A Chaotic Phenomenon in the Power Swing Equation Umesh G. Vaidya R. N. Banavar y N. M. Singh March, Abstract Existence of chaotic dynamics in the classical swing equations of a power system of three interconnected generators is shown. By incorporating damping in the swing equation we have extended prior work which proves chaotic behaviour for a conservative two degree of freedom (three machine ) swing equation. The occurrence of a chaotic phenomenon entails complex nonperiodic behaviour. Melnikov's technique developed for dissipative systems is used to show the presence of such chaotic dynamics. Introduction The presence of chaos and Arnold diusion (varied form of chaos) in the swing equation of power systems has been proved in [] and []. In both these cases, the swing equation is modeled in a Hamiltonian framework. In this paper wehave proved the phenomenon of chaos with damping incorporated in the swing equation. We prove the existence of Smale-birkho horseshoe chaos in the dynamics of the swing equation using the method of Melnikov. The outline of the paper is as follows. In section II we review the model of the swing equation. We then determine the time depandant constraint equations whichthepower angle and the angular velocity of the generator have to satisfy in the presence of of the damping term. In section III we show that by using a proper coordinate transformation these time depandant constraint equations can be converted into time independent constraint equations. In section IV we consider a special case of three interconnected generators. Inertia constant, Graduate student, University of California, Santa Barbara y Department of Mechanical and Aerospace Engineering, UCLA, Los Angeles, CA 995. ravi@artegal.seas.ucla.edu Email:
power rating and damping constant of the three generators are properly scaled. We use the same scaling as used in [] except for the scaling of the damping constant. In section V we review Melnikov's Integral technique which is used to detect the presence of transversal intersection between the stable and the unstable manifold. In section VI we include the results of F. M. A. Salam [5] which extends this Melnikov Integral technique to dissipative systems. Finally in section VII we prove the presence of horseshoe chaos in the swing equation. Mathematical Model of Three Machine Power Swing Equation Consider a power system model of synchronous generators with a load conguration as shown in g.. The assumptions which are made in setting up the mathematical model of the power swing equation are as follows ) A synchronous machine is represented by a constant voltage behind its transient reactance in other words it is assumed that ux linkages are constant during the transient period. Flux decay and voltage regulation are not taken into consideration. ) Damping power is assumed to be proportional to slip velocity and is thus assumed to be proportional to mechanical friction. Under these assumption the motion of the i th generator is described by the following dierential equations. where M i = Inertia constant D i = Damping constant P m i = Constant mechanical power input P e i = Electrical power output d dt i =! i ;! R () M i d dt! i + D i! i = P m i ; P e i () i = Angle of internal complex voltage or the torque angle of the i th machine! i =Therotor angular velocity of the i th machine! R =The reference frequency of the power system The electrical power of the i th machine is the sum of the power dissipated by its short circuit
G G Load G Figure : Three Interconnected Generators conductance and the power delivered to the transmission line connecting the machines under consideration to the other machine of the power system P e i = G ii E i + i= i6=j E i E j Y ij cos( i ; j ; ij ) i =:: () where E i is the internal voltage of the i th machine Y ij = Transfer admittance magnitude between internal nodes i and j ij = Transfer admittance phase angle between internal node i and j G ii =Total admittance at the internal node of generator i For most power systems the terms G ij := Y ij cos ij (i 6= j) are negligible and only the transfer suspectances B ij := Y ij sin ij have to be taken into consideration. Making this assumption and assuming that all the generators are uniformly damped i.e. D i M i = for i = to and with certain algebraic simplication we have i= M i i (t) = e;t ; i= + i= i= M i i () + M i! i (t) =e ;t M i! i () ; M! R t + i= i= M i! i () (4) M i! i () (5) The above two constraint equations (4) and (5) are time depandant. These time depandant constraint equations can be converted into time independent constraint equations by making
