INTERNATIONAL J OURNAL OF M ULTIDISCIPLINARY S CIENCES AND ENGINEERING , VOL . 5, NO. 11, NOVEMBER 2014 Modeling of Multiport Systems via a ModeRevealing Transformer Ehsan Elahi, Hamid Keivani and Alireza Alesaadi Abstract—Employing an admittance matrix in the frequency domain by rational functions for power transformers is a wellknown method which improves calculation efficiency. This model must be passive in order to avoid unstable time domain simulations. All of the methods have made efforts to overcome the problem of preserving the passivity of the final model. In this paper a similarity transformation matrix which better reveals the eigenvalues of the admittance matrix is presented. The chosen transformation preserves the passivity and symmetry of the original data. Keywords— Electromagnetic Transients, Network Equivalent, Passivity Enforcement, Pole-Residue Model and Vector Fitting I. INTRODUCTION F requency-dependent modelling of linear devices and systems is widely applied in the technical fields of power systems, microwave systems, and high-speed electronic systems [1]. A large number of methods have been proposed for construction of a network equivalent. These methods can be classified into time domain and frequency domain methods [2]. The solution in the frequency domain is essentially aimed at the identification of rational functions that approximate the admittance matrix of the external system seen from boundary bus(es). Among the various methods of fitting, the vector fitting (VF) has proved its efficiency and accuracy in different applications [3]. VF is essentially a robust reformulation of the sanathanan – koerner iteration [4], using rational basis functions instead of polynomials and pole relocation instead of weighting [5]. The model identification involves rational fitting of the given terminal responses, leading to models that can be included in EMTP-type programs by recursive convolution [6] or by a lumped electrical network [7]. This work was supported in part by the Islamic Azad University, Kazeroon Branch, Kazeroon, Iran. E. Elahi is with the Department of Electrical Engineering, Kazeroon Branch, Islamic Azad University, Kazeroon, Iran, (Ccorresponding Author; Email: [email protected]). H. Keivani is with the Department of Electrical Engineering, Kazeroon Branch, Islamic Azad University, Kazeroon, Iran (Email: [email protected]). A. Alesaadi is with the Department of Electrical Engineering, Kazeroon Branch, Islamic Azad University, Kazeroon, Iran (Email: [email protected]). [ISSN: 2045-7057] Past experiences has shown that simulations involving a fitted admittance matrix Y can sometimes lead to an unstable simulation, even though the elements of Y have been fitted using stable poles only [8]. In the other words, although the model obtained via vector fitting has guaranteed stable poles, the model may result in unstable simulations because the passivity of the model is not assured. Thus assess the passivity characteristics of the model is needed. For this purpose, it is common practice to calculate the eigenvalues of a Hamiltonian matrix which is associated with a state-space formulation of the model [9]. A difficulty with the Hamiltonian matrix is its size. Its dimension is two time the number of system states, which makes the computation of eigenvalues time-consuming for large models [10]. A second disadvantage is that eigenvalues corresponding to crossover frequencies are not exactly imaginary as they are associated with a small real part (noise). In the case of multi terminal problems, the output response is sometimes strongly depend on the distribution of the applied input on the terminals. Such behavior is often observed in the low-frequency range, due to large differences between shortcircuit and open-circuit modal characteristics. Direct application of standard modeling techniques to such cases gives a model with inaccurate representation of the small modes. This problem is addressed by the method known as modal vector fitting (MVF) [11] and modal perturbation (MP) [12] but comes at the cost of long computation times and increased memory requirements. In this paper, we introduce a new technique for the multiport modeling of systems with large differences in modal characteristics which overcome the high computational costs of MVF. We show how to compute a suitable transformation matrix using the given data only, and how to preserve the essential information of passivity and symmetry. Then transform the extracted model back to the physical domain via the inverse information. The advantages of the new approach are demonstrated for frequency-dependent network equivalent (FDNE) modeling and wideband modeling of transformers, focusing on accuracy and computational efficiency [14]. II. INPUT-OUTPUT TRANSFER FUNCTION The modelling starts from a given port-admittance matrix Y(s), which defines the relation between port voltage v and current i. This matrix can be obtained via calculations or measurements www.ijmse.org 21 INTERNATIONAL J OURNAL OF M ULTIDISCIPLINARY S CIENCES AND ENGINEERING , VOL . 