Saturated reactor compensator modelElectromagnetic transient formulation for main- andauxiliary-core winding systems

Prof. W. Derek Humpage, BSc, PhDT.T. Nguyen, BE, PhD

Indexing terms: Power systems and plant, Transients, Power transmission and distribution

Abstract: The paper develops a particularly directform of model for reactive-power compensatorsthat are based on saturated reactors. It is one inwhich each winding section of each limb of themain- and auxiliary-core structures is representedindividually together with its core nonlinearmagnetic-circuit characteristics. Depending on theprecise internal configuration, there may be 30-50individual winding sections. The complete modelis of a transient form, and it is shown in the paperhow it may be combined with a powerful z-planeelectromagnetic transient formulation for networksystems in which a saturated reactor compensatoris one element. As it is in general form, the com-pensator model makes provision for representingthe particular main- and auxiliary-core systems ofany given design such as those of the twin triplerand triple tripler compensators. Particular casesare selected by data items. Both the model itselfand its programming implementation are com-pletely general. The validity of the model devel-oped is examined on the basis of comparisonsbetween computer-simulation solutions and site-test recordings of the main compensator variablesfollowing the energisation of a compensator-terminated line section. In one particular aspect ofthe correlation, computer simulation reproducesclosely the subharmonic oscillations arising fromthe slope-correcting capacitors and the complexsaturating magnetic system of a twin triplercompensator.

List of principal symbols

Model for one limb of compensatorvbj> hj = vectors of winding voltages and currents,

respectivelyLbj = matrix of winding inductancesvnj> hj = vectors of nodal voltages and currents,

respectivelyC: = connection matrix

fnj{n — 1) = vector of previous valuesBnj = nodal coefficient matrix in the time domain

Paper 5258C (P7, P9), first received 29th May and in revised form 4thNovember 1986The authors are with the Department of Electrical and ElectronicEngineering, The University of Western Australia, Crawley, WesternAustralia 6009

k, I = row and column indices for Bnj

h = integration step lengthg =h/2m = total number of windingsn, = total number of distinct nodes

Network modelvNin\ ' N M = vectors of nodal voltages and currents,

respectively, at step nYN = nodal admittance matrixfiM — i) = vector of previous values

Compensator main winding system

vc(ri), ic(n) = vectors of nodal voltages and currents,respectively, at step n

Yc = nodal admittance matrixfc[n — 1) = vector of previous values/ = total number of limbs in main winding

system

Main winding system and network model together

vs(n), is(n) = vectors of nodal voltages and currents atstep n

Ys = nodal admittance matrixY's = intermediate matrix used in forming Ys

fs(n ~~ j) = vector of previous values

Auxiliary winding system

vA(n), iA{n) = vectors of nodal voltages and currents atstep n

YA = nodal admittance matrixfA(n — 1) = vector of previous values

Complete compensator and network model togetherv(n), i(n) = vectors of nodal voltages and currents at

step n= nodal admittance matrix= vector of previous values= F " 1

= -Zf(n-j)

YAn-j)ZF(n -j)

Nonlinearitiesv\A.n) = vector of voltages across nonl inear branch

elements'M( W ) = vector of currents th rough nonl inear branch

elements*PM = vector of flux linkages

Overall solution

RMU\iiny] = vector of residual functionsJMUIA11)]

= Jacobian matr ix

IEE PROCEEDINGS, Vol. 134, Pi. C, No. 3, MAY 1987 245

= iteration count 2 Degree of representation in transient analysis

Subscripts and superscripts

Subscripts Q and R denote sets of node pairs betweenwhich nonlinear magnetising branch elements are con-nected. Superscript t identifies a matrix or vector trans-pose.

1 Introduction

The representation in computer analysis of reactive-power compensators based on saturated reactors [1, 2]has been developed progressively and reported in pre-vious papers [3-6], and models have been derived for usein evaluations of steady-state, dynamic, and electromag-netic transient operating modes.

This paper is devoted to modelling methods in electro-magnetic transient analysis. In particular, the form ofmodelling is one in which the complete system of mainand auxiliary windings on separate multilimb cores isrepresented directly. A formulation is first developed forthe winding configuration of one limb. This lends itself todevelopment in general form for any specified number ofwindings or winding sections. Once developed, it isapplicable to both the main windings and to the auxiliarywindings. From the representation for an individual limb,a complete model for the main core is built up takinginto account the interconnections between windingsthroughout the main winding system. Exactly the sameprocedure is followed for the auxiliary core. The two arethen combined to give a complete compensator model.

