Research ArticleFixed-Time Feedback Control of the Hydraulic TurbineGoverning System

Caoyuan Ma ,1,2 Chuangzhen Liu ,1,2 and Xuezi Zhang 1,2

1Jiangsu Province Laboratory of Mining Electric and Automation, China University of Mining and Technology, Xuzhou,Jiangsu 221116, China2School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China

Correspondence should be addressed to Caoyuan Ma; mcaoyuan@cumt.edu.cn

Received 25 May 2018; Revised 20 June 2018; Accepted 25 June 2018; Published 1 August 2018

Academic Editor: Diyi Chen

Copyright © 2018 Caoyuan Ma et al. 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.

Dealing with convergence time and dealing with steady state are two of the most challenging problems in the field of stability of thehydraulic turbine governing system. In this paper, we solve these two challenging problems by designing fixed-time feedbackcontrollers. The design of the controllers is based on the fixed-time theory and backstepping method. Compared with theexisting controllers for fixed-time, finite-time, and other techniques, the designed controllers make the maximum convergencetime of the system unaffected by the initial state. The convergence time is also shorter. Meanwhile, they are continuousand do not include any sign function, and hence, the chattering phenomenon in most of the existing results is overcomevia nonchattering control. In addition, they give the system better stability and robustness to disturbances. Finally, thenumerical simulation results in this paper will contribute to a better understanding of the effectiveness and superiority ofthe proposed controllers.

1. Introduction

Hydropower is a renewable energy source and it has highpotential for economically viable utilization [1]. The pastfew decades have witnessed the rapid development ofhydropower energy, which supplements or often replacesdecommissioned coal-fired power plants [2]. The system ofsuch a powerful hydropower plant including penstocksystems, water turbines, generators, regulators, and loads, isso complex that it is difficult to control [3]. However, thestability of a hydropower system plays an important role inthe stability of the whole power system and the plantsthemselves [4]. The hydraulic turbine governing system isone of the most important parts of a hydropower plant,which plays key roles in maintaining safety, stability, andeconomical operation of the hydropower plant [5]. It is alsoa multiparameter, high-dimension complex system withnonlinear, time-variant, and nonminimum phase characters[6]. Therefore, the study of the hydraulic turbine governingsystem is of great importance. In this regard, a special kindof nonlinear control technique, which combines the fixed-

time control theory, feedback control theory, and backstep-ping method, is introduced into the controller design of thehydraulic turbine governing system in this paper.

In the past decades, many advanced control techniqueshave been applied to the controller design of the hydraulicturbine governing system, such as PID control [7, 8],sliding-mode control [9], nonlinear control [10–12], fuzzycontrol [13], fault tolerant control [14], predictive control[15, 16], finite-time control [17], and fixed-time control[18–20]. These control methods have important theoreticaland practical significance for the control of a hydraulicturbine governing system, but they also have some problemsin overcoming the long-term operation of the system inreal-time applications. For instance, the nonlinear natureof a water turbine and the constantly varying load makesthe gain schedule of the PID controller difficult to design,limiting the operating range. Sliding-mode control is a quickand powerful variable-structure approach governing thenonlinear system via a predefined switch sliding surface.However, a common drawback of the conventional sliding-mode control is the chattering phenomenon, due to the use

HindawiComplexityVolume 2018, Article ID 8767158, 9 pageshttps://doi.org/10.1155/2018/8767158

of a discontinuous sign function. The nonlinear control istargeted, and each nonlinear control strategy is only suitablefor solving some special nonlinear system control problems.Fuzzy control is difficult to adapt to the requirements of alarge-scale adjustment, and it needs constant adjustment ofthe control rules and parameters. The convergence time offinite-time control is heavily dependent on the initial valuesof considered systems, which makes the maximum conver-gence time not fixed for different initial values. Moreover,not all the initial values of the systems are available in prac-tice. These drawbacks prohibit the practical application offinite-time control [21]. In order to overcome the shortcom-ings of finite-time control, fixed-time control is proposed in[22]. The convergence time of fixed-time control can becompletely determined by the parameters of a closed-loopsystem. Based on the advantages of fixed-time control, thiscontrol strategy has been applied to some systems [23–29].It is worth mentioning that the sign function is indispensablefor the controllers in [23–29]. It is well known that the signfunction always introduces the chattering phenomenon tothe system state and control signal, which damages equip-ment and induces undesirable effects [30]. Therefore, howto design continuous controllers without a sign function tomake the system achieve fixed-time stability more rapidlyhas become the goal of this paper.

