Robust fixed-time sliding mode control for fractional-order nonlinear hydro-turbine governing system

Renewable Energy 139 (2019) 447e458

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Renewable Energy journal homepage: www.elsevier.com/locate/renene

Robust fixed-time sliding mode control for fractional-order nonlinear hydro-turbine governing system Sunhua Huang a, Bin Zhou a, *, Siqi Bu b, Canbing Li a, Cong Zhang a, Huaizhi Wang c, Tao Wang a a b c

College of Electrical and Information Engineering, Hunan University, Changsha 410082, China Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong College of Mechatronics and Control Engineering, Shenzhen University, Shenzhen 518060, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 1 June 2018 Received in revised form 17 January 2019 Accepted 16 February 2019 Available online 20 February 2019

This paper proposes a nonlinear sliding mode controller for effective fixed-time stabilization of a hydroturbine governing system (HTGS). The HTGS is a highly nonlinear, multi-variable coupled and nonminimum phase system to maintain frequency stability by regulating the generation output in response to load variations. The state trajectories of HTGS under random load disturbances exhibit unstable behaviours on the rotor angle, rotor speed, and guide vane opening, which would affect the stable operation of hydroelectric stations. In the proposed approach, the fractional calculus technique is adopted to model the nonlinear dynamics of fractional-order hydro-turbine governing system (FOHTGS). In order to validate the system stabilization and convergence within a bounded time, a robust sliding mode controller is proposed to force the FOHTGS to reach the equilibrium point based on the fixed-time stability and Lyapunov stability theories. The proposed controller can guarantee the superior stabilization of FOHTGS with the bounded and quantifiable convergence time, and thus overcome the drawback of finite-time controllers whose convergence characteristics are heavily dependent on the initial conditions. Finally, numerical simulations have been implemented to confirm the effectiveness and superior performance of the fixed-time controller compared with the existing finite-time control methods. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Hydraulic energy Hydro turbine Fractional-order modelling Fixed-time stability Renewable energy

1. Introduction Hydro energy is the largest renewable energy source for global electricity generation, accounting for about 71% of total renewable energy generation in the world [1]. According to the annual world hydropower statistics in Ref. [2], the total installed capacity of worldwide hydroelectricity has increased from 770 GW in 2006 to 1245 GW in 2016. Compared with wind and solar energy, hydroelectricity is a reliable, flexible, and cost-effective energy generation technology, with the advantages of high energy efficiency and easily stored in reservoir, and it can be used for frequency regulation, peak load shaving and emergency reserve in smart grid [3,4]. Due to its great potentials on social, economic and environmental benefits, hydroelectricity will promote and orient the sustainable development of global renewable energy supply, and a lot of government policies are established for supporting the hydraulic

* Corresponding author. E-mail address: [email protected] (B. Zhou). https://doi.org/10.1016/j.renene.2019.02.095 0960-1481/© 2019 Elsevier Ltd. All rights reserved.

energy deployment and utilization, especially in developing countries [5e7]. Hydro-turbine governing system (HTGS) is an essential portion in the hydroelectric station, which is composed of an electric generator, a penstock system, a hydro turbine and a hydraulic servo system [8]. Therefore, the modelling and control of HTGS is significant to ensure the system frequency stability and the stable operation of hydroelectric stations. HTGS is a highly nonlinear, multi-variable coupled and nonminimum phase system, and it may exhibit the complex dynamic behaviors with sudden, aperiodic, or stochastic phenomena under the non-rated operating conditions [9]. The dynamical complexity of HTGS has been formulated in Ref. [10] as various nonlinear models using the integer-order calculus. Nevertheless, the state variables of HTGS usually have hereditary characteristics with the history-dependent effects, and the dynamic stability of HTGS is modelled in Ref. [11] by fractional derivatives to form the fractional-order hydro-turbine governing system (FOHTGS). Furthermore, it has been found in Refs. [9,12] that the HTGS would express nonlinear coupling dynamics with chaotic and unstable

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S. Huang et al. / Renewable Energy 139 (2019) 447e458

