電汽熱動專業(yè)外文翻譯---電力系統(tǒng)混沌現(xiàn)象的建模與仿真（英文）

1、Applied Soft Computing 11 (2011) 103–110Contents lists available at ScienceDirectApplied Soft Computingjournal homepage: www.elsevier.com/locate/asocModeling and simulation of chaotic phenomena in electrical power system

2、sDeepak Kumar Lal, K.S. Swarup ?Department of Electrical Engineering, Indian Institute of Technology, Madras, Chennai 600036, Indiaa r t i c l e i n f oArticle history:Received 23 December 2007Received in revised form 30

3、 October 2009Accepted 15 November 2009Available online 18 November 2009Keywords:Nonlinear systemChaosHoff bifurcationDouble scroll equationDynamical systemLimit setsPower system instabilitya b s t r a c tModeling and sim

4、ulation of nonlinear systems under chaotic behavior is presented. Nonlinear systemsand their relation to chaos as a result of nonlinear interaction of different elements in the system arepresented. Application of chaotic

5、 theory for power systems is discussed through simulation results. Sim-ulation of some mathematical equations, e.g. Vander Pol’s equation, Lorenz’s equation, Duffing’s equationand double scroll equations are presented. T

6、heoretical aspects of dynamical systems, the existence ofchaos in power system and their dependency on system parameters and initial conditions using com-puter simulations are discussed. From the results one can easily u

7、nderstand the strange attractor andtransient stages to voltage collapse, angle instability or voltage collapse and angle divergence simultane-ously. Important simulation results of chaos for a model three bus system are

8、presented and discussed.© 2009 Elsevier B.V. All rights reserved.1. IntroductionChaotic phenomena have been drawing extensive attention invarious fields of natural science [1]. Recent developments in non-linear syst

9、em theories allow one to understand and analyze severalcomplex behaviors in power systems. Nonlinear phenomena suchas bifurcation and chaos in power systems has been observed in thepower system networks during the past f

10、ew years [2]. Disturbancesin power system causes change in parameters which result in thesystem exhibiting chaotic behavior. When chaos breaks, it entersinto different instability modes, which causes the power systems to

11、exhibit instability which needs to be avoided. Most of the physicalsystems in nature are nonlinear and as a result powerful mathemat-ical tools are required for analysis [3,4]. It is desirable to make linearassumptions w

12、henever a compromise can be obtained between thesimplicity of analysis and accuracy of results.Chaotic phenomena are one type of un-deterministic oscillationexisting in deterministic systems. They are related to random,

13、con-tinuous and bounded oscillation and not dynamically stable andmay face serious problems from an operation view point. The Hoffbifurcation and chaos limit the load-ability of the power systemand are unwanted phenomena

14、 [5]. For their complexity, mecha-nism of chaotic phenomena is very little known up to now. Thereis no generally accepted definition of chaos. Hence is called strange? Corresponding author at: Department of Electrical En

15、gineering, Indian Instituteof Technology, Madras, Electrical Science Block (ESB) 245-D, Chennai 600036, TamilNadu, India. Tel.: +91 44 2257 4440; fax: +91 44 2257 4402.E-mail address: swarup@ee.iitm.ac.in (K.S. Swarup).a

16、ttractor. Discovery of chaos enhances our understanding of com-plex and unpredictable behaviors arising from a wide variety ofsystems in engineering and sciences, mainly in nonlinear systemsresearch. Also, study on chaot

17、ic phenomena is one important partof power system stability studies [6,7]. In this paper the numeri-cal simulation of the mathematical relations for chaos occurring inpower systems have been simulated. The behavior of th

18、e systemunder various operating conditions is presented.The paper is organized as follows. Theoretical formulation andmathematical representation of chaos is given in Section 2. Section3 provides the steady-state behavio

19、r of nonlinear systems. Model-ing of chaotic behavior in power systems is described in Section 4.Section 5 provides the implementation aspects of the chaos. Chaosand instability in power systems is provided in Section 6.

20、 Importantconclusions are given in Section 7.2. Nonlinear dynamical systemsThree types of dynamical systems are presented with some use-ful facts from the theory of differential equations [1,8].2.1. Autonomous dynamical

21、systemsAn nth order autonomous dynamical system is defined by thestate equation˙ x = f (x) x(t0) = x0 (1)where ˙ x = dy/dt and x(t) ∈ ? are the state at time t and f. ? → ? iscalled the vector field.1568-4946/$ – see fro

22、nt matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.asoc.2009.11.001D.K. Lal, K.S. Swarup / Applied Soft Computing 11 (2011) 103–110 105Fig. 1. Limit cycle behavior of a nonlinear system. (a) Under damp

23、ed system for |x|? 1 with stable limit cycle. (b) Over-damped system |x|? 1 with unstable limit cycle.The phase trajectories for this equation will be diverged awayfrom limit cycle, and hence it will indicate the limit c

24、ycle will beunstable.3.3. Quasi-periodic solutionsA quasi-periodic solution is one that can be written as a sum ofperiodic functionsx(t) = ?ihi(t) (11)where hi has minimal period Ti and frequency fi = (1/Ti). Furthermore

25、, there exist a finite set of base frequencies {f1, f2, f3, . . ., fp}with the following properties:(i) it is linearly independent; that is, there does not exist a non-zero set of integers {k1, k2, k3, . . ., kp}; such t

26、hat k1f1 + k1f1 +k1f1 + · · · + kpfp.(ii) It forms a finite integral base for fi; that is, for each i, fi = k1f1 +k1f1 + k1f1 + · · · + kpfp for some integers {k1, k2, k3, . . ., kp}.In othe

27、r words, a quasi-periodic waveform is the sum of periodicwaveforms each of whose frequency is one of the various sums anddifferences of a finite set of base frequencies. Note that the basefrequencies are not uniquely def

28、ined, but that p is. A quasi-periodicsolution with p base frequencies is called p-periodic.Fig. 2. Two-dimensional trajectory of chaos in Lorenz system (for ? = 10, ? = 28, ˇ = 8/3). (a) Two dimensional corresponding to