Design of fuzzy sliding mode controller for hydraulic turbine regulating system via input state feedback linearization method

Energy 93 (2015) 173e187

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Energy journal homepage: www.elsevier.com/locate/energy

Design of fuzzy sliding mode controller for hydraulic turbine regulating system via input state feedback linearization method Xiaohui Yuan a, Zhihuan Chen a, Yanbin Yuan b, *, Yuehua Huang c a

School of Hydropower and Information Engineering, Huazhong University of Science and Technology, 430074 Wuhan, China School of Resource and Environmental Engineering, Wuhan University of Technology, 430070 Wuhan, China c College of Electrical Engineering and New Energy, China Three Gorges University, 443002 Yichang, China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 February 2015 Received in revised form 31 July 2015 Accepted 7 September 2015 Available online xxx

The HTRS (hydraulic turbine regulating system) plays an important role in hydropower electricity generating and safe operation of water turbine. In this paper, a novel approach to the LFC (load frequency control) is presented for the HTRS system. This approach combines sliding mode control with fuzzy logic control, where the robustness of the controller is guaranteed by a predefined sliding surface and chattering phenomenon is alleviated by the fuzzy logics. The dynamic model of a hydropower plant is developed with the consideration of inner perturbations and external noises of this system. Based on input state feedback linearization method, the relationship between reference output and control output is established. Simulations of an example HTRS system respect to the dynamical behaviors analysis without controller, fixed point stabilization, periodic orbit tracking and robustness test against random noises have been carried out by using the optimal PID (proportionaleintegralederivative) controller, conventional SMC (sliding mode controller) and proposed FSMC (fuzzy sliding mode controller) for evaluating the validity and effectiveness of different controllers. The results indicated that the proposed FSMC controller was excellent from the standpoint of system performance and stability for LFC control of the nonlinear HTRS system with uncertainties. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Hydraulic turbine regulating system Input state feedback linearization Sliding mode control Fuzzy logic control Load frequency control

1. Introduction In recent years, the structure of generation unit systems has been changing dramatically. This is primarily due to the rapid development of some renewable energy sources, such as the hydropower energy, wind energy, etc. which supplement or often replace decommissioned coal-fired power [1]. In order to take full advantage of these sustainable energy sources, the reliability and efficiency of renewable energy generator plants become important topics in researches [2e4]. HTRS (Hydraulic turbine regulating system) is a control system of HGU (hydroelectric generator unit) that governs the rotor speed of water turbine according to the setpoint of output power and setpoint of speed [5]. The main task of HTRS system is to adjust the power output to the grid and to track the frequency of the grid in general. As the HGU typically operates in different status, such as starting, outage and operating in parallel with power grid, and the important parameters in HTRS system are

* Corresponding author. Tel.: þ86 13627242159. E-mail address: [email protected] (Y. Yuan). http://dx.doi.org/10.1016/j.energy.2015.09.025 0360-5442/© 2015 Elsevier Ltd. All rights reserved.

dependent on time and working conditions, it is still a challenge problem to build a suitable prototype model and design proper control rules until now. A lot of research efforts have been devoted to the modeling of HTRS system and its components over the past decades. For instance, a model was studied for a plant with severely leaky wicket gate in 2004 [6]. In 2005, a hill chart look-up table was added to the water turbine output power and another look-up table was added to transform the real gate position to ideal gate position in the equation and the first order filter block was used to include unsteady effects from gate movement [7]. A modification on the Kundur's turbine model was made in Ref. [8], where the novel model is obtained through the frequency response tests. Elastic and non-elastic models [9,10] for conduit system have been investigated in the long pipeline system and short pipeline system, respectively. Chen et al. [11] provided the nonlinear dynamical system model associated with a surge tank. Pennacchi et al. [12] developed water turbine model in accordance with the response of an actual Francis turbine. Various completely HTRS models were available in Refs. [13e15], etc. These models described the characters of complicated HTRS system in different ways and tried to

