Magnetic Cores Modeling for Ferroresonance Computations using the Harmonic Balance Method

Transcription

1 1 Magnetic Cores Modeling for Ferroresonance Computations using the armonic Balance Method N. A. Janssens, Senior Member, IEEE Abstract-- The determination of the ris of ferroresonance in actual V or MV networs and the design of damping devices need the use of accurate models. In this frame, the nonlinear parts of the circuits, i.e. the magnetic cores, require a special attention. This paper describes a modeling of the magnetization curve and the core losses appropriate for ferroresonance computations using the harmonic balance method. Tests performed in order to get the parameters are discussed. Index Terms Ferroresonance, armonic balance method, igh Voltage, ysteresis, Magnetization, Modeling, Saturation, Voltage Transformer F I. INTRODUCTION ERRORESONANCE is a nonlinear phenomenon occurring in electrical circuits involving at least one or several saturable reactors, capacitances and a power supply. Often, the saturable reactors are inductive potential transformers. This phenomenon is characterized by the possible existence of several stable regimes. These regimes may be either periodic, with a base frequency equal to the power supply frequency or to a sub multiple of it, pseudoperiodic or chaotic. The harmonic balance method, which is a particular case of the Galerin method, has proven powerful for the study of periodic, but also pseudo-periodic regimes, in the above mentioned circuits [e.g. 1-5]. For this method, a limited Fourier series is used to represent the periodic (or pseudoperiodic behavior of the state variables. For instance, the expression of the magnetic flux in the magnetizing branch of the nonlinear inductance is : φ( t Φ + Φ, c cos ( ω t + Φ, s sin( ω t (1 ( K The corresponding Fourier coefficients for the current in this branch are given by integration from the magnetic i φ of the inductance : characteristic ( N. Janssens is with ELIA, Belgian Transmission System Operator, 15, Rodestraat, 163 Linebee, Belgium ( noel.janssens@elia.be and with the University of Louvain at Louvain La Neuve, Belgium. T I, c i( φ( t cos( ω tdt T ( and with similar expressions for the sine terms I, s and for the DC component I. The coefficients I, c, I, s, I are nonlinear functions of all the coefficients Φ, c, Φ, s, Φ. The harmonic balance method consists in introducing the limited Fourier series in the differential equation of the circuit and forcing to zero the contributions to each considered harmonic component. So, an algebraic set of nonlinear equations in the Fourier coefficients is obtained and may be solved by using a general purpose routine. The linear part of the circuit may be represented by its Thevenin equivalents (voltage source E and complex impedance Z for the different frequencies of the set K. The use of Thevenin equivalents allows reducing the number of equations to be solved : there is only one (complex equation for each harmonic component for each nonlinear component, instead of one equation for each harmonic component for each reactive element (linear or nonlinear. It also facilitates the choice of an initial approximation of the solution to be introduced in the computation program. The use of the harmonic balance method may be extended to find directly the domain limits in some parameter space of the various inds of ferroresonant regimes. This extension consists in adding one equation to the above mentioned algebraic set equating to zero the determinant of the Jacobian of the harmonic balance equations relative to a set K (not necessarily identical to the set K [1-4].. II. BASIC MODELING PRINCIPLES To get accurate results when studying a practical situation, all the relevant components of the system must be modeled adequately. In particular, the nonlinear reactors need a special attention because the ferroresonance phenomenon pushes these elements in high saturation beyond the validity domain of the usually available data. The present paper is devoted to the presentation of a saturable reactor modeling able to get

2 quantitative accurate results when used to study practical situations. The iron core magnetization characteristic has of course a great influence, since the ferroresonance phenomena are strongly related to this nonlinear function. Very often, the question to be answered for estimating the ris of ferroresonance or designing a damping circuit is : what is the lower bound of the voltage source for which a given ind of ferroresonant regime exists? This question has an energetic bacground : to what extent can the voltage source bring enough energy to compensate the losses of a ferroresonant regime? From computation and tests, it may be concluded that the system losses have a great importance on the lower limit of the voltage source interval where a given ferroresonant regime exists. A precise modeling of these losses is of prime importance. It