A principal feature of the overall model is that of itsdirect relationships to the actual winding system. A prin-cipal feature of its derivation is that of its modular formin which the basic module in analysis is that of a singlelimb having an arbitrary number of winding sections.Interconnections between windings can be expressed con-veniently in connection-matrix form. Depending on theprecise magnetic-circuit structure, up to 50 winding sec-tions may be represented individually in a single com-pensator model. There are interconnections betweensections on the same limb and also between winding sec-tions on different limbs. Provision is made for represent-ing interconnections in general form so that the coreconfiguration and winding interconnections of any givencompensator design, such as those of the twin tripler andtriple tripler compensators, may be specified in data formin analysis. The model developed is an entirely generalone and is applicable to any particular compensatordesign based on saturated reactors.

Some representative transient solutions from themodel are given for compensator energisation. An exami-nation is made of the validity of the model on the basis ofcorrelations between available test results and the resultsof computer simulation.

A very recent paper [7] has developed a model inwhich all sections of the main winding system are rep-resented individually, and it has reported studies ofswitching transients based on the model. The indepen-dent work of this paper provides an electromagnetic tran-sient formulation for the main and auxiliary windingsystems together. It shows how this can be incorporatedin a complete network system model of which a saturatedreactor compensator is one element, and it assesses thevalidity of the formulation derived in terms of recordingsfrom extensive field tests. No previous paper has relatedto these advances in modelling and computer analysismethods.

In matching the degree of compensator representation tothe requirements of particular studies and evaluations asthey arise, it seems permissible to recognise different cate-gories of electromagnetic transient analysis.

In one of these categories, interest focuses on thepropagation of surges in the windings of the compensatorand its relationship to the surge stressing of winding insu-lation. For studies of wave propagation in windingsinternal to the compensator, a basic requirement is thatof representing the distributed form of the inductance,capacitance, and resistance of individual windings. Asrelated to wave transit times in windings, transientperiods are typically 10-20 us in duration. The incidentsurge waveshape for which wave propagation is to beinvestigated is specified. Surge propagation evaluationsfocus on the compensator rather than on the network towhich the compensator is connected.

The almost converse category is one in which theprincipal emphasis is on transient wave propagationthroughout a network system of which one or more com-pensators are part. There is then corresponding emphasison network modelling and especially on transmission-linerepresentations in distributed-parameter form. Where thewave transit times of compensators are short in compari-son with those of other network branches, the prioritiesof analysis are usually met best by representing com-pensators by lumped circuit elements rather than in dis-tributed form. This coincides with the generally acceptedpractice of representing transformers in lumped formwhen representations of them are included in networktransient analysis.

Clear distinction can usually be drawn along theselines between analysis requirements which lead tolumped forms of compensator model, on the one hand,and those in which inductance, capacitance, and resist-ance distributions in windings are represented on theother.

Of these two principal categories of transient evalu-ation, it is that of the analysis of networks which includessaturated reactor compensators as network branches towhich the developments of the paper most directly relate.Compensator-winding representations are therefore inlumped form. With that basis, the paper seeks to developa rigorously based model in which the complex windingsystem of a saturated reactor compensator is representeddirectly. A model of this detail represents a considerablecomputing overhead in analysis and evaluation. To lowerthis, some form of reduced model could be sought. Oneof the immediate applications of the model of this paperis that of providing a reference for quantifying the degreeof representation that less detailed models may be able toachieve.

In summary, the investigations of this paper have beenundertaken to fulfil the following needs:

(a) to provide a detailed model for saturated reactorcompensators in a form that can be included in numeri-cal electromagnetic transient analysis of integrated net-works which include this form of compensator

(b) to provide an initial reference model from whichreduced representations may be derived in lowering com-puting overheads in analysis

(c) to provide a model which, from transient analysisin the time domain and subsequent transformation to thefrequency domain using the Fourier transform, allows thecharacteristics of compensators to be investigated over awide range of frequencies.

