Sudden 3-phase short-circuit characteristics of turbinegenerators from design data using electromagnetic

field calculationsA.Y. Hannalla, M.Sc. Ph.D., and D.C. MacDonald, B.Sc. (Eng.), Ph.D.

Indexing terms: Turbogenerators, Electromagnetics

Abstract: The transient performance of turbine alternators following a sudden 3-phase short circuit isobtained from design data using electromagnetic field calculations. The 3-phase stator windings are replacedby equivalent current sheets lumped into the stator slots, and the 2-axis theory is used to calculate the short-circuit currents in the stator windings. A 2-dimensional analysis of the field over a half pole pitch is made andthe electromagnetic problem is solved numerically. The nodal method is used to give a relationship betweencurrent and magnetic field at each node. Currents in the stator windings are taken to depend on rotationalvoltages, but the currents in the field winding, damper wedges and the solid rotor depend on the rate ofchange of the magnetic field in addition to the constant source current in the field winding. This leads to aformulation of governing electromagnetic equations coupled with circuit conditions. In the numericalformulation several conductivity matrices are defined to allow for the evaluation of the various currents andthe electromagnetic field distribution, at each instant, simultaneously. Induced eddy currents, currents in themachine windings and the flux distributions are obtained step-by-step using the predictor-corrector method.The problem is nonlinear, the permeabilities of the iron in the various regions being dependent on theresultant field pattern; the problem is solved iteratively using Newton-Raphson and sparsity techniques. Themachine performance is obtained during the subtransient and a part of the transient periods for armature andfield currents. They are compared with the experimental results on a 333 MVA turbine generator.

List of symbols

AaDeEHIiJLLom,nNPRrStx,y8(Ao)

CO

eaV

a

= magnetic vector potential= area associated to a node= area of stator slots= voltage= vector of errors= defined matrix= vector of currents= current= current density= length= overhang leakage reactance= number of nodes= number of turns= number of pole pairs= derivative of E— resistance= defined matrix= time= Cartesian co-ordinates= incremental value of a function= conductivity matrix= flux linkage= angular speed= angle= conductivity= reluctivity= partial derivative

Subscriptscd,qf

= associated to a node= direct and quadrature exis= field

i,],krTsms.c.

eff1,2

= indices= rotational action= transformer action= stator, source= maximum= short circuit= effective value= partioned or damper and field

Superscript= notation for a detailed vector or matrix

Paper 756C, first received 31st October 1979 and in revised form28th February 1980Dr. Hannalla was formerly a consultant with the RutherfordLaboratory, Chilton Didcot, Oxon, England. He is now with theElectrical Engineering Department, Faculty of Engineering, AinShams University, Cairo, Egypt. Dr. MacDonald is with the Elec-trical Engineering Department, Imperial College, London SW7 2BT,England

IEEPROC, Vol. 127, Pt. C, No. 4, JULY 1980

1 Introduction

The accepted theory of synchronous machines using circuitequations is based on the assumption of linear machinecharacteristics and constant inductances. Saturation hasbeen allowed for by choosing the value of each inductancewhich is most representative of the operating conditions,but a better approximation is obtained when mutualinductances are made functions of the fluxes on the rotoraxis. Eddy currents in damper-wedges and solid rotors havebeen represented by a rotor impedance measured over arange of frequency. These methods are reasonablysuccessful but are less accurate in solid rotors and wheresaturation is very significant. In particular, the subtransientparameters of large solid-iron rotors are obtained on a semi-empirical basis, and with the vast development and increaseof the size of turbine generators have become unsatisfactory.Therefore there is a need for a general method in which themachine geometry and other parameters may be fullyconsidered, and by which the essential transient andsubtransient machine parameters may be calculated over awide range of saturation.