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approximate the real model exclusively, but it is interesting to note that most of these methods are based on the equilibrium point around stability region and keep a blind eye for the uncertain external noises and inner perturbations. Accompanied with the development of HTRS system modeling, the controllers of the system have also experienced significant improvements. As we known, the power output of hydropower plant will vary with power network demand and produces variations in turbine rotational speed. The rotational speed of the turbine is checked by varying the wicket gate position which controls the flow through the turbine and subsequently torque. In the past, the wicket gate position was usually regulated with the PID (proportionaleintegralederivative) controller due to its simplicity and ease of implementation in practice [16e18]. Integral, proportional and derivative feedback controller is based on the past (I), present (P) and future (D) error. Its use ensures a faster wicket gate position response by providing transient gain reduction/increase. However, the nonlinear nature of water turbine and the constantly varying load makes the gain schedule of PID controller difficult to design, which limits the operating range, thus many advanced control approaches have been developed for the nonlinear HTRS system in the recent years, such as fuzzy control [19], predictive control [20], and self-tuned control [21]. Unfortunately, most of the above controller designs were based on linear model and do not guarantee the robustness to the system perturbations due to the wear and tear. In this regard, SMC (sliding mode controller) is introduced into the controller design of HTRS system. SMC controller is a quick and powerful variable structure approach that governing the nonlinear systems via a predefined switch sliding surface. There are numerous applications of SMC controller used in industries and the main advantage of this type of controller is its strong robustness against model uncertainties and parameter variations [22,23]. With the development of SMC techniques in practice, there gradually present the applications of SMC controller used in HTRS system [24e26]. However, a common drawback of conventional SMC controllers is the chattering phenomenon, due to the use of discontinuous sign function. Numerous techniques have been proposed to minimize chattering effect, such as selecting smooth approximation like sat, tanh and other continuous control laws to instead the sign function [22e24]. One of the most commonly used solutions is to combine FLC (fuzzy logic control) into SMC control [27,28], which overcomes the weakness of the sign function by the fuzzy logic so that the badly chattering phenomenon can be easily eliminated. Motivated by the above discussions, a novel mathematical model of HTRS system with the considerations of inner perturbations and external noises is presented, and the input state feedback linearization method is used to construct the relationship of reference output and control output. To ensure the stability of this model, an ingenious controller combining fuzzy logic control and sliding mode control is proposed, where the robust behavior of HTRS system with controller is guaranteed, and the chattering phenomenon can be attenuated. Compared with PID controller and a common used SMC controller for an example HTRS system simulation, the designed FSMC (fuzzy sliding mode controller) controller has a desirable response output. Furthermore, the dynamical analysis is introduced to study the stability of HTRS system, such as bifurcation maps, Lyapunov exponents, phase diagrams, power spectrums, etc. Many interesting phenomena have been observed. The rest of the paper is organized as follows. Section 2 briefly introduces the modeling of HTRS system with uncertainties. In Section 3, input state feedback linearization method of HTRS system, FSMC controller design, and the robustness analysis are discussed in details, where the application of FSMC controller in HTRS

system is formulated and constructed in Section 4. The comparative simulations are designed and the results are discussed in Section 5. The conclusions are summarized in Section 6 and acknowledgment is given in the end. 2. Model of HTRS system A simple scheme of hydropower station plant is shown in Fig. 1. The entire system model can be constructed by combining individual dynamic models of conduit penstock, electric-hydraulic servo system, water turbine and generator set [5,18]. As showed in Fig. 1 for a hydropower station system, running water from the reservoir flows into the scroll casing after passing through diversion system, and promotes the turbine to rotate. Then the power generator is driven for the joint of the shaft coupling. In order to keep the frequency, wicket gate positions are regulated by the oil-servo controlled by the speed governor. In this section, a kind of affine model of HTRS system is studied and sections of the model are illuminated, respectively. 2.1. Conduit penstock model According to the hydromechanics theory, the head and flow equation between two sections of conduit system can be deduced as [11]:




Hp ðsÞ Qp ðsÞ




¼

coshðrDxÞ sinhðrDxÞ=zc

zc sinhðrDxÞ coshðrDxÞ




Hq ðsÞ Qq ðsÞ

 (1)

where subscripts q and p are the symbols of upstream and downstream section of penstock in Fig. 1, respectively. Dx ¼ L means the penstock length, r and zcp are the composite equations of parameffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ters of the penstock. r ¼ MCs2 þ RCs, zc ¼ r=ðCsÞ. M; R; C among the equation can be written as:

M¼

Q0 ; gAH0

C¼

gAH0 ; v2 Q0

R¼

fQ02 gDA2 H0

Fig. 1. A simple scheme of hydropower station plant.

(2)

X. Yuan et al. / Energy 93 (2015) 173e187

where A; f ; D are parameters of the penstock. They represent area, head loss and diameter of the conduit system, respectively; H0 and Q0 are hydraulic head and turbine flow in penstock; v and g denote the wave pressure velocity and gravitational acceleration constant. Assuming that the water head loss is negligible, r and zc can be written as follows:

r¼

1 vQ0 s; zc ¼ 2hw ; hw ¼ v 2gAH0

(3)

With the hydraulic friction losses being trivial and Hq ðsÞ ¼ 0 (tunnel connects with the reservoir directly), the head and flow function is simplified as:

where em ¼ eqy eh =ey  eqh is named the turbine performance parameter, Gh ðsÞ is transfer function of conduit penstock system. The results can be written in state-space form as:

x_1 ¼ x2 ; x_2 ¼ x3 ; x_3 ¼ a0 x1  a1 x2  a2 x3 þ y

(10)

where x1 , x2 and x3 are intermediate variables, and the turbine torque output mt is deduced as:

mt ¼ b3 y þ ðb0  a0 b3 Þx1 þ ðb1  a1 b3 Þx2 þ ðb2  a2 b3 Þx3 (11)

Hp ðsÞ sinhðrDxÞ ¼ zc tanhðrDxÞ ¼ zc coshðrDxÞ Qp ðsÞ

(4)

It can be seen from Eq. (4) that the water hammer transfer function is a nonlinear hyperbolic tangent function, which was inconvenient to use and should be expanded by series. Substituting (3) into (4), the equation for the water strike transfer function Gh ðsÞ at point p is obtained as:

Pn

i¼0

Gh ðsÞ ¼ 2hw tanh ð0:5Tr sÞ ¼ 2hw P n

2iþ1

ð0:5Tr sÞ ð2iþ1Þ!

i¼0

(5)

ð0:5Tr sÞ2i ð2iÞ!

