k 2 V L is the normal-polarising signal and k p V p is the exter- Table 3: Source data nal, or crosspolarising signal. 8.2 Protection application model The model system on which the investigations of the paper are based is given in outline form in Fig. 12. The principal data of the model is summarised in Tables 2-4. Table 2: Principal transmission line data Nominal voltage 220 kV Line length 80 km Construction single circuit Conductor self impedance (9.7 +y'45.4) Q Interconductor mutual impedance (5.2+/19.4) Q Positive-phase-sequence impedance (4.5 +/26.1) Q Zero-phase-sequence impedance (20.1 +/84.2) Q Fault level 8134 MVA Positive-phase-sequence impedance (0.41 +/5.95) Q Zero-phase-sequence impedance (0.44 +/3.58) Q Table 4: Capacitor voltage transformer data Upper capacitor Lower capacitor Voltage divider ratio Voltage ratio of intermediate transformer Tuning inductance Burden Nominal secondary voltage Ferro-resonance suppressor 8.4 x 1 0 " 6 F 31 MO- 6 F 0.213 426 0.78 mH 160 VA at 0.9 pf lagging 63.5 V — Correspondence VALIDITY OF DESCRIBING FUNCTION ANALYSIS FOR 3-PHASE POWER SYSTEMS Indexing terms: Power systems and plant, Mathematical tech- niques Abstract: The applicability of the describing function (DF) in the study of jump resonance for 3-phase coupled systems is examined. Before contemplating the use of a describing function, the 3-phase systems should first be modelled in the block-diagram form, as is done for single-phase systems, and then the linear elements follow- ing the nonlinearity should be tested for their low-pass character- istics. The method is illustrated by two specific circuits. 1 Introduction The describing-function (DF) technique has been found to be a very effective tool in investigating the ferro-nonlinear oscillations of power systems at fundamental frequency. In terms of the block-diagram representation of a system, the DF approach requires that the input to the nonlinearity should be sinusoidal, and that the feedback paths to the nonlinear block must contain elements with low-pass fil- tering characteristics so as to attenuate the higher harmo- nics contained in the output from the nonlinear block. This condition is usually adequately met in single-phase systems, suceptible to jump resonance, and the technical literature contains ample evidence in this respect. But, very limited references are available on coupled 3-phase systems suceptible to ferro-nonlinear oscillations [1-3]. Based on the concept of frequency domain DF, Barakat and Hirst [3] have presented an iterative technique for predicting the jump resonance of 3-phase systems that are not amenable to single-phase approximation and require that the true form of circuit configuration be retained. They remark that 'it is not easy to say whether the harmonic filtering condi- tion is met adequately by 3-phase ferro-nonlinear net- works, due to interaction of coil currents and core fluxes between phases within the transformer. These interactions give rise to crosscoupling between feedback paths in the block diagram. The work described in the paper does not resolve this question'. On the a priori assumption that the transformer limbs can be modelled in terms of separate DFs, the necessary filtering effects are produced in the circuit and the validity of their assumptions is supported from the practical results obtained with an experimental 224 transformer. For the sake of generality, it is desirable that a theoretical justification should exist to advocate the use of DFs for the nonlinear transfer characteristics when applying them to 3-phase coupled systems. Here the extent of the applicability of the DF to the analysis of 3-phase nonlinear systems is examined for fundamental frequency oscillations. 2 Illustrative circuits 2.1 Transformer feeder circuit The system chosen is the same transformer feeder circuit (Fig. 1) that has been used by Barakat and Hirst [3]. The method applied is straightforward and is applicable to any Fig. 1 feeder Equivalent circuit of a 3-phase series compensated transformer practical 3-phase coupled system. In terms of Laplace transforms, the circuit equations for phase a of Fig. 1 are E l =R l I l R. l sI l +— l - Substituting eqn. 2 in eqn. 1, we obtain G l {s)E l = A a + GJLs)I. where R l J 0) (2) (3) (4a) -^- L s 2 + 1 + K a \ IEE PROCEEDINGS, Vol. 130, Pt. C, No. 4, JULY 1983
Transcript
k2 VL is the normal-polarising signal and kp Vp is the exter- Table 3: Source datanal, or crosspolarising signal.
8.2 Protection application modelThe model system on which the investigations of the paperare based is given in outline form in Fig. 12. The principaldata of the model is summarised in Tables 2-4.
CorrespondenceVALIDITY OF DESCRIBING FUNCTIONANALYSIS FOR 3-PHASE POWERSYSTEMS
Indexing terms: Power systems and plant, Mathematical tech-niques
Abstract: The applicability of the describing function (DF) in thestudy of jump resonance for 3-phase coupled systems is examined.Before contemplating the use of a describing function, the 3-phasesystems should first be modelled in the block-diagram form, as isdone for single-phase systems, and then the linear elements follow-ing the nonlinearity should be tested for their low-pass character-istics. The method is illustrated by two specific circuits.
