be constitu ted by bothpoint-to-point and multi-terminal HVDC linksas shown inFigure 2. Such a system con figuration can possibly arise whenthe proposed Baltic Ring and East-West Europe HVDCprojects are completed and integrated with the existing HVDClinks in the southern region of Scandinavia. The work s in [7]-[13] mainly analyzed the chain-type multi-infeed HVDCsystem configuration but generally, the proposedmethodologies are independent of system configuration.

Baltic-Scandinaviamulti-infeed HVDClinks

1 1 1 1 1

West

2.2

pI I I ]East- West Europe multi-terminal HVDC link

Figure 2: A gene ral multi-infeed HVDC system

Adequacy of System Models and Assum ptions

Generally for preliminary investigations on fundamental-frequency voltage and power stability of ACDCinterconnections, t is adequate to represent the AC systemas aconstant Thevenin voltage source with an equivalent short-circuit impedance. This is deemed ustified due to the assumedslower voltage response in the AC system (related to theexcitation control of synchronous machines) compared withthat of the primary DC controls. Moreover for such a quasi-static system model as shown in Figure 1, the dynamicsassociated with the outer-loop DC controls and DCtransmission system are typically neglected since these arealso assumed to have little impact on the voltage/powerstability limits. However there had been recent interestsconceming the impact of these system dynamics on powerstability limits [IO] [13]. Subsequently a HVDC system

model with representationof these system dynam icsas

shownin Figure 3 was proposed together with a dynam ic approach in[IO] to systematically investigate their impact on powerstability limits.As would be seen in section5 the validity of

the quasi-static assumptions critically depends on the dynamicmodel of th e synchronous machine and excitation system.

Figure 3: qYnamic modelof a single-infeed H V K configuation

3. AN ANALYTICAL FRAMEWORK

As discussed in the preceding sections, the newly emergingHVDC system configurations and concems regardingadequacy o f system modeling assumptions had motivated theworks in [7]-131. The analytical methodologies proposed inthose works can be categorized into a general analyticalframework as shown in Figure 4 which accounts for the staticand dynamic approaches to address these new needs andconcerns. However, the works in [7]-[13] mainly focused onquasi-static approaches and th e small-signal analysis techniqueaspect of the dynamic approaches. Large-signal disturbanceanalysis of multi-infeed HVDC systems is included in theanalytical framework for completeness, but is not covered inthis paper. Such analysis may be found elsewhere in theliterature as in [5]-[6]. Therefo re the discuss ions on theanalytical methodologies for multi-infeed HVDC systems inthe following sections will focus only on those particular areasof the analytical frameworkas mentioned above.

STATIC

wI-p-iYNAMIC

I -Figure 4: Framework or voltage andpow er stabiliw analysisofmulti-

infied HVDC gs t ems

4. STATIC APPROACHES

4.1 Sensitivity Techniques

A numb er of analysis meth ods [1]-[3] based on sensitivityconcepts had been proposed for analyzing the voltage/powerstability of weak single-infeed HVD C systems but the v oltagestability factor and maximum available power concepts hadbeen the m ost widely accepted.

The underlying assumption for these static approaches is thatthe instability phenomenon occurs in a very fast time frame

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that the ACDC system dynamics have no appreciable effect.The stability of the ACDC system is thus essentiallydetermined by the system algebraic states that govern thepower flow in the system, and the sensitivity of these systemstates to small changes in controlling system quantities is usedas a m easure of system stability.

4.2 Voltage Sensitivity

The concept of V oltage Stability Factor(VSF) in an A C D Cinterconnection was first proposed by Hammad in[ 2 ] for thesystem model of Figure1 as

AUVSF = . .( 1

In essence this VSF is the sensitivity of the incrementalconverter AC bus voltage, AU , o the incremental change inthe injected reactive power, AQ t the same bus. The VSFindicates the relative stabilityof the system - an increasingpositive VSF indicates a corresponding increase in systemvoltage sensitivity to a small change in injected reactivepower. Ultimately at the system stability limit, the VSFbecomes infinite and the system is unstable for negativeVSF.

AQ

Taking cue from this concept for the single-infeed HVDCsystem, it was extended to analyze the voltage stability ofmulti-infeed HVDC systems in[ 7 ] , 1 3 ] .This entailed use ofthe power flow Jacobian of the system model which is multi-dimensional in the case of multi-infeed HVD C systems. U singa m athematical technique knownas modal analysis the powerflow Jacobian can be reduced and then decomposed intoitsmodal equivalentas

- A 0AQ = - .

