Development of z-transform electromagnetic transientanalysis methods for multinode power networks

Prof. W.D. Humpage, B.Sc, Ph.D., K.P. Wong, M.Sc, Ph.D., and T.T. Nguyen, B.E.

Indexing terms: Power transmission, Electromagnetic transient, Z-Transforms

Abstract: The paper continues and extends the recent development of z-transform electromagnetic transientanalysis in power systems as reported in a companion paper. Whereas the original development was based onthe configuration of a single transmission line and a simplified equivalent-source model, the present paperextends this to the more general case of multinode-power-transmission networks. In those terms, theformulation developed in the present paper provides for distributed-parameter transmission-line models withseries-path nonlinear frequency dependence, together with lumped-parameter circuit elements of linear andnonlinear forms, in any arbitrary combination and interconnection. This general electromagnetic transientanalysis formulation is complemented by means of representing any circuit-breaker switching sequence andany short-circuit fault constraint. An assessment of the inherent accuracy and stability of the multinodez-transform solution scheme derived is made from comparisons with solutions from formal frequency-domainanalysis.

List of principal symbols

YN>ZN = nodal admittance and impedance matrices

for complete networkMx, M2 — modal transformation matricesR,L,C = resistance, inductance, and capacitance

matrices of lumped-parameter branch sets^(/n)> i(Jn) ~ vectors of voltage and current variablesRe,J = residual vector and Jacobian matrix in

Newton-Raphson sequence for nonlinearbranch sets

r = total number of node sets in networkk,j — node-set identifiersa, /3, 7 = parameters in representing nonlinear magnet-

isation characteristicAt — trapezoidal rule step lengthtn — current time step in recursive time-domain

analysis

At the branch level, superscript p identifies phase variablevectors and matrices.

Superscript t identifies a matrix or vector transpose

1 Introduction

A companion paper1 reports a new form of power-systemelectromagnetic transient analysis in the derivation of whichthe z-transform has a unique part. The present paper firstextends the z-plane formulation from that for a singletransmission-line model on which the original work wasbased to the more general case of multinode power net-works. It then closely examines the inherent accuracy andstability of this generalised analysis scheme.

Beginning with the z-plane formulation for a single trans-mission line, an equation system is assembled in the presentpaper for any arbitrary interconnection of distributed-parameter multiconductor transmission line models.Transformation from the separate modal co-ordinates ofindividual transmission circuits into a common axis systemfor a complete network leads to a formulation in which

Paper 989C, first received 2 1st January and in revised form 14thJuly 1980The authors are with the Department of Electrical & ElectronicEngineering, University of Western Australia, Crawley, WesternAustralia 6009

IEEPROC. Vol. 127, Pt. C, No. 6, NOVEMBER 1980

phase, or conductor, variables are used throughout. There-after, lumped-parameter elements, as in series and shuntforms of reactive-power compensation, and also nonlinearelements as in transformer representations, are brought intoanalysis directly in phase-variable form. Means ofrepresenting any given circuit-breaker switching sequence,and any form of short-circuit fault constraint, thencomplement this general electromagnetic transientformulation.

In providing the detailed assessment of the inherentaccuracy and stability that seems essential in the continuingdevelopment of z-transform-analysis methods, closecomparisons are made between solutions from the generalformulation of the present paper and those of establishedFourier-transform methods.2"4

2 Recursive sequence for single-transmission line

For a single transmission line terminating in node sets k and/, the successive stages of transformation in the originalwork lead to the recursive sequence in the time domain

= vzk(tn.j)

= Vzj(tn.j)

(1)

(2)

Voltage and current vectors at the node-set terminations ineqns. 1 and 2 are in the modal axes,5'6 and derivationsleading to the constant matrix Zh and to the previous-value vectors vzk (tn _;) and vzi{tn _;) are given in the firstpaper.1

In preparing for a nodal assembly of individualtransmission-line equations within a complete networkmodel, eqns. 1 and 2 are rearranged to

= Ylvk{tn)-izj{tn.j)

where

= Zf

and

izk(tn-j)=

(3)

(4)

(5)

-/) (6)

;) (7)