1 Introduction
The describing-function (DF) technique has been found tobe a very effective tool in investigating the ferro-nonlinearoscillations of power systems at fundamental frequency. Interms of the block-diagram representation of a system, theDF approach requires that the input to the nonlinearityshould be sinusoidal, and that the feedback paths to thenonlinear block must contain elements with low-pass fil-tering characteristics so as to attenuate the higher harmo-nics contained in the output from the nonlinear block.This condition is usually adequately met in single-phasesystems, suceptible to jump resonance, and the technicalliterature contains ample evidence in this respect. But, verylimited references are available on coupled 3-phase systemssuceptible to ferro-nonlinear oscillations [1-3]. Based onthe concept of frequency domain DF, Barakat and Hirst[3] have presented an iterative technique for predicting thejump resonance of 3-phase systems that are not amenableto single-phase approximation and require that the trueform of circuit configuration be retained. They remark that'it is not easy to say whether the harmonic filtering condi-tion is met adequately by 3-phase ferro-nonlinear net-works, due to interaction of coil currents and core fluxesbetween phases within the transformer. These interactionsgive rise to crosscoupling between feedback paths in theblock diagram. The work described in the paper does notresolve this question'. On the a priori assumption that thetransformer limbs can be modelled in terms of separateDFs, the necessary filtering effects are produced in thecircuit and the validity of their assumptions is supportedfrom the practical results obtained with an experimental
224
transformer. For the sake of generality, it is desirable thata theoretical justification should exist to advocate the useof DFs for the nonlinear transfer characteristics whenapplying them to 3-phase coupled systems. Here the extentof the applicability of the DF to the analysis of 3-phasenonlinear systems is examined for fundamental frequencyoscillations.
2 Illustrative circuits
2.1 Transformer feeder circuitThe system chosen is the same transformer feeder circuit(Fig. 1) that has been used by Barakat and Hirst [3]. Themethod applied is straightforward and is applicable to any
Fig. 1feeder
Equivalent circuit of a 3-phase series compensated transformer
practical 3-phase coupled system. In terms of Laplacetransforms, the circuit equations for phase a of Fig. 1 are
El =RlIl
R.
lsIl +—l-
Substituting eqn. 2 in eqn. 1, we obtain
Gl{s)El = Aa + GJLs)I.
where
Rl J
0)
(2)
(3)
(4a)- ^ - L s2 + 1 +
Ka \
IEE PROCEEDINGS, Vol. 130, Pt. C, No. 4, JULY 1983
and
Ra
oo then
1
LxC1s2 + R1C1s+
i -i- *
(46)
G-(s) = 7
CM = L, + ^CjS^
(5a)
(5b)
Equations similar to eqn. 3 exist for phases b and c,namely,
G2(s)E2 = Gb(s)Ib
and
3 = Ac + Ge(s)Ie
(6)
(7)
The G terms of eqns. 6 and 7 have the same form as thosefor eqn. 3, but with respective parameters for the phasesconsidered.
If we adopt the simplified magnetic circuit model ofNakra and Barton [4] for a 3-limb transformer, allowing asingle air path for zero-sequence flux, we can resolve thewinding currents as follows:
= I ma +
h = + to
+ h (8)
where Ima, Imb and Imc are the magnetising current com-ponents for the fluxes to set up in respective iron paths ofthe limbs, and Io is the excitation for the zero-sequenceflux path.
Considering the saturation characteristics of iron limbsonly, let the nonlinear relationships between the flux link-ages with the windings and excitation currents, setting upthe core fluxes, be expressed as
We get two more relations from the simplified magneticequivalent circuit [4] of a 3-limb transformer, namely,
Aa
and
A Ac = Ao
Ao = Lo/o / O (10)
where Lo is the constant inductance associated with zero-sequence path.
Eqn. 3 and eqns. 6-9 can be represented by the blockdiagram of Fig. 2 which illustrates the extent and nature ofcoupling between phases of the circuit of Fig. 1. It isevident from Fig. 2 that, if the transfer functions Ga, Gb
and Gc possess low-pass filtering characteristics, there is noreason to discard the modelling of the limbs by separateDFs under the same assumptions that are permitted forsingle-phase systems. Compared with single-phase systems,the situation will in no way be worse for Fig. 1, as thenonsinusoidal output from the nonlinearity will be accom-panied by an additional fundamental frequency com-ponent (Ao/Lo) before passing through the linear elements.
