Dynamic sliding mode control of pitch blade wind turbine using sliding mode observer

Abstract

In a wind turbine (WT), the maximum power can be achieved using a suitable and smooth signal, which should be applied to the pitch angle of the blades (PABLE). On the contrary, the uncertainties of the WT models cause the fatigue due to the mechanical stresses. To overcome these two problems, dynamic sliding mode control (D-SMC) is used because it is robust against uncertainties and can suppress the chattering by providing smooth signals. In D-SMC, an integrator is located before the actuator, as a low-pass filter, to suppress the high-frequency chattering. Then, the states number of the overall augmented system is one more than the states number of the actual system. To control such an augmented system, the added state variable needs to be estimated and hence, a novel sliding mode observer (SMO) is proposed. A trusty comparison is also presented using the conventional sliding mode control (C-SMC) with the proposed SMO. To implement D-SMC and C-SMC, a new state feedback is applied to the turbine at first. Therefore, a linear model with uncertainty is obtained, where its input is the PABLE. Lyapunov theory is used to proof the stability of the proposed SMO, D-SMC, and also the C-SMC. The presented comparison demonstrates the advantages of the D-SMC with respect to the C-SMC in removing the chattering and simplicity in concept and in implementation.

Keywords

Wind turbine pitch angle control sliding mode observer sliding mode control chattering

Introduction

As a result of increasing global warming due to the fossil fuels, many considerable efforts are focused on using renewable viable energies such as solar or wind energy (Burton et al., 2001; Song et al., 2000), which are clean and accessible worldwide (Carlin et al., 2003; Manwell et al., 2002). However, wind energy has many attentions because of economical preference of the wind turbine (WT) (Bianchi et al., 2007; Camblong, 2008). A WT converts the wind energy to the rotational torque for the generator. Two kinds of WT are developed in industries: WT with fixed speed (WTFS) (Rahimi and Parniani, 2009; Sumper et al., 2009) and WT with variable speed (WTVS) (Poultangari et al., 2012). The WTFS cannot capture the maximum power of the wind in various wind speeds (Poultangari et al, 2012) and hence, WTVS is developed in recent years (Oh et al., 2015). The strategy to obtain the maximum power of the wind by WTVS, is based on dividing the operation performance using the rated speed of the wind (Jabbari Asl and Yoon, 2016). Below this rated wind speed, the torque of generator is controlled for the maximum power point tracking (MPPT) (Ardjal et al., 2018), and above this rated wind speed, the pitch angle of the blades (PABLE) is used as the input control (Asgharniaa et al., 2018). There are also two others critical wind speeds: cut-out and cut-in wind speeds (Barambones et al., 2019; Sitharthan et al., 2019). In the outside of the interval between cut-out and cut-in, the WT would be shut down due to the economic reasons and fatigue damages respectively (Su et al. 2020).

Many PABLE regulators are proposed in the region between cut-out and rated wind speeds (Asgharniaa et al., 2018; Boukhezzar et al., 2007; Camblong, 2008; Chen et al., 2019; Gao and Gao, 2016; Hatami et al., 2016; Lasheen et al., 2020; Oh et al., 2015; Rahimi and Asadi, 2019; Yuan et al., 2020; Yuan and Tang, 2017). On the contrary, the more important challenges of WT are the mechanical stresses (Abolvafaei and Ganjefar, 2019) and uncertainties (Jiao et al., 2020). To remove the mechanical stresses, it is necessary to overcome the mechanical drivetrain uncertainties using a powerful robust controller (Abolvafaei and Ganjefar, 2019). Unfortunately, some proposed controllers are not robust for example, some simple nonlinear feedback controllers are proposed by Boukhezzar et al. (2007). Or classical controllers such as PID (proportional–integral–derivative) which are proposed by Gao and Gao (2016); Asgharniaa et al. (2018); Yuan et al. (2020). A PID with adaptive self-tuning regulator (STR) is constructed by Hatami et al. (2016), and gain scheduled PID is designed by Yuan and Tang (2017) or proportional–integral (PI) scheme of controller can be shown by Rahimi and Asadi (2019). Moreover, the combination of adaptive and PI controller has been designed by Lasheen et al. (2020).

However, among all of these methods, sliding mode control (SMC) is a powerful approach in facing mechanical stresses due to its invariance property (Abolvafaei and Ganjefar, 2019). Invariance is the most important property of SMC, which is stronger than robustness (Perruquetti and Pierre-Barbot, 2002; Slotine and Li, 1991). The invariance is achieved when the system states are on the sliding surface (Karami-Mollaee et al., 2011). Therefore, many research works are focused on using the sliding mode pitch control (Cetrini et al., 2019; Colombo et al., 2020; Golnary and Moradi, 2018, 2019; Nayeh et al., 2020; Yin et al., 2019). In the most of these works, SMC suffers from the chattering as a disadvantage phenomenon, which is due to the using of Signum function (Karami-Mollaee et al., 2011; Perruquetti and Pierre-Barbot, 2002; Slotine and Li, 1991). To remove the chattering, some methodologies are proposed in literature (Lee and Utkin, 2007): boundary layer sliding mode control (BL-SMC) (Chen et al., 2005; Fuh, 2008), adaptive boundary layer sliding mode control (ABL-SMC) (Allamehzadeh and Cheung, 2004; Zhang, 2016), higher-order sliding mode control (HO-SMC) (Cucuzzella et al., 2015; Nonaka et al., 2015), and dynamic sliding mode control (D-SMC) (Chen et al., 2007; Karami-Mollaee et al., 2011; Koshkouei et al., 2005). However, in BL-SMC and ABL-SMC, the invariance, as the most property of SMC, cannot be reserved (Karami-Mollaee et al., 2011; Perruquetti and Pierre-Barbot, 2002; Slotine and Li, 1991). HO-SMC needs to be the higher-order derivatives of system model, which should be estimated using some observers (Levant, 1993, 2005; Plestan et al., 2008; Shtessel et al., 2008; Yang et al., 2017). In D-SMC, an integrator, such as a low-pass filter, is located between the controller and the system model to smooth the effect of discontinuous Signum function (Chen et al., 2007). Then, due to the dynamic behavior of this integrator, the system states are increased. To control such an augmented system (system and integrator), the system model should be estimated via a capable observer (Karami-Mollaee et al., 2011). Then in HO-SMC, system model differentiations are needed and in D-SMC just added state would be estimated and this is the preference of D-SMC.

