Research ArticleAnalysis on Nonlinear Dynamic Characteristic of SynchronousGenerator Rotor System

Xiaodong Wang 12 and Caiqin Song 3

1School of Management Science and Engineering Shandong Normal University Jinan 250014 China2School of Astronautics Harbin Institute of Technology Harbin 150001 China3School of Mathematical Sciences University of Jinan Jinan 250022 China

Correspondence should be addressed to Xiaodong Wang wangxdsdnueducn and Caiqin Song songcaiqin1983163com

Received 18 October 2018 Revised 3 December 2018 Accepted 10 December 2018 Published 1 January 2019

Academic Editor Marcelo Messias

Copyright copy 2019 Xiaodong Wang and Caiqin Song This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper focuses on the swing oscillation process of the synchronous generator rotors in a three-machine power systemWith thehelp of bifurcation diagram time history phase portrait Poincare section and frequency spectrum the complex dynamical behav-iors and their evolution process are detected clearly in this power systemwith varying perturbation related parameters and differentsystem parameters Furthermore combining the qualitative and quantitative characteristics of the chaotic motion different pathsleading to chaos coexisting in this system have been found The Wolf method has been introduced to calculate the correspondinglargest Lyapunov exponent which is used to verify the occurrence of chaotic motionThese results obtained in this paper will con-tribute to a better understanding of nonlinear dynamic behaviors of synchronous generator rotors in a three-machine power system

1 Introduction

When power system suffers disturbances it can cause param-eters to change This will result in the system exhibitingabundant nonlinear dynamic behaviors including chaoticbehavior Chaotic oscillation in power systemmay cause volt-age collapse and even catastrophic blackouts Consequentlychaotic oscillation in the power system often makes a greatthreat to the stability of the power grid [1 2]

Discovery of chaos can enrich our understanding ofcomplex and unpredictable nonlinear behaviors arising inpower system therefore it is an important part of powersystem stability research In the last decades the nonlineardynamic behaviors including chaotic oscillation in powersystem have attracted much attention of the researchers Forsingle-machine infinite bus (SMIB) power system Wei etal [3 4] examined how a Gaussian white noise affectedthe dynamic behaviors of power system by means of arandom Melnikov method Zhu and Mohler [5] made aHopf bifurcation analysis with subsynchronous resonance(SSR) Nayfeh et al [6 7] investigated the period-doubling

bifurcations chaotic motions and unbounded motions (lossof synchronism) on a single-machine quasi-infinite bussystem Duan et al [8] studied the bifurcations associatedwith subsynchronous resonance of a SMIB power systemwith series of capacitor compensation Chen et al [9] studiedthe chaotic control and identification problem of a SMIBpower system where the power of the machine was assumedto be a simple harmonic quantity Moreover in order todiscuss bifurcations of periodic orbits and homo-(hetero-)clinic orbits of dynamical systems Melnikovrsquos method hasbeen effectively applied in the classical SMIB power systemmodels [10ndash14]

For a two-machine power system under some particularconditions Yuan and Sun [15] studied the occurrence ofchaotic phenomenon by using Melnikovrsquos method Ueda etal [16] investigated a two-generator electric power systemmodel by considering that the infinite busmaintains a voltageof fixed amplitude with a small periodic fluctuation in thephase angle

For a three-machine power system Majidabad et al[17] designed two novel nonlinear fractional-order sliding

HindawiComplexityVolume 2019 Article ID 3603172 14 pageshttpsdoiorg10115520193603172

2 Complexity

G2 G3G1

L1 L2 L3

1 2 3

4Infinite Bus

Figure 1 Three-machine infinite bus power system

mode controllers and applied them in a three-machine powersystem with two types of faults Jiang et al [18] proposed aWeierstrass-based numerical method for computing damp-ing torque of a three-machine power system during tran-sient period to determine appropriate and accurate dampingterms for power system dynamic simulation The bifurcationphenomena in a power system with three machines andfour buses were investigated by applying bifurcation theoryand harmonic balance method in [19] In addition transientstability analysis of a three-machine nine bus power systemwas carried out by considering a three-phase fault at busbars7 and 4 with the effect of various fault-clearing times [20]

Although the chaotic oscillations in simple power system(especially for SMIB) have been studied with many mathe-matical models and theories to the best of our knowledgethe chaotic oscillation research of three-machine powersystem with power disturbance is not systemic and thoroughenough In order to deal with the issue mentioned above thispaper provides an efficient analysis method of assessing thedynamic impacts of critical system parameters on dynamiccharacteristics of the power system The dynamic charac-teristics of the power system model with varying differentsystem parameters and perturbation related parameters areinvestigated for the first time

The synchronous generators are the main source ofenergy for the power system and they are also the core ofthe entire grid The motivation of this paper is to detectthe effect of several critical system parameters on the non-linear dynamic characteristics of synchronous generators inthree-machine power system Firstly the swing equationsdescribing the motions of the synchronous generator rotorsare established Based on the swing equations a more in-depth research on the dynamical properties of the system hasbeen carried out In addition the Wolf method is introducedto calculate the largest Lyapunov exponent which enablesus to identify the bifurcation points and to analyze thedynamic characteristics of the system influenced by thepower disturbance With the aid of bifurcation diagramtime history phase portrait Poincare section and frequencyspectrum the detailed numerical simulations are also carriedout

The remainder of this paper is organized as follows InSection 2 the swing equations describing the motions of thesynchronous generator rotors are formulated In Section 3 inorder to investigate the effects of the system parameters andperturbation related parameters on the dynamic character-istics of the system some numerical simulations are carriedout In Section 4 some discussions have been made Finallyconcluding remarks of this paper are presented in Section 5

2 System Configuration and Modelling

Synchronous motor as the significant equipment is the keyto studying the dynamic characteristic of the power systemConsider a general three-machine infinite bus power systemas shown in Figure 1 the swing equations describing thegenerator rotors are written as120575119894 = 1205961198942119867119894120596119904119894 = 119875119898119894 minus 119863119894120596119894 minus 119875119890119894

(1)

119875119890119894

= 1198642119894119866119894119894

+ 1198641198943

sum119895=1119895 =119894

119864119895 [119866119894119895 cos (120575119894 minus 120575119895) + 119861119894119895 sin (120575119894 minus 120575119895)]

119894 = 1 2 3

(2)

Here 120575119894 is the rotor angle of the ith generator 120596119904 is the syn-chronous speed 120596119894 is the deviation between the synchronousspeed and the rotor angular velocity119867119894 is the generator rotorinertia119875119898119894 is themechanical input power to the ith generator119875119890119894 is the electromagnetic power and 119863119894 is the damping ofthe ith generator 119864119894 is the magnitude of voltage behind thetransient reactance of the ith generator 119866119894119894 119861119894119894 are the self-conductance and self-admittance of the ith node 119866119894119895 119861119894119895 arethe mutual conductance and mutual admittance of the ithnode

Complexity 3

062 0625 063 0635 06402

025

03

035

04

045

05

VB1

1

(a)

062 0625 063 0635 064minus05

minus04

minus03

minus02

minus01

0

01

02

VB1

Larg

est L

yapu

nov

Expo

nent

(b)

Figure 2 (a) Bifurcation diagram (b) Largest Lyapunov exponent spectrum

Taking into account the practical engineering applicationan assumption is made about the impedance angle theelectromagnetic power 119875119890119894 is denoted in the format as

119875119890119894 = 1198642119894119866119894119894 + 119864119894

3

sum119895=1119895 =119894

119864119895119861119894119895 sin (120575119894 minus 120575119895) 119894 = 1 2 3 (3)

Therefore it can be seen that the expression of everyelectromagnetic output power is a multivariable functionrelated to the relative angle between its own rotor angle andthe other two generator angles

Considering that the generators are connected with theinfinite bus node combining (1) and (3) let 119875119894 = 119875119898119894 minus 1198642119894119866119894119894the following equation can be obtained120575119894 = 120596119894

2119867119894120596119904119894 = 119875119894 minus 119863119894120596119894 minus 119864119894

3

sum119895=1119895 =119894

119864119895119861119894119895 sin (120575119894 minus 120575119895)

minus 1198641198941198811198611198611198944 sin (120575119894 minus 120575119861)

119894 = 1 2 3

(4)

119881119861 is the voltage of the busbar 120575119861 is the phase angle of thebusbar and 1198611198944 is the admittance of the ith machine to theinfinite busbar Due to the fluctuation of reactive and activepower the fluctuation of infinite bus voltage is caused in thefollowing form

119881119861 = 1198811198610 + 1198811198611 cosΩ119905 (5)where 1198811198610 is the magnitude of the voltage of the busbar 1198811198611is the disturbancemagnitude of the voltage of the busbar andΩ is the disturbance frequency

3 Numerical Simulations

In order to get the effects of the critical system parameters onthe dynamic characteristics of this power system the fourth-order Runge-Kutta method is employed in the following

computations According to the qualitative and quantitativecharacteristics of chaotic oscillations we try to find out theroad to chaos and to understand the oscillation response pro-cess of the synchronous generator rotor In the followingwiththe aid of bifurcation diagram time history phase portraitPoincare map and frequency spectrum the characteristicsof swing oscillation of each generator rotor are illustratedIn addition the Wolf method is introduced to calculate thelargest Lyapunov exponent which enables us to identify thebifurcation points and different oscillation states [21]

31 Influence of Disturbance Amplitude of Infinite Bus VoltageThe calculation parameters of the considered system are H1= 137 H2 = 2 H3 = 137 D1 = 0008 D2 = 0008 D3 = 0008120596s = 120120587 P1 = 1 P2 = 05 P3 = 1 E1 = 127 E2 = 127 E3= 127 119881B0 = 1 Ω = 2 B12 = 01 B13 = 06 B23 = 1 B14 = 2B24 = 155 and B34 = 2 In this subsection the influences ofthe infinite bus voltage V1198611on the dynamic behaviors of thethree-machine power system are presented The bifurcationdiagram is plotted in Figure 2(a) for different values of 119881B1to show the responses of the system under different infinitebus voltage It can be seen that the oscillation response of thesystem leads to chaotic motion through the period-doublingbifurcation and the chaotic parameter region is very narrowEspecially after the parameter crosses the chaotic parameterregion the system exhibits a rotating motion orbit andthen the generator loses synchronization and in this casethe system is unstable According to the Wolf method theLyapunov exponent is calculated and the correspondingLyapunov exponent spectrum is shown in Figure 2(b) withvarying values of 119881B1 In the case of 119881B1 lt 06385 thelargest Lyapunov component is always negative which iscorresponding to periodic motion window in the bifurcationdiagramThis proves that system swing oscillation starts withperiodic motion When 119881B1 = 06315 119881B1 = 06369 and119881B1 = 06382 the largest Lyapunov component will changefrom negative to zero and finally to positive It indicates thatthe period-doubling bifurcation behavior occurs If 119881B1 =06385 the largest Lyapunov component is positive This

4 Complexity

01

02

03

04

05

06

1

80 100 120 14060t

(a)

minus02

minus01

0

01

02

1

04 045 050351

(b)

minus02

minus01

0

01

02

1

02 04 06 08 101

(c)

000200400600801

012014

Am

plitu

de1 2 3 4 50

Frequency

(d)

Figure 3 Period 1 motion (1198811198611 = 02) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

proves that in the system chaotic motion occurs ComparingFigures 2(a) and 2(b) it can be found that 119881B1 has the samevalue at every bifurcation point

For the purpose of clearly displaying the evolving processof swing oscillation as the infinite bus voltage 1198811198611 changesFigures 3ndash9 show the time history phase portrait Poincaremap and Frequency spectrum In the case of 1198811198611 = 02the dynamic behavior of the system is periodic 1 motionForm 1198811198611 = 05 to 1198811198611 = 063 it is worthy to point out thatthe dynamic behaviors are also periodic 1 motion but thereexist many super-harmonic components in the frequencyspectrum As shown in Figures 6ndash8 in the case of 1198811198611 =0635 1198811198611 = 0637 and 1198811198611 = 06382 the system presentsperiodic 2 oscillation periodic 4 oscillation and periodic8 oscillation respectively In addition the response of thesystem is chaotic motion at 1198811198611 = 064 (shown in Figure 9)In this case the corresponding largest Lyapunov exponentis 00082 it is further confirmed that chaotic oscillation hasoccurred

32 Influence of Damping In this section the damping onthe dynamic characteristics of the rotor system is discussedFor the sake of comparison herein choose the dampingcoefficient as the bifurcation parameter the other parametervalues are the same as the ones in the previous section Thebifurcation diagram and the corresponding largest Lyapunovexponent diagram are illustrated in Figures 10(a) and 10(b)respectively

From Figure 10 it can be seen that as the dampingcoefficient increases the dynamic responses of the system

undergo an inversed bifurcation process namely chaoticmotion997888rarrperiodic 8 motion997888rarrperiodic 4 motion997888rarrperi-odic 2motion997888rarrperiodic 1 motionTherefore increasing thedamping coefficient can prevent the occurrence of chaoticmotion Figures 11ndash14 have given the time history phaseportrait Poincare map and frequency spectrum under dif-ferent values of damping These figures have clearly shownthe evolution process of the dynamic response of the systemas the damping changes

33 Influence of the Basic Voltage 119881B0 Due to the active andreactive power of the load change it may cause a changein the basic value of the infinite bus voltage Thus it alsoaffects the dynamic behavior of the system Choose 1198811198610 asthe bifurcation parameter Figure 15 shows the bifurcationdiagram and the corresponding largest Lyapunov exponentof the dynamic response as 1198811198610 changes from small to largeFrom Figure 15 it can be seen that the dynamic response ofthe system also presents an inversed bifurcation process

34 Influence of the Perturbation Frequency It is interestingto note that at Ω = 196 a new route to chaos via quasi-periodic torus rupture in this system has been observed As1198811198611 changes from small to large Figures 16 and 17 have givenbifurcation diagram and corresponding largest Lyapunovexponent diagram respectively Moreover the results showthat the swing oscillation process is periodic quasi-periodicchaotic and out-of-step In order to show the evolution pro-cess more clearly the time history phase diagram Poincaresection and frequency spectrum are plotted (Figures 18ndash21)

Complexity 5

0

02

04

06

08

1

1

15 20 25 30 35 4010t

(a)

minus15

minus1

minus05

0

05

1

1

04 06 08 1021

(b)

minus15

minus1

minus05

0

05

1

1

01 02 03 04 05 0601

(c)

0

005

01

015

02

025

03

Am

plitu

de

5 100Frequency

(d)

Figure 4 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

25 30 35 4020t

(a)

minus10

minus5

0

5

10

1

05 1 15 201

(b)

minus5

0

5

1

02 04 06 08 101

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 5 Period 1 motion (1198811198611 = 063) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

6 Complexity

minus05

0

05

1

15

2

1

50 55 60 6545t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus3

minus2

minus1

0

1

2

1

01 02 03 04 05 0601

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 6 Period 2 motion (1198811198611 = 0635) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

1

15 20 25 3010t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus3

minus2

minus1

0

1

2

3

1

01 02 03 04 05 0601

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 200Frequency

(d)

Figure 7 Period 4 motion (1198811198611 = 0637) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 7

0

05

1

15

2

1

80 90 100 11070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 035 04 045 05 0550251

(c)

0

01

02

03

04

05

Am

plitu

de

2 4 6 8 100Frequency

(d)

Figure 8 Period 8 motion (1198811198611 = 06382) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

100 150 20050t

(a)minus05 0 05 1 15 2 25

minus15

minus10

minus5

0

5

10

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

005

01

015

02

025

03

Am

plitu

de

5 10 150Frequency

(d)

Figure 9 Chaotic motion (119881119861 = 064) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

8 Complexity

0009 001 0011 0012 00130008D1

02

03

04

051

(a)

minus05

minus04

minus03

minus02

minus01

0

01

larg

est L

yapu

nov

Expo

nent

0009 001 0011 00120008D1

(b)

Figure 10 (a) Bifurcation diagram (b) Largest Lyapunov exponent

minus05

0

05

1

15

2

1

30 40 50 60 70 8020t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 035 04 045 050251

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 11 Period 8 motion (D=000835) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

More concretely in the case of 1198811198611 = 05 the dynamicresponse is period 1 motion (shown in Figure 18) When1198811198611 = 07 the dynamic response is in almost periodicmotion the corresponding Poincare section is a circle (shownin Figure 20) In the case of 1198811198611 = 078 there existschaotic motion (Figure 21) After the parameter 1198811198611 passesthrough chaotic region the synchronous generator losessynchronization

4 Discussions

The chaotic dynamics may appear in power system whichmake a great threat to the stability of the power systemThere-fore many researchers pay much attention to the dynamic

behavior and chaotic mechanism especially for SMIB powersystem [3ndash9 22] From the point of nonlinear dynamic anal-ysis a few works have addressed the dynamic characteristicabout three-machine power system subjected to load distur-bance In this paper a relatively deep and systematic study ofdynamic response in three-machine power system subjectedto load disturbance has been carried out This paper aimsto demonstrate the complete transition process of differentdynamic behaviors by combining qualitative and quantitativeanalysis Comparing the results about simple power system(eg SMIB power system) [9ndash15] a new route to chaos viaquasi-periodic torus rupture has been found Moreover itis worthy to point out that in our considered system thereexist many super-harmonic components in the frequency

Complexity 9

minus05

0

05

1

15

2

1

35 40 45 50 5530t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 12 Period 4 motion (D=00085) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

75 80 85 90 95 10070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus6

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 13 Period 2 motion (D=0009) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

10 Complexity

minus05

0

05

1

15

2

1

280 285 290 295 300275t

(a)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(b)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(c)

0

01

02

03

04

05

06

Am

plitu

de5 10 150

Frequency

(d)

Figure 14 Period 1 motion (D=001) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

1 1005 101 1015 102 102502

03

04

05

VB0

3

(a)