Motivated by the above discussions, to ensure the stabil-ity of a hydraulic turbine governing system, a set of ingeniouscontrollers combining fixed-time control and feedbackcontrol is proposed. Compared with all the above controlstrategies, the designed fixed-time feedback controllers notonly make the system steady in a fixed time, but also elimi-nate the chattering phenomenon of the system. In addition,they make the system stabilize faster and be more robust todisturbances. The rest of this paper is organized as follows.In Section 2, the model of the hydraulic turbine governingsystem is presented and some necessary definitions andlemmas are given. In Section 3, fixed-time feedback control-lers are designed. Numerical simulations and theoreticalanalyses are presented to show the effectiveness of thecontrollers in Section 4. Finally, in Section 5, conclusionsare given.

2. Preliminaries

2.1. Model of Hydraulic Turbine Governing System. Themathematical model of the hydraulic turbine governingsystem with the elastic water hammer has been widelystudied in [12, 15, 31]. In this paper, the focus is to usean advanced control strategy to give the system betterquality. Therefore, we use the nonlinear model of thehydraulic turbine governing system which is based onthe previous work as follows:

x1 = x2,x2 = x3,x3 = −a0x1 − a1x2 − a2x3 + y,δ = ω0ω,

ω = 1Tab

mt −Dω − Pe ,

y = yTy

, 1

where

Pe =Eq′Vs

xdΣ′sin δ + δ0 + v2s

2xdΣ′ − xqΣxdΣ′ − xqΣ

sin 2δ + 2δ0 ,

mt = b3y + b0 − a0b3 x1 + b1 − a1b3 x2 + b2 − a2b3 x3,

xdΣ′ = xd′ + xT + 12 xL,

xqΣ = xq + xT + 12 xL,

b0 =24ey

eqhhwT3r

,

b1 =24eeyeqhT

2r

,

b2 =3ey

eqhhwTr,

b3 =eeyeqh

,

a0 =24

eqhhwT3r

,

a1 =24T2r

,

a2 =3

eqhhwTr

2x1, x2, and x3 denote intermediate state variables, δ denotesthe deviation of the generator rotor angle, ω denotes the devi-ation of the generator rotor speed, y denotes the deviation ofthe guide vane opening. The meanings of the other symbolsin the system are shown in the Nomenclature.

2.2. Definition and Lemmas. In this section, for theconvenience of analysis, we present the basic definition offixed-time stability first and then introduce some usefullemmas which are necessary for controller design.

Definition 1 (see [22]). Consider the following nonlineardynamic system.

x = f x , 3where x ∈ Rn is the system state and f is a smooth nonlinearfunction. If there exists a fixed convergence time T0, which isindependent of the initial condition and satisfies

limt→T0

x t = 0, 4

2 Complexity

then this nonlinear dynamic system is said to be fixed-time stable.

Lemma 1 (see [22]). Consider the following. If there exists acontinuous radically unbounded function V Rn → R+ ∪ 0 ,such that

(1) V x = 0⇔x = 0,(2) for some α, β, ϕ = 1 − 1/2γ, φ = 1 + 1/2γ, and γ > 1,

any solution x t satisfied the inequality

D ∗V x t ≤ −αVϕ x t − βVφ x t , 5

where D∗V x t denotes the upper right-handderivative of the function V x t , then the originis globally fixed-time stable and the followingestimate holds

T x0 ≤ Tmax =πγ

αβ,  ∀x0 ∈ Rn 6

Lemma 2 (see [32]). If x1, x2,… , xn ≥ 0, Then, the followingtwo inequalities hold.