the following coordinate transformation.! i!! i =! i ; e;t M i= M i! i () (6) i! i = i +! R t ; M ; M i= i= M i! i () + e;t M M i i () i= M i! i () (7) With this coordinate transformation and removing \ constraints equation (4 )(5) can be written as follows " the swing equations ()() and the M i _! i = P i ; i= i6=j Scaling Parameters _ i =! i (8) y ij sin( i ; j ) ; D i! i for i =::: (9) i= i= M i i = () M i! i = () We assume that the machines and have large inertias compared to machine. A parameter " is used to express the relative size of the inertia as M = M " M = M " and M = M () All the overbarred quantities are of the same order. The coupling between machine and is larger than the couplings of or to machines which is expressed as B = B " B = B and B = B () Similarly the external power P of generator is proportionally smaller than that of generator and, expressed as P = P " P = P " and P = P (4) 4
Since we have assume that all the generators are uniformly damped i.e. Substituting equation () in equation (5) gives The damping of the generators are scaled as D M = D M = D M = (5) "D M = "D M = D M = (6) D = D " D = D " and D = D (7) With this scaling of damping constants we obtain following relation between the scaled quantities. D D = D = = (8) M M M and the swing equations can be rewritten as _ i =! i i =::: (9) M _! = P ; B sin( ; ) ; " B sin( ; ) ; D! () M _! = P ; B sin( ; ) ; " B sin( ; ) ; D! () M _! = P ; B sin( ; ) ; B sin( ; ) ; D! () since i= M i i = = ; M M ; M M () Let := M M and := M M Substituting the constant and in equation (). = ; ; " (4) P i B Let := i ij := ij M i M i Substituting these constants in the swing equation and the value of from equation (4) _ =! (5) 5
_! = ; sin( + + " ) ; " sin( ; ) ;! (6) _ =! (7) _! = ; sin( ; ) ; sin( + + " ) ;! (8) For the special case when " = which implies that generator and have innite inertia constant and power capacity, the swing equation reduces to _ =! (9) _! = ; sin( + ) ;! () _ =! () _! = ; sin( ; ) ; sin( + ) ;! () Equation (9) and () can be solved independently for and! : This dierential equation is similar to a nonlinear damped pendulum with a nonzero driving torque and is well studied (ref.[4]). In particular for j j < j j the equation has a stable equilibrium point given by = = sin ; ( )! =. Assuming that under the unperturbed condition generator is at the stable equilibrium point. Under the perturbed condition the states of generator move slightly away from equilibrium point. (t) under this perturbed condition is given by the following equation. (t) = + A(t) sin(t +)+O( ) () where > is a small perturbation parameter and and are constant and depends upon the initial state under the perturbed condition and =( ; 4 ( + )) = :A(t) is bounded by some constant N and is continuous in time t (; ). We further assume that A(t)! ast!we have assume this particular form of A(t) since we know that the solution under the perturbed condition will be an exponentially decaying sinusoidal which tends to the equilibrium point as t!: Substituting equation () in equation () and simplifying results in _ =! (4) _! = ; sin( ; ) ; sin( + ) ;! + fa(t) sin(t +)g( cos( ; ) ; cos( + )) (5) 6
If we choose > anddene such that cos := cos ; cos( ) sin := sin + sin( ) then the swing equation becomes _ =! (6) _! = ; sin( ; ) ;! +fa(t)sin(t +)g( cos( ; ) ; cos( + )) (7) For = (t) = for all t and equation (6) and (7) is given by _ =! _! = ; sin( ; ) ;! (8) We further assume that value of = (critical value). At this critical value of there exists a homoclinic orbit to saddle point (! ) connecting stable manifold of the saddle point to the unstable manifold (ref. [4]). where = ; sin ; ( ) ; and! = We are interested in the dynamics of the third generator after perturbation. Under unperturbed condition we have assumed that this generator has a phase portrait which consists of a homoclinic orbit. We wish to determine whether there is a transversal intersection of the perturbed stable and unstable manifold. If these manifolds intersect once then they will intersect innite number of times. This situation is very well studied and there is a large literature concerning the consequences of such intersections (ref. []). We have used the Melnikov technique to detect whether or not there is transversal intersection. 4 Melnikov Integral Material for this section and the next section is taken from reference [5] and [7]. For the sake of continuity and completeness we adopt a brief description. Consider a dierential equation of the form _x = f(x)+g(x t) (9) Where > is a small perturbation parameter and x R :g(x t)iscontinuous and bounded by a constant M in time t (; ) and x R : Assume that the unperturbed system 7
