Analysis of a symmetrically stabilized three-phase oscillator and some of its applications

Abstract

A nonlinearly stabilized three-phase oscillator model is treated in the present work. It is shown analytically and demonstrated by a computer solution of the equations that the oscillator equations possess a relatively large region, where stability of solutions is assured. All trajectories initiating or reaching this region are proved to approach a limit cycle solution. The three variables <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_1, x_2</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_3</tex> , representing the final Stable solution versus time, vary in time in a way similar to that of the three voltages of a balanced three-phase power generating System in steady state. The analysis of the relatively complicated third-order nonlinear system is made possible by transforming the original three-phase variables <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_1, x_2</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_3</tex> to new variables (introduced by Daboul) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S, M</tex> , and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\phi</tex> . The above oscillator has been applied earlier for the representation of power systems. The present thorough analysis of the model increases the authors' confidence that such representation of power systems is dependable.