states on the mechanical torque, guide vane opening of hydroturbine, rotor angle and rotor speed of generator, and the stability of FOHTGS is influenced by stochastic load disturbances. Therefore, it is necessary to develop a robust controller to stabilize the chaotic and nonlinear vibration of FOHTGS with random load disturbances. So far, various modern control methods, including nonlinear PID control [13], sliding mode control [14,15], predictive control [16], and fuzzy control [17], have been applied for the stabilization and suppression on nonlinear and chaotic vibrations of various fractional-order renewable energy models. Chen et al. [11] presented a nonlinear fractional-order model of Francis HTGS with complex penstocks, and a basic law of bifurcation points of FOHTGS with a change in the fractional order can be found. Zhang et al. [16] investigated a nonlinear predictive control method to suppress the six-dimensional HTGS under stochastic disturbances. Wang et al. [17] proposed a robust Takagi-Sugeno fuzzy controller for the stability problem of FOHTGS. Nevertheless, these control methods in Refs. [11e17] can only ensure the asymptotic stability of FOHTGS, and would affect the stable operation of hydroelectricity stations if system vibrations cannot be effectively suppressed within the prescribed time. Consequently, various finite-time control methods (FTCMs) have been used in Refs. [19e22] to effectively stabilize the FOHTGS within a specified time under stochastic load disturbances. However, the finite-time controller in Ref. [19] exhibits the low nadir with large-amplitude oscillations in the state trajectories, while the long convergence time of the FTCM is resulted in Ref. [20]. Moreover, the convergence characteristics of FTCMs are readily affected by the initial conditions of FOHTGS, and the convergence time would increase unboundedly with the growth of initial system states [23]. In order to overcome this disadvantage of FTCMs, this paper aims to investigate a robust sliding mode control for the fixed-time stability of FOHTGS, in which the convergence characteristics can be exactly evaluated with independence of initial system states. In this paper, the fractional calculus technique is introduced to model the complex dynamics of HTGS system. Based on the dissipation and nonlinear characteristics of FOHTGS, the state and phase trajectories of FOHTGS under small-signal disturbances may exhibit unstable behaviors on its rotor angle, rotor speed, and guide vane opening, which would affect the stable operation of hydroelectric stations. Hence, a novel fixed-time sliding mode controller is proposed to enforce the FOHTGS for achieving the equilibrium point within an upper bounded time based on the sliding mode control and fixed-time stability theories. The main contributions of the proposed methodology can be summarized as follows: 1) Caputo fractional-order derivatives are adopted for modelling the evolutionary dynamics of FOHTGS to investigate the nonlinear and irregular oscillations under stochastic load disturbances; 2) The Proof is presented for the fixed-time stability of FOHTGS with the proposed controller based on the Lyapunov stability theory; 3) The proposed fixed-time controller can guarantee the superior stabilization of FOHTGS with the bounded convergence time which is independent on initial system conditions. Finally, numerical simulations have been implemented with different disturbances to confirm the effectiveness and superior performance of the proposed controller compared with the existing FTCMs. The remainder of this work is organized as below: The fractional-order modelling of HTGS is formulated in Section 2, and the dynamic behaviors of FOHTGS are studied in Section 3. In order to suppress the nonlinear and chaotic vibrations of FOHTGS under stochastic load disturbances, a robust sliding mode controller is proposed, and the Proof for the fixed-time stability of FOHTGS is presented in section 4. The numerical simulation is implemented in Section 5. Finally, the conclusion is presented in Section 6.

2. Modelling of FOHTGS 2.1. Definitions and properties of fractional-order calculus

Definition 1. [24]: Firstly, the Caputo fractional-order derivative is defined as,

C a t 0 Dt xðtÞ

¼

8 ðt > > 1 > > > < Gðn  aÞ

t0

> > > > > :

xn ðtÞ ðt  tÞqþ1n

dt

n  1
n

d xðtÞ dt n

(1)

a¼n

where a represents the fractional order of FOHTGS, and Gð$Þ is the Z ∞ t t1 et dt.

gamma function which is defined as GðtÞ ¼

0

Property 1. [19]: The following formula is valid for Caputo derivative, RL;C

DðRL;C Df ðxðtÞÞÞ ¼ RL;C Df ðxðtÞÞ

(2)

Lemma 1. [24]: Considering the following fractional incommensurate-order system Dai¼f (x(t)); 0
jargðli Þj > p=ð2ZÞ

(3)

where li are all the roots of the system as,

i   h det diag lZq1 ; lZq2 ; /lZqn  J * ¼ 0 where J* ¼

 

vf  vxx* ;

(4)

f ¼ [f1, f2 /, fn]T; ai ¼ ni =mi ; Z is the least

common multiple of the denominators mi of ai ; The greatest common divisor of mi and ni is 1, i.e., (mi ; vi ) ¼ 1; mi ; vi 2Z þ ; i ¼ 1, 2, 3. /n:

2.2. Fractional-order model of HTGS As shown in Fig. 1, the structure of a hydroelectric station consists of a reservoir, a hydro-turbine- generator unit and a penstock [8]. The electric generator is driven by the hydro-turbine based on the joint of shaft coupling. In order to maintain the rotor speed of generator, an oil hydraulic servomechanism controlled by the governor is used to regulate the wicket gates. Therefore, the HTGS can be divided into four parts: electric generator, penstock system, hydro energy system, and hydraulic servo system. The structure diagram of the HTGS is shown in Fig. 2. The mathematical expression of hydro turbine [6] can be

Fig. 1. General physical structure of a hydroelectricity station.