( = ) has a homoclinic orbit x (t): The unperturbed system is given by the following dierential equation _x = f(x) (4) Let x be the hyperbolic saddle point of the unperturbed dierential equation. We denote the stable and unstable manifold of x by x s u (t) (s-stable, u-unstable). By the invariant manifold theorem [6] there exists a solution x s and x u of the perturbed system with the following limiting behaviour. lim t! xs (t) =x and lim t!; xu (t) =x where x = x + O() (4) The stable and the unstable manifold of the perturbed system can be written as x u (t t )=x (t ; t )+x u (t t )+O( ) (4) x s (t t )=x (t ; t )+x s (t t )+O( ) (4) where t is arbitrary initial time. Inserting these equations into (9) we obtain the following rst variational equations _x u s = D x f(x (t ; t ))x u s (t t )+g(x (t ; t ) t) (44) where D x f(x (t;t )) is the Jacobian matrix of f evaluated at x (t;t )i.e. on the homoclinic orbit. The two solutions of the stable and the unstable manifold of the perturbed system dier by d (t t ): = x u (t t ) ; x s (t t ) = (x u (t t ) ; x s (t t )) + O( ) (45) The separation between the stable and the unstable manifold at time t = t is given by D (t t ):=hn d i (46) where h: :i denotes the innerproduct. D is a projection of d along the normal N to the unperturbed orbit as shown in g.. The normal to the unperturbed homoclinic orbit is given by N(t t ):= @ ;f (x (t ; t )) f (x (t ; t )) A (47) 8
x (t-t ) N(t,t ) x γ u x ((t,t ) x s x (t,t x (t-t ) d(t-t ) Figure : Perturbed Stable and Unstable Manifold where f =(f f )sod (t t ) can be written as follows. D (t t )= jf (x (t;t )j (f(x (t ; t )(x u (t t ) where is the wedge operator and is dened as follows x = @ x x ;x s (t t )) + O( ) (48) A and y = @ y y A then xy = x y ; x y Let Then M(t ):=f(x ())(x u (t t ) ; x s (t t )) (49) D(t t )= jf(x ()j M(t )+O( ) (5) M(t) = f(x (t ; t ))(x u (t t ) ; x s (t t )) = f(x (t ; t ))x u (t t ) ; f(x (t ; t ))x s (t t ) (5) Let and u (t t ):=f(x (t ; t ))x u (t t ) (5) s (t t ):=f(x (t ; t ))x s (t t ) (5) where equation (5) is valid for time interval t (; t ] and equation (5) for t [t ): Dierentiating equation (5) and equation (5) we obtain. d dt u (t t )=D x f(x (t ; t )_x x u (t t ) +f(x (t ; t )) _x u (t t ) (54) 9
Since x (t) is the solution of the unperturbed dierential equation (4) we have _x (t ; t )= f(x (t ; t )): We also know that _x u = D x f(x (t ; t ))x u (t t )+g(x (t ; t ) t) Substituting these two results in equation (54) we obtain This after simplication reduces to d dt u (t t )=D x f(x (t ; t )f(x (t ; t ))x u (t t ) +f(x (t ; t ))(D x f(x (t ; t ))x u (t t ) +g(x (t ; t ) t)) (55) _ u (t t )=TraceD x f(x ) u (t t )+f(x )g(x t) (56) Similarly _ s (t t )=TraceD x f(x ) s (t t )+f(x )g(x t) (57) Let a(t ; t ) : = T raced x f(x ) b(t ; t ) : = f(x )g(x t) Then _ u s (t t )=a(t ; t ) u s (t t )+b(t ; t ) (58) Using the results of F. M. A. Salam (ref. [5]) yields the Melnikov formula as M(t )= u (t t ) ; s (t t ) (59) = Z ; = 8 < Zt : exp 4 ; Z 8 < Zt : exp 4 ; ; t a(s ; t )ds 9 = 5 b(t ; t )dt f(x (t)g(x (t) t+ t ) T raced x f(x (s)ds 9 = 5 dt (6) where in the last step we performed the change of integration variables t ; t! t:
5 Chaotic Dynamics in Swing Equation _ =! (6) _! = ; sin( ; ) ;! +fa(t)sin(t +)g( cos( ; ) ; cos( + )) (6) Let cos( ; ) ; cos( + )=Ksin( ; C) This equation is of the form _! = ; sin( ; ) ;! f(x) = g(x t) = +fa(t)sin(t +)g(k sin( ; C)) (6) _x = f(x)+g(x t) where x = @! A @! ; sin( ; ) ;! A @ fa(t)sin(t +)g(k sin( ; C)) TraceD x f = ; x (t) = @ (t)! (t) where x (t) isthehomoclinic orbit to the saddle point x =(! ) let s =(t + t ) A f(x )g(x (t) s)=! (t)a(s)sin((s)+) Z t exp 4 ; T raced x f(x (s))ds5 = e t A K sin( (t) ; C) (64) (65)