1 1005 101 1015 102 1025minus05

minus04

minus03

minus02

minus01

0

01

02

VB0

Larg

est L

yapu

nov

Expo

nent

(b)

Figure 15 (a) Bifurcation diagram (b) Largest Lyapunov exponent

spectrum of the swing oscillation response Attention shouldbe paid to these new phenomena in engineering practice Inaddition for three-machine power system our future workwill investigate the mechanism of chaotic oscillation occur-rence by improving Melnikov analysis such that effectivemeasures can be taken in time to avoid system collapse

5 Conclusions

This paper has investigated the effects of critical systemparameters on dynamic characteristics of synchronous gen-erator rotors in a three-machine power system subjected

to load disturbance The swing equations describing themotions of the synchronous generator rotors are estab-lished Based on these swing equations with the help ofbifurcation diagrams largest Lyapunov exponent spectrumsphase portraits Poincare map and frequency spectrum theinfluence of system parameters on dynamic behaviors isshown clearly The Wolf method is introduced to calculatethe largest Lyapunov exponent which is used to verify theoccurrence of chaotic motion Moreover different pathsleading to chaos coexisting in this system have been foundThey are period-doubling cascading bifurcations to chaosinduced by changing the infinite bus voltage magnitude and

Complexity 11

05 055 06 065 07 07502

03

04

05

06

07

08

09

VB1

3

Figure 16 Bifurcation diagram

05 055 06 065 07 075minus05

minus04

minus03

minus02

minus01

0

01

VB1

Larg

est L

yapu

nov

Expo

nent

Figure 17 Largest Lyapunov exponent

minus02

0

02

04

06

08

1

3

16 17 18 19 2015t

(a)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(b)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(c)

0

005

01

015

02

Am

plitu

de

20 30 40 50 60 7010Frequency

(d)

Figure 18 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

12 Complexity

minus04minus02

002040608

112

3

28 30 3226t

(a)

minus6

minus4

minus2

0

2

4

6

3

02 04 06 0803

(b)

minus4

minus3

minus2

minus1

0

3

045 05 055 06 065043

(c)

0

005

01

015

02

Am

plitu

de

10 20 30 40 500Frequency

(d)

Figure 19 Quasi-periodic motion (1198811198611 = 06) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus04

minus02

0

02

04

06

08

3

21 22 23 24 25 2620t

(a)

minus3

minus2

minus1

0

1

2

3

3

03 035 04 0450253

(b)

minus2

minus15

minus1

minus05

0

05

3

0355 036 0365 037 0375 0380353

(c)

0

002

004

006

008

Am

plitu

de

10 20 30 400Frequency

(d)

Figure 20 Quasi-periodic motion (1198811198611 = 07) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 13

minus1

minus05

0

05

1

15

3

10 20 30 40 50 600t

(a)

minus30

minus20

minus10

0

10

20

30

3

050 1 15minus05minus13

(b)

minus30

minus20

minus10

0

10

20

30

3

04 06 08 1023

(c)

0005

01015

02025

03035

04

Am

plitu

de10 20 30 400

Frequency

(d)

Figure 21 Chaotic motion (1198811198611 = 078) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

quasi-periodic torus rupture to chaos induced by changingthe disturbance frequency of the infinite bus voltage Theseresults will contribute to a better understanding of the non-linear dynamic behaviors of synchronous generator rotors inthree-machine power system

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This research is supported byHigher Educational Science andTechnology Program of Shandong Province China (Grantno J18KA235) Shandong Provincial Natural Science Foun-dation China (Grants nos ZR2016AP06 ZR2018QA005ZR2018BA018 and ZR2018BA021) and National NaturalScience Foundation of China (Grants nos 11501246 and61703251)

References

[1] P M Anderson and A A Fouad Power System Control andStability Wiley-IEEE Press New York NY USA 2nd edition2002

[2] H Ma F Min and Y Wang ldquoNonlinear dynamic analysis andsurface sliding mode controller based on low pass filter forchaotic oscillation in power system with power disturbancerdquoChinese Journal of Physics vol 56 no 5 pp 2488ndash2499 2018

[3] D Q Wei and X S Luo ldquoNoise-induced chaos in single-machine infinite-bus power systemsrdquo EPL (Europhysics Letters)vol 86 no 5 Article ID 50008 2009

[4] D Q Wei B Zhang D Y Qiu and X S Luo ldquoEffect of noiseon erosion of safe basin in power systemrdquo Nonlinear Dynamicsvol 61 no 3 pp 477ndash482 2010

[5] W Zhu R Mohler R Spee W Mittelstadt and D Maratuku-lam ldquoHopf bifurcations in a SMIB power system with SSRrdquoIEEETransactions on Power Systems vol 11 no 3 pp 1579ndash15841996

[6] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaosand instability in a power system Subharmonic-resonant caserdquoNonlinear Dynamics vol 2 no 1 pp 53ndash72 1991

[7] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaos andinstability in a power system -Primary resonant caserdquoNonlinearDynamics vol 1 no 4 pp 313ndash339 1990

[8] X Duan J Wen and S Cheng ldquoBifurcation analysis foran SMIB power system with series capacitor compensationassociated with sub-synchronous resonancerdquo Science ChinaTechnological Sciences vol 52 no 2 pp 436ndash441 2009

[9] H K Chen T N Lin and J H Chen ldquoDynamic analysiscontrolling chaos and chaotification of a SMIB power systemrdquoChaos Solitons amp Fractals vol 24 no 5 pp 1307ndash1315 2005

[10] L F C Alberto and N G Bretas ldquoApplication of Melnikovrsquosmethod for computing heteroclinic orbits in a classical SMIBpower system modelrdquo IEEE Transactions on Circuits and Sys-tems I Fundamental eory and Applications vol 47 no 7 pp1085ndash1089 2000

14 Complexity

[11] W N Zhang and W D Zhang ldquoChaotic Ocillation of aNonlinear Power Systemrdquo Applied Mathematics Mechanics vol20 no 10 pp 1175ndash1182 1999

[12] L Zhou and F Chen ldquoChaotic dynamics for a class of single-machine-infinite bus power systemrdquo Journal of Vibration andControl vol 24 no 3 pp 582ndash587 2018

[13] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005

[14] X Wang Y Chen G Han and C Song ldquoNonlinear dynamicanalysis of a single-machine infinite-bus power systemrdquoAppliedMathematical Modelling vol 39 no 10-11 pp 2951ndash2961 2015

[15] B Yuan and Q H Sun ldquoChaos in the multi-machine powersystemrdquo Automation of Electric Power Systems vol 19 no 2 pp26ndash31 1995

[16] Y Ueda Y Ueda H B Stewart and R H Abraham ldquoNonlinearresonance in basin portraits of two coupled swings under peri-odic forcingrdquo International Journal of Bifurcation and Chaosvol 8 no 6 pp 1183ndash1197 1998

[17] S S Majidabad H T Shandiz and A Hajizadeh ldquoNonlinearfractional-order power system stabilizer for multi-machinepower systems based on sliding mode techniquerdquo InternationalJournal of Robust and Nonlinear Control vol 25 no 10 pp1548ndash1568 2015

[18] N Jiang and H-D Chiang ldquoDamping Torques of multi-machine power systems during transient behaviorsrdquo IEEETransactions on Power Systems vol 29 no 3 pp 1186ndash1193 2014

[19] Y Chang X Wang and D Xu ldquoBifurcation Analysis of aPower System Model with Three Machines and Four BusesrdquoInternational Journal of Bifurcation and Chaos vol 26 no 5 pp165ndash182 2016

[20] M A Salam M A Rashid Q M Rahman and M RizonldquoTransient stability analysis of a three-machine nine bus powersystem networkrdquo Engineering Letters vol 22 no 1 pp 1ndash7 2014

[21] A Wolf J B Swift and H L Swinney ldquoDetermining Lyapunovexponents froma time seriesrdquoPhysicaDNonlinear Phenomenavol 16 no 3 pp 285ndash317 1985

[22] M A Salam ldquoTransient stability analysis of a power systemwith one generator connected to an infinite bus systemrdquoInternational Journal of Sustainable Energy vol 33 no 2 pp251ndash260 2014

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

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Submit your manuscripts atwwwhindawicom

Complexity 3

062 0625 063 0635 06402

025

03

035

04

045

05

VB1

1

(a)

062 0625 063 0635 064minus05

minus04

minus03

minus02

minus01

0

01

02

VB1

Larg

est L

yapu

nov

Expo

nent

(b)

Figure 2 (a) Bifurcation diagram (b) Largest Lyapunov exponent spectrum

Taking into account the practical engineering applicationan assumption is made about the impedance angle theelectromagnetic power 119875119890119894 is denoted in the format as

119875119890119894 = 1198642119894119866119894119894 + 119864119894

3

sum119895=1119895 =119894

119864119895119861119894119895 sin (120575119894 minus 120575119895) 119894 = 1 2 3 (3)

Therefore it can be seen that the expression of everyelectromagnetic output power is a multivariable functionrelated to the relative angle between its own rotor angle andthe other two generator angles

Considering that the generators are connected with theinfinite bus node combining (1) and (3) let 119875119894 = 119875119898119894 minus 1198642119894119866119894119894the following equation can be obtained120575119894 = 120596119894

2119867119894120596119904119894 = 119875119894 minus 119863119894120596119894 minus 119864119894

3

sum119895=1119895 =119894

119864119895119861119894119895 sin (120575119894 minus 120575119895)

minus 1198641198941198811198611198611198944 sin (120575119894 minus 120575119861)

119894 = 1 2 3

(4)

119881119861 is the voltage of the busbar 120575119861 is the phase angle of thebusbar and 1198611198944 is the admittance of the ith machine to theinfinite busbar Due to the fluctuation of reactive and activepower the fluctuation of infinite bus voltage is caused in thefollowing form

119881119861 = 1198811198610 + 1198811198611 cosΩ119905 (5)where 1198811198610 is the magnitude of the voltage of the busbar 1198811198611is the disturbancemagnitude of the voltage of the busbar andΩ is the disturbance frequency

3 Numerical Simulations

In order to get the effects of the critical system parameters onthe dynamic characteristics of this power system the fourth-order Runge-Kutta method is employed in the following

computations According to the qualitative and quantitativecharacteristics of chaotic oscillations we try to find out theroad to chaos and to understand the oscillation response pro-cess of the synchronous generator rotor In the followingwiththe aid of bifurcation diagram time history phase portraitPoincare map and frequency spectrum the characteristicsof swing oscillation of each generator rotor are illustratedIn addition the Wolf method is introduced to calculate thelargest Lyapunov exponent which enables us to identify thebifurcation points and different oscillation states [21]

31 Influence of Disturbance Amplitude of Infinite Bus VoltageThe calculation parameters of the considered system are H1= 137 H2 = 2 H3 = 137 D1 = 0008 D2 = 0008 D3 = 0008120596s = 120120587 P1 = 1 P2 = 05 P3 = 1 E1 = 127 E2 = 127 E3= 127 119881B0 = 1 Ω = 2 B12 = 01 B13 = 06 B23 = 1 B14 = 2B24 = 155 and B34 = 2 In this subsection the influences ofthe infinite bus voltage V1198611on the dynamic behaviors of thethree-machine power system are presented The bifurcationdiagram is plotted in Figure 2(a) for different values of 119881B1to show the responses of the system under different infinitebus voltage It can be seen that the oscillation response of thesystem leads to chaotic motion through the period-doublingbifurcation and the chaotic parameter region is very narrowEspecially after the parameter crosses the chaotic parameterregion the system exhibits a rotating motion orbit andthen the generator loses synchronization and in this casethe system is unstable According to the Wolf method theLyapunov exponent is calculated and the correspondingLyapunov exponent spectrum is shown in Figure 2(b) withvarying values of 119881B1 In the case of 119881B1 lt 06385 thelargest Lyapunov component is always negative which iscorresponding to periodic motion window in the bifurcationdiagramThis proves that system swing oscillation starts withperiodic motion When 119881B1 = 06315 119881B1 = 06369 and119881B1 = 06382 the largest Lyapunov component will changefrom negative to zero and finally to positive It indicates thatthe period-doubling bifurcation behavior occurs If 119881B1 =06385 the largest Lyapunov component is positive This

4 Complexity

01

02

03

04

05

06

1

80 100 120 14060t

(a)

minus02

minus01

0

01

02

1

04 045 050351

(b)

minus02

minus01

0

01

02

1

02 04 06 08 101

(c)

000200400600801

012014

Am

plitu

de1 2 3 4 50

Frequency

(d)

Figure 3 Period 1 motion (1198811198611 = 02) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

proves that in the system chaotic motion occurs ComparingFigures 2(a) and 2(b) it can be found that 119881B1 has the samevalue at every bifurcation point

For the purpose of clearly displaying the evolving processof swing oscillation as the infinite bus voltage 1198811198611 changesFigures 3ndash9 show the time history phase portrait Poincaremap and Frequency spectrum In the case of 1198811198611 = 02the dynamic behavior of the system is periodic 1 motionForm 1198811198611 = 05 to 1198811198611 = 063 it is worthy to point out thatthe dynamic behaviors are also periodic 1 motion but thereexist many super-harmonic components in the frequencyspectrum As shown in Figures 6ndash8 in the case of 1198811198611 =0635 1198811198611 = 0637 and 1198811198611 = 06382 the system presentsperiodic 2 oscillation periodic 4 oscillation and periodic8 oscillation respectively In addition the response of thesystem is chaotic motion at 1198811198611 = 064 (shown in Figure 9)In this case the corresponding largest Lyapunov exponentis 00082 it is further confirmed that chaotic oscillation hasoccurred

32 Influence of Damping In this section the damping onthe dynamic characteristics of the rotor system is discussedFor the sake of comparison herein choose the dampingcoefficient as the bifurcation parameter the other parametervalues are the same as the ones in the previous section Thebifurcation diagram and the corresponding largest Lyapunovexponent diagram are illustrated in Figures 10(a) and 10(b)respectively

From Figure 10 it can be seen that as the dampingcoefficient increases the dynamic responses of the system

undergo an inversed bifurcation process namely chaoticmotion997888rarrperiodic 8 motion997888rarrperiodic 4 motion997888rarrperi-odic 2motion997888rarrperiodic 1 motionTherefore increasing thedamping coefficient can prevent the occurrence of chaoticmotion Figures 11ndash14 have given the time history phaseportrait Poincare map and frequency spectrum under dif-ferent values of damping These figures have clearly shownthe evolution process of the dynamic response of the systemas the damping changes

33 Influence of the Basic Voltage 119881B0 Due to the active andreactive power of the load change it may cause a changein the basic value of the infinite bus voltage Thus it alsoaffects the dynamic behavior of the system Choose 1198811198610 asthe bifurcation parameter Figure 15 shows the bifurcationdiagram and the corresponding largest Lyapunov exponentof the dynamic response as 1198811198610 changes from small to largeFrom Figure 15 it can be seen that the dynamic response ofthe system also presents an inversed bifurcation process

34 Influence of the Perturbation Frequency It is interestingto note that at Ω = 196 a new route to chaos via quasi-periodic torus rupture in this system has been observed As1198811198611 changes from small to large Figures 16 and 17 have givenbifurcation diagram and corresponding largest Lyapunovexponent diagram respectively Moreover the results showthat the swing oscillation process is periodic quasi-periodicchaotic and out-of-step In order to show the evolution pro-cess more clearly the time history phase diagram Poincaresection and frequency spectrum are plotted (Figures 18ndash21)

Complexity 5

0

02

04

06

08

1

1

15 20 25 30 35 4010t

(a)

minus15

minus1

minus05

0

05

1

1

04 06 08 1021

(b)

minus15

minus1

minus05

0

05

1

1

01 02 03 04 05 0601

(c)

0

005

01

015

02

025

03

Am

plitu

de

5 100Frequency

(d)

Figure 4 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

25 30 35 4020t

(a)

minus10

minus5

0

5

10

1

05 1 15 201

(b)

minus5

0

5

1

02 04 06 08 101

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 5 Period 1 motion (1198811198611 = 063) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

6 Complexity

minus05

0

05

1

15

2

1

50 55 60 6545t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus3

minus2

minus1

0

1

2

1

01 02 03 04 05 0601

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 6 Period 2 motion (1198811198611 = 0635) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

1

15 20 25 3010t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus3

minus2

minus1

0

1

2

3

1

01 02 03 04 05 0601

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 200Frequency

(d)

Figure 7 Period 4 motion (1198811198611 = 0637) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 7

0

05

1

15

2

1

80 90 100 11070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 035 04 045 05 0550251

(c)

0

01

02

03

04

05

Am

plitu

de

2 4 6 8 100Frequency

(d)

Figure 8 Period 8 motion (1198811198611 = 06382) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

100 150 20050t

(a)minus05 0 05 1 15 2 25

minus15

minus10

minus5

0

5

10

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

005

01

015

02

025

03

Am

plitu

de

5 10 150Frequency

(d)

Figure 9 Chaotic motion (119881119861 = 064) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

8 Complexity

0009 001 0011 0012 00130008D1

02

03

04

051

(a)

minus05

minus04

minus03

minus02

minus01

0

01

larg

est L

yapu

nov

Expo

nent

0009 001 0011 00120008D1

(b)

Figure 10 (a) Bifurcation diagram (b) Largest Lyapunov exponent

minus05

0

05

1

15

2

1

30 40 50 60 70 8020t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 035 04 045 050251

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 11 Period 8 motion (D=000835) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

More concretely in the case of 1198811198611 = 05 the dynamicresponse is period 1 motion (shown in Figure 18) When1198811198611 = 07 the dynamic response is in almost periodicmotion the corresponding Poincare section is a circle (shownin Figure 20) In the case of 1198811198611 = 078 there existschaotic motion (Figure 21) After the parameter 1198811198611 passesthrough chaotic region the synchronous generator losessynchronization