The concepts of observation, estimation, and identification are applicable in a wide ranges in control systems such as model identification (Davila et al., 2005), state observer (Liu et al., 2013; Xia et al., 2011), disturbance observer (Fuyang et al. 2016), and parameters estimation (Butt et al., 2013). Then, the main objective of using an observer is the estimation of the unknown systems states or unmeasurable systems parts, which is generally done using known system inputs and outputs as a feedback (Benchaib et al., 1999; Butt et al., 2013; Davila et al., 2005; Fuyang et al., 2016; Liu et al., 2013; Xia et al., 2011; Xiong and Saif, 2001). As the SMC, the preference of the sliding mode observer (SMO) is its robustness and its invariance in facing to the disturbances and the uncertainties (Benchaib et al., 1999; Fuh, 2008; Rahnavard et al., 2018). The proposed SMO observer is very simpler than of the SMO by Rahnavard et al. (2018). Moreover, the stability of SMO by Chakrabarty et al. (2018) is proved under some conditions, but convergence of the proposed SMO is proved without any challenges.

Motivated by the above discussions, due to the wind speed variations above the rated wind speed, PABLE controllers are inevitable. Two important challenges are available: drivetrain uncertainties and mechanical stresses. In fact, the uncertainties lead to the mechanical stresses. Therefore, using a robust controller is necessary to overcome on the uncertainties. Among all of the proposed controllers in the literature, SMC is robots and invariance with respect to the uncertainty. But SMC suffers from the chattering, which damages to the mechanical parts. Therefore, some methodologies are proposed for chattering suppression: BL-SMC, ABL-SMC, HO-SMC, and D-SMC. Based on the disadvantage problems of BL-SMC, ABL-SMC, and HO-SMC, we propose D-SMC method for the PABLE control of WT to reliable prevent chattering while preserve invariance property. Then we can remove and can overcome on the mechanical stresses. Moreover, to implement this controller and to identify the plant model, a new structure of SMO is constructed. Thereafter, the theory of Lyapunov is used for stability proof of D-SMC and SMO.

Therefore, the proposed approach is demonstrated in six sections. At first, the preliminaries about the WT model are presented in section “WT model and structure.” Then, the proposed structure of the SMO and its stability are constructed in section “The proposed SMO design.” Therefore, in section “PABLE control–based D-SMC design,” the proposed PABLE controller–based D-SMC structure and comparison with C-SMC are provided. To show the preference of the proposed method, some simulations are presented in section “Simulation results.” Finally, section “Conclusion” is devoted to the conclusion.

WT model and structure

A WT consists of three subsystems: aerodynamic, mechanical drivetrain, and generator (Abolvafaei and Ganjefar, 2019; Nayeh et al., 2020). This is illustrated in Figure 1, and each of them are explained as follows:

Figure 1.

The WT complete model.

The model of aerodynamic

The $P_{a}$ or the captured power by the WT with the blades length $r$ , is dependent to the air density $ρ$ , the swept surface by the blades $π r^{2}$ , the power coefficient $C_{p}$ , and the wind speed $v (t)$

P_{a} = \frac{ρ π r^{2} v^{3}}{2} C_{p} (β, λ)

(1)

In which, $β$ is the pitch blades and $λ$ is the linear velocity of the tip blades divided by wind speed (known as the tip speed ratio)

λ = \frac{r ω_{r}}{v}

(2)

where $w_{r}$ is the rotor side angular velocity of the blades turbine. According to the equation (1), power coefficient is dependent to the $β$ and $λ$ , commonly described using the following equation (Abolvafaei and Ganjefar, 2019)

\begin{matrix} C_{p} (β, λ) = (\frac{d_{1} d_{2}}{λ_{i}} - d_{1} d_{3} β - d_{1} d_{4}) e^{- \frac{d_{5}}{λ_{i}}} + d_{6} λ \\ \frac{1}{λ_{i}} = \frac{1}{λ + 0.08 β} - \frac{0.035}{β^{3} + 1} \end{matrix}

(3)

such that (Abolvafaei and Ganjefar, 2019)

\begin{matrix} d_{1} = 0.5176, d_{2} = 116, d_{3} = 0.4 \\ d_{4} = 5, d_{5} = 21, d_{6} = 0.0068 \end{matrix}

(4)

and $T_{a}$ , that is, the generated torque by the wind (rotor torque) can be calculated as follows

T_{a} = \frac{P_{a}}{w_{r}} = \frac{ρ π r^{3} v^{2}}{2 λ} C_{p} (β, λ)

(5)

The mechanical drivetrain model

The drivetrain is the mechanical part of the WT, which can be modeled using one-mass, two-mass, three-mass, or six-mass (Rigatos et al., 2019). Nevertheless, the two-mass model can also show the transient response and steady-state response in the presence of controller (Boukhezzar and Siguerdidjane, 2010). The two-mass mechanical drivetrain, which is depicted in Figure 2, is described by the following equations (Boukhezzar and Siguerdidjane, 2010; Jiao et al., 2019)

\begin{matrix} J_{r} {\overset{\cdot}{w}}_{r} = - K_{r} w_{r} + T_{a} - T_{ls} \\ J_{g} {\overset{\cdot}{w}}_{g} = - K_{g} w_{g} - T_{g} + T_{hs} \end{matrix}

(6)

Figure 2.