In the last decade, many workers1"3 showed theimportance of accurate determination of flux distributionfor obtaining the steady-state performance of the machine.Erdelyi,1 Demerdash2 and Silvester3 have developed thefinite difference and element techniques as numericalmethods for producing steady state flux patterns, similar tothat shown in Fig. 1, allowing for the machine geometry

213

0143-7046/80/04213 + 08 $01-50/0

and the nonlinear iron characteristics. 2-dimensionalanalysis of the machine cross-section has been consideredand end effects have been taken into account separately.

The work has been extended to simple machine transientapplications4'5 in which either stator or rotor transients areconsidered separately. Thus the problem becomes similar tothat of transient heat flow,6 but has the additionalcomplications of varying permeability and the action ofdistributed windings.

In this paper a first attempt is made to solve the, transient-field problem following a sudden 3-phase shortcircuit at the machine terminals from which transient andsubtransient parameters might be determined. Thecalculation of short-circuit current is a simple conditionwhen compared with a general transient state especiallywith the rather sweeping assumptions made here. As isshown below, the a.c. component of the stator current isobtained remarkably accurately. This work should beviewed as a stepping-stone to further work and fullercalculation. Particularly, the calculation of transient fieldcurrent is poor, and a better solution is required.

Here, the most significant effects have been representedat successive instants in time. It has been important to:

(a) allow for stator and rotor transient currents undercircuit terminal conditions

(b) take into account the relative motion between statorand rotor windings with minimal computations

(c) define the problem with as few nodes as possible(d) use a method for the field solution which converges

rapidly(e) make maximum use of sparsity techniques to

minimise the computational time(f) obtain maximum accuracy for values at the nodes

because these determine the currents at successive instants.The last four aims are met", as shown in Reference 7, byformulating the 2-dimensional problem in terms ofmagnetic vector potential A, and normal current density / .The nodal method7 is used to formulate flux-currentrelationships at each grid node. The use of a Newton-Raphson method with sparsity techniques gives a rapidlyconvergent scheme. The first two aims are met by replacingthe 3-phase windings by two equivalent current sheetssinusoidally distributed in stator slots and applying the2-axis theory. A significant reduction in the computational

Fig. 1 Steady-state open-circuit flux distribution for field currentwhich gives 0-85 p.u. voltage of a 4-pole 333 m.v.a. turbinealternator

time is achieved by formulating the numerical problem suchthat the voltage arising from the rate of change of the fluxlinking the stator coils is ignored and half-pole-pitch analysissuffices.

2 Definition of problem and assumptions made

As a first attempt at such a calculation, every opportunityhas been taken to simplify the problem. 2nd-order effectshave been neglected, and the calculation has had the limitedaim of obtaining the a.c. component of the 3-phase statorcurrents. The assumptions are as follows:

(i) Rotor speed is constant(ii) The stator winding is represented by d and g-axis

sinusoidally-distributed windings (similar to those used ingeneralised machine theory) which are located in the statorslots. Direct current in these windings gives approximatelysinusoidal-current distribution. The applied voltages at theterminals of these windings have rotational and transformercomponents and resistance drops

ed = co\jjq + pu>d + idra (1)

eq = — coi//d +pcoQ +/Qra (2)

If stator resistance is neglected (being much less than theleakage reactance) and the transformer voltages are ignored,only d-axis flux is present and

eq = - < o ^ d (3)

Flux is symmetrical about the <i-axis and the field needonly be obtained over half a pole pitch. The d-axis currentis then a slowly changing unidirectional quantity and inthese circumstances the calculation is freed from the needto follow a 50 Hz waveform.

(iii) The slot leakage reactance of a sinusoidallydistributed axis winding is that of a single phase of thestator, the contribution of the other phases being takeninto account. The space-harmonic leakage terms areneglected, being small for a large machine with many slotsper pole. The end leakage is taken into account separately.A further assumption arises from the neglect of transformervoltages in the stator. In a real, sudden, short-circuit, statorcurrents rise to their maxima in the first half cycle,rotational and transformer voltages acting together. Here,that interaction cannot occur, and an equivalent startingcondition is imposed. At the instant to of short circuit,stator current is allowed to jump to its full initial value, andcurrents are put in the surface of the rotor to preserve theremaining rotor currents and fluxes at their previous tovalues.

