Surge diverter and transformer nonlinearities inz-transform electromagnetic transient analysis

in power systemsProf. 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 analysis, Z-transforms

Abstract: The paper reports developments by which the nonlinear characteristics of surge diverters may berepresented in z-transform methods of electromagnetic transient analysis in power-transmission networks.For this purpose, it is shown how surge-diverter equations may be combined with those for network nodes towhich surge diverters are connected and solved using a Newton-Raphson sequence. As'surge^iiverter pro-tection is often associated with transformers, the analysis scheme is then extended to include the non-linearities of transformer magnetisation characteristics. The analysis method developed is a general andcomprehensive one. Within a multinode network formulation, it provides for surge-diverter connections toany number of network nodes, and it allows for any number of transformer branches. In these terms, thework of the paper further advances the recently developed forms of electromagnetic transient analysis inpower systems based on the z-transform.

List of principal symbols

Z-transform formulation%(^))'iv(?n) = vectors of nodal voltages and currents in

phase-variable form for a multinode net-work.

ZN = constant nodal impedance matrixVP (tn /) — nodal vector of previous valuesm = ratio of wave-transit time to step-interval

vg(tn),vd(tn)

vu(tn)

Surge diverter model— surge-diverter arc-gap and nonlinear resistor

voltage components, respectively= spark-over voltage threshold

id(tn) = surge-diverter currentkd = scaling-constant of nonlinear resistorvG(tn)>vD(tn) — vectors of gap and nonlinear resistor

voltages for a set of s diverters(D(AI) = vector of surge-diverter currents for a set of

s diverters= diagonal matrix of nonlinear resistor con-

stantsK

Nonlinear analysis formulationx = composite vector of unknowns/K(x), /T(x) = residual functions for surge diverters and

transformers, respectivelyJ(x) = Jacobian matrixp = Newton-Raphson iteration count

Node set identifiersM = nodes for which nodal current injections

are knownK = nodes to which surge diverters are con-

nectedT = nodes to which transformer shunt branches

are connectedL — nodes remaining after M, K and T node

segregations

1 Introduction

Several papers [1—4] have reported means by which the non-linear characteristics of surge diverters might be represented,and recent work by Cornick and Lucas [5] has shown how

Paper 1185C, first received 2nd May and in revised form 6th August1980The authors are with the Department of Electrical & ElectronicEngineering, University of Western Australia, Crawley, WesternAustralia 6009

IEEPR0C, Vol. 128, Pt. C, No. 2, MARCH 1981

typical representations can be included in power-transmission-line electromagnetic transient- analysis. In these latter pro-posals [5] , a Thevenin form of representation for networknodes to which surge diverters are connected is first derived interms of equivalent internal-voltage and indicial admittancefunctions. In turn, the network indicial admittance for anynominated node is formed from initial frequency-domainanalysis and inverse Fourier transform evaluations. Theequations of this equivalent-network representation thencombine with those of a surge-diverter model in nonlineartime-convolution analysis the iterative sections of which aresolved by a Newton-Raphson sequence. The paper [5] showshow both active-arc and controlled-arc forms of surge diverter[6] may be represented in this analysis scheme.

In independent work [7,8] , a new electromagnetictransient formulation based on the z-transform has been-developed, and reference has been made in this developmentto procedures by which nonlinear circuit elements may be-represented. Using these procedures, transformer-magnetisationcharacteristics have been represented in previous work [8],following which it is now required in the continuing develop-ment of z-transform analysis to investigate the extension ofnonlinear circuit modelling to surge diverters. This is theessential purpose of the present paper.

Whereas the recent proposals [5] allow a single point ofsurge-diverter application within a network to be examined, itwas required in the present work to make provision for surge-diverter connections to any number of network nodes.Moreover, as surge-diverter protection is often used at trans-former nodes, it was further required to include both surge-diverter and transformer models in the one formulation. Inachieving this, the analysis methods of the paper are bothgeneral and comprehensive. As proposals other than theoriginal ones [7,8] by the authors do not appear to have beenmade for z-transform electromagnetic transient analysis inpower systems, no previously published work has referred tothe developments that the present paper seeks to report.