379

0143- 7046/80/06379 + 07 $01-50/0

3 Multinode formulation

3.1 Transformation to common axis-system

There are pairs of equation sets of the form of those ineqns. 6 and 7 for each transmission line in the network forwhich a complete model is to be formed. Each is formed inits own modal co-ordinates.s>6 In bringing individualtransmission-line relationships together in a completenetwork formulation, it is first required to transform fromthe separate modal domains into some common axissystem. The most obvious common system into which totransform is that of conductor variables: there is aninteresting correspondence here between the transform-ations required in the present formulation and those ofmultimachine system dynamic analysis.7 Denoting vectorsof conductor voltages and currents by vp and ip, respect-ively, these are related to the corresponding modal variablesby the modal transformation matrices Mx and M2 in

»£(/„) = Mxvk{tn)

i{(tn) = M2ih(tn)

(8)

(9)

Using these transform relationships is closely similar to thestep in single-circuit z-transform analysis,1 and also in time-convolution analysis,8'9 when modal solution variables inthe time domain are finally transformed to conductorvariables. The transformation is made at a particularfrequency. The main frequency dependencies in theformulation are taken fully into account in forming F(z)and ZQ(Z),1 whereas the frequency dependence of Mx andM2 is discounted in the relationships of eqns. 8 and 9, andthe real parts of Mx and M2 are used in them. Using therelationships in eqns. 3 and 4 subject to' this then gives

-iPh{tn-j) (10)

(11)

(12)

(13)

(14)

if(tn) =

where •

Yf = M2YtMTl

izhitn-j) = M2izk(tn.j)

izj(tn-j) = M2izj(tn.j)

3.2 Nodal assembly

In the common phase-axes of eqns. 10 and 11, individualtransmission-line-equation sets in the time domain nowcollate directly to the nodal scheme:

'jv('n) = YN^(tn)-'p(tn-j) (15)

If there is a total of r transmission lines in the network

'"£('») = [<7(U]f [/f (tn)]l • • • , [iPr(tn)Y (16)

<(tn) = K(/J]fK(O]f.--,K(O]f 07)

i'pitn-i) = [iP{tn-i)YAiP{tn-i)V,--.,[iP{tn_j)}t

(18)For the diagonal sub-blocks of the YN matrix

YNQc,k) = YkJ (19)

where s is the set of node sets with direct node connectionsto node set A:.

For the off-diagonal sub-blocks

YN(k,f) = -YkJ (20)

In evaluating electromagnetic transient propagationthroughout the network for any given switching of short-circuit fault condition, one or more subvectors in vN(tn)will be specified. For clarity, we avoid giving the corre-sponding partitioned form of eqn. 15, but it will later beuseful to be able to refer, following this, to the inverse formof eqn. 15. We use for this

for

'N

(21)

(22)

(23)

andvp(tn-i) - YNlip(tn-j)

3.3 Lumped-parameter circuits

For lumped-parameter branch sets, as in series- and shuntforms of reactive-power compensation, time-domainequation sets are formed directly in conductor variablesusing the trapezoidal rule, and the state equations.10 For astep-length of At, the branch relationships have the generalform

iifin) = **/{«£(*„)-«f (*„)}+/?/(*„ - A0 (24)

Particular cases of this general form follow from:(a) resistor branch sets

= 0

(25)

(26)

Rkj is the phase-variable resistance matrix for the branchinterconnections between node sets k and/.(b) inductor branch sets

a' =ifU"fPj(tn-At) = ip

kj(tn-At)

(27)

(28)

Lkj is the phase-variable inductance matrix for branchesbetween node sets k and/,(c) capacitor branch sets

(29)

fPj(tn-At) = -i

-vf(tn-At)} (30)

Ckj is the phase-variable capacitance matrix for branchesbetween node sets k and/.

The lumped parameter branch equation of eqn. 24 is ina form in which it can directly augment the nodal schemefor distributed parameter circuits leading to eqn. 15. If, forexample, a lumped-parameter set is connected between the

380 IEEPROC, Vol. 127, Pi. C, No. 6, NOVEMBER 1980

k node-set and earth, the nodal equations for this setbecome

the same. Equating these gives

YkJvf(tn)

(31)

With similar steps for any particular connection of lumped-parameter elements, eqn. 15 remains the general form forany mix of distributed- and lumped-parameter circuits. Inparticular, Ykj in eqn. 20 is given by theaPj coefficient forseries-connected lumped-parameter branches, whereas it iszero in the case of a distributed-parameter circuit.

3.4 No nlinear circuit elemen ts

It is often necessary to include in analysis the non-linearities of power transformer magnetising branches, and,where they arise, those of static reactive-powercompensators. For these purposes, we consider a set of non-linear inductor elements with direct phase connections toany network node-set k. We discount losses, as these areeasily brought into the formulation subsequently, using thelumped-parameter branch sets of eqns. 24—31.