In the subsequent short circuit this surface current, thefield current, the damping circuit current and the statorcurrent (id) decay together. The correct rate of decay ofstator current is obtained if the sum of the rotor effects iscorrectly represented. However, the rotor currents aredistorted because the surface current has not been allowedto flow in its natural rotor paths initially, but was imposedarbitrarily. The 50 Hz component in the field is absent,stator transformer voltages being ignored.

3 Formulation of governing equations for steady stateand transient conditions

In a 2-dimensional analysis of the magnetic field problemin electrical machines, the field is expressed in terms of themagnetic vector potential A and the current density J. Both

214 IEEPROC, Vol. 127, Pt. C, No. 4, JULY 1980

become scalar quantities, and in Cartesian co-ordinates theelectromagnetic field in a region of reluctivity v andconductivity a is governed by

8 8A\ 8 8A8x I 8x 8y \ 8y

8A-J+o —

8t (4)

subject to a Dirichlet boundary condition along the polarcentre line and the inner and outer circumferentials, and aNeuman boundary condition along the interpolar axis.

The numerical solution of the above problem is obtainedby dividing a half pole pitch of the machine cross-sectioninto triangular elements8 within each of which theconductivity and permeability are assumed uniform. Gridnodes are defined on the vertices of the grid triangles andthe nodal method is applied at each grid node in tern. Thenodal method7 relates values of A at a node to values of ,4at neighbouring nodes and the current associated with thatmode. Thus

(S)(A) = (I) (5)

where S is a matrix and its entries depend on the griddimensions and the permeabilities of the grid elements. Thevector / of currents has entries dependent on the circuitconditions and the location of the node in the grid. In thefollowing Sections the vector of currents is examinedaccording to the circuit conditions.

3.1 Initial steady-state open circuit

Before the switching operation, the machine is operating onno load. The flux distribution depends on the field current.Thus the current vector of eqn. 5 has zero entries for allgrid nodes except for nodes in the field winding. The currentattributed to a particular field node is determined by a gridorthogonal to the element sides. Thus with Nf turns perpole and area ac associated with node / the currentattributed to it is5

(6)

The number of nodes per halfpole in the field windingbeing nf.

3.2 Final steady-state short circuit9

A long time after the sudden short circuit on the 3-phasewindings the problem becomes the determination of a pointon the steady-state short-circuit characteristic, correspondingto a given field current. Therefore, the currents to the gridnodes finally become the steady-state field currents to fieldnodes (given by eqn. 6), and currents to stator windings,obtainable from rotational voltages.

The current attributed to a stator node (on the statorcurrent sheet Section 2) at angle 6 from the pole centre linedue to a short circuit current of a peak value ie c is

h = !>sin (7)

as. is the area enclosed by the orthogonal grid around nodei, 8 9 is the angle between adjacent nodes circumferentially,when the nodes are chosen along the sides of slotscorresponding to half a slot pitch, and ns is the number ofnodes in each stator slot.

The voltage induced in the stator winding eq is given byeqn. 3, and i//d, the direct-axis flux linkage, is expressed interms of A at the nodes of the stator current sheet

1LAta, tin 0-86+£itmem (8)

where Leff is the effective magnetic length of the statorconductors, D is the area of the stator slots over half a polepitch and ms is the number of nodes in the stator windingsin the same region. Lo is the overhang leakage inductanceper phase of the stator windings,10 p is the number of polepairs. For short-circuited stator windings eQ = 0 and fromeqns. 3 and 8, the peak value of the short-circuit current is

(9)

The current vector of eqn. 5 can be expressed in the form

(/) = (7 ' ) - (Aa) r 04) (10)

where / ' is the vector of currents in the field nodes, ineqn. 6, and zero elsewhere. (Aa)r represents the conductivitymatrix for the stator windings for rotational voltages. It hasthe form

(Aa)r =(0) (0)

(0) (Ac/)

The nonzero entries Aa' relate only to nodes in the statorwindings. The entries to the Aa' matrix are given fromeqns. 7 and 9 as

Aa'u =\

fl-.a,. sin 6: sin 6i/LoD I s' S} J' (12)

where / and / are the numbers of any two nodes in thestator winding.

