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Application of Vector Fitting to State Equation Representation of Transformers for Simulation Of 834 IEEE Transactions on Power Delivery, Vol. 13, No. 3, July 1998 APPLICATION OF VECTOR FITTING TO STATE EQUATION REPRESENTATION OF T FOR SIMULATION OF ELECTROMAGNETIC TRANSIE Bjsrn Gustavsen" Adam Semlyen Department of Electrical and Computer Engineering University of Toronto Toronto, Onta.rio, Canada M5S 3G4 * On leave from the Norwegian Electric Power Research Institute (EFI), Trondheim, Norway. Abstract - This paper presents an efficient methodology for transient model can give unstable simulation results, depending on the modeling of power transformers based on measured or calculated connected network. frequency responses. The responses are approximated with rational In this paper we show that the recently developed method of functions using the method of vector fitting. The resulting model is vector fitting [3] is well suited for finding the needed rational dynamically stable since only negative poles are used, and inputatput function approximations. This method produces a state equation stability is assured by a novel adjusting procedure. The methodology is realization (SER) by solving an overdetermined set of equations as used to formulate two different transformer models: I)a full phase a least squares problem. There is no need for predefining partial domain model realized column-by-column, and 2) a model utilizing fractions, and there are inherent limitations only if a fit of very modal diagonalization. The application of the models is demonstrated order is desired for frequency responses with many peaks. for a 41 0 MVA generator step up transformer. high We use vector fitting to formulate two transformer models based 1 INTRODUCTION on the short circuit admittance matrix Y : The transient behavior of power transformers is very difficult 1. Fitting Y column-by-column in the phase domain to model accurately. The currentlvoltage characteristics at the 2. Applying modal decomposition to 3 x 3 blocks of Y terminals are strongly frequency dependent, and usually involve resonance phenomena. As this behavior cannot be deduced from Dynamic stability is achieved by forcing all poles to lie in the the power frequency data provided by the manufacturer, one has to left half plane. Input-output stability is ensured by adding a acquire the needed frequency responses either from detailed correction to the resulting fit. The correction is established by calculations based on internal geometry, or from terminal examining the eigenvalues of the fitted Y as function of measurements. Such frequency responses constitute the basis for frequency. model formulation. The methodology is demonstrated for a 410 MVA two- The frequency responses can be included in the time domain winding star-delta connected transformer. calculations by means of numerical convolution, but the resulting In this work we neglect any nonlinear phenomena, such as model would be computationally very time consuming. Therefore, saturation effects and hysteresis that may have significance at one will normally seek to replace the original frequency responses lower frequencies. If nonlinear effects need to be taken into by rational function approximations, as this enables a more account they have to be added on externally to the model as in [2]. efficient time domain implementation. Rational function 2 METHODOLOGY approximations can in principle be calculated by fitting rational polynomials to the data [ 11. However, the inherent ill-conditioning 2.1 Model formulation limits this method to low order approximations. Also, some poles of the resulting fit may turn out to be unstable. A way of overcoming these problems is by iteratively fitting partial fractions to the data [2].This methodology is complicated in the sense that the frequency responses must be scrutinized in advance to decide how many and which type of fractions to use (real or complex conjugate pairs). Also, the success of the method is dependent on the starting values used in the iteration. A sufficient requirement for the resulting transformer model to I I Ivn give a stable simulation is that all of its poles lie in the left half plane (dynamic stability), and that all eigenvalues of the resulting Fig. 1 n-terminal transformer admittance matrix lie in the right half plane (input-output stability). Many of the proposed fitting procedures hlfill the first A transformer will in the following be considered as an n-terminal requirement, but the last one can in practice only be assessed after linear component, as shown in figure 1. The current/voltage the fitting has been carried out. The problem of input-output relations at the terminals can in the frequency domain be stability is highly relevant in the case of transformers, as the expressed by the admittance matrix Y : internal losses are very small. Thus, even a quite accurately fitted i=Yv (1) where i and v are vectors of length n, and Y is an nxn complex matrix. It follows from (1) that the columns of Y can be measured one-by-one