Dynamic equivalents for transient stability studies of electrical power systems J Machowsk, Technical Un=verslty of Warsaw, ul Koszykowa 75, Gmach Mechan&t pok 3a, 00-661 Warszawa, Poland Along wtth the growth m stze of eleetrwal power systems, attentton ts mereasmgly bemg given to vartous reduction techntques of dynamtc system models for transient stabthty studtes One of these ts the topologwal reductton that includes coherency-based aggregatton A theorem from the nonhnear vtbrattons theory ts apphed to dertve analytwal coherency-crtterta for a classwal model of the system A stmphfied algortthm of coherency recognttton ts gtven and results for a sample system are shown Important properttes o/ topologtcal equtvalents are proved and a relauonshtp to modal reductton ts pointed out Keywords short-term system dynamics, transtent stabthty, dynamic equivalent I. Introductmn Electrical power systems, for transient stability studies, are described by a set of nonhnear differential and algebraic equations x = f(x,y) (1) 0 = g(x, y) that describe the dynamic response of synchronous machines and transmission networks Disturbances, such as faults and their clearances, are represented in equation (1) by the relevant changes of coefficients in the right hand side of the equation Equation (1) IS solved numerically m the parti- tioned or simultaneous way I Contemporary electrical power systems are so large that solving equation (1) numeri- cally has for a long time been an expensive process even with modern computers Hence, extensive efforts are being made to develop techniques to reduce this set of equations to a single smaller equivalent The techniques that have been developed can be classified as modal, physical and topologi- cal approaches The modal approach attempts modal reduc- tion of the hnearIzed system model by considering the controllability, observabihty and time-scale properties of Recewed 5 March 1982, Rev=sed 26 November 1984 various system modes It has a good theoretical basis, but certain disadvantages &scourage its wider application to large systems Using a physical approach the number of state variables are reduced by the correct choice of the model type for each system component according to its role in the given fault The topological approach consists of the decomposition of the system into subsystems and their aggregation Such operaUons can considerably reduce the number of both differentml and algebraic equations Additionally, selected load-nodes can be ehmlnated to decrease further the number of algebraic equations Topo- logical equivalents are homogeneous, i e they can be repre- sented In terms of typical power-system components, so they can be implemented in standard computer programs for power-system simulation Topological equwalents strictly preserve the dynamic properties of the original system if coherence is taken as the criterion for the decomposition Unfortunately, coherency recognition IS a difficult task that consists of predicting a property of the trajectory of nonhnear equations before they are solved So far, a few methods of coherency recognition have been produced Lee and Schweppe 2 have been investigating coherency m a statistical way, and have reached the conclu- sion that coherency is caused by specific relatlonshxps between some system parameters An algorithm tor coher- ency recogmtlon has been suggested Vorapay 3, on the basis of the equal-area method, has derived a coetficlent as a funcUon ot system parameters prior to the short-ctrcmt occurrence and the parameters during Its duration This coefficient has been used to select coherent generators Spalding et al 4 have observed that coherency Is also related to coordinates of the closest unstable equlhbnum point of equation (1) The procedure based on that observation has been described Ohsawa and HayaskIs have suggested utilizing a property of the Lya- punov function It has also been observed that the llnearIzed model of the system can be useful for coherency recogni- tion Results of the simulation of a simplified linear model have been utilized to select coherent generators 6 The Vol 7 No 4 October 1985 0142-0615/85/040215-10 © 1985 8utterworth & Co (Publishers) Ltd 215