3.3 Tran si en t period

In the transient period there are four distinct types ofcurrents in the nodes:

(a) Zero currents are associated with the nodes in theairgap or the laminated stator iron which do not lie on acommon boundary between the stator winding and the ironor the airgap and the rotor surface.

(b) In the solid iron or the damper windings eddycurrents are induced dependent on the rate of change of thefield, dA/dt. For a linear variation of current density withineach element, the current associated with each node isrelated to the rate of change of the field at that node andthe surrounding nodes. It has the form.11

1

13 A

oA

where Nm is the peak number of turns in the current sheet(1-35 N8Kw\j2), N8 is the turns per pole-pair per phase, 7 = 1 .

(13)

IEEPROC, Vol. 127, Pt. C, No. 4, JULY 1980 215

where K and K + 1 are two successive elements surroundingnode /' and joining it with node /. A and oeff are the areaand effective conductivity of a triangle, m is the number ofnodes surrounding node /.

(c) In the field winding the current is uniform but variesfrom one instant to the next. The current has twocomponents:

(i) steady-state component Ifs from the exciter (may bezero if the field terminals are short circuited)

(ii) a component IfT which depends on the total inducedvoltage from the rate of change of the flux linking thewinding and the circuit resistance.

Thus for a symmetrical distribution above the pole centreline, the current IfT is

IfT — —2NfL

Rfnf nf

/

1=1

dAdt

(14)

where L is the magnetic effective length of the fieldconductor and Rf is the circuit resistance. The currentattributed to a field node i is given by eqn. 6, in which Ifbecomes

= If.+Llfs fT(15)

(d) In the stator slots, the stator windings are replacedby an equivalent current sheet distributed sinusoidally inthe circumferential direction and uniform in the radialdirection over the stator slots (Section 2). The transientshort-circuit current is obtained from the 2-axis eqns. 1 and2. The neglect of pxj; terms and the armature resistancedrop (Section 2) gives for 3-phase short-circuited windingsi//d = 0 and 4JQ = 0. Therefore the governing eqns. 1 and 2reduce to eqn. 3. Accordingly the stator short-circuitcurrent at any instant during the transient period is givenby eqn. 9. The behaviour of all four regions during thetransient period is expressed in general by

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

(Ac

but the zero regions are more closely specified to coincidewith those defined for (Aa)T in eqn. 17. Thus

(Aa)r == „ „ „ (18)

4 Iterative solution and time stepping

Before the switching operation on the 3-phase windings,and at the end of the transient period, the flux distributionsare obtained by solving the steady-state open- and short-circuit problems.

In the transient period a solution is sought for eqn. 4subject to the current distribution eqn. 16. The (Aa)matrices are fully determined. In a step-by-step method thepredictor-corrector method is used to estimate values ofAt+6t at the new instant t 4- 5 fin the terms of the solution atthe previous instant. The first estimate for values of At+st

are

8t (19a)

which is used to start the iterative solution, and then usesthe second-order approximation for values of dA/dt att + St.

dAdt

dAdt

(19b)

Here the problem is nonlinear because the elements of theS-matrix are dependent on the iron permeabilities which arefunctions of the flux densities, and hence nonlinear func-tions of the resultant ^-distribution. A solution of theproblem is obtained when a vector of errors E given by

dA'— )-(Ao)r(A) (16)