is to be noted that, when the voltage is not close to these interval limits, the waveforms are not very sensitive to these losses. Consequently, a validation solely based on waveform comparisons is not very relevant. For the time simulation of the system, a static magnetic hysteresis model, lie those described in [6,7] may be used, associated with a conductance to represent the eddy current losses and series resistances and leaage inductances. Such a model could also be used for the harmonic balance method and proved to provide good results for a low voltage laboratory test [8]. owever, the computation time may be greatly reduced by using a univocal function i ( φ to represent the magnetic characteristic. In this case, the terms of the Jacobian may be expressed by the harmonic components of d i ( ( ( φ( t i φ t [8]. For instance, let us consider the term d φ I 1, c Φ, c successively : I 1, c Φ T, c T T 1 i with 1,, 1 T d i φ d φ Φ, c ( φ cos( ω t cos( 1ω tdt I 1 1+, c + I 1, c. Using (1, we obtain cos( 1ω tdt Section IV will pay attention to the measurement of the magnetic characteristic. In order to have a clean convergence of the iteration process, the function i ( φ must be continuous. On the other hand, the modeled magnetic characteristic must be very close to the measured curve. For this purpose, an analytical expression lie, for instance, a polynomial will, in general, show discrepancies with respect to the real curve or exhibit oscillations in the ( i φ plane. Therefore, we found more adequate to use a parabolic spline, consisting of successive parabola segments with a continuous slope at the nodes. Section IV will show an example of such a modeling. Besides the magnetization curve, another important aspect is the modeling of the core losses. During the computation process to solve the algebraic system of equations, at the beginning of each iteration, a set of Fourier coefficients is given from the previous iterations (for the first iteration, the user or an auxiliary routine provides an initial point. This gives a flux (and voltage behavior of the core magnetizing branch. From there, it is possible to determine the value of two linear conductances G F and G such that the losses in these conductances are equivalent respectively to the eddy current losses and the hysteresis losses according to an appropriate model. Doing so, the waveforms will differ slightly from those obtained using a model to be used for a time simulation. owever, these differences will be very small considering that the cores are made of soft magnetic materials with limited losses and that the saturation has a much larger effect on the waveform. The important point is that, globally for the whole oscillation period, the core losses are accurately modeled. The computation of the conductances G F and G from basic data is developed in section III and an example of core losses as a function of the saturation is shown in section IV. III. MAGNETIC CORE LOSSES Experimental studies have shown that the power W dissipated in silicon steel plates, for a distorted wave and in the frequency range under interest here, can be written as the sum of two terms relative to the hysteresis lossesw and the eddy current losses W F : W W + W F (3 of the form : W f φ (4 w loops ( U ( min,φ max F WF rms The symbols W (upper case letters designate the powers (energy per second while the w (lower case designate the energy dissipated per cycle. In (4, the sum deals with all the hysteresis loops (major and minor, f is the fundamental W (5 frequency of the oscillation, φ min and φ max are the extreme values of the flux. In (5, U rms is the r.m.s. voltage across the magnetizing branch. A. Eddy current losses We suppose that the function W F giving the eddy currents losses for a sine wave of the flux (directly related to the voltage at the grid frequency f as a function of its pea value φ max is nown. The associated eddy current conductance G F ( φ max is then related to W F by : ( ( ( π f max W φ F φmax GF φmax (6 Let us now consider a distorted wave characterized by its frequency components { Φ } (pea values where the numbers may be fractions. The pea value φ max of a sine wave corresponding to the same r.m.s. value of the voltage is given by :