For inductor branches connected between node set kand earth, the vector of flux-linkages, vpfe, is related to themagnetising currents in vector ih by

= /{/?(*„)}In turn, the nodal voltages at set k are given by

<(tn) = £{/(/£(*„))}t

Using the trapezoidal rule with step length At:

+ ~{vp

In more compact notation

(32)

(33)

= fy) +/;

where

= \f)vp(tn-At)+f{i

(34)

(35)

i(tn-At)}(36)

Prior to the connection of the nonlinear elements, thenode-set ^-voltages are given from the inverse form of thenetwork equations in Section 3.2, eqn. 21, by

(37)

On now making the connection of the nonlinear inductorelements to the network, v% (tn) as used in the inductorequations, and, in particular in eqn. 35, and vp(tn) ineqn. 37, and as formed from the network equations, will be

£ ) /{/J('n)}-^)/;('n "A*) = ZkkiP{tn)

r

+ I ZkJif(tn)+v%(tn.j)/;i (38)

In solving this nonlinear set, we use the Newton-Raphsonsequence

- I'"1 « (tn)}] Q~l[Re K (39)

in which q denotes the iteration count.The residual vector, Re {i% (tn)} in eqn. 39, is given by

Re{iph(tn)} = (47

- Zhk ip

k (tn) - l-£\ fp (tn -At) (40)

- I Zkjip(tn)-vp

k(tn.j)

The Jacobian matrix in eqn. 39 is formed from

'kk (41)

Very often, the magnetising characteristics commonlyarising can be represented closely by

f(im) = aOm + i"o) + 13 tan" ' {y(im + i0)} (42)

The parameters a, 0, 7 are easily found in providing a matchto a given magnetising characteristic: /0 represents the pre-transient remanence level. When the nonlinearity isrepresented in this way

(43)

Elements of the Jacobian matrix in eqn. 41 are then formedfrom eqn. 43 in terms of the basic magnetisation charac-teristic data it requires.

On achieving convergence in solving eqn. 39, the overallsolution sequence returns with this solution to the net-work equations from which the remaining nodal voltagesare then calculated. Although a single branch set has herebeen considered, the development clearly applies for anynumber of nonlinear element sets. The Jacobian matrix inthe Newton-Raphson algorithm is of order 3 Nn where Nn isthe total number of nonlinear inductor branch sets to beincluded in nonlinear analysis.

3.5 Circuit-breaker closing sequences

As the solution based on the formulation of the paper iswholly in the time domain, any specified circuit-breakerclosing sequence is easily incorporated in analysis. Insequential pole closure, for example, elements of the

IEEPROC, Vol. 127, Pt. C, No. 6, NOVEMBER 1980 381

voltage vector ^ ( / n ) are known at, and subsequent to,given closing instants. Voltages specified in this way arethen partitioned from the set of eqns. 15 in Section 3.2, andthe remaining nodal voltages are calculated in terms ofthem. To include trapped charge requires no more than acalculation of initial-condition values with specified pre-switching nodal voltages.

3.6 Short-circuit fault constraints

As the solution is throughout in terms of phase variables,any shunt-fault constraint is introduced by a matrix of faultselector admittances, Yf. For a single-phase-to-earth faulton phase V , the phase V diagonal element of Yf is set to aspecified fault path conductance, while remaining diagonaland off-diagonal element values are set to zero. Theprocedure is similar for any given combination of faultedconductors. For a b—c phase-to-phase fault clear of earth:

Yt =

0 0 0

0 Gf -Gf

0 -Gf Gf

(44)

Once formed for any given fault type, the fault selectormatrix augments the diagonal submatrix of the networkmatrix YN for the node set to which fault constraints are tobe applied. Sequential or evolving faults require only acontrol of the Yf matrix during the course of solution.