S. Huang et al. / Renewable Energy 139 (2019) 447e458

449

Fig. 2. Functional diagram of HTGS model.

described as follows,



Q ¼ Q ðH; n; YÞ Mt ¼ Mt ðH; n; YÞ

Gt ðsÞ ¼ ey (5)

where Q, Mt, Y, H, and n represent the amount of hydro flow, mechanical torque, guide vane opening, hydro-turbine head and speed, respectively. By employing the first-order Taylor expansion for Eq. (5), with the main servomotor Yk to replace guide vane opening Y, the dynamic characteristics of hydro turbine system can be expressed as follows,



q ¼ eqx x þ eqy y þ eqh h mt ¼ emx x þ emy y þ emh h

(6)

where q, y, h, mt, and x denote the corresponding deviations of Q, Y, vq t H, Mt and n, respectively; eqh ¼ vm ; eqx ¼ vq vx and eqy ¼ vy represent vh

the partial derivatives of water flow with respect to hydro-turbine

t ; head, hydro-turbine speed, and guide vane, respectively; emh ¼ vm vh

emx ¼

vmt vx ,

t and emy ¼ vm vy represent the partial derivatives of me-

chanical torque with respect to hydro-turbine head, hydro-turbine speed and guide vane, respectively. The transfer function of hydro turbine and penstock system [17] can be combined and expressed as follows,

Gt ðsÞ ¼ eqy

Gh ðsÞeh 1 þ eGh ðsÞ þ ey ¼ ey 1  eqh Gh ðsÞ 1  eqh Gh ðsÞ

(7)

where Gh(s) is the transfer function of water hammer; ey is the firstorder partial derivative value of torque with respect to wicket gate; e is the intermediate variable. In addition, Gh (s) and e can be respectively expressed as follows,

1  eTw s 1  eqh Gh ðsÞ

(11)

According to Eqs. (7)e(11), the output torque of HTGS can be obtained as follows [17], 

mt ¼

  eey Tw 1  mt þ ey y  ðu  yÞ eqh Tw Ty

(12)

Based on the dynamics of synchronous generator [19], the mathematical expression of generator rotor function can be obtained as follows,

8 > > < d ¼ u0 u

(13)

 1 > > ½mt  me  F u :u ¼ Tab

where d, u, F, me, and Tab denote the generator rotor angle, the variation of generator rotor speed, the damping factor of generator, the three electromagnetic torque, the inertia time constant of generator, respectively. The electromagnetic torque is equal to the electromagnetic power under the stable operation of HTGS [3], as follows,

me ¼ Pe

(14)

here, the electromagnetic power of generator, Pe, can be formulated as follows [19],

Pe ¼

x0 P  xq P Eq0 Vs Vs2 d sin d þ sin 2 d x0 P 2 x0 P xq P d

(15)

d

0

eqh eh e¼  eqh ey Gh ðsÞ ¼

HA ðsÞ ¼ 2hw thð0:5Tr sÞ QA ðsÞ

(8)

xq P can be respectively described as follows,

(9)

where hw is the characteristic coefficient of penstock; Tr is the phase length of water hammer wave. As the penstock is relatively short in the small and medium-size hydroelectric stations, the pressure penstock system can be described by the rigid water hammer function. Hence, the transfer function in Eq. (9) can be modelled based on the Taylor formula as follows,

Gh ðsÞ ¼ Tw s

where Vs and Eq are the voltage at infinity bus and the transient 0 internal voltage of armature, respectively. In addition, x P and

(10)

where Tw represents the inertia time constant of pressure water diversion system. By taking Eq. (10) into Eq. (7), the following expression can be obtained,

8 1 > 0 0 > < xd P ¼ xd þ xT þ xL 2 > 1 > P : ¼ xq þ xT þ xL xq 2

d

(16)

0

where xq and xd represent the synchronous reactance of quadrature-axis and transient reactance of direct-axis, respectively; xL and xT denote the reactance of transmission line and short-circuit reactance of transformer, respectively. The dynamics of the hydraulic servo system can be expressed as follows [17],

Ty

dy þy¼u dt

(17)

where u represents the output of governor, and Ty is the response

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matrix J* on the equilibrium point can be calculated using Eq. (21), as follows,

time of major engager relay. Because of the important historical dependence of hydraulic servo system, the fractional calculus model is utilized to describe the function with remarkable historical reliance. Inspired from Ref. [19], the dynamic model of fractional-order hydraulic servo system is formulated as follows,

Da y ¼

1 ðu  yÞ Ty

2

0 6  6 E0 V 2 0 V x  xqS s 6 6  q0  s dS 6 x x0dS xqS dS 6 6  J ¼6 6 0 6 6 6 6 4 0

(18)

Based on Eqs. (5)e(18), the variables d, u, mt, y are replaced by x1, x2, x3, x4, and the dynamic model of FOHTGS can be expressed as shown below,

8  > x1 ¼ u0 x2 > > > 0 1 > > 0 P >  xq P > Eq0 Vs >  1 @ Vs2 xd > > sin 2x1 A x3  Fx2  0 P sin x1  > > x2 ¼ T 2 x0d P xq P > xd ab <   >  eey Tw 1 > > > x  x ¼ þ e x þ x y 4 3 3 > > eqh Tw Ty 4 > > > > > > > > Da x ¼  1 ðx  uÞ : 4 Ty 4