(/') is the vector of imposed source current. Its entries areto field nodes from the current Ifs. (Aa)-r is the conductivitymatrix resulting from the rate of change of the field. It hasthe form

(Aa)T =

(0) 0 0 0

0 (Aat) 0 0

0 0 (Aa2) 0

0 0 0 (0)

(17)

where Aat and Aa2 represent the behaviour of the eddy-current region and the component of the field winding dueto the rate of change of the field IfT. The entries to theAox matrix are given by the coefficients of dA/dt ineqn. 13 and the entries to the Aa2 matrix are given fromeqns. 6 and 14. The first and last zero diagonal matricesrepresent the nonconducting and stator-winding regions,respectively. The (Aa)r matrix in eqn. 16 is that of eqn. 11,

(20)

reaches a minimal acceptable value. All values in the aboveequation are at time t + 8t except as indicated. ApplyingNewton-Raphson technique on eqn. 20, the improvedvalues of At+$t at the ( / + l ) t h iteration from the / thiteration are

where (R)j is the derivative of (E) with respect to (A) andhas the form

2_

dt(22)

(//) being the first derivative of (S)(A) with respect to 04)and its evaluation is explained in Reference 12. (//) and(Aa) are symmetrical and a best numbering scheme isnecessary to keep the sparsity of the (H) and (R) matrices.Eqn. 2 indicates an inversion, but this is avoided using

216 IEEPROC, Vol. 127, Pt. C, No. 4, JULY 1980

Jennings method,13 which gives the inversion multiplication(R)'1 (E) directly.

5 Initial value problem

5. / Discontinuity of the equations

Under no-load conditions, the field is governed by eqn. 5 inwhich the currents are distributed only at field nodes givenby eqn. 6. However, under transient conditions the fielddistribution is governed by eqn. 5, but with currents to allnodes given by eqn. 16. This equation contains the samevector of field current to the field nodes (/') besides twoadditional vectors of current:

(a) field-winding and eddy-current nodes due to the rateof change of the flux linkage

(b) stator nodes due to rotational voltage.If the vector of current distribution at t = 0" is keptunaltered at t = 0+ this implies

dA( A a ) T | — \~(Ao)r(A) = 0

Since the two vectors in the above equation are displaced(eqns. 17 and 18), both dA/dt at field and eddy currentnodes, and A to the stator nodes, are zero, which does notgive a start to the transient solution. This situation arisesfrom the neglect of p\jjd in the stator voltage eqns.

5.2 Definition of a new boundary condition

A solution is sought which allows initial subtransientcurrent to occur. Here it is assumed that the direct-axisshort-circuit current can suddenly change its value fromzero (open-circuited winding at t = 0") to its maximumvalue at t — 0+, which is again possible with the neglect ofthe p\\j term in the stator region. The induced currents inthe solid rotor iron and damper and the additional fieldcurrent are kept zero to start with. This implies that thefield distribution is frozen in the rotor region at theswitching instant while the field in the airgap and statorregions is free to change. Therefore the solution of themagnetic-field problem under no-load conditions defines anew boundary on the rotor surface for the flux patternbeyond the rotor surface at the time t = 0+.

5.3 Initial sub transien t curren t

The electromagnetic field problem at t = 0+ is confined tothe evaluation of the magnetic field distribution in thestator and airgap regions subject to Dirichlet and Neumanboundary conditions (defined in Section 3) and themagnetic vector potential distribution on the rotor surface.Thus by applying the nodal method at each grid node onthe stator and airgap gives

= (Ao)r(A) (24)

where (Si) is a part of the S' matrix which has nodes ofpotential A{ on the rotor surface and its rows aredistributed to nodes in the airgap region and connected tothe rotor surface by the grid lines. (S') is a part of the Smatrix corresponding to nodes in the airgap and statorregions. The solution of the above equation gives (at t =0+) a new distribution to values of A in the airgap andstator regions as shown in Fig. 2. This new flux pattern isobtained simultaneously with the peak value of the initial

subtransient current, i"d in the stator winding (calculatedfrom eqn. 9).