The two-mass mechanical drivetrain model.

Moreover, the braking torque in the low-speed shaft is as follows

T_{ls} = K_{ls} (w_{r} - w_{ls}) + B_{ls} (θ_{r} - θ_{ls})

(7)

Such that, $w_{g}$ and $w_{r}$ are angular velocity, $J_{g}$ and $J_{r}$ are moment of inertia, $K_{g}$ and $K_{r}$ are external damping, all for generator and rotor, respectively. Moreover, $T_{g}$ and $T_{a}$ are the generator torque output and rotor torque output, respectively, which are applied to the drivetrain as the inputs. Then, $w_{ls}$ and $θ_{ls}$ are angular velocity and angular speed of deviation in the low-speed shaft, and $θ_{r}$ is the rotor side speed of deviation. Finally $B_{ls}$ and $K_{ls}$ are the shaft stiffness and the low-speed shaft damping, respectively. The ideal gearbox ratio would be as follows

n_{g} = \frac{w_{g}}{w_{r}} = \frac{T_{ls}}{T_{hs}}

(8)

Now using the equation (8) and the second part of equation (6), one can write

J_{g} (n_{g} {\overset{\cdot}{w}}_{r}) = - K_{g} (n_{g} w_{r}) - T_{g} + (\frac{T_{ls}}{n_{g}})

(9)

{n_{g}}^{2} J_{g} {\overset{\cdot}{w}}_{r} = - {n_{g}}^{2} K_{g} w_{r} - n_{g} T_{g} + T_{ls}

(10)

Then, adding this equation to the first part of equation (6) the following equation is archived (Abolvafaei and Ganjefar, 2019).

J_{t} {\overset{\cdot}{w}}_{r} = - K_{t} w_{r} + T_{a} - n_{g} T_{g}

(11)

where $J_{t} = J_{r} + {n_{g}}^{2} J_{g}$ and $K_{t} = K_{r} + {n_{g}}^{2} K_{g}$ . In fact, all the parameters are transferred to the low-speed shaft (Abolvafaei and Ganjefar, 2019).

The generator dynamic model

The output torque produced by generator, that is, $T_{g}$ , is dependent to the dynamic model of generator. For example, a first-order model would be as follow, nevertheless the higher-order model for the generator model can be considered

{\overset{\cdot}{T}}_{g} = \frac{T_{ref} - T_{g}}{τ_{g}}

(12)

where $T_{ref}$ is the reference torque and $τ_{g} = 15$ is the generator time constant. Since we have focused on PABLE control, the generator torque reference is set to $T_{ref} = T_{rated}$ . Moreover, the produced output power which is delivered to the grid can be written as follow, where efficiency of generator is $η_{g}$

P_{g} = η_{g} w_{g} T_{g}

(13)

The proposed SMO design

The purpose is to calculate a smooth pitch angle $β$ to have a desired rotor angular velocity. To this end, we calculate the derivative of power coefficient equation (3) with respect to parameter $β$

\begin{matrix} \frac{d C_{p}}{d β} = \frac{\partial C_{p}}{\partial λ_{i}} \frac{\partial λ_{i}}{\partial β} + \frac{\partial C_{p}}{\partial β} \\ = (- \frac{c_{1} c_{2}}{λ_{i}^{2}} + \frac{c_{1} c_{2} c_{5}}{λ_{i}^{3}} - \frac{c_{1} c_{3} c_{5} β}{λ_{i}^{2}} - \frac{c_{1} c_{4} c_{5}}{λ_{i}^{2}}) \\ \times (\frac{- 0.08}{{(λ + 0.08 β)}^{2}} + \frac{0.035}{{(β^{3} + 1)}^{2}}) e^{- \frac{c_{5}}{λ_{i}}} - c_{1} c_{3} e^{- \frac{c_{5}}{λ_{i}}} \end{matrix}

(14)

Hence, the Taylor series of equation (5) around the optimal operating point $(β_{opt}, λ_{opt})$ would be as

\begin{matrix} T_{a} = \frac{ρ π r^{3} v^{2}}{2 λ_{opt}} \frac{d C_{p}}{d β} |_{β = β_{opt}, λ = λ_{opt}} (β - β_{opt}) \\ + Oth . T + H . O . T \end{matrix}

(15)

where $Oth . T$ and $H . O . T$ denote the other terms and higher-order terms, respectively. Therefore, the linear model of equation (11) will be as follows

\begin{matrix} {\overset{\cdot}{w}}_{r} = - \frac{K_{t}}{J_{t}} w_{r} - \frac{n_{g}}{J_{t}} T_{g} + \frac{T_{a}}{J_{t}} \\ = - \frac{K_{t}}{J_{t}} w_{r} - \frac{n_{g}}{J_{t}} T_{g} \\ + \frac{ρ π r^{3} v^{2}}{2 J_{t} λ_{opt}} \frac{d C_{p}}{d β} |_{β = β_{opt}, λ = λ_{opt}} (β - β_{opt}) + Δ \end{matrix}

(16)

and the unknown uncertainty is as

Δ = \frac{Oth . T + H . O . T}{J_{t}}

(17)

Due to the convergence of the Taylor series around the operating points, the uncertainty $Δ$ is bounded, that is

| Δ | \leq η

(18)

Now, the following state feedback with a new input signal $u \in R^{1 \times 1}$ , and an arbitrary positive scalar $a$ is used

β = \frac{(\frac{k_{t}}{J_{t}} - a) w_{r} + (\frac{n_{g}}{J_{t}}) T_{g} + u}{\frac{ρ π r^{3} v^{2}}{2 J_{t} λ_{opt}} \frac{d C_{p}}{d β} |_{β = β_{opt}, λ = λ_{opt}}} + β_{opt}

(19)

Then, the system equation (16) can be rewritten as

{\overset{\cdot}{w}}_{r} = - a w_{r} + u + Δ

(20)

