Systematic Inclusion of Stator Transients in Reduced Order Synchronous Machine Models

While the answers to these questions may illustrate the distinction between an engineer and a mathematician, the complexity of modern problems makes engineering judgement more and more difficult. The technique presented in this paper is designed to give a systematic approach to the formulation of reduced order models based on the physical phenomena which allows the reduction. Once the physical phenomenon is identified, the derivation and possible improvement of the reduced order model is obviated. The formal name for the technique is singular perturbation. While the technique has been applied in power system analysis in recent years the following application to stator transients has further academic as well as practical promise. 2) Stator Transients and the Reduced Order Models The paper illustrates the proposed technique through an example of a short circuited synchronous machine repre¬ sented in d q coordinates by three electrical equations (stator !-1-,-,-,-,-, -90 -60 -30 30 90 0 flux linkages plus field flux linkage) and two mechanical 60 (angle and speed). Let the two stator flux linkages //Degrees equations be in the vector z and the two mechanical variables plus field linkage be in the vector x. The equations are manipulated Fig. 2. Radial induction components in various cross sections flux into a form where a small parameter (1/60) is multiplied times (compare Fig. 1). the stator transient derivatives as, x = f(x,z) (1) ez=g(xfz, e). (2) 83 SM 494-2 June 1984, pp. 1348-1354 Systematic Inclusion of Stator Transients in Reduced Order Synchronous Machine Models P. W. Sauer, Member, IEEE, S. Ahmed-Zaid, and M. A. Pai, Senior Member, IEEE University of Illinois at Urbana-Champaign, Urbana, IL Abstract.This paper illustrates a systematic approach to the inclusion of stator transients in a reduced order synchronous machine model. The technique provides a mathematically consistent approach to the derivation of the reduced order model so that a means for capturing the fundamental dy¬ namics is made possible. The method appears to be applicable to a large class of problems where stator or network transients are traditionally neglected. 1) Introduction The analysis of electrical machine and power system dy¬ namics has advanced to a level where large scale multimachine problems are often considered. The dynamic equations which describe the electrical and mechanical responses to transients are frequently derived in advanced undergraduate or first year graduate study. The theory of electromechanics and circuits is reasonably well established for the systematic derivation of these models. There is however a considerable theoretical gap between these mathematical high order models and the frequently used reduced order models. This gap has been narrowing as more rigorous attention is given to the stator transients. In the classroom, students typically want to know how frequency can be assumed constant in the network and stator while speed dynamics are computed. Another sticky question is why constant flux linkages do not imply constant current and hence constant electrical torque. IEEE Power Engineering Review, June 1984 When the equations are examined in the limit that e tends to zero, the following reduced order model is presented as a quasi-steady state model: x = f{x,z) (3) O=0(x,z,O). (4) The algebraic equation (4) is essentially the phasor equation traditionally used in models where the "p^" transients are neglected. Note that the above formulation does not imply that z constant, rather ez 0 because e is considered zero. This approach avoids the contradiction that the system is in electrical steady state while mechanical transients are com¬ puted. The paper presents numerical comparisons between the exact solution of (1 ) and (2) and the reduced order model of (3) and (4). The large errors are corrected in a systematic bound¬ ary layer analysis which recovers the lost stator transients. = 3) Summary and Discussion Stator transients are frequently neglected in the study of interconnected machine dynamics. It is not obvious that this has an insignificant affect on the slow system response. Indeed there are many cases where stator transients must be considered. The technique presented in this paper facilitates the recovery of stator transients at a cost potentially less than full integration of the exact model. In addition, several sticky issues associated with neglecting stator transients that fre¬ quently plague many students, faculty and practicing engi¬ neers have been considered. In the technique presented, there is nothing inconsistent about allowing flux linkages, currents and speeds to vary since the algebraic equations are the result of analyzing the fast subsystem under the asymptotic limit of e tending to zero rather than the derivatives to zero. In addition, the discontinuity of initial conditions has been eliminated through the boundary layer correction. The technique presented is based on a large volume of sound mathematical theory. One of the possible future appli¬ cations of this theory involves the correction of the slow subsystem as well as the fast through higher order singular perturbations. Since the reduced order models created by this technique are automatically derived from this theory, they can be improved to virtually any degree of accuracy desired while requiring less computational effort than a full exact model. 47