transform methods. Depending on the parameters and fre-quency summation ranges chosen, the methods do ofcourse give slightly different results; but the claim that thez-transform technique is the more accurate of the two isnot strictly valid. It can be shown that over the frequencyrange 0—>fe ^ 03/(2nAt) there is good correspondencebetween the z-transform warped-frequency-response func-tions and the actual power-system-response functions.Some further improvement in the correspondence canactually be achieved over part of the frequency range byusing a technique known as pre-warping; but frequenciesnear and beyond the Nyquist frequency are still incorrectlyattenuated and phase modified. Furthermore, in accord-ance with the sampling theorem, all components above theNyquist frequency are folded and appear as lower-frequency baseband time-domain aliases that can lead toerrors. The degree of error which occurs depends on thestrength of the spectral components near and beyond theNyquist frequency. In some studies, the spectral density inthe latter region is such that the resulting error in z-transform analysis is small, and one obtains reasonablecorrespondence between z and frequency transformmethods; but extreme care has to be taken to ensure thatthis situation is preserved when the system responsechanges as a consequence of, for example, changing systemfault operating conditions, source and line configurationsor fault clearing sequences. In as. far as it is possible togeneralise, if the highest significant frequency associatedwith a particular study can be safely predetermined and ismade to correspond to the value fe ^ 0.3/(2nAt) ~ 0.3fs/(2n), then reasonable correspondence between theresponse functions and their warped versions over theimportant frequency range 0—>fe helps considerably inreducing errors. Adoption of such a criterion effectivelymeans that in z-transform-type analyses the sampling fre-quency fs should be approximately 20 times the highestsignificant frequency expected, i.e. fs ^ 20/e.

The latter considerations are particularly relevant to thecomputing requirements to effect accurate extended solu-tions in protective gear evaluation, where the signalsrequired are effectively band limited (to typically 2 kHz) byvirtue of transducer and prefiltering arrangements. In suchstudies, there is little point in calculating other than theband-limited version of the primary system responses,because components above typically 2kHz are subse-quently eliminated.

Nevertheless, it is necessary to ensure that the primarysystem response evaluations do not involve time-domainaliases within the band of interest. When using frequency-domain methods, one simply arranges for the truncationfrequency to correspond to the Nyquist rate, and this auto-matically recovers the band-limited form of the responsewithout leading to time-domain aliasing. Any frequenciesabove the Nyquist rate are not included in the frequency-domain integral summation and cannot be folded down toappear as lower-frequency aliases. Accurate frequencyfunction mapping is of course implicit to frequency-domain methods, and, in accordance with the samplingtheorem, a sampling frequency of 4 kHz can be used toaccurately recover the band-limited form. For the abovementioned reasons, the corresponding frequency necessaryto avoid both baseband aliasing and frequency functionwarping errors in z-transform analyses would be approx-imately 40 kHz. A determination of the various systemfrequency-response functions is common to all methods ofanalysis and the computational effort required to evaluatethese often accounts for a very significant proportion ofthe total computing time required. Furthermore, when

using the recursive form of the inverse Fourier transform,as opposed to the much less efficient full frequency-domainsolution, the significantly reduced sampling rate which isusable enables typical extended solution studies to beeffected very efficiently.

Comparisons between methods of analysis can of coursebe misleading in that much depends on the precise struc-ture of the computer programs. However, our experienceshows that the most significant determining factor inrespect of CPU times is the time required to calculate thesystem frequency-response functions. For example, if, asseems to be the case in the papers by Prof. Humpage et al.,the time to calculate the latter is ignored; the CPU time toexecute the frequency-domain single-line-section programof Reference C is currently 11 s (ICL 2980) for a 250 msobservation time. Such a requirement would seem to becomparable, on an equitable basis, to the correspondingrequirement for z-transform analysis as outlined in Table 1of paper 990C.

A.T. JOHNSR.K. AGGARWAL

M.A. MARTINA. BARKER

15th April 1981

Power Systems LaboratorySchool of Electrical EngineeringUniversity of BathClaverton Down, Bath BA2 7AYEngland

Dr. Johns and his colleagues raise interesting points inrelation to some of the aspects of transmission lineresponse function definition in z-transform electromagnetictransient analysis methods. There has been wide interest inthese developments.

