Application of Vector Fitting to High Frequency Transformer Modeling

Transcription

1 Application of Vector Fitting to High Frequenc Transformer Moeling Bjørn Gustavsen SINTEF Energ Research, N-76 Tronheim, Norwa ( Abstract This paper escribes a proceure for frequenc epenent moeling of power transformers from measure terminal characteristics. The amittance matrix is measure in the frequenc omain using a network analzer an a eicate measurement setup. Subsequent approximation of the amittance matrix with rational functions using Vector Fitting gives an EMTP-tpe compatible moel suitable for transient stuies. The approach is emonstrate for a -wining rectifier transformer an for the calculation of internal voltages in a wining. It is shown that that highl accurate results can be obtaine Kewors Power sstem transients, transformer overvoltages, frequenc epenence, moeling, rational functions, sstem ientification. I. INTROUCTION The calculation of transferre overvoltages through transformers requires that the frequenc epenent behavior of the transformer be taken into account. The moeling can in principle be carrie out b calculating an electrical network base on the transformer geometr [],[] but this approach requires etaile information about the transformer construction, which is usuall proprietar to the manufacturer. Another proceure is to assume a certain circuit topolog an then ientif the circuit elements using measurements [],[]. An alternative proceure is to measure the amittance matrix Y(s) with respect to the transformer terminals. Subsequent rational approximation of Y gives a realization which can be inclue in EMTP-tpe programs b a network equivalent or b convolutions []-[]. This approach gives a terminal equivalent onl, meaning that internal overvoltages cannot be compute. The voltages on taps can still be calculate as long as the are available for measurements. It is note that the resulting moel is linear an so it cannot simulate non-linear effects in the transformer. However, above a few khz the transformer essentiall behaves as a linear component. This paper outlines a proceure for terminal moeling of transformers, base on measurements in the frequenc omain. The proceure is base on a special measurement setup [9] an rational approximation b the Vector Fitting approach []. The approach is emonstrate for a wining rectifier transformer, an for internal overvoltage calculation of a wining assembl. All measurements were performe uring a -ear visit at the Universit of Stuttgart, German. II. RATIONAL APPROXIMATION OF LINEAR MULTI- TERMINAL COMPONENTS The terminal behavior of a linear component can be characterize b its voltage/current relationship efine b the amittance matrix, Y I() s = Y() s V() s () A moel suitable for EMTP-tpe simulation programs can be obtaine b approximating Y with rational functions. Several proceures have been applie for solving this nonlinear least squares problem, incluing polnomial fitting in [], Levenberg-Marquart in [6], an Vector Fitting [] in [7]-[]. In this paper results prouce b a matrix application of Vector Fitting are presente. The matrix application [] prouces an approximation in the form of resiue matrices an a common set of guarantee stable poles which are real or come in complex conjugate pairs. The realization is on the form N Y () s = Rm + + se () s a m= m In aition is prouce the alternative form Y () s C( si A) B se = + + () In orer to guarantee a stable simulation, passivit nees to be enforce. A proceure for passivit enforcement is shown in []. III. MEASUREMENT PROCEURE The elements of Y can be measure one-b-one using a network analzer an a current sensor. It follows from () that element Y jj can be measure b appling a p.u. voltage to terminal j with the remaining terminals groune. The current flowing from the voltage source into terminal j then equals Y jj. Element Y ij equals the current flowing from groun into terminal i. A ver careful measurement setup is require in orer to obtain a sufficientl accurate result. Fig. shows the measurement setup which comprises a network analzer, a connection box with built-in current sensor, an measurement cables between the connection box an the transformer. The results shown in this paper were obtaine using a preliminar version of this setup, which is escribe in full etail in [9]. All measurement were one on the connection box, thus proucing a moel of the transformer an the measurement cables. Reference [9] shows how to mitigate the effect of the measurement cables, but this was not one