2L where Tr ¼ 2Dx v ¼ v is water hammer time constant. Typically, there are two models, n ¼ 0; 1, as:

Gh ðsÞ ¼ Tw s

175

ðn ¼ 0Þ

(6)

a0 ¼ e

where b2 ¼ e

3ey qh hw Tr

24

3 qh hw Tr

; a1 ¼ 24 ;a ¼ e T2 2 r

3

qh hw Tr

24ey 3 qh hw Tr

; b0 ¼ e

24em ey ; eqh Tr2

; b1 ¼ 

e e

; b3 ¼  emqhy .

2.3. Generator model In this paper, assuming the generator connecting to an infinite bus through a transmission line, then the model of generator is described by a third-order differential equation set as [29]:

8 _ > < d ¼ xt ðmt  me  Dxt Þ=Ta x_t ¼    . > : E_ 0 ¼ E  E0  z  z0 *i Td f d d q d

(12)

where:

Gh ðsÞ ¼

1 T 3 s3 þ 1 T s 2 r 2hw 481 r 2 2 8Tr s þ 1

ðn ¼ 1Þ

(7)

where Tw ¼ hw Tr ¼ ðQ0 LÞ=ðgH0 AÞ, Eqs. (6) and (7) represents two common used kinds of water hammer models: the first model is called rigid water hammer model and the second model is called elastic water hammer model that is employed in this paper.

2.2. Water turbine model with elastic water hammer For a small perturbation around the rated point, the equation of turbine is shown as [7,25]:



mt ¼ ex xt þ ey y þ eh h q ¼ eqx xt þ eqy y þ eqh h

(8)

where mt and q represent turbine torque and turbine flow, six transfer constants of water turbine eh ¼ vmt =vh; ex ¼ vmt =vx; ey ¼ vmt =vy; eqh ¼ vq=vh; eqx ¼ vq=vx; eqy ¼ vq=vy are the partial derivatives of turbine torque and flow respect to turbine speed xt (it equals to the generator speed once water turbine is incorporated into grid), guide vane opening y and head h. As the hydro unit speed changes little, the speed deviation Dxt ¼ 0, Then the transfer function of the turbine and conduit penstock is:

Gt ðsÞ ¼ ey

Tr

Vs 0 Vs2 > > > : me ¼ pe yz0 Eq sin d þ 2 d

! 1 1  sinð2dÞ zq z0d

(13)

In this study model, d is the rotor angle; me is output electrical torque (external load), D is the damper coefficient, Ta is the generator mechanical time constant, Ef is field voltage, Eq0 is the interval voltage of armature, zd is the direct axis reactance, z0d is the direct axis transient reactance, Td is the field winding time constant, id is the direct axis currents, pe is generator electromagnetic power, Vs is voltage of the infinite bus, zq is the quadrature axis reactance. 2.4. Electric-hydraulic servo system model The electric-hydraulic servomotor, as the actuator of turbine, is used to amplify the control signal and provide enough power to operate the guide vane [30]. Its transfer function is:

y_ ¼ ðu  yÞ Ty

(14)

where Ty is the servo time constant, u is the controller output. 2.5. Integrated HTRS model

24em 3 3 2 24 ey em s  hw Tr s þ Tr2 s  hw Tr3 1 þ em Gh ðsÞ ¼¼  , 1  eqh Gh ðsÞ eqh s3 þ 3 s2 þ 242 s þ 24 3 eqh hw Tr

 . 8 > id ¼ Eq0  Vs cos d z0d > > <

eqh hw Tr

(9)

According to the above analysis and with the assumption that hydraulic friction losses are trivial and the speed deviation is negligible, the IMWU (integrated model with uncertainties) is written as:

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8_ x1 ¼ x2 þ d1 > > > > > > x_2 ¼ x3 þ d2 > > > > > > x_3 ¼ a0 x1  a1 x2  a2 x3 þ y þ d3 > > > > > 1 > > > y_ ¼ ðu  yÞ þ d4 > > T > y > > < _ d ¼ x t þ d5 IMWU : > " > > > 1 Vs V2 > > > x_t ¼ b3 y þ ðb0  a0 b3 Þx1 þ ðb1  a1 b3 Þx2 þ ðb2  a2 b3 Þx3  0 Eq0 sin d  s > > T 2 z a > d > > > " # > > >   Eq0  Vs cos d > 1 > _0 > Ef  Eq0  zd  z0d * þ d7 Eq ¼ > > > Td z0d > > > : Y ¼ xt

where d1  d7 are the bounded uncertainties (includes modeling errors caused by the assumption, un-modeled nonlinearities, inner perturbations, and external noises, etc.), Y is turbine rotational speed, which is also term with system output.

FðxÞ ¼

DðtÞ ¼

!