To construct the chattering-less D-SMC, the following sliding surface with desired signal $w_{rd}$ is defined

s = λ_{1} (w_{r} - w_{rd}) + λ_{2} ({\overset{\cdot}{w}}_{r} - {\overset{\cdot}{w}}_{rd})

(21)

Coefficients $λ_{i} : i = 1, 2$ are selected such that the sliding surface zero dynamics are stable, which means that the states $w_{r}$ and ${\overset{\cdot}{w}}_{r}$ converge to $w_{rd}$ and ${\overset{\cdot}{w}}_{rd}$ , respectively in the case of $s = 0$ . Unfortunately, the variable ${\overset{\cdot}{w}}_{r}$ is not available due to the uncertain function $Δ$ . Hence, a new SMO structure is proposed to solve this problem. Consider the following proposed SMO for the system equation (20)

{\begin{matrix} \overset{\cdot}{z} (t) = - z (t) + μ_{1} (t) \\ \overset{\cdot}{\hat{w}} (t) = + z (t) - a {\hat{w}}_{r} + u + η + μ_{2} (t) \end{matrix}

(22)

where $μ_{i} : i = 1, 2$ are the observer inputs and ${\hat{w}}_{r}$ is the estimation of the $w_{r} (t)$ and $z \in R^{1 \times 1}$ is the auxiliary signal. The convergence and stability of this observer are proved in the following theorem.

Theorem 1

The observer error estimation $e = w_{r} - {\hat{w}}_{r}$ converges to zero if the observer inputs are selected as follow

\begin{matrix} μ_{1} = e \in R^{1 \times 1} \\ μ_{2} = sign (e) \in R^{1 \times 1} \end{matrix}

(23)

Proof

The observer error as a dynamic system can be written as follows

\overset{\cdot}{e} (t) = - z (t) - ae - η + Δ - μ_{2} (t)

(24)

Consider the following Lyapunov candidate function

V (t) = \frac{1}{2} e^{2} (t) + \frac{1}{2} z^{2} (t)

(25)

Therefore

\overset{\cdot}{V} = e (t) \overset{\cdot}{e} (t) + z (t) \overset{\cdot}{z} (t)

(26)

Using equation (24) and the first part of the observer equation (22), one can write

\begin{matrix} \overset{\cdot}{V} = e (- z - ae - η + Δ - μ_{2}) + z (- z + μ_{1}) \\ = - ez - a e^{2} - e η + e Δ - e μ_{2} - z^{2} + z μ_{1} \\ = - ez - a e^{2} - e η + e Δ - esign (e) - z^{2} + ze \\ \leq - z^{2} - a e^{2} - e η + e | Δ | - | e | \\ = - z^{2} - a e^{2} - e (η - | Δ |) - | e | \leq - z^{2} - a e^{2} \end{matrix}

(27)

Consequently, both error signal $e$ and auxiliary signal $z$ converge to zero. Hence, the proof is completed.

PABLE control–based D-SMC design

In this section, for the system described by equation (20), a new proposed D-SMC is constructed to suppress chattering and then, C-SMC is also designed to show the effective performance of the D-SMC. The validity of the comparison is comprehensive due to using the same proposed observer of equation (22) in both approaches.

The proposed D-SMC

Based on the previous sections in designing the state feedbacks and the SMO, the implemented diagram of the proposed approach is illustrated in Figure 3, which the description of each block is made inside it. Now we proceed to construct the proposed D-SMC.

Figure 3.

The implemented diagram of the proposed structure.

Using the observer equation (22), the sliding surface equation (21) can be as follows

\begin{matrix} s = λ_{1} ({\hat{w}}_{r} - w_{rd}) + λ_{2} ({\overset{\cdot}{\hat{w}}}_{r} - {\overset{\cdot}{w}}_{rd}) \\ = λ_{1} {\hat{w}}_{r} + λ_{2} {\overset{\cdot}{\hat{w}}}_{r} - λ_{1} w_{rd} - λ_{2} {\overset{\cdot}{w}}_{rd} \end{matrix}

(28)

Then we have

\begin{matrix} \overset{\cdot}{s} = λ_{1} {\overset{\cdot}{\hat{w}}}_{r} + λ_{2} {\overset{\cdot\cdot}{\hat{w}}}_{r} - λ_{1} {\overset{\cdot}{w}}_{rd} - λ_{2} {\overset{\cdot\cdot}{w}}_{rd} \\ = λ_{1} {\overset{\cdot}{\hat{w}}}_{r} + λ_{2} (\overset{\cdot}{z} - a {\overset{\cdot}{\hat{w}}}_{r} + \overset{\cdot}{u}) - λ_{1} {\overset{\cdot}{w}}_{rd} - λ_{2} {\overset{\cdot\cdot}{w}}_{rd} \end{matrix}

(29)

\overset{\cdot}{s} = (λ_{1} - a λ_{2}) {\overset{\cdot}{\hat{w}}}_{r} + λ_{2} \overset{\cdot}{z} + λ_{2} \overset{\cdot}{u} - λ_{1} {\overset{\cdot}{w}}_{rd} - λ_{2} {\overset{\cdot\cdot}{w}}_{rd}

(30)

From the observer equation (22), one can write

\begin{matrix} \overset{\cdot}{s} = (λ_{1} - a λ_{2}) (z - a {\hat{w}}_{r} + u + η + μ_{2}) \\ + λ_{2} (- z + μ_{1}) + λ_{2} \overset{\cdot}{u} - λ_{1} {\overset{\cdot}{w}}_{rd} - λ_{2} {\overset{\cdot\cdot}{w}}_{rd} \end{matrix}

(31)

Theorem 2

The following input control signal for the system equation (20) causes reaching to the sliding surface equation (21) in finite time

\begin{matrix} \overset{\cdot}{u} = - \frac{(λ_{1} - a λ_{2}) (z - a {\hat{w}}_{r} + u + η + μ_{2}) + λ_{2} (- z + μ_{1})}{λ_{2}} \\ + \frac{λ_{1} {\overset{\cdot}{w}}_{rd} + λ_{2} {\overset{\cdot\cdot}{w}}_{rd} - k_{1} sign (s) - k_{2} s}{λ_{2}} \end{matrix}

(32)

where $k_{1}$ and $k_{2}$ are positive constants coefficients and $k_{1}$ is the switching gain.

