Iterative time-scale separation of synchronous machines M. S. Mahmoud zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED Electronics and Communications Department, Faculty of Engineering, Cairo University, Giza, Egypt A. S. Al-Fuhaid Electrical and Computer Engineering Department, Kuwait University, PO Box 5969, Kuwait F. A. Saleh Computer Centre, Kuwait University, PO Box5969, Kuwait (Received May 1986; revised August 1986) A scheme for iterative separation of time-scales in linear singularly- perturbed systems is developed and applied to synchronous machines, Both the current and flux-linkage dynamic models are considered and it is shown that they are linearly related for small disturbances. Simulation studies on a typical system have indicated the usefulness of time-scale separation schemes in anaiysing the dynamic behaviour of synchronous machines. Keywords: iteration, synchronous machines, flux-linkage, simula- tion Mathematical models of electric power systems are non- linear and generally display several interacting phenom- ena of various dynamic responses. For the case of a small disturbance, it is often desirable to seek linearized models. To analyse the dynamic behaviour of such models, it would seem beneficial to describe the linear- ized power models in terms of reduced-order models.’ Recent results have indicated the significance of multi- time-scale theory* in developing reduced models operat- ing at different time-scales which provide a good approx- imation to the full-order models. Previous work3 has focused on the analysis of the effects of neglecting stator transients in machine per- formance. The technique of asymptotic modal decomposition has been utilized in output feedback con- trol design with high gains.4 Subsequently, others5 have used singular perturbation theory with a single para- meter on two time-scales to further clarify the meaning of several classical results. Their work has centred around the electrical part of the synchronous machine model. 0307-904X/87/02132-08/$03.00 0 1987 Butterworth & Co. (Publishers) Ltd In this paper, an iterative scheme of time-scale separ- ation is developed and applied to linearized models of synchronous machines. Both the flux-linkage and cur- rent models are considered and it is shown that the two are linearly related. Therefore the flux-linkage model is focused on. Using typical data, it is concluded that the rotor angle, rotor angular speed and the field flux linkage are slow variables, sufficiently separated from the dq components of the stator and rotor flux linkages as fast variables. Simulation results have indicated the usefulness of the scheme developed in analysing the models of synchronous machines for small disturbances. Development of dynamic models The starting point in developing dynamic models of a synchronous generator connected to an infinite bus is the generator’s equivalent circuit. Figure I depicts the generator’s equivalent circuit using the so-called dq reference frame.6 It should be noted that this circuit is applicable when using physical quantities such as Appl. Math. Modelling, 1987, Vol. 11, April 133