246 IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

3 Compensator winding system

Modelling in this paper takes as its starting point thedevelopment of a winding-system primitive. From Fig. 1,this may be formed most directly for the winding config-uration of one limb. For the primitive to be applicable to

limb

From eqn. 3,

winding 1on limb j

winding 2on limb j

v3nj

(4)

Fig. 1 Main- and auxiliary-core system

any limb of either the main core or the auxiliary core, thegeneral case is that of a system of any specified number ofindividual windings.

In the diagrammatic interpretation of Fig. 1, the main-core system of a saturated reactor compensator has, alto-gether, '/' limbs. The auxiliary core has V limbs. Eachlimb of either the main core or the auxiliary core has anindependently specified number of windings. For the pur-poses of the winding layout of Fig. 1, the intercon-nections between windings are not shown. A procedureby which interconnections between windings on the samelimb of the main core, and also on different limbs, may berepresented in analysis is developed in Section 4.3.Exactly the same procedure applies to the auxiliary core.It seems helpful in the step-by-step development of themodel to deal first with the main core and then to turn tothe auxiliary core.

4 Model development

4.1 Branch form of representation for one limbOne limb of the main-core structure is shown in Fig. 2.This limb is identified by '/. There are 'm' separate wind-ings on the limb. Currents in the windings are denoted by(ihj)1, (ibJ)

2, ••-, (ibj)m- Voltages across the windings are

denoted by (vbjY, (vbJ)2, . . . , (vbJ)

m. Vectors ibj and vbj aredefined in terms of these variables using

If winding resistances and capacitances are discounted,and Lbj is the matrix of the self inductances of windingsand the mutual inductances between them, and p is thedifferential operator in the time domain, then

vbj = Lbjpibj

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

(3)

windingon limb

m

j

m 1'bj f .

1 I2m

V b ii

2m-1

' nj

i 2m-1nj

2m-1

1 -I vO nj

2m

Fig. 2 Winding configuration for one limb of main core

4.2 Interconnections between windingsAccording to the interconnections between the windingson limb;, winding voltage variables (vbJ)

1, (vbj)2,..., (vbj)

m

are related to the nodal variables identified in Fig. 2. Forl imb; , the nodal voltages are (vnj)

1, (vnj)2, ..., (vnj)

2m.Vector vnJ is now defined in terms of these so that

[vnJy = (vnj)\ (vnj)2,..., (vnJ)

2m (5)

The nodal currents of Fig. 2 are similarly grouped intovector inj where

\2m(6)

Interconnections between windings are now representedin connection-matrix form. The connection matrix isdenoted by Cj. Rows of Cj relate to winding voltages(vbj)

1, (vbJ)2, . . . , (vbJ)

m. Cj therefore has 'w' rows. Thecolumns of Cj are defined from the number identificationof winding terminals as in Fig. 2. These are the nodes of a

247

nodal-voltage analysis formulation. Some of the nodes ofthe primitive of Fig. 2 are interconnected in an actualcompensator configuration. As a result, pairs of nodeswith direct connections between them merge to formsingle nodes. If, for any specified winding connections,there are n, distinct nodes, connection matrix C, has n,columns. The vector of nodal voltages vnj together withthe vector of nodal currents inj each have n, elements.

Each row of Cj has in it two nonzero elements. One ofthese has the value 1, and the other has the value —1.These entries in Cj define each winding voltage in turnfrom the difference between the voltages at the nodes towhich the winding is connected. If, for example, there arethree windings on the limb of Fig. 1 so that m = 3, thenode identification following that of Fig. 2 will be 1, 2,. . . , 6. If nodes 4 and 5 are directly connected there willbe five distinct nodes. When merged to form a singlenode, nodes 4 and 5 can conveniently be identified by '4'.It is also helpful to renumber the last node so that it isidentified by '5'. With the connection between terminalsof the second and third windings there are five distinctnodes, with the number identification 1, 2, . . . , 5. Theconnection matrix Cj is in this case given by

1 2 3 4 5

1 - 11 - 1

1 - 1(7)

With an assembly of connection matrix Cj along theselines, winding voltages in vector vbj are related to nodalvoltages in vector vnj by

*>bj = Cj VnJ (8)

Nodal currents in vector inj are related to winding cur-rents in vector ibj by

inj = LCjJibj (9)

On using the relationships of eqns. 8 and 9 in eqn. 4,

pinj = lCJ]t[Lbj]-iCjvnj (10)

or

(11)

where

jLbj}-lCj (12)

Eqn. 11 gives the nodal form of the differential equationsfor one limb of a complete multilimb core configuration.