4 Some aspects of accuracy and stability

4.1 General

A consideration of the accuracy that different forms ofelectomagnetic transient analysis can provide suggests thatthe accuracy of the formulation of the present paper mightmost directly be evaluated in terms of comparisons withsolutions from frequency-domain analysis. In the form firstderived,2"4 frequency-domain methods may have highercomputing-time overheads than those of time-con-volution,8'9 direct time-domain,10 or the travelling-wave

'lattice,17'18 but they can provide a rigorous analysis

-*-

method of high solution accuracy. Some limitations how-ever arise, if computing times are to be kept within reason-able bounds, in both the network configuration and theswitching sequences to which they might best be applied.Perhaps the most basic test configuration, but one forwhich frequency-domain methods can directly be used, isthat of Fig. la. The accuracy of all methods of electro-magnetic transient analysis is tested most stringently inapplication studies in which the time-rates-of-change ofsolution variables are highest, hi the formulation of the firstpaper,1 the source from which energisation takes place is ofa lumped-parameter form, and whereas this is widely used inovervoltage studies, other cases can lead to more demandinganalysis conditions. In particular, where there aredistributed-parameter circuits behind the point of circuit-breaker switching, the interaction between transient wavepropagation in these and in the circuit energised from themcan lead to significantly higher rates-of-change of variablesthan when energisation is from a lumped-parameter source.The configuration of Fig. \a allows the effects of travelling-wave interactions to be investigated. Rates-of-change ofvariables are further accentuated when there is preswitchingtrapped charge on the incoming circuit.

In addition to line energisation and the switching over-voltages to which it gives rise, the electromagnetictransients initiated by short-circuit fault on-set, or bycircuit-breaker operations in fault clearance, can havepronounced pulse-type components. Here, the rates-of-change in solution may well be higher than in lineenergisation. It is feasible to carry out short-circuit faultanalysis in the frequency domain for the system of Fig. \b,and this configuration is one that allows conditions whichare representative of short-circuit fault analysis moregenerally to be examined.

4.2 Energisa tion from dis tribu ted-parame ter sou rce

For the purposes of comparative study each transmissionline in the system of Fig. la is 160km in length;transmission line data are summarised in Reference 1. InFig. 2 is shown the z-transform solution for the voltagewaveform in phase V at the open-circuited end of thesecond transmission line following simultaneous poleclosure in circuit-breaker energisation at C. At the instantof closure, charge equivalent to peak system voltage istrapped on the second circuit. Fig. 2 gives the general formof the voltage solution in one phase over a time periodtypical of those in switching over-voltage evaluations. Theeffects of trapped charge are clear, as are the considerablyhigher rates-of-change in solution when compared withthose in the companion paper.1 Of particular interest inrelation to solution accuracy are the effects of wave

-X-

Fig. 1 Elemental test configurations

S = equivalent lumped-parameter source modelL = distributed-parameter multiconductor line modela Circuit breaker C closes at time t— 0 in line energisation. Trapped

charge on incoming transmission line is equivalent to the pre-switching voltage distribution: Va =— 10p.u.Kb= 0-5 p.u., Vc —0-5 p.u.

b Electromagnetic transient propagation is initiated by 3-phase-to-earth fault on-set at F

382

Fig. 2 Switching overvoltage transient in line energisation

System and trapped charge distribution of Fig. \a

IEEPROC, Vol. 127, Pt. C, No. 6, NOVEMBER 1980

reflections at the interconnection between the source andthe first transmission line in Fig. 2 at 1-7ms, 28ms, and3 9 ms and to later ones of lower amplitude separated bytwice the wave transit time of the line. On an enlargedscale, and over a 4 ms solution period, z-transform andfrequency-domain solutions are superimposed in Fig. 3.This enlargement allows the initial response, including theearly discontinuities, to be examined in detail and closecomparisons to be made between the two solutions.

Although the z-transform solution retains high waveformaccuracy, there is some loss of definition in the frequency-domain solution when the solution rates-of-change arehighest. Here, the frequency separation is 200 rad/s, theupper frequency limit is 10s rad/s, the step length ininverse Fourier-transform evaluations is 20/us, and thedamping factor2'3 is 250. The Lanczos sigma factor isincluded to filter Gibbs-type oscillations from solution.Inevitably, the sigma factor will influence the realisablesolution accuracy, especially as the frequency spectrum ofparticular transient conditions increases. To examine thiseffect, the sigma factor is removed from the frequency-domain solution sequence leading to the waveform ofFig. 4. The oscillatory components that the sigma factor isintended to suppress are now present, but, apart from this,frequency-domain and z-transform solutions are now moreclosely comparable.