' ai

where A6 = 7 [AQ]and A Q = T I [AUIUI are the incrementalmodal reactive power and voltage, respectively, andAi s thei-th eigenmode. q, 4 are the left row and right columneigenvector matrices resulting from the modal decompositionof the power flow reduced Jacobian,r] q iare the eigenvectorscorresponding o the i-th eigenmode.AP, Q are the vectors ofincremental active and reactive power, respectively, of theconverter AC buses. For a multi-infeed HVD C system withNnumber of HVDC links, there will beN eigenmodes but theone with the minimum value determines how close a systemoperating point is to voltage instability. At the voltage stabilityboundary, this critical eigenmode,in, has zero value and fornegative values the system becom es unstable. Intuitively thesecharacteristics ofk n n ( 2 ) is akin to the VSF in 1) for thesingle-infeed case. It can thus be interpretedas the ModalVoltage Sensitivity F actor M S O or the multi-infeed HVDCsystem. A detailed mathematical derivation of the abovemethodology is given in[ 7 ] ,[ 131.

Based on the above technique, a general and comprehensivemethod for voltage stability analysis of multi-infeed HVDCsystems was developed in [ 7 ] , [ 1 3 ] .The method was furtherdeveloped in [ 8 ] , [ 1 3 ] through use of participation factors,computed from the left and right eigenvectors correspondingto the critical mode, to determine the critical system location

in a multi-infeed HVDC system wh ere the most severe systemphenomena are expected to occur, and for evaluating theeffectiveness of remedial actions implemented. The converterAC bus w ith the largest participation factor is the critical bus,meaning that it has the largest involvement in the voltageinstability. Consequently itis also the m ost effective locationfor implementing remedial measures.

4.2.1 Practical Examples

Here an example is given to illustrate the practical applicationof the analytical methodas described in the p receding section.

AC/DC system

new AC interconnectionACsystem 2

e-*pVew HVDC link

ESCR2

Figure 5: Example case of a multi-infeed HVDC system

3 2 SCRI=I. . , . ~ ~ ; , ( r a t e d ) = ~ . f i

: unstable

2.40 0 5 I I 5Figure 6: ESCR-z,, voltage stability graph

(a) Determining vol tage s tabi l i ty margin s

Figure 5 depicts an example system planning scenario wherean existing AC system2 is in the neighborhood of ACDC

system 1 .A new HVDC linkis planned for AC system2 andwith that an interconnection with A C D C system1 is alsoenvisaged, perhaps to export the then excess power of AC D Csystem 2 to 1 In this situation the system planner is interestedto know the voltage stability of the integrated ACDC system,as affected by th e coupling im pedance,z12, nd the EffectiveShort C ircuit Ratio,ESCR, (see [7] or definition) of the newlyestablished AC DC system2. The voltage stability boundaryfor the system model of Figure 5 is thus determined in theESCR-z12parameter spaceas hose parameter values for which

in in ( 2 ) s zero, as shown in Figure6 . Thus the system wouldbe voltage stable at an operating pointP1 and voltage unstableat P2 to a small electrical disturbance at any one of the ACconverter buses, as verified by nonlinear time-domainsimulations given by Figure7 and 8 , respectively.

b) Determ ining critical system locations

Practical multi-infeed HVDC systems may com prise threeormore HVD C links. In such situations, when oneor more of theconstituent ACD C interconnections are electrically weak and

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As explained earlier, another aspect of the works in[7]-[13]was to analyze multi-infeed HVDC systems fiom a dynamicapproach as outlined in the analytical framework in Figure4.

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In line with this, these works had proposed dynamic system 0.96

The proposed dynamic methods in[lo]-[13] are based on amathematical representation of the power system bya set ofdifferential-algebraicequations (DAE) given by;

..... 5 )y f(Y,X,PU)dt0 = d Y * , PI ..... 6)

. * - QMPC BCR=l.4)-

h . .-d I p 4

where y E Y E f l m re the differential and algebraicstates of the system, respectively.p E Y? are the systemparameters. Nonli near function s andg describe the dynamicbehavior of the power system and relationship between thedifferential-algebraicstates, respectively.

Applying the mathematical model given by 5)- 6) to thesingle or multi-infeed HVDC system, the dynam ics of powerand voltage stability in these system configurations areinvestigated,as described in the following.

5.1 Dynamic M aximum Available Power

The impact of system dynamics on the power stability ofHVDC systems can be investigated with the dynamic systemmodel of Figure3 whose mathematical representationis of theform given by 5)- 6) (describing equations are given in[IO][13]). Based on this dynamic system model, the DC power-

current relationship is thus derived. This is done using time-domain simulation with a transient stability programasdistinguished from using only steady-state equations to derivethe maximum power curve for the quasi-static system m odel.The resulting relationship is referred to as the DynamicMaximum Power Curve (DMPC) and the limiting DC powerdeliverable is called theDynamic Maximum Available P ower( D M P ) .