3 3 φ Φ (7 max ence, by taing : G Φ F GF (8 an expression of WF GF Urms of the form (5 is obtained corresponding to the data (6 for a sine wave. Measurements on magnetic cores showed that a polynomial of the third degree is able to represent the function G F with a sufficient accuracy. B. ysteresis losses The purpose is to determine a conductance G valid for any periodic evolution using easily obtainable data. These data are : a the hysteresis losses for a symmetric evolution with respect to the origin for a pure sine applied voltage. These φ max losses are expressed using a conductance ( G function of the maximum flux of the evolution for the grid frequency f ω π. The loss for one cycle is related to the conductance by the relation : ( φ π ω φ G ( φ w max max max (9 For practical cases, the representation of the function G by a third order polynomial has been found to be sufficiently accurate. b the smallest value φ of the half amplitude (pea to pea of a closed loop such that both extremes are located on the limit cycle. This value illustrates the speed to approach to the other side of the limit cycle after a change in the sense of the flux and current variation. This value may be estimated by the examination of the hysteresis loops. For lac of this, one may choose half of the flux of a point located in the saturation nee. To represent a magnetic evolution, we will use the Preisach model, taing the flux as independent variable and assuming that the Gire coefficient is infinite [8 pp 8-88]. In this frame, a periodic evolution will be composed of a set of closed hysteresis loops. For a given periodic evolution of the flux (available at the beginning of each iteration when using the harmonic balance method, a first computation step is to associate all the extreme values by pairs, according to the mechanism of the Preisach model. The loss for one cycle is the sum of the losses for each individual loop. The loss of one loop is a function of its minimum and maximum values, expressed by introducing a function w 1 : w w1 in,φmax (1 In the following, we will use an alternative formulation maing use of the mean flux φ m and the half width φ d of the loops : w w φ, φ (11 ( m d Since, in general, the loss for unsymmetrical loops is not nown, we must restrict ourselves to the use of the φ max function ( w given by (9 for symmetrical loops. We will assume that : a for φ d > φ (large loops, the extreme values are located on the limit cycle. ence, 1 w1 in, φmax ( w ax w in 1 or w, φd ( w + φd w φd (1 b for φ d < φ (small loops, we will assume that w is the product of the loss of a symmetrical loop and a function f to tae the unsymmetrical character into account : ( φ φ w ( φ f ( φ w The continuity of the loss for φ d φ defines the function f : m, d d m (13 ( 1 f (14 w ( φ φ ( φ φ m + w m w ( φ φ d φ, the function w ( φ m, φd c for is continuous. owever, its partial derivative with respect to φ d is not. In order to have a clean convergence of the iterative computations including this model, in a small band of φ d around φ, an interpolation using a third order polynomial is used in order to have the continuity of the function and its derivative. Finally, having defined the function w ( φ, φ m d for all loops by means of the available data as explained here above, the conductance G is given, for an evolution whose fundamental frequency is f, by : G π f f Φ loops w ( φ, φ IV. MEASUREMENTS FOR PARAMETER IDENTIFICATION m d (15 The ferroresonance phenomena are associated, in general, with a very high saturation of the magnetic cores. ence, the magnetic flux spreads largely outside the iron cores. As a consequence, the core and windings design (location, shape, as well as the transformer tan and the internal shields play a role in the high saturation range. Therefore, the best way to measure the model parameters is to use a complete device fed through the V primary side. Of course, this requires measurements in a V laboratory. If this is not possible, measurements can be done on a magnetic core fitted with two low voltage windings (one for the magnetizing current, the second one for the voltage and flux measurement. For the high saturation region, the asymptotic behavior of the magnetization curve may be approached by a straight line obtained as follows : the slope is the transformer V primary winding in air minus the primary lea inductance ; the flux offset for zero current corresponds to the full magnetization of the core. Some interpolation must

4 4 be done between the highest flux and current obtained by the low voltage measurements and the asymptotic straight line of the magnetic characteristic. owever, the extrapolation of the losses in the high saturation region may lead to significant errors. The results given below relate to a voltage transformer designed for the 45 V voltage level. The measurements were done in a V laboratory by feeding the device either by the primary or by the secondary winding. Special care was taen for the measurements : shielding of the connection cables between the measurement points and the recording devices, digital synchronous recording with a sampling rate of 5 z (1 points per period, same input impedance of the various channels, chec of the transmission delay of the channels. The losses were also measured using a wattmeter. For unsaturated periodic evolutions, the measurement given by the wattmeter matched very well with the computation based on the voltage and current waveforms. owever, for highly saturated evolutions, the power factor was about. and the error on the measurement of the wattmeter reached about 5 %. Since the primary