4 Discussions

The chaotic dynamics may appear in power system whichmake a great threat to the stability of the power systemThere-fore many researchers pay much attention to the dynamic

behavior and chaotic mechanism especially for SMIB powersystem [3ndash9 22] From the point of nonlinear dynamic anal-ysis a few works have addressed the dynamic characteristicabout three-machine power system subjected to load distur-bance In this paper a relatively deep and systematic study ofdynamic response in three-machine power system subjectedto load disturbance has been carried out This paper aimsto demonstrate the complete transition process of differentdynamic behaviors by combining qualitative and quantitativeanalysis Comparing the results about simple power system(eg SMIB power system) [9ndash15] a new route to chaos viaquasi-periodic torus rupture has been found Moreover itis worthy to point out that in our considered system thereexist many super-harmonic components in the frequency

Complexity 9

minus05

0

05

1

15

2

1

35 40 45 50 5530t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 12 Period 4 motion (D=00085) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

75 80 85 90 95 10070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus6

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 13 Period 2 motion (D=0009) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

10 Complexity

minus05

0

05

1

15

2

1

280 285 290 295 300275t

(a)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(b)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(c)

0

01

02

03

04

05

06

Am

plitu

de5 10 150

Frequency

(d)

Figure 14 Period 1 motion (D=001) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

1 1005 101 1015 102 102502

03

04

05

VB0

3

(a)

1 1005 101 1015 102 1025minus05

minus04

minus03

minus02

minus01

0

01

02

VB0

Larg

est L

yapu

nov

Expo

nent

(b)

Figure 15 (a) Bifurcation diagram (b) Largest Lyapunov exponent

spectrum of the swing oscillation response Attention shouldbe paid to these new phenomena in engineering practice Inaddition for three-machine power system our future workwill investigate the mechanism of chaotic oscillation occur-rence by improving Melnikov analysis such that effectivemeasures can be taken in time to avoid system collapse

5 Conclusions

This paper has investigated the effects of critical systemparameters on dynamic characteristics of synchronous gen-erator rotors in a three-machine power system subjected

to load disturbance The swing equations describing themotions of the synchronous generator rotors are estab-lished Based on these swing equations with the help ofbifurcation diagrams largest Lyapunov exponent spectrumsphase portraits Poincare map and frequency spectrum theinfluence of system parameters on dynamic behaviors isshown clearly The Wolf method is introduced to calculatethe largest Lyapunov exponent which is used to verify theoccurrence of chaotic motion Moreover different pathsleading to chaos coexisting in this system have been foundThey are period-doubling cascading bifurcations to chaosinduced by changing the infinite bus voltage magnitude and

Complexity 11

05 055 06 065 07 07502

03

04

05

06

07

08

09

VB1

3

Figure 16 Bifurcation diagram

05 055 06 065 07 075minus05

minus04

minus03

minus02

minus01

0

01

VB1

Larg

est L

yapu

nov

Expo

nent

Figure 17 Largest Lyapunov exponent

minus02

0

02

04

06

08

1

3

16 17 18 19 2015t

(a)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(b)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(c)

0

005

01

015

02

Am

plitu

de

20 30 40 50 60 7010Frequency

(d)

Figure 18 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

12 Complexity

minus04minus02

002040608

112

3

28 30 3226t

(a)

minus6

minus4

minus2

0

2

4

6

3

02 04 06 0803

(b)

minus4

minus3

minus2

minus1

0

3

045 05 055 06 065043

(c)

0

005

01

015

02

Am

plitu

de

10 20 30 40 500Frequency

(d)

Figure 19 Quasi-periodic motion (1198811198611 = 06) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus04

minus02

0

02

04

06

08

3

21 22 23 24 25 2620t

(a)

minus3

minus2

minus1

0

1

2

3

3

03 035 04 0450253

(b)

minus2

minus15

minus1

minus05

0

05

3

0355 036 0365 037 0375 0380353

(c)

0

002

004

006

008

Am

plitu

de

10 20 30 400Frequency

(d)

Figure 20 Quasi-periodic motion (1198811198611 = 07) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 13

minus1

minus05

0

05

1

15

3

10 20 30 40 50 600t

(a)

minus30

minus20

minus10

0

10

20

30

3

050 1 15minus05minus13

(b)

minus30

minus20

minus10

0

10

20

30

3

04 06 08 1023

(c)

0005

01015

02025

03035

04

Am

plitu

de10 20 30 400

Frequency

(d)

Figure 21 Chaotic motion (1198811198611 = 078) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

quasi-periodic torus rupture to chaos induced by changingthe disturbance frequency of the infinite bus voltage Theseresults will contribute to a better understanding of the non-linear dynamic behaviors of synchronous generator rotors inthree-machine power system

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This research is supported byHigher Educational Science andTechnology Program of Shandong Province China (Grantno J18KA235) Shandong Provincial Natural Science Foun-dation China (Grants nos ZR2016AP06 ZR2018QA005ZR2018BA018 and ZR2018BA021) and National NaturalScience Foundation of China (Grants nos 11501246 and61703251)

References

[1] P M Anderson and A A Fouad Power System Control andStability Wiley-IEEE Press New York NY USA 2nd edition2002

[2] H Ma F Min and Y Wang ldquoNonlinear dynamic analysis andsurface sliding mode controller based on low pass filter forchaotic oscillation in power system with power disturbancerdquoChinese Journal of Physics vol 56 no 5 pp 2488ndash2499 2018

[3] D Q Wei and X S Luo ldquoNoise-induced chaos in single-machine infinite-bus power systemsrdquo EPL (Europhysics Letters)vol 86 no 5 Article ID 50008 2009

[4] D Q Wei B Zhang D Y Qiu and X S Luo ldquoEffect of noiseon erosion of safe basin in power systemrdquo Nonlinear Dynamicsvol 61 no 3 pp 477ndash482 2010

[5] W Zhu R Mohler R Spee W Mittelstadt and D Maratuku-lam ldquoHopf bifurcations in a SMIB power system with SSRrdquoIEEETransactions on Power Systems vol 11 no 3 pp 1579ndash15841996

[6] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaosand instability in a power system Subharmonic-resonant caserdquoNonlinear Dynamics vol 2 no 1 pp 53ndash72 1991

[7] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaos andinstability in a power system -Primary resonant caserdquoNonlinearDynamics vol 1 no 4 pp 313ndash339 1990

[8] X Duan J Wen and S Cheng ldquoBifurcation analysis foran SMIB power system with series capacitor compensationassociated with sub-synchronous resonancerdquo Science ChinaTechnological Sciences vol 52 no 2 pp 436ndash441 2009

[9] H K Chen T N Lin and J H Chen ldquoDynamic analysiscontrolling chaos and chaotification of a SMIB power systemrdquoChaos Solitons amp Fractals vol 24 no 5 pp 1307ndash1315 2005

[10] L F C Alberto and N G Bretas ldquoApplication of Melnikovrsquosmethod for computing heteroclinic orbits in a classical SMIBpower system modelrdquo IEEE Transactions on Circuits and Sys-tems I Fundamental eory and Applications vol 47 no 7 pp1085ndash1089 2000

14 Complexity

[11] W N Zhang and W D Zhang ldquoChaotic Ocillation of aNonlinear Power Systemrdquo Applied Mathematics Mechanics vol20 no 10 pp 1175ndash1182 1999

[12] L Zhou and F Chen ldquoChaotic dynamics for a class of single-machine-infinite bus power systemrdquo Journal of Vibration andControl vol 24 no 3 pp 582ndash587 2018

[13] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005

[14] X Wang Y Chen G Han and C Song ldquoNonlinear dynamicanalysis of a single-machine infinite-bus power systemrdquoAppliedMathematical Modelling vol 39 no 10-11 pp 2951ndash2961 2015

[15] B Yuan and Q H Sun ldquoChaos in the multi-machine powersystemrdquo Automation of Electric Power Systems vol 19 no 2 pp26ndash31 1995

[16] Y Ueda Y Ueda H B Stewart and R H Abraham ldquoNonlinearresonance in basin portraits of two coupled swings under peri-odic forcingrdquo International Journal of Bifurcation and Chaosvol 8 no 6 pp 1183ndash1197 1998

[17] S S Majidabad H T Shandiz and A Hajizadeh ldquoNonlinearfractional-order power system stabilizer for multi-machinepower systems based on sliding mode techniquerdquo InternationalJournal of Robust and Nonlinear Control vol 25 no 10 pp1548ndash1568 2015

[18] N Jiang and H-D Chiang ldquoDamping Torques of multi-machine power systems during transient behaviorsrdquo IEEETransactions on Power Systems vol 29 no 3 pp 1186ndash1193 2014

[19] Y Chang X Wang and D Xu ldquoBifurcation Analysis of aPower System Model with Three Machines and Four BusesrdquoInternational Journal of Bifurcation and Chaos vol 26 no 5 pp165ndash182 2016

[20] M A Salam M A Rashid Q M Rahman and M RizonldquoTransient stability analysis of a three-machine nine bus powersystem networkrdquo Engineering Letters vol 22 no 1 pp 1ndash7 2014

[21] A Wolf J B Swift and H L Swinney ldquoDetermining Lyapunovexponents froma time seriesrdquoPhysicaDNonlinear Phenomenavol 16 no 3 pp 285ndash317 1985

[22] M A Salam ldquoTransient stability analysis of a power systemwith one generator connected to an infinite bus systemrdquoInternational Journal of Sustainable Energy vol 33 no 2 pp251ndash260 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Complexity

01

02

03

04

05

06

1

80 100 120 14060t

(a)

minus02

minus01

0

01

02

1

04 045 050351

(b)

minus02

minus01

0

01

02

1

02 04 06 08 101

(c)

000200400600801

012014

Am

plitu

de1 2 3 4 50

Frequency

(d)

Figure 3 Period 1 motion (1198811198611 = 02) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

proves that in the system chaotic motion occurs ComparingFigures 2(a) and 2(b) it can be found that 119881B1 has the samevalue at every bifurcation point

For the purpose of clearly displaying the evolving processof swing oscillation as the infinite bus voltage 1198811198611 changesFigures 3ndash9 show the time history phase portrait Poincaremap and Frequency spectrum In the case of 1198811198611 = 02the dynamic behavior of the system is periodic 1 motionForm 1198811198611 = 05 to 1198811198611 = 063 it is worthy to point out thatthe dynamic behaviors are also periodic 1 motion but thereexist many super-harmonic components in the frequencyspectrum As shown in Figures 6ndash8 in the case of 1198811198611 =0635 1198811198611 = 0637 and 1198811198611 = 06382 the system presentsperiodic 2 oscillation periodic 4 oscillation and periodic8 oscillation respectively In addition the response of thesystem is chaotic motion at 1198811198611 = 064 (shown in Figure 9)In this case the corresponding largest Lyapunov exponentis 00082 it is further confirmed that chaotic oscillation hasoccurred

32 Influence of Damping In this section the damping onthe dynamic characteristics of the rotor system is discussedFor the sake of comparison herein choose the dampingcoefficient as the bifurcation parameter the other parametervalues are the same as the ones in the previous section Thebifurcation diagram and the corresponding largest Lyapunovexponent diagram are illustrated in Figures 10(a) and 10(b)respectively

From Figure 10 it can be seen that as the dampingcoefficient increases the dynamic responses of the system

undergo an inversed bifurcation process namely chaoticmotion997888rarrperiodic 8 motion997888rarrperiodic 4 motion997888rarrperi-odic 2motion997888rarrperiodic 1 motionTherefore increasing thedamping coefficient can prevent the occurrence of chaoticmotion Figures 11ndash14 have given the time history phaseportrait Poincare map and frequency spectrum under dif-ferent values of damping These figures have clearly shownthe evolution process of the dynamic response of the systemas the damping changes

33 Influence of the Basic Voltage 119881B0 Due to the active andreactive power of the load change it may cause a changein the basic value of the infinite bus voltage Thus it alsoaffects the dynamic behavior of the system Choose 1198811198610 asthe bifurcation parameter Figure 15 shows the bifurcationdiagram and the corresponding largest Lyapunov exponentof the dynamic response as 1198811198610 changes from small to largeFrom Figure 15 it can be seen that the dynamic response ofthe system also presents an inversed bifurcation process

34 Influence of the Perturbation Frequency It is interestingto note that at Ω = 196 a new route to chaos via quasi-periodic torus rupture in this system has been observed As1198811198611 changes from small to large Figures 16 and 17 have givenbifurcation diagram and corresponding largest Lyapunovexponent diagram respectively Moreover the results showthat the swing oscillation process is periodic quasi-periodicchaotic and out-of-step In order to show the evolution pro-cess more clearly the time history phase diagram Poincaresection and frequency spectrum are plotted (Figures 18ndash21)

Complexity 5

0

02

04

06

08

1

1

15 20 25 30 35 4010t

(a)

minus15

minus1

minus05

0

05

1

1

04 06 08 1021

(b)

minus15

minus1

minus05

0

05

1

1

01 02 03 04 05 0601

(c)

0

005

01

015

02

025

03

Am

plitu

de

5 100Frequency

(d)

Figure 4 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

25 30 35 4020t

(a)

minus10

minus5

0

5

10

1

05 1 15 201

(b)

minus5

0

5

1

02 04 06 08 101

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 5 Period 1 motion (1198811198611 = 063) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

6 Complexity

minus05

0

05

1

15

2

1

50 55 60 6545t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus3

minus2

minus1

0

1

2

1

01 02 03 04 05 0601

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 6 Period 2 motion (1198811198611 = 0635) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

1

15 20 25 3010t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus3

minus2

minus1

0

1

2

3

1

01 02 03 04 05 0601

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 200Frequency

(d)

Figure 7 Period 4 motion (1198811198611 = 0637) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 7

0

05

1

15

2

1

80 90 100 11070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 035 04 045 05 0550251

(c)

0

01

02

03

04

05

Am

plitu

de

2 4 6 8 100Frequency

(d)

Figure 8 Period 8 motion (1198811198611 = 06382) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

100 150 20050t

(a)minus05 0 05 1 15 2 25

minus15

minus10

minus5

0

5

10

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

005

01

015

02

025

03

Am

plitu

de

5 10 150Frequency

(d)

Figure 9 Chaotic motion (119881119861 = 064) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

8 Complexity

0009 001 0011 0012 00130008D1

02

03

04

051

(a)

minus05

minus04

minus03

minus02

minus01

0

01

larg

est L

yapu

nov

Expo

nent

0009 001 0011 00120008D1

(b)

Figure 10 (a) Bifurcation diagram (b) Largest Lyapunov exponent

minus05

0

05

1

15

2

1

30 40 50 60 70 8020t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 035 04 045 050251

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 11 Period 8 motion (D=000835) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

More concretely in the case of 1198811198611 = 05 the dynamicresponse is period 1 motion (shown in Figure 18) When1198811198611 = 07 the dynamic response is in almost periodicmotion the corresponding Poincare section is a circle (shownin Figure 20) In the case of 1198811198611 = 078 there existschaotic motion (Figure 21) After the parameter 1198811198611 passesthrough chaotic region the synchronous generator losessynchronization

4 Discussions

The chaotic dynamics may appear in power system whichmake a great threat to the stability of the power systemThere-fore many researchers pay much attention to the dynamic

behavior and chaotic mechanism especially for SMIB powersystem [3ndash9 22] From the point of nonlinear dynamic anal-ysis a few works have addressed the dynamic characteristicabout three-machine power system subjected to load distur-bance In this paper a relatively deep and systematic study ofdynamic response in three-machine power system subjectedto load disturbance has been carried out This paper aimsto demonstrate the complete transition process of differentdynamic behaviors by combining qualitative and quantitativeanalysis Comparing the results about simple power system(eg SMIB power system) [9ndash15] a new route to chaos viaquasi-periodic torus rupture has been found Moreover itis worthy to point out that in our considered system thereexist many super-harmonic components in the frequency

Complexity 9

minus05

0

05

1

15

2

1

35 40 45 50 5530t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 12 Period 4 motion (D=00085) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

75 80 85 90 95 10070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus6

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 13 Period 2 motion (D=0009) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

10 Complexity

minus05

0

05

1

15

2

1

280 285 290 295 300275t

(a)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(b)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(c)

0

01

02

03

04

05

06

Am

plitu

de5 10 150

Frequency

(d)

Figure 14 Period 1 motion (D=001) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

1 1005 101 1015 102 102502

03

04

05

VB0

3

(a)

1 1005 101 1015 102 1025minus05

minus04

minus03

minus02

minus01

0

01

02

VB0

Larg

est L

yapu

nov

Expo

nent

(b)

Figure 15 (a) Bifurcation diagram (b) Largest Lyapunov exponent

spectrum of the swing oscillation response Attention shouldbe paid to these new phenomena in engineering practice Inaddition for three-machine power system our future workwill investigate the mechanism of chaotic oscillation occur-rence by improving Melnikov analysis such that effectivemeasures can be taken in time to avoid system collapse

5 Conclusions

This paper has investigated the effects of critical systemparameters on dynamic characteristics of synchronous gen-erator rotors in a three-machine power system subjected

to load disturbance The swing equations describing themotions of the synchronous generator rotors are estab-lished Based on these swing equations with the help ofbifurcation diagrams largest Lyapunov exponent spectrumsphase portraits Poincare map and frequency spectrum theinfluence of system parameters on dynamic behaviors isshown clearly The Wolf method is introduced to calculatethe largest Lyapunov exponent which is used to verify theoccurrence of chaotic motion Moreover different pathsleading to chaos coexisting in this system have been foundThey are period-doubling cascading bifurcations to chaosinduced by changing the infinite bus voltage magnitude and

Complexity 11

05 055 06 065 07 07502

03

04

05

06

07

08

09

VB1

3

Figure 16 Bifurcation diagram

05 055 06 065 07 075minus05

minus04

minus03

minus02

minus01

0

01

VB1

Larg

est L

yapu

nov

Expo

nent

Figure 17 Largest Lyapunov exponent

minus02

0

02

04

06

08

1

3

16 17 18 19 2015t

(a)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(b)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(c)