〠n

i=1xni ≥ 〠

n

i=1xi

η

, 0 < η ≤ 1,

〠n

i=1xθi ≥ n1−θ 〠

n

i=1xi

θ

, θ > 1

7

3. Fixed-Time Feedback Controllers Design

From (1), it is known that when the system is running to thepoint P(0,0,0,0,0,0), the system returns to its original state ofstability. In order to quickly stabilize the system to theequilibrium point P, the fixed-time controllers uω and uyare added to the fifth subsystem and sixth subsystem of thesystem (1), respectively, and the controlled system can bedescribed as follows:

x1 = x2,x2 = x3,x3 = −a0x1 − a1x2 − a2x3 + y,δ = ω0ω,

ω = 1Tab

mt −Dω − Pe + uω,

y = 1Ty

−y + uy

8

For the uncertain system (1), in order to make thehydraulic turbine governing system stabilize in fixed time,we design the controllers which are based on the fixed-time

control theory, feedback control theory, and backsteppingmethod as follows:

uy = −k1y − k2ym/n − k2y

p/q,

uω = −ω0δ −mt −Dω − Pe

Tab− k3ω0ω −

mk4ω0ω

nδ m−n /n

−pk4ω0ω

qδ p−q /q − k3 ω + k3δ + k4δ

m/n + k4δp/q

− k5 ω + k3δ + k4δm/n + k4δ

p/q m/n

− k5 ω + k3δ + k4δm/n + k4δ

p/q p/q,

9

where k1 > 0, k2 > 0, k3 > 0, k4 > 0, and k5 > 0, and m, n, p,and q are positive odd integers satisfying m < n and p > q;thus, the hydraulic turbine governing system is fixed-time stable.

Here, we will give the design process of the fixed-timefeedback controllers uy and uω of the hydraulic turbine gov-erning system step by step. Consider the Lyapunov function:

V1 t = y2

2 10

Differentiating V1 t along the solution of (8) givesthe following:

V1 t = yy = yTy

−y + uy 11

According to (11), in order to achieve fixed-time stabilityfor the V1 t , the actual fixed-time feedback control law isobtained as follows:

uy = −k1y − k2ym/n − k2y

p/q, 12

where −k2ym/n − k2yp/q is to ensure the fixed-time stability of

V1 t and −k1y is to speed up the stability of V1 t .To bring (12) into V1(t), the following can be obtained:

V1 t = yTy

−y − k1y − k2ym/n − k2y

p/q

= −1Ty

y2 + k1y2 + k2y

m+n /n + k2yp+q /q

≤ −k2Ty

y2m+n /2n + y2

p+q /2q

= −k2Ty

y2

2

m+n /2n· 1

2− m+n /2n

+ y2

2

p+q/2q· 1

2−p+q/2q

= −k2Ty

· 2 m+n /2nV m+n /2n1 t −

k2Ty

· 2p+q/2qVp+q/2q1 t ,

13

3Complexity

where α1 = k2/Ty · 2 m+n /2n, β1 = k2/Ty · 2 p+q /2q, ϕ1 = m +n /2n and φ1 = p + q /2q According to Lemma 1, weknow that the sixth subsystem of (8) is stable in fixed time

t1 =Tyπqn

k2 pn −mq · 2 qm+pn−2qn /4qn , 14

which means that the system state variable y satisfies thefollowing relation y = 0 when t ≥ t1. And

x1 = x2,x2 = x3,x3 = −a0x1 − a1x2 − a2x3,δ = ω0ω,

ω = 1Tab

mt −Dω − Pe + uω

15

For the system (15), the desired controllers uω can beworked out by the following steps.

Step 1. Define the tracking errors as follows: e1 = δ,e2 = ω − ωd, in which ωd = −k3δ − k4δ

m/n − k4δp/q.