2 Z-transform electrogmagnetic transient analysis

2.1 Multinode formulation

Time-domain sequences derived from z transform analysis fora single transmission line [7] collate for an interconnection oflines in a complete network to the nodal form [8]:

vN(tn) = ZNiN(tn) + vp(tn_j) (1)

In eqn. 1, vN(tn) and iN(tn) are vectors of network nodalvoltages and currents, respectively, at time step tn in recursive

0143-7046/81/02063 + 07 $01.50/0 63

time-domain analysis. Vector z>p(/„_,) is formed from solutionvalues in steps prior to that at tn, and ZN is a constant nodalimpedance matrix. In the assembly of the network equationset in eqn. 1, equations for each transmission line in thenetwork are initially most directly formed in their own modalaxes. These are then transformed for each transmission lineinto conductor variables in which form they may be combineddirectly in an overall network formulation. On this basis,vectors vN(tn) and /#(£„) are expressed in terms of phasevariables.

2.2 Node-set partitioning

Nodes for which voltage functions are specified in electro-magnetic transient analysis are replaced by injected nodalcurrents. These are now grouped into vector iM(tn), so thatiM(fn) is the vector of forcing inputs for which analysis iscarried out, and is known at each time step in solution. Nodesto which surge-diverter models are later to be connected(Section 4) are grouped, and the corresponding network nodalcurrent denoted by iK(tn). The nodal currents remaining arethen grouped into vector iL(tn). With this segregation andgrouping, the complete network set of eqn. 1 partitions to

VMp(tn-j)

LL

mSurge-diverter models are included in z-transform electro-magnetic transient analysis by combining the networkequations for the surge-diverter nodes with surge>diverterequations and solving the nonlinear system to which this leadsat each time step in analysis. During the iterative sequence thatprovides this solution, variables of the remaining partitions ofeqn. 2 remain constant. In extracting^the.network equationsfor the surge-diverter nodes from the compound form ofeqn. 2 for use in Section 4 with surge divecter/equaticms, it istherefore convenient to use.trie-form

= ZKKiK(tn)

in which

= ZKMiM(tn) + ZKLiL.(tn)

(3)

(4)

As formed in eqn. 4 and used in eqn. 3,vKO is constant duringthe iterative solution of the nonlinear equations of surge1

diverters when combined with the network equations for thenodes to which surge diverters are connected.

3 Surge diverter characteristics

3.1 Equivalent single-unit diverter for modelling

Whereas a complete surge diverter is formed from a serieschain of arc gaps and nonlinear resistor elements, as in Fig. la,a multiunit diverter of this kind is represented'in analysis inthe equivalent single unit form of Fig. \b\ Grading resistorsused to equalise the voltage distribution across series-connected arc gaps, do not therefore arise in the. equivalentform to be included in line-transient studies.

3.2 Nonlinear resistor

Surge-diverter current *d(/n) at any time interval th for whichthe diverter is conducting is related to the voltage across thenonlinear resistor, vd(tn) by

The index 7 in eqn. 5 typically satisfies

3 < 7 < 6

It is not proposed in the present investigations to represent theforms of hysteresis in nonlinear resistors to which referencehas previously been made [3].

3.3 Active- and control I ed-arc gaps

Characteristics by which active-arc gaps may be represented intransient analysis are summarised in Fig. 2. For clarity in thedevelopment of the analytical formulation, these characteristicsare used throughout. The procedure for representing con-trolled arc gap based on characteristics corresponding to thoseof Fig. 2 is closely similar to that here derived for the specificcase of active-arc gaps.

