Effect of AC system impedance on theregulation and, hence, the stability margin

of HVDC schemesR. Yacamini, M.Sc, C.Eng., M.I.E.E., and A.I. Taalab, M.Sc, Ph.D.

Indexing terms: Load and voltage regulation, Power systems and plant, Stability

Abstract: In the paper, a mathematical analysis of how the AC system network parameters, including theharmonic filter impedances, affect the regulation of an HVDC scheme, is given. The regulation and, in particu-lar, the sign of the regulation will be one of the parameters which will affect the stability of the overall scheme.The paper shows how increasing the impedance at both the rectifier and inverter busbars will either improve ordegrade the overall system stability margin, the rectifier impedance improving and the inverter degrading. Theimportance of a correct assessment of the AC system damping angles is emphasised, by showing the effect on thestability margin and on the correct rating of reactive-compensation equipment. The paper aims to demonstratethe effect of the impedances by carrying out a parametric analysis, which will be helpful to system designengineers in setting up the system which they must model to determine overall stability.

List of symbols

Vs = RMS value of infinite busbar phase voltageVf = RMS value of filter (or convertor) busbar phase

voltageZs = equivalent system impedance at 50 Hz\j/ = AC system impedance angle0 = displacement angle/i = fundamental component of the AC current

referred to the primary of the convertor trans-former

VY = fundamental component of voltage referred tothe primary of the convertor transformer

Xc = harmonic filter reactance at 50 HzXt = convertor transformer reactanceXs, Rs= AC system reactance and resistance, respectivelyIc = fundamental current in the filterIs = fundamental current in the system impedanceVf = RMS value of the phase voltage across the filtera = firing or delay anglefj. = overlap angle

1 Introduction

When comparisons are made between different HVDCschemes and the relative stability is being assessed, oneparameter which is widely used as a comparative feature isthe short-circuit ratio (SCR) at the inverter busbar. Theability of a DC link to operate with an SCR as low as twoor three is often taken as an indication of the quality of thecontrol system. As important as the SCR, however, is theimpedance angle (or damping angle) of the AC system intowhich the DC link will be feeding, this being a measure ofthe resistive component of the network. It has been stated[1] that increasing the damping (or reducing the dampingangle) improves the stability of the system considerably.However, no explanation of how reducing the AC systemdamping angle improves the stability and regulation of anHVDC scheme has been reported. It has also been stated[2] that the harmonic filters, together with any shuntcapacitance, effectively increase the impedance of thesystem at the fundamental frequency. This impliesincreased regulation effects and increased inverter control

Paper 3943C (P9, PI 1, C6), received 26th October 1984Mr. Yacamini is with the Department of Engineering, University of Aberdeen,Marischal College, Aberdeen AB9 IAS, United Kingdom, and Dr. Taalab is withMenofia University, Egypt

difficulties, i.e. the effective negative resistance at the inver-ter is increased. An analytical method is needed to showthe effect of the filter capacitance and other componentson the stability of the scheme. Therefore, it is of particularinterest to offer a simplified mathematical analysis to showhow the A.C system damping angle \jj, impedance magni-tude Zs and the size of the filter capacitance affect the sta-bility margin of an HVDC scheme.

The purpose of this paper is therefore, first, to showhow regulation at the rectifier and inverter busbars can behelpful in determining whether a system will be morestable or less stable, i.e. how it will affect the stabilitymargin. Equations have been developed which will accu-rately describe the regulation; these can then be used tocarry out the parametric analysis involving system imped-ance, system damping and harmonic filter size. In carryingout this analysis, it is assumed that there is an operatingsystem, and that, for example, the control system feedbackgains would be held constant during any changes of ACsystem parameters. It is not therefore intended that themethod described in this paper should be used to deter-mine whether the system is stable or not. The method is,however, useful at the early stages of system design intrying to determine the worst case for modelling either bycomputer or simulator. If the convertor controls are to beconsidered, there are two different modelling approacheswhich can be used. An example of the mathematicalapproach using small signal stability analysis can be foundin Reference 3, for example. An example of the physicalmodelling approach which shows the effect of all majorsystem parameters and treats the scheme as a multi-variable control system can be found in Reference 4. Bothof these papers come to the same general conclusions, i.e.that the stability margins were, in general, enhanced for alower short-circuit ratio at the rectifier and higher short-circuit ratio at the inverter. It was during the work andpreparation of the latter paper that the authors felt that amore descriptive explanation of the effect was needed,resulting in this paper.