winding has tens of thousands of turns, its capacitance may not be neglected. Figure 1 shows the flux current recording for the nominal voltage applied to the primary. The current I 1 is the primary current, the flux being obtained by integration of the secondary voltage. It may be seen that the behavior of this inductive voltage transformer is essentially capacitive. Taing into account the primary winding capacitance (modeled by a shunt capacitance in parallel with the magnetizing branch, the flux - corrected current I 1 c exhibits a more usual hysteresis loop shape. In order to determine the magnetization curve and the losses, an adjustable sine voltage was applied to the primary winding. Its amplitude covered the interval from zero to about 3 times the nominal voltage. Primary voltage and current waveforms were recorded as well as the secondary and tertiary winding voltage. Figure shows the relation between the maximum flux and maximum current of these periodic waveforms, either by integrating the secondary voltage to obtain the flux (circle marers, or the tertiary voltage (square marers. The first one is higher than the second one because the secondary winding is located closer to the primary than the tertiary. Similar tests were done by feeding the voltage transformer through the secondary winding. In this case, the capacitance of the primary bushing loaded greatly the voltage transformer. The current in the magnetizing branch is then obtained by the difference between the secondary and the primary current. ence, for evolutions in the unsaturated region, the precision on the result is rather bad. The magnetic characteristic obtained with the secondary winding feeding and using the tertiary flux is also shown on figure (cross marers. Figure 3 illustrates the construction of the parabolic spline modeling the magnetization curve. The step curve represents the slope (current vs. flux of the broen line of figure obtained for the primary winding feeding and the secondary winding flux. Nodes (stars on fig. 3 are chosen, usually among the data points, in order to build a broen line whose integral approximates closely the magnetization curve. Starting from an initial slope, this broen line is constructed by a Fig. 1. Magnetic flux current evolution flux current Fig.. Magnetization curve progression from a zero flux in the direction of the saturated region. In the small flux region, the broen line does not attempt to approximate the step function, since the hysteresis effect induces a lowering of the slope that is to be neglected in the frame considered here. The primary winding resistance R 1 and lea inductance L 1 with respect to the secondary and tertiary windings may be computed by regression : from the recordings relative to the primary winding feeding, the parameters R 1, α and 1, α L, where α or 3 according to the reference to the secondary of the tertiary winding, are chosen in order to minimize the integral : T d i J u R i L 1 u dt 1 1, α 1 1, α α dt Figure 4 shows the results for the inductance L 1 (circle marers for L 1, and square marers for L 1, 3. It may be seen

5 5 that the lea inductance increases with the saturation level. This results from the dispersion of the flux outside the core for a high saturation level. It may also be seen that the leaage of 1 75 series inductance secondary or tertiary voltage Fig. 4. Primary lea inductance.1.9 shunt conductance secondary or tertiary voltage Fig. 5. Shunt conductance for the modeling of the core losses Fig. 3. Construction of the parabolic spline the primary winding with respect to the tertiary winding is greater than with respect to the secondary winding, because this last one is closer to the primary winding. The recorded waveforms allow to compute the losses. From the primary current and the secondary or tertiary voltage, the core losses may be computed as a function of the voltage amplitude. The ratio of the losses and the square of the voltage give an equivalent conductance G. It may be seen on figure 5 (circle marers for the secondary voltage, square marers for the tertiary voltage that this conductance is far from being constant. V. CONCLUSIONS The computation of the ris of ferroresonance and the design of damping devices for practical situations need an accurate modeling of the system. In particular, due to the high saturation level reached in the iron cores, the saturable reactors and transformers require a special attention. In this frame, the most important aspects of these components are the saturation curve and a good representation of the losses. A suitable model for the magnetization curve is a parabolic spline, made of a succession of parabola segments with a continuous derivative. This allows a large flexibility to reproduce very closely the experimental data. On the other hand, the continuous character of its derivative leads to a clean convergence process of the ferroresonant regimes computations.

