The algebra of the ideal doubly-fed induction generator

Three-phase, doubly-fed induction generators (DFIGs) are key constituents of power conversion plants. Modeling and control of a DFIG become relevant to deal with intermittency in primary energy supply (e.g., the wind velocity field) and with uncertainty in the load. An ideal DFIG is modeled by inductance matrices which relate electric and magnetic quantities. The well-known Blondel-Park transformation allows a change to a reference frame, with respect to which the machine governing equations simplify remarkably. Said transformation was deduced from first principles by the authors in a previous conference paper, where an exponential representation of the corresponding one parameter group was obtained as well as its unique infinitesimal generator. Herewith the focus is on the algebraic properties of the mutual (rotor-to-stator) inductance matrix and of the infinitesimal generator, respectively. Left zero divisors of the former matrix are determined, then a differentiation formula and an exponential representation are given. A recurrent formula for the powers of the infinitesimal generator is provided. The results provided herewith remarkably simplify all calculations involving the machine model, including power balance.