2

0 6 1:08 6 ¼4 0 0

314 0:1053 0 0

u0

0

F  Tab

1 Tab

0

1  eqh Tw

0 0:0526 2:5 0

0

0 0 0 7 7 16:5 5 10

l39 þ 10l30 þ 99=38  l29 þ 495=19  l20 þ 161207=475  l19 10 9 þ322414=95  l þ 4239=5  l þ 8478 ¼ 0 (23) Therefore, the roots of Eq. (23) are substituted into Eq. (3) as below,

0:05p  minfjargðli Þjg ¼ 0:1571 > 0 i

3. Dynamic characteristics of FOHTGS

·

·

·

(20)

2 1 1  ¼ 17:72 < 0 ¼0  9 0:5  0:8 0:1 Based on Eq. (20), it is shown that the FOHTGS is dissipative. In order to study the nonlinear characteristics of FOHTGS, the Jacobian matrix of FOHTGS can be achieved from Eq. (19) as shown below,

2

0 6  6 E0 V cosðx Þ 2 0 Vs xdS  xqS cosð2x1 Þ s 6 1 6 q  6 x0dS x0dS xqS 6 6 J¼6 6 0 6 6 6 6 4 0

u0

0

F  Tab

1 Tab

0

1  eqh Tw

0

0

0

3

7 7 7 7 0 7 7 7 eey 7 7 eqh Tw Ty 7 7 7 1 7 5  Ty

For the dynamic model of FOHTGS in Eq. (19), its fractional orders are set as ða1 ; a2 ; a3 ; a4 Þ ¼ ð1; 1; 1; 0:9Þ and the equilibrium point is O ¼ ð0; 0; 0; 0Þ. Thus, the elements in Jacobian

(24)

This implies that, for fractional orders ða1 ; a2 ; a3 ; a4 Þ ¼ ð1; 1; 1; 0:9Þ, the FOHTGS is instable. In order to study the dynamic behaviors of FOHTGS, the state and phase trajectories of FOHTGS are given in Figs. 3e6. As illustrated in Fig. 3(a)e(d), the state trajectories of FOHTGS with the initial conditions [0.01, 0.01 0.01, 0.01] can still recover to a stable state after a long period of convergent oscillations, especially for the generator rotor angle and the variation of generator rotor speed. It can also be found in Fig. 4(a)e(d) that the FOHTGS can converge to the equilibrium point (0, 0, 0, 0). Because the initial states of FOHTGS are not far from the equilibrium point (0, 0, 0, 0), the dynamics of FOHTGS can be depicted accurately after linearization. On the other hand, the dissipation of FOHTGS would be revealed when system states are far from the equilibrium point. Nevertheless, this long period of oscillations in Figs. 3e4 would

For the dissipation property of FOHTGS, the divergence degree of Eq. (19) is dissipative if and only if VV < 0, and it can be described mathematically as follows, ·

7 7 7 7 0 7 7 7 eey 7 7 eqh Tw Ty 7 7 7 1 7 5  Ty

Due to the fractional form (a1 , a2 , a3 , a4 ) ¼ (1, 1, 1, 9/10), Z is selected as 10 from Lemma 2. Z and J* are substituted into Eq. (4), and Eq. (23) can be calculated as,

where the model parameters from Ref. [19] are set as: u0 ¼ 314, 0 0 Eq ¼ 1.35, xqS ¼ 1.474, xdS ¼ 1.15, Tab ¼ 9.0s, F ¼ 2.0, Tw ¼ 0.8s, Vs ¼ 1.0, Ty ¼ 0.1s, eqh ¼ 0.5, u ¼ 0, e ¼ 0.7, ey ¼ 1.0, a ¼ 0.9, respectively.

vx1 vx vx vx F 1 1 þ 2 þ 3 þ 4 ¼0   Tab eqh Tw Ty vx1 vx2 vx3 vx4

3

(22)

(19)

VV ¼

3

0

(21)

cause great threat to the stable operation of hydroelectricity station and smart grids [9]. When the initial conditions of FOHTGS are set as [x1, x2, x3, x4] ¼ [0.1, 0.1 0.1, 0.1], it is shown from Figs. 5(a) and (b) and

S. Huang et al. / Renewable Energy 139 (2019) 447e458

451

Fig. 3. State trajectories of the FOHTGS with the initial conditions. [x1, x2, x3, x4] ¼ [0.01, 0.01 0.01, 0.01].

6(a)e(b) that the generator rotor angle and the rotor speed are gradually moving away from the equilibrium point after a period of small oscillations, and the FOHTGS cannot maintain the system stability. In addition, as shown in Figs. 5(c), (d) and 6(c), (d), the mechanical torque and the guide vane opening can return to the vicinity of the equilibrium point. Consequently, it can be concluded from Figs. 3e6 that the FOHTGS is a highly nonlinear system, and a nonlinear robust controller should be developed to stabilize the FOHTGS under stochastic load disturbances. Moreover, the HTGS would be affected by the random load disturbances [19]. Based on the fractional-order model in Eq. (19), the differential dynamics of FOHTGS with the random load disturbances can be modelled as follows,

