Steady-state stability of an HVDC systemusing frequency-response methods

R. Yacamini, M.Sc, C.Eng., M.I.E.E., and A.M.I. Taalab, M.Sc, Ph.D.

Indexing terms: Convenors, Power systems and plant, Power transmission and distribution, Control systems

Abstract: An HVDC scheme is an amalgamation of both AC and DC components, the behaviour of which isoften difficult to describe because of the interdependence of many of the parameters. The response of the DCsystem is dependent on the AC system and vice versa. The basic scheme controls also include both rectifier andinverter controls, which are basically separate but interdependent. The complete scheme could be described as anonlinear multivariable control system which is sensitive to external disturbances. The paper describes theanalysis of such an HVDC scheme using a simulator model and frequency-response techniques. The resultsdescribed constitute a sensitivity analysis of the system to the principal parameters which affect the steady-statestability, such as the inverter short-circuit ratio and AC-system damping, as well as the sensitivity to controlparameter changes.

1 Introduction

Several authors have, in the past, described the stabilityanalysis of various parts of an HVDC scheme, notably thecontrols and the convertors. These methods have normallyused computer programs to calculate the effect of differentparameters and, as such, have involved simplifications of amathematical or analytical type. The literature contains noanalysis of an HVDC scheme as an entity, where the entirescheme, including the AC systems and the DC line, havebeen subjected to stability analysis.

An HVDC scheme is a large-scale, multivariable system,which is also both nonlinear and suffers from external dis-turbances. As such, the analysis becomes extremely com-plicated and uncertain. Modelling techniques where all thecontrols, all the nonlinearities, and all the disturbances,can be included simultaneously are therefore of consider-able value in determining stability. With such a model,there is no need to make approximations, other than thosewhich are inherent in any model, which can be virtuallyeliminated by experience and good technique.

This paper, therefore, describes for the first time theapplication of frequency response methods to an HVDCscheme. The use of these techniques gives informationabout the system which will transform it into a scalarmodel. The paper highlights the effect of the impedanceand damping angle of the networks, into which the HVDClink is feeding by carrying out a parameter sensitivity anal-ysis. (The description of the methods used have recentlybeen collected in the publication by Macfarlane [1], wheresome 40 papers are included, which cover the entire field.)

The measurements were carried out on an HVDC simu-lator. The closed-loop frequency response was deduced fora transmission system consisting of a rectifier with con-stant current (CC) control, an inverter with constantextinction angle (CEA) control, and a DC transmissioncable. The open-loop frequency response was found fromthe closed-loop response, using a Nichols' chart and theNyquist diagram plotted of the open-loop response. Theeffect of varying different system parameters on the relativestability of the DC transmission system can then be assess-ed by plotting a Nyquist diagram with only one of themany variables changed at a time.

Paper 2585C, first received 23rd November 1982 and in revised form 18th April1983

Mr. Yacamini is with the Department of Engineering, University of Aberdeen.Marischal College, Aberdeen AB9 IAS, Scotland, and Dr. Taalab is with theFaculty of Engineering and Technology, Menofia University Shebin El-Kom,Egypt. Both authors were formerly with UMIST

An accurate Nyquist diagram can be used to determinethe range of gain values over which the system will bestable, and the frequency (or frequencies) at which it willtend to oscillate as it becomes unstable. Step-response testswere also carried out to show the effect of different systemparameters on the closed-loop transient response; in parti-cular, on the control amplifier circuit parameters at thedesign stage. When it came to optimisation of control par-ameters, small variations of some of these parameters hadlittle measured effect on the step response; however, theeffect could be distinguished using frequency-responsemeasurements, thus suggesting that frequency-responsetechniques should be used in conjunction with the stepresponse to optimise control parameters.