Of the numerous earlier methods that have so far beendeveloped and applied, the discussion appears only to referto those of conventional frequency-domain analysis. Excel-lent though these may be, for certain purposes, their limi-tations are widely known and generally agreed. Given theavailability of other methods, it seems hard to recommendthem without major reservations for the particular applica-tion which the discussion seems to emphasise; that of someaspects of the testing of some forms of protective gear. Inany event, while this might justifiably be considered tohave a place within the total range of application of elec-tromagnetic transient analysis methods, it will be plainfrom papers 988C, 989C and 990C, and perhaps also frompapers 1186C and 1185C [Proc. IEE, 1981, 128, (2), pp.55-69], that z-transform analysis is of much wider applica-tion. It is applicable to use in all of the distinguishableareas of applied electromagnetic transient analysis inpower systems. It was developed from that basis as ageneral method for general application. Within this, itseems very likely that the new method has special merit inthe primary system representation of composite dynamicsimulation for protection signal-processing studies, and incomputer-based forms of protection-logic validation pro-cedures [E, F], particularly when compared with conven-tional frequency-domain analysis. But the sole emphasis inthe discussion on this narrow application area seems to bein marked contrast to the wide scope of z-transform anal-ysis in practice, and to the wider subject area of which ithas become a part.

The role of the forward-impulse response and surgeimpedance functions in z-transform methods is closelysimilar to that in time-convolution methods to which the

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

discussion, surprisingly, makes no reference. The overallaccuracy of the final solutions in the time domain has adependence on the closeness with which functions aredefined, which is similar to that in the earlier convolutionschemes. It might have been expected that paper 990Cwould have brought that clearly into notice. However, asthe frequency content of transients varies widely through-out the solution period, the relationship between the accu-racy of response definition and the overall accuracy ofsolutions in the time domain is not a simple one. It seemsto be essential to acknowledge that when commenting onthe accuracy of response definition.

The discussion refers to the premise on which functionsare defined. Coefficients of the rational-fraction forms oftransmission-line functions are found from a frequency-weighted least-squares minimisation, in which the errorfunction is formed from the modulus of the rational-fraction function and specified values for it found from lineparameters. On finding coefficient values from this pro-cedure, and then using them in the rational-fractionexpression, the modulus response is accurately reproduced.By the Hilbert transform, the phase response is also rep-resented accurately, although it is not included explicitly inthe error function on which minimisation is based. Equallywell, coefficients could be found from the minimisation ofan error function based on the real part of transmission-line responses, following which the imaginary part is accu-rately reproduced by the rational-fraction expression. Realand imaginary parts, or modulus and phase, are not inde-pendent; and only one, in either case, is necessary in theminimisation process. It should also be clear that the errorfunction can easily be extended to include both real andimaginary parts, or both modulus and phase; but when theHilbert transform conditions are fulfilled, one of either pairis redundant. In Fig. B is shown the magnitude and phasecharacteristic of the earth-mode forward-impulse responsefor the transmission-line construction of paper 989C.Further to indicate the accuracy of the phase response, itsnonlinear component is shown on an enlarged scale of Fig.C. While this component contributes to the overall charac-teristic by no more than about 5% of the total, it remainsclosely defined without its being directly reflected into theminimisation sequence from which function coefficients arefound. From the development of paper 989C, it should beclear that it is only the nonlinear part which is representedin the rational-fraction form of the impulse-response func-tion. The frequency range is unrestricted in choosing thehigh frequency at which the slope of the phase character-istic is formed, to give the earth-mode transit time to

1.0 2.0 3.0frequency, rad/s x1

4.0 5.0

Fig. B Modulus and phase components of earth-mode forward impulseresponse

= response calculated from transmission-line parametersx x x = response from rational-fraction expression following s-plane to z-plane

transformation• • • • | = response from rational-fraction expression synthesised directly in the

z-plane

which the time-domain sampling interval At is laterrelated. Although the assertions of the discussion are mis-taken, there does appear to be value in a close investiga-tion [H] of the Hilbert transform relationships in theirapplication to transmission-line response functions in thez-plane, and, from this, in providing confirmation of thebasis of accurately synthesising total response functionsfrom their modulus components only.

