IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 12, DECEMBER 2005 811

Chaotic and Subharmonic Oscillations of aNonlinear Power SystemXingwu Chen, Weinian Zhang, and Weidong Zhang

Abstract—In order to analyze complex oscillations with a largedeviation for a nonlinear nonautonomous power-transmissionsystem, heteroclinic and subharmonic bifurcations are discussedby technically computing Melnikov functions with the residue of acomplex function and elliptic integrals, which gives a condition ofparameters for chaotic oscillation and one for periodic oscillation.We describe the three-dimensional geometric structures of theseparameter regions and the geometric relations among them. Ac-cording to these regions, numerical simulations are implementedto demonstrate chaotic phenomena and subharmonic oscillations.

Index Terms—Chaos, heteroclinic bifurcation, nonlinear oscilla-tion, power system, subharmonic bifurcation.

I. INTRODUCTION

POWER-transmission systems oscillate periodically whenthey run normally. However, sometimes they perform ir-

regularly [1], [2], oscillating persistently with no definite am-plitude and no definite frequency, or oscillating suddenly, ran-domly and chaotically. Since such morbid oscillations are veryharmful to the security of power systems, great efforts are madein researching complex oscillations of power systems both intheory and practice [3]–[5].

Those morbid or complex oscillations actually reflect thenonlinearity of practical power systems. More and more ex-periments exhibit their rich nonlinear phenomena. Usually,nonlinear oscillations are displayed very explicitly in systemsof large power transmission, where large deviation (i.e., mo-tion far from equilibria) might occur. Recently, bifurcationphenomena and the caused nonlinear oscillations have beenstudied extensively, e.g., in [1], [8]–[16]. The Hopf bifurcationwas discussed in [9] and [10]. Quasi-periodic motions wereinvestigated in [11] for the single-machine infinite-bus (SMIB)system. Chaotic phenomena were observed numerically in [12]for a power system composed of two generator buses and a loadbus and confirmed by calculation of its Lyapunov exponentsand its broad-band spectrum. Reference [1] showed that voltagecollapse can occur “prior” to a saddle-node bifurcation anda chaotic blue sky bifurcation. The well-known Melnikov’smethod [17] is an effective way to discuss bifurcations ofperiodic orbits and homo-(hetero-) clinic orbits for dynam-ical systems. For a classical SMIB power-system model, thismethod was applied in [8] to give conditions under which the

Manuscript received September 4, 2004; revised January 30, 2005, and May31, 2005. This work was supported by NSFC (China), TRAPOYT, and ChinaMOE. This paper was recommended by Associate Editor J. L. Moiola.

X. Chen and W. Zhang are with the Department of Mathematics, SichuanUniversity, Chengdu 610064, China (e-mail: matzwn@sohu.com).

W. Zhang is with Chongqing Power College, Chongqing 400053, China.Digital Object Identifier 10.1109/TCSII.2005.853512

Fig. 1. Simple connection system.

heteroclinic orbit persists with small perturbations. Being atwo-dimensional autonomous system, the considered SMIBmodel has no possibilities to oscillate chaotically.

In order to investigate bifurcations from which chaos andsubharmonic oscillations are caused, consider a simple powersystem as shown in Fig. 1, where 1 and 2 are equivalent gen-erators of systems 1 and 2, respectively, and are equivalentmain adaptors of systems 1 and 2, respectively, is the charge,

is the switch, and is the linkwire. Such a system qualita-tively exhibits important aspects of behaviors of multimachinesystems and is relatively simple to study [18]. We hope to dis-play our analysis methods and computation techniques clearlyby discussing it.

Let be relative angle between voltagesof generators of the two systems at time . As given in [2, p. 18,eq. (1.51)], it satisfies

(1)

where are equivalent rotational inertia and equivalentdamp coefficient, respectively, is the maximum electricpower transmitting from system 1 to 2, is the machinerypower of generator 1. Without the last term ,i.e., , the system is the standard SMIB system [8].Considered in [2], and are the amplitude and frequencyof perturbation, which is the synthetic effect either from bigcharges which produce strong anti-electromotive force peri-odically or from errors caused by machinery rotation in thegenerators.

In this paper, we use Melnikov’s method to analyze the powersystem (1) for its heteroclinic bifurcations and subharmonicbifurcations. By computing curves of energy for its “ideal”system [i.e., (1) with vanished , and ] and the residueof a complex function for Melnikov functions technically, weobtain conditions for heteroclinic bifurcations and chaotic be-haviors. Furthermore, we calculate elliptic integrals for anotherMelnikov functions to show both inner periodic oscillationsand outer ones arising from subharmonic bifurcations. Then,we synthesize heteroclinic bifurcation conditions and sub-harmonic bifurcation conditions to describe relations among

, and geometrically in a three-dimensional parameterspace. Numerical simulations are implemented to demonstratechaotic phenomena and subharmonic oscillations, and finallythe practical implications of the analysis results are explainedin the section of conclusions for power systems.