Eqns. Ab and 5b show that the low-pass action of Ga
depends on the values of the external parameters, namelyLl, Ct and Rl} and iron-loss equivalent resistance Ra;similarly with Gb and Gc. The circuit considered here is aseries compensated transmission system, and the presenceof capacitance is essential for jump resonance to occur. If,by eqn. Ab or 5b, it is found that the line parameters aresuch as to yield sufficiently attenuated values of Ga, Gb andGc for the harmonics compared with the fundamental, theN terms of eqn. 9 may then be interpreted as DFs. For thesame system parameters and symmetrical capacitance, asgiven in Reference 3, Table 1 presents the frequency-dependent values of Ga, Gb and Gc by eqn. Ab, up to the7th harmonic. Calculations for still higher frequenciesshowed continuous fall in gain.
The values of G terms in Table 1 clearly indicate that,for the given parameters, filtering conditions are met with
Table 1 : Frequency-dependent values of linear complex gainsin the feedback paths of Fig. 1
Wb turn/A bWb turn/A
(9)
where the N terms stand for some nonlinear functions.
Fig. 2 Block-diagram representation of Fig. 1
IEE PROCEEDINGS, Vol. 130, Pt. C, No. 4, JULY 1983
by the circuit configuration of Fig. 1, thereby justifying theuse of the DF concept by Barakat and Hirst [3]. In mostpractical cases of interest, the L terms will probably besmall enough not to interfere with the filtering effect of theR and C terms.
2.2 Potential transformers on an isolated systemFig. 3a shows a very typical 3-phase circuit configuration[5-7], which under no approximation can be treatedphase-wise for ferro-resonance study even for a crude esti-mate. This is a star-connected bank of single-phase poten-tial transformers, with grounded-neutral and open-deltasecondary, connected on an ungrounded system for earth-fault detection. It has been experienced [5-7] that, for acertain zero-sequence capacitance of the system, such atransformer connection is able to sustain ferro-nonlinearoscillation at fundamental, or at multiple or submultiple
225
frequencies, following switching operations or temporaryground faults.
Fig. 3b shows the block-diagram representation of thecircuit with linear gains that are defined by the followingequations:
Table 2: Frequency-dependent values of linear complex gainsin the feedback paths of Fig. 3d
h s2 + (Rc + rjs
G0(s) =
s2 + (Rc + rjs
and Gb(s) = Ge(s) = Ga(s) and G^s) = G2(s) = G3(s).Derivation of the above is omitted, because it follows
the same line of procedure as the circuit of Fig. 1, namelysetting up of equations based on Kirchhoff's EMF andcurrent laws, elimination of certain variables from theseequations and rearranging the resulting equations in suit-able form for block-diagram representation.
Experiments were conducted with 400/110 V, 50 Hz,laboratory-type potential transformers on the 110 V side.Line-to-line voltage was taken as 110 V for the reasonsstated in Reference 5, and Co was made equal to 20 fiF.With this value of capacitance, the response was found tobe predominantly at fundamental frequency. Table 2 pre-sents the frequency-dependent gains of the linear elementsGo and Ga up to the 7th harmonic, and for higher harmo-nics gains were less. Table 3 compares the measured valuesof the phase and the system neutral-to-ground voltageswith the predicted values obtained by numerical solution
of the governing equations of the circuit. These are calcu-lated from the block diagram of Fig. 3, for fundamentalfrequency (s = jcoj, using the concept of describing func-tion for the nonlinear saturation characteristic of the core.A quintic-type nonlinearity, of the form im = at Xm + a5 l
5m,
was assumed to represent an instantaneous saturationcharacteristic [8] of the core. Although not a very closeagreement between prediction and observation can benoted, a reasonable assessment of the likely situation hasbeen obtained. Besides the assumptions underlying the DFconcept, deviation can be attributed to the assumed modelof the saturation curve and to the manner of arriving atthis model from the RMS saturation curve [8].
3 Conclusion
Like the circuits of Figs. 1 and 3, a number of 3-phasecoupled circuits have been examined by the author for thelinear transfer functions following the nonlinearity in theblock-diagram representation of the coupled systems. Ithas been found, for the cases examined, that the lineartransfer functions are of the form
G(S) =F2(s)
Fig. 3 Grounded star/open delta connected bank of potential trans-formers on an ungrounded balanced sourcea Circuit diagramb Block-diagram representation of the circuit
and the order of the polynomial F2(s) has always been atleast one more than that of F^s), thereby indicating thelikelihood of the fall in gain of G(jco) with increase infrequency. This would obviously depend on the relativevalues of the parameters in the circuit. The accuracy ofinformation obtained from applying the DF techniquewould depend upon how well G{jco) would attenuate withhigher frequencies.