5.4 Significance of the new boundary on the overallsolution

At time t = 0, the solution of the magnetic-field problemhas been obtained in two regions: stator and airgap, androtor, iron and damper. Each satisfies separately theelectromagnetic eqn. 4, with the addition of rotationalvoltage effects. They are coupled with a common potentialdistribution A on the rotor surface. However, nodes on therotor surface do not satisfy eqn. 4 and give a vector oferrors £ to all nodes on the rotor surface. The values of theerrors and their distribution form a distributed surface-current sheet which appears at the switching instant of thesudden 3-phase short circuit. Thus the transient problem isaltered to the decay with spread of the rotor current sheetwhich becomes a source of transient currents and fluxdistributions.

(23) 5.5 Definition of a boundary value problem

The solution of the transient-field problem starts with fieldand current distributions at t = 0+. The magnetic-fielddistribution has been obtained, as explained above, in rotor,airgap and stator regions separately. The currents aredistributed in

(a) the field windings from the source voltages(b) the stator windings from rotational voltages(c) rotor surface-current sheet which is determined by

the error in the solution of the magnetic-field equation.Induced eddy currents in the solid iron, wedges, and theadditional field current owing to the rate of change of the'field, start from zero.

5.6 Rotor surface current sheet and the transien t problem

A first-order approximation for values of dA/dt has beenused to start the numerical solution. This approximationovercomes the difficulties that arise with a sudden changein values of dA/dt on the rotor surface, and has been foundto overcome oscillations in the numerical solution thatarises with the predictor-corrector method (Reference 8).The problem has been solved iteratively as given inSection 4. However, an instability has been found in thenumerical solution at the start of the solution of thetransient problem. This is related to the use of the Newton-Raphson technique which only gives fast convergence forsmall changes in the field problem at each time step. Thisproblem has been solved by considering small time stepsand reducing the effect of the currents on the rotor surfaceof the field distribution during the iterative solution. Thishas been achieved by considering the equation

)j- 0-5 (25)

instead of eqn. 21 in the iterative process, and by increasingthe number of iterations in the initial time stop.

6 Modelling of the iron characteristic

In representing the change of reluctivity of iron with fluxdensity it is necessary to choose a model that follows theB/H curve accurately and also allows iteration to take placewith minimal computational effort. Silvester et al.u

showed that modelling of the iron characteristic using thereluctivity against the flux density squared is much better

IEEPROC, Vol. 127, Pt. C, No. 4, JULY 1980 217

than using the B/H curve directly. Here, curves for v, bv/b (B)2

and b2v/b(B2)2 against B2 are used in the solution. Fastconvergence is found by accurate evaluation of the first-and second-order derivative curves of v against B2. Thusfourth-order polynomials are used at high and low values of

B2 and these regions are joined by an exponentialexpression. The method of evaluating the variousparameters of the above mentioned curve from theexperimental results is given in Reference 14.

7 Application of the method

The method has been applied to a 4-pole 333 m.v.a. turbinegenerator to calculate the sudden 3-phase short-circuit

Fig. 2 Flux distribution just after sudden 3-phase short circuit onterminals of the turbine-alternator

Fig. 5 Eddy-current density distribution 0-001 s after switchingoperation of a sudden 3-phase short circuit on terminals of theturbine-alternator

Fig. 3 Flux distribution 0-006 is after switching operation of asudden 3-phase short circuit on terminals of the turbine-alternator

Fig. 6 Eddy current density distribution 0-006 s after switchingoperation of a sudden 3-phase short circuit on terminals of theturbine-alternator

Fig. 4 Flux distribution 0-059 s after switching operation of asudden 3-phase short circuit on terminals of the turbine-alternator

Fig. 7 Eddy current density distribution 0-059 s after theswitching operation of a sudden 3-phase short circuit on theterminals of the turbine-alternator