Proof

Substituting equation (32) into equation (31) results in

\overset{\cdot}{s} = - k_{1} sign (s) - k_{2} s

(33)

Considering the Lyapunov function $V = 0.5 s^{2}$ , then

\begin{matrix} \overset{\cdot}{V} = s \overset{\cdot}{s} = s (- k_{2} s - k_{1} sign (s)) \\ = - k_{2} s^{2} - k_{1} | s | \leq - k_{1} | s | \end{matrix}

(34)

Consider the finite reaching time to the sliding surface, that is, $t_{f}$ , then one can easily to show that $t_{f} \leq | s (0) | / k_{1}$ (Karami-Mollaee et al., 2011). Note that by substitution of observer inputs from equation (23) we have

\begin{matrix} \overset{\cdot}{u} = - \frac{(λ_{1} - a λ_{2}) (z - a {\hat{w}}_{r} + u + η + μ_{2}) + λ_{2} (- z + μ_{1})}{λ_{2}} \\ + \frac{λ_{1} {\overset{\cdot}{w}}_{rd} + λ_{2} {\overset{\cdot\cdot}{w}}_{rd} - k_{1} sign (s) - k_{2} s}{λ_{2}} \end{matrix}

(35)

The C-SMC design

Consider the following sliding surface

s = λ_{1} (w_{r} - w_{rd})

(36)

Using the observer equation (22), we have

\begin{matrix} \overset{\cdot}{s} = λ_{1} {\overset{\cdot}{w}}_{r} - λ_{1} {\overset{\cdot}{w}}_{rd} = λ_{1} (z - a {\hat{w}}_{r} + u + η + μ_{2}) - λ_{1} {\overset{\cdot}{w}}_{rd} \\ = λ_{1} z - a λ_{1} {\hat{w}}_{r} + λ_{1} u + λ_{1} η + λ_{1} μ_{2} - λ_{1} {\overset{\cdot}{w}}_{rd} \end{matrix}

(37)

Theorem 3

The following input control signal for the system equation (20) causes reaching to the sliding surface equation (36) in finite time

\begin{matrix} u = - \frac{λ_{1} z - a λ_{1} {\hat{w}}_{r} + λ_{1} η + λ_{1} μ_{2} - λ_{1} {\overset{\cdot}{w}}_{rd}}{λ_{1}} \\ - \frac{k_{1} sign (s) + k_{2} s}{λ_{1}} \end{matrix}

(38)

where $k_{1}$ and $k_{2}$ are positive constants coefficients and $k_{1}$ is the switching gain.

Proof

By substitution of equation (38) into equation (39) one can result that

\overset{\cdot}{s} = - k_{1} sign (s) - k_{2} s

(39)

Considering the Lyapunov function $V = 0.5 s^{2}$ , then

\begin{matrix} \overset{\cdot}{V} = s \overset{\cdot}{s} = s (- k_{2} s - k_{1} sign (s)) \\ = - k_{2} s^{2} - k_{1} | s | \leq - k_{1} | s | \end{matrix}

(40)

\begin{matrix} u = - \frac{λ_{1} z - a λ_{1} {\hat{w}}_{r} + λ_{1} η + λ_{1} sign (e) - λ_{1} {\overset{\cdot}{w}}_{rd}}{λ_{1}} \\ - \frac{k_{1} sign (s) + k_{2} s}{λ_{1}} \end{matrix}

(41)

The reference of rotor angular velocity

As it is seen, according to the Figure 4, the operation of a WTVS is divided to the four regions based on the wind speeds: cut-in, cut-out and rated. In the first region, below the cut-in wind speed, the turbine would be shut down based on the economic criterion. In the second region, between the cut-in wind speed and rated wind speed, the turbine pitch blades are fixed and generator torque or generator power would be controlled. In this case, the rotor speed is increasing to obtain the maximum of power coefficient. In the third region, between the rated wind speed and cut-out wind speed, the generator power will be set to its rated. In this region, PABLE would be increased to reduce the rotor speed. Finally, in the fourth region, above the cut-out wind speed, the turbine would be shut down again to protect it from the fatigue damages (Oh et al., 2015).

Figure 4.

The operation regions of WTVS.

In this paper, we have focused on pitch angle control in region three, whereas the rotor angular velocity should be reduced with the increase of wind speed. Therefore, the reference of rotor angular velocity will be as follows

w_{rd} = w_{rated} - w_{rated} \frac{v - v_{rated}}{v_{cut - out} - v_{rated}}

(42)

Such that based on equation (2), the maximum of rotor angular velocity can be written as

w_{rated} = \frac{λ_{opt} v_{rated}}{r}

(43)

We use the two-mass 5MW WT in National Renewable Energy Laboratory (NREL) located at Colorado, with the parameters of Table 1 (Bossanyi et al., 2010).

Table 1.

The parameters of WTVS.

Parameter	Value	Unit
$T_{rated}$	$4.309 \times 10^{4}$	$N \times m$
$v_{c u t - i n}$	$3$	$m / s$
$v_{rated}$	$11.4$	$m / s$
$v_{cut - out}$	$25$	$m / s$
$β_{opt}$	$0$	°
$λ_{opt}$	$7.55$	$rad$

Simulations results

For a reliable comparison, two simulations are done using D-SMC and C-SMC, both based on the same SMO and MATLAB software with step time of 0.01. In both simulations, the feedback parameter is selected as $a = 100$ . In addition, all the observer initial values are set to zero, that is, ${\hat{w}}_{r} = 0$ , $z (0) = 0$ . This is to show the performance convergence of the proposed SMO. Due to using NREL WT in the proposed two approaches, the following parameters in Table 2 are used (Bossanyi et al., 2010).