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812 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 12, DECEMBER 2005

II. SIMPLIFIED SYSTEM AND IDEAL SYSTEM

Let . With the change of variables, , and , (1)

can be simplified as

(2)

where , and. In order to apply the theory in [17], we take

a small parameter such that ,where , replace with for convenience and rewrite(2) into the form

For a given , and , the values of arelarge as is chosen very small.

Consider its ideal system first, i.e., , and all vanish,i.e., the system , where . By periodic symmetry it suf-fices to consider it on the interval of the -axis. Being aHamiltonian system, it has center and two saddles

and . The Hamiltonian energy ofsystem is and orbits ofsystem are described by , where .It describes a pair of heteroclinic orbits connecting and

for , a family of periodic orbits around the centerinside for , and another family of periodic or-

bits outside for . The closed forms of these orbitsare obtained (as shown in the Appendix)

(3)

(4)

where , , and are elliptic functions [19]. For each

, has period

and have period

III. HETEROCLINIC BIFURCATIONS AND CHAOS

Consider system with and investigate how the pa-rameters and affect the heteroclinic orbits. As in [17],we calculate the Melnikov functions along

(5)

A heteroclinic orbit exists if one of has a zero. Chaoticoscillation occurs when the heteroclinic intersection betweenthe stable manifold and the unstable manifold is transversal.The transversality depends on whether the zero ofis simple, i.e., the derivative .

Theorem 1: (i) If the parameters , and satisfy

(6)

then, for a sufficiently small , system is chaotic in thehorse-shoe type. (ii) If condition (6) is an equality, then, near

, there is a bifurcation value at which quadratic hetero-clinic tangencies occur in system . (iii) If in (6) is replacedwith , then no heteroclinic orbits are preserved.

Proof: Difficulties arise from computation of .Clearly . Computing residues, we get

Then, from (5)

For , we similarly obtain

Thus, has zeros if and only if. In addition, the

zero is simple if and only if , i.e.,. Therefore, the strict inequality (6)

is obtained. Similarly, solving zeros of from, we get another

inequality

(7)

Since , and are all nonnegative in practice, condition (7)is stronger than (6). Thus, under (6), system is chaotic in thehorse-shoe type. On the other hand, both Melnikov functionsremain away from zero under the condition of (iii). So, the resultof (iii) is obvious.

Under condition (ii), the Melnikov function hasquadratic zeros at and (for integers ) because

,

, , , and. The result of (ii) follows from [17, Th.

4.5.4].

IV. SUBHARMONIC BIFURCATIONS

Here, we discuss system with for persistence ofperiodic orbits whose periods are multiples of , theperiod of the perturbation.

In order to investigate inside the heteroclinic loop we discussthe case , and compute the Melnikov functions forthe resonant periodic orbits. The resonance condition is

(8)

where are positive integers. It is easy to check that func-tion , defined in Section II, is monotonic in . Hence, a unique

CHEN et al.: CHAOTIC AND SUBHARMONIC OSCILLATIONS OF NONLINEAR POWER SYSTEM 813

can be determined by (8). As in [17], the sub-harmonic Melnikov function on is

(9)

where , ,, and Computing

elliptic integrals we getand

or is even

and is odd

where , and. Here, and are the complete elliptic inte-

gral of the first kind and the Legendre’s incomplete elliptic in-tegral of the second kind respectively (seen in [19]). From (9)

(10)

The definition of implies that . Define

(11)

Theorem 2: (i) If parameters and satisfy

(12)

then for sufficiently small there is a subharmonic orbit oforder (period ) near . (ii) If is oddand parameters and satisfy

(13)

then for each there is a bifurcation value near at whichsaddle-nodes of periodic orbits occur. (iii) If in (12) is re-placed with , then near there are no subharmonicorbits having period for all integers .

Proof: From (10) and (11), we see easily that (12) isa sufficient and necessary condition for to havea simple zero. Reference [17, Th. 4.6.2] implies the resultof (i). When the condition of (iii) is satisfied, the Melnikovfunction has no zeros and the result of (iii) is ob-vious. On the other hand, under (13) the functionhas quadratic zeros at and for integers be-cause , ,

,, when and is odd. Here, we note

that is independent of . Reference [17, Th. 4.6.3]implies the result of (ii) since for allintegers , as shown in Section V.

Similarly, we discuss outside the heteroclinic loop.The resonance condition of periodic orbits is the same as in (8)and is determined in uniquely by (8).The corresponding Melnikov function on is

where

,

, , and

other

Here , and. Similarly, we obtain the following results.