When function coefficient values are found, first, in thes-plane and then transformed to the z-plane, the bilineartransformation of paper 989C is one which always guar-antees that the coefficients lead to stable functions; but itintroduces into z-plane functions a widely known distor-tion error. This appears to be of the greatest concern to thediscussors. But while interest appears to be limited to thederivation of the coefficients in the rational-fraction formsof transmission-line responses, it will be clear that this canlead only to a restricted consideration of distortion inmapping from the s-plane to the z-plane. The effect canonly be evaluated conclusively in terms of that which anal-ysis is in total required to provide: electromagnetic tran-sient solutions in the time domain. Solutions are shown inFigs D and E from a line-energisation study in which z-transform and frequency-domain solutions are compared.In so far as it is possible to isolate the effects of distortionerror arising from s-plane to z-plane transformation, it canbe seen in the amplitude difference in the phase 'b' solutionat certain points in the solution period. In a representativesolution, this is the only consequence of the error sourcewhich the discussion suggests is very significant. For a verywide range of practical analysis conditions, the position isclosely similar to that of Figs. D and E. In any case, thesynthesis of transmission line functions directly in thez-plane avoids s-plane to z-plane transformation alto-gether, and thereby removes the basis of the greater part ofthe discussion. Comprehensive procedures for directz-plane function synthesis have been reported [G], andthese lead to the high accuracy of transmission-lineresponse function definition which Figs. B and C confirm.

When a step length is specified in the time-domainrecursive sequences for solution, a frequency is defined interms of it at which responses in the frequency domain areperiodic. Responses given by the rational-fraction expres-sions are repeated in the frequency domain with thisdefined frequency. It is a very basic aspect of analysis tochoose the step length so that the frequency content of anygiven transient condition to be evaluated is contained inthe first period of the responses. The discussion, however,

2.4

2.0

1.6

1.2

0.8en

0.4

1.0 2.0 3.0frequency, rad/s x

4.0 5.0

Fig. C

A =

B =

Nonlinear component of earth-mode phase response

response calculated from transmission-line parameters in s-plane mini-misationresponse calculated from transmission-line parameters in z-plane mini-misationresponse from rational-fraction expression following s-plane to z-planetransformationresponse from rational-fraction expression synthesised directly in thez-plane

IEE PROCEEDINGS, Vol. 130, Pt. C, No. 4, JULY 1983 229

appears to emphasise this periodicity, and gives unusedparts of the responses in Fig. A. The response diagrams ofFig. A are intriguing in that the greater part of them iswithout relevance when related to the requirements inchoosing the time-domain sampling interval. When trans-formation is made directly from the frequency domain inits discrete form to the discrete time domain, as in numeri-cal frequency-domain analysis, responses are periodic inboth the frequency domain and in the time domain. One ofthe principal reasons for transforming from the frequencydomain, first, into the z-plane, as an intermediate trans-form step, is that of thereby retaining frequency in its con-tinuous form. Finally transforming from the z-plane to thetime domain then leads to a response which repeats onlyas the time axis tends to infinity. It seems that there maybe a fundamental misconception in the discussion in rela-

tion to this. In numerical frequency-domain analysis,errors arise from its discrete form when transforming tothe time domain. There are no corresponding time-domainerrors in z-transform analysis and the solution time isunrestricted. This particular advantage of z-transformanalysis can of course be of special significance in thoseapplications in which there are requirements for extendedsolution times. In relation to short-circuit fault conditions,on which the discussion largely concentrates, as the solu-tion sequences in z-transform analysis are wholly in thetime domain, fault-arc path nonlinearities, evolving faults,and the invariably sequential form of closing on to a fault,are directly represented in analysis. To these should ofcourse be added the more basic range of requirementswhich frequency-domain methods should be able to fulfil.When the discussion suggests that the structure of individ-

1-480

-640-

-800L-

Fig. D Transmission-line energisationfrequency-domain solutionz-transform solution

Transmission-line energisationfrequency-domain solutionz-transform solution

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

ual programs can make comparisons between differentanalysis methods misleading, it would be especially mis-leading to seek to explain the differences betweenfrequency-domain and z-transform methods in these terms.

When the discussion refers to transmission-line impulse-response and surge-impedance function definition, itshould perhaps be mentioned that this is carried out at thetime of line parameter calculations. Response functioncoefficients are thereafter held with other line data. For agiven transmission line, they are evaluated only once; and,on a CDC Cyber 72/73, the computing time requirement isabout 2.0 s. In the transient solution itself, z-transformanalysis appears to have computing time requirementswhich are several times lower than those quoted in thediscussion.

The discussion has offered several points, some of whichhave been addressed in recently published work extendingthe above studies. However, the discussion and this replyhave provided a useful opportunity for referring further tosome basic aspects of z-plane electromagnetic transient-analysis methods.