2 for the results shown in the current paper. kv kv 6 kv Fig. -wining transformer The 6x6 amittance matrix was measure with respect to the kn kv wining, giving linearl space samples between khz an MHz. The 6 measure elements are shown in fig.. Fig. Measurement setup Amittance [S] IV. MEASUREMENT VALIATION It is recommene to make aitional measurements so as to valiate the accurac of the measure Y-matrix. A first test is to verif that Y is smmetric. In aition, one shoul make measurements with ifferent terminal conitions. E.g., one can measure the open circuit voltage ratio between winings, from high to low, an from low to high. For a -wining transformer this gives x matrices V LH an V HL, respectivel. Next, one can calculate V LH an V HL from Y an compare with the irectl measure quantities. The calculation is one as follows. Consier a - wining transformer partitione into x blocks: I H YHH YHL V = H I L YLH YLL VL From () we get HL HH HL () V = Y Y () V = Y Y (6) LH LL LH V. APPLICATION TO -WINING TRANSFORMER Fig. elements of Y Smmetr was enforce for Y, an the resulting accurac was assesse b calculating the voltage ratio between the kv wining an the kv wining b () an (6). The voltage ratio was next measure an a irect comparison was mae. Fig. shows the calculate an the measure voltage ratio from high to low (9 elements). It is seen that a ver goo agreement has been obtaine, incluing the small elements. Calculate A. Measurements As an example we consier a MVA -wining rectifier transformer, see fig.. Because the transformer was available for onl a ver short time, it was ecie to create a moel with respect to onl two winings, with the terminals of the thir wining ( kv) being open. The neutral of the kv wining was groune uring the measurements. 6 8 Fig. from high to low (V LH )

3 Fig. shows the corresponing result for the voltage ratio from low to high. Again, a goo agreement has been achieve. Calculate Conition number Y HH Y LL 6 Frequenc [Hz] 6 8 Fig. from low to high (V HL ) Fig. 6 shows an expane view of the low frequenc area in fig.. A significant iscrepanc can be observe below khz. This eviation is cause b the kv wining being ungroune. This causes the submatrix Y HH in () to get a high conition number (ratio between largest an smallest singular value) ue to a ver small zero sequence current at low frequencies, thus causing the measurement errors to be magnifie b the matrix inversion. Fig. 7 shows the conition number of submatrices Y HH an Y LL as function of frequenc. Reference [] shows how to overcome this accurac problem b separate measurement an moeling of the zero-sequence sstem. That metho was however not available at the time when the measurements on the -wining transformer were carrie out. Fig. 7 Conition number for Y HH an Y LL B. Rational approximation The Matrix Fitter [] was applie to Y using 6 poles per column, with iterations an enforcement of smmetr. Each element in Y was in the fitting process weighte with the inverse of its magnitue. The resulting approximation is shown in fig. 8. It can be seen that the rational approximation gives a ver goo agreement with the measurements. Magnitue [p.u.] Original Approximation Calculate Fig. 8 Rational approximation of Y Fig. 6 from low to high (V HL ) C. Time omain valiation The rational approximation was inclue in an EMTPtpe program as a Norton equivalent with histor sources upate using convolution with trapezoial integration. Fig. 9 shows a case where terminal of the HV-wining is energize b a step voltage with the other terminals of that wining groune. The applie voltage was measure an taken as input in the simulation b an ieal voltage source. Fig. compares the measure an simulate voltages on terminals,,6 on the low-voltage wining. A ver goo agreement can be observe, incluing the small voltage on terminal.

4 + kv kv V V Voltage [kv] V V Fig. 9 Excitation of high voltage wining..6.8 Voltage [kv] V xv xv xv Time [ms] Fig. an simulate responses Fig. shows a case where a step voltage is applie to terminal of the low voltage wining with terminal open an terminal 6 groune. All terminals of the high-voltage wining are open. The applie voltage was measure an taken as input in the simulation b an ieal voltage source. Fig. compares the measure an simulate voltages on terminals,, of the high-voltage wining. Although a significant eviation emerges over time, the simulation captures the major characteristic in terms of peak value, attenuation, an funamental frequenc components. kv kv Time [ms] Fig. an simulate responses VI. APPLICATION TO WINING ASSEMBLY In principle, the voltage at an point in a transformer can be inclue in the moel b introucing an aitional row an column in Y, provie that the point is available for measurements. Some points, e.g. taps, will never be connecte to an external network. In such situations it is more efficient to calculate the voltage on that point using the voltage transfer function between the transformer terminals an the aitional point []. For a two-wining transformer, the voltage V x is calculate as Vx = H V (7) where H is a x6 row matrix an V is a 6x column vector containing the terminal voltages. As an example a free-staning wining was consiere which containe an internal cliner to represent the capacitive coupling to the core, see fig.. Measurement taps were available so that the voltage at internal points coul be measure. Fig. Excitation of low voltage wining Fig. Internal point (b) in wining Fig. shows the measure transfer function / from the top to the internal point as obtaine b a network analzer with logarithmic space frequenc samples. Also is shown a th orer rational function approximation calculate b Vector Fitting. A ver goo agreement can be observe.