4 References
1 OCUMURA, K., and KIJIMA, A.: 'Non-linear oscillation of three-phase circuits', Electr. Eng. Jpn., 1976, 96, pp. 106-112
2 TEAPE, J.W., SLATER, R.D, SIMPSON, R.R.S., and WOOD, W.S.:'Hysteresis effects in transformers, including ferroresonance', Proc.IEE, 1976,123, (2), pp. 153-158
3 BARAKAT, E.E., and HIRST, D.E.: 'Susceptibility of 3-phase powersystems to ferro-nonlinear oscillations', Proc. IEE, 1979, 126, (12), pp.1295-1300
4 NAKRA, H.L., and BARTON, T.H.: 'Three-phase transformer tran-sients', IEEE Trans., 1974, PAS-93, pp. 1810-1819
226 IEE PROCEEDINGS, Vol. 130, Pt. C, No. 4, JULY 1983
5 PETERSON, H.A.: 'Transients in power systems' (Wiley, New York,1951)
6 STANBURY, A.J., and CURRIE, C.V.: Terro-resonance experiencedat Lake Coleridge power station', New Zealand Eng., 1973, pp. 317-322
7 GERM AY, N., MASTERO, S., and VROMAN, J.: 'Review of ferro-resonance phenomenon in high-voltage power systems and presen-tation of a voltage transformer model for predetermining them',CIGRE, Paris, Paper 33-18, 1974
8 PRUSTI, S., and RAO, M.V.S.: 'New method for predetermination oftrue saturation characteristics of transformers and nonlinear reactors',IEE Proc. C, Gen., Trans. & Distrib., 1980, 127, (2), pp. 106-110
Department of Electrical EngineeringIndian Institute of TechnologyKharagpur 721302West Bengal, India
S.N. BHADRA
SPTC 180
Z-TRANSFORM TRANSIENT ANALYSIS INPOWER SYSTEMS
Prof. Humpage and his co-authors have presented inter-esting papers (988C, 989C and 990C [ /££ Proc. C, 1980,127, (6), pp. 370-394]) on various aspects of the use ofz-transforms for power-system transient analysis. The tech-nique is of x>bvious importance as a possible alternative toother well established validated frequency-domainmethods [A-C] currently employed as part of the protec-tive gear design and industrial testing activities. However,it is perhaps worthwhile to note that, although in conceptattractive, the z-transform method possesses some impor-tant limitations which may perhaps not be immediatelyapparent from the existing publications. When applyingthe z-transform, the accuracy obtainable is heavily depen-dent on the ability to accurately transform various systemfrequency response functions from the frequency to thez-plane. As part of this process, low-order polynomialcurve fitting to the response functions exp {— ln{co)l] andZ0{s\ as detailed in Section 5 of paper 988C, is employed.However, fitting is applied only to the magnitude of theresponse functions | exp {— ln{oS)l) | and | Z0(s) — Zoc |. This,however, does not ensure that the associated phaseresponses are correctly emulated. For example, the corre-spondence between both the amplitude and phase responseof the earth-mode function exp { — Xn{(o)l} for a 160 km400 kV line, as obtained when using the polynomialapproximation method proposed in paper 988C, is shownin Fig. A. Although close matching to the amplitude of theresponse is obtained, the associated phase response{— <t>n{co)l) is seen to be in error.
A further very significant error, to which no reference ismade in the papers, arises through the use of the bilineartransform applied to the polynomial expressions (eqn. 68of paper 988C). The equivalence indicated in the latterequation is not valid for all frequencies, as evidenced bywriting the function 2(z — l)/(z + l)At in the alternativeformy2{tan (caAt/2)}/At. In applying such equivalences, thefrequency responses are distorted. This effect is one that isknown as frequency warping [D]. A further consequenceof the technique employed is that frequency-domain alia-sing occurs so as to cause both amplitude- and phase-response functions to fold about the Nyquist frequencyfJ2 = l/2Af. In effect, the frequency/^ in the z-transformsimulated responses corresponds to an actual system fre-quency of infinity. The extent to which the latter effectscause distortion of both phase and amplitude responses,when using the time step of 50 fis in obtaining the solu-tions presented in Section 6 of paper 988C, is also shownin Fig. A. Similar distortion arises in respect of the systemsurge-impedance functions.
The effect of the foregoing errors is manifested in theform of incorrectly simulated wave attenuation and velo-cities, and many of the differences observed when compar-ing z-transform with frequency-domain derived responses
are attributable to such errors. Against this background,care has to be taken when comparing, as for example inSection 4.2 of paper 989C, the accuracy of z and frequency
(i)10 15 20 25
angular frequency,rad/s x!0A30
-360L
Fig. A Correspondence between actual system response functions andtheir z-transform approximations
(') Typical amplitude-response function= actual value exp [—a(w)f]= polynomial approximation= z-transform warped approximation to exp [ — i(a>)f]
(ii) Associated phase-response function= actual value — pn(co)l
Time step = 50 //s (as used in paper 988C; corresponding angular Nyquistfrequency = 2n x 10* rad/s; response functions are for 160 km, 400 kV line (earthmode)
IEE PROCEEDINGS, Vol. 130, Pt. C, No. 4, JULY 1983 227