8 > > > x1 ¼ u0 x2 þd1 ðtÞ > 0 1 > > 0P P > > Eq0 Vs > 1 @ Vs2 xd xq > > x2 ¼ sin2x1 A þd2 ðtÞ x3 Fx2  0 P sinx1  > 0 > Tab 2 x Px P > xd < q d   >  eey Tw 1 > > > x3 ¼ x3 þey x4 þ x4 þd3 ðtÞ > > eqh Tw Ty > > > > > > 1 > > : Da x4 ¼  ðx4 uÞþd4 ðtÞ Ty (25) The state trajectories of FOHTGS in model Eq. (25) are implemented through numerical calculations and presented in Fig. 7 when d1(t) ¼ rand(1), d2(t) ¼ 0.1rand(1), d3(t) ¼ 0.5rand(1), d4(t) ¼ 0.9rand(1) [17]. It can be found from Fig. 7(a) that the state trajectory of generator rotor angle is fast away from the equilibrium

point. As shown in Fig. 7(b), although the state trajectory of rotor speed variation diverges slowly, it eventually moves away from the equilibrium point. Besides, the state trajectories of mechanical torque and guide vane opening have strong chattering phenomena as confirmed in Fig. 7(c) and (d). The numerical results also indicate that the FOHTGS appears nonlinear and irregular vibrations on the rotor angle, speed, mechanical torque and guide vane opening, which has significant influence on the stable operation of hydroelectricity station.

4. Fixed-time stability and control for FOHTGS In this study, a robust fixed-time sliding mode controller is proposed for the FOHTGS with random disturbances, and the Proof for the fixed-time stability of FOHTGS under the proposed controller is presented. In order to facilitate the mathematical analysis, the unified form of FOHTGS with random disturbances is formulated as,

Da xðtÞ ¼ f ðx; tÞ þ dðtÞ þ uðtÞ where

a ¼

½1;

1;

1; T

½u1 ðtÞ; u2 ðtÞ; u3 ðtÞ; u4 ðtÞ ;

(26) 0:9; x ¼ dðtÞ ¼

½x1 ; x2 ; x3 ; x4 T ;

uðtÞ ¼

½ d1 ðtÞ; d2 ðtÞ; d3 ðtÞ; d4 ðtÞT ,

and f ðx; tÞ ¼ ½f ðx1 ; tÞ; f ðx2 ; tÞ; f ðx3 ; tÞ; f ðx4 ; tÞT denote the state vectors, the controller output, random disturbances and the model function of FOHTGS, respectively; d(t) is considered as jd(t)jx, x > 0. The sliding surface of FOHTGS is presented as,

452

S. Huang et al. / Renewable Energy 139 (2019) 447e458

Fig. 4. Phase trajectories of the FOHTGS with the initial conditions. [x1, x2, x3, x4] ¼ [0.01, 0.01 0.01, 0.01].

 sðtÞ ¼ Da1 x þ D1 k1 x þ k2 jxjs satðxÞ

(27)

where sðtÞ ¼ ½s1 ; s2 ; s3 ; s4 T are the sliding surfaces; k1 ; k2 ; g are the given parameters, with k1 > 0; k2 > 0; 0 < s < 1; satð$Þ is the saturation function which can be defined as,

 satðwÞ ¼

signðw=kÞ; jwj > L w=k; jwj  L

(28)

where L is a given positive constant [18]. The following equation can be obtained when the state trajectories of FOHTGS in Eq. (26) achieve the sliding surface in Eq. (27),

sðtÞ ¼ 0

and

·

sðtÞ ¼ 0

(29)

On the foundation of Eqs. (27) and (29) and Eq. (2) in Property 1, the dynamic characteristics of the sliding surface in Eq. (27) can be formulated, as shown below,

 · sðtÞ ¼ Da x þ k1 x þ k2 jxjs satðxÞ ¼ 0

(30)

Then, the sliding mode system of FOHTGS can be achieved,

 Da x ¼  k1 x þ k2 jxjs satðxÞ

In order to validate the stability of sliding mode dynamic system in Eq. (31) and the effectiveness of the proposed fixed-time sliding mode controller for FOHTGS in Eq. (26), the Lyapunov stability Lemma and fixed-time stability theories are adopted as follows. Lemma 2. [19]: It is assumed that xðtÞ ¼ 0 is the equilibrium point of fractional-order dynamic system D a ¼ f ðx; tÞ, a2ð0; 1; f ðx; tÞ can satisfy the Lipschitz condition. Then, the fractional-order system is asymptotically stable if Lyapunov function V ðt; ðx; tÞÞ can satisfy,

a1 kxkb  Vðt; xðtÞÞ  a2 kxk ·

V ðt; xðtÞÞ  a3 kxk

(32)

where k$krepresents an arbitrary norm; a1, a2, a3 and b denote the specified positive constants. Lemma 3. [25]: It is assumed that x(t)¼0 is the equilibrium point of Da ¼ f ðx; tÞ, a2ð0; 1; the fractional-order system is asymptotically stable if Lyapunov function V ðt; xðtÞÞ and class-K functions mi ð i ¼ 1; 2; 3Þ can satisfy,

m1 kxk  Vðt; xðtÞÞ  m2 kxk C t 0 DVðt; xðtÞÞ  m3 kxk

(33)

(31)

It can be concluded that the sliding surface s(t) of FOHTGS in Eq. (27) would converge to zero once the dynamic system in Eq. (31) is stable.