3.4 Equations for a set of s diverters

For the equivalent single-diverter unit of Fig. \b

and

(6)

id(tn) = kdvj(tn) (7)

These equations extend directly to where a total of s surge-

va(t)

Fig. 1 Multiunit surge diverter and equivalent form for analysis

1.00.8

Q; 0.6jP1 0.4

0.2

0 0A 0.8 1.2 1.6time.ms

1 10 100 1000time, ps

2.0-

1.2

0 0.5 1.0 1.5 2.0 0 0.4 0.8 1.2v d . P u

= kdvj{tn) (5)Fig. 2 Typical characteristics of active-arc surge diverter

Vu is spark-over voltage

64 IEEPROC, Vol. 128, Pt. C, No. 2, MARCH 1981

diverter units are to be represented in analysis at s/3 diverterlocations.

The vector form of eqns. 6 and 7 is

(8)

(9)

tn) = VG(tn)+VD(tn)

I'DM = KDv&tn)

In eqns. 8 and 9,

vUtn) = vn(tn), vf2(tn), .

vUtn) = vgl(tn), VnM,

vb(tn) = vdl(tn), vd2(tn), . . . vds(tn)

[v&tn)V = v]{(tn); vJi(fn) . . . v%(

KD = d i a g { * d l , kd2 . . . A:ds}

vfs(tn)

vgs(tn) (11)

(12)

03)(14)

4. Surge diverters in z-transform electromagnetic transientanalysis

4.1 Combining surge diverter and network equations

When diverter models are connected to nodes in node-set K,

From eqns. 8 and 15:

vD(tn) = vK(tn)-vG(?n)

(16)

(17)

The diverter currents during periods of conduction are thenformed from eqns. 9 and 17 using

iD{tn) = KD{vK(tn)-vG(tn)y

The vector on the RHS of eqn. 18 is given by

K i ( O - t > * i ( O ] 7 '

(18)

(19)

From the network equations for the nodes to which surgediverters are connected,

(20)

(21)

VKM = -ZKKiD(pn) +

On using eqn. 18 in eqn. 20,

vK(tn) = -ZKKKD{vK(tn)-vG(tn)y

The requirement now reduces to solving eqn. 21 for vK(tn)given individual gap profiles in vG(tn) and vKO from thenetwork equations of Section 2.2. Adopting the Newton-Raphson method for this purpose, a residual function,fK{vK(tn)}, is first formed from

D {%(*„) -vG{tn)Y-vKO

(22)

The Jacobian matrix of partial derivatives is now given by

From the residual function of eqn. 22,

J(vK(tn)} = u +AZKKKDB

IEEPROC, Vol. 128, Pt. C, No. 2, MARCH 1981

(24)

where

A = d i a g { 7 , , 7 2 , . . . 7 s }

B = di

(25)

(26)

Nodal voltages in vK(tn) during periods for which surgediverters conduct are then found from

(27)

4.2 Solution sequence

The z-transform nodal equation set of eqn. 2 is solved insuccessive time intervals ty, t2,.. . tn, and at each step nodal-voltage values in vector vK(tn) are checked against thethreshold settings of the surge diverters to which they relate.When one or more settings are exceeded, the correspondingnodal voltages are calculated using the iterative sequenceof eqn. 27. In addition the time step at which each diverterbegins to conduct marks the commencement of each gap-voltage characteristic. The progression through each charac-teristic is then updated at each itime stejp in solution. Onachieving a solution for vK(tn) using eqn. il,,iK{tn) is foundfrom the surge-diverter relationships in eqn. .18, followingwhich vM(tn) and v L ( v ) are calculated from the M and Lpartitions of eqn. 2. The solution sequence .proceeds in thisway when one'or 'more surge diverters are in conductingmodes,'and reverts to the unconstrained z-plane formulationof eqn. 1 when all diverters have cleared. The nodes in set Kfor which either the network equations or the network equa-tions combined with surge-diverter equations provide a sol-ution changes throughout the solution period as individualdiverters enter or leave a conducting mode.

