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)

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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.

A new and efficient method for calculating the load flow solution of weakly meshed transmission and distribution systems is presented. Its essential advantages over a previous approach' are the following: (1) It uses active and reactive powers as flow variables rather than complex currents, thus simplifying the treatment of P,V buses and reducing the related computational effort to half; (2) It uses an efficient tree labeling technique which also contributes to the computational efficiency of the procedure; (3) It uses an improved solution strategy, thereby reducing the burden of mismatch calculations which is an important component of the solution process. Results of tests with 30,243,1380, and 4130 bus systems are given to illustrate the performance of the proposed method.