#

(15)

1 1  sinð2dÞ  Dxt þ d6 zq z0d

€ ¼ b3 u þ FðxÞ þ DðtÞ Y Ta Ty

(16)

where:

! # ! ! " b2  a2 b3 Db3 b DVs2 1 1 Vs sin d zd 0 zd  2  3 yþ   E   1 V cos d sinð2dÞ  E s f 0 Ta Ta Ty z0d q z0d Ta Td z0d Ta 2Ta2 zq zd " " # # DEq0 Vs a0 ða2 b3  b2 Þ Dða0 b3  b0 Þ b0  a0 b3  a1 ðb2  a2 b3 Þ Dðb1  a1 b3 Þ þ 2 0 sin d þ þ  x1 þ x2 Ta Ta Ta2 Ta2 Ta zd 3 2 " #  0  0 2 2 V E cos d V cosð2dÞ z  z b1  a1 b3  a2 ðb2  a2 b3 Þ D ðb2  a2 b3 Þ D s q q d 5xt þ   2þ x3  4 s Ta xq z0d Ta z0d Ta Ta2

(17)

b0  a 0 b3 b  a1 b3 b  a2 b3 b d1 þ 1 d2 þ 2 d3 þ 3 d4 Ta Ta Ta Ta 0 1   Vs Eq0 cos d Vs2 cosð2dÞ z0d  zq Ad5  D d6  Vs sin d d7 þ d_ 6 @ þ Ta Ta zq z0d Ta z0d Ta z0d

3. Design of a fuzzy sliding mode controller for the uncertain HTRS system In this section, the synthesis of FSMC controller based on the input state feedback linearization for the uncertain HTRS system is presented. 3.1. The input state feedback linearization of HTRS system The main control task of HTRS system is to ensure the speed output of Y (i.e. xt ) is similar to the desired trajectory output Yd (i.e. xtd ) as much as possible. In IMWU model (i.e. system (15)), it is seen that there is no direct relationship between the system output Y and controller output u, thus the input state feedback linearization method is adopted to construct the linear model of HTRS system. In accordance with the IMWU model, differentiate Y_ respect to the time, there gets:

3.2. Fuzzy sliding mode controller design for the HTRS system Sliding mode control is one of the effective nonlinear robust control approaches, due to its high robustness and availability. However, the use of sign function in the control input signal gives rise to a chattering effect. The finite time delays and limitations of practical control systems hinder the implementation of such problematic control signals into the real-world systems. Nevertheless, the problem is resolved by using the fuzzy logic scheme [27]. Added with a proper expert experience, the fuzzy logic scheme can come up with an appropriate reaching law for the HTRS system. 3.2.1. Configuration of SMC controller The fundamental concepts of sliding mode control theory can be found in Refs. [31,32]. Generally, there are two steps for sliding mode control design as follows.

X. Yuan et al. / Energy 93 (2015) 173e187

177

First step is to determine the sliding mode hyper-plane. Taking into account the control objective of HTRS system is to force the compensating speed output xt to track the reference value xtd , we design a sliding mode manifold with partial actions as:

s ¼ le þ e_

e ¼ xtd  xt ;

(18)

and a common used switching item is defined a function of sliding mode hyper-plane as:

usw ¼

Ta Ty hsgnðsÞ b3

(19)

where l and h are two constants, the value of h is bigger than the upper bound of DðtÞ, (i.e. h > Dr ; Dr ¼ max DðtÞ), the definition of t: 0/∞ sgnðsÞ is described as:

8 <1 sgnðsÞ ¼ 0 : 1

s>0 s¼0 s<0

(20)

Second step is to determine approaching control law. Differentiate the sliding mode manifold s with respect to time, there have:

s_ ¼ € e þ le_

(21) If ðs is NZÞ;

Substituting Eq. (18) to Eq. (21), it obtains:

s_ ¼ € xtd 

b3 u  FðxÞ  DðtÞ þ le_ ¼ 0 Ta Ty

(22)

Then the equivalent control ueq is deduced as:

ueq

Ta Ty ¼ ½€ x þ le_  FðxÞ b3 td

Ta Ty ¼ ½€ x þ le_  FðxÞ þ hsgnðsÞ b3 td

(23)

(24)

where ueq drive the system toward a specified sliding mode surface (approaching phase) and usw maintain the system's motion in the neighborhood of this surface(sliding mode phase).

3.2.2. Fuzzy sliding mode controller design The robustness of SMC controller is guaranteed by the switching control term usw , but the use of usw is easy to cause the chattering phenomenon of the controlled system (see Fig. 2). In Fig. 2, it is seen that while the system is approaching the sliding mode surface, as the function of discontinuous control term usw , it will across this surface and arrive at the region that s < 0, which makes the control output u changes accordingly, and then the system is go back to s ¼ 0 with the force of ueq , and across the surface again. This movement rolls in cycles and formulates the chattering effect, which is the main shortcoming of conventional SMC controller. In order to weaken this badly chattering phenomenon, usw should be smaller with fuzzy rules with satisfying the robust requirement condition of the controlled system while it across the sliding mode surface. On the basis of previous descriptions, it shows that the switching control item usw compels the state trajectory toward this surface, thus it is reasonable to give the fuzzy logic rules as follows:

If ðs is ZÞ;

  then u is ueq ;

  then u is ueq þ usw ;

(25)

(26)

where fuzzy sets Z and NZ denote zero and nonzero, and the logic rule laws of fuzzy sling mode controller is deduced as:

u ¼ ueq þ m,usw

and the resulting sliding mode control law is valued as:

u ¼ ueq þ usw

Fig. 2. The sliding mode orbits of SMC controller.

(27)

where m is the fuzzy adjustment parameter, and the fuzzy idea of it is given as [28]:

If ðs is NÞ then ðm is PÞ; If ðs is ZÞ then ðm is ZÞ; If ðs is PÞ then ðm is PÞ;

(28)

where N and P denote negative and positive fuzzy set. The membership functions of N and P for Eq. (28) should be selected overlap, to ensure the available of defuzzification. In the case of m ¼ 0, there gets u ¼ ueq , which means the orbits of the system is on the sliding mode surface.