Then, their derivatives can be written as e1 =δ = ω0ω, e2 = ω − ωd , in which ω is the virtualcontroller and ωd is the virtual stabilizationfunction. Then, the derivative of e1 can bewritten as

e1 = ω0 16

A Lyapunov candidate function V2 t for sta-bility analysis is defined as follows:

V2 t = e212 17

Differentiating V2 t with respect to time,there is

V2 t = e1e1

= ω0 e1e2 − k3e21 − k4e

m+n /n1 − k4e

p+q /q1

18

From (18) we can know V2 t ≤ 0 when e2 = 0.Step 2. Then, going one step ahead provides the candi-

date Lyapunov function V3 t as follows:

V3 t =V2 t + e222 19

Differentiating V3 t with respect to time yields

V3 t =V2 t + e2e2

= ω0 e1e2 − k3e21 − k4e

m+n /n1 − k4e

p+q /q1

+ e2 ω − ωd

= ω0 −k3e21 − k4e

m+n /n1 − k4e

p+q /q1

+ e2 ω0e1 +1Tab

mt −Dω − Pe + uω

+ k3ω0ω + mk4ω0ω

n· δ m−n /n

+ pk4ω0ω

q· δ p−q /q

20

In order to achieve fixed-time stability for the V3 t , theactual fixed-time feedback control law is obtained as

uω = −ω0δ −1Tab

mt −Dω − Pe − k3ω0ω −mk4ω0ω

nδ m−n /n

−pk4ω0ω

qδ p−q /q − k3e2 − k5e

m/n2 − k5e

p/q2

21

To bring uω into V3 t the following can be obtained:

V3 t = ω0 −k3e21 − k4e

m+n /n1 − k4e

p+q /q1

− k3e22 − k5e

m+n /n2 − k5e

p+q /q2

≤ −k4ω0em+n /n1 − k5e

m+n /n2 − k4ω0e

p+q /q1 − k5e

p+q /q2

≤ −km e m+n /n1 + e m+n /n

2 − km e p+q /q1 + e p+q /q

2

= −kme212

m+n /2n 12

− m+n /2n

+ e222

m+n /2n 12

− m+n /2n

− kme212

p+q /2q 12

− p+q /2q

+ e222

p+q /2q 12

− p+q /2q

= −km2 m+n /2n e212

m+n /2n+ e22

2

m+n /2n

− km2 p+q /2q e212

p+q /2q+ e22

2

p+q /2q

≤ −km2 m+n /2nV m+n /2n3 t − 2kmV

p+q /2q3 t ,

22

4 Complexity

where km =min k4ω0, k5 , α2 = km2 m+n /2n, β2 = 2km, ϕ2 =m + n /2n, and φ2 = p + q /2q. According to Lemma 1,we know that V3 t is stable in fixed time.

t2 =πqn

km pn −mq · 2 m−n /4n , 23

which means V3 t = 0 when t ≥ t2. It also means e1 = 0,e2 = 0, δ = 0, and ω = 0, and the system reaches steadystate. To sum up, when t ≥ t3, the hydraulic turbine govern-ing system is stable under the action of fixed-time feedbackcontrollers uω and uy, where t3 = t1 + t2. In other words, thesystem is stable in the fixed time.

4. Simulation Results and Analyses

In this section, the simulation results are provided to verifythe validity and effectiveness of the proposed fixed-time feed-back control strategy. Based on some published papers, thevalue of system parameters are selected from [18] and thesystem parameters and controller parameters of this paperare shown in Table 1.

Figures 1(a)–1(c) display the transient responses of thesystem state variables δ, ω, and y after the controllers uωand uy are applied to the hydraulic turbine governing system,respectively. From Figure 1, it can be seen that when thesystem is coupled with uω and uy, the system state variabley reaches steady state at 0.11 s, and the system state variablesδ and ω reach steady state at 0.18 s, simultaneously. Theconvergence time of δ, ω, and y is much smaller than thetheoretical estimate t3 = 4 634 s, which is calculated by thevalues of the parameters of the fixed-time feedback control-lers. The simulation results show that the system can reachthe steady state rapidly under the action of fixed-time feed-back controllers uω and uy, and the control effect is achieved.