218 IEEPROC, Vol. 127, Pt. C, No. 4, JULY 1980

characteristic for the excitation current which gives anopen-circuit voltage of 0-85 p.u. The steady-state initialconditions of the open-circuited machine give the fluxdistribution shown in Fig. 1. The flux distribution just afterthe switching instant is shown in Fig. 2. The differencebetween these two distributions is in the airgap and statorregions. The saturation level is high in the top part of thestator teeth nearest to the airgap (in Fig. 2) and the fluxlines are mainly crossing the stator teeth. The value of theinitial subtransient current gave an error between calculatedand measured subtransient reactance of 4%.

The flux and eddy current density distributions at threeselected instants from the start of the transient are shownin Figs. 3, 4, 5, 6 and 7. The variation of the induced eddy-current density distributions with time shows that

(a) The induced eddy currents are mainly on the rotorsurface during the initial transient period.

(b) The induced eddy currents spread mainly over theteeth where the transients are also in the field circuit.

(c) Eddy currents spread over the rotor body.The induced eddy currents affect the flux patterns duringthe subtransient and transient periods as shown in Fig. 2, 3and 4. The final steady-state condition, steady-state shortcircuit, is shown in Reference 9. xd was obtained to within2% of the measured value.9

008 012 016times

02 024

Fig. 8 Experimental results of 3-phase armature currents followingsudden 3-phase short circuit

envelopes of armature currents in the three phases

30

20

10

0 001 002 003 0<K 0O5 0-06 007 008 009time,s

Fig. 9 Comparison between computed and experimental results ofdirect-axis current

experimental results for the three phasescomputed results

IEEPROC, Vol. 127, Pt. C, No. 4, JULY 1980

The envelopes of the experimental results of therecorded three-phase armature currents are shown in Fig. 8.It is found that the direct-axis current derived from the a.c.component of the various phases is slightly variable through-out the transient period. It is related to the accuracy of themeasurements and the drawing of the envelope. Thus acomparison between the computed direct-axis current andthat obtained from the experimental results of the threephases is shown in Fig. 9. It is found that the computedvalues of the direct-axis current closely follow the averageexperimental values, with a maximum error of less than 5%.

The field current shows some difference betweencomputed and experimental results as shown in Fig. 10.This difference arises because:

(i) the neglect of p\p terms removes the a.c. componentof the field current

(ii) the sudden change in the stator current at t% hasbeen allowed to produce instantaneously a current sheet onthe rotor surface without producing any effect on theinitial rate of change of the flux in the field winding. Thusa part of the transient component is also absent from thefield current.The above computed results were obtained using a grid oftriangles with 322 nodes. The initial flux distribution at thestart of the transient period has been obtained using thesame program for steady-state calculations, but with thenew boundary condition in Section 5.2, as defined fromthe open-circuit field calculations. The transientcalculations for 50 points took a c.p.u. time of 1690 s(28min) on an IBM 192 computer. It is found necessary to•reduce the time steps and increase the number of iterationswhen there is a rapid variation of the flux and currents.Such a situation appears in the initial subtransient period.If this is not done the convergence is poor, and could.be-unstable, especially in the first time step as explained in thetext. Thus 25 iterations were used at the first time stepfollowed by 20 iterations for the successive time steps togive a maximum error of 10~4 in the solution of theequations.

The program can be used for different initial fieldcurrents according to which the saturation of the iron varies,and consequently transient performance could be obtainedas a function of the initial field current.

8 Conclusion

A method for predicting the sudden 3-phase a.c. short-circuit currents of turbine generators from designinformation has been presented. The accuracy with whichthe stator current is obtained is remarkable in view of theapproximations made, although the calculated currentclearly has a very fast initial time constant which is notpresent in the curves obtained from test results.