Theorem 3: (i) If parameters , and satisfy

(14)

then for sufficiently small there is a subharmonic orbitof order (period ) near . (ii) If (14) is anequality, then near there is a bifurcation value at whichsaddle-nodes of periodic orbits occur. (iii) If in (14) is re-placed with , then near there are no subharmonicorbits having period .

The Melnikov function on is given by

where is replaced by . Similarly, we obtain the following.Theorem 4: (i) If parameters , and satisfy

(15)

then for sufficiently small there is a subharmonic orbit oforder (period ) near . (ii) If condition (15)is an equality, then near there is a bifurcation value atwhich saddle-nodes of periodic orbits occur. (iii) If in (15) isreplaced with , then near there are no subharmonicorbits of period .

V. PARAMETER REGIONS AND NUMERICAL SIMULATIONS

Our theorems give criteria to judge whether a power system ofthe form (1) oscillates chaotically or subharmonically. The suf-ficiently small in those theorems, which was adopted techni-cally, does not affect our judgement. In fact, the simplification inSection II gives relations , ,

, . Condition (6) in Theorem 1is then equivalent to

(16)where is reduced in the ratio. Similarly, conditions (12), (14),and (15) in Theorems 2–4 are, respectively, equivalent to

(17)

(18)

(19)

814 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 52, NO. 12, DECEMBER 2005

Fig. 2. Pyramid � (h = 2).

Fig. 3. Pyramid � (0 < h < 2).

Fig. 4. � (h > 2) near .

Fig. 5. � (h > 2) near .

We geometrically describe those regions of parametersin the first octant of (i.e.,

), in which the power systemoscillates chaotically or subharmonically. Referring to [8] and[14], we choose the data

, and in(1). Then, (16)–(19) can be simplified,respectively, to

(20)

(21)

(22)

(23)

Let , and be regions of parameters defined by(20)–(23), respectively. They are all pyramids showed schemat-ically in Figs. 2–5.

Fig. 6. Relations of Figs. 2 –5.

By the definitions of , and, those regions have the following relations (seen Fig. 6).

from the lower side of in as .from the upper side of in

from the lower of in and from thelower of in as .

Continuing this work, we show the chaotic phenomenaand subharmonic oscillations. Choose the data

in (1). Calculate the period. We can check that (20) hold,

but (21) does not hold. i.e., parameters fall outside pyramidbut in pyramid . Our theorems imply that (1) oscillates

chaotically but has no subharmonic orbits of period . Simu-lating (1), we observe chaos in Fig. 7.

With the choice of , (20) does nothold, but (21) holds, i.e., parameters fall outside pyramid butin pyramid . By our theorems, (1) has a subharmonic orbit ofperiod but no chaos occurs. we observe a -periodic orbitin Fig. 8.

VI. CONCLUSION

In this paper, the Melnikov’s method is applied to ana-lyze nonlinear oscillations of power system (1). Regarding

as parameters we give criteria (16)–(19), for hete-roclinic bifurcations and subharmonic ones. Those bifurcationsindicate vicissitudes of the voltage state of the power systemin operation among regular oscillations of different periodsor between those regular oscillations and the morbid chaoticone as the damp coefficient , the machinery powerand the amplitude of perturbation varying. Those criteriaare geometrically described as different regions in the spaceof , that therefore suggests a mechanism for thedetection and control of such vicissitudes. Those regions ofparameters reveal the relations among for chaoticoscillation and subharmonic one of the voltage, indicate anexplicit direction for improving the management and enableus to design an appropriate controller [6], [7] so as to con-fine the change of period, avoid the chaos, or stabilize themorbid voltage oscillation for the power system. To our bestof knowledge, most known results on practical applications ofthe Melnikov’s method are given for two parameters, but oursis three. Our analysis results are demonstrated by numericalsimulations.

For the special case that , our Theorem 1 also impliesthe results on preservation of heteroclinic orbits given in [8]for a classical SMIB system. Additionally, Theorems 2–4 giveconditions for the SMIB system to preserve periodic orbits.

CHEN et al.: CHAOTIC AND SUBHARMONIC OSCILLATIONS OF NONLINEAR POWER SYSTEM 815

Fig. 7. (a) Chaos in (�; !; t). (b) Chaos in (�; !). (c) Chaos in (�; t). (d) Chaos in (!; t).

Fig. 8. Orbit of period 3T .

APPENDIX

In Section II, closed forms of orbits of system are solvedfrom the differential equation .It is easy to solve the equation for the pair of hetero-clinic orbits , where . For , the equation

is equivalent to .

With the change of variables , it fol-

lows that By the def-

initions of elliptic functions, we see that

and , and therefore,(3)–(5) are obtained.

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