Arnold Diffusion in the Swing Equations of a Power System

We present an application of the theory of Arnold diffusion to interconnected power systems. Using a Hamiltonian formulation, we show that Arnold diffusion arises on certain energy levels of the swing equations model. The occurrence of Arnold diffusion entails complex nonperiodic dynamics and erratic transfer of energy between the subsystems. Conditions under which Arnold diffusion exists in the dynamics of the swing equations are found by using the vector-Melnihov method. These conditions become analytically explicit in the caSe when some of the subsystems undergo relatively small oscillations. Perturbation and parameter regions are found for which Arnold diffusion occurs. These regions allow for a class of interesting systems from the point of view of power systems engineering. I. INTR~DUCTI~N W E APPLY the results on Arnold diffusion of Holmes and Marsden [26] (see also [l, sect. 41) to power systems [45]. We note that the results in fact apply to all systems of the forced pendulum family such as interconnected power systems employing the swing equations model, coupled Josephson junction circuits with negligible dissipation, a Josephson junction driven by a direct current source (plus a small alternating current) coupled to two (respectively one) nonlinear oscillators, and coupled mechanical pendulums. The precise calculations are carried out here for a dynamical model of interconnected power systems. In the dynamical behavior of a large interconnected power system, the question of transient stability is often considered. This concerns the system’s behavior following a sudden fault (such as short circuit) or a large impact (such as lightning). The transient stability is precisely the Lyapunov stability in a state-space formulation of a simplified differential equations model possessing multiple equilibria. Let the dynamics be given by i = f(x) and let x0 be a stable equilibrium point which is presumably “closest” to the prefault equilibrium point (see [32], [9]). The transient stability problem is to determine whether a given point in the state space belongs to the region of stability of this stable equilibrium point. Thus the transient stability problem leads to an investigation of the region of stability of a given stable equilibrium point [28], [33], [13], Manuscript received March 14, 1984. This work was supported in part by DOE Contracts DE-AT03-82 ER 12097 and DE-ASOl-78ER29135, and by the National Science Foundation under Grant ECS-811213. F. M. A. Salam was with the Universit 94720. He is now with the Department o Y of California, Berkeley, CA Mechanical Engineering and Mechanics, Drexel University, Philadelphia, PA. .J. E. Marsden and P. P. Varaiya are with the De artment of Electrical Engineering and Computer Sciences, University o P CA 94720. Cahfomia, Berkeley, [14], [18]. Many studies of transient stability [28], [33], [13], [14], [18] have been conducted exploiting a first integral of the differential equation as a Lyapunov (energy) function. Kopell and Washburn [29] were the first to show the presence of chaotic motion in the classical swing equations model of power systems for a 2-degree-of-freedom system (3 generators). Their work is based on the original Melnikov method for vector fields (see Holmes [24]) and the energy function was not exploited to locate the energy levels where chaos resides. Here we show the presence of Arnold diffusion in the (n 2 3)-degree-of-freedom Hamiltonian system (with constraints) of the classical model. In the case when (n = 2) only horseshoes are present. This case is analogous to the one obtained by Kopell and Washburn except that we also specify the energy levels on which chaos resides, an advantage of exploiting the energy function. The paper is organized in the following way. In Section II we summarize the key result of Holmes and Marsden [26]. Section III contains some motivation and the derivation of the swing equations model. In Section IV we consider specific choices of parameter ranges to simplify the model before applying the results of Section II. In this section we also study the Hamiltonian formulation of the swing equations. They form a 2n degree of freedom system with two time-independent constraints. In Section V we show that the conditions of Section II can be satisfied for a large choice of parameters. These conditions can be simplified if all but one of the subsystems undergo small oscillations. This case is discussed in Section VI. Conclusions and suggestions for future work are collected in Section VII. II. ARNOLD DIFFUSION IN HAMILTONIAN SYSTEMS In this section we summarize the results of Holmes and Marsden [26] for Hamiltonian systems with n-degrees of freedom (n > 3). These results extend the work of Arnold [ll], for more discussion of Arnold diffusion, see [l]. Problem Statement Consider the unperturbed Hamiltonian system H’(q, P, 2, Y) = F(q, p)+G(x’, y’) (2-l) where F is a Hamiltonian which possesses a homoclinic orbit (q, p) associated with a hyperbolic saddle point qo, po. Let x be the energy constant of this orbit, i.e., F(ij, p) = h. The parameters (q, p, I, y) are assumed to be 0098-4094/84/0800-0673$01.00 01984 IEEE 614 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-31, NO. 8, AUGUST 1984 canonical coordinates on a 2( n + 1)-dimensional symplectic fold. Let manifold P; q and p are real and x’ = (xi,. . . , xn), y’ = (VI,* *, y,) are n-vectors. We assume that in a certain region of the state space a canonical transformation to action-angle coordinates (6,; . ., a,,,, Zi; . ., I,,) can be found such that the system (2.1) takes the form HO(q,P,d,I)=F(q>p)+ i G,(Zi) (2.4 i=l where G,(O) = 0, for all j and c2j(zj)=$o, for Ij > 0. J (2.3) Applying the reduction procedure (see Holmes and Marsden [25], [26]), we solve Ho = h for Z,, thereby eliminating the action Z,. We also replace the time variable by the 2a-periodic angle 9,. Then the equations Gj(lj) = h, i3j=~j(zj)8n+‘#j(o), j=l;**,n-I 4= 409 P = PO (2.4) describe an (n 1)-parameter family of invariant (n l)dimensional tori T( h,, . . . , h,-,). For a fixed set of h . .*, hnel, the torus T(h,;*., h,-,) is connected to itielf by the n-dimensional homoclinic manifold Gj( Ij) = h, lYj=slj(zj);m+ly, l< j<n-7 q=4(%-+Y) 2 p=p(8n-tY;). (2.5) This manifold consists of the coincident stable and unstable manifolds of the torus T(h,; . 0, h,-l), i.e., W(T(h,;.., h,-1)) = W”(Tth,,. -3 h,-1)). The perturbed problem considered here has the following form