In this part, in order to prove the validity and superiorityof the fixed-time feedback controllers proposed in this paper,we compared the control effect of the fixed-time feedbackcontrollers and the feedback controllers. From Section 3, weknow that the fixed-time feedback controllers are composedof two parts of the fixed-time control part and the feedbackcontrol part, and the feedback control part can also bringthe hydraulic turbine governing system to a stable state intheory. The feedback controllers used for comparison arethe feedback control part of the fixed-time feedback

controllers. In order to make a fair comparison betweenfixed-time feedback control and feedback control, we selectthe same initial conditions and system parameters. Theparameters of the feedback controllers are exactly the sameas those of the feedback control part of the fixed-time feed-back controllers and these parameters have been given inTable 1. Figures 2(a)–2(c) show the dynamic responses ofthe system state variables under the action of fixed-time feed-back controllers and feedback controllers. From Figure 2, itcan be found that the performance of the system statevariables δ, ω, and y with fixed-time feedback controllers ismuch better than the performance of the system state vari-ables δ, ω, and y with the feedback controllers. Under theaction of fixed-time feedback controllers, the system statevariables δ, ω, and y are stable at 0.18 s, 0.18 s, and 0.11 s,respectively. And under the action of feedback controllers,the system state variables δ, ω, and y are stable at 1.08 s,1.08 s, and 0.25 s, respectively. The fixed-time controllersstabilize the nonlinear system faster than feedback control-lers do. In other words, compared with the feedback control,the fixed-time feedback control has better control effect onthe hydraulic turbine governing system.

In order to explore the effect of different initial conditionson system stability under the action of fixed-time feedbackcontrollers, we compared the responses of three differentinitial conditions of the hydraulic turbine governing system.In this section, the initial values of the hydraulic turbinegoverning system are shown as follows: S1= (0.02, 0.02,0.02, 0.02, 0.02, 0.02), S2= (0.05, 0.05, 0.05, 0.05, 0.05, 0.05),and S3= (0.08, 0.08, 0.08, 0.08, 0.08, 0.08). From Figure 3,we can see that as the initial values increase, the convergencetime gradually increases when the system is coupled with uωand uy, but they all do not exceed the theoretical estimatet3 = 4 634 s. And no matter how the initial value changes,the system achieves the ideal steady state. That is to say,the simulation results are consistent with the theoreticalderivation and the actual situation.

Figures 4(a)–4(c) show the dynamic responses of thesystem state variables when the system is subjected to exter-nal disturbances. In order to verify the robustness of thecontrolled system to disturbance, this paper gives δ and y adisturbance in 0.5 s and 1 s, respectively. From Figure 4, itcan be found that the system achieves steady state underthe action of fixed-time feedback controllers before 0.5 s.When the system is disturbed in 0.5 s, δ and ω quickly

Table 1: The system and controllers parameters.

Parameter Value Parameter Value Parameter Value Parameter Value

D 0.5 E′q 1.35 x′d〠 1.15 ey 1

e 0.7 h 2 xq〠 1.474 Tab 8

eqh 0.5 Ty 0.1 Tr 1 Vs 1

ω0 314 δ0 30 xL 0.08 k1 2.5

k2 0.5 k3 0.05 k4 0.05 k5 1

m 11 n 15 p 11 q 7

5Complexity

increase, whose maximum deviations reach to 2.51 and 0.05,respectively. But under the action of the controller uω, thestate of y has not changed. As time goes on, δ and ω gradually

3

2

1

0

−1

−2

0.0 0.1 0.2

Time (s)

0.3 0.4 0.5

�훿

(a)

0.06

0.04

−0.04

0.0 0.1 0.2 0.3

Time (s)

0.4 0.5

0.02

−0.02

0.00

�휔

(b)

0.05

0.04

0.03

0.02

0.01

0.00

0.0 0.1 0.2 0.3

Time (s)

0.4 0.5

�훾

(c)

Figure 1: The response of the system variables δ, ω, and y withfixed-time feedback controllers.

