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HVDC POWER & VOLTAGE STABILITY 2014

ABSTARCT

Traditionally HVDC links have been built as single point to- point AC/DC interconnections,

or single-in feed HVDC systems as they are commonly knon! Hoever in recent years, as

the use of HVDC transmission continues to develop situations have arisen here multiple

HVDC links terminate in close pro"imity in a common AC system area! These emerging

system configurations, generally knon as multi-in feed HVDC systems, are particularly

inherent in regional poer systems predominantly interconnected by HVDC links! Already

such configurations are apparent in the #candinavian and $orth %uropean poer systeminterconnection! %lsehere in rapidly industriali&ing regions such as #outh America and the

A#%A$ 'Association of #outh %ast Asian $ations(, multi-in feed HVDC systems potentially

arise!

A phenomenon that is of great concern in the planning and operation of HVDC systems is

voltage/poer stability! )n many cases this has been limiting for the operation of a link during

eak AC system conditions! Also in AC systems ithout HVDC links voltage stability has

been of concern in the poer industry during recent years and a number of methods and

tools for studying this phenomenon have been developed!

Despite the fact that many aspects of the voltage stability problem are identical for HVDC

systems and *pure+ AC systems, much of the ork has been made focusing on only one ofthese to system types! This paper provides an attempt to overcome this and a number of

methods traditionally mainly applied to AC systems ill be applied to various types of

HVDC poer systems! The paper is a summary of ork that has been done by the authors

during the last years and much of the ork has been reported earlier in refs! . 01. of the

paper!

The paper starts ith a presentation and discussion of the models used and their limitations!

#ome of the concepts in modal analysis of voltage stability are revieed and this method is

then applied to multi-in feed HVDC systems!

2y this method stability margins can be calculated and critical system locations can be

identified! )n the analysis of single-in feed HVDC systems 3a"imum 4oer Curves '34C(

have been e"tensively used! This concept is e"tended to multi-in feed HVDC systems and it

provides additional insight into the analysis! )n the paper the findings gained by the analytical

tools are verified by time domain simulations!

The methods described in the previous paragraph are based on static, or rather 5uasi-static,

models! )t has been shon that these methods correctly capture and model important aspects

of voltage/poer stability, but not all!

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C6$T%$T1.INTRODUCTION2

2.SYSTEM MODELINGISSUES..5

3.QUSAI STEADY STATE

METHOD..6

4.DYNAMICMETHOD.14

5.CONCULSION.23

6.REFERENCE

..24

1. INTRODUCTION

Traditionally HVDC links have been built as single point to-point AC/DC interconnections,

or single-in feed HVDC systems as they are commonly knon! Hoever in recent years, as

the use of HVDC transmission continues to develop situations have arisen here multiple

HVDC links terminate in close pro"imity in a common AC system area 0.!

These emerging system configurations, generally knon as multi-in feed HVDC systems, are

particularly inherent in regional poer systems predominantly interconnected by HVDC

links! Already such configurations are apparent in the #candinavian and

$orth %uropean poer system interconnection! %lsehere in rapidly industriali&ing regions

such as #outh America and the A#%A$ 'Association of #outh %ast Asian $ations(, multi-in

feed HVDC systems potentially arise!

)n the past, voltage and poer instability have been e"perienced in electrically eak single-in

feed HVDC systems, i!e! hen the HVDC link is terminated at an ACDept.of EEE

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system location of lo short-circuit capacity relative to the poer rating of the HVDC link!

These problems have traditionally been studied using analytical methods based

on 5uasi steady-state assumptions 7.-8.!

9or multi in feed HVDC systems, adverse AC/DC interactions resulting in voltage and poer

instability are also e"pected to arise hen one or more of the constituent AC/DC

interconnections are electrically eak, similar to

the situation for single-in feed HVDC systems! #ince the multi-in feed HVDC system is an

outgroth of the single in feed case, many similarities e"ist beteen them! Thusthe e"isting analytical concepts and tools developed for single-in feed HVDC systems may be

adapted for the analysis of multi-in feed HVDC systems!

$evertheless, multi-in feed HVDC systems are of recent origin and distinct from their

single-in feed counterpart, ne analytical concepts and tools are therefore also needed to

investigate the associated voltage and poer instability problems! This paper presents these

ne analytical concepts and a comprehensive tool for analysis of these nely emerging

HVDC system configurations!To date these analytical concepts and tools, irrespective have

only been based on 5uasi steady-state assumptions! This paper also addresses voltage and

poer stability in HVDC systems from a dynamic approach in line ithrecent keen poer

industry interest in this conte"t!