At the design stage of a proposed system, insufficientinformation about AC system impedance, or lack of con-fidence in the available information, would tend to encour-age the use of pessimistic assumptions for SCR anddamping angle. Typically, an angle of 75°-85° has beenused for simulator studies. The actual impedance anglemay be less than this, which means the actual system willbe more damped and the system response will be animprovement on that obtained in the study. How the value

202 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 4, JULY 1985

of the AC impedance angle, at both rectifier and inverterends, affects the stability and regulation of an HVDCscheme is shown in the following Section.

2 Effect of regulation on convertor impedance andstability margin

In this analysis, the AC network feeding the convertor maybe represented, for frequencies close to normal frequencyof say 50 Hz, by an equivalent inductance Ls in series withan equivalent resistance Rs, with a capacitance C, con-nected in parallel to the terminals of the convertor station.This capacitance included in the AC harmonic filters mayprovide reactive power typically equal to about half thereactive power consumed. The circuit diagram of this sim-plified representation of the AC network is shown in Fig.1. The phasor diagram which relates voltages, currents anddisplacement factor </> of the inverter AC network is givenin Fig. 2.

I,

Fig. 1

load

AC system representation

The effect of the AC system network parameters on thestability margin of an HVDC scheme can be establishedusing the derivative where Vf is the AC phasevoltage at the inverter AC busbar, and 7t is the fundamen-

DC line. As the DC side current changes, so the AC sidecurrent changes, resulting in voltage changes at the ACbusbar which further increases the negative resistanceeffect. The amount of this negative resistance thereforedepends on the value of the voltage drop due to com-mutation and fundamental voltage regulation. The effect ofthe system impedance on the commutation voltage drop isof little importance [5], as it can be shown that the idealharmonic filter provides a perfect sinusoidal waveform atthe convertor busbar. However, the main effect of the ACsystem impedance is on the fundamental AC voltage dropor convertor busbar voltage regulation. Increasing regula-tion, as shown by the parameter dVfldlx, will give a highernegative resistance effect and therefore more tendencytowards instability.

At the rectifier end, an increase of the DC line currentcauses a reduction of the rectifier DC voltage, as it does atthe inverter. However, reduction of the voltage at the recti-fier tends to reduce the current in the link; in other words,provides an inherent damping to the original disturbance.This means that, as dVfjdlx increases, the amount ofinherent damping also increases, which results in animprovement of the stability margin.

To summarise, therefore, a high dVj/dl^ at the rectifieris desirable; at the inverter it is undesirable.

3 Derivation of the AC system voltage equationsfor the inverter

With reference to the phasor diagram of Fig. 2, the deriv-ative of the AC busbar voltage Vf with respect to the fun-damental component of current 7X is derived and given inAppendix 8. The result of this derivation is rewritten as

dVf - <A) - Y cos <t>\ ^ J - — il/) — —r sin

X. X,

(1)

^¥1

The displacement angle 4> can be expressed in terms of thefiring angle a and the overlap angle pL, as shown in Appen-dix 8.2:

<f> ^ (a 4- O.Sfi) (2)

Rcl.

Fig. 2 Phasor diagram for the inverter AC system

tal component of the AC current referred to the primary ofthe convertor transformer. The method used to examinethe effect of a particular parameter is to fix all other par-ameters, except the one under investigation; this is thenchanged gradually and the ratio dVfldlx calculated aftereach change. If, at the inverter station, dVfldlx increaseswhen a certain parameter is increased, then the effect ofincreasing this parameter will be to reduce the stabilitymargin of the total scheme as follows:

At the inverter station, the normal mode of control is tokeep the extinction angle constant, which, because ofoverlap effects and reduction in the AC busbar voltage,results in the DC voltage at the inverter terminals reducingas the DC line current increases. Therefore, at low fre-quencies, and to changes in DC current, the inverter effec-tively presents a negative resistance characteristic to the

As far as the overlap angle is concerned, the system imped-ance Zs is of little importance [4]. This agrees with theresult of a test which was made by observing the wave-forms on a simulator. Therefore, the equation whichrelates the overlap angle, the DC current Id, the convertortransformer leakage inductance Lf, the nominal ACbusbar voltage Vf, and the extinction angle y at the inver-ter end may be written as