4.3 Representative solutions

In Figs. 3-5 are shown representative solutions in the case of asurge-diverter application at one end of a 160 km, 400 kVsingle-circuit line. The line is energised from a lumped-parameter source at one end and is open-circuited at the otherend. Source model and transmission-line data are summarisedin Appendix 9.\ and 9.2, and surge-diverter parameters aregiven in Appendix 9 3 .

^. 800.00§> 640.00§ 480.00£ 320.005 160.00gO.OOp -160.008j -320.00"S.-480.00

-640.0C3.0.70| 056-3 0.42£0.28I 0.14"So.ooen

3

\

1.00 2.00 ^A

3.00V 4.00 5.00\9.00 /

10.00/

10.00

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00time, ms

Fig. 3 Line energisation transients in phase 'a'

160 km, 400 kV transmission linevoltage waveform in absence of surge diverters

65

5 Combining surge diverter and transformer nonlinearities

5.1 Equation formulation

In cases where surge diverters provide surge protection fortransformers, as in transformer feeders, it will often berequired to include in analysis the nonlinearities of trans-former magnetisation characteristics in addition to those ofsurge diverters. For this purpose, eqn. 2 is now furtherpartitioned to

^"MM ^MK ^MT ^ML

ZRM ZKK ZKT ZKL

ZTM ZTK ZJT ZTL

Z 7 7 7LM &LK ^LT t'LL

(28)

VT(tn)

3.00 4.00V/ 5.00 6.00 7.00 8.00 9.00 10.00time, ms

1.00 2.00 3.00 4.00 500 600 7.00 8.00 9.00 10.00time ms

Fig. 4 Line energisation transients in phase 'b'

160 km, 400 kV transmission linevoltage waveform in absence of surge diverters

300.00200.00100.00-

I 200.00

| 0.00•g -100.00.8 -200.00

t'°-300.00

-400.00-500.00

<-600.00« 0.00| -0.14<-> -0.28

I -°A 2| -0.56% -0.70-

6.00 8.00.V

1.00 2.00 3.00 4.00 5.00 ,'~\ !Mj\rf 900 10.00, ' ' time,ms

2.003.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

time . ms

Fig. 5 Line energisation transients in phase 'c'

160 km, 400 kV transmission linevoltage waveform in absence of surge diverters.

66

In eqn. 28, subscript T identifies the transformer node set.Corresponding to eqn. 3, the network equations for the surgerdiverter and transformer-node groups are isolated and used inthe form

(29)

(30)vTO

and

vT(tn) = ZTKiK{tn)

where

%o = ZKMiM{tn) + ZKLiL(tn) + vKP(tn -j

vTO = ZTMiM(tn) + ZTLiL{tn) + vTP(tn .j)

On connecting surge diverters to nodes in set K

iK(tn) = ~KD [vK{tn) - vG(tn)] i

(32)

(33)

For the transformer magnetising branches connected to nodesin set T, and using the development of Appendix 9.4,

= VT(tn)+fTP (34)

in which vector / T P is formed from solution values in stepsprior to tn. Replacing the nodal current vector /"x(^n) m

eqn. 29 by the expression for /^(/n) in eqn. 33 gives

%('*) = -ZKKKD{VK{tn)-VG{tnW +ZKTiT(tn)

(35)

Similarly, vT(tn) from the transformer relationships of eqn. 34,together with the expression for surge-diverter currents ineqn. 33, substitute into eqn. 30 to give

= -ZTKKD{VK{tn)-VG{tn)Y

+ ZTTiT(tn) + vTO + fTP (36)

It is now convenient to combine the surge-diverter nodal-voltage vector VK(tn) and the transformer-magnetising-currentvector iT(tn) into composite-solution vector, x, using

~vK(tn)x = |

Residual functions fK(x) andand 36:

and

(37)

now follow from eqns. 35

— ZKTiT(tn) — vKO

fT(x) = f{iT(tn)} + ZTKKD{vK{tn)-vG{tn)Y

— Zj^ixitn) — vTO ~/TP

Vector functions /K(X) and fo(x) now combine to

14*)}

The Jacobian matrix is then given by

fix) =

/(*) =

(38)

