IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, No. 2, February 1981 AN ANALYSIS OF INTERAREA DYNAMICS OF MULTI-MACHINE SYSTEMS J.R. Winkelman, J.H. Chow, B.C. Bowler General Electric Company Schenectady, New York 12345 ABSTRACT The slow coherency concept is introduced and an algorithm is developed for grouping machines having identical slow motions into areas. The singular perturbation method is used to separate the slow variables which are the area center of inertia vari- ables and the fast variables which describe the inter- machine oscillations within the areas. The areas obtained by this method are independent of fault locations. Three types of simulation approximations illustrated on a nonlinear 48 machine system indicate the validity of this algorithm. 1. INTRODUCTION This paper presents a systematic procedure for grouping the machines of a power system into areas. The concept of an area is based upon the observation that in postfault transients only some machines close to the fault location respond with fast intermachine oscillations, while other machines more distant from the fault swing together in groups with "in phase" slow motion. Our approach is to define areas by grouping the machines which lexhibit this slow co- herency phenomenon. Allowing the machines in the same area to differ in their fast dynamics makes it possi- ble to retain the same area grouping for different fault locations. The resulting conceptual simplifi- cations and computational savings are significant in simulation and planning studies when many contin- gencies need to be examined. The notion of slow coherency is expressed in the following way. If we consider the r slowest modes of the system's response to any fault, then machines "i" and "j" are slowly coherent if the difference of their angles x.(t) and x.(t) i i; (1. 1) xi(t) - xj(t) = zij (t) contains none of the r slowest modes. This definition disregards differences of the fast dynamics of ma- chines within the same area. In contrast to the more conventional definitions of coherency [1-6], which require that the total angular difference z. .(t) be within a specified tolerance, here the toleranc'A is specified only for the slow modes in z. (t). B. Avramovic, P.V. Kokotovic University of Illinois Urbana, Illinois 61801 Our approach to grouping machines starts with the linearized electromechanical model without damping, and separates its slow and fast modes using the so called dichotomic transformation from the singular perturbation technique [7]. The dichotomic transfor- mation matrices L and M define a set of physically meaningful state variables. In the ideal slow coher- ency case the dichotomic L is a "grouping" matrix, whose elements are zeros and ones, and the state variables of the fast subsystem are machine angle differences within areas. On the other hand, the matrix M, which separates the slow subsystem, defines the slow variables as the area centers of inertias [1,2,4]. In a nonideal case we search for a dicho- tomic L whose elements are close to zeros and ones. This results in areas which contain machines that are near-coherent in their slow modes. The slow interarea dynamics and the fast intra area dynamics are suitable for two time scale analysis of power systems by the singular perturbation method. This method is applicable to systems in the so called state separable form dt =f(t,n,t) , t(t ) =to dt 0 8 dE1 = g(t,fl,t) , q(t ) = n dt (1.2) (1.3) where t and n represent the "slow" states and the "fast" states of the system respectively, and E is a small positive parameter which accounts for small time constants, inverses of high gain coefficients, small inertias, etc. If the separation between time scales in (1.2) and (1.3) is large, 8 will be small and may be approximated by 8=0. The model (1.2) and (1.3) with 8=0 then defines the quasi-steady-state t (t), q (t) as s dt (t) dt = f(5 S , t) I (t) = 0 = g(ts, rs, t) (1.4) (1.5) where the differential equations for q have been reduced to algebraic or transcendental equations. In (1.2), (1.3) the variables t are predominantly slow, that is, t(t) t- (t), while the variables 0(t) contain a significant fast component 0(t) - rl (t) which becomes infinitely fast as &->O. For application of the singular perturbation method it is necessary to express the system dynamics in the form (1.2) (1.3). 80 SM 533-0 A paper recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the IEEE PES Summer Meeting, Minneapolis, Minnesota, July 13-18, 1980. Manuscript submitted February 4, 1980; imade available for printing April 21, 1980. System models which describe fast and slow phe- nomena do not always appear in this form. For exam- ple, the electromechanical model using individual machine speeds and angles as the state variables does not exhibit this slow-fast separation. A new set of state variables which brings the model to the form (1.2), (1.3) are the interarea motions which represent the "slow" states t, and intra area motions of the machines within an area which represent the "fast" states n in (1.2), (1.3). ©C) 1981 IEEE 754