4.3 Form for numerical solutionIn the numerical integration sequence based on the trape-zoidal rule, integrable variables at the nth step in solutionin vector Jt(n) are found from

x(n) = x(n - 1) + g\j)x(n) + px(n - 1)] (13)

In eqn. 13, g = h/2 where h is the integration step length.Substituting from eqn. 11 into this general form gives

'„/«) = ' „ > - 1) + 9lLnJ]-l\ynj{n) + vnj{n - 1)] (14)

Eqn. 14 now rearranges to the form

inj(n) = Bnjvnj{n)+fnj{n-l) (15)

for

Bnj = g\_LnjYl (16)

fnj{n - 1) = inj(n - 1) + g[Lnj] " \ / n - 1) (17)

248

Eqn. 15 is now in a form that can be combined with atime-domain equation system for the network to whichthe compensator is connected. Still only one limb of thecompensator is considered. Rows and columns of the Bnj

matrix can conveniently be identified by indices k and /,respectively, so that elements of the matrix are denotedby BnJ{k, I) for k = 1,2,..., n} and / = 1, 2, . . . , nJt whererij is the number of distinct nodes in the representation oflimb'/.

Having established the steps for one limb, these applywithout change for the remaining limbs of the completecompensator.

4.4 Network modelPrevious work [8, 9] has established a comprehensivenetwork model for use in numerical electromagnetic tran-sient analysis leading to the nodal form

M) +fN(n ~j) (18)

In eqn. 18, i^n) and v^ri) are the vectors of nodal cur-rents and nodal voltages, respectively, at the nth sam-pling interval in the time domain. YN is the nodalcoefficient matrix, and ff^n — j) is a vector of previousvalues. The structure and derivation of YN are developedin the earlier work [9].

4.5 Combining compensator and network modelsWith a single compensator connected to one phase-nodeset in the network model, eqn. 18 extends to

U w J = L yd ») M»- )JIn eqn. 19, ic(n) and v^ri) are vectors of compensatorcurrent and voltage nodal variables at sampling intervaln. Yc is the nodal coefficient matrix of the compensatormodel, and fdn — 1) is a vector of variables at the sam-pling interval immediately prior to the current interval.Connecting the compensator to the network alters thediagonal submatrix of YN for the network nodes to whichthe compensator is connected. To the submatrix of YN forthese nodes is added the partition of Yc for the com-pensator branches connected to them. Following thisaddition, the network nodal matrix is denoted by YNC.The previous-value vector in the network equations nowalso has contributions from the compensator model forthe nodes to which the compensator is connected. Withthese contributions, the previous-value vector is denoted

With YNC formed, the compound matrix of eqn. 19 isassembled by considering the compensator susceptancematrix for one limb at a time. Distinct nodes throughoutthe main winding system are identified by number code.Compensator terminal nodes that are connected to phasenodes in the network have the same identification as thatinitially assigned in the network model. The internalcompensator nodes remaining have separate and uniqueidentification.

If the compound matrix of eqn. 19 is denoted by Ys

then

n = (20)

In making provision for the assembly of Ys from thestarting point of YNC being known, the intermediate

1EE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

matrix Y$ is formed where

Ys• • • [ ' • . ] (21)

If elements of Ys and Y's are denoted by Y^i, j) andY'siU j), respectively, elements corresponding to one limbof the compensator are included using

Y^i, j) = Y^(i, j) + Bnj{k, /) (22)

Indices i and j correspond to the number identification ofthe distinct nodes internal to the main winding system.

The procedure of eqn. 22 is repeated for successivelimbs of the compensator until the complete matrix isassembled. In the present model, possible couplingbetween windings on separate limbs is discounted. Wherethis is thought to be required and the relevant coefficientsof coupling are available, interlimb coupling is easily rep-resented at this stage of model assembly. Only the low-order dimensions of the Bnj matrix arise at each step.Using a sparsity methodology, nonzero elements only ofYs are stored in a compacted-list form. Main- andauxiliary-core winding resistances may be included inmodel derivation by adding a term to the winding resist-ance matrix Rbj and the winding current vector ibj in eqn.3 of Section 4.1. This is then present in the subsequentsteps of the development. However, substitution into thetrapezoidal rule would then have to be made prior to theconnection-matrix derivation of Section 4.2. Alternative-ly, given the likely contribution of winding resistances,they may be included as a separate set of equivalentseries-connected branches external to the winding system.Test-data relationships between compensator activepower losses and phase currents allow resistance valuesto be found for use in this equivalent form of representa-tion. On completion of the assembly of the Ys matrix andthe related steps in forming previous-value vectors, theequation system for the network and compensatortogether is

is(n) = Ysvs(n) +fs(n -j)

where

(23)