W.D. HUMPAGE

Department of Electrical and Electronic EngineeringUniversity of Western AustraliaCrawley, Western Australia 6009

References

A JOHNS, A.T., and AGGARWAL, R.K.: 'Digital simulation of faultede.h.v. transmission lines with particular references to very-high-speedprotection', Proc. IEE., 1976, 123, (4), pp. 353-359

B JOHNS, A.T., and AGGARWAL, R.K.: 'Performance of high-speeddistance relays with particular reference to travelling-wave effects',ibid., 1977, 124, (7), pp. 639-646

C JOHNS, A.T., EL-NOUR, M., and AGGARWAL, R.K.: 'Per-formance of distance protection of e.g.v. feeders utilising shunt reactorarrangements for arc suppression and voltage control', IEE Proc. C,Gen., Trans. & Distrib., 1980, 127,(5), pp. 304-316

D GOLDEN, R.M.: 'Digital filter synthesis by sampled data transform-ation', IEEE Trans., 1968, AU-16, pp. 331-329

E HUMPAGE, W.D., WONG, K.P., AL-DABBAGH, M.H., andMUKHTAR, E.S.: 'Dynamic simulation of high-speed protection',Proc. IEE, 1974, 121, (6), pp. 474-480

F HUMPAGE, W.D., and WONG, K.P..: 'Some aspects of the dynamicresponse of distance protection', Trans. Inst. Eng. Aust. Electr. Eng.,1979, EE15, pp. 122-129

G HUMPAGE, W.D., WONG, K.P., and NGUYEN, T.T.: 'z-plane syn-thesis of response functions and interpolators in z-transform electro-magnetic transient analysis in power systems', IEE Proc. C, Gen.,Trans. & Distrib., 1982, 129, (2), pp. 101-106

H HUMPAGE, W.D., WONG, K.P., and NGUYEN, T.T.: 'Hilberttransform in impulse functions of power, transmission-line electro-magnetic transient model, IEE Proc. C, Gen., Trans. & Distrib., 1983,130, (5), (to be published)

DTC 104C

Editorial comment

The above correspondence was first received on the 15thApril and a reply obtained from the authors on the 6thMay 1981, with a revised version on the 21st December1981. The final forms of both discussion and reply werereceived on the 16th May 1983.

Normally the editors would like to publish discussionon papers within a few months of their publication. In thiscase, the editors very much regret the delay in publicationof this correspondence and reply, due to complexities indealing with the technical points involved; but, neverthe-less, they hope that this delayed publication will still be ofvalue to readers in this important area of research andapplication.

B.J. CORYM.C. RALPH

AUTOMATIC GENERATION CONTROLOF INTERCONNECTED POWERSYSTEMS USING VARIABLE-STRUCTURECONTROLLERS

We wish to comment on the following aspects of paper1513C [IEE Proc. C, 1981, 128, (5), pp. 269-279].

Most simulation studies dealing with the load-frequencycontrol problem [A-D], whether using conventional con-troller or linear optimal controller, consider two equalinterconnected areas with steam generation. In all thesestudies, the control law assumed for both areas is also thesame because both areas have the same parameters as wellas the same capacity. The authors of this paper do notprovide any reason for using different constants for thetwo areas, even though they are studying the 'problem ofan interconnected power system consisting of two(identical) steam plants (subsystems)' (Section 5 of thepaper). If the same constants are used for both areas asdone by other investigators, results shown in Figs. 3-6 willbe different.

Secondly, there seems to be a flaw in the argument inobservation (v) Section 6 of the paper. In the first sentenceof this paragraph, the authors state that 'to avoid unneces-sary small random load fluctuations, the conventional con-troller is preserved in the VSS controller ...'. In the nextsentence it is stated that this modifed portion of the VSScontroller, i.e. conventional controller, becomes functionalonly when the absolute value of the system frequency devi-

ation exceeds a prespecified value, e = 0.001 Hz. Funda-mentally, it seems incorrect because one is interested inreducing the frequency deviation and tie-line-powerdeviation during the transient period when the error islarge, and changes in control strategies are required to bemade for this period. The control strategy should, there-fore, be just the opposite to that stated by the authors.

The VSS controller should become functional onlywhen the absolute value of the system frequency deviationexceeds a prespecified value. In that case, the dynamicresponse with the VSS controller may well be quite differ-ent from that shown in the paper.

ASHOK KUMARO P . MALIK

University of Calgary2500 University Drive NWCalgary, AlbertaCanada T2N 1N4

We would like to respond to the points raised by thediscussers in the following way:(a) The gains for the VSS controllers were determinedby a trial-and-error approach to yield the best dynamicperformance for the specific disturbance. It has been foundthat the system dynamic response is not sensitive to varia-tions in the gains. Symmetric controllers' gains were usedin our recent work [E-G].

IEE PROCEEDINGS, Vol. 130, Pt. C, No. 4, JULY 1983 231