0

005

01

015

02

Am

plitu

de

20 30 40 50 60 7010Frequency

(d)

Figure 18 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

12 Complexity

minus04minus02

002040608

112

3

28 30 3226t

(a)

minus6

minus4

minus2

0

2

4

6

3

02 04 06 0803

(b)

minus4

minus3

minus2

minus1

0

3

045 05 055 06 065043

(c)

0

005

01

015

02

Am

plitu

de

10 20 30 40 500Frequency

(d)

Figure 19 Quasi-periodic motion (1198811198611 = 06) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus04

minus02

0

02

04

06

08

3

21 22 23 24 25 2620t

(a)

minus3

minus2

minus1

0

1

2

3

3

03 035 04 0450253

(b)

minus2

minus15

minus1

minus05

0

05

3

0355 036 0365 037 0375 0380353

(c)

0

002

004

006

008

Am

plitu

de

10 20 30 400Frequency

(d)

Figure 20 Quasi-periodic motion (1198811198611 = 07) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 13

minus1

minus05

0

05

1

15

3

10 20 30 40 50 600t

(a)

minus30

minus20

minus10

0

10

20

30

3

050 1 15minus05minus13

(b)

minus30

minus20

minus10

0

10

20

30

3

04 06 08 1023

(c)

0005

01015

02025

03035

04

Am

plitu

de10 20 30 400

Frequency

(d)

Figure 21 Chaotic motion (1198811198611 = 078) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

quasi-periodic torus rupture to chaos induced by changingthe disturbance frequency of the infinite bus voltage Theseresults will contribute to a better understanding of the non-linear dynamic behaviors of synchronous generator rotors inthree-machine power system

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This research is supported byHigher Educational Science andTechnology Program of Shandong Province China (Grantno J18KA235) Shandong Provincial Natural Science Foun-dation China (Grants nos ZR2016AP06 ZR2018QA005ZR2018BA018 and ZR2018BA021) and National NaturalScience Foundation of China (Grants nos 11501246 and61703251)

References

[1] P M Anderson and A A Fouad Power System Control andStability Wiley-IEEE Press New York NY USA 2nd edition2002

[2] H Ma F Min and Y Wang ldquoNonlinear dynamic analysis andsurface sliding mode controller based on low pass filter forchaotic oscillation in power system with power disturbancerdquoChinese Journal of Physics vol 56 no 5 pp 2488ndash2499 2018

[3] D Q Wei and X S Luo ldquoNoise-induced chaos in single-machine infinite-bus power systemsrdquo EPL (Europhysics Letters)vol 86 no 5 Article ID 50008 2009

[4] D Q Wei B Zhang D Y Qiu and X S Luo ldquoEffect of noiseon erosion of safe basin in power systemrdquo Nonlinear Dynamicsvol 61 no 3 pp 477ndash482 2010

[5] W Zhu R Mohler R Spee W Mittelstadt and D Maratuku-lam ldquoHopf bifurcations in a SMIB power system with SSRrdquoIEEETransactions on Power Systems vol 11 no 3 pp 1579ndash15841996

[6] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaosand instability in a power system Subharmonic-resonant caserdquoNonlinear Dynamics vol 2 no 1 pp 53ndash72 1991

[7] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaos andinstability in a power system -Primary resonant caserdquoNonlinearDynamics vol 1 no 4 pp 313ndash339 1990

[8] X Duan J Wen and S Cheng ldquoBifurcation analysis foran SMIB power system with series capacitor compensationassociated with sub-synchronous resonancerdquo Science ChinaTechnological Sciences vol 52 no 2 pp 436ndash441 2009

[9] H K Chen T N Lin and J H Chen ldquoDynamic analysiscontrolling chaos and chaotification of a SMIB power systemrdquoChaos Solitons amp Fractals vol 24 no 5 pp 1307ndash1315 2005

[10] L F C Alberto and N G Bretas ldquoApplication of Melnikovrsquosmethod for computing heteroclinic orbits in a classical SMIBpower system modelrdquo IEEE Transactions on Circuits and Sys-tems I Fundamental eory and Applications vol 47 no 7 pp1085ndash1089 2000

14 Complexity

[11] W N Zhang and W D Zhang ldquoChaotic Ocillation of aNonlinear Power Systemrdquo Applied Mathematics Mechanics vol20 no 10 pp 1175ndash1182 1999

[12] L Zhou and F Chen ldquoChaotic dynamics for a class of single-machine-infinite bus power systemrdquo Journal of Vibration andControl vol 24 no 3 pp 582ndash587 2018

[13] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005

[14] X Wang Y Chen G Han and C Song ldquoNonlinear dynamicanalysis of a single-machine infinite-bus power systemrdquoAppliedMathematical Modelling vol 39 no 10-11 pp 2951ndash2961 2015

[15] B Yuan and Q H Sun ldquoChaos in the multi-machine powersystemrdquo Automation of Electric Power Systems vol 19 no 2 pp26ndash31 1995

[16] Y Ueda Y Ueda H B Stewart and R H Abraham ldquoNonlinearresonance in basin portraits of two coupled swings under peri-odic forcingrdquo International Journal of Bifurcation and Chaosvol 8 no 6 pp 1183ndash1197 1998

[17] S S Majidabad H T Shandiz and A Hajizadeh ldquoNonlinearfractional-order power system stabilizer for multi-machinepower systems based on sliding mode techniquerdquo InternationalJournal of Robust and Nonlinear Control vol 25 no 10 pp1548ndash1568 2015

[18] N Jiang and H-D Chiang ldquoDamping Torques of multi-machine power systems during transient behaviorsrdquo IEEETransactions on Power Systems vol 29 no 3 pp 1186ndash1193 2014

[19] Y Chang X Wang and D Xu ldquoBifurcation Analysis of aPower System Model with Three Machines and Four BusesrdquoInternational Journal of Bifurcation and Chaos vol 26 no 5 pp165ndash182 2016

[20] M A Salam M A Rashid Q M Rahman and M RizonldquoTransient stability analysis of a three-machine nine bus powersystem networkrdquo Engineering Letters vol 22 no 1 pp 1ndash7 2014

[21] A Wolf J B Swift and H L Swinney ldquoDetermining Lyapunovexponents froma time seriesrdquoPhysicaDNonlinear Phenomenavol 16 no 3 pp 285ndash317 1985

[22] M A Salam ldquoTransient stability analysis of a power systemwith one generator connected to an infinite bus systemrdquoInternational Journal of Sustainable Energy vol 33 no 2 pp251ndash260 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 5

0

02

04

06

08

1

1

15 20 25 30 35 4010t

(a)

minus15

minus1

minus05

0

05

1

1

04 06 08 1021

(b)

minus15

minus1

minus05

0

05

1

1

01 02 03 04 05 0601

(c)

0

005

01

015

02

025

03

Am

plitu

de

5 100Frequency

(d)

Figure 4 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

25 30 35 4020t

(a)

minus10

minus5

0

5

10

1

05 1 15 201

(b)

minus5

0

5

1

02 04 06 08 101

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 5 Period 1 motion (1198811198611 = 063) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

6 Complexity

minus05

0

05

1

15

2

1

50 55 60 6545t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus3

minus2

minus1

0

1

2

1

01 02 03 04 05 0601

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 6 Period 2 motion (1198811198611 = 0635) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

1

15 20 25 3010t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus3

minus2

minus1

0

1

2

3

1

01 02 03 04 05 0601

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 200Frequency

(d)

Figure 7 Period 4 motion (1198811198611 = 0637) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 7

0

05

1

15

2

1

80 90 100 11070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 035 04 045 05 0550251

(c)

0

01

02

03

04

05

Am

plitu

de

2 4 6 8 100Frequency

(d)

Figure 8 Period 8 motion (1198811198611 = 06382) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

100 150 20050t

(a)minus05 0 05 1 15 2 25

minus15

minus10

minus5

0

5

10

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

005

01

015

02

025

03

Am

plitu

de

5 10 150Frequency

(d)

Figure 9 Chaotic motion (119881119861 = 064) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

8 Complexity

0009 001 0011 0012 00130008D1

02

03

04

051

(a)

minus05

minus04

minus03

minus02

minus01

0

01

larg

est L

yapu

nov

Expo

nent

0009 001 0011 00120008D1

(b)

Figure 10 (a) Bifurcation diagram (b) Largest Lyapunov exponent

minus05

0

05

1

15

2

1

30 40 50 60 70 8020t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 035 04 045 050251

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 11 Period 8 motion (D=000835) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

More concretely in the case of 1198811198611 = 05 the dynamicresponse is period 1 motion (shown in Figure 18) When1198811198611 = 07 the dynamic response is in almost periodicmotion the corresponding Poincare section is a circle (shownin Figure 20) In the case of 1198811198611 = 078 there existschaotic motion (Figure 21) After the parameter 1198811198611 passesthrough chaotic region the synchronous generator losessynchronization

4 Discussions

The chaotic dynamics may appear in power system whichmake a great threat to the stability of the power systemThere-fore many researchers pay much attention to the dynamic

behavior and chaotic mechanism especially for SMIB powersystem [3ndash9 22] From the point of nonlinear dynamic anal-ysis a few works have addressed the dynamic characteristicabout three-machine power system subjected to load distur-bance In this paper a relatively deep and systematic study ofdynamic response in three-machine power system subjectedto load disturbance has been carried out This paper aimsto demonstrate the complete transition process of differentdynamic behaviors by combining qualitative and quantitativeanalysis Comparing the results about simple power system(eg SMIB power system) [9ndash15] a new route to chaos viaquasi-periodic torus rupture has been found Moreover itis worthy to point out that in our considered system thereexist many super-harmonic components in the frequency

Complexity 9

minus05

0

05

1

15

2

1

35 40 45 50 5530t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 12 Period 4 motion (D=00085) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

75 80 85 90 95 10070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus6

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 13 Period 2 motion (D=0009) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

10 Complexity

minus05

0

05

1

15

2

1

280 285 290 295 300275t

(a)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(b)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(c)

0

01

02

03

04

05

06

Am

plitu

de5 10 150

Frequency

(d)

Figure 14 Period 1 motion (D=001) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

1 1005 101 1015 102 102502

03

04

05

VB0

3

(a)

1 1005 101 1015 102 1025minus05

minus04

minus03

minus02

minus01

0

01

02

VB0

Larg

est L

yapu

nov

Expo

nent

(b)

Figure 15 (a) Bifurcation diagram (b) Largest Lyapunov exponent

spectrum of the swing oscillation response Attention shouldbe paid to these new phenomena in engineering practice Inaddition for three-machine power system our future workwill investigate the mechanism of chaotic oscillation occur-rence by improving Melnikov analysis such that effectivemeasures can be taken in time to avoid system collapse

5 Conclusions

This paper has investigated the effects of critical systemparameters on dynamic characteristics of synchronous gen-erator rotors in a three-machine power system subjected

to load disturbance The swing equations describing themotions of the synchronous generator rotors are estab-lished Based on these swing equations with the help ofbifurcation diagrams largest Lyapunov exponent spectrumsphase portraits Poincare map and frequency spectrum theinfluence of system parameters on dynamic behaviors isshown clearly The Wolf method is introduced to calculatethe largest Lyapunov exponent which is used to verify theoccurrence of chaotic motion Moreover different pathsleading to chaos coexisting in this system have been foundThey are period-doubling cascading bifurcations to chaosinduced by changing the infinite bus voltage magnitude and

Complexity 11

05 055 06 065 07 07502

03

04

05

06

07

08

09

VB1

3

Figure 16 Bifurcation diagram

05 055 06 065 07 075minus05

minus04

minus03

minus02

minus01

0

01

VB1

Larg

est L

yapu

nov

Expo

nent

Figure 17 Largest Lyapunov exponent

minus02

0

02

04

06

08

1

3

16 17 18 19 2015t

(a)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(b)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(c)

0

005

01

015

02

Am

plitu

de

20 30 40 50 60 7010Frequency

(d)

Figure 18 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

12 Complexity

minus04minus02

002040608

112

3

28 30 3226t

(a)

minus6

minus4

minus2

0

2

4

6

3

02 04 06 0803

(b)

minus4

minus3

minus2

minus1

0

3

045 05 055 06 065043

(c)

0

005

01

015

02

Am

plitu

de

10 20 30 40 500Frequency

(d)

Figure 19 Quasi-periodic motion (1198811198611 = 06) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus04

minus02

0

02

04

06

08

3

21 22 23 24 25 2620t

(a)

minus3

minus2

minus1

0

1

2

3

3

03 035 04 0450253

(b)

minus2

minus15

minus1

minus05

0

05

3

0355 036 0365 037 0375 0380353

(c)

0

002

004

006

008

Am

plitu

de

10 20 30 400Frequency

(d)

Figure 20 Quasi-periodic motion (1198811198611 = 07) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 13

minus1

minus05

0

05

1

15

3

10 20 30 40 50 600t

(a)

minus30

minus20

minus10

0

10

20

30

3

050 1 15minus05minus13

(b)

minus30

minus20

minus10

0

10

20

30

3

04 06 08 1023

(c)

0005

01015

02025

03035

04

Am

plitu

de10 20 30 400

Frequency

(d)

Figure 21 Chaotic motion (1198811198611 = 078) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

quasi-periodic torus rupture to chaos induced by changingthe disturbance frequency of the infinite bus voltage Theseresults will contribute to a better understanding of the non-linear dynamic behaviors of synchronous generator rotors inthree-machine power system

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This research is supported byHigher Educational Science andTechnology Program of Shandong Province China (Grantno J18KA235) Shandong Provincial Natural Science Foun-dation China (Grants nos ZR2016AP06 ZR2018QA005ZR2018BA018 and ZR2018BA021) and National NaturalScience Foundation of China (Grants nos 11501246 and61703251)

References

[1] P M Anderson and A A Fouad Power System Control andStability Wiley-IEEE Press New York NY USA 2nd edition2002

[2] H Ma F Min and Y Wang ldquoNonlinear dynamic analysis andsurface sliding mode controller based on low pass filter forchaotic oscillation in power system with power disturbancerdquoChinese Journal of Physics vol 56 no 5 pp 2488ndash2499 2018

[3] D Q Wei and X S Luo ldquoNoise-induced chaos in single-machine infinite-bus power systemsrdquo EPL (Europhysics Letters)vol 86 no 5 Article ID 50008 2009

[4] D Q Wei B Zhang D Y Qiu and X S Luo ldquoEffect of noiseon erosion of safe basin in power systemrdquo Nonlinear Dynamicsvol 61 no 3 pp 477ndash482 2010

[5] W Zhu R Mohler R Spee W Mittelstadt and D Maratuku-lam ldquoHopf bifurcations in a SMIB power system with SSRrdquoIEEETransactions on Power Systems vol 11 no 3 pp 1579ndash15841996

[6] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaosand instability in a power system Subharmonic-resonant caserdquoNonlinear Dynamics vol 2 no 1 pp 53ndash72 1991

[7] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaos andinstability in a power system -Primary resonant caserdquoNonlinearDynamics vol 1 no 4 pp 313ndash339 1990

[8] X Duan J Wen and S Cheng ldquoBifurcation analysis foran SMIB power system with series capacitor compensationassociated with sub-synchronous resonancerdquo Science ChinaTechnological Sciences vol 52 no 2 pp 436ndash441 2009

[9] H K Chen T N Lin and J H Chen ldquoDynamic analysiscontrolling chaos and chaotification of a SMIB power systemrdquoChaos Solitons amp Fractals vol 24 no 5 pp 1307ndash1315 2005

[10] L F C Alberto and N G Bretas ldquoApplication of Melnikovrsquosmethod for computing heteroclinic orbits in a classical SMIBpower system modelrdquo IEEE Transactions on Circuits and Sys-tems I Fundamental eory and Applications vol 47 no 7 pp1085ndash1089 2000

14 Complexity

[11] W N Zhang and W D Zhang ldquoChaotic Ocillation of aNonlinear Power Systemrdquo Applied Mathematics Mechanics vol20 no 10 pp 1175ndash1182 1999

[12] L Zhou and F Chen ldquoChaotic dynamics for a class of single-machine-infinite bus power systemrdquo Journal of Vibration andControl vol 24 no 3 pp 582ndash587 2018

[13] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005

[14] X Wang Y Chen G Han and C Song ldquoNonlinear dynamicanalysis of a single-machine infinite-bus power systemrdquoAppliedMathematical Modelling vol 39 no 10-11 pp 2951ndash2961 2015

[15] B Yuan and Q H Sun ldquoChaos in the multi-machine powersystemrdquo Automation of Electric Power Systems vol 19 no 2 pp26ndash31 1995

[16] Y Ueda Y Ueda H B Stewart and R H Abraham ldquoNonlinearresonance in basin portraits of two coupled swings under peri-odic forcingrdquo International Journal of Bifurcation and Chaosvol 8 no 6 pp 1183ndash1197 1998

[17] S S Majidabad H T Shandiz and A Hajizadeh ldquoNonlinearfractional-order power system stabilizer for multi-machinepower systems based on sliding mode techniquerdquo InternationalJournal of Robust and Nonlinear Control vol 25 no 10 pp1548ndash1568 2015

[18] N Jiang and H-D Chiang ldquoDamping Torques of multi-machine power systems during transient behaviorsrdquo IEEETransactions on Power Systems vol 29 no 3 pp 1186ndash1193 2014

[19] Y Chang X Wang and D Xu ldquoBifurcation Analysis of aPower System Model with Three Machines and Four BusesrdquoInternational Journal of Bifurcation and Chaos vol 26 no 5 pp165ndash182 2016

[20] M A Salam M A Rashid Q M Rahman and M RizonldquoTransient stability analysis of a three-machine nine bus powersystem networkrdquo Engineering Letters vol 22 no 1 pp 1ndash7 2014