Definition 2. [26e28]: It is assumed that the system in Eq. (26) can be finite-time stable at the equilibrium point within the bounded convergence time T(x0), the fractional-order system is considered to be

S. Huang et al. / Renewable Energy 139 (2019) 447e458

453

Fig. 5. State trajectories of the FOHTGS with the initial conditions. [x1, x2, x3, x4] ¼ [0.1, 0.1 0.1, 0.1].

fixed-time stable, and the limited time constant Tmax satisfies T(x0)
[29,30]: Considering the system as follows,

·

y ¼ bym=n  gyp=q ; yð0Þ ¼ y0

(34)

where b; g > 0; p, q, m and n all are positive odd integers satisfying pn. The system in Eq. (34) is fixed-time stable at the equilibrium point, and the upper bounded convergence time can be achieved by,

T<

1

n 1 q þ b mn g qp

(35)

can be formulated as,

 Da VðtÞ ¼ signðxÞDa x ¼ signðxÞ  k1 x  k2 jxjs satðxÞ s ¼ k1 jxj  k2 jxj signðxÞ  satðxÞ

Using the satð$Þ function in Eq. (28), the equations can be yielded as,

signðsÞsatðsÞ ¼ signðsÞsignðs=LÞ ¼ 1

ðjsj > LÞ

signðsÞsatðsÞ ¼ signðsÞ  s=L ¼ jsj=L > 0 ðjsj  LÞ

Theorem 1. It is assumed that the sliding surface of the proposed controller is formed as Eq. (27), the sliding mode dynamic system in Eq. (31) would converge to zero. Proof. The Lyapunov function is set as follows,

VðtÞ ¼ jxj

(36)

The fractional-order derivative of Lyapunov function in Eq. (36)

(38) (39)

Based on Eqs. (37)e(39), the following fractional-order equation is obtained,

Da x ¼ k1 jxj  k2 jxjs signðxÞ  satðxÞ  k1 jxj < 0 Remark 1. Compared with the finite-time stability, the fixed-time stability can ensure the system stabilization within a bounded time and would be affected by the influence of initial conditions.

(37)

(40)

The Proof is completed. Based on Lemma 3, it is indicated that the sliding surface in Eq. (27) would converge to 0. According to Theorem 1, it is shown that the designed sliding surface in Eq. (27) is appropriate. A nonlinear controller uðtÞ which can guarantee the x(t) to reach and remain on the sliding surface sðtÞ ¼ 0 is designed as follows: Firstly, the equivalent controller of FOHTGS in Eq. (26) should be designed. Based on Eqs. (26) and (30), the equivalent controller ueq(t) can be formulated as follows,

 ueq ðtÞ ¼  f ðx; tÞ  dðtÞ þ k1 x þ k2 jxjs satðxÞ

(41)

In order to enforce the system to effectively reach the sliding surface, it is necessary to design the reaching controller ur(t). Thus, the switching controller is designed as follows,

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S. Huang et al. / Renewable Energy 139 (2019) 447e458

Fig. 6. Phase trajectories of the FOHTGS with the initial conditions. [x1, x2, x3, x4] ¼ [0.1, 0.1 0.1, 0.1].

derivative is expressed as,

ur ðtÞ ¼ k3 sigðsÞm=n  k4 sigðsÞp=q

(42)

  where k3, k4 > 0; sigð$Þd ¼ $jd sign($).

V ðtÞ  signðsÞs

Based on Eqs. (41) and (42), the sliding-mode control rules are proposed as,

Based on Eq. (30) and Eq. (26), the following equation can be obtained,

 uðtÞ ¼ ueq ðtÞ þ ur ðtÞ ¼  f ðx; tÞ þ k1 x þ k2 jxjs satðxÞ m=n p=q  k4 sigðsÞ dðtÞ  k3 sigðsÞ

·

(43)

Because the random disturbances in Eq. (43) are unknown (i.e. d(t) is unknown), the sliding-mode controller in Eq. (43) is impracticable. Hence, a feasible sliding-mode controller can be devised in Eq. (44),

 uðtÞ ¼ ueq ðtÞ þ ur ðtÞ ¼  f ðx; tÞ þ k1 x þ k2 jxjs satðxÞ x  k3 sigðsÞm=n  k4 sigðsÞp=q

(44)

where jd(t)jx, x > 0. Theorem 2. For the fractional-order model Eq. (26) with the proposed control scheme in Eq. (44), the system state trajectories can converge to the sliding surface sðtÞ ¼ 0 within the fixed time limited by,

1 n 1 q þ T< k3 m  n k4 q  p

(45)

·

·  · V ðtÞ ¼signðsÞs ¼ signðsÞ Da x þ k1 x þ k2 jxjs satðxÞ g ¼ s f ðx; tÞ þ dðtÞ þ uðtÞ þ k1 x þ k2 jxj satðxÞ

(46)

(47)