00 002 O0A0O6 0 0801012 016 02 024time.s

Fig. 10 Comparison between computed and experimental resultsof field current

experimental recordx x x x computed values

219

As the results obtained are so near the measured values,the approximations made can be considered reasonable. Inparticular, the conditions in the stator are not as consideredin this solution (fluxes being trapped by the stator circuits)and consequently the permeabilities there will not becorrect. This confirms the impression that it is thebehaviour of the rotor which dictates the performance.However, even there, the currents are not correct (the a.c.component being absent). The accuracy of the resultprobably follows from a good representation of the short-circuit instant, open-circuit rotor flux being suddenly putinto the stator leakage paths givingx'd. The accuracy of thestator time constants must arise from a faithful decay ofthe rotor flux.

It is hoped to follow this work with a fuller calculationof short-circuit performance. It will then be possible to seehow it is that the alternating component of id, neglectedhere, appears to have little effect on the slowly decayingvalues.

9 Acknowledgments

Thanks are due to A.B.J. Reece of GEC Ltd., Stafford,England who provided details of the 333 MVA generator,and to D.B. Thomas and C.W. Trowbridge of RutherfordLaboratory, Chilton, Didcot, England for the financialsupport of the project and the facilities offered in theRutherford Laboratory.

10 References

1 FUCHS, E.F., and ERDELYI, E.A.: 'Determination of waterwheel alternator steady state reactances', IEEE Trans., 1972,PAS-91, pp. 2510-2527

2 DEMERDASH, N.A., and HAMILTON, H.B.: 'Simulation for

design purposes of magnetic fields in turbo alternators withasymmetrical and symmetrical rotors', ibid., 1972, PAS-91,pp. 1992-1998

3 CHARI, M.C.K., and SILVESTER, P.: 'Analysis of turboalternator magnetic field by finite elements',ibid., 1971.PAS-90,pp. 454-464

4 HANNALLA, A.Y., and MACDONALD, D.C.: 'Numericalanalysis of transient field problems in electrical machines', Proc.IEE, 1976,123, (9), pp. 893-898

5 HANNALLA, A.Y., and MACDONALD, D.C.: 'Comparison ofstator and rotor induced transients in solid rotor turbinegenerators', IEEE Trans., 1979, MAG-15, pp. 1485-1487

6 FLATABQ, N.: 'Transient heat conduction problems in powercables solved by the finite element method', ibid., 1973, PAS-92,pp. 56-63

7 HANNALLA, A.Y., and MACDONALD, D.C.: 'A nodal methodfor the numerical solution of transient field problems in electricalmachines', ibid., 1975, MAG-11, pp. 1544-1546

8 HANNALLA, A.Y., and MACDONALD, D.C.: 'The solution oftransient fields by the nodal method'. Proceedings of theComumag Conference on the computations of magnetic fields,March-April 1976, pp. 381-388

9 HANNALLA, A.Y., and MACDONALD, D.C.: 'Steady stateshort-circuit characteristics of turbine generators from designdata using field calculations',. Presented at the IEEE Power Eng.Society Meeting in New York February 1980, Paper F80-207-1,to be published in IEEE Trans, on Power Apparatus & Systems

10 ALGER, P.L.: 'Induction machines' (Gordon and Breach, 1970,2nd edn.)

11 CARPENTER, C.J.: 'Finite-element network models and theirapplication to eddy-current problems', Proc. IEE. 1975, 122,(4) pp. 455-462

12 SILVESTER, P., CABAYAN, H.S., and BROWNE, B.T.:'Efficient technique for finite element analysis of electricmachines', IEEE Trans., 1973, PAS-92, pp. 1274-1281

13 JENNINGS, A.: A compact scheme for the solution of symmetriclinear simultaneous equations', Comput. J., 1966. pp. 281-285

14 HANNALLA, A.Y.: 'Transient analysis of solid rotor turbo-alternators allowing for saturation and eddy currents', Ph.D.thesis, University of London, 1975

220 IEE PROC, Vol. 127, Pt. C, No. 4, JULY 1980