Stable Manifold Saddle Point Unstable Manifold Figure : Poincare section showing intersection of Stable and Unstable Manifold Substituting equation (65) and (64) in the general formula of Melnikov integral (6) we get following Melnikov integral for the power swing equation. R ; M(t )=! A(s) sin((s)+)k sin( (t) ; C)e t dt (66) let M (t) =! (t)a(s)sin(t)sin( (t) ; C)e t and M (t) =! (t)a(s)cos(t)sin( (t) ; C)e t M(t )=K +K Z 4 ; Z 4 ; M (t)dt5 cos((t )+) M (t)dt5 sin((t )+) (67) If M(t ) = and dm (t ) 6= then the stable and unstable manifold of the saddle point dt intersect transversally at t = t : If the stable and the unstable manifold intersect once then they must intersect innite number of time as shown in g.. Although the Melnikov integral is dicult to compute it is possible to detect whether M(t ) has zero crossing without computing the integral. The above Melnikov integral resembles a Fourier integral and we can utilize this resemblance to analyze the zero crossing of M(t): Dene F (t) :=A(t + t )! (t)sin( (t) ; C)e t (68)
Then M(t ) can be written as follows M(t ) = Kfcos(t +) R sin(t +) R ; ; F (t)sin(t)dt + F (t) cos(t)dtg (69) Either M(t )=twice in one period and M (t ) 6= where M(t )= or M(t ) = for all t: The latter is possible only if Z ; F (t) sin(t)dt = If F (t) isintegrable over (; ) then the Fourier transform ^F () = Z ; Z ; F (t) cos(t)dt = (7) F (t)e it dt is analytic in : Hence either ^F () = for all or ^F () = for at most discrete value of : The former case is possible only if F (t) = for all t: Thus we conclude that if F (t) given by (68) is not identically zero and is integrable over (; ) thenfor all except a discrete set of frequencies the Melnikov integral has two zeros in one period and M (t ) 6= for M(t ) = : Consequently for these frequencies and suciently small the stable and unstable manifolds intersect transversally. So our aim is to show that integral of F (t) exists and nite and is not identically zero. We now need to show that e t! (t) is nite both as t! : Since! (t) is a bounded smooth function with! (t)! as t! it is enough to show e t! (t)! fast enough as t! for the integral to be nite. Now! (t) isthe component of x (t) the homoclinic orbit of the unperturbed system. Hence for values of t close to the rate of approach of! (t) to the saddle point x is given by the linearization of the vector eld of the unperturbed system at the saddle point x =( ): i.e. the eigenvalues of 4 ; cos( ; ) ; where := < eigenvalues 5 4 ( ; ) = ; ( ) 5 + ; ( ; ) = = =
s u = ; ( +4 ( ; )) = s = ; ; ( +4 ( ; )) = < ; < (7) u = ; +( +4 ( ; )) = > (7) Where s is the negative (stable) eigenvalue and u is the positive (unstable) eigenvalue of the saddle point x : As t!,! (t) approaches as e st while as t! ;,! (t) approaches as e ut. Thus the quantity! (t)e t is of the order exp( s + )t as t! and exp( u + )t as t! ;: From (7) and (7) it follows that! (t)e t goes to zero exponentially as t!: (t) is the solution of equation (9) and (t)! as t!. Hence the integrals in the brackets are nite and well dened. 6 Conclusion Using the Melnikov technique for dissipative systems we have proved the phenomena of chaotic dynamics in the swing equation of a power systems. We have shown the presence of horseshoe chaos for = but the conclusion will hold true for small variation of around : It will be interesting to study the behaviour of the system with the explicit form of A(t): References [] F. M. Salam, J. Marsden, and P. Varaiya \Arnold Diusion in the Swing Equation of a Power System", IEEE Trans. on Circuits and syst., Vol. CAS-, No.8, 67-687, August 984 [] Nancy Kopell and Robert B. Washburn, \Chaotic Motions in the Two-Degree of Freedom Swing Equation". IEEE Trans. on Circuits and Systems, Vol. CAS-9, No., 78-745, [] S. Wiggins, Introduction to applied Nonlinear Dynamical Systems and Chaos. Springer- Verlag 99 [4] A. A. Andronov and C. E. Chaikin, Theory of Oscillation. Princeton. NJ. :Princeton Univ. Press [5] Fathi M. A. Salam,\ The Melnikov Technique for the Highly Dissipative Systems" SIAM J. Applied Mathematics, Vol. 47, April 987. 4
[6] Hirsch M. W., Pugh C. C., and Shub, M. Invariant Manifolds, Springer Lecture Notes in Mathematics, Berlin bf58, Springer-Verlag (977). [7] J. H. Sanders and F. Verhulst. Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag. 5