Id = \—L {cos y - cos (y + fi)}2coLt

(3)

where

N = convertor transformer turns ratio

co = nominal frequency, rad/s

The fundamental component of the convertor AC current7X in terms of the DC current is 7X = yj6ljn. The deriv-ative of the displacement angle <p (eqn. 2) with respect to 7X

is

d<j> djtx + 0.5/i) djlSO - (ji + y) + 0.5/*)

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 4, JULY 1985 203

assuming the extinction angle y is constant, then

diy ~ ~ Inldifferentiating Id in eqn. 3 with respect to \i results in

dld J^NVr— = sin (y + n)d\i 2coLt

therefore

dlx J6NV,~d^ = 0JSx2coLt

Sm(y + ^

Substitute from eqn. 5 into eqn. 4, then

# 0.78 x 2 x a)Lt

,. = — 0.5

Table 1: System parameters

Parameter Per-unit value(4) DC current ld

Magnitude of ACsystem impedance Zs

AC voltage V,

AC filter reactanceX,. at 50 Hz

= 0.3

(5)0.085

= 3.257

Convertor transformerleakage reactance Xt

Fundamental component 0.78of transformersecondary current/ ,=0.78/d

sin (y +

= -0.318NVf sin (y +

(6)

Substitute from eqn. 6 into eqn. 1 gives the final form ofdVf/dIx as

Using these parameters and substituting into eqn. 7,first using \p = 65°, the value of dVf/dIx was found to be0.096 p.u.

The first term of the numerator of eqn. 7 will be muchsmaller than the second term; therefore, for the sake of

sin (</> — y) — — cos-O.318X,

NVf sin (y +

I Z2 { Z-^—L + ZA cos (0 — J/0 — —£ sin

-sin c ° s & -") - 1 1 sin (7)

It is not a straightforward task to assess the effect ofvarying each of the parameters on the right-hand side ofthe above equation on the ratio dVf/dlv Therefore, it isconvenient to calculate the magnitude of the dVj/d^ byvarying, over a defined range, the parameter which is ofconcern. For example, the AC system damping angle ^was varied between 65° to 90°, with three different valuesof filter reactance, namely, Xc = 2.26 p.u., 3.25 p.u. and4.52 p.u., and also with three different values of systemimpedance Zs = 0.15 p.u., 0.3 p.u. and 0.45 p.u. (theassumed base quantities used were the DC power and DCvoltage). The other nominal system parameters used areshown in Table 1.

0.24

0.2 3

0.22

0.21

0.20

0.19

d. 0.18

-"" 0.17

t 016

0.15

0.U

0.13

0.12

0.11

0.10

0.09'60 65 70 75 80 85 90inverter AC network damping angle if, degrees

Fig. 3 Relationship between AC network damping angle ip, dVf/dI, andsize of AC filter at the inverter end

Zs = 0.3 p.u., <f> = 158°a Xc = 2.26 p.u.; b Xc = 3.25 p.u.; c Xc = 4.52 p.u.

simplicity, the first term can be neglected with respect tothe second, without noticeable loss of accuracy. On thisbasis, the range of calculations were carried out and theresults plotted in Figs. 3 and 4.

In Fig. 3 it can be seen that the magnitude of dVfjdl^increases as the damping angle increases. As the filter sizeincreases (Xc decreases), dVfldlx also increases. In Fig. 4 itcan be seen that, as the system impedance Zs increases, thedVfjdl^ mean higher values of AC voltage regulation andless stability margin. It should be emphasised that theanalysis adopted in this Section does not cater for theeffect of the convertor control action on stability.

0.480.45

0.42

0.39

0 3 6

0.33

6. °-3 0

^ 0.27

•? 0.24

021

0.18

0.15

012

0.09

0.06

0.03

- V S -'60 65 70 75 80 85 90

inverter AC network damping angle -f, degreesFig. 4 Relationship between the AC network damping angle, ij/,and magnitude of network impedance Zs at the inverter endXc = 3.25 p.u, <j> = 158°u Z s = 0.45 p.u.; b Zs = 0.3 p.u.; c Zs = 0.15 p.u.

204 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 4, JULY 1985

However, increasing damping angle, increasing filter sizeand increasing system impedance all make the systemmore unstable.