Based on Eq. (44), there is, ·  V ðtÞ ¼ signðsÞ f ðx; tÞ þ dðtÞ þ uðtÞ þ k1 x þ k2 jxjs satðxÞ  ¼ signðsÞ f ðx; tÞ þ dðtÞ  f ðx; tÞ þ k1 x þ k2 jxjs satðxÞ 

þk1 x þ k2 jxjs satðxÞ  x  k3 sigðsÞm=n  k4 sigðsÞp=q   ¼ signðsÞ dðtÞ  x  k3 s  k3 sigðsÞm=n  k4 sigðsÞp=q    signðsÞ  k3 sigðsÞm=n  k4 sigðsÞp=q þ jdðtÞj  jxj

(48)

Owing to jd(t)jx, one gets,

  · V ðtÞ  signðsÞ  k3 sigðsÞm=n  k4 sigðsÞp=q þ jdðtÞj  jxj    signðsÞ  k3 sigðsÞm=n  k4 sigðsÞp=q (49)

Proof. The Lyapunov function is chosen as VðtÞ ¼ jsj and its time

Due to sign(s)  sign(s)¼1, it obtains,

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455

Fig. 7. State trajectories of FOHTGS with random disturbances.

  · V ðtÞ  signðsÞ  k3 sigðsÞm=n  k4 sigðsÞp=q   ¼ signðsÞ  k3 jsjm=n signðsÞ  k4 jsjp=q signðsÞ

Thus, the state trajectories of FOHTGS with random disturbances jdðtÞj  x would converge to the stable state within a fixed-time as demonstrated in Eq. (51).

(50)

¼ k3 jsjm=n  k4 jsjp=q ¼ k3 V m=n  k4 V p=q

5. Numerical simulations

Therefore, the state trajectories of FOHTGS with random disturbances would converge to the sliding surface s(t)¼0 based on Lemma 2, and the state variable vector x(t) can be asymptotically stable. Based on Lemma 4, the FOHTGS would converge to zero in the fixed-time, and the convergence time of FOHTGS with rodman disturbances is upper limited by,

T<

1 n 1 q þ k3 m  n k4 q  p

In order to verify the effectiveness, robustness and superior performance of the proposed fixed time sliding mode controller, the comparative numerical studies are thoroughly implemented in this section. The FOHTGS with stochastic disturbances is presented as follows,

(51)

8  > > > x1 ¼ u0 x2 þ d1 ðtÞ > 0 1 > > 0 P > 2 x P  xq > Eq0 Vs >  d 1 V > s @x3  Fx2  > x2 ¼ sin 2x1 A þ d2 ðtÞ sin x1  > > > Tab 2 x0d P xq P x0d P <   >  eey Tw 1 > > > x3 ¼  x3 þ ey x4 þ x4 þ d3 ðtÞ > > eqh Tw Ty > > > > > > 1 > > : Da x4 ¼  x4 þ d4 ðtÞ Ty

(52)

456

S. Huang et al. / Renewable Energy 139 (2019) 447e458

Fig. 8. Comparative performances of FOHTGS with the proposed controller and existing FTCMs.

where a ¼ (1, 1, 1, 0.9). d(t)¼(0.1rand(1), 0.1rand(1), 0.1rand(1), 0.1rand(1)).

3 u0 x 2 0 1 7 6 0 P 7 6 1 2 x P  xq Eq0 Vs d V 6 s @x3  Fx2  A7 6 0 P sin x1  0 P P sin 2x1 7 7 6 Tab 2 xd xq xd 7 6 7 6   f ðx; tÞ ¼ 6 7 eey Tw 1 7 6 7 6  x3 þ ey x4 þ x4 7 6 e T T y qh w 7 6 7 6 5 4 1  x4 Ty 2

(53) From Eqs. (27) and (44), the parameters are selected as: k1 ¼ 300; k2 ¼ 300; s ¼ 0:4; k3 ¼ 50; k4 ¼ 20; x1 ¼ 0:1; x2 ¼ 0:1; x3 ¼ 0:1; x4 ¼ 0:1; m ¼ 6; n ¼ 5; p ¼ 9; q ¼ 10. The sliding mode controller with the sliding surface are formulated as follows,

 si ðtÞ ¼ Dai 1 xi þ D1 k1 xi þ k2 jxi js satðxi Þ

(54)

 ui ðtÞ ¼  f ðxi ; tÞ þ k1 xi þ k2 jxi js satðxi Þ  xi  k3 sigðsi Þm=n  k4 sigðsi Þp=q i ¼ 1; 2; 3; 4 (55) With the proposed nonlinear controller and sliding surface in Eqs. (54) and (55), the state trajectories of FOHTGS are numerically calculated and shown in Fig. 8. It can be found that the irregular and nonlinear oscillations of FOHTGS with random disturbances have