Table 2.

The parameters of the mechanical drivetrain.

Notation	Value	Unit
$r$	$21.62$	$m$
$ρ$	$1.308$	$kg / m^{3}$
$J_{r}$	$3.25 \times 10^{5}$	$kg \times m^{2}$
$J_{g}$	$34.4$	$kg \times m^{2}$
$K_{r}$	$27.36$	$(N \times m) / (rad / s)$
$K_{g}$	$0.2$	$(N \times m) / (rad / s)$
$K_{ls}$	$9.5 \times 10^{3}$	$(N \times m) / rad$
$B_{ls}$	$2.691 \times 10^{5}$	$(N \times m) / (rad / s)$
$n_{g}$	$43.165$	$Scalar$

The wind speed has been shown in Figure 5 with the white noise and mean value 16 and maximum disturbance of 5. Note that this is between 11.4 and 25, that is, in region 3: $v_{rated} = 11.4 < v (t) < v_{cut - out} = 25$ . Moreover, Figure 6 shows the reference of rotor speed in the region 3 denoted by equation (42). To show the convergence of Taylor series and its small bound value, the ratio of $T_{a}$ in equation (5) divided by $J_{t}$ is drawn in Figure 7. Hence, the bound $η$ in the observer equation (22) cannot produce a critical conditions.

Figure 5.

The time series of the wind speed profile.

Figure 6.

Reference of rotor speed in region 3 of WTVS.

Figure 7.

Verification of the Taylor series convergence.

Example 1: The D-SMC proposed approach

To have a surface with stable zero dynamics, sliding coefficients are selected as $λ_{1} = 5$ , $λ_{2} = 1$ . Moreover, there is a trade of in choosing the parameter $k_{1}$ . A large $k_{1}$ increases the switching gain and decreases reaching to the sliding surface, that is, $t_{f}$ and vice versa. Therefore, we choose a small value for $k_{1}$ , and to compensate it, another bigger value is considered as $k_{2}$ . Then, we choose $k_{1} = 2$ and $k_{2} = 5$ . Furthermore, to calculate the input control signal from equation (35), we need its initial value, selected as $u (0) = 0$ . The results are presented in Figures 8 –11.

Figure 8.

Rotor angular velocity and its estimation tracking desired trajectory in D-SMC.

Figure 9.

Finite-time zero convergence of the sliding surface in D-SMC.

Figure 10.

PABLE of the WT in D-SMC.

Figure 11.

Wind speed, rotor angular velocity, and PABLE of WT in D-SMC.

Figure 8 shows the angular velocity of rotor and its estimation using the proposed SMO. This figure shows the good tracking performance of reference. Figure 9 presents the zero convergence of the sliding surface in finite time, which also is without chattering. Note that, the system states are adhered to the sliding surface using the Signum function. The sliding surface is free of uncertainty and disturbance, which leads to invariance property. In Figure 10 the pitch angle of the turbine blades is shown, which also is the input control signal of the system or the controller output. The pitch angle is between 0 and 90° and is without chattering. In Figure 11, all the wind speed, rotor speed, and pitch angle are drawn. We can see when the wind speed is decreased (for example at time 10.2 second), the WT input or the pitch angle is reduced to its optimal value, that is, to zero or $β_{opt} = 0$ . This would be caused as the rotor speed is increased because in this case, the wind speed may be reduced to the rated wind speed. On the contrary, if the wind speed is increased (for example at time 14 second), the pitch angle goes toward 90° which is causes the rotor speed reduced toward cut-out value and then, the turbine should be stopped to protect the turbine from the fatigue damages.

Example 2. The C-SMC design

To show the validity and preference of the D-SMC approach, all the parameters are as the example 1. Hence, the sliding surface coefficient are selected as $λ_{1} = 5$ and also $k_{1} = 2$ and $k_{2} = 5$ . The results are presented in Figures 12 till 14. As the previous example, the rotor speed converges to the desired signals and is tracked by the observer state with zero error, which is shown in Figure 12. The sliding surface is shown in Figure 13. The sliding surface converges to zero in finite time also. However, the chattering effect can be observed in pitch angle from Figure 14. Note that this signal cannot be applied to the actuator blades due to its mechanical parts. Moreover, from the power consumption point of view, the input control signal is very important. So the chattering not only can damage the mechanical parts of the system but can also cause power consumption. From comparison of these figures, one can see the good performance of the proposed SMO. Nevertheless, in the viewpoint of the controller design, the D-SMC has better performance from C-SMC in chattering suppression. Moreover, the simplicity of D-SMC versus C-SMC is noticeable in design and in concept.

Figure 12.

Rotor angular velocity and its estimation tracking desired trajectory in C-SMC.

Figure 13.

Finite-time zero convergence of the sliding surface in C-SMC.

Figure 14.

PABLE of the WT in C-SMC.

Conclusion

In this paper, two new approaches are presented to extract the maximum power of WTs. To this end, PABLE controller is developed using D-SMC and conventional sliding mode control (C-SMC). To show the superiority of D-SMC, the same proposed SMO is used in both controllers. To have a valid comparison, all other parameters are also selected to be equivalent, such as sliding surface coefficients and observer parameters. The comparison results show that in D-SMC, chattering is removed completely, but chattering is remained in C-SMC. From the power consumption point of view, the input control signal is very important. Then, the chattering not only can damage to the mechanical parts of the system but can also cause power consumption. To implement controllers and observer, a state feedback is applied to the WT at first, which leads to a linear model with uncertainty and one input as the PABLE. Then, the closed-loop stability of the developed D-SMC and C-SMC combined with SMO is also represented using Lyapunov stability theory. Finally, the design procedure shows the simplicity of the D-SMC in concept and in realization.