In a large multivariable system, such as an HVDCscheme, there are many system parameters which can bevaried, consequently, numerous cases can be studied.However, the parameters which were chosen as being ofinterest for this paper were the damping angle if/ of the ACnetwork into which the DC transmission is feeding, theinverter AC system impedance Zs, the rectifier firing anglea, the inverter extinction angle y, the DC line length andthe derivative feedback gain of the rectifier current control.

A summary of the methods previously used to assesssystem steady-state stability and a description of themethod used for this paper are given in the following twoSections.

2 Stability analysis of HVDC schemes

The convertor bridge and its controls constitute a nonlin-ear discontinuous type of control system. The discontin-uity comes from the fact that the input signal to theconvertor bridge controls the output at discrete instants oftime, at a sampling rate which corresponds to the firinginstants, which have a fixed sequence. This means that thebandwidth of the control loop is limited, owing to thepresence of the static converter as a control element in theloop. It has long been established that static convertorscan be made to follow a sinusoidal input signal if thefrequency of the latter is less than half the firing frequency[2]. For practical applications, therefore, it is possible totreat the convertor bridge as a continuous element, rep-resented by its incremental DC gain and a dead time cor-responding to half the interval between two firing instants.Fortunately, owing to the normal high accuracy, inte-grating characteristic of the control amplifier, the band-width of the constant current-control loop is reduced evenfurther; therefore, the interesting range of input frequency

194 IEE PROCEEDINGS, Vol. 130, Pt. C, No. 4, JULY 1983

may be considered very much less than half the samplingfrequency. For example, for a 6-pulse convertor, with/0 =50 Hz, half the sampling frequency is 3/0 , and the inter-esting range shown by the frequency response is from zeroto about 2/0. The nonlinearity of the control loop is theresult of the relation between the incremental variation ofthe control amplifier output signal (the amplified errorsignal), and the corresponding variation in the convertoroutput DC voltage. Mathematical procedures for the exacttreatment of non-linear elements are difficult to carry out;thus, an approximate linear model is often used instead. Instudying the stability of controlled convertors, a small-displacement linear model has been used since Busem-mann [3] studied the hunting of rectifiers operating underconstant-current control. In this, he represented the recti-fier by a constant gain and ignored the commutation angleas well as the sampling action of the convertor. Otherauthors [4] modelled the rectifier as a pure sampler and,using small perturbation analysis, were able to derive aninfinite series for the critical gain of the closed-loop systemat half the sampling (firing) frequency.

Control action was considered in a study of HVDC linkstability, as reported in Reference 5. In this reference, theconvertor is represented by a constant gain and the inversecosine firing system used to obtain a linear relationshipbetween the control voltage and the convertor output. Inmodern HVDC schemes, the inverse cosine controller isnot used. In Reference 6, a stability study was carried outusing theoretically calculated Nyquist diagrams, based ona previously calculated transfer function. In this study,although the control amplifier action was considered, itwas represented by a simplified transfer function with pro-portional gain control. Reference 7 describes a study whichused a discrete model to study the stability of controlledrectifiers, using Z-transform and root-locus techniques todetermine the onset of instability and system stabilityboundaries. This model takes into account the effect of thecommutating reactance and the sampling nature of therectifier. In these references, the indepenent phase controlfiring system and proportional gain control amplifier wereused. The effect of a voltage controlled oscillator (VCO)firing system in the control loop of an HVDC scheme wassimulated, and an investigation of the stability of anHVDC link between weak AC system studied using themodulation technique based on sinusoidal carrier func-tions to determine an approximate transfer function. Arecent publication [8] gives another critical assessment ofconvertor stability analysis and a fuller list of references.

All methods described in the literature hitherto, tostudy the stability of HVDC systems are based on simpli-fied representations, with different degrees of simplificationof the convertor, the DC link, the AC system impedance,and the control transfer function. Also, the techniques gen-erally used were based on predicting the onset of stability;that is, determining only whether the system is stable ornot and giving no information about stability margin. In amore comprehensive study of the type described in thispaper, all parameters of the AC system and the controlcircuits in the two terminals, as well as those of the DCline, are considered. This will give a method of studyingthe stability and dynamics of an HVDC scheme whichshould be better than mathematical approximationmethods. For this reason, a frequency-response analysis onan HVDC simulator was carried out. This type of analysisis the closest approximation that can be made to real lifeand includes a real control system and a well modelledconvertor and system representation. It will be more usefulthan using digital computer models and could be used for

adjusting the control system parameters at the commis-sioning stage of a real scheme.