(24)

(25)

(26)

4.6 Compensator auxiliary windingsFor each limb of the auxiliary winding system, there is avector equation of the form of eqn. 15. Distinct nodes inthe auxiliary winding representation are identified bynumber code. Points of connection between auxiliarywindings and main windings share a common identifica-tion. Otherwise, all node identification in the networkmodel, in the main winding representation, and in theauxiliary winding representation, is unique.

The auxiliary winding system extends eqn. 23 to

«» »» L ( 2 7 )

In eqn. 27, iA(n) and vA(n) are the vectors of current andvoltage variables, respectively, at the nth sampling inter-val in the time domain, and fA(n — 1) is a vector ofprevious-step values. YA is formed from the susceptancematrices of the form of eqn. 15 but is now formed forindividual limbs of the auxiliary windings. The partition

of YA for the nodes of the auxiliary windings that areconnected to nodes of the main winding system adds tothe diagonal block of Ys for these nodes to give the YSA

matrix of eqn. 27. There are contributions also to theprevious-value vector f^n —j). The vector with thecontribution from the auxiliary windings is denoted byfsAin-j).

Exactly the same procedure as that of Section 4.5 isnow used to assemble the compound matrix of eqn. 27using the coefficient matrices of auxiliary winding limbstogether with the previous-value vector. Using, for eqn.27,

WAY = Psto]1, UA(AY

WAY = MAY, MAY[/(«-7)]' = [(/s>-;)]

Y=di*g{YSA,YA}

(28)

(29)

(30)

(31)

The final equation system for solution is then

i(n)=Yv(n)+f(n-j) (32)

or

v{n) = Zi(n) + F(n - j) (33)

for

Z=Y! (34)

and

F(n-j)=-Zf(n-j) (35)

5 Nonlinearities

5.1 System partitioningTo the linear model of eqn. 33 are now added the nonlin-earities of the magnetising characteristics of each of themain and auxiliary winding cores. Very high reactorwinding current levels may possibly lead to the causalsaturation of the yoke but that is not represented here.The magnetisation characteristic of each individual limbthat is central to compensator operation is representeddirectly. For this purpose, one nonlinear branch is nowintroduced between a selected node pair for each mainwinding limb and for each auxiliary winding limb. Thesepairs of nodes are now defined in the complete systemmodel of eqn. 33. If one of each pair is in the set Q andthe second one of each is in the set R, voltages at thesenodes are available from partitions of eqn. 33 in the form

'LQ

Zc

Z*

'LR

'QR

'iL{n)

'Q(H)

MA-+

'FL(n-jYFQ{n-j)

-FR(n-j).

(36)

In eqn. 36, subscript L denotes the set of nodes in thecomplete system remaining after partitioning out thosenodes involved in the representation of the nonlinearcharacteristics of the compensator.

From eqn. 36,

vQ(n) = ZQLiL(ri) + ZQQiQ(n) + ZQRiR(n) + FQ{n -j) (37)

vR(n) = ZRLiL{n) + ZRQiQ(n) + ZRRiR{n) + FR(n -j) (38)

Collecting terms other than those in iQ(n) and iR{n),

vQ(n) = ZQQ iQ{n)

vR(n) = ZRQ iQ(n)

ZQR iR(n)

ZRR iR(n)

VQ0(n - j)

VRO(n - j)

(39)

(40)

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987 249

where

" ~j) = ZQLiL{n) + FQ(n -j)

" -j) = ZRLiL{n) + FR(n - j)

(41)

(42)

A diagrammatic interpretation of the inclusion of nonlin-earities in the compensator model is shown in Fig. 3.

(n)

linear partof

completesystem

node setQ

( n )

node setR

Fig. 3 Representation of compensator nonlinearities

Nonlinear magnetisation characteristics are specifiedin terms of the currents and flux linkages in the nonlinearbranches of Fig. 3. Nonlinear branch currents aregrouped into vector iM. The corresponding voltages areformed in vector vM where

vM = dVJdt (43)

Elements of vector iM are related to corresponding ele-ments of *FM by specified magnetisation characteristics.