(39)

(40)

(41)

Differentiating the residual functions of eqns. 38 and 39 gives

J(x) =u + AZKKKDB , — ZKT (42)

r KB \ C Z

IEE PROC, Vol. 128, Pt. C, No. 2, MARCH 1981

In eqn. 42, diagonal matrices A and B are as defined in eqns.25 and 26 and

C = ( v nn (43)

The solution vector'**' at iteration p is now found from theNewton-Raphson formula

5.2 Solution procedure

Removing the transformer node set from eqn. 28 and setting/L(fn) = 0 where there are no further constraints to besatisfied in node-set L gives,

= ZMMiM(tn) + ZMKiK{tn)

+ ZMTiT(tn) + vMP(tn _,)

= ZKMiM{tn) + ZKKiK(tn)

= ZLMiM(tn)+ZLKiK(tn)

(45)

(46)

i) + vLP{tn-j) (47)

At the first time step in solution, tx, vM(tx), VK(tx )> and vL(tx)are found from eqns. 45—47 for /'M(^I) known and initial-condition values iK(to), iT(to), vMP(to), vKP(to), and vLP(to).If no values in vK{tx) exceed surge-diverter spark-over settings,the lower partition of eqn. 44 is solved to give #Y(^i) drawingon the available solution values vM(tx), vK(tx), and vL(tx).Knowing the transformer magnetising currents in ir(tx), trans-former voltages in vT(tx) are then available from the trans-former relationships of eqn. 34. All solution values are thenknown at the first step. The procedure is repeated in successivetime steps until solution values in vK(tn) exceed one or moresurge-diverter spark-over settings. When this occurs, theNewton-Raphson sequence of eqn. 44 is used to solve forthose nodal voltages in vK(tn) that are in excess of diverterthreshold settings, together with transformer magnetisingcurrents in iT(tn). Transformer voltages in vT(tn) are thenfound from eqn. 34. For nodes in set K at time periods forwhich surge diverters connected to them are not conducting,nodal voltages are found from the corresponding partition ofeqn. 46. For the remaining nodes in set K, those for whichsurge diverters connected to them are in a conducting mode,nodal voltages are found from the particular rows representingthem in eqn. 44. The precise combination of equations solved inNewton-Raphson analysis varies during solution as differentcombinations of surge diverters extinguish or begin toconduct. Using inverse tangent functions in representing trans-former magnetisation characteristics, as in Appendix 9.4, theiterative section of the overall solution typically requires nomore than three or four iterations to converge.

5.3 Represen ta five solu tions

For the 400 kV transmission line of Section 4.3 and Figs. 3 -5 ,but now terminated in a 400/275 kV transformer open-circuited on the 275 kV side, voltage waveforms on the 400 kVside of the transformer following simultaneous-phase energis-ation at the end of the line remote from that of transformerconnection, are shown in Fig. 6. Transformer nonlinearitiesinfluence the phase V waveform only marginally for theconditions considered, whereas the greater asymmetry inphases '&' and 'c' progressively leads to saturation and to acorresponding effect on phase '&' and 'c' voltage waveforms.The peak overvoltage for the particular transient condition ofFig. 6, however, remains unaffected by transformer non-

linearities. The phase 'c' in-rush current waveform of thetransformer is shown in Fig. 7. The combined effect oftransformer and surge-diverter characteristics are shown inFig. 8 for which the surge diverter parameters are those ofTable 2 iri Appendix 9 3 , and the parameters of the trans-former model are summarised in Appendix 9.4.