3.3. Robustness analysis of HTRS system with FSMC controller For the stability and robustness analysis of system external dynamics of xt , we define a sliding mode surface as below:

s ¼ le þ e_ ¼ 0

(29)

Here, we select a Lyapunov function as [31]:

1 VðxÞ ¼ sT s 2

(30)

whose time-derivative is:

_ _ ¼ sð€ xt þ leÞ V_ ¼ ss_ ¼ sð€ e þ leÞ xtd  €  b ¼s € xtd  FðxÞ  3 u  D þ le_ Ta Ty Substituting (40) into (43), we can get:

(31)

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X. Yuan et al. / Energy 93 (2015) 173e187

_ xtd þ le_  FðxÞ þ h*sgnðsÞÞ  D þ leÞ V_ ¼ sð€ xtd  FðxÞ  ð€











¼ sð  mhsgnðsÞ  DÞ ¼ Ds  mh s  D s  mh s



¼ ðD  mhÞ s

(32)

As the stability of fuzzy control system is guaranteed, which means mðtÞhðtÞ  DðtÞ > 0, thus it can be deduced that V_  0, which implies the attractiveness of sliding surface (30). Moreover, since lim ss_ ¼ 0, the surface is invariant, when the sliding mode occurs s/0 on the surface, then s ¼ s_ ¼ 0, thus the tracking behavior of the system external dynamics is governed by the following equations:

x_td  x_t ¼ lðxtd  xt Þ

(33)

Because l is a positive constant, xt exponentially converges to xtd , which is proved that FSMC controller can control the system to any point or periodic orbits with proper parameter selection. 4. Application of FSMC controller in the HTRS system

At present, the parallel PID controller and its variants are still the most often used control law in HTRS system. The equation of PID controller is shown as follows:

u ¼ kp*e þ ki*

e dt þ kd*e_

u ¼ ueq þ m,usw ¼

Ta Ty ð€ xtd þ ce_  FðxÞ þ mhsgnðsÞÞ b3

where m ¼ 1, FSMC controller is degraded into a conventional SMC controller. The structure of HTRS system with designed SMC/FSMC controller is depicted in Fig. 5. The analysis of complex dynamics in HTRS system without controller, fixed point stabilization, periodic orbit tracking and robustness against random noises experiment simulations are conducted respectively, to verify the validity and effectiveness of the proposed FSMC strategy throughout the later simulation results and discussions.

(34)

To verify the effectiveness and validity of the proposed FSMC controller for the nonlinear HTRS system, HTRS system (i.e. IMWU model) is simulated under MATLAB/SIMULINK environment. The parameters selected for this system are listed as:

Hydraulic system parameters : hw ¼ 0:64; em ¼ 0:7; ey ¼ 1:0; eqh ¼ 0:5; Ty ¼ 0:1; Tr ¼ 0:2;

0

where kp, ki, kd are the proportional gain, integral gain and derivative gain of PID controller, respectively. The performance of PID controllers depends on the values of the controller parameters. Conventionally, the designer, who attempts to find an excellent controller solution, manually tunes these parameters, which depends on the experience of the designer. If the designer is not experienced this process can become tedious and time consuming. Besides that, there is no guarantee that the designed solution will perform satisfactorily as the tune process depends on the qualitative judgment of the designer. A solution to this problem is to use optimization techniques that tune such parameters automatically, such as genetic algorithm [33], gravitational search algorithm [34,35], etc. In Ref. [17], there adopts an improved PSO (particle swarm optimization) to tune the gains of PID controllers for the linear HTRS system and uses the ITAE (integral of time-multiplied absolute value of error) index to evaluate the optimal response performance in a finite time T, which is defined as:

ZT ITAE ¼

tjeðtÞjdt

(36)

5. Simulations and discussions

4.1. Overview of PID controllers used in the HTRS system

Zt

functions of set N, Z and P respect to the fuzzy input s and output m are depicted in Fig. 4. In accordance with proposed fuzzy sliding mode controller design scheme, the FSMC controller design based on the HTRS system is shown as:

(35)

Generator system parameters : zd ¼ 0:95; z0d ¼ 1:8; D ¼ 2; Vs ¼ 1:0; Ef ¼ 1:83; Ta ¼ 9:0; The relative error eðtÞ ¼ xtd ðtÞ  xt ðtÞ is defined to evaluate the tracking performance between the compensating speed xt and its reference output xtd . Without loss of generality, the uncertainties and the initial state of IMWU model is selected as follows:

Uncertainties : d1 ¼ 0:02sinðtÞ;

d2 ¼ 0:01sinð0:5tÞ;

d3 ¼ 0:02sinð0:5tÞ; d4 ¼ 0:01sinðtÞ; d5 ¼ 0:02cosðtÞ; d6 ¼ 0:01cosð0:5tÞ; d7 ¼ 0:01sinðtÞ; Initial state : x1 ð0Þ ¼ x2 ð0Þ ¼ x3 ð0Þ ¼ yð0Þ ¼ dð0Þ ¼ xt ð0Þ ¼ Eq0 ð0Þ ¼ 0:01 The model with respect to the HTRS system with FSMC controller has been illustrated in Fig. 5 and the performance of proposed FSMC controller will be analyzed in the following cases.