It is now required to replace iQ(n) and iR(n) in eqns. 41and 42 by i^n) using

iM(n) = iM (44)

iM{n) = -iQ(n) (45)

On substituting from eqns. 44 and 45 into eqns. 39 and40,

vQ(n) = - ZQQ iM(n) + ZQR iM{n)

vR(n) = - ZRQ ijn) + ZRR iM(n)

From Fig. 3

vjn) = vQ{n) - vR{n)

vQ0(n - j)

vR0(n - j)

(46)

(47)

(48)

Using the expressions for vQ(n) and vR(n) in eqns. 46 and47 in eqn. 48 gives

jfi) = ZMiM(n) + vMO(n -j)

where

and

— ZR

(49)

(50)

(51)

5.2 Nonlinear characteristicsThe nonlinearities to be represented are in the vectorform:

4* = / • ( * * ) (52)

From eqn. 43 and using the trapezoidal rule for numeri-cal integration,

« - 1)]

From eqn. 53 and using a for l/g,

for

fjn - 1) = -aVJn - 1) + vjn - 1)

(53)

(54)

(55)

Equating the expressions for vM(n) in eqns. 49 and 54gives

0 = aVM - ZM iM +fM(n-l)- vMO(n - ;) (56)

As in previous work [9], the nonlinear magnetisationcharacteristic is represented in the form

VMj = <xj arctan {j3/im,- + iOj)} + y/imj + iOj)

f o r ; = l , 2, . . . , ( / + c) (57)

In this representation, a,-, ^, and y} are to be found fromthe nonlinear characteristic in graphical form, and iOj

makes provision for representing remanence.Eqn. 56 is now solved by the Newton Raphson algo-

rithm to give ij^n). Using p for the iteration count, theiteration sequence is

UMY = VMY'1 - {JMUMY'T'RMUMY-1

(58)

where the residual function /?Af[/M(n)] is from eqn. 56:

The Jacobian matrix /M[iM(n)] is formed from

(59)

(60)

Differentiating the residual function of eqn. 59 to give theJacobian matrix of eqn. 60 leads to

for; = 1,2, . . . , ( / + c) (61)

6 Overall solution

The equation system for the network and compensatortogether is that of

v(n) = Zi(n) + F(n - j) (62)

An initial steady-state solution derived from the earlierformulation [6] provides starting values and initial pre-vious values for the solution of eqn. 62 for given networksource infeeds as specified in elements of i(n). A solutionfor network nodal voltages then follows. Included inthese are voltages in the compensator main and auxiliarywinding systems in vectors vR(n) and vQ{n). Using thesevectors, the residual function vector /?M[*M(n)] ls formedtogether with the Jacobian matrix JM[i\t(n)\- The coremagnetising current components in vector ifjji) are thenfound from the iteration sequence

(63)

250 IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

Solution values in iM(ri) are then used in vector i(n) finallyto form the nodal voltages of the formulation in vectorv(ri) using eqn. 62 and now with the nonlinearities of themagnetising circuits taken into account.

Usually, no more than four or six iterations arerequired in the solution of eqn. 63.

7 Model validation

A system interconnection including a saturated reactorcompensator and for which comprehensive test record-ings are available, is shown in outline in Fig. 4. The satu-

A B132kV 220kV

C D22OkV 132kV

CB 326km

•29-5kV

fixed- valuereactor

Fig. 4 Test configuration

A delta connectionLLJ unearthed star connection

slope -correcting "T

circuit

saturated reactor

Cc Z ±

Fig. 5 Saturated reactor with slope-correcting circuit

Cs slope-correcting capacitorCB bypass capacitorLB bypass inductorRB bypass damping resistorA delta connection

rated reactor compensator together with its harmonicfilters terminate a 326 km section of 220 kV line. Thecompensator slope-correcting circuit is that of Fig. 5.Among the test conditions for which recordings are avail-able are those of energising the compensator-terminatedline by closing circuit breaker CB. In the tests, the fault

level at A in Fig. 4 was 1000 MVA. System data is sum-marised in Section 11. Transformers in Fig. 4, includingtheir magnetisation characteristics, are represented as inearlier work [10].