been effectively suppressed, while the existing FTCMs in Refs. [19,20] cannot completely eliminate the chattering phenomenon. Furthermore, with the Proof in Eq. (51), the upper bound of the convergence time can be constrained as T < 0.5 s, and the stabilization time of FOHTGS under the proposed controller is limited within 0.2 s in Fig. 8. For the in-depth investigation on the robustness of the proposed fixed-time controller, the stochastic disturbances of FOHTGS are introduced and set as: d(t)¼(rand(1), 0.1rand(1), 0.5rand(1), 0.9rand(1)). In Eqs. (27) and (44), the parameters are selected as: k1 ¼ 300; k2 ¼ 300; s ¼ 0:4; k3 ¼ 50; k4 ¼ 20; x1 ¼ 1; x2 ¼ 0:1; x3 ¼ 0:5; x4 ¼ 0:9; m ¼ 6; n ¼ 5; p ¼ 9; q ¼ 10. Fig. 9 illustrates the state trajectories of FOHTGS under the proposed controller. Compared with the existing FTCMs, it is obvious from Figs. 8 and 9 that the proposed fixed-time control performs the best with higher nadir and convergence rate, and the chattering phenomenon of FOHTGS is significantly weakened under different random disturbances. In order to investigate the effects of initial conditions on the system convergence characteristics under different control methods, Fig. 10 illustrates the settling time of FOHTGS with the increase of initial states under the proposed control method and two exiting FTCMs. It can be found that the convergence time under FTCMs in Refs. [19,20] increases unboundedly with the growth of the initial conditions, while the convergence property under the proposed fixed-time controller is upper restrained and its upper limit of convergence time is 0.118s. It can thus be concluded that the proposed controller can effectively suppress the nonlinear and chaotic vibrations of FOHTGS within the prescribed time, and the convergence time can be exactly estimated and is independent of initial conditions. Moreover, in order to further investigate the robustness of the

S. Huang et al. / Renewable Energy 139 (2019) 447e458

457

Fig. 9. Comparative performances of FOHTGS under the proposed controller and existing FTCMs.

Fig. 10. Effects of logarithmic norm of initial states on system convergence time with a ¼ 0.9.

proposed fixed-time control method on system convergence characteristics with different fractional orders of a, Figs. 11 and 12 show the comparative settling time of two FOHTGS models with a ¼ 0.95 and 0.99 when the initial system state value increases, respectively. From Fig. 10 (a ¼ 0.9), Fig. 11 (a ¼ 0.95) and Fig. 12 (a ¼ 0.99), it can be found that, with the same initial values, the convergence time under FTCMs in Refs. [19,20] increases when the fractional order a increases, and the convergence time would also increase unboundedly with the growth of initial conditions. Contrastively, the convergence property under the proposed fixedtime controller can be upper restrained and its upper limits of the

Fig. 11. Effects of logarithmic norm of initial states on system convergence time with a ¼ 0.95.

convergence time are 0.120s (a ¼ 0.95) and 0.122s (a ¼ 0.99), respectively. It can thus be concluded that the convergence property of the proposed controller can be exactly estimated for the FOHTGS under different fractional order a. 6. Conclusions In this paper, a fractional-order sliding mode controller is proposed for the stabilization and suppression of FOHTGS within the fixed bound time. Comparative simulations on the FOHTGS with the proposed fixed-time controller and the existing FTCMs have

458

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Fig. 12. Effects of logarithmic norm of initial states on system convergence time with a ¼ 0.99.

been implemented to confirm the effectiveness, robustness and superior performance of the developed controller. The conclusions of this investigation are presented as follows: 1) The state and phase trajectories of FOHTGS exhibit the nonlinear and irregular oscillations which will have a great impact on the stable operation of hydropower stations under stochastic load disturbances; 2) The state trajectories of FOHTGS which show that the irregular and nonlinear oscillations of FOHTGS with random disturbances have been effectively suppressed by fixed-time control method, and the fixed-time control performs the best with higher nadir and convergence rate, and the chattering phenomenon of FOHTGS is significantly weakened compared with FTCMs; 3) The upper limit of system convergence time can be exactly evaluated and is independent of initial conditions. Acknowledgment The authors gratefully acknowledge the support of the National Natural Science Foundation of China (51877072, 51507056), the Natural Science Foundation of Hunan Province (2017JJ3019), and the Hunan Strategic Industries Scientific and Technological Project under Grant 2017GK4028. References [1] World energy council. https://www.worldenergy.org/data/resources/ resource/hydropower/. [2] World hydropower statistics. https://www.hydropower.org/worldhydropower-statistics. [3] W.J. Yang, P. Norrlund, L. Saarinen, J.D. Yang, W.C. Guo, W. Zeng, Wear and tear on hydro power turbineseInfluence from primary frequency control, Renew. Energy 87 (2016) 88e95. [4] W.C. Guo, J.D. Yang, Modeling and dynamic response control for primary frequency regulation of hydro-turbine governing system with surge tank, Renew. Energy 121 (2018) 173e187. [5] J. Liu, J. Zuo, Z. Sun, G. Zillante, X. Chen, Sustainability in hydropower development d a case study, Renew. Sustain. Energy Rev. 19 (2013) 230e237. [6] D. Kumar, S.S. Katoch, Environmental sustainability of run of the river hydropower projects: a study from western Himalayan region of India, Renew.

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