[21] A Wolf J B Swift and H L Swinney ldquoDetermining Lyapunovexponents froma time seriesrdquoPhysicaDNonlinear Phenomenavol 16 no 3 pp 285ndash317 1985

[22] M A Salam ldquoTransient stability analysis of a power systemwith one generator connected to an infinite bus systemrdquoInternational Journal of Sustainable Energy vol 33 no 2 pp251ndash260 2014

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

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Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

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AnalysisInternational Journal of

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Complexity

minus05

0

05

1

15

2

1

50 55 60 6545t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus3

minus2

minus1

0

1

2

1

01 02 03 04 05 0601

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 6 Period 2 motion (1198811198611 = 0635) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

1

15 20 25 3010t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus3

minus2

minus1

0

1

2

3

1

01 02 03 04 05 0601

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 200Frequency

(d)

Figure 7 Period 4 motion (1198811198611 = 0637) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 7

0

05

1

15

2

1

80 90 100 11070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 035 04 045 05 0550251

(c)

0

01

02

03

04

05

Am

plitu

de

2 4 6 8 100Frequency

(d)

Figure 8 Period 8 motion (1198811198611 = 06382) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

100 150 20050t

(a)minus05 0 05 1 15 2 25

minus15

minus10

minus5

0

5

10

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

005

01

015

02

025

03

Am

plitu

de

5 10 150Frequency

(d)

Figure 9 Chaotic motion (119881119861 = 064) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

8 Complexity

0009 001 0011 0012 00130008D1

02

03

04

051

(a)

minus05

minus04

minus03

minus02

minus01

0

01

larg

est L

yapu

nov

Expo

nent

0009 001 0011 00120008D1

(b)

Figure 10 (a) Bifurcation diagram (b) Largest Lyapunov exponent

minus05

0

05

1

15

2

1

30 40 50 60 70 8020t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 035 04 045 050251

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 11 Period 8 motion (D=000835) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

More concretely in the case of 1198811198611 = 05 the dynamicresponse is period 1 motion (shown in Figure 18) When1198811198611 = 07 the dynamic response is in almost periodicmotion the corresponding Poincare section is a circle (shownin Figure 20) In the case of 1198811198611 = 078 there existschaotic motion (Figure 21) After the parameter 1198811198611 passesthrough chaotic region the synchronous generator losessynchronization

4 Discussions

The chaotic dynamics may appear in power system whichmake a great threat to the stability of the power systemThere-fore many researchers pay much attention to the dynamic

behavior and chaotic mechanism especially for SMIB powersystem [3ndash9 22] From the point of nonlinear dynamic anal-ysis a few works have addressed the dynamic characteristicabout three-machine power system subjected to load distur-bance In this paper a relatively deep and systematic study ofdynamic response in three-machine power system subjectedto load disturbance has been carried out This paper aimsto demonstrate the complete transition process of differentdynamic behaviors by combining qualitative and quantitativeanalysis Comparing the results about simple power system(eg SMIB power system) [9ndash15] a new route to chaos viaquasi-periodic torus rupture has been found Moreover itis worthy to point out that in our considered system thereexist many super-harmonic components in the frequency

Complexity 9

minus05

0

05

1

15

2

1

35 40 45 50 5530t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 12 Period 4 motion (D=00085) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

75 80 85 90 95 10070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus6

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 13 Period 2 motion (D=0009) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

10 Complexity

minus05

0

05

1

15

2

1

280 285 290 295 300275t

(a)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(b)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(c)

0

01

02

03

04

05

06

Am

plitu

de5 10 150

Frequency

(d)

Figure 14 Period 1 motion (D=001) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

1 1005 101 1015 102 102502

03

04

05

VB0

3

(a)

1 1005 101 1015 102 1025minus05

minus04

minus03

minus02

minus01

0

01

02

VB0

Larg

est L

yapu

nov

Expo

nent

(b)

Figure 15 (a) Bifurcation diagram (b) Largest Lyapunov exponent

spectrum of the swing oscillation response Attention shouldbe paid to these new phenomena in engineering practice Inaddition for three-machine power system our future workwill investigate the mechanism of chaotic oscillation occur-rence by improving Melnikov analysis such that effectivemeasures can be taken in time to avoid system collapse

5 Conclusions

This paper has investigated the effects of critical systemparameters on dynamic characteristics of synchronous gen-erator rotors in a three-machine power system subjected

to load disturbance The swing equations describing themotions of the synchronous generator rotors are estab-lished Based on these swing equations with the help ofbifurcation diagrams largest Lyapunov exponent spectrumsphase portraits Poincare map and frequency spectrum theinfluence of system parameters on dynamic behaviors isshown clearly The Wolf method is introduced to calculatethe largest Lyapunov exponent which is used to verify theoccurrence of chaotic motion Moreover different pathsleading to chaos coexisting in this system have been foundThey are period-doubling cascading bifurcations to chaosinduced by changing the infinite bus voltage magnitude and

Complexity 11

05 055 06 065 07 07502

03

04

05

06

07

08

09

VB1

3

Figure 16 Bifurcation diagram

05 055 06 065 07 075minus05

minus04

minus03

minus02

minus01

0

01

VB1

Larg

est L

yapu

nov

Expo

nent

Figure 17 Largest Lyapunov exponent

minus02

0

02

04

06

08

1

3

16 17 18 19 2015t

(a)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(b)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(c)

0

005

01

015

02

Am

plitu

de

20 30 40 50 60 7010Frequency

(d)

Figure 18 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

12 Complexity

minus04minus02

002040608

112

3

28 30 3226t

(a)

minus6

minus4

minus2

0

2

4

6

3

02 04 06 0803

(b)

minus4

minus3

minus2

minus1

0

3

045 05 055 06 065043

(c)

0

005

01

015

02

Am

plitu

de

10 20 30 40 500Frequency

(d)

Figure 19 Quasi-periodic motion (1198811198611 = 06) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus04

minus02

0

02

04

06

08

3

21 22 23 24 25 2620t

(a)

minus3

minus2

minus1

0

1

2

3

3

03 035 04 0450253

(b)

minus2

minus15

minus1

minus05

0

05

3

0355 036 0365 037 0375 0380353

(c)

0

002

004

006

008

Am

plitu

de

10 20 30 400Frequency

(d)

Figure 20 Quasi-periodic motion (1198811198611 = 07) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 13

minus1

minus05

0

05

1

15

3

10 20 30 40 50 600t

(a)

minus30

minus20

minus10

0

10

20

30

3

050 1 15minus05minus13

(b)

minus30

minus20

minus10

0

10

20

30

3

04 06 08 1023

(c)

0005

01015

02025

03035

04

Am

plitu

de10 20 30 400

Frequency

(d)

Figure 21 Chaotic motion (1198811198611 = 078) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

quasi-periodic torus rupture to chaos induced by changingthe disturbance frequency of the infinite bus voltage Theseresults will contribute to a better understanding of the non-linear dynamic behaviors of synchronous generator rotors inthree-machine power system

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This research is supported byHigher Educational Science andTechnology Program of Shandong Province China (Grantno J18KA235) Shandong Provincial Natural Science Foun-dation China (Grants nos ZR2016AP06 ZR2018QA005ZR2018BA018 and ZR2018BA021) and National NaturalScience Foundation of China (Grants nos 11501246 and61703251)

References

[1] P M Anderson and A A Fouad Power System Control andStability Wiley-IEEE Press New York NY USA 2nd edition2002

[2] H Ma F Min and Y Wang ldquoNonlinear dynamic analysis andsurface sliding mode controller based on low pass filter forchaotic oscillation in power system with power disturbancerdquoChinese Journal of Physics vol 56 no 5 pp 2488ndash2499 2018

[3] D Q Wei and X S Luo ldquoNoise-induced chaos in single-machine infinite-bus power systemsrdquo EPL (Europhysics Letters)vol 86 no 5 Article ID 50008 2009

[4] D Q Wei B Zhang D Y Qiu and X S Luo ldquoEffect of noiseon erosion of safe basin in power systemrdquo Nonlinear Dynamicsvol 61 no 3 pp 477ndash482 2010

[5] W Zhu R Mohler R Spee W Mittelstadt and D Maratuku-lam ldquoHopf bifurcations in a SMIB power system with SSRrdquoIEEETransactions on Power Systems vol 11 no 3 pp 1579ndash15841996

[6] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaosand instability in a power system Subharmonic-resonant caserdquoNonlinear Dynamics vol 2 no 1 pp 53ndash72 1991

[7] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaos andinstability in a power system -Primary resonant caserdquoNonlinearDynamics vol 1 no 4 pp 313ndash339 1990

[8] X Duan J Wen and S Cheng ldquoBifurcation analysis foran SMIB power system with series capacitor compensationassociated with sub-synchronous resonancerdquo Science ChinaTechnological Sciences vol 52 no 2 pp 436ndash441 2009

[9] H K Chen T N Lin and J H Chen ldquoDynamic analysiscontrolling chaos and chaotification of a SMIB power systemrdquoChaos Solitons amp Fractals vol 24 no 5 pp 1307ndash1315 2005

[10] L F C Alberto and N G Bretas ldquoApplication of Melnikovrsquosmethod for computing heteroclinic orbits in a classical SMIBpower system modelrdquo IEEE Transactions on Circuits and Sys-tems I Fundamental eory and Applications vol 47 no 7 pp1085ndash1089 2000

14 Complexity

[11] W N Zhang and W D Zhang ldquoChaotic Ocillation of aNonlinear Power Systemrdquo Applied Mathematics Mechanics vol20 no 10 pp 1175ndash1182 1999

[12] L Zhou and F Chen ldquoChaotic dynamics for a class of single-machine-infinite bus power systemrdquo Journal of Vibration andControl vol 24 no 3 pp 582ndash587 2018

[13] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005

[14] X Wang Y Chen G Han and C Song ldquoNonlinear dynamicanalysis of a single-machine infinite-bus power systemrdquoAppliedMathematical Modelling vol 39 no 10-11 pp 2951ndash2961 2015

[15] B Yuan and Q H Sun ldquoChaos in the multi-machine powersystemrdquo Automation of Electric Power Systems vol 19 no 2 pp26ndash31 1995

[16] Y Ueda Y Ueda H B Stewart and R H Abraham ldquoNonlinearresonance in basin portraits of two coupled swings under peri-odic forcingrdquo International Journal of Bifurcation and Chaosvol 8 no 6 pp 1183ndash1197 1998

[17] S S Majidabad H T Shandiz and A Hajizadeh ldquoNonlinearfractional-order power system stabilizer for multi-machinepower systems based on sliding mode techniquerdquo InternationalJournal of Robust and Nonlinear Control vol 25 no 10 pp1548ndash1568 2015

[18] N Jiang and H-D Chiang ldquoDamping Torques of multi-machine power systems during transient behaviorsrdquo IEEETransactions on Power Systems vol 29 no 3 pp 1186ndash1193 2014

[19] Y Chang X Wang and D Xu ldquoBifurcation Analysis of aPower System Model with Three Machines and Four BusesrdquoInternational Journal of Bifurcation and Chaos vol 26 no 5 pp165ndash182 2016

[20] M A Salam M A Rashid Q M Rahman and M RizonldquoTransient stability analysis of a three-machine nine bus powersystem networkrdquo Engineering Letters vol 22 no 1 pp 1ndash7 2014

[21] A Wolf J B Swift and H L Swinney ldquoDetermining Lyapunovexponents froma time seriesrdquoPhysicaDNonlinear Phenomenavol 16 no 3 pp 285ndash317 1985

[22] M A Salam ldquoTransient stability analysis of a power systemwith one generator connected to an infinite bus systemrdquoInternational Journal of Sustainable Energy vol 33 no 2 pp251ndash260 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 7

0

05

1

15

2

1

80 90 100 11070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 035 04 045 05 0550251

(c)

0

01

02

03

04

05

Am

plitu

de

2 4 6 8 100Frequency

(d)

Figure 8 Period 8 motion (1198811198611 = 06382) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

100 150 20050t

(a)minus05 0 05 1 15 2 25

minus15

minus10

minus5

0

5

10

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

005

01

015

02

025

03

Am

plitu

de

5 10 150Frequency

(d)

Figure 9 Chaotic motion (119881119861 = 064) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

8 Complexity

0009 001 0011 0012 00130008D1

02

03

04

051

(a)

minus05

minus04

minus03

minus02

minus01

0

01

larg

est L

yapu

nov

Expo

nent

0009 001 0011 00120008D1

(b)

Figure 10 (a) Bifurcation diagram (b) Largest Lyapunov exponent

minus05

0

05

1

15

2

1

30 40 50 60 70 8020t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 035 04 045 050251

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 11 Period 8 motion (D=000835) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

More concretely in the case of 1198811198611 = 05 the dynamicresponse is period 1 motion (shown in Figure 18) When1198811198611 = 07 the dynamic response is in almost periodicmotion the corresponding Poincare section is a circle (shownin Figure 20) In the case of 1198811198611 = 078 there existschaotic motion (Figure 21) After the parameter 1198811198611 passesthrough chaotic region the synchronous generator losessynchronization

4 Discussions

The chaotic dynamics may appear in power system whichmake a great threat to the stability of the power systemThere-fore many researchers pay much attention to the dynamic

behavior and chaotic mechanism especially for SMIB powersystem [3ndash9 22] From the point of nonlinear dynamic anal-ysis a few works have addressed the dynamic characteristicabout three-machine power system subjected to load distur-bance In this paper a relatively deep and systematic study ofdynamic response in three-machine power system subjectedto load disturbance has been carried out This paper aimsto demonstrate the complete transition process of differentdynamic behaviors by combining qualitative and quantitativeanalysis Comparing the results about simple power system(eg SMIB power system) [9ndash15] a new route to chaos viaquasi-periodic torus rupture has been found Moreover itis worthy to point out that in our considered system thereexist many super-harmonic components in the frequency

Complexity 9

minus05

0

05

1

15

2

1

35 40 45 50 5530t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 12 Period 4 motion (D=00085) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

75 80 85 90 95 10070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus6

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 13 Period 2 motion (D=0009) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

10 Complexity

minus05

0

05

1

15

2

1

280 285 290 295 300275t

(a)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(b)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(c)

0

01

02

03

04

05

06

Am

plitu

de5 10 150

Frequency

(d)

Figure 14 Period 1 motion (D=001) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

1 1005 101 1015 102 102502

03

04

05

VB0

3

(a)

1 1005 101 1015 102 1025minus05

minus04

minus03

minus02

minus01

0

01

02

VB0

Larg

est L

yapu

nov

Expo

nent

(b)

Figure 15 (a) Bifurcation diagram (b) Largest Lyapunov exponent

spectrum of the swing oscillation response Attention shouldbe paid to these new phenomena in engineering practice Inaddition for three-machine power system our future workwill investigate the mechanism of chaotic oscillation occur-rence by improving Melnikov analysis such that effectivemeasures can be taken in time to avoid system collapse

5 Conclusions

This paper has investigated the effects of critical systemparameters on dynamic characteristics of synchronous gen-erator rotors in a three-machine power system subjected

to load disturbance The swing equations describing themotions of the synchronous generator rotors are estab-lished Based on these swing equations with the help ofbifurcation diagrams largest Lyapunov exponent spectrumsphase portraits Poincare map and frequency spectrum theinfluence of system parameters on dynamic behaviors isshown clearly The Wolf method is introduced to calculatethe largest Lyapunov exponent which is used to verify theoccurrence of chaotic motion Moreover different pathsleading to chaos coexisting in this system have been foundThey are period-doubling cascading bifurcations to chaosinduced by changing the infinite bus voltage magnitude and

Complexity 11

05 055 06 065 07 07502

03

04

05

06

07

08

09

VB1

3

Figure 16 Bifurcation diagram

05 055 06 065 07 075minus05

minus04

minus03

minus02

minus01

0

01

VB1

Larg

est L

yapu

nov

Expo

nent

Figure 17 Largest Lyapunov exponent

minus02

0

02

04

06

08

1

3

16 17 18 19 2015t

(a)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(b)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(c)

0

005

01

015

02

Am

plitu

de

20 30 40 50 60 7010Frequency

(d)

Figure 18 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

12 Complexity

minus04minus02

002040608

112

3

28 30 3226t

(a)

minus6

minus4

minus2

0

2

4

6

3

02 04 06 0803

(b)

minus4

minus3

minus2

minus1

0

3

045 05 055 06 065043

(c)

0

005

01

015

02

Am

plitu

de

10 20 30 40 500Frequency

(d)

Figure 19 Quasi-periodic motion (1198811198611 = 06) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus04

minus02

0

02

04

06

08

3

21 22 23 24 25 2620t

(a)

minus3

minus2

minus1

0

1

2

3

3

03 035 04 0450253

(b)

minus2

minus15

minus1

minus05

0

05

3

0355 036 0365 037 0375 0380353

(c)

0

002

004

006

008

Am

plitu

de

10 20 30 400Frequency

(d)

Figure 20 Quasi-periodic motion (1198811198611 = 07) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 13

minus1

minus05

0

05

1

15

3

10 20 30 40 50 600t

(a)

minus30

minus20

minus10

0

10

20

30

3

050 1 15minus05minus13

(b)

minus30

minus20

minus10

0

10

20

30

3

04 06 08 1023

(c)

0005

01015

02025

03035

04

Am

plitu

de10 20 30 400

Frequency

(d)

Figure 21 Chaotic motion (1198811198611 = 078) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

quasi-periodic torus rupture to chaos induced by changingthe disturbance frequency of the infinite bus voltage Theseresults will contribute to a better understanding of the non-linear dynamic behaviors of synchronous generator rotors inthree-machine power system

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This research is supported byHigher Educational Science andTechnology Program of Shandong Province China (Grantno J18KA235) Shandong Provincial Natural Science Foun-dation China (Grants nos ZR2016AP06 ZR2018QA005ZR2018BA018 and ZR2018BA021) and National NaturalScience Foundation of China (Grants nos 11501246 and61703251)