Two approaches to derive the DMPC were proposed in [lo][13]. One approach is to operate the system initially in steady-state under nominal conditions and subsequently ramping theDC current up and down to obtain the corresponding dynamicresponse of the DC power delivered by the converter. Theresulting DC power-current relationship isknown as thenominal-one DMPC. Another approach is similar to the

nominal-one but with the converter initially unloaded. The DCcurrent is then similarly ramped up dynamically to a highervalue. This resulting relationshipis referred to as the nominal-zero DMPC. Since the DMPC is derived under dynamicconditions, a consideration is the D C current ramp upor down

rate. Two categories are generally defined, i.e.slow ramp ratecorresponding to the gradual and manual operator action tochange the DC power order, andfast ramp rate correspondingto the spontaneous and immediate DC power modulationaction.

Using the methodology as described above, an example ofnominal-one, fast DMPC's is shown in Figure 10. For slowD M P C S , it can be also shown as in [lo], [13] that thenominal-one and zero slowDMPC yield a lower maximumavailable power than their quasi-static counterpart. Thisimplies that system dynam ics impact negatively on the powerstability limit derived under quasi-static assumptions. Figure10 shows fast DMPC's derived for exciters with fast andslow response. It is seen that theDMPC's initially closelyfollow plot h which is the equivalent quasi-staticMPC(QMPC) for a dynamic situation where the excitation systemhas not yet responded. This implies that the DC power-currentrelationship derived under quasi-static conditions is valid forthe dynamic conditions during this initial time period.However, beyond this initial period the fastDMPC's departfrom plot h and migrate towards plotg This im plies that theexcitation system has startedto respond since plotg representsan equivalent QMPC for a dynamic situation where theTheveninAC bus voltage is ideally maintained constant. Thusin practical situations, the basic quasi-static assumptions

crucially depend on the excitation system response speed andthe associated DMPC is effective in the interven.ing regionbetween the two equivalentQMPCs, i.e. plotsg and h.

5 2 Nonlinearity-caused Voltage Insta bilit y

Recent works [ll] [13], have investigated the effects ofnonlinear dynamics on the voltage stability of H VDC systems.There the HVDC system is treatedas a parameter dependentsystem of DAE of the form given by 5)- 6) which defines itsunderlying qualitative dynamical structure. When thisdynamical structure changes qualitatively under parametervariation, the system is said to undergobijiwcation and loss ofsystem stability may be associated with this phenomenon.Local bifurcations in which the dynamical structure changesqualitatively in the neighborhood of an operating point, are of

particular interest in this work. In this respect the DAE of theHVDC system are linearized and reduced to a form given by;

..... 7)y AAydt

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1.18-

za l . 1 6 -

s

9 1 1 -

1.14-

1.12-

108 0.k 0.98 ; 1.02 1.04 l.b 6Figure I I : Nonlinear limit cycle

where A = - g,gi f , is the dy namic state matrix. , g,,f are the Jacobian submatrices comprising partial derivativeswith respect to they, x as indicated by the subscripts. ThoughA is linearized its eigenvalues can detect the types of nonlinearbifurcationas described below;

Saddle-node bifurcation: an eigenvalue of A crossing theorigin axis u nder parameter variation.

Hopf b i f i r ca t i on : A com plex eigenvalue pair ofA crossing theimaginary axis as system parameters vary. A Hopf bifurcationis characterized by periodic orbits (limit cycles) emanatingfrom equilibra at the bifurcation point. The periodic orbits canbe supe rc r i t i c a l or subcr i t ical . A subcritical Hopf bihrcationoccurs when an unstable limit cycle coalesces witha stableequilibrium point. A supercritical Hopf bifurcation occurswhen a stable limit cycle coalesces with an unstableequilibrium point.

Based on the nonlinear dynamical methodology describedabove, analytical expressions for the existence of saddle-nodeand Hopf bifurcations may be derived for the HVDC systemmodel of Figure 3 with various system components anddevices incorporated and also assuming typical constantThevenin AC voltage sources. From these analyticalexpressions it can be shown[11] [13] that aperiodic voltagecollapse through saddle-node bifurcation under quasi-static

conditions is equivalent to that under dynmaic conditions.Oscillatory voltage instability through Hopf b ifurcationis alsoa feasible voltage collapse mechanism for HV DC system s.Avariety o f nonlinear phenomena may also be discovered fromthese analytical expressionsas in [ll], [13]. An example isgiven in Figure 1 1 for the system model of Figure3 with along HVDC cable. Here, the system exhibits stable limit cycleor supercritical behavior when the HVDC cable capacitance,which is the bifurcation parameter is increased to a valuecoinciding with the Hopf bifurcation boundary.

