Structure-preserving model reduction with applications to power system dynamic equivalents

Abstract

Consider systems that have an intrinsic mathematical representation of the form: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A_{n}x^{(n)}+ \cdots + A_{0}x = Bu</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x^{(i)}</tex> is the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</tex> th derivative of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x^{(i)}(0)= 0, i = 1, \cdots , n-1</tex> . The matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A_{i}</tex> may possess certain properties. The problem is to construct a reduced model having the same form with corresponding matrices <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\bar{A}_{i}</tex> of smaller dimension and possessing the same properties. An algorithm for this structure-preserving model reduction is presented. An algorithm for constructing an approximate reduced model, called <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon</tex> -structure preserving reduced model is also presented, together with the error bounds. The application of the method to power system dynamic equivalents is described.