4 Rectifier regulation and stability

An equation identical to eqn. 7 can be derived for the ACnetwork at the rectifier end, on the assumption of a con-stant rectifier firing angle a. The difference between themlies in the values of the parameters at both ends. Themost significant difference is in the displacement angle (f)which can be seen by comparing the phasor diagrams ofFigs. 2 and 5. The rectifier AC network (not shown) is rep-resented by an identical configuration to Fig. 1. Using theparameters of the rectifier AC network and substitutingagain in eqn. 7, a similar set of curves to those of the inver-ter end can be obtained; these are plotted and given inFigs. 6 and 7. It can be seen from these results that increas-ing both the system impedance Zs and the filter size (or

Fig. 5 Phasor diagram for the rectifier AC system

0.32

0.30

0.28

0.26

0.24

0.22

0.20

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

060 65 70 75 80 85 90rectifier AC network damping angle ifr, degrees

Fig. 6 Relationship between AC network damping angle ip, dV^/dl, andfilter size at the rectifier endZ5 = 0.3 p.u., <f> - 21°a Xc = 2.26 p.u.; b Xc = 3.25 p.u.; c Xc= 4.52 p.u.

The conclusions for both rectifier and inverter ACsystem effects agree with those shown by other methods inReferences 3 and 4.

0.54

0.51

0.48

0.45

0.42

0.39

0.36

0.33

0.30

0.27

0.24

0.21

0.18

0.15

0.12

0.09

0.06

0.03

060 65 70 75 80 85 90

rectifier AC network damping angle ifr,degrees

Fig. 7 Relationship between AC network damping angle,the magnitude of AC impedance Zs at the rectifier endXc = 3.25 p.u., 4> = 21°aZ5 = 0.45 p.u.; b Zs = 0.3 p.u.; c Z, = 0.15 p.u.

^ and

rectifier characteristic

line DCvoltage

operating point

invertercharacteristic

-current margin

line DCcurrent( I o rectifier)

Fig. 8 Steady-state characteristics of rectifier and inverter controls

reducing Xc), increases the magnitude of dVfld!x. Increas-ing the damping angle \p reduces the dVf/dl^ which is theopposite effect to that at the inverter end. Higher values ofdVf/dIi at the rectifier end mean higher values of regula-tion and better stability margin. In the discussion of thestability of any HVDC scheme it is necessary to define thesteady-state VJld characteristic which shows the co-ordination of the rectifier and inverter characteristics. Thisis shown in Fig. 8.

5 Impedance angle and reactive-compensatorrating

The improvement to the regulation at the busbars of theinverter, with a reduction of the damping angle ij/, can alsobe appreciated directly from the phasor diagram of Fig. 2.For instance, if the damping angle \p is decreased from 90°to 54°, then the locus of phasor Vs will move from point Pto point K of Fig. 2, and the magnitude is reduced from

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 4, JULY 1985 205

the value which is represented by the line extendedbetween points 0 and P to the value represented by OK.Then the difference between Vs and Vf, which is pro-portional to the regulation, is reduced and the regulationimproved.

The effect that the specification of damping angle canhave on the rating of the reactive compensator can bedemonstrated by considering the following example.Suppose that an AC system is wrongly assumed to have animpedance angle of 90°. Considering the same example asbefore, the phasor Vs will have magnitude OP as illustratedin Fig. 9. The phasor of Vs could be shifted by increasing

T/T=90°

2 BOWLES, J.P.: 'Alternative techniques and optimisation of voltageand reactive power control at HVDC converter stations'. IEEE Con-ference on overvoltages and compensation on integraded AC/DCsystems, Winnipeg, Canada, July 1980

3 SUCENA-PAIVA, J.P., and FRERIS, L.L.: 'Stability of a DC trans-mission link between weak AC systems', Proc. IEE, 1974, 121, (6), pp.508-515

4 YACAMINI, R., and TAALAB, A.M.I.: 'Steady-state stability of anHVDC system using frequency-response methods', IEE Proc. C,Gener., Trans. & Distrib., 1983, 130, (4), pp. 194-200

5 ARRILLAGA, J., HARKER, B.J., and TURNER, K.S.: 'Clarifying anambiguity in recent AC-DC load-flow formulations', ibid., 1980, 127,(5), pp. 324-325

6 CORY, B.J. (Ed.): 'High voltage direct current converters and systems'(Macdonald, 1965)

XsIs

K Vs

Fig. 9 Phasor diagram for the inverterTo show increase in reactive compensation necessary if \\i is wrongly assumed as 90°instead of 54° (as in Fig. 2)

the value of Ic, and hence reducing the magnitude of thesystem current /s from OW to OX. If Ic is chosen tochange Vs from OP to OK, as in the preceding Figure, thisdemonstrates that the effect of the 54° damping angle canbe achieved by increasing Ic by about 40%. Thus, if asystem which in fact has an impedance angle of 54° isassumed to have an angle of 90°, the reactive compensatorsize would have to be increased by 40% to give the sameinverter busbar regulation. This could be a significantportion of the cost of the total scheme.

