2. 山东交通学院信息科学与电气工程学院 济南 250357
2. School of Information Science and Electrical Engineering, Shandong Jiaotong University, Jinan 250357, China
Recent years wind energy conversion systems have advanced rapidly, due to the increasing concern for the environment pollution caused by the traditional energy sources. Through tracking the changing wind speed by adjusting the shaft speed, the variable speed wind turbines (VSWTs) enable the turbine to operate at its maximum power coefficient over a wide range of wind speeds [1]. To achieve the major task of power efficiency maximization, the VSWT should track the maximum power point despite wind variations [2].
Originally, the VSWT control system is designed based on the linear system theory [3][5]. The linear control encountered difficulties since VSWT system is a complex and highly nonlinear system with strong coupling features and uncertainty in both the aerodynamics and the electrical parts. To account for the nonlinear behavior, and to deal with the recognized problem of parameter variations, various advanced control strategies have been proposed, such as neural networks [6], adaptive control [7], feedback linearization technique [8], predictive control [9] and sliding mode approach [10][11]. Among these control strategies, sliding mode control (SMC) has proven to be very robust with respect to system parameter variations and external disturbances, and thus suitable for realizing VSWT system control. However, the standard firstorder sliding mode (FOSM) control generally shows significant drawback of chattering phenomenon. Some key components in the wind turbine (WT), such as gear box, would get damaged by the abrupt commutation in forces and torques [12].
Instead of influencing the first order time derivative of sliding variable, the secondorder sliding mode (SOSM) control acts on the second order derivatives of the sliding surfaces, which can help reduce the chattering phenomenon and provide higher sliding precision. Particularly, the SOSM supertwisting algorithm has been an effective way of controlling the VSWT system, since it only requires measurement of the sliding variable without using information about derivatives of the sliding constraint. A number of contributions on this control strategy have recently appeared for the realization of maximum power point tracking (MPPT) of VSWT system. Valenciaga et al.[13] obtained the control objective of wind energy conversion optimization of brushless doubly fed reluctance machine (BDFRM) based VSWT. Beltran et al. [14] presented a SOSM controller that facilitates the generator torque tracking the optimal torque, and maximized the power extraction of doubly fed induction generator (DFIG) based VSWT. Evangelista et al. [15] synthesized a supertwisting sliding mode control with variable gain, and showed a better performance in terms of mechanical loads and power tracking. However, in the aforementioned literatures, the SOSM control mainly concentrated on the electrical side dynamic control. For achieving the control objective of maximum power extraction, the generator torque was set to track the optimal aerodynamic torque, and it means that the energy loss caused by friction and mechanical rotational inertia in the transmission system was neglected in such circumstance. In other words, the captured power from wind was assumed to be equal to generator output power. This assumption is not always consistent with the actual situation, especially in turbulent wind scenarios.
Therefore, this paper aims to investigate a new SOSM control method for the DFIG based VSWT system. A complete dynamic model of the VSWT system is established by integrating both aerodynamic and electrical parts, together with parameter uncertainties and perturbations. The control objective is to make the wind turbine rotor speed track the desired speed (the speed that is given by a MPPT) in spite of system uncertainties, and maintain the rotor current at rated value which can minimize the reactive power. This control strategy can provide a faster control action as the wind speed variations can be reflected instantaneously and significantly on the rotor speed reference signal, and yield more energy. Besides, by adopting quadratic Lyapunov function [16], the range of control parameters is obtained to guarantee the finite time convergence of the SOSM control system.
The article is organized as follows. The modeling of the VSWT system is presented in Section 2. Section 3 explains the problem formulation and the detailed control strategy. The comparative simulation results for the 1.5 MW threeblade DFIGbased WT system are investigated in Section 4. Finally, the conclusion is stated in Section 5.
