~ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA IEEE Transactions on Energy Conversion, Vol. 12, No. 2, June 1997 118 zyxwvutsrqp Data Translation and Order Reduction for Turbine- Generator odels Used in Network Studies I.KAMWA, MEMBER, IEEE zyxwvutsrqp lnstitut de recherche d'Hydro-Que'bec IREQ , 1800 Lionel-Boulet Varennes, Que'bec, Canada JlX IS1 ABSTRACT - This paper develops suitable procedures for trans- lating high-order networks into conventional stability constants. The approach is computer-oriented and imposes no restriction on the number of equivalent damper windings per axis. Optimum order reduction is used to demonstrate that second-order networks have a bandwidth limited to between 1 and 5 Hz while third-order models may extend this bandwidth beyond 100 Hz. The latter are therefore mandatory for subsynchronous resonance and harmonic studies. In the course of this investigation, the need for a precise definition of the so-called stability constants arises and it is conclu- ded that only a second-order SSFR-based model yields constants that are consistent with the latest, short-circuit test bound, IEEE standard definitions. I. INTRODUCTION The standstill frequency response (SSFR) method, now incorporated in an IEEE standard (No 115), allows equivalent circuits with multiple rotor circuits in the d- and q-axes to be determined by direct fitting of test data [l]. In contrast with the time-domain approach, SSFR-based networks of suitably high order have proven very effective in representing turbine-genera- tor dynamics in a bandwidth ranging from 0 to well above 120 Hz zyxwvutsrqpo [6]. The emergence of high-order networks creates a new need for analytical procedures allowing equivalent circuits to be translated with more than two rotor circuits into dynamic reac- tances and time constants. With few exceptions [ 11, the currently available literature on this subject oftcn restricts the number of rotor circuits to two or three [3,4], justifying this by the fact that either higher-order networks are unlikely to happen in reality or their increased complexity makes them unsuitable for network studies. Part of this belief lies in the well known fact that second-order network models seem able to predict most time-domain responses of the machine, including responses to sudden-shorl-circuit, line-swit- ching, small excitation steps, etc. However, comparisons with measurements are generally made from the stability perspective only, based on envelope or power swing records whose band- widths rarely exceed 5 Hz [5]. Actually, recent advances in commended and approved by by the lEEE Electric Machinery Committee of the IEEE Power Engineering Socle& for presentation at the 1996 IEEUPES Summer Meeting, July 28 - August 1, 1996, in Denver, Colorado. Manuscript submitted December 27, 1995, made available for printing May 10, 1996. M. FARZANEH, SENIOR MEMBER, IEEE De'partement des sciences applique'es Universite' du Qukbec b Chicoutimi Chicoutimi, Que'bec, Canada G7H 2B1 Higher-order models may be required in any instance where the transmission system model includes frequency dependency, as in most EMTP or harmonic penetration studies. The dynamic range of the machine model should then be at least as wide as that of the transmission model. As time-domain tests are known to be unsuitable for identifying models with a dynamic range spanning more than three decades (e.g. 1 to 1000 Hz or 0.1 to 100 Hz), the SSFR method seems, so far, to be the only one able to fulfil this need. This paper is about translating high-order, SSFR-based equi- valent circuits into dynamic constants in the form of In pursuing this main objective, the order extension was found to introduce ambiguity in the con- ventional meaning of the dynamic constants. Alternative defini- tions, hopefully more precise and not directly linked to the short-circuit test, are then proposed. They emphasize the pheno- menological meaning of each dynamic constant rather than the means of measuring it. The technique of minimum-Hankel norm order reduction [ 101 is applied to the equivalent circuits of actual turbine-genc- rators in order to establish clearly, and for the first time we believe, the direct links that exist between the order and band- width of a model. Our results point out that, with a bandwidth limited to 1 Hz, a second-order representation is, a at best, a rough representation of the two machines investigated. id, T'd, zyxwvut xnd, T''d , etc. 11. FROM EQUIVALENT CIRCUITS TO OPERATIONAL INDUCTANCES If we assume wm = 0, the two equivalent circuits in Fig. 1 are decoupled, with the following short-circuit impedances: where zyxwv wn is the rated frequency and w , , the rotor speed in p.u. The so-called operational impedances, xd(s), xq(s), X ~ S ) and G(s) are defined as follows: 0885-8969/97/$10.00 zyxwvu 0 1996 IEEE