)n particular the impact of dynamic system modeling on common industry assumptions

based on 5uasi steady state conditions are e"amined! Also the nely emerging

issue of nonlinear dynamics in poer systems are addressed in the conte"t of HVDC systems

particularly here poer system devices such as nonlinear loads and long HVDC cables are

increasingly becoming common! :sing a nonlinear dynamical theoretic methodology, the

role of these devices in the aperiodic and oscillatory voltage collapse mechanisms of HVDC

systems are also demonstrated in this paper!

2. SYSTEM MODELING ISSUES

2.1 Modeling Emerging Sysem Con!ig"r#ions

A typical system model to represent HVDC systems for 5uasi steady-state stability analysis is

as shon in 9igure

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0! This is a simplified single-in feed HVDC system model that can capture many of the

important system phenomena in physical AC/DC interconnections!

Figure 1: Classical simplified model of a single-in feed HVDC Configuration 2ased on this

system model, analytical concepts such as3a"imum Available 4oer 'MAP(, Voltage

#ensitivity9actor 'VSF( and Control #ensitivity )nde" 'CSI( havebeen proposed 7.-8.!

Hoever, recently emerging multi-infeed HVDC systems orldide as e"plained in the

previous section have motivated neer system models to be proposed for use in voltage and

poer stability analysis .-;., 07.-01.! 6ne possible type of multiinfeed situation is the

ring-type configuration here a multi-terminal HVDC system rings a large city to in


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4erhaps a most general multi-in feed HVDC system configuration ould ultimately be one

constituted by point-to-point as ell as multi-terminal HVDC links! This scenario could

possibly come about ith the reali&ation of the proposed 2altic =ing and %ast->est %urope

HVDC pro


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@enerally for preliminary investigations on fundamental fre5uency voltage/poer stability of

HVDC systems, it is ade5uate to represent the AC system as a constant Thevenin voltage

source ith an e5uivalent short-circuit impedance! This is deemed


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The concept of voltage stability or sensitivity can be similarly e"tended to the multi-infeed

situation using a mathematical techni5ue knon as modal analysis ., 01.! #tarting from the

ell knon acobian of the $eton-=aphson poer flo solution, flo solution,

acobian$% for the multi-infeed HVDC system can be obtained by assuming that there is no

active poer in


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'.1.2 0#ri%iion #%ors

:sing the left and right eigenvectors corresponding to an eigen value, the participation factor

4ki of converter AC bus k in the i-th eigen mode may be defined asE

here G ki, I ik are the k-th element of the right column and left ro eigenvector,

respectively, of the i-th eigen mode! 4hysically, I ik is a measure of the activity of the

converter AC bus k in the i-th voltage variation mode! G ki is the eighting of the

contribution of this activity, and so their product is a measure of the net participation of the

converter AC bus k in the i-th voltage variation eigen mode.

The bus participation factors computed from the critical mode provide information on the

critical system location of voltage instability! The converter AC bus ith the largest

participation factor is the critical bus, meaning that it has the largest involvement in thevoltage instability! Conse5uently it is also the most effective location for implementing

remedial measures K., 01.!

F()*%$ 6+ E,-/0$ -$ - /0-() $-%(

'.1.' 0r#%i%#l E#m&les

3odal analysis as described in the preceding sections can be applied as a comprehensive

planning tool as illustrated in the practical e"amples belo!

'a( Determining voltage stability margins

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9igure depicts an e"ample system-planning scenario here an AC system 7 is e"isting in

the neighborhood of AC/DC system 0! A ne HVDC link is planned for AC system 7 and

ith that an interconnection ith AC/DC system 0 is also envisaged, perhaps to e"port the

then e"cess poer of AC/DC system 7 to 0! )n this situation the system planner is interested

to kno the voltage stability of the integrated AC/DC system, as affected by the coupling

impedance, &07, and the %#C= of the nely established AC/DC system 7! :sing modal

analysis, the voltage stability boundary for the system model of 9igure can be determined in

the %#C=-&07 parameter space as shon in 9igure ! Thus the system ould be voltagestable at an operating point 40 and voltage unstable at 47 to a small system disturbance at any

one of the AC converter buses, as verified by nonlinear time-domain simulations shon in

9igure K and ;, respectively!