6 6 When using the harmonic balance method, the core losses may be modeled by two linear conductances, one for the eddy currents and the other for the hysteresis losses. Their value is adapted at each iteration of the Fourier coefficients computation to tae into account the amplitude and the waveform of the magnetic flux behavior. This modeling succeeded to get an accurate determination of the domains of existence in some parameter space of the various ferroresonant regimes for V and MV systems, as shown in [, 3, 5]. VI. REFERENCES [1] N. Janssens, «Calcul des zones d'existence des régimes ferrorésonants pour un circuit monophasé», IEEE Canadian Communications and Power Conference, Montréal 18- Oct. 1978, Cat No 78 C REG 7, pp [] N. Janssens, A. Even,. Denoël, P-A. Monfils, «Determination of the ris of ferroresonance in high voltage networs. Experimental verification on a 45 V voltage transformer», Sixth International Symposium on igh Voltage Engineering, New Orleans, Aug 8 - Sep 1, 1989, paper 11.3 [3] N. Janssens, V. Vanderstoct,. Denoël, P-A. Monfils, «Elimination of temporary overvoltages due to ferroresonance of voltage transformers : design and testing of a damping system», CIGRE Session 199, Paris, Report 33-4 [4] N. Janssens, Th. Van Craenenbroe, D. Van Dommelen, F. Van De Meulebroee, «Direct calculation of the stability domains of threephase ferroresonance in isolated neutral networs with groundedneutral voltage transformers», IEEE Trans. on Power Delivery, Vol 11, n 3, , July (presented at the IEEE PES Summer Meeting 1995, Portland, paper 95 SM 4- PWRD [5] T. Van Craenenbroec, D. Van Dommelen, C. Stucens, N. Janssens, P.A. Monfils, «armonic balance based bifurcation analysis with full scale experimental validation», IEEE 1999 Transmission & Distribution Conference, New Orleans, USA, 1999, CD-ROM (6 pages [6] N. Janssens «Mathematical modelling of magnetic hysteresis», Proceedings of the COMPUMAG Conference on the computation of magnetic fields, Oxford, 31 March - April 1976, pp [7] N. Janssens, «Static models of magnetic hysteresis», IEEE Transactions on Magnetics, Vol MAG-13 n 5, Sept 1977 pp [8] N. Janssens, «ystérésis magnétique et ferrorésonance. Modèles mathématiques et application aux réseaux de puissance», PhD thesis, UCL (Université Catholique de Louvain, Belgium, June 1981 [9] T. Naata, Y. Ishihara, M. Naano, «Iron losses of silicon steel core produced by distorted flux», Electrical Eng. in Japan, Vol 9, n 1, 197, pp 1- VII. BIOGRAPY Noël Janssens (S 1 was born in Louvain, Belgium, in e is Electrical Engineer from the University of Louvain (1971 and obtained a Ph.D degree in 1981 (modeling of magnetic hysteresis and study of ferroresonance. From 1981 to 1983, he wored at ACEC (Charleroi as head for R&D in the On Load Tap Changer department. From 1978 to 1981 and from 1984 to 1995, he was with Laborelec, where his main fields of interest were the modeling, simulation and control of power systems. Since 1996, he is at the Belgian National Dispatching. e is also teaching at the University of Louvain (Louvain La Neuve in the Electrical Engineering Department.