References

[1] P M Anderson and A A Fouad Power System Control andStability Wiley-IEEE Press New York NY USA 2nd edition2002

[2] H Ma F Min and Y Wang ldquoNonlinear dynamic analysis andsurface sliding mode controller based on low pass filter forchaotic oscillation in power system with power disturbancerdquoChinese Journal of Physics vol 56 no 5 pp 2488ndash2499 2018

[3] D Q Wei and X S Luo ldquoNoise-induced chaos in single-machine infinite-bus power systemsrdquo EPL (Europhysics Letters)vol 86 no 5 Article ID 50008 2009

[4] D Q Wei B Zhang D Y Qiu and X S Luo ldquoEffect of noiseon erosion of safe basin in power systemrdquo Nonlinear Dynamicsvol 61 no 3 pp 477ndash482 2010

[5] W Zhu R Mohler R Spee W Mittelstadt and D Maratuku-lam ldquoHopf bifurcations in a SMIB power system with SSRrdquoIEEETransactions on Power Systems vol 11 no 3 pp 1579ndash15841996

[6] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaosand instability in a power system Subharmonic-resonant caserdquoNonlinear Dynamics vol 2 no 1 pp 53ndash72 1991

[7] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaos andinstability in a power system -Primary resonant caserdquoNonlinearDynamics vol 1 no 4 pp 313ndash339 1990

[8] X Duan J Wen and S Cheng ldquoBifurcation analysis foran SMIB power system with series capacitor compensationassociated with sub-synchronous resonancerdquo Science ChinaTechnological Sciences vol 52 no 2 pp 436ndash441 2009

[9] H K Chen T N Lin and J H Chen ldquoDynamic analysiscontrolling chaos and chaotification of a SMIB power systemrdquoChaos Solitons amp Fractals vol 24 no 5 pp 1307ndash1315 2005

[10] L F C Alberto and N G Bretas ldquoApplication of Melnikovrsquosmethod for computing heteroclinic orbits in a classical SMIBpower system modelrdquo IEEE Transactions on Circuits and Sys-tems I Fundamental eory and Applications vol 47 no 7 pp1085ndash1089 2000

14 Complexity

[11] W N Zhang and W D Zhang ldquoChaotic Ocillation of aNonlinear Power Systemrdquo Applied Mathematics Mechanics vol20 no 10 pp 1175ndash1182 1999

[12] L Zhou and F Chen ldquoChaotic dynamics for a class of single-machine-infinite bus power systemrdquo Journal of Vibration andControl vol 24 no 3 pp 582ndash587 2018

[13] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005

[14] X Wang Y Chen G Han and C Song ldquoNonlinear dynamicanalysis of a single-machine infinite-bus power systemrdquoAppliedMathematical Modelling vol 39 no 10-11 pp 2951ndash2961 2015

[15] B Yuan and Q H Sun ldquoChaos in the multi-machine powersystemrdquo Automation of Electric Power Systems vol 19 no 2 pp26ndash31 1995

[16] Y Ueda Y Ueda H B Stewart and R H Abraham ldquoNonlinearresonance in basin portraits of two coupled swings under peri-odic forcingrdquo International Journal of Bifurcation and Chaosvol 8 no 6 pp 1183ndash1197 1998

[17] S S Majidabad H T Shandiz and A Hajizadeh ldquoNonlinearfractional-order power system stabilizer for multi-machinepower systems based on sliding mode techniquerdquo InternationalJournal of Robust and Nonlinear Control vol 25 no 10 pp1548ndash1568 2015

[18] N Jiang and H-D Chiang ldquoDamping Torques of multi-machine power systems during transient behaviorsrdquo IEEETransactions on Power Systems vol 29 no 3 pp 1186ndash1193 2014

[19] Y Chang X Wang and D Xu ldquoBifurcation Analysis of aPower System Model with Three Machines and Four BusesrdquoInternational Journal of Bifurcation and Chaos vol 26 no 5 pp165ndash182 2016

[20] M A Salam M A Rashid Q M Rahman and M RizonldquoTransient stability analysis of a three-machine nine bus powersystem networkrdquo Engineering Letters vol 22 no 1 pp 1ndash7 2014

[21] A Wolf J B Swift and H L Swinney ldquoDetermining Lyapunovexponents froma time seriesrdquoPhysicaDNonlinear Phenomenavol 16 no 3 pp 285ndash317 1985

[22] M A Salam ldquoTransient stability analysis of a power systemwith one generator connected to an infinite bus systemrdquoInternational Journal of Sustainable Energy vol 33 no 2 pp251ndash260 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Complexity

0009 001 0011 0012 00130008D1

02

03

04

051

(a)

minus05

minus04

minus03

minus02

minus01

0

01

larg

est L

yapu

nov

Expo

nent

0009 001 0011 00120008D1

(b)

Figure 10 (a) Bifurcation diagram (b) Largest Lyapunov exponent

minus05

0

05

1

15

2

1

30 40 50 60 70 8020t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 035 04 045 050251

(c)

0

01

02

03

04

05

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 11 Period 8 motion (D=000835) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

More concretely in the case of 1198811198611 = 05 the dynamicresponse is period 1 motion (shown in Figure 18) When1198811198611 = 07 the dynamic response is in almost periodicmotion the corresponding Poincare section is a circle (shownin Figure 20) In the case of 1198811198611 = 078 there existschaotic motion (Figure 21) After the parameter 1198811198611 passesthrough chaotic region the synchronous generator losessynchronization

4 Discussions

The chaotic dynamics may appear in power system whichmake a great threat to the stability of the power systemThere-fore many researchers pay much attention to the dynamic

behavior and chaotic mechanism especially for SMIB powersystem [3ndash9 22] From the point of nonlinear dynamic anal-ysis a few works have addressed the dynamic characteristicabout three-machine power system subjected to load distur-bance In this paper a relatively deep and systematic study ofdynamic response in three-machine power system subjectedto load disturbance has been carried out This paper aimsto demonstrate the complete transition process of differentdynamic behaviors by combining qualitative and quantitativeanalysis Comparing the results about simple power system(eg SMIB power system) [9ndash15] a new route to chaos viaquasi-periodic torus rupture has been found Moreover itis worthy to point out that in our considered system thereexist many super-harmonic components in the frequency

Complexity 9

minus05

0

05

1

15

2

1

35 40 45 50 5530t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 12 Period 4 motion (D=00085) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

75 80 85 90 95 10070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus6

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 13 Period 2 motion (D=0009) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

10 Complexity

minus05

0

05

1

15

2

1

280 285 290 295 300275t

(a)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(b)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(c)

0

01

02

03

04

05

06

Am

plitu

de5 10 150

Frequency

(d)

Figure 14 Period 1 motion (D=001) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

1 1005 101 1015 102 102502

03

04

05

VB0

3

(a)

1 1005 101 1015 102 1025minus05

minus04

minus03

minus02

minus01

0

01

02

VB0

Larg

est L

yapu

nov

Expo

nent

(b)

Figure 15 (a) Bifurcation diagram (b) Largest Lyapunov exponent

spectrum of the swing oscillation response Attention shouldbe paid to these new phenomena in engineering practice Inaddition for three-machine power system our future workwill investigate the mechanism of chaotic oscillation occur-rence by improving Melnikov analysis such that effectivemeasures can be taken in time to avoid system collapse

5 Conclusions

This paper has investigated the effects of critical systemparameters on dynamic characteristics of synchronous gen-erator rotors in a three-machine power system subjected

to load disturbance The swing equations describing themotions of the synchronous generator rotors are estab-lished Based on these swing equations with the help ofbifurcation diagrams largest Lyapunov exponent spectrumsphase portraits Poincare map and frequency spectrum theinfluence of system parameters on dynamic behaviors isshown clearly The Wolf method is introduced to calculatethe largest Lyapunov exponent which is used to verify theoccurrence of chaotic motion Moreover different pathsleading to chaos coexisting in this system have been foundThey are period-doubling cascading bifurcations to chaosinduced by changing the infinite bus voltage magnitude and

Complexity 11

05 055 06 065 07 07502

03

04

05

06

07

08

09

VB1

3

Figure 16 Bifurcation diagram

05 055 06 065 07 075minus05

minus04

minus03

minus02

minus01

0

01

VB1

Larg

est L

yapu

nov

Expo

nent

Figure 17 Largest Lyapunov exponent

minus02

0

02

04

06

08

1

3

16 17 18 19 2015t

(a)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(b)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(c)

0

005

01

015

02

Am

plitu

de

20 30 40 50 60 7010Frequency

(d)

Figure 18 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

12 Complexity

minus04minus02

002040608

112

3

28 30 3226t

(a)

minus6

minus4

minus2

0

2

4

6

3

02 04 06 0803

(b)

minus4

minus3

minus2

minus1

0

3

045 05 055 06 065043

(c)

0

005

01

015

02

Am

plitu

de

10 20 30 40 500Frequency

(d)

Figure 19 Quasi-periodic motion (1198811198611 = 06) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus04

minus02

0

02

04

06

08

3

21 22 23 24 25 2620t

(a)

minus3

minus2

minus1

0

1

2

3

3

03 035 04 0450253

(b)

minus2

minus15

minus1

minus05

0

05

3

0355 036 0365 037 0375 0380353

(c)

0

002

004

006

008

Am

plitu

de

10 20 30 400Frequency

(d)

Figure 20 Quasi-periodic motion (1198811198611 = 07) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 13

minus1

minus05

0

05

1

15

3

10 20 30 40 50 600t

(a)

minus30

minus20

minus10

0

10

20

30

3

050 1 15minus05minus13

(b)

minus30

minus20

minus10

0

10

20

30

3

04 06 08 1023

(c)

0005

01015

02025

03035

04

Am

plitu

de10 20 30 400

Frequency

(d)

Figure 21 Chaotic motion (1198811198611 = 078) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

quasi-periodic torus rupture to chaos induced by changingthe disturbance frequency of the infinite bus voltage Theseresults will contribute to a better understanding of the non-linear dynamic behaviors of synchronous generator rotors inthree-machine power system

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This research is supported byHigher Educational Science andTechnology Program of Shandong Province China (Grantno J18KA235) Shandong Provincial Natural Science Foun-dation China (Grants nos ZR2016AP06 ZR2018QA005ZR2018BA018 and ZR2018BA021) and National NaturalScience Foundation of China (Grants nos 11501246 and61703251)

References

[1] P M Anderson and A A Fouad Power System Control andStability Wiley-IEEE Press New York NY USA 2nd edition2002

[2] H Ma F Min and Y Wang ldquoNonlinear dynamic analysis andsurface sliding mode controller based on low pass filter forchaotic oscillation in power system with power disturbancerdquoChinese Journal of Physics vol 56 no 5 pp 2488ndash2499 2018

[3] D Q Wei and X S Luo ldquoNoise-induced chaos in single-machine infinite-bus power systemsrdquo EPL (Europhysics Letters)vol 86 no 5 Article ID 50008 2009

[4] D Q Wei B Zhang D Y Qiu and X S Luo ldquoEffect of noiseon erosion of safe basin in power systemrdquo Nonlinear Dynamicsvol 61 no 3 pp 477ndash482 2010

[5] W Zhu R Mohler R Spee W Mittelstadt and D Maratuku-lam ldquoHopf bifurcations in a SMIB power system with SSRrdquoIEEETransactions on Power Systems vol 11 no 3 pp 1579ndash15841996

[6] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaosand instability in a power system Subharmonic-resonant caserdquoNonlinear Dynamics vol 2 no 1 pp 53ndash72 1991

[7] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaos andinstability in a power system -Primary resonant caserdquoNonlinearDynamics vol 1 no 4 pp 313ndash339 1990

[8] X Duan J Wen and S Cheng ldquoBifurcation analysis foran SMIB power system with series capacitor compensationassociated with sub-synchronous resonancerdquo Science ChinaTechnological Sciences vol 52 no 2 pp 436ndash441 2009

[9] H K Chen T N Lin and J H Chen ldquoDynamic analysiscontrolling chaos and chaotification of a SMIB power systemrdquoChaos Solitons amp Fractals vol 24 no 5 pp 1307ndash1315 2005

[10] L F C Alberto and N G Bretas ldquoApplication of Melnikovrsquosmethod for computing heteroclinic orbits in a classical SMIBpower system modelrdquo IEEE Transactions on Circuits and Sys-tems I Fundamental eory and Applications vol 47 no 7 pp1085ndash1089 2000

14 Complexity

[11] W N Zhang and W D Zhang ldquoChaotic Ocillation of aNonlinear Power Systemrdquo Applied Mathematics Mechanics vol20 no 10 pp 1175ndash1182 1999

[12] L Zhou and F Chen ldquoChaotic dynamics for a class of single-machine-infinite bus power systemrdquo Journal of Vibration andControl vol 24 no 3 pp 582ndash587 2018

[13] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005

[14] X Wang Y Chen G Han and C Song ldquoNonlinear dynamicanalysis of a single-machine infinite-bus power systemrdquoAppliedMathematical Modelling vol 39 no 10-11 pp 2951ndash2961 2015

[15] B Yuan and Q H Sun ldquoChaos in the multi-machine powersystemrdquo Automation of Electric Power Systems vol 19 no 2 pp26ndash31 1995

[16] Y Ueda Y Ueda H B Stewart and R H Abraham ldquoNonlinearresonance in basin portraits of two coupled swings under peri-odic forcingrdquo International Journal of Bifurcation and Chaosvol 8 no 6 pp 1183ndash1197 1998

[17] S S Majidabad H T Shandiz and A Hajizadeh ldquoNonlinearfractional-order power system stabilizer for multi-machinepower systems based on sliding mode techniquerdquo InternationalJournal of Robust and Nonlinear Control vol 25 no 10 pp1548ndash1568 2015

[18] N Jiang and H-D Chiang ldquoDamping Torques of multi-machine power systems during transient behaviorsrdquo IEEETransactions on Power Systems vol 29 no 3 pp 1186ndash1193 2014

[19] Y Chang X Wang and D Xu ldquoBifurcation Analysis of aPower System Model with Three Machines and Four BusesrdquoInternational Journal of Bifurcation and Chaos vol 26 no 5 pp165ndash182 2016

[20] M A Salam M A Rashid Q M Rahman and M RizonldquoTransient stability analysis of a three-machine nine bus powersystem networkrdquo Engineering Letters vol 22 no 1 pp 1ndash7 2014

[21] A Wolf J B Swift and H L Swinney ldquoDetermining Lyapunovexponents froma time seriesrdquoPhysicaDNonlinear Phenomenavol 16 no 3 pp 285ndash317 1985

[22] M A Salam ldquoTransient stability analysis of a power systemwith one generator connected to an infinite bus systemrdquoInternational Journal of Sustainable Energy vol 33 no 2 pp251ndash260 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 9

minus05

0

05

1

15

2

1

35 40 45 50 5530t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 12 Period 4 motion (D=00085) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus05

0

05

1

15

2

25

1

75 80 85 90 95 10070t

(a)

minus15

minus10

minus5

0

5

10

1

0 05 1 15 2 25minus051

(b)

minus6

minus4

minus2

0

2

4

6

1

03 04 05021

(c)

0

01

02

03

04

05

06

Am

plitu

de

5 10 15 20 250Frequency

(d)

Figure 13 Period 2 motion (D=0009) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

10 Complexity

minus05

0

05

1

15

2

1

280 285 290 295 300275t

(a)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(b)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(c)

0

01

02

03

04

05

06

Am

plitu

de5 10 150

Frequency

(d)

Figure 14 Period 1 motion (D=001) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

1 1005 101 1015 102 102502

03

04

05

VB0

3

(a)

1 1005 101 1015 102 1025minus05

minus04

minus03

minus02

minus01

0

01

02

VB0

Larg

est L

yapu

nov

Expo

nent

(b)

Figure 15 (a) Bifurcation diagram (b) Largest Lyapunov exponent

spectrum of the swing oscillation response Attention shouldbe paid to these new phenomena in engineering practice Inaddition for three-machine power system our future workwill investigate the mechanism of chaotic oscillation occur-rence by improving Melnikov analysis such that effectivemeasures can be taken in time to avoid system collapse

5 Conclusions

This paper has investigated the effects of critical systemparameters on dynamic characteristics of synchronous gen-erator rotors in a three-machine power system subjected

to load disturbance The swing equations describing themotions of the synchronous generator rotors are estab-lished Based on these swing equations with the help ofbifurcation diagrams largest Lyapunov exponent spectrumsphase portraits Poincare map and frequency spectrum theinfluence of system parameters on dynamic behaviors isshown clearly The Wolf method is introduced to calculatethe largest Lyapunov exponent which is used to verify theoccurrence of chaotic motion Moreover different pathsleading to chaos coexisting in this system have been foundThey are period-doubling cascading bifurcations to chaosinduced by changing the infinite bus voltage magnitude and

Complexity 11

05 055 06 065 07 07502

03

04

05

06

07

08

09

VB1

3

Figure 16 Bifurcation diagram

05 055 06 065 07 075minus05

minus04

minus03

minus02

minus01

0

01

VB1

Larg

est L

yapu

nov

Expo

nent

Figure 17 Largest Lyapunov exponent

minus02

0

02

04

06

08

1

3

16 17 18 19 2015t

(a)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(b)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(c)

0

005

01

015

02

Am

plitu

de

20 30 40 50 60 7010Frequency

(d)

Figure 18 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

12 Complexity

minus04minus02

002040608

112

3

28 30 3226t

(a)

minus6

minus4

minus2

0

2

4

6

3

02 04 06 0803

(b)

minus4

minus3

minus2

minus1

0

3

045 05 055 06 065043

(c)

0

005

01

015

02

Am

plitu

de

10 20 30 40 500Frequency

(d)