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'b( Determining Critical system locations

9igure 1 depicts another e"ample here the multi-infeed HVDC system has three HVDC

links! )f more than one of the constituent HVDC links are electrically eak and reactive

poer compensation e5uipment, for e"ample a static var compensator '#VC(, is re5uired to

provide voltage stability support, a 5uestion arises as to here the e5uipment should be

installed! >ith the use of participation factors it is possible to identify the most effective or

critical converter AC bus to install the #VC! )n this e"ample, ith the given sample systemparameters 'see 9igure 1( and %ffective #hort Circuit =atios '%#C=(, bus 0 has the largest

bus participation factor 'see 9igure 00( and thus the most effective system location.

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This is seen in 9igure 00 from the stable voltage response to a system disturbance hen the

#VC is installed at bus 0! )n comparison, the system is voltage unstable to the same

disturbance hen the same #VC is installed at bus 1 as seen from the voltage response in

9igure 0B!

1!7 4oer #tability

The concept of 3a"imum Available 4oer '3A4( as first introduced by Ainsorth et! al!

in 7. using a system model as shon in 9igure 0 ith the converter operating in constant

e"tinction angle 'C%A( control mode! This is the ma"imum DC poer the converter is

capable of delivering to the AC system corresponding to a DC current )3A4! A further

increase in the DC current )d beyond )3A4 actually results in a decrease in the DC poer

4d! This is due to the larger percentage decrease in the converter AC bus voltage as compared

ith the increase in )d, resulting in a net decrease in 4d! #uch a phenomenon corresponds

ith unstable system behavior, thus the 3A4 condition determines the poer stability limit

of the AC/DC interconnection! 3athematically this point corresponds to the condition thatE

1!7!0 3a"imum Available 4oer

The concept of ma"imum available poer applied to single-infeed HVDC systems can be

e"tended to the multi-infeed case by using a similar approach to the one described in section

1!0!0 ;., 01.! )n this case the poer flo e5uations for the converter AC buses are

augmented ith those for the associated DC buses, neglecting all other non-converter ACDept.of EEE

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buses! Applying the ell-knon $eton-=aphson poer flo solution for these e5uations

the AC/DC acobian is thus obtained! This is reduced to a DC acobian dc= for the multi-

infeed HVDC system by eliminating all the converter AC buses based on the assumption that

there are no active/reactive poer in


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&ero it is seen that the

multi-infeed HVDC system reaches a system condition akin to the 3A4 condition defined in

'( for the single-infeed case! Thus 'K( and ';( may also be used to define the poer stability

limit of the multi- infeed HVDC system! %5uation 'K( defines a direct though constrained

incremental DC poer-current relationship for a single DC link in the multi-infeed HVDC

system, much alike the single-infeed case such that this approach is knon as #tandard

3a"imum 4oer Curve '#34C( in ;.,01., hereas e5uation ';( defines an unconstrained

modal relationship and is knon as the 3odal 3a"imum 4oer Curve '334C( in ;., 01.!

The to approaches yield poer stability boundaries of the multi-infeed HVDC system that

are non-e5uivalent since they are derived under different system conditions, i!e! &%-($

- *&%-($ DC *%%$& 7-%(-&( % $ SM8C - MM8C $%$/$&(7$0:.)n a practical conte"t, the #34C poer stability boundary applies to a situation here one

constituent HVDC link of the multi-infeed HVDC system operates in constant- poer control

mode hile the others are in constant-DC current control, hereas the 334C poer stability

boundary applies to a situation here all the constituent HVDC links operate in constant-

poer control mode! The non-coincident poer stability boundaries is illustrated by a

practical e"ample shon in 9igure 07 using sample system parameters for the system model

of 9igure ! )t is noted that the 334C poer stability boundary gives a more conservative

estimate of the system stability region as compared ith the #34C boundary! This is

physically intuitive since the 334C method allos more degrees of freedom for DC current

variation to cause the system to become unstable!

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F()*%$ 12+ MM8C - SM8C /'$% &-;(0(&: )%-/#

8! $%> A$AMNT)CAM 3%TH6D6M6@)%#O DN$A3)C 3%TH6D#

As e"plained in section 7!7, the 5uasi-static methods for voltage/poer stability analysis of

single or multi-infeed HVDC systems neglect the various dynamics present in a real poer

system! Hoever there had been recent industry concerns regarding the impact of system

dynamics on the stability limits of HVDC systems ?., .! )n line ith these concerns, recent

orks 0B., 00., 01., had proposed dynamic system models and methods to address these

concerns systematically!

These dynamic methods are based on a mathematical representation of the poer system by a

set of differential-algebraic e5uations 'DA%( given byE

here m are the differential and algebraic states of the system,

respectively! P p are the system parameters! $onlinear functions f and g describe the

dynamic behavior of the poer system and relationship beteen the differential-algebraic

states, respectively!