3

2

1

0

−1

−2

−30.0 0.2 0.4 0.6

Time (s)

Fixed-time feedbackFeedback

0.8 1.0

�훿

(a)

�휔

0.06

0.04

0.02

0.00

−0.02

−0.04

−0.060.0 0.2 0.4 0.6

Time (s)

0.8 1.0

Fixed-time feedbackFeedback

(b)

0.05

0.04

0.03

0.02

0.01

0.00

0.0 0.2 0.4 0.6

Time (s)

0.8 1.0

Fixed-time feedbackFeedback

�훾

(c)

Figure 2: Comparison of converging speed of fixed-time feedbackand feedback controllers.

6 Complexity

reach the steady state, which indicate that the hydraulicturbine governing system remains in a stable state underthe appropriate control of the fixed-time feedback controlleruω. When the system is disturbed in 1 s, y gradually decreased

5

4

3

2

1

0

−1

−2

−3

−40.0 0.1 0.2

Time (s)0.3 0.4 0.5

0.2

0.0

−0.20.10 0.12 0.14 0.16

S1S2S3

�훿

(a)

�휔

0.10

0.08

0.06

0.04

0.02

0.00

0.0 0.1 0.2Time (s)

0.3 0.4 0.5

−0.02

−0.04

−0.06

−0.08

−0.0040.000

0.09 0.12 0.15

S1S2S3

0.004

(b)

0.08

0.06

0.04y

0.02

0.00

0.0 0.1 0.2Time (s)

0.3

0.004

0.002

0.000

0.08 0.12

S1S2S3

0.4 0.5

(c)

Figure 3: The response of the system with the different S.

3

2

1

0

−1

−2

0.0 0.3 0.6Time (s)

0.9 1.2 1.5

�훿

(a)

�휔

0.06

0.04

0.02

0.00

−0.02

−0.04

0.0 0.3 0.6 0.9Time (s)

1.2 1.5

(b)

y

0.05

0.04

0.03

0.02

0.01

0.00

0.0 0.3 0.6Time (s)

0.9 1.2 1.5

(c)

Figure 4: The response of the system with the differentdisturbances.

7Complexity

from the disturbance and eventually stabilized to zero. Thereis no change in the state of δ and ω in the controlled system.Therefore, the simulation results are consistent with thetheoretical derivation and show that the system has goodrobustness to external disturbances under the action offixed-time feedback controllers.

5. Conclusions

The control problem of a nonlinear hydraulic turbine gov-erning system is studied in this paper. In order to give thesystem better quality, the fixed-time feedback controllerswere designed by combining fixed-time control and feedbackcontrol and compared with the feedback controllers. Thedesigned controllers are without sign function, which meansthe chattering phenomenon will not appear in the system.The fixed-time feedback control inherits the advantages ofthe fixed-time control, so that the maximum convergencetime of the system is not affected by the initial state of thesystem. The introduction of feedback control of the fixed-time feedback control makes the controlled system achievea steady state faster. The fixed-time feedback controllers alsogive the system robustness to external disturbances. Finally,simulations were carried out in the presence of various situ-ations to evaluate the effectiveness of the proposed fixed-time feedback controllers, where the results demonstratedthe effectiveness and superiority of the proposed controllers.

Nomenclature

D: Generator damping coefficiente: Intermediate variableeqh: First-order partial derivative value of flow rate with

respect to water headey: First-order partial derivative value of torque with

respect to wicket gateEq′: The transient internal voltage of armaturehω: The characteristic coefficient of the pipelinemt: The deviation of the mechanical torque of the

hydro-turbinePe: The electromagnetic power of generatorTy: The engager relay time constantTr: The reflection time of the penstockTab: The mechanical starting timexd′: The direct axis transient reactancexq: The quartered axis reactancexT: Short circuit reactance of the transformerxL: Reactance of the electric transmission linexd∑′ : Direct axis transient reactancexq∑: Quadrature axis reactanceVs: The voltage of infinite busδ0: Initial generator rotor angleω0: Synchronous angular speed.

Conflicts of Interest

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

Authors’ Contributions

Caoyuan Ma and Chuangzhen Liu conceived and designedthe experiments. Chuangzhen Liu and Xuezi Zhang per-formed the experiments and analyzed the data. ChuangzhenLiu and Xuezi Zhang wrote the paper. Caoyuan Ma andChuangzhen Liu contributed equally to this work.

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