Figure 19 Quasi-periodic motion (1198811198611 = 06) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus04

minus02

0

02

04

06

08

3

21 22 23 24 25 2620t

(a)

minus3

minus2

minus1

0

1

2

3

3

03 035 04 0450253

(b)

minus2

minus15

minus1

minus05

0

05

3

0355 036 0365 037 0375 0380353

(c)

0

002

004

006

008

Am

plitu

de

10 20 30 400Frequency

(d)

Figure 20 Quasi-periodic motion (1198811198611 = 07) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 13

minus1

minus05

0

05

1

15

3

10 20 30 40 50 600t

(a)

minus30

minus20

minus10

0

10

20

30

3

050 1 15minus05minus13

(b)

minus30

minus20

minus10

0

10

20

30

3

04 06 08 1023

(c)

0005

01015

02025

03035

04

Am

plitu

de10 20 30 400

Frequency

(d)

Figure 21 Chaotic motion (1198811198611 = 078) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

quasi-periodic torus rupture to chaos induced by changingthe disturbance frequency of the infinite bus voltage Theseresults will contribute to a better understanding of the non-linear dynamic behaviors of synchronous generator rotors inthree-machine power system

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This research is supported byHigher Educational Science andTechnology Program of Shandong Province China (Grantno J18KA235) Shandong Provincial Natural Science Foun-dation China (Grants nos ZR2016AP06 ZR2018QA005ZR2018BA018 and ZR2018BA021) and National NaturalScience Foundation of China (Grants nos 11501246 and61703251)

References

[1] P M Anderson and A A Fouad Power System Control andStability Wiley-IEEE Press New York NY USA 2nd edition2002

[2] H Ma F Min and Y Wang ldquoNonlinear dynamic analysis andsurface sliding mode controller based on low pass filter forchaotic oscillation in power system with power disturbancerdquoChinese Journal of Physics vol 56 no 5 pp 2488ndash2499 2018

[3] D Q Wei and X S Luo ldquoNoise-induced chaos in single-machine infinite-bus power systemsrdquo EPL (Europhysics Letters)vol 86 no 5 Article ID 50008 2009

[4] D Q Wei B Zhang D Y Qiu and X S Luo ldquoEffect of noiseon erosion of safe basin in power systemrdquo Nonlinear Dynamicsvol 61 no 3 pp 477ndash482 2010

[5] W Zhu R Mohler R Spee W Mittelstadt and D Maratuku-lam ldquoHopf bifurcations in a SMIB power system with SSRrdquoIEEETransactions on Power Systems vol 11 no 3 pp 1579ndash15841996

[6] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaosand instability in a power system Subharmonic-resonant caserdquoNonlinear Dynamics vol 2 no 1 pp 53ndash72 1991

[7] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaos andinstability in a power system -Primary resonant caserdquoNonlinearDynamics vol 1 no 4 pp 313ndash339 1990

[8] X Duan J Wen and S Cheng ldquoBifurcation analysis foran SMIB power system with series capacitor compensationassociated with sub-synchronous resonancerdquo Science ChinaTechnological Sciences vol 52 no 2 pp 436ndash441 2009

[9] H K Chen T N Lin and J H Chen ldquoDynamic analysiscontrolling chaos and chaotification of a SMIB power systemrdquoChaos Solitons amp Fractals vol 24 no 5 pp 1307ndash1315 2005

[10] L F C Alberto and N G Bretas ldquoApplication of Melnikovrsquosmethod for computing heteroclinic orbits in a classical SMIBpower system modelrdquo IEEE Transactions on Circuits and Sys-tems I Fundamental eory and Applications vol 47 no 7 pp1085ndash1089 2000

14 Complexity

[11] W N Zhang and W D Zhang ldquoChaotic Ocillation of aNonlinear Power Systemrdquo Applied Mathematics Mechanics vol20 no 10 pp 1175ndash1182 1999

[12] L Zhou and F Chen ldquoChaotic dynamics for a class of single-machine-infinite bus power systemrdquo Journal of Vibration andControl vol 24 no 3 pp 582ndash587 2018

[13] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005

[14] X Wang Y Chen G Han and C Song ldquoNonlinear dynamicanalysis of a single-machine infinite-bus power systemrdquoAppliedMathematical Modelling vol 39 no 10-11 pp 2951ndash2961 2015

[15] B Yuan and Q H Sun ldquoChaos in the multi-machine powersystemrdquo Automation of Electric Power Systems vol 19 no 2 pp26ndash31 1995

[16] Y Ueda Y Ueda H B Stewart and R H Abraham ldquoNonlinearresonance in basin portraits of two coupled swings under peri-odic forcingrdquo International Journal of Bifurcation and Chaosvol 8 no 6 pp 1183ndash1197 1998

[17] S S Majidabad H T Shandiz and A Hajizadeh ldquoNonlinearfractional-order power system stabilizer for multi-machinepower systems based on sliding mode techniquerdquo InternationalJournal of Robust and Nonlinear Control vol 25 no 10 pp1548ndash1568 2015

[18] N Jiang and H-D Chiang ldquoDamping Torques of multi-machine power systems during transient behaviorsrdquo IEEETransactions on Power Systems vol 29 no 3 pp 1186ndash1193 2014

[19] Y Chang X Wang and D Xu ldquoBifurcation Analysis of aPower System Model with Three Machines and Four BusesrdquoInternational Journal of Bifurcation and Chaos vol 26 no 5 pp165ndash182 2016

[20] M A Salam M A Rashid Q M Rahman and M RizonldquoTransient stability analysis of a three-machine nine bus powersystem networkrdquo Engineering Letters vol 22 no 1 pp 1ndash7 2014

[21] A Wolf J B Swift and H L Swinney ldquoDetermining Lyapunovexponents froma time seriesrdquoPhysicaDNonlinear Phenomenavol 16 no 3 pp 285ndash317 1985

[22] M A Salam ldquoTransient stability analysis of a power systemwith one generator connected to an infinite bus systemrdquoInternational Journal of Sustainable Energy vol 33 no 2 pp251ndash260 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Complexity

minus05

0

05

1

15

2

1

280 285 290 295 300275t

(a)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(b)

minus15

minus10

minus5

0

5

10

1

05 1 15 201

(c)

0

01

02

03

04

05

06

Am

plitu

de5 10 150

Frequency

(d)

Figure 14 Period 1 motion (D=001) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

1 1005 101 1015 102 102502

03

04

05

VB0

3

(a)

1 1005 101 1015 102 1025minus05

minus04

minus03

minus02

minus01

0

01

02

VB0

Larg

est L

yapu

nov

Expo

nent

(b)

Figure 15 (a) Bifurcation diagram (b) Largest Lyapunov exponent

spectrum of the swing oscillation response Attention shouldbe paid to these new phenomena in engineering practice Inaddition for three-machine power system our future workwill investigate the mechanism of chaotic oscillation occur-rence by improving Melnikov analysis such that effectivemeasures can be taken in time to avoid system collapse

5 Conclusions

This paper has investigated the effects of critical systemparameters on dynamic characteristics of synchronous gen-erator rotors in a three-machine power system subjected

to load disturbance The swing equations describing themotions of the synchronous generator rotors are estab-lished Based on these swing equations with the help ofbifurcation diagrams largest Lyapunov exponent spectrumsphase portraits Poincare map and frequency spectrum theinfluence of system parameters on dynamic behaviors isshown clearly The Wolf method is introduced to calculatethe largest Lyapunov exponent which is used to verify theoccurrence of chaotic motion Moreover different pathsleading to chaos coexisting in this system have been foundThey are period-doubling cascading bifurcations to chaosinduced by changing the infinite bus voltage magnitude and

Complexity 11

05 055 06 065 07 07502

03

04

05

06

07

08

09

VB1

3

Figure 16 Bifurcation diagram

05 055 06 065 07 075minus05

minus04

minus03

minus02

minus01

0

01

VB1

Larg

est L

yapu

nov

Expo

nent

Figure 17 Largest Lyapunov exponent

minus02

0

02

04

06

08

1

3

16 17 18 19 2015t

(a)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(b)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(c)

0

005

01

015

02

Am

plitu

de

20 30 40 50 60 7010Frequency

(d)

Figure 18 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

12 Complexity

minus04minus02

002040608

112

3

28 30 3226t

(a)

minus6

minus4

minus2

0

2

4

6

3

02 04 06 0803

(b)

minus4

minus3

minus2

minus1

0

3

045 05 055 06 065043

(c)

0

005

01

015

02

Am

plitu

de

10 20 30 40 500Frequency

(d)

Figure 19 Quasi-periodic motion (1198811198611 = 06) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus04

minus02

0

02

04

06

08

3

21 22 23 24 25 2620t

(a)

minus3

minus2

minus1

0

1

2

3

3

03 035 04 0450253

(b)

minus2

minus15

minus1

minus05

0

05

3

0355 036 0365 037 0375 0380353

(c)

0

002

004

006

008

Am

plitu

de

10 20 30 400Frequency

(d)

Figure 20 Quasi-periodic motion (1198811198611 = 07) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 13

minus1

minus05

0

05

1

15

3

10 20 30 40 50 600t

(a)

minus30

minus20

minus10

0

10

20

30

3

050 1 15minus05minus13

(b)

minus30

minus20

minus10

0

10

20

30

3

04 06 08 1023

(c)

0005

01015

02025

03035

04

Am

plitu

de10 20 30 400

Frequency

(d)

Figure 21 Chaotic motion (1198811198611 = 078) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

quasi-periodic torus rupture to chaos induced by changingthe disturbance frequency of the infinite bus voltage Theseresults will contribute to a better understanding of the non-linear dynamic behaviors of synchronous generator rotors inthree-machine power system

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This research is supported byHigher Educational Science andTechnology Program of Shandong Province China (Grantno J18KA235) Shandong Provincial Natural Science Foun-dation China (Grants nos ZR2016AP06 ZR2018QA005ZR2018BA018 and ZR2018BA021) and National NaturalScience Foundation of China (Grants nos 11501246 and61703251)

References

[1] P M Anderson and A A Fouad Power System Control andStability Wiley-IEEE Press New York NY USA 2nd edition2002

[2] H Ma F Min and Y Wang ldquoNonlinear dynamic analysis andsurface sliding mode controller based on low pass filter forchaotic oscillation in power system with power disturbancerdquoChinese Journal of Physics vol 56 no 5 pp 2488ndash2499 2018

[3] D Q Wei and X S Luo ldquoNoise-induced chaos in single-machine infinite-bus power systemsrdquo EPL (Europhysics Letters)vol 86 no 5 Article ID 50008 2009

[4] D Q Wei B Zhang D Y Qiu and X S Luo ldquoEffect of noiseon erosion of safe basin in power systemrdquo Nonlinear Dynamicsvol 61 no 3 pp 477ndash482 2010

[5] W Zhu R Mohler R Spee W Mittelstadt and D Maratuku-lam ldquoHopf bifurcations in a SMIB power system with SSRrdquoIEEETransactions on Power Systems vol 11 no 3 pp 1579ndash15841996

[6] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaosand instability in a power system Subharmonic-resonant caserdquoNonlinear Dynamics vol 2 no 1 pp 53ndash72 1991

[7] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaos andinstability in a power system -Primary resonant caserdquoNonlinearDynamics vol 1 no 4 pp 313ndash339 1990

[8] X Duan J Wen and S Cheng ldquoBifurcation analysis foran SMIB power system with series capacitor compensationassociated with sub-synchronous resonancerdquo Science ChinaTechnological Sciences vol 52 no 2 pp 436ndash441 2009

[9] H K Chen T N Lin and J H Chen ldquoDynamic analysiscontrolling chaos and chaotification of a SMIB power systemrdquoChaos Solitons amp Fractals vol 24 no 5 pp 1307ndash1315 2005

[10] L F C Alberto and N G Bretas ldquoApplication of Melnikovrsquosmethod for computing heteroclinic orbits in a classical SMIBpower system modelrdquo IEEE Transactions on Circuits and Sys-tems I Fundamental eory and Applications vol 47 no 7 pp1085ndash1089 2000

14 Complexity

[11] W N Zhang and W D Zhang ldquoChaotic Ocillation of aNonlinear Power Systemrdquo Applied Mathematics Mechanics vol20 no 10 pp 1175ndash1182 1999

[12] L Zhou and F Chen ldquoChaotic dynamics for a class of single-machine-infinite bus power systemrdquo Journal of Vibration andControl vol 24 no 3 pp 582ndash587 2018

[13] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005

[14] X Wang Y Chen G Han and C Song ldquoNonlinear dynamicanalysis of a single-machine infinite-bus power systemrdquoAppliedMathematical Modelling vol 39 no 10-11 pp 2951ndash2961 2015

[15] B Yuan and Q H Sun ldquoChaos in the multi-machine powersystemrdquo Automation of Electric Power Systems vol 19 no 2 pp26ndash31 1995

[16] Y Ueda Y Ueda H B Stewart and R H Abraham ldquoNonlinearresonance in basin portraits of two coupled swings under peri-odic forcingrdquo International Journal of Bifurcation and Chaosvol 8 no 6 pp 1183ndash1197 1998

[17] S S Majidabad H T Shandiz and A Hajizadeh ldquoNonlinearfractional-order power system stabilizer for multi-machinepower systems based on sliding mode techniquerdquo InternationalJournal of Robust and Nonlinear Control vol 25 no 10 pp1548ndash1568 2015

[18] N Jiang and H-D Chiang ldquoDamping Torques of multi-machine power systems during transient behaviorsrdquo IEEETransactions on Power Systems vol 29 no 3 pp 1186ndash1193 2014

[19] Y Chang X Wang and D Xu ldquoBifurcation Analysis of aPower System Model with Three Machines and Four BusesrdquoInternational Journal of Bifurcation and Chaos vol 26 no 5 pp165ndash182 2016

[20] M A Salam M A Rashid Q M Rahman and M RizonldquoTransient stability analysis of a three-machine nine bus powersystem networkrdquo Engineering Letters vol 22 no 1 pp 1ndash7 2014

[21] A Wolf J B Swift and H L Swinney ldquoDetermining Lyapunovexponents froma time seriesrdquoPhysicaDNonlinear Phenomenavol 16 no 3 pp 285ndash317 1985

[22] M A Salam ldquoTransient stability analysis of a power systemwith one generator connected to an infinite bus systemrdquoInternational Journal of Sustainable Energy vol 33 no 2 pp251ndash260 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 11

05 055 06 065 07 07502

03

04

05

06

07

08

09

VB1

3

Figure 16 Bifurcation diagram

05 055 06 065 07 075minus05

minus04

minus03

minus02

minus01

0

01

VB1

Larg

est L

yapu

nov

Expo

nent

Figure 17 Largest Lyapunov exponent

minus02

0

02

04

06

08

1

3

16 17 18 19 2015t

(a)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(b)

minus4

minus2

0

2

4

3

02 03 04 05 06013

(c)

0

005

01

015

02

Am

plitu

de

20 30 40 50 60 7010Frequency

(d)

Figure 18 Period 1 motion (1198811198611 = 05) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

12 Complexity

minus04minus02

002040608

112

3

28 30 3226t

(a)

minus6

minus4

minus2

0

2

4

6

3

02 04 06 0803

(b)

minus4

minus3

minus2

minus1

0

3

045 05 055 06 065043

(c)

0

005

01

015

02

Am

plitu

de

10 20 30 40 500Frequency

(d)

Figure 19 Quasi-periodic motion (1198811198611 = 06) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus04

minus02

0

02

04

06

08

3

21 22 23 24 25 2620t

(a)

minus3

minus2

minus1

0

1

2

3

3

03 035 04 0450253

(b)

minus2

minus15

minus1

minus05

0

05

3

0355 036 0365 037 0375 0380353

(c)

0

002

004

006

008

Am

plitu

de

10 20 30 400Frequency

(d)

Figure 20 Quasi-periodic motion (1198811198611 = 07) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 13

minus1

minus05

0

05

1

15

3

10 20 30 40 50 600t

(a)

minus30

minus20

minus10

0

10

20

30

3

050 1 15minus05minus13

(b)

minus30

minus20

minus10

0

10

20

30

3

04 06 08 1023

(c)

0005

01015

02025

03035

04

Am

plitu

de10 20 30 400

Frequency

(d)

Figure 21 Chaotic motion (1198811198611 = 078) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

quasi-periodic torus rupture to chaos induced by changingthe disturbance frequency of the infinite bus voltage Theseresults will contribute to a better understanding of the non-linear dynamic behaviors of synchronous generator rotors inthree-machine power system

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This research is supported byHigher Educational Science andTechnology Program of Shandong Province China (Grantno J18KA235) Shandong Provincial Natural Science Foun-dation China (Grants nos ZR2016AP06 ZR2018QA005ZR2018BA018 and ZR2018BA021) and National NaturalScience Foundation of China (Grants nos 11501246 and61703251)

References

[1] P M Anderson and A A Fouad Power System Control andStability Wiley-IEEE Press New York NY USA 2nd edition2002

[2] H Ma F Min and Y Wang ldquoNonlinear dynamic analysis andsurface sliding mode controller based on low pass filter forchaotic oscillation in power system with power disturbancerdquoChinese Journal of Physics vol 56 no 5 pp 2488ndash2499 2018

[3] D Q Wei and X S Luo ldquoNoise-induced chaos in single-machine infinite-bus power systemsrdquo EPL (Europhysics Letters)vol 86 no 5 Article ID 50008 2009

[4] D Q Wei B Zhang D Y Qiu and X S Luo ldquoEffect of noiseon erosion of safe basin in power systemrdquo Nonlinear Dynamicsvol 61 no 3 pp 477ndash482 2010

[5] W Zhu R Mohler R Spee W Mittelstadt and D Maratuku-lam ldquoHopf bifurcations in a SMIB power system with SSRrdquoIEEETransactions on Power Systems vol 11 no 3 pp 1579ndash15841996