Comparisons are made in Figs. 6-8 between the com-pensator main winding current transients as recorded inthe tests and those from computer simulation based onthe model developments of this paper.

For the same switching conditions as those of Figs.6-8, voltage transients across the compensator slope-correcting capacitors are shown in Figs. 9-11. The sub-harmonic modulation at a frequency of about 12 Hz isreproduced in the computer simulation. The sub-harmonic oscillatory mode here is well damped. There isno tendency towards a sustained subharmonic resonancecondition.

From among the extensive recordings of the site tests,the final recordings chosen for correlation purposes arethose of the current transients in the bypass resistor cir-cuits of Fig. 5. These are shown in Figs. 12-14. The sub-harmonic frequency of about 38 Hz is closely reproducedin the results of simulation together with the overall formof the transient. The oscillatory mode at 38 Hz is a stableone.

All solution variables of the complete system modelcorrelate closely with test results and similarly to the cor-relation of Figs. 6-14.

8 Conclusions

The developments of the paper seek to establish adetailed model for reactive-power compensators of thesaturated reactor kind. Each winding section throughoutthe main and auxiliary winding systems is representedindividually together with its nonlinear magnetising char-acteristics.

In modelling terms, the basic unit of the analysis deri-vation is that of a single limb and its windings. A tran-sient analysis formulation is derived in the time domainfor this basic compensator element. From this, a com-plete representation for the main-core system isassembled. A model for the auxiliary winding systemfollows from an identical assembly procedure. Sections4.5 and 4.6 of the paper show how the compensatormodel developed can be combined with a comprehensivez-plane electromagnetic transient analysis formulationwhich now appears to be coming into very wide practicalapplication. Comparisons between site-test recordingsfrom a particular transmission interconnection, whichincludes saturated reactor compensators, and solutionsfrom computer simulation confirm the validity of themodelling methods developed and their programmingimplementation. They provide the most complete formu-lation so far developed. The formulation is of a generalkind and is applicable to any particular core configura-tion as it arises. Particular designs are selected by dataitems only.

Based on the detailed model of the present paper,companion studies are being carried out to define a seriesof reduced representations. These include a model withlower computing overheads than the present model foruse in studies of switching overvoltages and related elec-tromagnetic transient evaluations; a model for use in thestudy of dynamic operating conditions related to possiblemodes of instability; a model for use in harmonic propa-gation studies [11]; and a model for use in studies ofsteady-state unbalanced operating conditions.

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987 251

9 Acknowledgments

The authors wish to thank the State Energy Commissionof Western Australia for permission to include in thepaper test recordings from measurements on the Com-mission's network. Dr. Nguyen also gratefully acknow-

ledges the award of a Research Fellowship from theCommission. The authors are grateful to the West Aus-tralian Regional Computing Centre for their co-operation in the development and running of theirprogram implementing the analysis formulation of thepaper.

5000n

-2500J

time.ms

Fig. 6 Main winding current transient in phase 'a'

Test configuration of Fig. 4; line section energisation at B in Fig. 4from test recordingsfrom simulation

2500-

oQ.

-2500-

Fig. 7 Main winding current transient in phase 'b'

Key as for Fig. 6

2500-

oCL

0 - -

-2500-

Fig. 8 Main winding current transient in phase 'c'

Key as for Fig. 6

252 1EE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

200

Fig. 9 Voltage transient across slope-correcting capacitor in 'a-b' phase pair

Test configuration of Fig. 4; slope-correcting circuit of Fig. 5from test recordingsfrom simulation

Fig. 10 Voltage transient across slope-correcting capacitor in 'b-c' phase pair

Key as for Fig. 9

10 References

1 THANAWALA, H.L., KELHAM, W.O., and WILLIAMS, W.P.:The application of static shunt reactive compensators in conjunc-tion with line series capacitors to increase the transmission capabil-ities of long a.c. lines'. CIGRE, Report 31-09, 1976

2 FRIEDLANDER, E.B.: Transient reactance effects in static shuntreactive compensators for long a.c. lines', IEEE Trans., 1976,PAS-95, (5), pp. 1669-1680

3 THANAWALA, H.L., KELHAM, W.O., and CRAWSHAW, A.N.:'Static compensators using thyristor control with saturated or low-reactance reactors', GEC J. Sci. & Technoi, 1982, 48, (3), pp.163-169