0

In this paper, we adopt the same way to seek for the best gains of PID controller in the nonlinear HTRS system. The diagram of the HTRS system with PID controller is shown in Fig. 3. The detailed operators of PSO algorithm can be found in Ref. [17], and the structure and robustness analysis of PID controller used in the HTRS system has been verified in Refs. [11,18]. 4.2. The strategy of FSMC controller used in the HTRS system Similar to the current used PID controllers, the proposed FSMC/ SMC controller is also equipped in electric-hydraulic servo system. In accordance with an intensive preliminary test phase, the

5.1. The analysis of complex dynamics in HTRS system without controller When the HTRS system operates in rated condition, a small disturbance, such as the fluctuation of power grid system frequency, the uncertain load disturbance, the changed hammer pressure of the penstock, etc., will break the stability of entire control system, and a new operating state will be reached. Because of the nonlinearity of every part, there will be a transient complex process. In this section, we will investigate the dynamical behaviors of uncertain HTRS system (i.e. IMWU model) along with the changed hammer pressure of penstock system without controller.

X. Yuan et al. / Energy 93 (2015) 173e187

179

Fig. 3. The scheme of PID controller used in HTRS system tuning by PSO algorithm.

5.1.1. Bifurcation map The Bifurcation map is used to describe and analyze the complex dynamical characteristics of nonlinear systems as the system parameter varies. Bifurcation is the main route to the chaos from a stable state [22]. Apparently, it is known that as the water wave hammer time constant Tr in HTRS system is always changed with the frequent load fluctuation in grid, the running state of the system changes accordingly, which demonstrates that the system is disturbed by the load disturbance. To be concrete, we study the bifurcation diagram by plotting the maxima of coordinate xt with water hammer wave time constant Tr varying as shown in Fig. 6. The bifurcation control parameter is monitored in the range ½0:2; 2:6. From Fig. 6, it is seen that there is one important point (Tr ¼ 1:72), for the range 0:32 < Tr < 1:72, the orbit is intensive and

convergent; for the range 1:72 < Tr < 2:6, the orbit is moving along with nonlinear, projectable trajectory, which can preliminarily infer that the HTRS system may have a complicated dynamic character of chaos. In order to verify this assumption, Lyapunov exponents, phase diagrams and power spectrum are investigated in further. 5.1.2. Lyapunov exponents, phase diagrams and power spectrum In this part, various numerical results of phase diagrams, time wave plots, Lyapunov exponents and power spectrum were obtained, which further confirm the occurrence of chaos route depicted previously from Fig. 7. The LE (Lyapunov exponent), which determines the notion of predictability, is one of the most reliable quantitative indexes to justify whether there is chaotic behavior motion for the system. For a common chaotic system,

180

X. Yuan et al. / Energy 93 (2015) 173e187

Fig. 4. The membership function of fuzzy logic rule laws.

Fig. 5. The scheme of proposed SMC/FSMC controller used in HTRS system.

X. Yuan et al. / Energy 93 (2015) 173e187

181

25

20

x

t

15

10

5

0 0

0.5

1

Tr

1.5

2

2.5

(a) Bifurcation diagram of HTRS system with (0.2
9

4

8 7

3.5

6

3

5 x

t

xt

2.5

4

2 3

1.5

2

1

1

0.5 0 0.2

0

0.4

0.6

0.8

1 Tr

1.2

1.4

1.6

1.8

(b) Local amplification of bifurcation map with (0.32
-1 1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

Tr

(c) Local amplification of bifurcation map with (1.72
Fig. 6. The bifurcation map for HTRS system with Tr varying from 0.2 to 2.6.

there is just one positive LE value whereas a hyper-chaotic system possesses more than one positive LE [36]. In addition, a broadband noise-like power spectrum is the signature of a chaotic movement. From Figs. 6 and 7, it is known that the nonlinear dynamical behaviors of HTRS system are considerable complicated, and the system goes through a series of operating states. When Tr ¼ 0:8 and Tr ¼ 1:6, the system is in chaotic state, and when Tr ¼ 2:4, the system is in hyper-chaotic state (mainly owe to the blowout bifurcation movement of chaotic attractor [37]). As the hyperchaotic systems are characterized with more than one positive LE value, the dynamics of HTRS system has a more complicated chaotic attractor and expanded in multi-directions. Due to the existence of chaotic dynamics, when the HTRS operates on the chaotic condition, the occurrence of irregular oscillations is harmful to the stability and security of the whole system in the sense that the chaos is sensitive to the initial values and fractal dimensions. Consequently, it is necessary to design a suitable controller that makes the chaotic system or hyper-chaotic system back to a steady state, in order to ensure the system stability.

5.2. The analysis of complex dynamics in HTRS system with FSMC controller In this section, we will investigate the dynamical analysis results of HTRS system with FSMC controller. As the HTRS system has been proved to own complicated dynamic orbit, where chaotic behaviors and hyper-chaotic behaviors appear alternately while the parameter Tr changes on the basis of previous simulations, Tr ¼ 0:8, Tr ¼ 1:6 and Tr ¼ 2:4 are chosen as the representatives to testify the performance of the nonlinear HTRS system. In order to illustrate the effectiveness of the proposed controller, PID controller and conventional SMC controller are also employed to adjust the orbit of turbine speed relative deviation xt . The optimal parameter gains of PID controller with respect to the HTRS system are shown in Table 1 through the above-mentioned PSO approach and the obtained performance evaluations comparison is listed in Table 2. 5.2.1. Fixed point stabilization The HTRS system can be stabilized to arbitrary fixed points with proposed FSMC controller. Assume that the fixed point is chosen as xtd ¼ 1ðtÞ for a case, FSMC controller, SMC controller and PID

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l

l Fig. 7. The phase diagrams, time wave plots, Lyapunov exponents and power spectrum of the HTRS system with different Tr values.