From Figs. 3 and 4, it is clear that, with the frequency-domain parameters here summarised, the z-transformsolution is of the higher accuracy. It will furthermore beclear that the difference between frequency-domain andz-transform solutions close to the discontinuities arisingfrom travelling-wave reflections is related to truncationbeing required in frequency-domain analysis but not inz-transform methods. When the z-plane forward-responseand surge-impedance function in the z-plane are derived,the rational-fraction forms for each are matched to

> 1400I•£ 122012

200 240 280 320 360 V4O0time.ms

Switching overvoltage transient in line energisationSystem and trapped charge distribution of Fig. la

frequency domainz-transform

140012201040-860680500320"140-40r

-220--400 '

040 O80 120 166 200 240 2-80 3-20 3:6o7400time.ms

Fig. 4 Switching overvoltage transient in line energisation

System and trapped charge distribution of Fig. lafrequency domain with sigma factor excludedz-transform

frequency-domain functions in the range 0—10s rad/s. Thishas been deliberately chosen to coincide with the frequencyrange used in frequency-domain analysis. It is interesting toconsider the difference between the two in these terms. Infrequency-domain analysis, there is an abrupt cut-off in thefrequency plane when w = 10s rad/s: components ofhigher frequency are rejected. In z-transform analysis, theresponse functions F(z) and Z0(z), although specificallymatched in the range 0— 10s rad/s, are not in any wayterminated, and are generally valid, well beyond this range.There is considerable advantage in the particular form thatthe F(oS) and Z0(co) functions take: F(GJ)->-0, andZo (co) -> a constant value, at high frequency. Precisematching for frequencies in excess of 10s rad/s seemsunlikely to be required in many practical applications,although some detailed examinations of wave fronts in thefirst 1—2 ms of propagation might possibly require a rangeof 0—106 rad/s. Specified frequency ranges along theselines are easy to meet in z-transform analysis. But functionsmatched in a given range retain their general validity wellbeyond it. To the extent that frequency components intransient analysis are not expected outside of a specifiedrange, precise function values will not enter into thesolution, but the fact that they are not at any stageterminated is of considerable benefit in avoiding thegeneration of unwanted components in solution.

4.3 Short-circuit fault conditions

Electromagnetic transients in short-circuit fault operationare here considered for the case of a symmetrical3-phase-to-earth fault at F in the system of Fig. \b. From aseries of studies of different fault types it appears that, inthe context of the present studies, analysis conditions aremost testing in the 3-phase-to-earth fault case. The 400kVtransmission lines of Fig. \b are those of Fig. la, and bothlines are terminated by source models of the form given inthe earlier work.1 Z-transform solutions for phase Vvoltage and current waveforms at point P in Fig. \b areshown over a 10 ms period in Figs. 5 and 6, respectively.These give the general form of fault current and voltagesolutions for one phase. The short-circuit fault dis-continuity, and the pulse components in the voltage wave-form to which it gives rise, will severely test solutionaccuracy, but the waveform remains well-defined in thez-transform solution of Fig. 6.

From the general form of the solution, it is useful nowto turn to enlargements over the first 2 ms of the solutionperiod. This interval includes the most pronounced dis-continuities in the conductor voltage waveform, and theexceptionally high rates-of-change of solution variablesassociated with them. On this basis, Fig. 7 compares

600-530'460-3 90-320-2-50-1 80-

-110-0-40--0-30r 100 200 300 4O0 5O0 600 700 800 900 -K>00

time.ms

Fig. 5 Conductor current waveform following 3-phase faultinception

System of Fig. lb: z-transform solution at node P

IEEPROC, Vol. 127, Pt. Q No. 6, NOVEMBER 1980 383

z-transform and frequency-domain solutions where thefrequency-domain analysis parameters are those of Section4.2 and the sigma factor is included in the-solution method.Although retaining high accuracy when the rates-of-changeof solution variables are modest, the frequency-domainmethod is unable accurately to reproduce the short-duration voltage pulses. Eliminating the sigma factorgives the frequency-domain solution of Fig. 8. Althoughthis confirms that the filtering that the sigma factorimportantly provides limits the high-frequency response ofthe analysis method, there is still a significant differencebetween z-transform and frequency-domain solutions inintervals close to the major waveform discontinuities. Theexamination is continued by extending the frequency-rangein the frequency-domain analysis from 105 rad/s to106 rad/s. For this upper frequency limit, a frequency'separation of 2000 rad/s has been used, together with adamping factor of 1500 and a steplength in inverse Fouriertransforms evaluations of 20jus. This leads to thefrequency-domain solution of Fig. 9. The sigma factor isexcluded, and the increased upper frequency limit thenlargely avoids unwanted oscillatory components.Frequency-domain analysis now offers a similar accuracy tothat of z-transform analysis. No special or additionalmeasures are required to achieve the high accuracy of thez-transform scheme. The accuracy which the solutions andcomparisons of Fig. 9 confirm is the inherent accuracy ofthe z-transform method as originally developed.1 Thecomputing time of the z-transform method for the solutionof Fig. 9 is less than 1% of that when using the frequency-domain method.

