| This thesis presents efficient methods for constructing an equivalent of an immittance matrix of a passive electric network based on either infinite-dimensional theory or fitting of measured (calculated) frequency responses. The immittance matrix can be used: (1) To represent a frequency-dependent equivalent of a large electric network, e.g. an interconnected power system, with respect to prespecified terminal(s), in a wide frequency range for (a) analyses of electromagnetic transients, and (b) analyses and synthesis of controllers, or (2) To represent a frequency dependent equivalent of electric apparatus, e.g. a power transformer, in a wide frequency range, to investigate its terminal behaviour with respect to an external phenomenon (black-box representation).;Requirements for the equivalent immittance matrix to provide stable results are that (1) all its poles must lie in the left-half plane and, (2) it must represent a passive network. Immittance matrix of a passive network represents a positive real system. Necessary and sufficient conditions for an immittance matrix to represent a passive network, for both rational transfer matrices and state-space matrices, are presented. Based on developed transforms of these conditions, direct methods, for testing positive realness are deduced. The methods and the corresponding algorithms require only evaluation of a set of simple algebraic conditions. These provide alternative procedures to the application of positive real lemma. Based on these algorithms, positive realness of an equivalent is guaranteed.;Based on the following two approaches, a finite-dimensional, linear, and passive model of a network, represented by an immittance matrix, is deduced. (1) The first approach is based on solution of a special class of linear hyperbolic partial differential equations using infinite-dimensional theory. Passivity and stability conditions are ensured. (2) The second approach is based on approximation (fitting) of measured (calculated) frequency responses using either a rational immittance matrix or a state-space representation. Stability and passivity conditions for the model are imposed during optimization fitting process.;Applications of the developed approaches and corresponding algorithms are demonstrated based on obtaining frequency-dependent equivalents of (1) a large 500kV transmission system, and (2) a three-phase distribution class transformer. |