For the rectifier end, the reverse of the above two casescan be shown to be true by considering Fig. 5. A reductionin the damping angle assumed would require an increasedin the reactive-compensator current. In practice, reactivecompensators are more often used in the inverter AC ter-minals, which by their nature are weak.

6 Conclusions

The behaviour of an HVDC link and how it is affected bythe parameters of the AC systems to which it is connectedhave been described. The derived equations and resultingcalculations have been used to illustrate the destabilisingeffect of inverter AC system impedance and the stabilisingeffect of rectifier AC system impedance.

The paper also highlights the importance of properlyestimating the value of the damping angle of the ACnetwork at the design stage. This is essential in determin-ing the cost of the compensator equipment, as well asplaying a major part in determining the overall stability ofthe HVDC/AC systems.

7 References

1 CALVERLEY, T.E., and TURNER, A.B.: 'Limitation inherent in thestudy of proposed DC infeeds to weak AC systems during transients'.CIGRE Study Committee, 14th August 1979

8 Appendix

8.1 Derivation of the AC system voltage equationWith reference to the phasor diagram of Fig. 2, it is pos-sible to deduce the following relations directly:

y\ = V} + (ISZS)2 - 2VfIsZs cos OA + 180 - 0')

= V) + (IsZf + 2VfIsZs cos (</>' - to (1)

Is cos (f)' = IY cos (j) (2)

Is sin 0 ' = /x sin </> — —fX r

and also

I2 = (7X cos (j))2 + Oi sin (j)V }2

X,

(3)

(4)

Substituting from eqns. 2, 3 and 4 in eqn. 1 to eliminate 4>r

and 7S, and then arranging to obtain the final voltageequation as follows:

+ 2VfZI1 cos (0 - to - Y sin <j>\

( 7 \ 2 17l~x)

Los {4>-to-Y sin (5)

As Vs is constant (infinite-busbar voltage), then the partialdifferential of eqn. 5 is given as

4Z V+ (1 - sin «A) —^ dVf + 2Z2It dl,

+ cos (<£ - i/O - Y sin

cos (0 - «A) - | ^ sin

in (j) \2ZSIX dVf

dl,

1+ 2VfZsI1 -sin (0 - \j/) d(f) - Y cos 0 d<f>

206 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 4, JULY 1985

dVA 2V( 1 — —- J H — - (1 — sin \b)fl f\ Xj Xc

f tJ. ,x Zs • 1+ 2ZS/! cos (q) — y/) sin $L xc J

r r z+ d/x < 2 / i Z j + 2ZS Ky cos ((f> — ij/) — — l sin

I L Xc

Asin^-il/)-^ cos ^ ) I-=7- -

sin (0 - 0) - | i cos

dh

whereV.

zs

Xc

8.2 Approximate formula for displacement angle 0The full equation widely used in the calculation of dis-placement angle is

(f) = tanJsisin 2a — sin (ct + fi) + 2fi

(7)cos 2a — cos 2(a + \J) J

Reference 3 gives the approximation

</> = (a + 0.5/1) (8)

It is this latter equation which has been used in this paper

2Z.V,zs \"l— sin 0 I

7 \2

2 V r [ l ~ x )

= RMS value of the infinite-busbar phase voltage= equivalent system impedance at 50 Hz= AC system impedance angle= displacement angle= fundamental component of AC current referred to

primary of convertor transformer= harmonic filter reactance at 50 Hz= RMS value of phase voltage across the filter.

c o s (<j) — xjf) — —- s in (f)X.

to give equations which are of a manageable form. Theerrors that the approximation causes are of the order of 1°to 1.5°, for most practical values of firing angle a andoverlap angle fi. For example, if a = 15° and fi = 20° istaken to be typical, then eqn. 7 will give an angle of 26°,while eqn. 6 gives 25°. An error of 1° due to the simplifica-tion will have no significant effect on the substance of thepaper.

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 4, JULY 1985 207