Chaos, Solitons & Fractals 75 (2015) 50–61

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Chaos, Solitons & FractalsNonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier .com/locate /chaos

Modeling and stability analysis of a fractional-order Francishydro-turbine governing system

http://dx.doi.org/10.1016/j.chaos.2015.01.0250960-0779/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +86 181 6198 0277.E-mail address: diyichen@nwsuaf.edu.cn (D. Chen).

Beibei Xu, Diyi Chen ⇑, Hao Zhang, Feifei WangInstitute of Water Resources and Hydropower Research, Northwest A&F University, Yangling, Shaanxi 712100, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 October 2014Accepted 30 January 2015Available online 26 February 2015

In this paper, a fractional order mathematical model of a hydro-turbine governing systemis presented to analyze the dynamic stability of the hydro-turbine governing system in theprocess of operation. The fractional order hydro-turbine governing system is composed of ahydro-turbine and penstock system, a generator system and a hydraulic servo system. As apioneering work, we proposed a universal solution about the relationship of two para-meters in higher-degree equations according to the stability theorem of a fractional ordersystem. Based on the above theorem, we presented a variable law of stable regions of thefractional-order hydro-turbine governing system and analyzed the effect of various degreeof elastic water hammer on the stable regions of the parameters kd and kp with the increaseof fractional order a. The nonlinear dynamic behaviors of the system are also studied indetail. Finally, all of these results supply some basic theories for the running of ahydropower plant.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Hydropower, as a renewable, clean and cost-effectiveresource, is well developed in China. By the end of 2005,21 large hydropower plants with a gross installed capacityof 39.73 GW have been run in China. Moreover, there arealso 182 large and middle-scale hydropower plants witha gross installed capacity of 92.5 GW under construction[1]. Obviously, with the rapidly development of hydropow-er plants, hydropower plays an important role in maintain-ing the stability of electrical systems in China [1,2].However, because of lacking a comprehensive set of ser-vice and management system, how to maintain the sta-bility of a large hydroelectric station is a challengingprobelm [3–5]. For example, the great accident in theSayano-Shushenskaya hydroelectric power station in Rus-sia happened on 17 August 2008. Practically speaking,hydroelectric generator number 2 (HG2) suddenly

destroyed itself during operation and was thrown fromits position by water pressure [6]. As we all know, thehydro-turbine governing system is one of the most impor-tant parts of a hydroelectric station, and its running condi-tions directly affect the stability of hydroelectric stationsand electrical systems. Therefore, it is important and nec-essary to study the dynamics of a hydro-turbine governingsystem. For a long time, many scholars have establisheddifferent mathematical models of hydro-turbine governingsystems based on integer order calculus [7–13]. Forinstance, Liu and Liu [5] studied the stability of ahydropower plant with a linear turbine model in a steadystate. Avdyushenko et al. [14] made significant contribu-tions to the modeling of the hydro-turbine in the transientstate. Meanwhile, there are a lot of published papers aboutthe models of each individual part of the hydro-turbinegoverning system [15–19]. For example, an elastic modeland a nonelastic model based on first-order differentialequations for penstock systems have been studied in Refs.[7,14], respectively. However, owning to the complex ofnonlinear, time-variant and non-minimum phase of the

Nomenclature

y the incremental deviation of the guide vaneopening

GhðsÞ the transfer function of water hammerGtðsÞ the transfer function of the hydro-turbine and

penstock systeme the intermediate variableey the first-order partial derivative value of torque

with respect to wicket gateeqy the first-order partial derivative value of flow

rate with respect to turbine speedh the incremental deviation of the penstock sys-

temeh the first-order partial derivative value of the

torque with respect to water headeqh the first-order partial derivative value of flow

rate with respect to water headmt the deviation of the incremental torqueTr the length of the phase of the wave of water

hammerTw the inertia time constant of the penstock sys-

tem

u the control signald the rotor anglex the variation of the speed of the generatorx0 the rated angular speed of the generatorf 0 the rated frequency of the generatorD the damping factor of the generatorme the torque of the electrical loadPe the terminal active powerEq the transient internal voltage of the armatureVs the bus voltage at infinityx0dR the direct axis transient reactancexqR the quadrature axis reactancexT the short circuit reactance of the transformerxL the reactance of a electric transmission lineTy the major relay connecter response timer the reference inputki the integral gain of a PID controllerkd the differential gain of a PID controllerkp the proportional gain of a PID controller.

B. Xu et al. / Chaos, Solitons & Fractals 75 (2015) 50–61 51

governing system, integer order calculus is a typical algo-rithm with limited nature and not suitable to describe it.

Fractional order calculus has extended the conceptionof classical integer order calculus. In recent years, owningto the advantages of fractional order calculus in the strongdependence, memory and sufficient preponderance inmodeling viscoelastic substance [20–24], many research-ers have introduced it to model the mechanical behaviorsof polymers, gels foams and the rheology of soft matterand biological tissues [25–31]. In 2008, Machado et al.[32] stated that, while individual dynamics of each ele-ment has an integer-order nature, the global dynamicsreveal the existence of both integer and fractional dynam-ics. More importantly, Luo et al. [33] claimed that fraction-al order systems can simulate various real systems moreadequately than integer order ones and provide reliablemodeling tool in describing many real dynamical process-es. Therefore, it is necessary to try to build a fractional-order mathematical model to study the dynamic stabilityof a hydro-turbine governing system.

Motivated by the above discussions, a fractional ordermathematical model of a hydro-turbine governing systemis established, which is a bridge between an integer ordersystem and a fractional order system. Furthermore, weproposed a universal method to solve the stable region ofthe any two parameters of a fractional order system. Basedon the above method, a variable law of the stable regions ofthe system is presented with the increase of the fractionalorder a, and it extends the very narrow stable region ofinteger order to the stable region of fractional order. Final-ly, the dynamical behaviors of the above system includingfractional bifurcation diagrams, time waveforms, phaseorbits and power spectrums are studied in detail.

The organization of the rest paper is as follows. In Sec-tion 2, the definition of fractional order calculus and the

stability theorem of a fractional order system are present-ed. A novel mathematical model of a hydro-turbine gov-erning system is presented in Section 3. In Section 4, thedynamic behaviors of the fractional-order system are ana-lyzed in detail. Section 5 closes this paper.

2. Preliminaries

In this section, first, we give the definition of fractionalorder calculus. Second, a theorem about a fractional ordersystem is presented.

Definition [34]. Let f: [a, b]! R be a function, a be apositive real number, n be the integer satisfyingn� 1 6 a 6 n, and C be the Euler gamma function.Then

(1) The left and right Riemann-Liouville fractionalderivatives of order a of f ðxÞ are given as

aDax f ðxÞ ¼ 1

Cðn� aÞdn

dxn

Z x

aðx� tÞn�a�1f ðtÞdt

and

xDabf ðxÞ ¼ ð�1Þn

Cðn� aÞdn

dxn

Z b

xðt � xÞn�a�1f ðtÞdt;

respectively.

(2) The left and right Caputo fractional derivatives oforder a of f ðxÞ are given as

Z x

aCDa

x f ðxÞ ¼ 1Cðn� aÞ a

ðx� tÞn�a�1f ðnÞðtÞdt

52 B. Xu et al. / Chaos, Solitons & Fractals 75 (2015) 50–61

and

xCDa

bf ðxÞ ¼ 1Cðn� aÞ

Z b

xð�1Þnðt � xÞn�a�1f ðnÞðtÞdt;

respectively.

Theorem 1. [35] We consider the following fractionalorder system

DaX ¼ AX; ð1Þ

where A 2 Rn�n, x 2 Rn, xð0Þ ¼ x0; the n� 1 matrix adescribes the different fractional orders, a = ½a1;a2; � � � ;ai; � � � ;an�T : The above system is asymptotically stable ifand only if j argðkiÞj > qp=2 is satisfied for all eigenvalueski of the matrix A. Furthermore, this system is stable ifand only if j argðkiÞjP qp=2 is satisfied for all eigenvalueski of the matrix A and those critical eigenvalues that satisfythe condition j argðkiÞjP qp=2 have geometric multiplicityone.

3. Modeling of a hydro-turbine governing system

3.1. Fractional order mathematical modeling of the hydro-turbine and penstock system

From Ref. [36], the relationship between the deviationof the incremental torque and output power can beexpressed as

Pm ¼ mt þx; ð2Þ

where mt is the deviation of the incremental torque; x isthe variation of the speed of the generator.

A typical diagram of the hydro-turbine and penstocksystem is shown in Fig. 1 [36]. We suppose that thecross-sectional area of the penstock is constant. Then thetransfer function of the hydro-turbine and penstock sys-tem can be written as

GtðsÞ ¼ ey1þ eGhðsÞ

1� eqhGhðsÞ; ð3Þ

where eqh is the first-order partial derivative value of flowrate with respect to water head; e is the intermediate

Fig. 1. The linear model of the hydro

variable; ey is the first-order partial derivative value of tor-que with respect to wicket gate; GhðsÞ is the transfer func-tion of water hammer, and it can be described as

GhðsÞ ¼HAðsÞQ AðsÞ

¼ �2hwthð0:5TrsÞ; ð4Þ

where hw is the characteristic coefficient of the penstock;Tr is the length of the phase of the wave of water hammer.From Ref. [37], the transfer function of the penstock systemcan be rewritten as

GhðsÞ ¼ �2hw

148 T3

r s3 þ 12 Trs

18 T3

r s2 þ 1: ð5Þ

Substituting Eq. (5) into Eq. (3), the transfer functionbetween the incremental deviation of the guide vane open-ing y and the deviation of the incremental torque mt can berewritten as

GtðsÞ ¼ �ey

eqh

es3 � 3hwTr

s3 þ 24eT2

rs� 24

hwT3r

s3 þ 3eqhhwTr

s3 þ 24T2

rsþ 24

eqhhwT3r

: ð6Þ

From Eq. (6), the state space equations of the hydro-tur-bine and penstock system can be described as

_x1 ¼ x2

_x2 ¼ x3

_x3 ¼ �a0x1 � a1x2 � a2x3 þ y

8><>: ð7Þ

and

mt ¼ b3yþ ðb0 � a0b3Þx1 þ ðb1 � a1b3Þx2 þ ðb2

� a2b3Þx3; ð8Þ

where x1; x2 and x3 are state variables, a0 ¼ 24eqhhwT3

r;

a1 ¼ 24T2

r; ½a2 ¼ 3

eqhhwTr; b0 ¼ 24ey

eqhhwT3r; b1 ¼ � 24eey

eqhT2r; b2 ¼ 3ey

eqhhwTrand

b3 ¼ � eey

eqh.

Viscoelasticity exists in the water of the penstock in theprocess of operation, which has a great effect on the dynam-ic behaviors of the hydro-turbine. Considering the advan-tages of fractional order calculus in modeling viscoelasticsubstance, we try to introduce fractional order calculus tothe mathematical modeling of the hydro-turbine and pen-stock system. According to the definition of fractional ordercalculus, Eq. (7) can be rewritten as

-turbine and penstock system.

Fig. 2. The stable regions of kd and kp with the decrease of fractional order a.

B. Xu et al. / Chaos, Solitons & Fractals 75 (2015) 50–61 53

Dax1 ¼ x2

Dax2 ¼ x3

Dax3 ¼ �a0x1 � a1x2 � a2x3 þ y

8><>: ; ð9Þ

where a is the value of the fractional order; Da refers toCaputo fractional derivative operator with 0 < a < 2 .

3.2. Fractional order mathematical model of the generatorsystem

A second-order mathematical model of the generator isused to study the dynamic behaviors of the generator inthe operation of the hydropower plant in detail. The math-ematical equations are

_d ¼ x0x_x ¼ 1

Tab½mt �me � Dx�

(: ð10Þ

For the generator system, the spinning generator hasgreat inertia. Thus, its dynamic behaviors are dependenton the history. Owning to the advantage of fractional ordercalculus on the dependence of history, we also introducefractional order calculus to the mathematical modeling ofthe generator system. According to the definition of frac-tional order calculus, the fractional order mathematicalmodel of the generator can be described as

Dad ¼ x0xDax ¼ 1

Tab½mt �me � Dx�

(; ð11Þ

54 B. Xu et al. / Chaos, Solitons & Fractals 75 (2015) 50–61

where a is the value of the fractional order; Da refers toCaputo fractional derivative operator with 0 < a < 2; d isthe rotor angle; x is the variation of the speed of the gen-erator, x0 is the rated angular speed of the generator,x0 ¼ 2pf 0; D is the damping factor of the generator, andit is generally regarded as a constant. If the influence ofthe rotor speed on the torque is added to the damping fac-tor, the torque of the electrical load and the terminal activepower are equal to each other, i.e.

me ¼ Pe: ð12Þ

For the generator, the terminal active power can bedescribed as

Pe ¼E0qVs

x0dR

sin dþ V2s

2x0dR � xqR

x0dRxqRsin 2d ð13Þ

and

x0dP ¼ x0d þ xT þ 1

2 xL

xqP ¼ xq þ xT þ 1

2 xL

8><>: ; ð14Þ

where E0q is the transient internal voltage of the armature;Vs is the bus voltage at infinity; x0d is the direct axis tran-sient reactance; xq is the quartered axis reactance; xT isthe short circuit reactance of the transformer; xL is thereactance of the electric transmission line.

3.3. Fractional order mathematical model of the hydraulicservo system

The dynamic characteristics of a hydraulic servo system[37] can be got as

Tydydtþ y ¼ u; ð15Þ

where y is the incremental deviation of the guide vaneopening.

If the PID controller is active in the governing system,the output signal can be written as

u ¼ kpðr �xÞ þ ki

Z t

0ðr �xÞdt þ kd

ddtðr �xÞ; ð16Þ

where r is the reference input; ki is the integral gain of aPID controller; kd is the differential gain of a PID con-troller; kp is the proportional gain of a PID controller.For a steady state, we set the reference input of thespeed of the generator r as zero. Then the Eq. (15) canbe rewritten as

u ¼ kpðr �xÞ þ ki

Z t

0ðr �xÞdt þ kd

ddtðr �xÞ

¼ �kpx�ki

x0d� kd _x: ð17Þ

From Eq. (2) to Eq. (17), combining every parts of thegoverning system into an organic whole, the fractionalorder mathematical model of the hydro-turbine governingsystem can be described as

Dqx1 ¼ x2

Dqx2 ¼ x3

Dqx3 ¼ �a0x1 � a1x2 � a2x3 þ y

Dqd ¼ x0x

Dqx ¼ 1Tab

mt �E0qVs

x0dR

sin d� V2s

2x0

dR�xqR

x0dR

xqRsin 2d� Dx

h i

Dqy ¼ 1Ty�kpðr �xÞ � ki

x0d� kdDqx� y

� �

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

;

ð18Þ

where a is the value of the fractional order; Da refers toCaputo fractional derivative operator with 0 < a < 2 .

4. Nonlinear dynamic analyses

4.1. Stability of a fractional order system

According to theorem 1, the key to judging the stabilityof the Eq. (18) is calculating the eigenvalues of the coeffi-cient matrix of Eq. (18). However, from the group theoremof Galois, for an high dimensional equation

f ðxÞ ¼ bnxn þ bn�1xn�1 þ � � � þ bsxs � � � þ btxt þ � � �þ b1xþ b0; ð19Þ

there is no analytic solution.Here, we assume that the characteristic equation of the

Eq. (1) is

f ðkÞ ¼ bnkn þ bn�1k

n�1 þ � � � þ bsks � � � þ btk

t þ � � � þ b1k

þ b0;

If the fractional orders of the system satisfya1 ¼ a2 ¼ � � � ¼ an, we set k ¼ r cosðap2 Þ þ i sin ap

2

� �� �. The

any two parameters of the fractional order system can bewritten as

f ðr;aÞ ¼ bn r cosap2

� �þ i sin

ap2

� �� �� �n

þ bn�1 r cosðap2Þ þ i sin

ap2

� �� �� �n�1

þ � � � þ bs r cosap2

� �þ i sin

ap2

� �� �� �s

þ � � � þ btðr cosðap2Þ þ i sin

ap2

� �� �Þ

t

þ � � � þ b1r cosap2

� �þ i sin

ap2

� �� �þ b0 ¼ 0

ð20Þ

Then

realðf ðr;aÞÞ ¼ 0

imagðf ðr;aÞÞ ¼ 0

(: ð21Þ

From Eq. (20), one obtains

bs ¼ f ðb0; b1; � � � ; bs�1; bsþ1; � � � ; bt�1; btþ1; � � � ; bn; rÞ

bt ¼ gðb0; b1; � � � ; bs�1; bsþ1; � � � ; bt�1; btþ1; � � � ; bn; rÞ

(:

ð22Þ

Fig. 3. The motion law of bifurcation point with the increase of a.

Table 1The stable ranges of kd with the decrease of a.

Fractional order Stable range Fractional order Stable range Fractional order Stable range

a = 1.1 (0,0.7033) a = 1.0 (0.2378,3.731) a = 0.95 (0,5.266)a = 0.9 (0,6.63) a = 0.85 (0,7.68) a = 0.8 (0,9.352)a = 0.7 (0,11.64) a = 0.6 (0,14.57) a = 0.5 (0,17.00)a = 0.4 (0,19.15) a = 0.3 (0,21.05) a = 0.2 (0,22.81)

Fig. 4. The stable regions of kd � kp of different Tr for the system with fractional order a = 1.0, 0.95, 0.9, 0.85 and 0.8, respectively.

B. Xu et al. / Chaos, Solitons & Fractals 75 (2015) 50–61 55

where the values of s and t range from 1 to n-1,respectively. bs and bt are general variables. The stableregion of the any two parameters of the fractional ordersystem can be got by controlling the range of theparameter r.

The parameters in this paper are w0 = 314, Tab = 8.0,D = 0.5, E0q = 1.35, x0

dP = 1.15, x0

qP = 1.474, Ty = 0.1, Vs =

1.0, eqh = 0.5, ey = 1.0, e = 0.7, Tr = 1.0, hw = 2.0, r = 0. Initial

values are ½x1; x2; x3; d;w; y�T ¼ ð0;0;0;0;0;0ÞT . The Jacobianmatrix of Eq. (17) is

56 B. Xu et al. / Chaos, Solitons & Fractals 75 (2015) 50–61

0 1 0 0 0 0

0 0 1 0 0 0

�24 �24 �3 0 0 1

0 0 0 0 314 0

365 0 9

10405cosð2dÞ

16951 �27cosðdÞ

184�116

�740

�72kd 0 �9kd�10kdð405cosð2dÞÞ

16951 � 27cosðdÞ184 � 5

157ki5kd

8 �10kp7kd

4 �10

2666666666666664

3777777777777775

:

ð23Þ

From Eq. (22), we can get the characteristic equation as

9288þ 10166kþ 30kpkþ 30kdk2 � 42kpk

2 þ 2343k2

� 42kdk3 þ 767:1k3 þ 3:75kdk

4 � 1:75kpk4 þ 93:39k4

� 1:75kdk5 þ 13:06k5 þ k6 ¼ 0 ð24Þ

From theorem 1, the hydro-turbine governing systemremains stability if j argðkÞjP ap

2 . When ki = 1, we can getthe stable regions of kd and kp with the change of fractionalorder a. Fig. 2 shows the stable regions of kd and kp withthe decrease of fractional order a .

From Fig. 2, the fractional order a has a great effect onthe stable regions of kd and kp. With the decrease of a,bifurcation points shifted to the right. The curve, which is

Fig. 5. Bifurcation diagrams in (kd ; x) plane for the Francis hydro-turbine govea = 0.9; (d) a = 0.85; (e) a = 0.8.

composed of bifurcation points, tends to be a straight line.In other words, the stable regions of kd and kp increasegradually, and these results provide the higher probabil-ities of the stable ranges to satisfy the data of practicalengineering. From the point of view of engineering, thevalues of kd and kp are usually more than zero. Therefore,the stable regions of kd and kp disappear when a > 1.1. Inother words, the system is in an unstable state, which isharmful to the stability of the whole system.

To illustrate the effect of a on the bifurcation pointclearly, the motion law of bifurcation points with theincrease of fractional order a is presented in Fig. 3. FromFig. 3, the value of the bifurcation point decreases linearlywith the increase of a. The functional relation between kd

and a can be expressed as kd ¼ �24:77aþ 28:763. Mean-while, the stable region of kd is also presented in Table 1.

4.2. The influence of elastic water hammer on the stability ofthe fractional order system

The elastic water hammer, in a manner, affects thedynamic stability of the hydro-turbine governing systemwhen the length of a penstock is longer than 800 m.Because Tr reflects the intensity of elastic water hammer,

rning system with different fractional orders. (a) a = 1.0; (b) a = 0.95; (c)

Fig. 6. Time waveforms, phase orbits, and power spectrums of the hydro-turbine governing system with different kd when a = 1. (a1) Time waveform withkd = 3. (a2) Phase orbit with kd = 3. (a3) Power spectrum with kd = 3. (b1) Time waveform with kd = 3.8. (b2) Phase orbit with kd = 3.8. (b3) Power spectrumwith kd = 3.8. (c1) Time waveform with kd = 4.3. (c2) Phase orbit with kd = 4.3. (c3) Power spectrum with kd = 4.3. (d1) Time waveform with kd = 4.7. (d2)Phase orbit with kd = 4.7. (d3) Power spectrum with kd = 4.7.

B. Xu et al. / Chaos, Solitons & Fractals 75 (2015) 50–61 57

we chose it as the independent variable to study the sta-bility of the fractional order system with the decrease ofa. Similarly, the stable regions of kd and kp with differenta are presented in Fig. 4 when Tr = 1.0, 1.3, 1.5 and 2.0,respectively.

From Fig. 4, with the increase of Tr or a, the stableregions of kd and kp decrease gradually. More specifically,the enhancement to the power of the elastic water ham-mer and the increase of fractional order a both reducethe regions of stable operation.

4.3. Chaos and bifurcation

In this section, we will study the nonlinear dynamics ofthe fractional order hydro-turbine governing system. Here,

the time step is 0.02; initial values are ½x1; x2; x3; d;w; y�T ¼

ð0;0;0;0;0; 0ÞT ; kp = 2, ki = 1; the values of other para-meters are the same with those in Section 4.1. Therefore,we can get the bifurcation diagrams of the fractional orderhydro-turbine governing system (as shown in Fig. 5) witha = 1.0, a = 0.95, a = 0.9, a = 0.85 and a = 0.8, respectively.In addition, the parameter kd is taken as an independentvariable.

For a = 1.0, the bifurcation diagram of the fractionalorder system is shown in Fig. 5(a). From Fig. 5(a), when0.2378 < kd < 3.731, the output of x is zero, which meansthe system is in a stable state. When kd goes across3.731, the system loses its stability. Meanwhile, x showsa limit cycle, which is agree with the analysis of Section 4.1.With the increase of kd, the system enters into chaosthrough a kind of behaviors which is called multiple perioddoubling bifurcation. Note that the bifurcation point

Fig. 7. Time waveforms, phase orbits, and power spectrums of the hydro-turbine governing system with different kd when a = 0.9. (a1) Time waveform withkd = 6. (a2) Phase orbit with kd = 6. (a3) Power spectrum with kd = 6. (b1) Time waveform with kd = 7. (b2) Phase orbit with kd = 7. (b3) Power spectrum withkd = 7. (c1) Time waveform with kd = 7.4. (c2) Phase orbit with kd = 7.4. (c3) Power spectrum with kd = 7.4. (d1) Time waveform with kd = 7.6. (d2) Phase orbitwith kd = 7.6. (d3) Power spectrum with kd = 7.6.

58 B. Xu et al. / Chaos, Solitons & Fractals 75 (2015) 50–61

between multiple period doubling motion and chaoticmotion is kd = 4.544. In order to further analyze the charac-teristics of the system, typical characteristics at kd = 3,kd = 3.8, kd = 4.3 and kd = 4.7 are shown in Fig. 6 by usingphase orbits, time waveforms and power spectrums,respectively.

For kd = 3, from Fig. 6(a), we can learn that the deviationof the rotor speed x changes to zero after a short transi-tion. The phase orbit shows that the deviation of the rotorspeed x and the deviation of the rotor angle d both con-verge to a stable point. Moreover, the frequency of theincremental deviation of the guide vane opening y is about4.98HZ, and its peak value is about 0.0035. These resultsindicate that the hydro-turbine governing system is in astable state.

The responses of the hydro-turbine governing systemwith kd = 3.8 are shown in Fig. 6(b). From Fig. 6(b), thedeviation of the rotor speed is in period-1. A limit cycleexists in the phase orbit. Moreover, the frequency of theincremental deviation of the guide vane opening changesto 5.371HZ, and its peak value increases to 346.1.

The time waveform, the phase orbit and the powerspectrum of the hydro-turbine governing system withkd = 4.3 are shown in Fig. 6(c), respectively. From the timewaveform, we learn that the motion of the deviation of therotor speed x is periodical. A similar strange attractorexists in the phase orbit. All these results illustrate thatthe vibration of the system intensifies gradually.

The dynamic behaviors of the system at kd = 4.7 areshown in Fig. 6(d). From Fig. 6(d), the deviation of the rotor

Fig. 8. Time waveforms, phase orbits, and power spectrums of the hydro-turbine governing system with different kd when a = 0.8. (a1) Time waveform withkd = 8.6. (a2) Phase orbit with kd = 8.6. (a3) Power spectrum with kd = 8.6. (b1) Time waveform with kd = 9.4. (b2) Phase orbit with kd = 9.4. (b3) Powerspectrum with kd = 9.4. (c1) Time waveform with kd = 9.8. (c2) Phase orbit with kd = 9.8. (c3) Power spectrum with kd = 9.8. (d1) Time waveform withkd = 10. (d2) Phase orbit with kd = 10. (d3) Power spectrum with kd = 10.

B. Xu et al. / Chaos, Solitons & Fractals 75 (2015) 50–61 59

speed x is ceaseless change. A number of strange attrac-tors appear in the phase orbit. Now, the system is in achaotic vibration.

Through the analysis of the hydro-turbine governingsystem when a = 1.0, we find two special bifurcationpoints, which are 3.731 and 4.544, respectively. Moreover,when kd goes across a bifurcation point, all of theparameters of the system change a lot, and these changesare harmful to the dynamic stability of the hydro-turbinegoverning system.

For a = 0.9, the bifurcation diagram of the hydro-tur-bine governing system is illustrated in Fig. 5(c). FromFig. 5(c), the values of the two bifurcation points changeto 6.63 and 7.596, respectively. Moreover, it is agree with

the analysis in Section 4.1. Interestingly, with thedecrease of fractional order a, the range of multipleperiodic vibration decreases gradually. Typical character-istics at kd = 6, kd = 7, kd = 7.4 and kd = 7.6 are also shownin Fig. 7 by using phase orbits, time waveforms and pow-er spectrums, respectively.

As previously mentioned, for a = 0.8, the values of thebifurcation points change to 9.352 and 9.814, respectively.Moreover, it is agree with the analysis in Section 4.1. Fromthe point of view of engineering, with the decrease of frac-tional order a, the stable range of parameter kd increasesgradually. Typical characteristics at kd = 8.6, kd = 9.4,kd = 9.8 and kd = 10 are also shown in Fig. 8 by using phaseorbits, time waveforms and power spectrums, respectively.

60 B. Xu et al. / Chaos, Solitons & Fractals 75 (2015) 50–61

5. Conclusions and discussion

This paper pays attention to a fractional order mathe-matical modeling of a hydro-turbine governing system.First, we present a variable law of the stable regions ofthe system with the increase of fractional order a by usingthe stability theorem of fractional-order system. It is abridge between the stable region of an integer order sys-tem and a fractional order system. Second, the enhance-ment to the power of the elastic water hammer and theincrease of fractional order a both reduce the stableregions of the parameters kd and kp. Third, the dynamiccharacteristics of the hydro-turbine governing system fordifferent fractional order a are studied including fraction-al-order bifurcation diagrams, fractional-order time wave-forms, fractional-order phase orbits and fractional-orderpower spectrums.

Fractional order calculus is first introduced to the mod-eling of a hydro-turbine governing system, although thispaper studied only by using numerical simulations. In thefuture, the dynamical behaviors of the real system of ahydropower plan will be studied with nonlinear dynamicaltheory and fractional order calculus method. As we allknow, traditional linear mathematical models cannotdescribe the stability of complex systems very well. There-fore, we will attempt to establish novel nonlinear mathe-matical models to describe the dynamical characteristicsof the hydro-turbine governing system. Finally, consider-ing the problems of stability of the system in the transientstates, we will focus on novel mathematical models includ-ing fractional order calculus, which are more suitable tostudy the transient states.

Acknowledgements

This work was supported by the scientific researchfoundation of National Natural Science Foundation(51479173, 51109180), the National Science & TechnologySupporting Plan from the Ministry of Science & Technologyof P. R. of China (2011BAD29B08), the FundamentalResearch Funds for the Central Universities of Ministry ofEducation of China (201304030577), Northwest A&FUniversity Foundation, China (2013BSJJ095) and the scien-tific research foundation on water engineering of ShaanxiProvince (2013slkj-12) .

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