5, NO. 11, NOVEMBER 2014 (1) The admittance Y is a symmetric matrix, satisfying the symmetry property (2) (2) It is assumed that a rational model for Y has been identified which satisfies (2). This can be easily achieved by fitting the elements of Y using the pole relocating algorithm known as vector fitting (VF)[14]. This leads to a pole-residue model (3) which can be expanded into a model in standard state-space form (4). (3) (4) Physicality of the model leads to the following requirements: Y is a symmetric matrix. Hence, { }, D, and E are symmetric. D and E are real matrices. The pole and residues are real or come in complex conjugate pairs. The poles are in the left half plane [12]. The passivity constraint in terms of the transfer matrix (4) is equivalent to being Positive-Real (PR) which means that: (5) Where denotes complex conjugate. This constraint can be verified by ensuring that all the eigenvalues of the real section of the admittance matrix are nonnegative in the whole region of frequency. (6) Where: G(j ) = eal (Y(j )) III. MODE – REVEALING TRANSFORMATION A. Rationale Any nonsingular matrix T will preserve the eigenvalues of a matrix Y by the similarity transformation subjecting to rational modeling by VF and subsequent passivity enforcement with relative error control, one can therefore achieve that the small eigenvalues become more accurately represented. The model of must next be transformed back to the original (physical) basis to give a model representation for Y. B. Preservation of Symmetry and Passivity Diagonalizing Y gives (8) In the general case with a frequency-dependent T, it is not possible to fit the eigenvalues using a stable rational model. Instead, we will have to use a constant, real transformation matrix and apply it at all frequencies. This gives us an eigenvalue decomposition matrix at the identified frequency . To obtain a real matrix, we follow Brandwajns approach [16] of rotating each eigenvector by an angle to minimize its imaginary part in the least – squares sense. For each rotated eigenvector (9) The imaginary part Im (10) Differentiating (10) with respect to = 0 gives the final solution [16] [ISSN: 2045-7057] and solving for df ⁄ dθ (11) Since can present a minimum and maximum value, we evaluate (10) at and at to determine the correct value. Finally, we discard the imaginary part . A difficulty with the proposed approach so far is that the transformation will neither preserve the symmetry nor the passivity of Y. to overcome this problem, we will approximate with a real, orthogonal matrix Q. Since orthogonality implies , we obtain for (7) (12) Using the rule for the transpose of a matrix product, we obtain (7) By making a proper choice of T, it is possible to achieve that the small eigenvalues become more observable in the elements of than in Y. we will therefore refer to the transformation as a mode revealing transformation (MRT). By has norm (13) Which proves that the transformed matrix is symmetrical. Q can conveniently be chosen as the matrix Q of the QR factorization, where Q is orthogonal and R is an www.ijmse.org 22 INTERNATIONAL J OURNAL OF M ULTIDISCIPLINARY S CIENCES AND ENGINEERING , VOL . 5, NO. 11, NOVEMBER 2014 upper triangular matrix. However, in our case, we simply try to calculate a Q which is as close to T as possible. This problem is addressed by the orthogonal procrustes problem (14) Which seeks to calculate the orthogonal matrix Q which rotates B as close as possible into A.[15] With the obtained transformation matrix Q, we calculate the matrix using (7) which is fitted by the rational model (3) using vector fitting (VF). C. Passivity Enforcement: Fast Modal Perturbation In [17], it was proposed to perturb the rational model such that the eigenvalues of Y are perturbed in relation to their size. Diagonalizing Y gives (15) (16) (22) IV. WIDEBAND TRANSFORMER MODELING A. Frequency Sweep Measurements We use frequency sweep measurements for wideband modelling of power transformers. In this paper the 30 – KVA Define abbreviations and acronyms the first time they are distribution transformer described in [18] is used. See Fig. 1. Fig. 2 shows the measured elements of the Y using 348 logarithmically spaced samples between 100HZ and 1MHZ. B. Modeling Fig. 3 shows the elements of as obtained via a transformation matrix Q calculated at 100HZ. The sample plot shows a rational approximation of using N = 60 poles obtained by VF with inverse magnitude weighting. All elements are seen to be represented very accurately. Fig. 4 shows the eigenvalues of (which are identical to those of Y). The admittance matrix of the model has eigenvalues that are very close to the eigenvalues of the data. Postmultiplying (15) with T and taking first order derivatives gives for each eigenpair ( ) (17) Ignoring terms involving gives and replacing with (16) (18) Fig. 1. Two-winding transformer(30 KVA). The perturbation size is made inversely proportional to the eigenvalue size by using a weighting that is equal to the inverse of the eigenvalue magnitude [12] (19) Equation (19) is built for all modes i and is used as a replacement for (16). The matrix diagonalization is introduced in order to reduce the number of free variables, leading to FMP. D. Transformation Back to the Physical Domain The extracted model is finally transformed back into the physical domain by the inverse transformation of (7), i.e., (20) Application of (20) gives the final mode as Fig. 2. Elements of Y. (21) With [ISSN: 2045-7057] www.ijmse.org 23 INTERNATIONAL J OURNAL OF M ULTIDISCIPLINARY S CIENCES AND ENGINEERING , VOL . 5, NO. 11, NOVEMBER 2014 Fig. 5. Measured versus Simulated voltages. Fig. 3. Elements of . Fig. 4. Eigenvalues of Y. C. Time Domain Simulation Fig. 5 shows an example of time – domain results for the given transformer. In a laboratory measurement, a step-like voltage was applied to terminal 1(Fig. 1) with terminal #2 and #3 grounded, and the voltage responses on terminals #4,#5, and #6 were recorded [18]. The measured voltage on terminal #1 was then applied as an ideal voltage source in an EMTPlike time-domain simulation using the rational model. It is observed that the model obtained via the new approach closely agrees with the measurement whereas the model extracted by the conventional approach has large errors. admittance matrix Y. The method is based on applying a similarity transformation matrix Q to Y to make the eigenvalues of Y more observable. The transformation matrix is calculated via eigenvalue decomposition, eigenvector rotations, and orthogonalization. 1) The transformed matrix remains symmetrical and it preserves the eigenvalues and the passivity characteristics of Y. 2) Applying conventional VF and passivity enforcement schemes to with inverse least – squares weighting gives a much better representation of the small eigenvalues of Y than what is achieved when applying these techniques directly to Y. 3) The extracted pole – residue model of is transformed back to the physical domain by applying the inverse transformation to each residue matrix individually, giving a model of . The new method was demonstrated for the frequency – domain modeling of a six – terminal distribution transformer using N = 60 poles. The extracted models were shown to provide a more accurate representation of the small eigenvalues of Y compared to a conventional approach based on direct application of VF Y to, thereby avoiding error magnification in applications with high impedance terminations. The new approach was also found to be about 10 times faster compared to the use of MVF. REFERENCES [1] [2] V. CONCLUSION [3] This paper has introduced a new, straightforward approach for terminal modeling of multiport systems. The approach is very useful when the system is characterized by a large ratio between the large and small eigenvalues of the terminal [4] [ISSN: 2045-7057] A. Semlyen, B. Gustavsen, “A half – size singularity test matrix for fast and reliable passivity assessment of rational models,” IEEE Trans, Power Del, Vol.24, NO.1, Jan 2009. B. Porkar, M. Vakilian, R. Iravani, S. 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Gustavsen, C.Heitz,“Fast realization of the modal vector fitting method for rational modeling with accurate representation of small eigenvalues,” IEEE Trans, Power Del, Vol. 24, NO. 3, pp. 1396-1405, July 2009. B. Gustavsen,“Fast passivity enforcement for pole – residue models by perturbation of residue matrix eigenvalues,” IEEE Trans, Power Del, Vol. 23, NO. 4, pp. 2278-2285, Oct 2008. B. Gustavsen,“Rational modeling of multiport systems via a symmetry and passivity preserving mode – revealing transformation,” IEEE Trans, Power Del, Vol. 29, NO. 1, Feb 2014. B. Gustavsen, A. Semlyen,“Rational approximation of frequency domain responses by vector fitting,” IEEE Trans, Power Del, Vol. 14, NO. 3, pp. 1052-1061, July 1999. G.H. Gloub, C.F. van Loan,“Matrix Computations,” 2nd ed, Baltimore, MD, USA: John Hopkins Univ, Press 1989. V. Brandwajn,“Modification of user’s instructions for MARTI SETUP,” EMTP Newslett, Vol. 3, NO. 1, pp. 76-80, Aug 1982. B. Gustavsen,“Passivity enforcement of rational models by modal perturbation ,” IEEE Trans, Power Del, to be published. B. Gustavsen,“Frequency – dependent modeling of power transformers with ungrounded windings,” IEEE Trans, Power Del, Vol. 19, NO. 3, pp. 1328-1334, Jul 2004. [ISSN: 2045-7057] Ehsan Elahi was born in Kazeroon, Iran, on September 20, 1989. He received his B.S. degree in Electrical Engineering from Islamic Azad University, Kazeroon Branch, Kazeroon, Iran, in 2011, and., M.S. degree in Electrical Engineering (Power Electronics) from Kazeroon University, Iran, in 2014. He is currently as a Lecturer in Department of Electrical Eng, Kazeroon Branch, Islamic Azad University, Kazeroon, Iran. His research areas of research interest are power systems, power systems analysis, transient, and control systems. www.ijmse.org 25

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