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Applying the mathematical model given by '0B(-'00( to the single or multi-infeed HVDC

system, the dynamics of poer and voltage stability in these system configurations are

investigated, as described in the folloing!

8!0 Dynamic 3a"imum Available 4oer

The impact of system dynamics on the poer stability of HVDC systems can be investigated

ith the dynamic system model of 9igure ? hose mathematical representation is of the formgiven by '0B(-'00( 'describing e5uations are given in 0B., 01.(! 2ased on this dynamic

system model, the DC poer-current relationship is thus derived!

This is done using time- domain simulation ith a transient stability program as distinguished

from using only steady-state e5uations to derive the ma"imum poer curve for the 5uasi-

static system model! The resulting relationship is referred to as the Dynamic 3a"imum

4oer Curve 'D34C( and the limiting DC poer deliverable is called the Dynamic

3a"imum Available 4oer 'D3A4(!To approaches to derive the D34C ere proposed in

0B., 01.!

6ne approach is to operate the system initially in steady-state under nominal conditions andsubse5uently ramping the DC current up and don to obtain the corresponding dynamic

response of the DC poer delivered by the converter! The resulting DC poer-current

relationship is knon as the nominal-one D34C! Another approach is similar to the

nominal-one but ith the converter initially unloaded! The DC current is then similarly

ramped up dynamically to a higher value! This resulting relationship is referred to as the

nominal-&ero D34C !

#ince the D34C is derived under dynamic conditions, a consideration is the DC current ramp

up or don rate! To categories are generally defined, i!e! slo ramp rate corresponding to

the gradual and manual operator action to change the DC poer order, and fast ramp rate

corresponding to the spontaneous and immediate DC poer modulation action!

:sing the methodology as described above, e"amples of nominal-one and &ero slo D34CQs

and fast D34CQs are derived as shon in 9igure 01 and 08, respectively! )n 9igure 01, for the

same sample system parameters it can be seen that the slo D34CQs 'plot a, b( are loer

than the corresponding Ruasi-static 3a"imum 4oer Curve 'R34C( 'plot c( derived from

using steady-state assumptions!

This implies that system dynamics impact negatively on the poer stability of the system

model such that the ma"imum poer achievable is less than the 5uasi-static case! 9igure 08

shos fast D34CQs derived for e"citers ith fast and slo response! )t is seen that the

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D34CQs initially closely follo plot h hich is the e5uivalent R34C for a dynamic situation

here the e"citation system has not yet responded, i!e! the R34C derived ith system

strength specified using aggregate of the synchronous machine reactance and AC system

impedance! This implies that the DC poer-current relationship derived under 5uasi-static

assumptions is valid for the dynamic conditions during this initial time period! Hoever,

beyond this initial period the fast D34CQs depart from plot h and migrate toards plot g!

This implies that the e"citation system has started to respond since plot g represents an

e5uivalent R34C for a dynamic situation here the Thevenin AC bus voltage is ideally

maintained constant, i!e! the R34C derived ith the system strength specified using only theAC system impedance and e"cluding the synchronous machine reactance! Thus in practical

situations, the basic 5uasi- static assumptions crucially depend on the e"citation system

response speed and synchronous machine model, and the associated D34C is effective in the

intervening region beteen the to e5uivalent R34CQs, i!e! plots g and h!

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S Hopf bifurcation O A comple" eigenvalue pair of A crossing the imaginary a"is as system

parameters vary! A Hopf bifurcation is characteri&ed by periodic orbits 'limit cycles(

emanating from e5uilibra at the bifurcation point! The periodic orbits can be supercritical or

subcritical! A subcritical Hopf bifurcation occurs hen an unstable limit cycle coalesces ith

a stable e5uilibrium point! A supercritical Hopf bifurcation occurs hen a stable limit cycle

coalesces ith an unstable e5uilibrium point! 2ased on the nonlinear dynamical methodology

described above, analytical e"pressions for the e"istence of saddle- node and Hopf

bifurcations may be derived for the HVDC system model of 9igure ? ith various systemcomponents and devices incorporated and also assuming typical constant Thevenin AC

voltage sources 'analytical e"pressions can be found in 00., 01.(! These system components

and devices are such as nonlinear loads or an #VC connected at the inverter AC bus, or a

long HVDC cable!