X. Yuan et al. / Energy 93 (2015) 173e187 Table 1 The obtained optimal gains of PID controller throughout PSO algorithm. Tr

The gains of PID controller kp

ki

kd

(a) The optimal gains of PID controller under fixed point stabilization 6.37562 3.13609 4.89912 Fixed point stabilization Tr ¼ 0:8 Tr ¼ 1:6 23.1529 2.55953 4.09104 Tr ¼ 2:4 18.6444 7.55206 5.74746 (b) The optimal gains of PID controller under period orbit tracking Periodic orbit tracking Tr ¼ 0:8 0.2976 1.8922 8.0590 Tr ¼ 1:6 5.0806 0 9.9511 Tr ¼ 2:4 5.7060 5.1251 8.8288 (c) The optimal gains of PID controller under random noise disturbances Random noise disturbances Tr ¼ 0:8 8.3643 9.1107 8.4357 11.7115 3.0285 9.9759 Tr ¼ 1:6 Tr ¼ 2:4 6.6370 5.4548 5.6910

183

parameter gains of PID controller are shown in Table 1(b), and the parameters of SMC controller and FSMC controller keep unchanged. The tracking results are presented in Fig. 9 and Table 2(b). Where three commonly used error performance indicators, i.e. IAE (integral of absolute error), ISE (integral of squared error), and ITAE, are considered. The definition of ITAE has given in Eq. (35), and other two error performance indicators in a finite time T are defined as:

ZT IAE ¼

jeðtÞjdt

(37)

e2 ðtÞdt

(38)

0

ZT ISE ¼ 0

controller are employed to adjust the response of uncertain HTRS system. The parameter gains of PID controller are listed in Table 1(a), and the parameters of SMC controller and FSMC controller are set as: l ¼ 80; h ¼ 100, where h satisfied the condition of Eq. (19). Fig. 8 shows the tracking results of this system with different controllers and three correspondence performance indicators are listed in Table 2(a). From Fig. 8 and Table 2(a), it is seen that compared with the optimal PID controller, SMC and FSMC controller achieve a better performance accuracy, which reflects the superiority of sliding mode technology for the uncertain HTRS system. Although SMC controller has a shorter rise time than the FSMC controller, there are many hackly surfaces on the orbit, which are termed chattering phenomenon. This phenomenon has a badly effect on the control accuracy of the nonlinear HTRS system, which breaks up the equilibriums of this system, and increases the fuel cost. On the other hand, FSMC controller can greatly weaken the chattering phenomenon with the satisfactions of the controller requirement, which drives the system to the fixed point smoothly. 5.2.2. Periodic orbit tracking The HTRS system can be stabilized to arbitrary periodic orbits with proposed FSMC controller. Here, take xtd ¼ sinðtÞ as an example. FSMC, SMC and PID controllers are employed to adjust the turbine speed orbit of the HTRS system as comparison. The

From Fig. 9 and Table 2(b), it is obvious that after a relative short transient period, the source speed output xt is almost sinusoidal with its reference output xtd by SMC and FSMC controller, and the tracking result by PID controller presents “out of synchronous” and “delay ” characters, which means there has a better dynamic response of HTRS system with SMC and FSMC controller under periodic orbit tracking. When compared three error performance indicators, computational results show that the response of the FSMC controller is better than that by the PID controller and SMC controller, which proves the superiority of proposed FSMC controller throughout blurring the gain of switching control item (i.e. usw ) while it across sliding mode surface.

5.2.3. Robustness against random noises A controller can be designed to achieve the favorable performance by possibly canceling all the system perturbations only when the exact model and external disturbance of the system which is to be controlled is known [38]. However, as the complex motion of the HTRS system, it is difficult to model the system exactly without unknown characters over a wide range of the operating status, thus in this part of simulations, random noises are chosen as a representative random impact adding to the IMWU model, The uncertainties added to this system is shown as follows, and the simulations with the unknown factors are activated at 0 s:

Table 2 The comparison of PID controller, SMC controller and FSMC controller. (a) The performance of three controllers under fixed point stabilization Controllers Tr ¼ 0:8 Tr ¼ 1:6 RT ST OP RT PID 1.1583 >6 33.1795 1.6379 SMC 0.6327 0.7829 1.2591 0.6334 FSMC 1.0230 1.2591 0.1506 1.0265 (b) The performance of three controllers under periodic orbit tracking Controllers Tr ¼ 0:8 Tr ¼ 1:6 IAE ITAE ISE IAE PID 1545.5 4223.6 1358.9 3023.4 SMC 2.9622 8.8821 0.0065 2.6980 FSMC 2.5604 7.6088 0.0048 2.3041 (c) The performance of three controllers under random noise disturbances Controllers Tr ¼ 0:8 Tr ¼ 1:6 RT ST OP RT PID 1.1072 >6 40.650 2.1069 SMC 0.6319 0.7832 0.1928 0.6334 FSMC 1.0221 1.2576 0.1673 1.0265