[6] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaosand instability in a power system Subharmonic-resonant caserdquoNonlinear Dynamics vol 2 no 1 pp 53ndash72 1991

[7] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaos andinstability in a power system -Primary resonant caserdquoNonlinearDynamics vol 1 no 4 pp 313ndash339 1990

[8] X Duan J Wen and S Cheng ldquoBifurcation analysis foran SMIB power system with series capacitor compensationassociated with sub-synchronous resonancerdquo Science ChinaTechnological Sciences vol 52 no 2 pp 436ndash441 2009

[9] H K Chen T N Lin and J H Chen ldquoDynamic analysiscontrolling chaos and chaotification of a SMIB power systemrdquoChaos Solitons amp Fractals vol 24 no 5 pp 1307ndash1315 2005

[10] L F C Alberto and N G Bretas ldquoApplication of Melnikovrsquosmethod for computing heteroclinic orbits in a classical SMIBpower system modelrdquo IEEE Transactions on Circuits and Sys-tems I Fundamental eory and Applications vol 47 no 7 pp1085ndash1089 2000

14 Complexity

[11] W N Zhang and W D Zhang ldquoChaotic Ocillation of aNonlinear Power Systemrdquo Applied Mathematics Mechanics vol20 no 10 pp 1175ndash1182 1999

[12] L Zhou and F Chen ldquoChaotic dynamics for a class of single-machine-infinite bus power systemrdquo Journal of Vibration andControl vol 24 no 3 pp 582ndash587 2018

[13] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005

[14] X Wang Y Chen G Han and C Song ldquoNonlinear dynamicanalysis of a single-machine infinite-bus power systemrdquoAppliedMathematical Modelling vol 39 no 10-11 pp 2951ndash2961 2015

[15] B Yuan and Q H Sun ldquoChaos in the multi-machine powersystemrdquo Automation of Electric Power Systems vol 19 no 2 pp26ndash31 1995

[16] Y Ueda Y Ueda H B Stewart and R H Abraham ldquoNonlinearresonance in basin portraits of two coupled swings under peri-odic forcingrdquo International Journal of Bifurcation and Chaosvol 8 no 6 pp 1183ndash1197 1998

[17] S S Majidabad H T Shandiz and A Hajizadeh ldquoNonlinearfractional-order power system stabilizer for multi-machinepower systems based on sliding mode techniquerdquo InternationalJournal of Robust and Nonlinear Control vol 25 no 10 pp1548ndash1568 2015

[18] N Jiang and H-D Chiang ldquoDamping Torques of multi-machine power systems during transient behaviorsrdquo IEEETransactions on Power Systems vol 29 no 3 pp 1186ndash1193 2014

[19] Y Chang X Wang and D Xu ldquoBifurcation Analysis of aPower System Model with Three Machines and Four BusesrdquoInternational Journal of Bifurcation and Chaos vol 26 no 5 pp165ndash182 2016

[20] M A Salam M A Rashid Q M Rahman and M RizonldquoTransient stability analysis of a three-machine nine bus powersystem networkrdquo Engineering Letters vol 22 no 1 pp 1ndash7 2014

[21] A Wolf J B Swift and H L Swinney ldquoDetermining Lyapunovexponents froma time seriesrdquoPhysicaDNonlinear Phenomenavol 16 no 3 pp 285ndash317 1985

[22] M A Salam ldquoTransient stability analysis of a power systemwith one generator connected to an infinite bus systemrdquoInternational Journal of Sustainable Energy vol 33 no 2 pp251ndash260 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Complexity

minus04minus02

002040608

112

3

28 30 3226t

(a)

minus6

minus4

minus2

0

2

4

6

3

02 04 06 0803

(b)

minus4

minus3

minus2

minus1

0

3

045 05 055 06 065043

(c)

0

005

01

015

02

Am

plitu

de

10 20 30 40 500Frequency

(d)

Figure 19 Quasi-periodic motion (1198811198611 = 06) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

minus04

minus02

0

02

04

06

08

3

21 22 23 24 25 2620t

(a)

minus3

minus2

minus1

0

1

2

3

3

03 035 04 0450253

(b)

minus2

minus15

minus1

minus05

0

05

3

0355 036 0365 037 0375 0380353

(c)

0

002

004

006

008

Am

plitu

de

10 20 30 400Frequency

(d)

Figure 20 Quasi-periodic motion (1198811198611 = 07) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

Complexity 13

minus1

minus05

0

05

1

15

3

10 20 30 40 50 600t

(a)

minus30

minus20

minus10

0

10

20

30

3

050 1 15minus05minus13

(b)

minus30

minus20

minus10

0

10

20

30

3

04 06 08 1023

(c)

0005

01015

02025

03035

04

Am

plitu

de10 20 30 400

Frequency

(d)

Figure 21 Chaotic motion (1198811198611 = 078) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

quasi-periodic torus rupture to chaos induced by changingthe disturbance frequency of the infinite bus voltage Theseresults will contribute to a better understanding of the non-linear dynamic behaviors of synchronous generator rotors inthree-machine power system

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This research is supported byHigher Educational Science andTechnology Program of Shandong Province China (Grantno J18KA235) Shandong Provincial Natural Science Foun-dation China (Grants nos ZR2016AP06 ZR2018QA005ZR2018BA018 and ZR2018BA021) and National NaturalScience Foundation of China (Grants nos 11501246 and61703251)

References

[1] P M Anderson and A A Fouad Power System Control andStability Wiley-IEEE Press New York NY USA 2nd edition2002

[2] H Ma F Min and Y Wang ldquoNonlinear dynamic analysis andsurface sliding mode controller based on low pass filter forchaotic oscillation in power system with power disturbancerdquoChinese Journal of Physics vol 56 no 5 pp 2488ndash2499 2018

[3] D Q Wei and X S Luo ldquoNoise-induced chaos in single-machine infinite-bus power systemsrdquo EPL (Europhysics Letters)vol 86 no 5 Article ID 50008 2009

[4] D Q Wei B Zhang D Y Qiu and X S Luo ldquoEffect of noiseon erosion of safe basin in power systemrdquo Nonlinear Dynamicsvol 61 no 3 pp 477ndash482 2010

[5] W Zhu R Mohler R Spee W Mittelstadt and D Maratuku-lam ldquoHopf bifurcations in a SMIB power system with SSRrdquoIEEETransactions on Power Systems vol 11 no 3 pp 1579ndash15841996

[6] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaosand instability in a power system Subharmonic-resonant caserdquoNonlinear Dynamics vol 2 no 1 pp 53ndash72 1991

[7] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaos andinstability in a power system -Primary resonant caserdquoNonlinearDynamics vol 1 no 4 pp 313ndash339 1990

[8] X Duan J Wen and S Cheng ldquoBifurcation analysis foran SMIB power system with series capacitor compensationassociated with sub-synchronous resonancerdquo Science ChinaTechnological Sciences vol 52 no 2 pp 436ndash441 2009

[9] H K Chen T N Lin and J H Chen ldquoDynamic analysiscontrolling chaos and chaotification of a SMIB power systemrdquoChaos Solitons amp Fractals vol 24 no 5 pp 1307ndash1315 2005

[10] L F C Alberto and N G Bretas ldquoApplication of Melnikovrsquosmethod for computing heteroclinic orbits in a classical SMIBpower system modelrdquo IEEE Transactions on Circuits and Sys-tems I Fundamental eory and Applications vol 47 no 7 pp1085ndash1089 2000

14 Complexity

[11] W N Zhang and W D Zhang ldquoChaotic Ocillation of aNonlinear Power Systemrdquo Applied Mathematics Mechanics vol20 no 10 pp 1175ndash1182 1999

[12] L Zhou and F Chen ldquoChaotic dynamics for a class of single-machine-infinite bus power systemrdquo Journal of Vibration andControl vol 24 no 3 pp 582ndash587 2018

[13] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005

[14] X Wang Y Chen G Han and C Song ldquoNonlinear dynamicanalysis of a single-machine infinite-bus power systemrdquoAppliedMathematical Modelling vol 39 no 10-11 pp 2951ndash2961 2015

[15] B Yuan and Q H Sun ldquoChaos in the multi-machine powersystemrdquo Automation of Electric Power Systems vol 19 no 2 pp26ndash31 1995

[16] Y Ueda Y Ueda H B Stewart and R H Abraham ldquoNonlinearresonance in basin portraits of two coupled swings under peri-odic forcingrdquo International Journal of Bifurcation and Chaosvol 8 no 6 pp 1183ndash1197 1998

[17] S S Majidabad H T Shandiz and A Hajizadeh ldquoNonlinearfractional-order power system stabilizer for multi-machinepower systems based on sliding mode techniquerdquo InternationalJournal of Robust and Nonlinear Control vol 25 no 10 pp1548ndash1568 2015

[18] N Jiang and H-D Chiang ldquoDamping Torques of multi-machine power systems during transient behaviorsrdquo IEEETransactions on Power Systems vol 29 no 3 pp 1186ndash1193 2014

[19] Y Chang X Wang and D Xu ldquoBifurcation Analysis of aPower System Model with Three Machines and Four BusesrdquoInternational Journal of Bifurcation and Chaos vol 26 no 5 pp165ndash182 2016

[20] M A Salam M A Rashid Q M Rahman and M RizonldquoTransient stability analysis of a three-machine nine bus powersystem networkrdquo Engineering Letters vol 22 no 1 pp 1ndash7 2014

[21] A Wolf J B Swift and H L Swinney ldquoDetermining Lyapunovexponents froma time seriesrdquoPhysicaDNonlinear Phenomenavol 16 no 3 pp 285ndash317 1985

[22] M A Salam ldquoTransient stability analysis of a power systemwith one generator connected to an infinite bus systemrdquoInternational Journal of Sustainable Energy vol 33 no 2 pp251ndash260 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 13

minus1

minus05

0

05

1

15

3

10 20 30 40 50 600t

(a)

minus30

minus20

minus10

0

10

20

30

3

050 1 15minus05minus13

(b)

minus30

minus20

minus10

0

10

20

30

3

04 06 08 1023

(c)

0005

01015

02025

03035

04

Am

plitu

de10 20 30 400

Frequency

(d)

Figure 21 Chaotic motion (1198811198611 = 078) (a) Time history (b) Phase portrait (c) Poincare map (d) Frequency spectrum

quasi-periodic torus rupture to chaos induced by changingthe disturbance frequency of the infinite bus voltage Theseresults will contribute to a better understanding of the non-linear dynamic behaviors of synchronous generator rotors inthree-machine power system

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This research is supported byHigher Educational Science andTechnology Program of Shandong Province China (Grantno J18KA235) Shandong Provincial Natural Science Foun-dation China (Grants nos ZR2016AP06 ZR2018QA005ZR2018BA018 and ZR2018BA021) and National NaturalScience Foundation of China (Grants nos 11501246 and61703251)

References

[1] P M Anderson and A A Fouad Power System Control andStability Wiley-IEEE Press New York NY USA 2nd edition2002

[2] H Ma F Min and Y Wang ldquoNonlinear dynamic analysis andsurface sliding mode controller based on low pass filter forchaotic oscillation in power system with power disturbancerdquoChinese Journal of Physics vol 56 no 5 pp 2488ndash2499 2018

[3] D Q Wei and X S Luo ldquoNoise-induced chaos in single-machine infinite-bus power systemsrdquo EPL (Europhysics Letters)vol 86 no 5 Article ID 50008 2009

[4] D Q Wei B Zhang D Y Qiu and X S Luo ldquoEffect of noiseon erosion of safe basin in power systemrdquo Nonlinear Dynamicsvol 61 no 3 pp 477ndash482 2010

[5] W Zhu R Mohler R Spee W Mittelstadt and D Maratuku-lam ldquoHopf bifurcations in a SMIB power system with SSRrdquoIEEETransactions on Power Systems vol 11 no 3 pp 1579ndash15841996

[6] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaosand instability in a power system Subharmonic-resonant caserdquoNonlinear Dynamics vol 2 no 1 pp 53ndash72 1991

[7] M A Nayfeh A M A Hamdan and A H Nayfeh ldquoChaos andinstability in a power system -Primary resonant caserdquoNonlinearDynamics vol 1 no 4 pp 313ndash339 1990

[8] X Duan J Wen and S Cheng ldquoBifurcation analysis foran SMIB power system with series capacitor compensationassociated with sub-synchronous resonancerdquo Science ChinaTechnological Sciences vol 52 no 2 pp 436ndash441 2009

[9] H K Chen T N Lin and J H Chen ldquoDynamic analysiscontrolling chaos and chaotification of a SMIB power systemrdquoChaos Solitons amp Fractals vol 24 no 5 pp 1307ndash1315 2005

[10] L F C Alberto and N G Bretas ldquoApplication of Melnikovrsquosmethod for computing heteroclinic orbits in a classical SMIBpower system modelrdquo IEEE Transactions on Circuits and Sys-tems I Fundamental eory and Applications vol 47 no 7 pp1085ndash1089 2000

14 Complexity

[11] W N Zhang and W D Zhang ldquoChaotic Ocillation of aNonlinear Power Systemrdquo Applied Mathematics Mechanics vol20 no 10 pp 1175ndash1182 1999

[12] L Zhou and F Chen ldquoChaotic dynamics for a class of single-machine-infinite bus power systemrdquo Journal of Vibration andControl vol 24 no 3 pp 582ndash587 2018

[13] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005

[14] X Wang Y Chen G Han and C Song ldquoNonlinear dynamicanalysis of a single-machine infinite-bus power systemrdquoAppliedMathematical Modelling vol 39 no 10-11 pp 2951ndash2961 2015

[15] B Yuan and Q H Sun ldquoChaos in the multi-machine powersystemrdquo Automation of Electric Power Systems vol 19 no 2 pp26ndash31 1995

[16] Y Ueda Y Ueda H B Stewart and R H Abraham ldquoNonlinearresonance in basin portraits of two coupled swings under peri-odic forcingrdquo International Journal of Bifurcation and Chaosvol 8 no 6 pp 1183ndash1197 1998

[17] S S Majidabad H T Shandiz and A Hajizadeh ldquoNonlinearfractional-order power system stabilizer for multi-machinepower systems based on sliding mode techniquerdquo InternationalJournal of Robust and Nonlinear Control vol 25 no 10 pp1548ndash1568 2015

[18] N Jiang and H-D Chiang ldquoDamping Torques of multi-machine power systems during transient behaviorsrdquo IEEETransactions on Power Systems vol 29 no 3 pp 1186ndash1193 2014

[19] Y Chang X Wang and D Xu ldquoBifurcation Analysis of aPower System Model with Three Machines and Four BusesrdquoInternational Journal of Bifurcation and Chaos vol 26 no 5 pp165ndash182 2016

[20] M A Salam M A Rashid Q M Rahman and M RizonldquoTransient stability analysis of a three-machine nine bus powersystem networkrdquo Engineering Letters vol 22 no 1 pp 1ndash7 2014

[21] A Wolf J B Swift and H L Swinney ldquoDetermining Lyapunovexponents froma time seriesrdquoPhysicaDNonlinear Phenomenavol 16 no 3 pp 285ndash317 1985

[22] M A Salam ldquoTransient stability analysis of a power systemwith one generator connected to an infinite bus systemrdquoInternational Journal of Sustainable Energy vol 33 no 2 pp251ndash260 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

14 Complexity

[11] W N Zhang and W D Zhang ldquoChaotic Ocillation of aNonlinear Power Systemrdquo Applied Mathematics Mechanics vol20 no 10 pp 1175ndash1182 1999

[12] L Zhou and F Chen ldquoChaotic dynamics for a class of single-machine-infinite bus power systemrdquo Journal of Vibration andControl vol 24 no 3 pp 582ndash587 2018

[13] X Chen W Zhang and W Zhang ldquoChaotic and SubharmonicOscillations of a Nonlinear Power Systemrdquo IEEE Transactionson Circuits and Systems II Express Briefs vol 52 no 12 pp 811ndash815 2005

[14] X Wang Y Chen G Han and C Song ldquoNonlinear dynamicanalysis of a single-machine infinite-bus power systemrdquoAppliedMathematical Modelling vol 39 no 10-11 pp 2951ndash2961 2015

[15] B Yuan and Q H Sun ldquoChaos in the multi-machine powersystemrdquo Automation of Electric Power Systems vol 19 no 2 pp26ndash31 1995

[16] Y Ueda Y Ueda H B Stewart and R H Abraham ldquoNonlinearresonance in basin portraits of two coupled swings under peri-odic forcingrdquo International Journal of Bifurcation and Chaosvol 8 no 6 pp 1183ndash1197 1998

[17] S S Majidabad H T Shandiz and A Hajizadeh ldquoNonlinearfractional-order power system stabilizer for multi-machinepower systems based on sliding mode techniquerdquo InternationalJournal of Robust and Nonlinear Control vol 25 no 10 pp1548ndash1568 2015

[18] N Jiang and H-D Chiang ldquoDamping Torques of multi-machine power systems during transient behaviorsrdquo IEEETransactions on Power Systems vol 29 no 3 pp 1186ndash1193 2014

[19] Y Chang X Wang and D Xu ldquoBifurcation Analysis of aPower System Model with Three Machines and Four BusesrdquoInternational Journal of Bifurcation and Chaos vol 26 no 5 pp165ndash182 2016

[20] M A Salam M A Rashid Q M Rahman and M RizonldquoTransient stability analysis of a three-machine nine bus powersystem networkrdquo Engineering Letters vol 22 no 1 pp 1ndash7 2014

[21] A Wolf J B Swift and H L Swinney ldquoDetermining Lyapunovexponents froma time seriesrdquoPhysicaDNonlinear Phenomenavol 16 no 3 pp 285ndash317 1985

[22] M A Salam ldquoTransient stability analysis of a power systemwith one generator connected to an infinite bus systemrdquoInternational Journal of Sustainable Energy vol 33 no 2 pp251ndash260 2014

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

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International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom