research article nonlinear dynamical analysis of hydraulic ... · model, the existence and...

Research ArticleNonlinear Dynamical Analysis of Hydraulic Turbine GoverningSystems with Nonelastic Water Hammer Effect

Junyi Li and Qijuan Chen

Power and Mechanical Engineering, Wuhan University, Wuhan, Hubei 430072, China

Correspondence should be addressed to Junyi Li; [email protected]

Received 27 March 2014; Revised 5 June 2014; Accepted 5 June 2014; Published 25 June 2014

Academic Editor: Hamid Reza Karimi

Copyright © 2014 J. Li and Q. Chen. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A nonlinear mathematical model for hydroturbine governing system (HTGS) has been proposed. All essential components ofHTGS, that is, conduit system, turbine, generator, and hydraulic servo system, are considered in the model. Using the proposedmodel, the existence and stability of Hopf bifurcation of an example HTGS are investigated. In addition, chaotic characteristicsof the system with different system parameters are studied extensively and presented in the form of bifurcation diagrams, timewaveforms, phase space trajectories, Lyapunov exponent, chaotic attractors, and Poincare maps. Good correlation can be foundbetween the model predictions and theoretical analysis. The simulation results provide a reasonable explanation for the sustainedoscillation phenomenon commonly seen in operation of hydroelectric generating set.

1. Introduction

Many hydropower plants have been built [1–3] worldwideto harness the energy of falling or running water for elec-tricity purpose. An important part of hydropower plantis the hydraulic turbine governing system (HTGS), whichserves to maintain safe, stable, and economical operationof hydropower generating unit [4]. The HTGS is in naturea complex nonlinear, multivariable, time-varying, and non-minimum phase system, which involves the interactionsbetween hydraulic system, mechanical system, and electricalsystem [5].The complex dynamic behaviors of the HTGS sig-nificantly influence the operation conditions of hydroelectricgenerating set. For instance, oscillatory problems in hydro-electric generating units were reported to be closely relatedto the possible Hopf bifurcation and chaotic oscillatorybehaviors in HTGS [6–9]. In the absence of a model to con-veniently predict the dynamic behaviors of the system,meansto address the practical operational issues of hydroelectricgenerating units, that is, sustained oscillation phenomenon[6, 10], have to be limited.

The literature review reveals that considerable researchefforts have been devoted to the modeling of each indi-vidual part of the HTGS [11]. For instance, elastic model

and nonelastic model [12, 13] for the conduit system havebeen adopted in long pipeline and short pipeline systems,respectively. In addition, various hydroturbine models werealso available in [10, 14–26]. While Sanathanan [14] claimedthat the output of hydroturbine was proportional to hydraulichead and volume flow, Hannett et al. [15] approximatedthis term by its first-order Taylor formula and Kishor et al.[16, 17] proposed six detailed expressions about these sixtransfer coefficients with turbine speed and head, whichwere shown to provide reasonably accurate predictions forHTGS with small disturbance. X. Liu and C. Liu [10] studiedsmall disturbance stability of hydropower plant with complexconduit with a linear turbine model, but they fell short ofincluding elastic water-hammer effects. Bakka et al. madesignificant contributions to modeling, simulation [18–22],and control [23–26] of wind turbine system, which stimulatesmodeling of hydroturbine in this paper.

In regards to the power generator, models with differentorders could also be found in [27–29]. While higher orderpower generator models typically represent better accuracy,the overall computational efficiency would be greatly sacri-ficed especially in case of the modeling of a fully coupledgoverning system. To compromise with the computationalefficiency, the power generator model has to be selected

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014, Article ID 412578, 11 pageshttp://dx.doi.org/10.1155/2014/412578

2 Journal of Applied Mathematics

carefully such that it provides accurate approximation withacceptable computational effort.

Aside from the abovementioned models for individualpart of the HTGS, investigations on bifurcation and chaos fortheHTGS at the system level were reported only in a few stud-ies [6, 30, 31]. For instance, Konidaris and Tegopoulos [31]investigated the oscillatory problems in hydraulic generatingunits; Mansoor et al. [6] successfully reproduced an oscilla-tory phenomenonwhose causes were difficult to be identifiedwith limited recorded data. He also proposed a methodologyto improve the stability of the control system. While thesestudies were instrumental in identifying and documentingthe oscillations in hydraulic generating units, neither of themnoticed the effects of Hopf bifurcation on the oscillatorybehaviors. Ling and Tao [30] first reported the influence ofbifurcation phenomenon on the sustained oscillations in hisstudy of Hopf bifurcation behavior of HTGS with saturation.However, his model was oversimplified by employing first-order model for the power generator and the PI governingsystem. Chen et al. [32] developed a novel nonlinear dy-namical model for hydroturbine governing system with asurge tank and studied exhaustively the influences of differentparameters for the first time. However, they failed to includeadequate theoretical analysis and description on bifurcation.Determination of the existence and detailed calculation ofHopf bifurcation were also not considered in their work.

The above literature review shows that numerous well-developed models for each individual part of HTGS areavailable, but investigations of bifurcation and chaotic oscil-lations in HTGS with a fully coupled model of the systemare rarely seen. As such, this paper aims to develop a fullycoupled nonlinear dynamical model for HTGS and investi-gate the bifurcation and chaotic oscillatory behaviors of theHTGS. In addition, a theorem for existence determination ofHopf bifurcation for four-dimensional nonlinear system hasbeen proposed for convenient prediction of the bifurcationof critical points, which would otherwise be impracticalespecially for high-dimensional systems due to tremendouscomputational demand by conventional analysis methods,that is, Lyapunov-Schmidt (L-S) method, center manifoldmethod [33, 34], or normal form theory [35].

There are three main contributions of this paper com-pared with prior works. First, a new four-dimensional fullycoupled nonlinear mathematical model of HTGS was pre-sented and the parameters were from a practical power sta-tion, which made the work more consistent with actualproject compared with [10, 30] work. Second, the theoremfor stability, Hopf bifurcation, and dynamic quality analysis offour-dimensional system that can avoid excessive and tediouscalculations was firstly introduced in the paper, providing anew approach for HTGS analysis and computation. Third,nonlinear dynamical behaviors of the above system withdifferent parameters were studied in detail and necessarynumerical simulation results were presented.

The paper is outlined as follows. First, the formulations ofa fully coupled nonlinear dynamical model will be presented.Next, the investigations of the Hopf bifurcation and chaoticbehaviors of the system will be described. Finally, a briefconclusion will be given.

2. Nonlinear Mathematical Model of HTGS

HTGS consists of five parts, that is, conduit system, hydro-turbine, governor, electrohydraulic servo system, and powergenerator. Model for each individual part has been welldeveloped. Water from reservoir enters tunnel first and thenflows through penstock before reaching turbine gate. Next,it flows into scroll casing to promote the hydroturbine torotate. The power generator and hydroturbine are connectedby a shaft coupling. Water that flows into the hydroturbinecan be regulated by wicket gates, which are controlled by thegovernor system. The governor system operates accordinglygiven the deviation between electric demand and developedtorque [35].

2.1. Conduit System Model. A no-elastic model [28] is em-ployed for the conduit system in this study. The unsteadyflow partial differential equations in pressure pipes can bedescribed as

Momentum equation: 𝜕𝐻𝜕𝑥

+1

𝑔𝐴

𝜕𝑄

𝜕𝑡+

𝑓𝑄2

2𝑔𝐷𝐴2= 0,

Continuity equation: 𝜕𝑄𝜕𝑥

+𝑔𝐴

𝑎2

𝜕𝐻

𝜕𝑡= 0,

(1)

where 𝐷, 𝑓, 𝐴 are parameters of the penstock. They denotethe diameter, head loss, and area of the pipeline, respectively.𝐻, 𝑄 are hydraulic head and turbine flow in penstock inoperating condition, 𝑎 is pressure wave velocity, and 𝑥 is thelength from upstream.

The head and flow equation between two sections ofpenstock can be deduced from (1). It can be described as [28]

[𝐻𝐴(𝑠)

𝑄𝐴(𝑠)

] = [

[

𝑐ℎ (𝑟Δ𝑥) − 𝑍𝑐𝑠ℎ (𝑟Δ𝑥)

−𝑠ℎ (𝑟Δ𝑥)

𝑍𝑐

𝑐ℎ (𝑟Δ𝑥)]

]

[𝐻𝐵(𝑠)

𝑄𝐵(𝑠)

] .

(2)

Subscripts 𝐵,𝐴 are symbols of upstream and downstreamsection of pipeline, respectively. 𝑟 and 𝑍

𝑐are the composite

equations of parameters of the penstock, 𝑟 = √𝐿𝐶𝑠2 + 𝑅𝐶𝑠,𝑍𝑐= 𝑟/𝐶𝑠.𝐿, 𝑅, 𝐶 can be written as

𝐿 =𝑄0

𝑔𝐴𝐻0

, 𝐶 =𝑔𝐴𝐻0

(𝑎2𝑄0), 𝑅 =

(𝑓𝑄2

0)

(𝑔𝐷𝐴2𝐻0). (3)

Providing that the head loss is negligible, 𝑟 and 𝑍𝑐can be

rewritten as follows:

𝑟 =1

𝑎𝑠, 𝑍

𝑐= 2ℎ𝑤, ℎ

𝑤=

𝑎𝑄0

(2𝑔𝐴𝐻0). (4)

With the hydraulic friction losses being trivial and𝐻𝐵(𝑠) = 0 (tunnel connects with reservoir directly), the head

and flow function is simplified as follows:

𝐻𝐴(𝑠)

𝑄𝐴(𝑠)

= −𝑍𝑐

𝑠ℎ (𝑟Δ𝑥)

𝑐ℎ (𝑟Δ𝑥)= −𝑍𝑐𝑡ℎ (𝑟Δ𝑥) . (5)

Journal of Applied Mathematics 3

It can be seen from (5) that the water hammer transferfunction is a nonlinear hyperbolic tangent function, whichwas inconvenient to use and should be expanded by series.Substituting (4) into (5), the equation for the water striketransfer function 𝐺

ℎ(𝑠) at point 𝐴 is obtained as

𝐺ℎ(𝑠) = −2ℎ

𝑤𝑡ℎ (0.5𝑇

𝑟𝑠)

= −2ℎ𝑤

∑𝑛

𝑖=0((0.5𝑇

𝑟𝑠)2𝑖+1

/ (2𝑖 + 1)!)

∑𝑛

𝑖=0((0.5𝑇

𝑟𝑠)2𝑖

/ (2𝑖)!)

.

(6)

Typically, there are the two models, 𝑖 = 0, 1, as follows:

𝐺ℎ(𝑠) = −2ℎ


𝑟𝑠) = −2ℎ

𝑤

(1/48) 𝑇3

𝑟𝑠3+ (1/2) 𝑇

𝑟𝑠

(1/8) 𝑇2𝑟𝑠2 + 1

,

(7)

𝐺ℎ(𝑠) = −2ℎ


𝑟𝑠) = −𝑇

𝑤𝑠, (8)

where 𝑇𝑟= 2Δ𝑥/𝑎, 𝑇

𝑤= 𝐿𝑄

0/𝐴𝑔𝐻

0. Equations (6) to (8)

represent three kinds of water hammer models: the first twomodels are called elastic water hammer model and the lastone is rigid water hammer model that is employed in thepaper. In the above equations, the pipeline is assumed to haveconstant cross-sectional area over the full length, which isimpossible in actual engineering. Relevant parameters in thefield are obtained by the following formulas:

𝐿 =

𝑛

∑

𝑖=1

𝐿𝑖, 𝐴 =

∑𝑛

𝑖=1𝐿𝑖

∑𝑛

𝑖=1(𝐿𝑖/𝐴𝑖), 𝑇

𝑟=

𝑛

∑

𝑖=1

2𝐿𝑖

𝑎𝑖

,

𝑇𝑤=

∑𝑛

𝑖=1(𝐿𝑖/𝐴𝑖) 𝑄0

𝑔𝐻0

(9)

where 𝐿𝑖, 𝐴𝑖, 𝑎𝑖denote the length, cross-sectional area, and

the velocity of pipeline 𝑖, respectively, and there are 𝑛 pipes intotal.

2.2. TurbineModel of RigidWater Hammer. For small pertur-bation around the rated operating point, the equation of theturbine can be represented as below:

𝑚𝑡= 𝑒𝑥𝑥 + 𝑒𝑦𝑦 + 𝑒ℎℎ,

𝑞 = 𝑒𝑞𝑥𝑥 + 𝑒𝑞𝑦𝑦 + 𝑒𝑞ℎℎ.

(10)

The six constants of hydroturbine 𝑒𝑥, 𝑒𝑦, 𝑒ℎ, 𝑒𝑞𝑥, 𝑒𝑞𝑦, 𝑒𝑞ℎ

are the partial derivatives of the torque and flow with respectto turbine speed, guide vane, and head, respectively. Theseconstants may vary as the operating point changes.

In the dynamic models of the turbine and conduit system[12, 14] where the relative deviation is used to represent thestate variables, the relationship between turbine torque andits output power is

𝑃𝑚

= 𝑚𝑡+ Δ𝜔. (11)

u y mtGt(s)1

1 + Tys

Servomotor Turbine and conduit system

Figure 1: Dynamic model of turbine and conduit system.

As the unit speed changes little, the speed deviationΔ𝜔 =0, leading to 𝑃

𝑚= 𝑚𝑡. Then the transfer function of the

turbine and conduit system is

𝐺𝑡(𝑠) = 𝑒

𝑦

1 + 𝑒𝐺ℎ(𝑠)

1 − 𝑒𝑞ℎ𝐺ℎ(𝑠)

, (12)

where 𝐺ℎ(𝑠) is transfer function of conduit system defined in

(6).In case of rigid water hammer, the coefficient 𝑛 in (6) is

0 and the dynamic equation for the turbine output torque, asshown in Figure 1, becomes

�̇�𝑡=

1

𝑒𝑞ℎ𝑇𝑤

[−𝑚𝑡+ 𝑒𝑦𝑦 −

𝑒𝑒𝑦𝑇𝑤

𝑇𝑦

(𝑢 − 𝑦)] . (13)

2.3. Generator Model. A synchronous generator [36], whichconnected to an infinite bus through a transmission line, isconsidered as the target system. The second-order nonlineardynamical model, after making standard considerations, canbe written as

̇𝛿 = 𝜔0𝜔,

�̇� =1

𝑇𝑎𝑏

(𝑚𝑡− 𝑚𝑒− 𝐾𝜔) ,

(14)

where 𝛿, 𝜔, 𝐾, and 𝑇𝑎𝑏denote the rotor angle, relative speed

deviation, damping coefficient, andmechanical starting time,respectively. The electromagnetic torque of the generator 𝑚

𝑒

is equal to its electromagnetic power 𝑃𝑒:

𝑚𝑒= 𝑃𝑒. (15)

The electromagnetic power can be calculated with thefollowing formula:

𝑃𝑒=

𝐸

𝑞𝑉𝑠

𝑥𝑑𝑥∑

sin 𝛿 +𝑉2

𝑠

2

𝑥

𝑑𝑥∑− 𝑥𝑞𝑥∑

𝑥𝑑𝑥∑

𝑥𝑞𝑥∑

sin 2𝛿, (16)

where the effects of speed deviations, damping coefficient,and torque variations are all included in the analysis of thegenerator dynamic characteristics

𝑥

𝑑∑= 𝑥

𝑑+ 𝑥𝑇+

1

2𝑥𝐿,

𝑥𝑞∑

= 𝑥𝑞+ 𝑥𝑇+

1

2𝑥𝐿.

(17)

Equations (14) to (17) are the simplified second-ordernonlinear generator model based on the turbine model of


rigid water hammer, which has been widely applied in non-linear controller design and stability analysis of HTGS. It isoften used for the stability characteristics and dynamic qual-ity analysis of HTGS from the perspective of power system.Higher order nonlinear generator model could be employedaccording to research needs.

2.4. Hydraulic Servo System Model. The servomotor, whichacts as the actuator, is used to amplify the control signals andprovide power to operate the guide vane. Its transfer functioncan be written as

𝐺𝑠(𝑠) =

1

1 + 𝑇𝑦𝑠, (18)

where 𝑇𝑦is the engager relay time constant.

2.5. GovernorModel. At present, a parallel PID controller [18,20, 22–26] is widely used in hydraulic turbine governors [27,30] in filed. Its transfer function is given as

𝐺2(𝑠) = (𝑘

𝑝+

𝑘𝑖

𝑠+ 𝑘𝑑𝑠) . (19)

Substituting (19) into (18), the results can be written instate-space form. At last, the mixed function can be obtainedas follows:

𝑑𝑦

𝑑𝑡=

1

𝑇𝑦

(−𝑘𝑝𝜔 − 𝑘𝑖∫Δ𝜔 − 𝑘

𝑑�̇� − 𝑦) . (20)

Based on the discussions above, the differential equationsthat coupled each individual part of the turbine nonlinearcontrol system can be written as

̇𝛿 = 𝜔0𝜔,

�̇� =1

𝑇𝑎𝑏

(𝑚𝑡− 𝐷𝜔 −

𝐸

𝑞𝑉𝑠

𝑥𝑑𝑥∑

sin 𝛿

−𝑉2

𝑠

2

𝑥


𝑥𝑑𝑥∑

𝑥𝑞𝑥∑

sin 2𝛿) ,

�̇�𝑡=

1

𝑒𝑞ℎ𝑇𝑤

(− 𝑚𝑡+ 𝑒𝑦𝑦

−𝑒𝑒𝑦𝑇𝑤

𝑇𝑦

(−𝑘𝑝𝜔 −

𝑘𝑖

𝜔0

𝛿 − 𝑘𝑑�̇� − 𝑦)) ,

̇𝑦 =1

𝑇𝑦

(−𝑘𝑝𝜔 −

𝑘𝑖

𝜔0

𝛿 − 𝑘𝑑�̇� − 𝑦) .

(21)

Equation (21) is the four-dimensional water-electrome-chanical coupled model of HTGS that integrates the turbinemodel of rigid water hammer, the water pipes linear model,and the nonlinear dynamic generator model. Compared tothe linearmodel, it could reflect the complex nonlinear natureproblem within the system much better. Equation (21) could

be applied to analyze and simulate the dynamic characteris-tics of the HTGS.

At the equilibrium point (0, 0, 0, 0), the following condi-tion has to be satisfied:

𝜔 = 0,

𝑚𝑡= 𝑒𝑦𝑦,

𝑦 = −𝑘𝑖

𝛿

𝜔0

,

𝑒𝑦𝑘𝑖

𝛿

𝜔0

+𝐸

𝑞𝑉𝑠

𝑥𝑑𝑥∑

sin 𝛿 +𝑉2

𝑠

2

𝑥


𝑥𝑑𝑥∑

𝑥𝑞𝑥∑

sin 2𝛿 = 0.

(22)

Equations (21) and (22) can be numerically solved toinvestigate the nonlinear behaviors of HTGS such as Hopfbifurcation points, bifurcation surface of PID adjustment co-efficients, time domain response waveforms of state variables,and Lyapunov exponent with MATLAB by using a variable-step continuous solver based on the four-order Runge-Kuttaformula. Time step is 0.01 in the study.

3. The Existence of Dynamic Hopf Bifurcation

3.1. Existence Determination of Hopf Bifurcation. For a four-dimensional nonlinear system, the criteria for the existence ofHopf bifurcation are given in the following theorem. Mean-while, the bifurcation value collection can also be determinedwhen Hopf bifurcation occurs.

Theorem 1. For a nonlinear system 𝑦 = 𝐹(𝑥, 𝜇), 𝑥 ∈ 𝑅4, 𝜇 ∈𝑅1 is the bifurcation parameter, 𝑥 = 0 is the equilibrium point,

and the Jacobi matrix characteristic polynomial at the equilib-rium point is

𝑓 (𝜆, 𝜇) = 𝜆4+ 𝑝3(𝜆) 𝜆3+ 𝑝2(𝜆) 𝜆2+ 𝑝1(𝜆) 𝜆 + 𝑝

0(𝜆) .

(23)

If 𝜇 = 0, the following conditions hold.(1) The coefficient 𝑝

𝑖> 0 (𝑖 = 0, 1, 2, 3),

𝑝3𝑝2𝑝1= 𝑝2

3𝑝0+ 𝑝2

1. (24)

(2)

𝑝

0̸= (

𝑝3𝑝0

𝑝1

−𝑝1

𝑝3

)𝑝3𝑝

1− 𝑝

3𝑝1

𝑝23

+𝑝

2𝑝1

𝑝3

, (25)

where ∗ = 𝑑∗/𝑑𝜇, when |𝜇| is sufficiently small, Hopf bifurca-tion exists in one side of 𝜇 = 0, and the period of limit cycle canbe described as follows:

𝑇 = 2𝜋√𝑝3

𝑝1

. (26)

This theorem is the direct algebraic criterion to determinethe existence of Hopf bifurcation for a four-dimensional system,which can avoid excessive and tedious calculations. However,


0

0.5 1 1.52 2.5 3 3.5

0

1

2

3

0

1

2

3

Kp

Kd

Ki

Figure 2: Bifurcation surface of 𝐾𝑝, 𝐾𝑖, 𝐾𝑑.

the theorem does not give bifurcation direction. Each point onthe surface that is determined by (24) is bifurcation criticalpoint when Hopf bifurcation occurs (the whole bifurcationcritical points constitute the entire bifurcation collection).For example, periodic oscillation phenomenon, namely, Hopfbifurcation, will occur, if the system parameters and the PIDparameters satisfy conditions 1 and 2 of the above theorem in acertain operating condition.The surface determines the scope ofthe system’s stability region also. In fact, from the perspective ofnonlinear dynamical systems structure stability analysis, whenthe PID parameters can meet with some certain conditions, thestructure stability of the system will change dramatically andthen comes to the system structure instability, which leads tocomplex nonlinear oscillations of the system.

3.2. Analysis and Simulation for Hopf Bifurcation. Taking apractical power plant, for example, setting parameters 𝜔

0=

314, 𝑇𝑤

= 0.8 s, 𝑇𝑎𝑏

= 9.0 s, 𝑇𝑦

= 0.1 s, 𝐷 = 2, 𝐸𝑞= 1.35,

𝑉𝑠= 1, 𝑥

𝑑𝑥∑= 1.15, 𝑥

𝑞𝑥∑= 1.474, 𝑒

𝑞ℎ= 0.5, 𝑒

𝑦= 1.0,

and 𝑒 = 0.7, respectively, analyzes theoretical calculation andsimulation in the paper based on the data. The followingsare the system Jacobi matrix characteristic polynomial coef-ficients at equilibrium point (0, 0, 0, 0):

𝑝3= 12.72 − 1.56 ∗ 𝐾

𝑑;

𝑝2= 2.78 ∗ 𝐾

𝑑− 1.56 ∗ 𝐾

𝑝+ 62.07;

𝑝1= 2.78 ∗ 𝐾

𝑝− 1.56 ∗ 𝐾

𝑖+ 434.15;

𝑝0= 2.78 ∗ 𝐾

𝑖+ 857.2.

(27)

It requires all the characteristic polynomial coefficients𝑝𝑖> 0 (𝑖 = 0, 1, 2, 3) by condition 1 when using Theorem 1

to determine the existence of Hopf bifurcation. For example,𝑝3> 0; then𝐾

𝑑< 8.17857, which means that the Hopf bifur-

cation area is limited to the range of𝐾𝑑< 8.17857. Similarly,

the ranges of𝐾𝑖and𝐾

𝑝can be obtained. After substituting 𝑝

𝑖

into (24), a three-parameter implicit polynomial is obtained,and a space bifurcation surface can be made according to thepolynomial, as shown in Figure 2.

All the points on the surface are Hopf bifurcation criticalpoints that would lead to the amplitude oscillation of the unit;stability region lies beneath the surface; PID parameters instability region ensure the stable and safe operation of units;the instability region is located above surface in certain area.

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3.53 4

Kp

Kd

Kd1(0.313)Kd2(2.036)

Limit cycle

Stability region

Ki = 3

Ki = 1

Ki = 0

Figure 3: Curves of bifurcation points with 𝐾𝑖= 0, 1, 3.

0 0.5 1 1.5 20

0.01

0.02

0.03

0.04

2.5

Kd

𝜔

Kd2(2.036)Kd1(0.313)

H1 H2

Figure 4: Bifurcation diagram of rotor speed 𝜔.

Values of the PID parameters that are above the surface oron the surface should be avoided when a PID or a gain-scheduling PID scheme is used in the system. Otherwise, theHopf bifurcation or undesirable oscillations may occur in thegoverning system. Figure 2 can be used to study the relation-ship between the stability of system and PID parameters, andstable PID parameters could be chosen from it directly.Whentaking 𝜇

𝑑= 𝐾𝑑− 𝐾𝑑∗ or 𝜇

𝑝= 𝐾𝑝− 𝐾𝑝∗ as the bifurcation

parameter, it can be calculated that the left hand side of (25) isequal to zero, while the right hand side is not; thus condition2 of the theorem holds.

According to Theorem 1, when |𝜇| is sufficiently small,Hopf bifurcation phenomenon of the system will take place.For example, when 𝐾

𝑖∗ = 1 and 𝐾

𝑝∗ = 2, the bifurcation

parameters 𝐾𝑑1∗ = 0.313 and 𝐾

𝑑2∗ = 2.036 can be calculated

from (24). 𝐾𝑑1∗ and 𝐾

𝑑2∗ can also be got by the intersection

values between the curve (𝐾𝑖= 1) and the line (𝐾

𝑝= 2), as

shown in Figure 3. The curve (𝐾𝑖= 0, 1, 3) is the boundary

between stability region and limit cycle; point that located inlimit cycle, for example,𝐾

𝑖= 1,𝐾

𝑑= 2.5, and𝐾

𝑝= 2, would

lead to the undesirable oscillation. Figure 3 is more intuitiveto tell the stale and unstable PID parameters compared withFigure 2.

Figure 4 is the bifurcation diagram of unit speed𝜔.When𝐾𝑝

= 2, 𝐾𝑖= 1, and 𝐾

𝑑is the bifurcation parameter, there

are two bifurcation points in the figure; that is, 𝐾𝑑1

= 0.313


Table 1: Changes of system properties with the bifurcation parameter𝐾𝑑.

Bifurcation parameter𝐾𝑑

𝐾𝑑< 𝐾𝑑1∗ 𝐾

𝑑= 𝐾𝑑1∗ = 0.313 𝐾

𝑑1∗ < 𝐾

𝑑< 𝐾𝑑2∗ 𝐾

𝑑= 𝐾𝑑2∗ = 2.036 𝐾

𝑑> 𝐾𝑑2∗

Jacobi matrixeigenvalues

Two complexconjugate eigenvalues,negative real part oneand positive real part

one

−2.46

−9.77

−4.5𝑒 − 5 ± 5.98𝑖

Two complexconjugate eigenvalueswith negative real

parts

−2.76

−6.80

−2.7𝑒 − 6 ± 6.77𝑖

Two complexconjugate eigenvalues,negative real part oneand positive real part

one

Property Stable limit cycle center of bifurcationin a supercritical stateStable focus, stable

domainCenter of bifurcationin a supercritical state Stable limit cycle

𝛿

t (s)

0

0

0.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.5

T = 1.053 s

(a)

T = 1.053 s𝜔 0

×10−3

5

−5

t (s)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(b)

Figure 5: Time waveforms of 𝛿 and 𝜔 with 𝐾𝑖= 1, 𝐾

𝑝= 2, and 𝐾

𝑑= 0.309.

and 𝐾𝑑2

= 2.036 as presented calculated by the Matcontsoftware. It is observed that the system is stable and systemstate variable𝜔will converge to the equilibrium point 0 when𝐾𝑑1

< 𝐾𝑑< 𝐾𝑑2; it will converge to a stable limit cycle when

𝐾𝑑< 𝐾𝑑1

or 𝐾𝑑> 𝐾𝑑2, and the amplitude of the limit cycle

oscillation corresponds to the ordinate value of limit cyclecurve. The bifurcation thresholds 𝐾

𝑑1and 𝐾

𝑑2in Figure 4

are consistent with the intersection values in Figure 3 andthe results calculated from (24) as shown in Figure 2. Table 1shows the properties of the system and the changes of Jacobimatrix eigenvalues with different 𝐾

𝑑. When the system is in

stable domain (𝐾𝑑1∗ < 𝐾

𝑑< 𝐾𝑑2∗), all eigenvalues of Jacobi

matrix have negative real parts and the system tends to bestable. As the parameter changes to the critical bifurcationpoint (𝐾

𝑑1∗ , 𝐾𝑑2∗), the eigenvalues of Jacobi matrix have

complex conjugate eigenvalues with zero (−4.5𝑒−5, −2.7𝑒−6)real part; that is, the system is in a supercritical state. When𝐾𝑑

< 𝐾𝑑1∗ or 𝐾

𝑑> 𝐾𝑑2∗ , the matrix has two complex

conjugate eigenvalues; one has a negative real part and theother has a positive real part, system state variables cannotconverge to the equilibrium point at this time, and the systemis in stable amplitude oscillation (limit cycle) in this region.

Table 1 provides theoretical 𝑒 explanation for the phe-nomenon that occurs in Figure 4. Dynamic behaviors ofnonlinear systems in critical bifurcation point on both sidescan be seen from Figure 4 and Table 1 clearly. Furthermore, itcan be obtained from simulation and theoretical analysis thatthe farther from the two bifurcation points (𝐾

𝑑1∗ = 0.313

and 𝐾𝑑2∗ = 2.036) and the nearer to middle place the 𝐾

𝑑

is, the quicker the convergence rate is and the more stablethe system is. In practical applications, value of𝐾

𝑑should be

between two critical places and the farther the better, which

will be verified by the simulation results in Figures 5, 6, 7,and 8.

Time domain response waveforms of 𝛿 and 𝜔 after sta-bilization can be seen in Figure 5 where the PID parametersare 𝐾

𝑖= 1, 𝐾

𝑝= 2, and 𝐾

𝑑= 0.309, which are in limit

cycle area in Figure 3, indicating that system state variables𝛿 and 𝜔 will converge to a stable limit cycle. The motion ofthe system at this value of adjustment coefficients is sustainedperiodic oscillation and the oscillations amplitude of 𝜔 is thelimit cycle curve ordinate value at point𝐾

𝑑= 0.309, as can be

seen from Figure 4. In addition, the oscillation cycle of 𝛿 and𝜔 at bifurcation point is about 1.053 s.Meanwhile, it can be gotthat 𝑝

3= 12.24, 𝑝

1= 438.15 easily from (27), so the period of

limit cycle𝑇 = 1.050 s can be calculated from (26) at theHopfbifurcation point, which is consistent with the simulationresult.These results indicate that the governing systemmakesa periodic vibration and further reveal that the system is notable to be stable. Similarly, the waveforms of 𝛿 and 𝜔 for thevalue of 𝐾

𝑑= 0.6, 0.8, 1.15, which are beneath the limit cycle

curve (𝐾𝑖= 1), namely, the stable parameters, are presented

in Figures 6 to 8. After a period of time, each state (𝛿 and𝜔) will converge to equilibrium point 0 and the system keepssteady at these values of adjustment coefficients; the biggest 𝛿and𝜔 value are 0.5, 0.01, respectively. It is observed that whenvalue of𝐾

𝑑is farther from the two critical points (𝐾

𝑑1= 0.313

and 𝐾𝑑2

= 2.036) and nearer to middle place (1.175), theconvergence rate is quicker and the system is more stable.

Figures 9 and 10 show the phase space orbits of the systemvariables. It can be observed that the closed orbits limit cyclesfor these parameters formed in a certain region in the phasespace orbit when 𝜇

𝑑= 0.002; thus 𝐾

𝑖= 1, 𝐾

𝑝= 2, and 𝐾

𝑑=

2.038. The system shows a diffused and nonlinear growth


𝛿 0

1

0.5

−0.5

−1

t (s)0 50 100 150 200 250

(a)

𝜔

−0.02

−0.01

0

0.01

0.02

t (s)0 50 100 150 200 250

(b)


𝑝= 2, 𝐾

𝑑= 0.6.

𝛿 0

1

0.5

−0.5

−1

t (s)0 50 100 150 200 250

(a)

𝜔

−0.02

−0.01

0

0.01

0.02

t (s)0 50 100 150 200 250

(b)


𝑝= 2, and 𝐾

𝑑= 0.8.

𝛿 0

1

0.5

−0.5

−1

t (s)0 50 100 150 200 250

(a)

𝜔

−0.02

−0.01

0

0.01

0.02

t (s)0 50 100 150 200 250

(b)


𝑝= 2, and 𝐾

𝑑= 1.15.

0

0.05

−0.05

−0.5−0.04

−0.02−1

10.5

0 00.02

0.04

𝛿 𝜔

mt

Figure 9: Phase orbit of 𝛿, 𝜔, and𝑚𝑡.

oscillations curve. Hopf bifurcation occurs and the systemtends to be a stable oscillation state where −0.06 < 𝑚

𝑡< 0.06,

−0.04 < 𝜔 < 0.04, −1 < 𝛿 < 1, −0.04 < 𝑦 < 0.04, indicatingthat the system is in sustained periodic oscillation state; thisvalue of the differential adjustment coefficient cannot beused in practical applications. Therefore, when the turbineregulating system uses PID (or gain scheduling PID) strategy,the values of adjustment coefficients should be beneath thelimit cycle curve. Otherwise, there may be a sustained and

𝜔

0

0 0

0.02

0.020.01

0.04

0.05

−0.04

−0.02

−0.01−0.02 −0.05

y

mt

Figure 10: Phase orbit of𝑚𝑡, 𝜔, and 𝑦.

stable oscillations, instead of tending to be stable, and thegoverning system would lose stability finally, leading to anunstable control.

4. Analysis and Simulation for Chaos

When 𝐾𝑑

> 2.5, with the continually increasing of 𝐾𝑑,

unit speed 𝜔 on bifurcation diagram remains as a singlerising curve until 𝐾

𝑑= 5.72. The bifurcation diagram gets


Kd

0.4

0.3

0.2

4.5 5 5.5 6 6.5 7 7.5 8 8.5

0.1

0

𝜔

Figure 11: Bifurcation diagram of 𝜔 with 𝐾𝑝= 2, 𝐾

𝑖= 1.

0

5 5.5 6 6.5 7 7.5 8

1

2

5.72

−4

−3

−2

−1

Kd

LE1

LE

LE2

LE3

LE4

Figure 12: Lyapunov exponent with 𝐾𝑑.

an intricate pattern when 𝐾𝑑

> 5.72, the system stateparameters converge neither to the equilibrium point norto a stable limit cycle, and it undergoes a random motionwithout any rules, namely, the chaotic motion.There are richand complex nonlinear dynamical behaviors in this area, asshown in Figure 11.

Lyapunov exponent is a quantitative indicator to measurethe systemdynamic behavior, which indicates the average rateof convergence or divergence of the system in phase spaceamong different adjacent tracks. The existence of chaoticdynamics for the system can be judged intuitively by thelargest Lyapunov exponent depending onwhether it is greaterthan 0 or not. The system chaotic motion can be obtainedby Lyapunov exponent with different 𝐾

𝑑, as presented in

Figure 12. When 𝐾𝑑

≦ 5.72, the largest Lyapunov exponentLEl is zero, indicating that the system is in periodic motionstate (limit cycle). In the vicinity of 5.72 ≦ 𝐾

𝑑≦ 6.0, LE1

changes between 0 and 1, and thus, the systemmovement statealternates between periodic motion and chaotic motion; LE1and LE2 are both greater than zero till 𝐾

𝑑= 8.0 and means

chaotic motion that leads to the unstable state of the controlsystem exists.

Figure 13 shows the system stability domain, limit cycles,and chaotic region of the system when 𝐾

𝑖= 1.0, There are

three different areas in the Fig, namely, the stable region,limit cycle, and chaos region, respectively; boundary betweenperiodic motion and chaotic zone is determined by the threePID parameters especially𝐾

𝑑when the maximum Lyapunov

exponent LEl is greater than zero.

Kp

Kd

Limit cycle

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

Stability region

Chaos region

Boundary

Figure 13: Stability domain, limit cycles, and chaotic region of 𝐾𝑝,

𝐾𝑑with 𝐾

𝑖= 1.

𝛿−0.1

−0.05−50

−5

𝜔

0

0 0.050.1

0

5

50

mt

Figure 14: The chaotic attractor with 𝐾𝑖= 1, 𝐾

𝑝= 2, and 𝐾

𝑑= 6.5.

𝛿

y

𝜔

−2

−50 −0.1

0

500

0

0.1

2

Figure 15: The chaotic attractor with 𝐾𝑖= 1, 𝐾

𝑝= 2, and 𝐾

𝑑= 6.5.

Chaotic motion of nonlinear systems can generate attrac-tors in the phase space with unique nature. The originalstable periodic motion turns to instability after entering intothe chaotic region. Chaotic attractor has a complex motioninternal with countless unstable periodic orbits studded init. Overall, the system trajectories are always stretching andvarying within a certain range and quite disordered in somepart, leading to the chaotic strange attractor. When𝐾

𝑝= 2.0,

𝐾𝑖= 1.0, and 𝐾

𝑑= 6.5, chaotic attractors that are similar to

the one with scroll structure in Chua’s circuit [37] generatein the 𝛿, 𝜔,𝑚

𝑡and 𝛿, 𝜔, 𝑦 phase space, as shown in Figures

14 and 15. It is observed that they have the characteristicof being globally bounded, but being local unstable, whichwould do great harm to the HTGS in operation, furtherreveals that the system would lose stability finally. Figure 16is the time domain response waveforms of 𝛿 and 𝜔, resultshows that the system makes a disordered and aperiodic


𝛿

−50

50

200 400 600 800 1000 1200 1400 1600 1800 2000

100

0

0

t (s)

(a)

𝜔

−0.06

−0.12

0.12

0.06

0

200 400 600 800 1000 1200 1400 1600 1800 20000

t (s)

(b)


𝑝= 2, and 𝐾

𝑑= 6.5.

oscillation, and behaviors of the system are hardly to bepredicted. The rotor angle 𝛿 and turbine speed 𝜔 have lefttheir original position largely, oscillating between −50 and 50−0.1 to 0.1, respectively, with a randomperiod, and the systemis in a chaos, which means that this value of the differentialadjustment coefficient should be avoided.

Poincare maps are shown in Figures 17 and 18. It can beseen that numerous intersections of the continuous trajectoryturn up in the Poincare maps on section of 𝜔 = 0 and𝑚

𝑡= 0.

However, they neither fill with the entire phase space nordistribute discretely in the 𝛿, 𝑦,𝑚

𝑡and 𝛿, 𝜔, 𝑦 phase space

but occur in a certain region frequently, constituting a pieceof dense point just like cloud. These results indicate that thesystem is in chaotic state and the system cannot tend to bestable. Figures 11, 12, 13, 14, 15, 16, 17, and 18 verify the chaoticand aperiodic movement of the system from mathematicalperspective.

5. Conclusion

In this paper, a fully coupled nonlinear mathematical modelfor HTGS has been developed. Extensive investigations of thenonlinear behaviors of an example HTGS were conductedusing the proposedmodel.The existence and stability ofHopfbifurcation were studied first. Both the simulation resultsand theoretical analysis predicted that Hopf bifurcationwould occur when the governor’s parameters satisfy certaincriteria. It was also found that the convergence rate andstability of the system were closely related to the bifurcationparameters, indicating that the PID adjustment coefficientparameters have to be carefully selected to guarantee thesafe and stable operation of hydroelectric generating unit.Relevant stability ranges of PID adjustment coefficients forthe exampleHTGSwere presented in the paper. In addition tothe study of Hopf bifurcation, chaotic behaviors of the systemwere also investigated extensively using different methods.It was found that in certain region of PID parameter space,chaos phenomenon would occur and lead to the chaoticoscillatory and unstable control of the system,which has beenidentified as a cause of decreased availability and reducedstability of the system. The above investigations on Hopfbifurcation and chaos phenomenon of the example HTGSclearly illustrated the complex nonlinear nature of the system.The simulation results based on the proposed model wereshown to have good correlation with the theoretical analysispredictions.Themodel presented in this study for the analysis

−3−4 −3

−2

−2

−1

−1

2

2

3

3 4

1

1

0

0

y

mt

Figure 17: Poincare map with 𝜔 = 0.

−3−0.08 −0.06 −0.04 −0.02

−2

−1

2

3

1

0

0 0.02 0.04 0.06 0.08 0.1

y

𝜔

Figure 18: Poincare map with𝑚𝑡= 0.

of the dynamic behaviors of HTGS can be further extendedto a more sophisticated system.

Some related topics, such as data-driven framework [38],chaos control, and Hopf Bifurcation in HTGS with nonlinearsaturation, would be considered in the future work to achievemore practical oriented results.

Nomenclature

𝐻(𝑠): Laplace transform of ℎ, p.u.𝑄(𝑠): Laplace transform of 𝑞, p.u.𝑄0: Initial turbine flow of turbine, m3/s

𝐻0: Initial hydraulic head of turbine, m

𝑔: Gravitational acceleration𝑙: Penstock length, m𝐴: Penstock area, m2𝐷: Penstock diameter, m


𝑓: Head loss coefficient, p.u.𝑎: Pressure wave velocity, m/s𝑇𝑟: Elastic time constant, s

𝑇𝑤: Water starting time, s

𝑦: Incremental deviation of guide vane/wicketgate position, p.u.

𝑒𝑥, 𝑒𝑦, 𝑒ℎ: Partial derivatives of turbine torque with

respect to head, guide vane, and speed, p.u.𝑒𝑞𝑥, 𝑒𝑞𝑦, 𝑒𝑞ℎ: Partial derivatives of the flow with respect tohead, guide vane, and turbine speed, p.u.

𝛿: Rotor angle, rad𝜔: Turbine/rotor speed, rad/s𝜔0: Base angular speed, rad/s

Δ𝜔: Speed deviation𝐸

𝑞: Transient electric potential of 𝑞-axis, p.u.

𝑉𝑠: Infinite bus voltage, p.u.

𝑥

𝑑𝑥∑: Transient reactance of 𝑑-axis, p.u.

𝑥

𝑞𝑥∑: Synchronous reactance of 𝑞-axis, p.u.

𝑥

𝑑: Direct axis transient reactance, p.u.

𝑥𝑞: Quadrature axis reactance, p.u.

𝑥𝑇: Transformer short circuit reactance, p.u.

𝑥𝐿: Transmission line reactance, p.u.

𝑇𝑎𝑏: Mechanical starting time, s

𝑚𝑡: Mechanical torque of turbine, N.m

𝑚𝑒: Electrical torque, N.m

𝐾: Damping factor, p.u.𝑃𝑒: Terminal active power

𝑃𝑚: Output power

𝑇𝑦: Engager relay time constant

𝐾𝑝: Proportional adjustment coefficient

𝐾𝑖: Integral adjustment coefficient

𝐾𝑑: Differential adjustment coefficient.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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