6 Conclusions

The analysis methods of the paper allow surge-diverter charac-teristics at any number of nodes within a power-networksystem to be included in electromagnetic transient analysis.Where required, the nonlinearities of transformer magnet-

80O00r

10.00

time. ms

- 600.001

Fig. 6 Effect of transformer characteristics on line energisationtransients

160 km, 400 kV linewith transformerwithout transformer

0.00-0.15

-0.30

-0.45

-0.60

-075

-0.90

-1.05

-1.20-

-1.35

-1.50

1.00time , ms

2.00 3.00 4.00 5.00 6.00 7.00 8.00 900 10.00

Fig. 7 Transformer current transient

160 km, 400 kV linewith transformer and surge diverters

IEEPROC, Vol. 128, Pt. C, No. 2, MARCH 1981 67

isation characteristics may also be represented in the one non-linear formulation. Nodes to which either surge diverters ortransformer magnetising branches are connected are partitionedfrom the time-domain network equations in reducing to aminimum the order of the nonlinear equation set to be solved.For a very wide range of transient operating conditions, thesolution of this set has been found to require no more thanthree or four iterations. It appears that the formulation andsolution procedure derived will achieve the lowest possiblecomputing time overheads of nonlinear surge diverter andtransformer representation in z-transform electromagnetictransient analysis. In particular, the computing time require-ments in solution as a whole are substantially less than thoseof time-convolution analysis when developed to providesimilar nonlinear representation. The formulations of thepaper further enhance and add to the comprehensiveness ofz-transform methods of electromagnetic transient analysisin power systems.

8 References

800-OOr

40000 -

> 0 00"o

g -40000-

-800 00-

60000

5 40000-

A• VX^AyA6!3 0 A , , '0-0°\800

8 00 " Y V lO'OO

time . ms

1000

•y time . ms

-SOOOCh

Fig. 8 Combined effect of transformer and surge diverter character-istics

160 km, 400 kV linewith transformer and surge diverterswithout transformer and surge diverters

7 Acknowledgments

The authors are grateful to the Australian Research GrantsCommittee for financial support and to the West AustralianRegional Computing Centre for running their programs. Thegenerous support of power systems research in the Departmentof Electrical and Electronic Engineering at the University ofWestern Australia by Professor A.R. Billings, Head of Depart-ment, together with his professional co-operation at all times,is gratefully acknowledged.

1 HUSBAND, D.W., MORSTYN, K., and DILLON, T.S.: 'Influence ofsurge diverters on over-voltage probability distributions', Proc IEE,1974,121,(5), pp. 361-365

2 TAYLOR, E.R., and MERRY, S.M.: 'Switching surge-duty onmodern arresters', IEEE Trans, 1971, PAS-90, pp. 1103-1111

3 CARROL, D.P., FLUGUM, R.W., KALB, J.W., and PETERSON,H.A.: 'A dynamic surge arrester model for use in power systemtransient studies', ibid., 1972, PAS-91, pp. 1057-1064

4 TOMINAGA, S., AZUMI, K., SHIBUYA, Y., IMATAKI, M.,FUJIWARA, Y., and NISHIDA, S.: 'Protective performance ofmetal oxide surge arresters based on dynamic characteristics', ibid.,1979, PAS-98, pp. 1860-1869

5 CORNICK, K.J., and LUCAS, J.R.: 'Inclusion of surge diverters insystem overvoltage calculations', Proc. IEE, 1979, 126, (12),pp. 1263-1268

6 SCHEI, A., and JOHANSSON, A.: 'Temporary over-voltage andprotective requirements for EHV and UHV arresters', CIGRE,Report 33-04, 1972, pp. 1-14

7 HUMPAGE, W.D., WONG, K.P., NGUYEN, T.T., and SUTANTO,D.: 'Z-transform electromagnetic transient analysis in powersystems', IEE Proc. C, Gen. Trans. & Distrib., 1980, 127, (6),pp. 370-378

8 HUMPAGE, W.D., WONG, K.P., and NGUYEN, T.T.: 'Developmentof z-transform electromagnetic transient analysis methods for multi-node power networks', ibid, 1980, 127, (6), pp. 379-385

9 BATTISSON, M.J., DAY, S.J., MULLINEUX, N., PARTON, K.C.,>and REED, J.R.: 'Calculations of switching phenomena in powersystems', ibid, 1967,114,(4), pp. 478-486