3 Frequency-response measurements

The possibility of using frequency-response methods tostudy the stability was tested by injecting a small sinu-soidal signal into the current-control loop at a certainpoint in the closed loop. The phase and amplitude of theoutput signal were measured at different points in the loop.The magnitude of the input signal was varied within acertain range (equivalent to a variation of DC current of 4to 5%). The phase shift of the output signal was found tobe almost constant at any particular frequency; i.e. it wasindependent of the magnitude. This suggests that theHVDC transmission system and its control loops could becategorised as a 'linear' or 'almost linear' system for smalldeviations around the operating point. The guidelines laiddown in Reference [9] are therefore met and it wasassumed that for this 'linear' or 'almost linear' system, thatfrequency-response analysis can be used with sufficientaccuracy to satisfy design objectives.

A transfer function analyser (TFA) was used to measurethe amplitude and phase of the test signal at the input andoutput of the control loop. The complex ratio of the inputand output was found at each test frequency. Stabilitymargin information (phase and gain margin) can then beobtained if the open-loop frequency response is available.

At first sight, a feasible way of obtaining the open-loopmeasurement directly would be to break the closed loop ofthe constant-current control at a convenient point and toinject the test signal at a summing point of one of theamplification stages: the response would be measured atthe output of the previous amplification stage. Unfor-tunately, the controls could not be tested with an openloop owing to control amplifier saturation, which forcedthe firing angle a to either of its two limits (amax or amin).This means that the DC current in the link would drop tozero or increase to its maximum limit. In both cases, thetest signal would then have no significant effect. To over-come the difficulty of measuring the open-loop response,the method was adopted whereby the closed-loop responsewas measured first, and then the open-loop response wasdeduced using the well known Nichols' chart.

To measure the closed-loop response, the test signal wasinjected at the input of the final stage of the control-amplifier circuit, the return signal was monitored at theoutput of the main amplifier. This is shown in the simpli-fied diagram of the DC transmission system in Fig. 1. Adescription of the firing correction and gamma correctiongenerators can be found in Reference 10 and its associatedpapers. The input to and the output from the TFA weretaken at the two auxiliary points M3 and M 4 . Injecting asinusoidal signal r at point M3 will give rise to change ofAe = G2 • r in the error signal, where G2 is the transferfunction of the loop compensator. This is better illustratedby the block diagram shown in Fig. 2 which represents thesystem when operated in the steady state, with the rectifieron constant current and the inverter on constantextinction-angle control. The change in the error signalwill give rise to change of Aa = G3 • Ae in the firing angle,where G3 is the transfer function of the firing-pulse gener-ator. For constant DC, variations of firing angle contributeto direct-voltage variations AVdr is given by AVd3 =G4 • Aa. For constant firing angle, the contribution to AVdr

is given by AVd2 = G5 • AIdr, where AVd2 represents thechange in voltage drop due to commutation reactance andAC system impedance. Deviation of the direct current A/r

IEE PROCEEDINGS, Vol. 130, Pt. C, No. 4, JULY 1983 195

compensator-function

firingcorrectiongen- cH_d

Ld dt

V correction

Fig. 1 Simplified diagram of H VDC transmission system and controlsZT = transformer impedance