Footnotes

Acknowledgements

The authors wish to express their gratitude to the Basque Government, through the project EKOHEGAZ (ELKARTEK KK-2021/00092), to the Diputación Foral de Álava (DFA), through the project CONAVANTER, and to the UPV/EHU, through the project GIU20/063, for supporting this work.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ali Karami-Mollaee

References

Abolvafaei

Ganjefar

(2019) Maximum power extraction from a wind turbine using second-order fast terminal sliding mode control. Renewable Energy 139: 1437–1446.

Allamehzadeh

Cheung

(2004) Optimal fuzzy sliding mode control with adaptive boundary layer. WSEAS Transaction on Systems 3(5): 1887–1892.

Ardjal

Mansouri

Bettayeb

(2018) Fractional sliding mode control of wind turbine for maximum power point tracking. Transactions of the Institute of Measurement and Control 41(2): 447–457.

Asgharniaa

Shahnazi

Jamali

(2018) Performance and robustness of optimal fractional fuzzy PID controllers for pitch control of a wind turbine using chaotic optimization algorithms. ISA Transactions 79: 27–44.

Barambones

Cortajarena

Calvo

, et al. (2019) Variable speed wind turbine control scheme using a robust wind torque estimation. Renewable Energy 133: 354–366.

Benchaib

Rachid

Audrezet

, et al. (1999) Real time sliding-mode observer and control of an induction motor. IEEE Transactions on Industrial Electronics 46(1): 128–138.

Bianchi

Battista

Mantz

(2007) Wind Turbine Control Systems: Principles, Modelling and Gain Scheduling Design. London: Springer.

Bossanyi

Wright

Fleming

(2010) Controller field tests on the NREL CART2 turbine. No. NREL/TP-5000-49085. Golden, CO: National Renewable Energy Lab (NREL).

Boukhezzar

Siguerdidjane

(2010) Nonlinear control of a variable-speed wind turbine using a two-mass model. IEEE on Energy Conversion 26(1): 149–162.

10.

Boukhezzar

Lupu

, et al. (2007) Multivariable control strategy for variable speed, variable pitch wind turbines. Renewable Energy 32(8): 1273–1287.

11.

Burton

Sharpe

Jenkins

, et al. (2001) Wind Energy Handbook. Hoboken, NJ: Wiley.

12.

Butt

Bhatti

Mufti

, et al. (2013) Modeling and online parameter estimation of intake manifold in gasoline engines using sliding mode observer. Simulation Modelling Practice and Theory 32: 138–154.

13.

Camblong

(2008) Digital robust control of a variable speed pitch regulated wind turbine for above rated wind speeds. Control Engineer Practice 16(8): 946–958.

14.

Carlin

Laxson

Muljadi

(2003) The history and state of the art of variable-speed wind turbine technology. Wind Energy: An International Journal for Progress and Applications in Wind Power Conversion Technology 6(2): 129–159.

15.

Cetrini

Cianetti

Corradini

, et al. (2019) On-line fatigue alleviation for wind turbines by a robust control approach. Electrical Power and Energy Systems 109: 384–394.

16.

Chakrabarty

Rundell

Zak

, et al. (2018) Unknown input estimation for nonlinear systems using sliding mode observers and smooth window functions. SIAM Journal on Control and Optimization 56(5): 3619–3641.

17.

Chen

Renn

(2005) Sliding mode control with varying boundary layers for an electro-hydraulic position servo system. International Journal of Advanced Manufacturing Technology 26(1-2): 117–123.

18.

Chen

Yang

Gao

, et al. (2019) A novel real-time feedback pitch angle control system for vertical-axis wind turbines. Journal of Wind Engineering and Industrial Aerodynamics 195: 104023.

19.

Chen

Yang

(2007) An LTR-observer based dynamic sliding mode control for chattering reduction. Automatica 43(6): 1111–1116.

20.

Colombo

Corradini

Ippoliti

, et al. (2020) Pitch angle control of a wind turbine operating above the rated wind speed: A sliding mode control approach. ISA Transactions 96: 95–102.

21.

Cucuzzella

Incremona

Ferrara

(2015) Design of robust higher order sliding mode control for micro grids. IEEE Journal on Emerging and Selected Topics in Circuits and Systems 5(3): 393–401.

22.

Davila

Fridman

Levant

(2005) Second-order sliding mode observer for mechanical systems. IEEE Transactions on Automatic Control 50(11): 1785–1789.

23.

Fuh

(2008) Variable-thickness boundary layers for sliding mode control. Journal of Marine Science and Technology 16(4): 288–294.

24.

Fuyang

Lei

Zhang

, et al. (2016) A novel nonlinear resilient control for a quadrotor UAV via backstepping control and nonlinear disturbance observer. Nonlinear Dynamics 85(2): 1281–1295.

25.

Gao

(2016) Pitch control for wind turbine systems using optimization, estimation and compensation. Renewable Energy 91: 501–515.

26.

Golnary

Moradi

(2018) Design and comparison of quasi continuous sliding mode control with feedback linearization for a large scale wind turbine with wind speed estimation. Renewable Energy 127: 95–508.

27.

Golnary

Moradi

(2019) Dynamic modelling and design of various robust sliding mode controls for the wind turbine with estimation of wind speed. Applied Mathematical Modelling 65: 566–585.

28.

Hatami

ans Moetakef-Imani

(2016) Innovative adaptive pitch control for small wind turbine fatigue load reduction. Mechatronics 40: 137–145.

29.

Jabbari Asl

Yoon

(2016) Adaptive control of variable-speed wind turbines for power capture optimisation. Transactions of the Institute of Measurement and Control 36(11): 1663–1672.

30.