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

11 BOWMAN, W.I., and McNAMEE, J.M.: 'Development of equivalentmatrix circuits for long untransposed transmission lines', IEEETrans, 1964, PAS-83, pp. 625-632

12 WEDEPOHL, L.M.: 'Application of matrix methods to the solutionof travelling-wave phenomena in polyphase systems', Proc. IEE1965, 110,(12), pp. 2200-2212

9 Appendix

9.1 Source model

In the simplified equivalent-source representation of Fig. 9,phase impedances are calculated from a specified fault level atthe terminals of the model, and the neutral impedance iscalculated from a specified ratio of source impedances in thepositive- and zero- phase sequences. For the purposes of thestudies of Sections 4 3 and 5.3, the fault level at the sourceend of the transmission line has been set to 10,000MVA,whereas the ratio of source impedances in the positive- andzero-phase-sequences is one in all cases.

Zso

Zsb

ITT

Fig. 9 Simplified equivalent source model

68 IEE PROC., Vol. 128, Pt. C, No. 2, MARCH 1981

9.2 Basic transmission line data

In Fig. 10 is shown the tower configuration and conductor,spacing of the 400 kV transmission line on which the testanalyses of Sections 4 3 and 5.3 have been based. Other basicline data are collected together in Table 1.

Table 2: Principal parameters of active arc-gap surge diverter

4 1 . 5 3

Fig. 10 400 k V transmission line conductor spacings (m)

Table 1 : Basic transmission line data

Number of circuits 1Number of conductors per phase 4Number of earth wires 1Conductor position symmetry noneConductor resistivity, fim 3.2 X 10"8

Earth-wire resistivity,.flm 2.69 X 10"Earth resistivity, fim 20.0Conductor strand diameter, cm 0.32Earth-wire strand diameter, cm 0.32Geometric mean diameter for 4-conductor bundle, cm 3054Outer diameter of earth wire, cm 2.86Number of effective strands in phase conductors 54Number of effective strands in earth wire 54

9.3 Surge diverter parameters

For the purposes of the test studies of Section 4.3, the surge-diverter gap-voltage profile of Fig. 2 has been represented inthree- linear segments, and the dependence of gap voltage ondiverter current has been discounted. Given this, the principalparameters of the surge-diverter model included in analysis arethose of Table 2.

Threshold setting of arc gapInitial gap voltageGap voltage when arc is establishedReseal settingTime to initiate arc formationTime to establish arc

1.4pu0.1 pu0.8 pu0.8 pu0.5 ms0.5 ms

9.4 Transformer magnetisation characteristic

For transformer magnetising branches connected betweennodes in set T and earth, transformer flux linkages in vectori//T are related to magnetising currents in vector iT by

M 0 = ST{h{t)}

Nodal voltages for node set Tare then given by

(48)

«*(0 = ^ [grihit)}] (49)

Using the trapezoidal rule with step length At in eqn. 49:

(50)

From eqn. 50:

/{MAi)} = vr(tn)+fTP

where

_ 2

and

(51)

(52)

fTP = vT(tn - At) +f{iT(tn ~ A0} (53)

Individual transformer-magnetisation characteristics can oftenclosely be represented by

f(im) = oc(im + io) + P tan"1 (a(/m + i0)} (54)

in which im is the transformer magnetising current and i0 is ameasure of pretransient remanence. The parameters a, (3, and aare easily found for any given magnetisation characteristic.When the nonlinearity is represented in this form

g g M - a * . . * . . . , (55)Elements of the diagnonal matrix C in eqn. 43 of Section 5.1are then formed from eqn. 55.

The magnetisation characteristic of the 400/275 kV,500 MVA transformer included in the test analyses of Sections4.3 and 5.3 is defined by eqn. 54. The following coefficientvalues have been used in the studies leading to Figs. 6,7, and8:a = 0,/0 =O,0 = 9OO,a= 1.67.

JEEPROC, Vol. 128, Pt. C, No. 2, MARCH 1981 69