ST 5.9437 0.7869 1.2678

OP 45.1281 0.1794 0.1469

Tr ¼ 2:4 RT 1.7441 0.6335 1.0271

ST >6 0.7876 1.2693

OP 55.0172 0.1794 0.1449

ITAE 8726.1 8.3713 7.0595

ISE 5074.0 0.0056 0.0040

Tr ¼ 2:4 IAE 3913.2 2.5128 2.1498

ITAE 12,248.2 8.1255 6.8638

ISE 9980.6 0.0053 0.0037

ST >6 0.7870 1.2681

OP 9.5676 0.1918 0.1634

Tr ¼ 2:4 RT 3.8202 0.6335 1.0274

ST >6 0.7876 1.2695

OP 18.339 0.1978 0.1503

Abbreviation: RT: rise time (s); ST: settle time (s); OP: overshoot percentage (%); IAE: integral of absolute error; ITAE: integral of time-multiplied absolute error; ISE: integral of squared error. The bold values are the best values obtained among the different controllers.

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Fig. 8. Comparison of PID, SMC and FSMC controllers under fixed point stabilization.

d1 ¼ 0:02sinðtÞ þ r1 ;

d2 ¼ 0:01sinð0:5tÞ þ r2 ;

d3 ¼ 0:02sinð0:5tÞ þ r3 ; d4 ¼ 0:01sinðtÞ þ r4 ; d5 ¼ 0:02cosðtÞ þ r5 ; d6 ¼ 0:01cosð0:5tÞ þ r6 ; d7 ¼ 0:01sinðtÞ þ r7 ; where: r1  r7 is bounded in a finite range ½0:01; 0:01. Similarity to the front simulation experiments, FSMC, SMC and PID controllers are employed to adjust the response of the HTRS system, The reference output is valued as: xtd ¼ 1ðtÞ. The gains of

PID controller are presented in Table 1(c) and the parameters of SMC and FSMC controller are not changed. The obtained turbine speed output xt is presented in Fig. 10 and Table 2(c) lists the performance indicators of three controllers. From Fig. 10 and Table 2(c), it is clear that FSMC controller performs better and is more capable to force the compensating speed to track its reference promptly with little chattering phenomenon when the random noise interrupts, which means the robustness of FSMC controller is stronger than that by SMC and PID controller for the chaotic and hyper-chaotic HTRS system.

X. Yuan et al. / Energy 93 (2015) 173e187

185

Fig. 9. Comparison of PID, SMC and FSMC controllers under periodic orbit tracking.

5.3. Discussions At present, most industrial systems are affected by the strongly or weakly system uncertainties. HTRS system is not an exception. As the effect of system external noises and inner perturbations, it is difficult to use accurate theoretical differential equations characterizing the dynamical behaviors of the nonlinear HTRS system with increasing capacity of hydro generation units nowadays, thus it is necessary to make a further research respect to the nonlinear property analysis of HTRS system with uncertainties and design a new type robustness controller to handle these uncertainties. Extensive investigations of nonlinear properties of an example HTRS system with uncertainties are conducted by using a proposed

seven-order model, which includes bifurcation map, Lyapunov exponents, phase diagrams and power spectrum, etc. These investigations make a new insight for the stability analysis of this complicated system and prove the existence of chaotic behaviors. For most hydropower plants, chaos is disadvantage to the stability of entire control system and harmful to the generator unit operating. Therefore, selecting a set of appropriate control rules and designing a reasonable chaotic controller is essential for the safety of the developing HTRS system. The objective of chaotic controller is to stabilize chaotic system to periodic orbits or equilibrium point by keeping or adjusting the parameters, while the system parameters cannot be changed or it must pay a terrible price for great changing. In this study, FSMC

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Fig. 10. Comparison of PID, SMC and FSMC controllers under random noise disturbances.

controller is used to regulate the HTRS system. Numerical simulations show that no matter disturbances or noises are imposed into the HTRS system, the proposed FSMC controller can quickly eliminate the unexpected chaotic behaviors and makes the system stabilized at the desired point or periodic orbits, which means that the nonlinear HTRS system assisted with FSMC controller has a strong robustness to the external disturbances and system perturbations and can be stabilized to any points and any periodic orbits. 6. Conclusions In this study, a novel nonlinear mathematic model for the HTRS system with the considerations of inner perturbations and external

noises is established by using the state-space equations. Input state linearization method is used to construct the direct relationship of the reference output and the control output. FSMC controller is then designed for this system to reject the influence of the inner perturbations and external disturbances. The FSMC controller was designed by combining SMC controller that approximating the optimal equivalent controller and the fuzzy robust controller for compensating the difference between the optimal control item and the switching control item. The fuzzy logic rules were derived from the sense of Lyapunov stability theorem to ensure the stability of HTRS system. Simulations were carried out in the presence of various uncertainties to evaluate the effectiveness of proposed FSMC controller, where the results demonstrated that the proposed

X. Yuan et al. / Energy 93 (2015) 173e187

controller was excellent from the standpoints of performance and stability for the wicket gate position control while compared with currently often used PID controller and conventional SMC controller, which denotes that the FSMC controller is more credible and suitable to handle the affine nonlinear model of the complicated HTRS system and may serve as an efficient alternative for the design of next-generation controllers for the HTRS system.

Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 51379080).

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