T = transformerZf = filter voltageZs = system impedanceVx = primary phase voltage of conv. transformer

is fed back to the main amplifier transfer function G1 as— Aid = G6 • AIdr, where G6 is the transfer function of thecombine DCCT, DCT, feedback and derivative feedbackamplifier. The total inverter direct voltage deviation AVdiis the sum of AVd^ and AVd5, where AVd5 = AIdx • G9 isthe deviation due to the AC-voltage drop and thecommutation-voltage drop produced by direct currentdeviation under constant extinction angle control. Thequantity AVd^ is due to the extinction angle deviation Ayf

resulting from the change in the commutation (overlap)period for a constant inverter firing angle. Deviation of theextinction angle is related to deviation in direct current bytransfer function G,, ; G,ft and G,, are the transfer func-

DCT DCCT

tions of the inverter bridge and the inverter firing pulsegenerator, respectively.

The DC line is represented by its 2-part parameters G7,G8, G13 and G14, where G8 = ldJVdr for Vdt = 0, G14 =

for Vdr = 0, G7 = Idr/Vdr for Vdt = 0 and G13 =i for Vdr = 0.

Some of the transfer functions in Fig. 2, namely, G4,G6, G9, G1 0, G12 cannot be easily calculated, as theyexpress response involving quantities on both the DC andAC side of the convertor. No attempt has been made tocalculate each individual frequency-dependent transferfunction in order to calculate overall frequency response.This is obtained from direct measurements on the HVDCscheme.

It should be emphasised that the closed-loop transferfunction depends on the points of measurement on thecontrol loop; however, the open-loop transfer function

M2

inverter

Fig. 2 Block diagram of HVDC system incorporating rectifier constantcurrent and inverter constant extinction angle controls

196

ref. source36 kHz

does not. With reference to Fig. 2, if the input signal r isinjected at point M 4 , and the return signal C is monitoredat point M2, then the closed-loop transfer function is givenby the complex ratio C'/r = G/(l + GC±), where G is theoverall forward gain and Gx is the feedback gain. However,if the return signal C is monitored at point M 3 , then theratio C/r = GGJ(l + GG^) is another value of the closed-loop transfer function with a unity feedback. Although twodifferent values of closed-loop transfer function areobtained by measuring the output signal at two differentpoints on the control loop, the two transfer functions havethe same open-loop transfer function of GG±. As theNichol's chart is usable for unity feedback systems, it wasconvenient to measure the output signal at point M3 andthe input signal at point M2 •

The ratio C/r was measured, and the open-loop transferfunction GGV was found at each test frequency using theNichols' chart. The results of these measurements are givenin the following Sections, to show the effect of differentsystem parameters on stability.

4 Relative stability study

Results of the frequency-response measurements are givenin this section for some selected system parameters. Theeffect of each of these parameters upon relative stability isshown on the Nyquist plot for three different values ofeach parameter, and on closed-loop frequency response interms of the resonant peak Mr and resonance frequencycor. The corresponding step of current response is alsogiven at each of these three values. All results presented inthis Section were obtained when the HVDC system wasrunning at rated direct current, with the rectifier on con-stant current control of control parameter Kx = 8 (dialposition), and with the inverter on CEA control with par-ameters K3 = 0.5, X4 = 0.25. (These values only havemeaning for the particular control system used on thesimulator. The comparative values however—as in allmodel tests—have true meaning.)

Two different techniques have been used to assess sta-bility. Step responses of the type used by practicaldesigners were carried out for the initial design of the con-trols and to partly optimise the system. The frequency-

IEE PROCEEDINGS, Vol. 130, Pt. C, No. 4, JULY 1983

I I

10 20 30 i.0 50 60 70 80 90Hz

f = 7Hz

•miniiiiiiiiiiiii '

I Ic 50ms

Fig. 3 Effect of derivative feedback on stabilitya Closed-loop frequency response

/ , = 20 Hz/ r = 24Hz/ r = 31 Hz

b Nyquist plotPHM = 11°, GM = 1.42PHM = 27°, GM = 2.85PHM = 43°, GM = 2.22

c Step response of direct current

response techniques described here are then used to finallyoptimise the control parameters and to carry out a sensi-tivity study. This type of study is more useful to the totalsystem designer and the supply authorities to whosesystem the HVDC is connected.