4.4 Ex tended solu tion periods

In frequency-domain analysis, the solution accuracy that

Fig. 6 Conductor voltage waveform following 3-phase faultinception

System of Fig. lft: z-transform solution at node P

600-520'440<360280'200'120'

-40<-120'-200-

0:20' 0:40' ^60 080 K> V2 14 1-6 1-8 20time.ms

Fig. 7 Initial fault-voltage waveform

System of Fig. 16: solutions at node Pfrequency domain, sigma factor included, upper-frequencylimit 10s rad/sz-transform

can be provided is related to the time-period over which thesolution is assured. The high accuracy of the solution ofFig. 10, for example, is sustained for only slightly longerthan the 2 ms period for which the solution is shown.Beyond this, aliasing effects lead to a rapid buildup oferrors which mask the solution. By comparison, there is nocorresponding limitation in z-transform analysis. Themethod is of high inherent stability. In principle, thisderives from the form of the transformation between thefrequency domain and the z-plane. As a passive distributed-parameter system is a stable system, a transmission-linemodel in the frequency domain is always stable. It is aproperty of the bilinear transformation (z — l)/(z + 1)that, in transformation, stability is always preserved. F(z)and Z0(z) are always stable system functions, and therecursive sequences deriving from them on taking inverse,z-transforms are therefore stable sequences. There istherefore no limitation in the solution time for which thez-transform time-domain sequences are evaluated. Z-transform analysis achieves simultaneously high accuracy

020 0-40 0-80 10 12 14 16 18 20

time.ms

Fig. 8 Initial fault-voltage wave-form

System of Fig. 16: solutions at node Pfrequency domain, sigma factor excluded, upper frequencylimit 10s rad/sz-transform

0-80 10 1-2 1-4 16 1-8 20time.ms

Fig. 9 Initial fault-voltage waveform

System of Fig. lft: solutions at node Pfrequency domain, sigma factor excluded, upper frequencylimit 106 rad/sz-transform

90/100

Fig. 10 Fault-voltage waveform in extended solution period

System of Fig. 1ft: z-transform solution at node P

384 IEEPROC, Vol. 127, Ft. C, No. 6, NOVEMBER 1980

and high stability for any solution time period. Both areinherent in the z-plane formulation.

A study application requiring extended solution times isthat of composite primary system/transducer/processingsystem simulation13 in the investigation of signal processingmethods for fault monitoring and protection purposes.Thoroughly to explore certain threshold conditions mayrequire solution times of up to 100 ms. The fault voltagewaveform for the system of Fig. \b is shown in Fig. 10for this extended period. Related to the stability ofz-transform methods, they lend themselves easily toautomatic time-interval control as the initially high rates-of-change of solution variables subside during the total solutiontime.

4.5 Other test conditions

Numerous other test-analysis conditions might be suggested,but particular reference should be made to those in whichthe nonlinearities of transformer magnetisation character-istics are included and to those where circuit-breaker closureis of a sequential-pole-closing form. Using the methods ofSections 3.4 and 3.5, both are represented directly inz-transform analysis, whereas as the frequency-domainmethods on which the comparisons of Sections 4.2—4.4have been based do not easily lend themselves to theseanalysis conditions in providing reference solutions. Infurther work,14 z-transform analysis is related to establishedconvolution methods, and here solutions from the twomethods are compared for sequential pole-closing andwhere transformer nonlinearities are represented.

5 Conclusions

The paper has shown that z-plane methods can lead to aparticularly direct, yet general and comprehensive, schemeof electromagnetic transient analysis for multinode powertransmission networks. The final equation system forsolution is wholly in the time domain, and is in terms ofconductor variables throughout. That form allows anycircuit-breaker switching-sequence or any short-circuit faultconstraint easily and directly to be represented. Nonlinearelements are included using the development of Section3.4, and although that relates principally to the non-linearities arising from transformer magnetisation charac-teristics, similar methods are clearly applicable whenrepresenting the nonlinear response of overvoltageprotection systems. Test analyses where inverse tangentfunctions are fitted to magnetisation characteristics inpower transformer models, confirm the rapid convergencethat this can give in the Newton-Raphson iteration analysisdeveloped.