2 VSWT System ModelingThe VSWT system is mainly composed of a turbine, a gearbox, and a generator, which are combined to convert wind energy into electric energy. According to different wind speeds, the VSWT system works at four different operating regions, as shown in Fig. 1. While the wind speed is below the cutin speed
Obviously, the control task in the partial load region is more challenging, since the dynamics in this region is quite complex. Thus the study in this paper concentrates on the modeling and control in this region. The mathematical model in this region consists of formulas accounting for the electrical dynamics of the rotor and the dynamics of the mechanical rotational speed, including also the damping and resistance uncertainties caused by longterm mechanical wear and ambient temperature changes [8], [14]. It is expressed as:
$ \begin{align} \label{eq1}\begin{cases} \dot {\omega }_w =\frac{T_w }{J}\frac{K+\Delta K}{J}\omega _w +\frac{3L_m \phi _s n_p n_g }{2L_s J}I_{rq} \\[2mm] \dot {I}_{rd} =\frac{L_s (R_r +\Delta R_r )}{L_m ^2L_r L_s }I_{rd} +\omega _1 I_{rq} n_p n_g \omega _w I_{rq} \\[1mm] \qquad\ \ +~\frac{L_s }{L_r L_s L_m ^2}U_{rd} \\[2mm] \dot {I}_{rq} =\frac{L_m \phi _s n_p n_g }{L_r L_s L_m ^2}\omega _w \omega _1 I_{rd} +\frac{L_s (R_r +\Delta R_r )}{L_m ^2L_r L_s }I_{rq} \\[1mm] \qquad\ \ +~n_p n_g \omega _w I_{rd} +\frac{L_s }{L_r L_s L_m ^2}U_{rq} \frac{L_m \phi _s \omega _1 }{L_r L_s L_m ^2} \end{cases} \end{align} $  (1) 
where
wind turbine rotor speed;  
the 

the 

the 

the 

the inertia of the combined rotating parts;  
turbine total external damping;  
gearbox ratio;  
stator flux;  
mutual inductance;  
stator leakage inductance;  
rotor leakage inductance;  
rotor resistance;  
synchronous speed;  
pole pair number. 
In order to concisely express (1), define
$ \begin{align*}&k_1 =\frac{1}{J}, \hspace{20mm} k_2 =\frac{K}{J} \\&\Delta k_2 ={\Delta K}{J}, \hspace{12.6mm}k_3 =\frac{3L_m \phi _s n_p n_g }{2L_s J}\\& k_4 =\frac{L_s R_r }{L_m ^2L_r L_s }, \hspace{6.1mm}\Delta k_4 =\frac{\Delta R_r L_s }{L_m ^2L_r L_s } \\ &k_5 =\omega _1, \hspace{20mm} k_6 =n_p n_g\\ & k_7 =\frac{L_s }{L_r L_s L_m ^2}, \hspace{6.5mm} k_8 =\frac{L_m \phi _s n_p n_g }{L_r L_s L_m ^2}\\& k_9 =\frac{L_m \phi _s \omega _1 }{L_r L_s L_m ^2} \end{align*} $ 
this leads to
$ \begin{align} \label{eq2} \begin{cases} \dot {\omega }_w =k_1 T_w (k_2 +\Delta k_2 )\omega _w +k_3 I_{rq} \\ \dot {I}_{rd} =(k_4 +\Delta k_4 )I_{rd} +k_5 I_{rq} k_6 \omega _w I_{rq} +k_7 U_{rd} \\ \dot {I}_{rq} =(k_4 +\Delta k_4 )I_{rq} k_5 I_{rd} +k_6 \omega _w I_{rd} +k_7 U_{rq}\\ \qquad\ \ +~k_8 \omega _w k_9. \end{cases} \end{align} $  (2) 
The VSWT system model (2) is composed of two parts, e.g., the turbine model and the DFIG model. The single mass model of the turbine is [8]:
$ \begin{align} \label{eq3} J\dot {\omega }_w =T_w K\omega _w n_g T_{em} \end{align} $  (3) 
where
The nonlinear characteristics of
$ \begin{align} \label{eq4} T_w =\frac{P_w }{\omega _w }=\frac{1}{2}\rho \pi R^3\frac{C_p (\lambda, \beta )}{\lambda }v^2 \end{align} $  (4) 
where
$ \begin{align} \label{eq5} \lambda =\frac{\omega _w R}{v}. \end{align} $  (5) 
The DFIG model in the synchronously rotating frame