The parameters under consideration were varied one ata time; the corresponding values are given in each casestudied. These cases are as follows.

4.1 Effect of derivative feedback gain /C> in therectifier current control

This feedback gain (the acceleration feedback) is thedominant control signal in determining system stability. Its

choice can be crucial in a real scheme. The effect ofincreasing derivative feedback gain (K2) on stability isshown in Fig. 3b, in which a Nyquist plot of the open-looptransfer function is plotted for K2 = 1, 2 and 3. Each of thethree plots indicates a stable system. However, the plot ofK2 = 1 indicates that the stability margin is less than inthe other two plots. Quantitively, the phase margin PHMof the solid-line plot is 11°, and the gain margin GM is1.42; however, the PHM for the dashed-line plot is 27° andthe GM is 2.85, and the PHM of the dotted-line plot is 43°and the GM is 2.22. From these results, it may be con-cluded that, as K2 increases, the PHM increases; however,a large phase margin causes a time delay which may havean undesirable effect on the system transient response. Thisis seen in the corresponding step response due to a step indirect current, as shown in Fig. 3c. The gain margin alsoincreases as K2 increases up to a certain value (K2 about2.5), beyond which it starts to decrease again. This can bededuced from the three Nyquist plots of Fig. 3b. Thereason for reducing the gain margin is the appreciableincrease of ripple magnitude in the control loop, which ledto instability if K2 = 6 was exceeded. On the other hand, areduction of K2 below a value of about 0.5 can lead toloop instability. An optimum value of K2 is the value atwhich the Nyquist plot reveals a phase margin ^ 30° anda gain margin ^ 3. These two values of phase and gainmargins are considered as basic figures in linear controlsystem design [9].

Two important features which can be obtained from theclosed-loop frequency-response plot (obtained directlyfrom the measurements) are shown in Fig. 3a. These arethe resonance peak Mr and the resonance frequency cor.The resonance peak is the maximum value of the magni-tude of the closed-loop frequency response, i.e. Mr = maxm

I C/r(joS) I, which is also used as a measure of relative sta-bility. The closed-loop frequency response is plotted for thethree mentioned values of K2. From these, it can be seenthat cor is increased in frequency and Mr is reduced, as thevalue of K2 is increased. It can be seen also that the band-width is increased with increasing K2.

It should be noted, in this Section, that the HVDCscheme was operated with a back-to-back link, with short-circuit ratio = 4.22 at the inverter end, Zs = 0 at the recti-fier end, rectifier firing angle a = 16°, time constant of therectifier current control T = 240 ms and inverter extinc-tion angle y = 16°.

4.2 Effect of rectifier firing angle a and inverterextinction angle y

The HVDC scheme was operated with the same systemparameters as in the above case, except K2, was set to avalue of 2. The closed-loop frequency response, Nyquistplot, and the step of direct current response are given inFigs. 4a, b and c, respectively. Each of these responses isgiven at three different values of rectifier firing anglealpha = 18°, 27° and 37°. From Fig. 4b, it can be con-cluded that, as alpha increases, the stability margindecreases. This important result agrees with previouslypublished results in this area. Increasing alpha shifts fr

towards higher values and increases the magnitude of Mr,as shown in Fig. 4a. The speed of system transientresponse also increases as alpha increases: this may beseen on the step of direct current response of Fig. 4c. Oneof the conclusions which can be drawn is that a rectifierstation which is run temporarily into a reduced number ofinverter bridges will be encroaching on the stabilitymargin.