The comparisons between z-transform and frequency-domain solutions of Section 4 can do much to confirm theaccuracy and stability that are inherent in z-transformmethods. Only by reducing the consequences of truncationto levels at which the filtering action of the sigma factor isnot required, and then removing it from the solutionprocedure, can frequency-domain methods providecomparable accuracy. Whereas in many applications, thisorder of accuracy is unlikely to be crucial, it is essential in

extensively evaluating the new solution method to examineits underlying accuracy and stability in detail, andespecially for exacting analysis conditions. Without anyspecial measures, the z-plane formulation is more thanequal to the tests of the present paper, and it combines thehigh inherent accuracy and stability which these confirmwith low computing-time requirements.

6 Acknowledgments

The authors are grateful to the Australian Research GrantsCommittee for financial support, and to the WestAustralian Regional Computing Centre for running theirprograms. The generous support of power systems researchin the Department of Electrical & Electronic Engineeringat the University of Western Australia by Professor A.R.Billings, together with his professional co-operation at alltimes, is gratefully acknowledged.

7 References

1 HUMP AGE, W.D., WONG, K.P., NGUYEN, T.T., andSUTANTO, D.: 'z-transform electromagnetic transient analysisin power systems', IEE Proc. Q Gen., trans. & dist., 1980, 127,(6), pp. 370-378

2 BATTISSON, M.J., DAY S.J., MULLINEUX, N., PARTON,K.C., and REED, J.R.: 'Calculation of switching phenomena inpower systems', Proc. IEE, 1967, 114, (4), pp. 478-486

3 BATTISSON, M.J., DAY, S.J., MULLINEUX, N., PARTON,K.C., and REED, J.R.: 'Some effects of the frequency depen-dence of transmission line parameters', ibid., 1969, 116, (7),pp. 1209-1216

4 BATTISON, M.J., DAY, S.J., MULLINEUX, N., PARTON, K.C.and REED, J.R.: 'Calculation of transients on transmission lineswith sequential switching', ibid., 1970, 117, (3), pp. 587-590

5 BOWMAN, W.I., and McNAMEE, J.M.: 'Development ofequivalent matrix circuits for long untransposed transmissionlines', IEEE Trans., 1964, PAS-83, pp. 625-632

6 WEDEPOHL, L.M.: 'Application of matrix methods to thesolution of travelling-wave phenomena in polyphase systems',Proc. IEE, 1965, 110, (12), pp. 2200-2212

7 HUMPAGE, W.D.: 'Structure for multi-node power systemdynamic analysis methods', ibid., 1973, 120, (8), pp. 853-859

8 BERGMANN, R. Ch.G., and PONSIOEN, P.J.M.: 'Calculation ofelectrical transients in power systems: untransposed transmissionline with frequency-dependent parameters', ibid., 1979, 126, (8),pp. 764-770

9 NGUYEN, T.T., WONG, K.P., and HUMPAGE, W.D.: 'Impulsesampling sequences in time-convolution electromagnetictransient analysis in power systems', Electr. Power Syst. Res. (inpress)

10 DOMMEL, H.W.: 'Digital computer solution of electromagnetictransients in single-and multi-phase networks', IEEE Trans,1969, PAS-88, (4), pp. 388-396

11 BICKFORD, J.P. and DOEPEL, P.S.: 'Calculation of switchingtransients with particular reference to line energisation', ProcIEE, 1967, 114, (4), pp. 465-477

12 BICKFORD, J.P. and RAHMAN, M.H.A.: 'Application oftravelling-wave methods to the calculation of transient-faultcurrents and voltages in power-system networks', IEE Proc. C,Gen., Trans. & Dist., 1980, 127, (3) pp. 153-168

13 HUMPAGE, W.D., and WONG, K.P.: 'Some aspects of thedynamic response of distance protection', Trans. I.E. Aust.,1979, EE15, (3), pp. 122-129

14 HUMPAGE, W.D., WONG, K.P., and NGUYEN, T.T.: Time-convolution and z-transform methods of electromagnetictransient analysis in power systems', IEE Proc. C, Gen., Trans.,& Dist., 1980, 127, (6), pp. 386-394

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