The paper describes a general methodology for the fitting of measured or calculated frequency domain responses with rational function approximations. This is achieved by replacing a set of Starting poles with an improved set of poles via a scaling procedure. A previous paper [5] has described the application of the method to smooth functions using real starting poles. This paper extends the method to functions with a high number of resonance peaks by allowing complex starting poles. Fundamental properties of the method are discussed and details of its practical imptementation are described. The method is demonstrated to be very suitable for fitting network equivalents and transformer responses. The computer code is in the public domain, available from the first author. 2 VECTOR FITTING BY POLE RELOCATION Consider the rational function approximation N f(s)= x L + d + s h (2) '-an The residues c, and poles a, are either real quantities or come in complex conjugate pairs, while d and h are real. The problem at hand is to estimate all coefficients in (2) so that a least squares approximation of f (s) is obtained over a given frequency interval. We note that (2) is a nonlinear problem in terms of the unknowns, because the unknowns a, appear in the denominator. Vector fitting solves the problem (2) sequentially as a linear problem in two stages, both times with known poles. Stage #l : Dole identification Specify a set of starting poles Z , in (2), and multiply f (s) with an unknown function ~(s). In addition we introduce a rational approximation for ~(s). This gives the augmented problem : Note that in (3) the rational approximation for ~(s) has the same poles as the approximation for ~(s)

This paper presents a transmission. line model for the simulation of electromagnetic transients in power systems. The model can be applied to both overhead lines and cables, even in the presence of a strongly frequency dependent transformation matrix and widely different modal time delays. This has been achieved through a phase domain formulation where the modal characteristics have been utilized in the approximation for the propagation matrix. High computational efficiency is achieved by grouping modes with nearly equal velocities, and by columnwise realization of the matrices for propagation and characteristic admittance.

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Abstract-A linear power system component can be included in a transient simulation as a terminal equivalent by approximating its admittance matrix by rational functions in the frequency domain. Physical behavior of the resulting model entails that it should absorb active power for any set of applied voltages, at any frequency. This requires the real part of to be positive definite (PD). We calculate a correction to the rational approximation of which enforces the PD-criterion to be satisfied. The correction is minimal with respect to the fitting error. The method is based on linearization and constrained minimization by Quadratic Programming. Examples show that models not satisfying the PD-criterion can lead to an unstable simulation, even though the rational approximation has stable poles only. Enforcement of the PD-criterion is demonstrated to give a stable result.