IEE PROCEEDINGS, Vol. 130, Pt. C, No. 4, JULY 1983 197

10 20 30 40 50 60 70 80 90Hz

-2

36,- >

28.-- / / sZb/

Fig. 4 £#ect of rectifier firing angle a. on stability

a Closed-loop frequency responsefr = 20 Hz/ r = 36Hz/ r = 40 Hz

b Nyquist plotPHM = 34°, GM =3.33PHM =30°, GM =2.1PHM = 20°, GM = 1.74

c Step response of direct currenta = 1 8 ° , . a = 27°, a = 37°

A similar investigation was carried out to show theeffect of inverter extinction angle y. Nyquist plots weredrawn for three different values of y = 16°, 26° and 32°(not included in paper). It can be concluded that the stabil-ity margin increases with increasing y, fr increases, Mr

decreases and, consequently, the system speed of responsedecreases as y increases. Running with minimum y will givea fast but less stable response.

4.3 Effect of DC line lengthThe effect of increasing the DC line length on the stabilityof the HVDC scheme was considered by examining threevalues of line length. The response was measured for differ-ent values of DC line model which represented lines of 33miles (53 km), 59 miles (95 km) and 124 miles (200 km),respectively. The results were obtained when the HVDCscheme was operated at the same parameters as the firstcase (Section 4.1) except that, in this case, K2 = 2,T = 158 ms. The Nyquist plot (not shown) reveals that asthe DC line length increases, the stability margin increases.A similar conclusion was obtained [5] from a study per-formed by applying Nyquist stability criterion to a mathe-matical model. The closed-loop response indicates that, theshorter the line length, the higher the values of fr and Mr.

The result shows a destabilising effect as the DC linelength decreases. In Reference 5, it is stated that the causeof obtaining a stabilising effect as the DC line lengthincreases, may be attributed to the increase in the totalattenuation with an increase in line length. Another factorwhich may be important is that reduction of the line length(effectively the line capacitance), brings the resonance ofthe link near to a low integer frequencies, especially thefundamental and second harmonic. The presence of non-theoretical harmonics on the DC side can have a signifi-cant effect on system stability [11, 12].

4.4 Effect of inverter AC impedance Zs

The most commonly quoted system parameter used tocompare different HVDC schemes is the short-circuit ratio(SCR) at the inverter AC busbar. In fact, the ability of aDC link to operate with an SCR as low as two or three isoften taken as an indication of the quality of the controlsystem. The effect of increasing Zs on reducing the steady-state stability of an HVDC scheme has been described inreferences 5 and 13.

The effect of three different values of Zs on the stabilityof an HVDC scheme was examined. These values of Zs

were inductive, with the impedance angle kept constant at89°. The frequency-response measurements were carriedout when the HVDC system was operated at the samesystem conditions as in the first case (Section 4.1) exceptthat K2 is set to 2. The Nyquist plot is given in Fig. 5b forZs = 0, Zs = 0.346 pu and Zs = 0.414 pu. From theseplots, it can be seen that the gain margin decreases as Zs

increases and the phase margin decreases at first and thenincreases again. From the closed-loop frequency responseof Fig. 5a, and the Nyquist plot, it can be seen neither theresonance frequency fr, nor the phase margin, has a linearrelationship with the system impedance. However, thereduction of the gain margin may be regarded as linearlyproportional with the increase of system impedance. Onthe other hand, the transient response tends to be sloweras Zs increases. This can be seen on the step response ofFig. 5c.

4.5 Effect of inverter A C-system clamping angle ipIn representing the system impedance for simulator andcomputer studies, the impedance angle \j/ (or dampingangle) is of particular importance. This impedance angle isused as a measure of the resistive components of thenetwork. It is suggested [14] that, in representing thedamping of an inverter station which has a substantialload relatively close to the AC busbars, an impedanceangle of 75° should be used. For a rectifier station fedsolely by generators with little local load, an impedanceangle of 85° should be used.