9rom these analytical e"pressions it can be shon 00., 01. that aperiodic voltage collapse

through saddle-node bifurcation under 5uasi-static conditions is e5uivalent to that under

dynamic conditions! 6scillatory voltage instability through nonlinear behavior of Hopf

bifurcation is also a feasible voltage collapse mechanism for HVDC systems! To illustrate

this, the system model of 9igure ? ith a long DC cable is used as an e"ample! 9rom theanalytical e"pressions, an admissible solution of the Hopf bifurcation boundary Croot0 may

be derived as shon in 9igure 0? for the C-%#C= parameter space! A system operating point

is chosen to be '%#C=b,Cb( hich is on the Hopf bifurcation boundary! The nature of system

behavior around this operating point cannot be determined by eigenvalue analysis since the

phenomena is essentially nonlinear! )t can hoever be demonstrated using time-domain

simulations ith a transient stability program

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U() $ DC -;0$ -/-(&-$ - $ ;(*%-&( /-%-$&$% F()*%$ 1.>42>3 $ '#(# ( - 0(&&0$ )%$-&$% - $ ;(*%-&( 7-0*$ C;9 (.$. C?C;.F% ( 7-0*$ C9 - 0-%)$ 8=>.>15 /*! % -00 8=>.>>2 /*! (&*%;-$-*$ $ :&$ & $7$0/ *&-($ (00-&( - #' ( F()*%$ 1< - 19%$/$&(7$0:. T#$$ (/0: -& - &-;0$ 0((& :0$ $,(& -%* $ %()(-0 &-;0$/$%-&() /(& '#(# #- &*%$ *&-;0$ *$% /-%-$&$% 7-%(-&( *# -& $:&$ ( -&&%-&$ & $ &-;0$ 0((& :0$ '#$ ;*/$ *&($ $$ /#-$/%&%-(& F()*%$ 16! % '(( (& F()*%$ 1@!. T#* $ H/ ;(*%-&( (*/$%%(&(-0. T#$$ 0($-% /#$$- -& ;$ /%$(&$ ;: $ 0($-%($$B*-&( )(7$ ;: $ :-( &-&$ -&%(, A ( 12!. A $$ ( F()*%$ 2>9 $%$/$ $ 0($-%($ :&$ ( &(00 *&-;0$ & $ -$ (&*%;-$

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3. CONCLUSION

T#( /-/$% /%7($ - 7$%7($' %$$&0: $7$0/$ B*-(&-&( - :-(70&-)$/'$% &-;(0(&: --0:( $)($ -//%/%(-&$ % $'0: $$%)()HDC :&$ ()*%-&(. I& '- #' #' $ B*-(&-&( $/& -$ *(&$ % $ ()0$($$ :&$ $0 *0 ;$ $,&$$ & $ #()#$%

($( HDC :&$ $0.

D:-( $ /%$$&$ #-7$ -0 ;$$ #' & ;$ -;0$ & -%$ $ $ /'$% (*&%: $% %$)-%() :&$ $00() -$B*-: - $ (/-& :&$ :-( &-;(0&: 0((&9 - '$00 - -(0(&-&() *-$&-0 *$%&-() $ 0$$% $&-;0(#$ (00-&%: 70&-)$ 00-/$ $#-( ( HDC :&$.

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4. REERENCES

[1] S$#&-9 M.9 $&. -0.9 T#$ $#-7(*% S$7$%-0 HDC L( T$%(-&() ( $S-$ L- A%$-9 C()%$ G$$%-0 S$(9 8-%(9 F%-$9 8-/$% 142>19 12.[2] A('% J. D.9 G-7%(07(9 A.9 T#--'-0-9 H. L.9 S&-&( - S:#%*

C/$-&( % HDC T%-(( C7$%&$% C$&$ & "$- ACS:&$9 C()%$ G$$%-0 S$(9 8-%(9 F%-$9 8-/$% 31>19 1.[3] H--9 A. E.9 S-$9 K.9 K-*$%0$9 J.9 A N$' A//%-# % $ A-0:( -S0*&( AC 0&-)$ S&-;(0(&: 8%;0$ -& HDC T$%(-09 8%$$() I&$%-&(-0 C$%$$ DC 8'$% T%-((9 M&%$-09 C---9 //. 1641@>9 J*$ 1


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[12] L$$9 H.A.9 D$(9 A$%9 G.9 M-0 A-0:( M*0&(I$$ HDCS:&$9 8%$$() $ I&$%-&(-0 8'$% E)($$%() C$%$$9S()-/%$9 0.19 //.31>3159 M-: 1@.[13] L$$9 H.A. D$(9 0&-)$ - 8'$% S&-;(0(&: A-0:( HDC S:&$9 8#D$(9 R:-0 I&(&*&$ T$#0):9 S'$$9 TRITAEES19 ISSN11>>16>@9M-%.1

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