by successively applying a sinusoidal voltage PE-673-PWRD-0-03-1997 A paper recommended and approved by to one of the terminals, with the other terminals connected to the IEEE Transmission and Distribution Committee of the IEEE Power ground. Engineering Society for publication in the IEEE Transactions on Power A transformer model could be obtained by representing each Deliveiy. Manuscript submitted October 18, 1996; made available for element of Y with a state equation realization (SER). However, printing March 26, 1997. the time domain simulations would be somewhat tedious as this 0885-8977/98/$10.00 0 1997 IEEE 835 model involves n2 SERs. The computational burden can be 2.2 Rational function approximation reduced a factor of about 2 by fitting Y columnwise using the by In model I we approximate each column of Y by a finte order same set of poles in each column [3]. This can easily be done Yi state equation realization (SER) using the method of vector fitting adopted in this paper. In principle, one could increase the computational speed even Yj = c (sP-A)--' B+D (5) more by applying modal diagonalization to Y, assuming a constant, real transformation matrix. However, this would lead to calculated. using the method of vect.or fitting developed in [3]. inaccurate results as the transformation matrix is in general both Similarly, in model 2 we use vector fitting to find a SER for each frequency dependent and complex. diagonal element in the blocks qy. Morched et al. [2] proposed to subdivide Y into 3x 3 blocks, A, described in [3], the fitting process involves two stages : one for each transformer winding. For a two-winding transformer, 1. Calculation of poles (A-matrix) from "starting poles" equation (1) can be written 2. Calculation of C with known poleis (B is normalized and fixed) As starting poles we use real, negative poles that are linearly [:I=[::: [:I=[::: :][:;] distributed over the frequency range: where the frequency where subscripts 1 and 2 denote winding 1 and 2, respectively. responses are to be fitted. The preference here for linearly spaced Then they averaged the diagonal and off-diagonal elements within poles over: logarithmically spaced poles as used in [3] is due to the each block, thus making each block a balanced matrix of the form responses appearing smoother when plotted on a linear frequency base. The: new set of poles calculated at stage 1 may in general (3) contain unstable poles. These are forced to be stable by switching [!t :] the sipof their real parts, before proceeding with stage 2. This allowed them to apply modal decomposition to each block, --2.3 Stabikation procedure using a constant, real transformation matrix. However, the use of balanced submatrices is not always Fitting the transformer frequency responses with stable poles is a acceptable : three-legged transformers may exhibit symmetry only necessary but not sufficient condit.ion for obtaining a stable about the middle leg, and transformers utilizing a delta or zigzag simulation. In addition, we must require that the transformer winding will have off-diagonal blocks with cyclic symmetry. For model behave as a passive network as seen from its terminals. instance, for the star-delta connected transformer investigated in From circuit theory it is known that passive network behavior this work we found Y to have the following symmetry properties : implies positive realness of all eigenvalues of the fitted admittance matrix Yjt(o).To ensure positive realness we use the following stabilization procedure, which contains two stages : --2.3.1 Stabilizing the measured data where 1 and 2 denote the star and delta winding, respectively. At each frequency point, diagonalize Y into its eigenvalues A The problem of cyclic off-diagonal blocks can be overcome by with eigenvectors T. All real parts of A that are negative (due tci using Fortescue transformation ( +-0 components). This measurement errors!) are replaced with zero. With the modified1 transformation will perfectly diagonalize cyclic blocks and eigenvalu.es and the original eigenve:ctor matrix, retrieve a more: approximately diagonalize blocks with mid-leg symmetry. realistic Y-matrix. Therefore, the model will also work if there is no star-delta or --2.3.2 Stabilizing the rational functionapproximation zigzag connection. Diagonal blocks are never cyclic, but may have a significant 1. From the §ER, calculate Y$t(o:) and its eigenvalues A(w) mid-leg symmetry. An analysis (see Appendix A) has shown that over the frequency range of interest. Find the smallest reall increased accuracy is achieved for such blocks when using ctp0 part (most negative) of each eigenvalue within this frequency transformation, instead of the prebalancing procedure used in [2]. range, and store in the diagonal m.atrix A,, . Replace al'L Thus, we propose the following two transformer models : positive elements in A,, with 0. Also, find the highest 1. Columnwise realization of Y (phase domain) frequency point a* above which all eigenvalues are positive real (if such a point exists).
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