198 IEE PROCEEDINGS, Vol. 130, Pt. C, No. 4, JULY 1983

10 20 30 40 50 60 70 80 90a Hz

or

Zs = Rs + jXs

where

R(coL)2

R =R2 + (coL)2

and

(coL)2 + 2wLR:

R2 + (coL)2

50 ms

Fig. 5 Effect on inverter AC system impedance on stability

a Closed-loop frequency responsefr = 20 Hz/ p =15Hz/, = 30 Hz

b Nyquist plotZ,= 0.414 puZ5 = 0Z, = 0.346 pu

c Step response of direct currentPHM = 31°, GM = 3PHM = 40°, GM = 2.7PHM = 51°, GM =4

The AC-system impedance could be represented simplyby a resistance in series with an inductance for a singlefrequency representation. However, it is required to rep-resent the system over a range of frequencies. An extensiveand complicated representation is given in Reference 15.However, for practical purposes, the representation givenin References 13 and 14 is considered adequate and wastherefore used in these studies.

In Fig. 6, an equivalent circuit of the inverter AC-system impedance is given. The impedance Zs of thisequivalent circuit can be expressed as

It can be seen, from this particular system impedancerepresentation, that an approximate constant impedanceangle V can be obtained for a low range of frequencies(say, from the fundamental to the fifth harmonic), which isthe important range for steady-state stability studies.

The effects of three different values of if/ on the stabilityof the HVDC system were examined, keeping the magni-tude of the impedance constant to give an SCR of 3.2. Thefrequency-response measurements were carried out whenthe HVDC system was running, with the same system con-ditions as in the first case, except that Zs is set to giveSCR = 3.2, and K2 is equal to 2.

conv.

-0D aninfinitebus

s ~(coL)2R . (coL)3 + 2coLR

+JR2 +J R2+ (coL)2

filter

Fig. 6 AC-system equivalent circuit

The Nyquist plot of the open-loop transfer function isgiven in Fig. 1b for ij/ = 71°, \f/ = 74.6° and if/ = 89°, fromwhich it can be directly deduced that the stability marginincreases as if/ decreases. This result confirms the dis-cussion presented in Reference 16. The closed-loop fre-quency response is given in Fig. la, in which fr increasesand Mr decreases as \jj decreases. The speed of the tran-sient response also decreases as \j/ decreases; this is madeclear by step response to the direct current change shownin Fig. 7c.

One of the conclusions which can be drawn is that, aslocal AC-system loads increase and if these are largelyresistive, then the stability margin of an HVDC scheme isimproved. These must be properly assessed at the initialdesign stage.

5 Conclusions

The results of a frequency-response analysis of a simulatedHVDC scheme are presented in this paper for the firsttime. The system studied was for a scheme operating withthe rectifier in constant-current control and the inverter onconstant extinction angle control. A physical simulation ofthis type allows all the nonlinearities and the interdepen-dence of control loops to be analysed in a way which ismore comprehensive than any mathematical or computedsimulation. The results given could apply equally to anactual system and, in fact, frequency-response analysis ofthis type could be carried out on a system at the finalcommissioning stage, to confirm stability margins. In prac-tice, it would be impossible to change many of the vari-ables in the same way as in this paper. The system studiedis a nonlinear multivariable control system.

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a 10 20 30 40 50 60 70 80 90Hz

75 90.52 y^- 4im

f=6Hz

Fig. 7 Effect of inverter AC-system damping angle \)/ on stability forSCR = 3

a Closed-loop frequency response/ r = 28 Hz/ r = 26Hzfr = 19 Hz

b Nyquist plotPHM = 44, GM = 3.6PHM = 39, GM = 3PHM = 29, GM = 2.8

c Step response of direct current^ = 71°, ^ = 74.6°, i/f = 89°

In particular, the paper shows the effect on stabilitymargin of two types of parameter. First, the HVDC andcontrol parameters such as rectifier and inverter firingangles, constant-current loop gains and derivative feed-back effects. Secondly, the effect of system parameters, suchas line or cable length and inverter AC busbar strengthand damping angle.

6 Acknowledgments

The authors would like to thank their colleague Dr. F.M.Hughes for his advice on the practicalities of frequency-response measurements.

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10

11

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