Jiao

Meng

Yang

, et al. (2019) Adaptive Continuous neural pitch angle control for variable-speed wind turbines. Asian Journal of Control 21(2): 1–14.

31.

Jiao

Yang

Fan

, et al. (2020) EWSE and uncertainty and disturbance estimator based pitch angle control for wind turbine systems operating in above-rated wind speed region. ASME Journal of Dynamic Systems, Measurement and Control 142(3): 031006.

32.

Karami-Mollaee

Pariz

Shanechi

(2011) Position control of servomotors using neural dynamic sliding mode. Journal of Dynamic Systems, Measurement, and Control 133(6): 141–150.

33.

Koshkouei

Burnham

Zinober

(2005) Dynamic sliding mode control design. IEE Proceedings-control Theory and Applications 152(4): 392–396.

34.

Lasheen

Elnaggar

Yassin

(2020) Adaptive control design and implementation for collective pitch in wind energy conversion systems. ISA Transactions 102: 251–263.

35.

Lee

Utkin

(2007) Chattering suppression methods in sliding mode control systems. Annual Review in Control 31(2): 179–188.

36.

Levant

(1993) Sliding order and sliding accuracy in sliding mode control. International Journal of Control 58(6): 1247–1263.

37.

Levant

(2005) Homogeneity approach to high-order sliding mode design. Automatica 41(5): 823–830.

38.

Liu

Kung

Chang

, et al. (2013) Observer-based adaptive sliding mode control for pneumatic servo system. Precision Engineering 37(3): 522–530.

39.

Manwell

McGowan

Rogers

(2002) Wind Energy Explained: Theory, Design and Applications. Hoboken, NJ: Wiley.

40.

Nayeh

Moradi

Vossoughi

(2020) Multivariable robust control of a horizontal wind turbine under various operating modes and uncertainties: A comparison on sliding mode and H-infinity control. Electrical Power and Energy Systems 115: 105474.

41.

Nonaka

Yamashita

Tsubakino

(2015) General scheme for design of higher-order sliding-mode controller. American Control Conference (ACC)1: 5176–5181.

42.

Park

Lee

(2015) Implementation of a torque and a collective pitch controller in a wind turbine simulator to characterize the dynamics at three control regions. Renewable Energy 79: 150–160.

43.

Perruquetti

Pierre-Barbot

(2002) Sliding Mode Control in Engineering. New York: Marcel Dekker.

44.

Plestan

Glumineau

Laghrouche

(2008) A new algorithm for high-order sliding mode control. International Journal of Robust and Nonlinear Control 18(4–5): 441–453.

45.

Poultangari

Shahnazi

Sheikhan

(2012) RBF Neural network based PI pitch controller for a class of 5-MW wind turbines using particle swarm optimization algorithm. ISA Transactions 51(5): 641–648.

46.

Rahimi

Asadi

(2019) Control and dynamic response analysis of full converter wind turbines with squirrel cage induction generators considering pitch control and drive train dynamics. Electrical Power and Energy Systems 108: 280–292.

47.

Rahimi

Parniani

(2009) Dynamic behavior and transient stability analysis of fixed speed wind turbines. Renewable Energy 34(12): 2613–2624.

48.

Rahnavard

Ayati

Hairi Yazdi

(2018) Robust actuator and sensor fault reconstruction of wind turbine using modified sliding mode observer. Transactions of the Institute of Measurement and Control 41(6): 1504–1518.

49.

Rigatos

Siano

Abbaszadeh

, et al. (2019) Nonlinear optimal control for wind power generators comprising a multi-mass drivetrain and a DFIG. Journal of the Franklin Institute 356(5): 2582–2605.

50.

Shtessel

Fridman

Zinober

(2008) Higher order sliding modes. International Journal of Robust and Nonlinear Control 18(4–5): 381–384.

51.

Sitharthan

Parthasarathy

Sheeba Rani

, et al. (2019) An improved radial basis function neural network control strategy-based maximum power point tracking controller for wind power generation system. Transactions of the Institute of Measurement and Control 41(11): 3158–3170.

52.

Slotine

JJE

(1991) Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice Hall.

53.

Song

Dhinakaran

Bao

(2000) Variable speed control of wind turbines using nonlinear and adaptive algorithms. Wind Engineering and Industrial Aerodynamics 85(3): 293–308.

54.

Dou

, et al. (2020) Experimental investigation of a novel vertical axis wind turbine with pitching and self-starting function. Energy Conversion and Management 217: 113012.

55.

Sumper

Gomis-Bellmunt

Sudri-Andreu

, et al. (2009) Response of fixed speed wind turbines to system frequency disturbances. IEEE Transaction on Power Systems 24(1): 181–192.

56.

Xia

Zhu

(2011) Back-stepping sliding mode control for missile systems based on an extended state observer. IET Control Theory and Application 5(1): 93–102.

57.

Xiong

Saif

(2001) Sliding mode observer for nonlinear uncertain systems. IEEE Transactions on Automatic Control 46(12): 2012–2017.

58.

Yang

Qin

Jiang

(2017) A modified super-twisting sliding mode control with inner feedback and adaptive gain schedule. International Journal of Adaptive Control and Signal Processing 31(3): 398–416.

59.

Yin

Zhang

Jiang

, et al. (2019) Adaptive robust integral sliding mode pitch angle control of an electro-hydraulic servo pitch system for wind turbine. Mechanical Systems and Signal Processing 133: 105704.

60.

Yuan

Tang

(2017) Adaptive pitch control of wind turbine for load mitigation under structural uncertainties. Renewable Energy 105: 483–494.

61.

Yuan

Chen

Tang

(2020) Multivariable robust blade pitch control design to reject periodic loads on wind turbines. Renewable Energy 146: 329–341.

62.

Zhang

(2016) Sliding mode-like fuzzy logic control with adaptive boundary layer for multiple-variable discrete. Journal of Intelligent Systems 25(2): 209–220.