4 BARTHOLD, L.O., BECKER, H., CLERICI, A., DALZELL, J.,MORAN, R.J., NORMAN, H.B., PEIXOTO, C.A.O., REICHERT,K., ROY, J.C., THOREN, B., and WILLIAMS, W.P.: 'Modelling ofstatic shunt VAR systems for systems analysis', Electra, 1977, 51, pp.45-74

5 COOPER, C.B., and YACAMINI, R.: 'Choice of analytical andmodelling methods for reactive compensation equipment', IEE Proc.C, Gen., Trans. & Distrib., 1981, 128, (6), pp. 402-406

6 HUMPAGE, W.D., WONG, K.P., NGUYEN, T.T., andMcLOUGHLIN, J.M.: 'Phase-variable modelling of saturatedreactor compensators with particular reference to long-distancetransmission', ibid., 1985,132, (5), pp. 237-247

7 DAVIES, A.E., SERENO, J.J.U., and GERMAN, D.M.: 'Modellingof saturated reactor compensator for system studies', ibid., 1985, 132,(6), pp. 307-311

8 HUMPAGE, W.D., WONG, K.P., NGUYEN, T.T., andSUTANTO, D.: 'Z-transform electromagnetic transient analysis inpower systems', ibid., 1980,127, (6), pp. 370-378

9 HUMPAGE, W.D., WONG, K.P., and NGUYEN, T.T.: 'Develop-ment of z-transform electromagnetic transient analysis methods formultinode power networks', ibid., 127, (6), pp. 379-385

10 HUMPAGE, W.D., WONG, K.P., and NGUYEN, T.T.: 'Surgediverter and transformer nonlinearities in z-transform electromag-

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987 253

200

Fig. 11 Voltage transient across slope-correcting capacitor in 'c-a' phase pair

Key as for Fig. 9

500-

oQ.

>--500-

200

time, ms

Fig. 12 Current transient in bypass resistor in 'a-b' phase pair

Test configuration of Fig. 4; bypass circuit of Fig. 5from test recordingsfrom simulation

500-

200

Fig. 13 Current transient in bypass resistor in 'b-c phase pair

Key as for Fig. 12

254 IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

1000-1

200

time, ms

Fig. 14 Current transient in bypass resistor in 'c-a' phase pair

Key as for Fig. 12

netic transient analysis in power systems', ibid., 1981, 128, (2), pp.63-69

11 HUMPAGE, W.D., WONG, K.P., NGUYEN, T.T., and DeLAETER, M.G.: 'Harmonic propagation in multi-voltage levelpower networks', J. Electr. & Electron. Eng. Aus., 1985, 5, (4), pp.278-285

11 Appendixes

11.1 Transmission-line dataThe basic data for the 220 kV transmission-line sectionshown in Fig. 4 is given in Table 1.

The transmission-line tower configuration is that ofFig. 15.

0.609

688

Table 1 : Transmission-line data

0.38

o ' o

-4.911

5 450

22.0

Fig. 15 Transmission-line tower configurationSpacing in metres

Number of conductors per phase 2Number of earth conductors 1Phase conductor resistivity 3.457 x 10~8 QmEarth conductor resistivity 1.887 * 10"7 QmGMD for 2-conductor phase bundle 12.63 cmOverall diameter of earth conductor 0.825 cmPhase conductor strand diameter 0.3 cmEarth conductor strand diameter 0.275 cmRelative permeability of earth conductor 31Number of strands in phase conductors 30Number of outer strands 18Number of strands in earth conductor 7Number of outer strands 6Earth resistivity 690 Qm

11.2 Transformer dataThe two 3-winding transformers in Fig. 4 are identical.Each has winding voltages of 220, 132 and 29.5 kV. Star-circuit equivalent reactances in per unit on a base of

100 MVA are

XH = 0.0529

XL= -0.0004

XT = 0.0921

H denotes the 220 kV windings, L denotes the 132 kVwindings, and T denotes the 29.5 kV windings.

11.3 Compensator dataThe reactor knee-point voltage is 30.8 kV, and the slopereactance in the positive phase sequence is 0.229 p.u. on abase of 100 MVA. With reference to Fig. 5, slope-correcting and bypass circuit parameters are given inTable 2.

Table 2: Slope-correcting and bypass circuit parameters

Element Value

s

CB

490 /vF584 /JF17.4 mH0.99 Q

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987 255