5 Magnitue [p.u.] Original Approximation Fig. Rational approximation of transfer function / Fig. shows with soli trace the measure voltage on terminal b when appling a near step voltage to terminal a. The measure voltage on terminal a was taken as a known quantit an the voltage on terminal b was simulate, shown with otte line. A ver goo agreement can be observe. Fig. 6 shows the initial part of the response. (All measurements were one using an active, high impeance voltage probe. The setup in fig. was not use). Voltage [V] Voltage [V] Time [µs] Fig. an simulate response Time [µs] Fig.6 Expane view of initial response VII. CONCLUSIONS This paper has emonstrate the suitabilit of Vector Fitting (VF) for high frequenc moeling of power transformers. A terminal moel was first obtaine b measuring the amittance matrix Y in the frequenc omain using a network analzer. Y was next subjecte to rational approximation using VF, thereb proucing a realization which is compatible with EMTP-tpe simulation programs. Application to a -wining transformer emonstrate the accurac of the proceure. It was also shown how to appl VF for the calculation of transferre overvoltages at open points/terminals. Both the VF routine an the matrix fitter routine are freel available from the author in Matlab coe. ACKNOWLEGMENTS This work was supporte in full b the European Commission uner the Fifth Frame Programme b a G Research Non Nuclear Energ Programme Marie Curie Fellowship. The author is inebte to Prof. Kurt Feser an the staff at the Institut für Energieübertragung un Hochspannungstechnik, Universit of Stuttgart, for proviing laborator facilities an support of this project. REFERENCES [] P. I. Fergesta an T. Henriksen, "Transient oscillations in multiwining transformers", IEEE Trans. Power Apparatus an Sstems, vol. 9, pp. -9, 97. [] R. C. egeneff, "A general metho for etermining resonances in transformer winings", IEEE Trans. Power Apparatus an Sstems, vol. 96, no., pp. -, March/april 977. [] C. Anrieu an. Boss, "A wie frequenc range moel for a MV/LV core transformer", Proceeings of IPST', pp. 8-86, June -8, Rio e Janeiro. [] T. Noa an S. Yokoama, "Accurate moeling of core-tpe transformers for electromagnetic transient stuies", IEEE Trans. Power eliver, vol. 7, no., pp , October. [] Q. Su, R.E. James an. Sutanto, "A z-transform moel of transformers for the stu of electromagnetic transients in power sstems", IEEE Trans Power Sstems, vol., no., Feb 99. [6] A. Morche, L. Marti, an J. Ottevangers "A high frequenc transformer moel for the EMTP", IEEE Trans. Power eliver, vol. 8 no., pp. 6-66, Jul 99. [7] B. Gustavsen an A. Semlen, "Application of Vector Fitting to the State Equation Representation of Transformers for Simulation of Electromagnetic Transients", IEEE Trans. Power eliver, vol., no., pp. 8-8, Jul 998. [8] M.J. Manahi an R. Thottappillil, "Transfer of lightning transients through istribution transformer circuits", Proceeings of ICLP', pp. -, September -6, Cracow. [9] B. Gustavsen, "Wie ban moeling of power transformers", paper accepte for publication in IEEE Trans. Power eliver. [] B. Gustavsen, Frequenc epenent moeling of power transformers with ungroune winings", paper accepte for publication in IEEE Trans. Power eliver. []B. Gustavsen an A. Semlen, "Rational Approximation of Frequenc omain Responses b Vector Fitting", IEEE Trans. Power eliver, vol., no., pp. -6, Jul 999. []B. Gustavsen, "Computer coe for rational approximation of amittance matrices", IEEE Trans. Power eliver, vol. 7, no., pp. 9-98, October. []B. Gustavsen an A. Semlen, "Enforcing Passivit for Amittance Matrices Approximate b Rational Functions", IEEE Trans. Power Sstems, vol. 6, no., pp. 97-, Februar. []A.S. Morche, L. Marti, R.H. Brierl an J.G. Lacke, "Analsis of internal wining stresses in EHV generator step-up transformer", IEEE Trans. Power eliver,vol., no., pp , April 996.