6. CONCLUSION

In this paper an overview of recently developed voltage andpower stability analysis methodologies appropriate for newly

emerging HVDC system configurations was provided in asingle analytical framework. These methodologies includeboth quasi-static and dynamic approaches that could addresssome of t he concem s regarding system modelling .adequacyand the impact of system dynam ics on stability limits.

REFERENCES

Ainsworth, J. D., G avrilovic, A., Th anawala, H. L., Static andSynchronous Compensations for HVDC TransmissionConverters Connected to Weak AC Systems, Cigre GeneralSession,Paris, France, Paper 31-01, 1980.

Hamm ad, A. E., Sadek, K., Kauferle, J., A New Approach forthe Analysis of and Solution of AC Voltage Stability Problemsat HVDC Terminal, Proceedings of International Conferenceon DC Power Trunstnission, Montreal, Canada, pp. 164-170,June 1984.

Nayak, 0 B., et. al., Control Sensitivity Indices for StabilityAnalysis of HVDC Systems,IEEE Transactions on PowerDelivery,Vol.10 No.4, pp.2054-2060,Oct.1995.

Szechtman, M., et. al, The Behavior of S everalHVDC LinksTerminating in the Same Load Area,Cigre General Session,Paris, France, Paper 14-201, 1992.

Bui, L. X., ood, V. K., Laurin, S., Dynamic Interactionsbetween HVDC Systems Connected to AC Buses in CloseProximity, IEEE Trunsuctions on Power Delivery, Vo1.6No.1, pp.223-230, January 1991.

Reeve, J., Lane-Smith, S. P., Multi-Infeed HVDC TransientResponse and Recovery Strategies,IEEE Trunsuctions on

Power Delivery,Vol.8 No.4, pp.1995-2001, October 1993.

Lee, H.A., Denis, Andersson,G., Voltage Stability Analysisof Multi-Infeed HVDC Systems, IEEE Trunsuctions onPower Delivery,V01.12 No.3, pp.1309-1316, July 1997.

Lee, H.A., Denis, Andersson,G., Use. f Participation Factorsin the Voltage Stability Analysis of Multi-Infeed HVDCSystems, IEEE Transuctions on Power Delivery,Vol. 13 No. 1,pp.203-211, Jan. 1998.

Lee, H.A., Denis, Andersson,G., Power Stability Analysis ofMulti-Infeed HVDC Systems, IEEE Trunsuctions on PowerDelivery,Vo1.13 No.3, pp.923-931, July 1998.

Lee, H.A., Denis, Andersson,G., Impact of Dynamic SystemModelling on the Power Stability ofHVDC Systems, IEEETransactions on Power Delivery, Vo1.14 No.4, pp.1417-1426,Oct. 1999.

Lee, H.A., Denis, Andersson,G., Nonlinear Dynamics inHVDC Systems, IEEE Trunsucrions on Power Delivery,V01.14 N0.4, pp.1427-1437, Oct. 1999.

Lee, H.A., Denis, Andersson,G., Voltage and Power Stabilityof HVDC Systems: Emerging Issues and New Methodologies,VII Synrposiutn of Specialists in Eleclric Operationul andExpunsion Plunning (SEPOPE), Curitiba, Brazil, May 2000.

Lee, H.A. Denis, Voltage and Power Stability Analysis ofHVDC Systems,Ph.D t hesis, Royal Institute of Technology,Swe&n, TRITA-EES-9801, ISSN -I 100-1607, Mar.1998.

AUTHORS BIOGRAPHY

Denis Lee Hau Aik: obtained his B.Eng degree from the NationalUniversity of Singa pore in 1984 and the Ph.D degree from th e RoyalInstilute of Technology, Sw eden in 1998. He w orks for the Sarawak

Electricity Supply Corporation, a State owned electr icity utility in the EastMalaysian state of Sarawak, as a power system planning engineer. Hiswork res ponsib ilities includes planning and network ana lysis of highvoltage transmission systems. His technical interests is primarily instability, control, and dynamics of power systems, particularly in thevoltage and power interactions ofA C E nterconnections.

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Methods for Voltage and Power Stabilty Analisys of Emerging HVDC System Confugurations

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