$ \begin{align} \label{eq6} \begin{cases} U_{sd} =0 \\ U_{sq} =\omega _1 \phi _{sd} =U_s \\ U_{rd} =R_r I_{rd} +\left( {L_r \frac{L_m ^2}{L_s }} \right)\dot {I}_{rd} \left( {L_r \frac{L_m ^2}{L_s }} \right)I_{rq} \omega _s \\ U_{rq} =R_r I_{rq} +\left( {L_r \frac{L_m ^2}{L_s }} \right)\dot {I}_{rq} +\left( {L_r \frac{L_m ^2}{L_s }} \right)I_{rd} \omega _s \\\qquad\ \ +~\frac{L_m \phi _S }{L_s }\omega _s \\ T_{em} =\frac{3}{2}n_p \frac{L_m \phi _S }{L_s }I_{rq}. \end{cases} \end{align} $  (6) 
Considering (4) and (5) in Section 2, the aerodynamic power that can be captured by a wind turbine is:
$ \begin{align} \label{eq7} P_w =\frac{1}{2}\rho \pi R^2C_p (\lambda, \beta )v^3. \end{align} $  (7) 
As the main control objective of the above VSWT system in the partial load region is to track the maximum power point and harvest more wind energy, the power conversion coefficient
$ \begin{align} \label{eq8} C_p \left(\lambda, \beta \right)=c_1 \left(\frac{c_2 }{\Lambda }c_3 \beta c_4 \right)\times e^{\frac{c_5 }{\Lambda }}+c_6 \lambda \end{align} $  (8) 
with
Actually, in the partial load region, the pitch angle
Considering Fig. 3 and the definition of
$ \begin{align} \label{eq9} \omega _{\rm opt} =\frac{\lambda _{\rm opt} v}{R}. \end{align} $  (9) 
In the DFIG based VSWT system (2), this maximum wind energy capture objective can be achieved by means of the rotor voltage regulation in the generator. The rotor voltage can also control the rotor current
$ \begin{align} \label{eq10} Q_s =\frac{3}{2}(U_{sd} I_{sq} +U_{sq} I_{sd} )=\frac{3U_s (\phi _s L_m I_{rd} )}{2L_s }. \end{align} $  (10) 
Thus the designed SOSM control system should accomplish two major objectives. One is to maximize power extraction, by controlling the rotor speed to track the optimal rotor speed
Here,
$ \begin{align} \label{eq11} I_{rdr} =\frac{U_s }{L_m \omega _1 }. \end{align} $  (11) 
In order to achieve the control objectives in the partial load region, the nonlinear SMC scheme is presented in this section. The schematic diagram of the DFIGbased WT system is shown in Fig. 4.
In designing a general SMC for the VSWT system, the tracking error between the actual rotor speed and the desired value
$ \begin{align} \label{eq12} \begin{cases} e_1 =\omega _w \omega _{\rm opt} \\ e_2 =I_{rd} I_{rdr}. \end{cases} \end{align} $  (12) 
Then the sliding variables are defined as follows:
$ \begin{align} \label{eq13} \sigma _1 &=ce_1 +\dfrac{{d}}{{d}t}e_1\notag \\ &=\dot {\omega }_w +c\omega _w c\omega _{\rm opt} \dot {\omega }_{\rm opt} \end{align} $  (13) 
where
$ \begin{align} \label{eq14} \sigma _2 =e_2 =I_{rd} I_{rdr}. \end{align} $  (14) 
Respecting the VSWT system model (2), the firstorder derivatives of sliding variables are
$ \begin{align} \dot {\sigma }_1 =&\ \ddot {e}_1 +c\dot {e}_1 \notag\\ =&\ k_1 \dot {T}_w +(ck_2 \Delta k_2 )\dot {\omega }_w +k_3 k_7 U_{rq}\notag\\ &\ddot {\omega }_{\rm opt} c\dot {\omega }_{\rm opt} +k_3 ( (k_4 +\Delta k_4 )I_{rq}\notag \\ & k_5 I_{rd} +k_6 \omega _w I_{rd} +k_8 \omega _w k_9 ) \notag\\ =&\ G_1 +k_3 k_7 U_{rq} \end{align} $  (15) 
$ \begin{align} \dot {\sigma }_2 =&\ \dot {e}_2 =\dot {I}_{rd} \dot {I}_{rdr} \notag\\ =&\ (k_4 +\Delta k_4 )I_{rd} \dot {I}_{rdr} +k_5 I_{rq} k_6 \omega _w I_{rq} +k_7 U_{rd}\notag \\ =&\ G_2 +k_7 U_{rd} \end{align} $  (16) 
where
$ \begin{align} G_1 =&\ k_1 \dot {T}_w +(ck_2 \Delta k_2 )\dot {\omega }_w \ddot {\omega }_{\rm opt} c\dot {\omega }_{\rm opt} \notag \\ & +k_3 \left( {(k_4 +\Delta k_4 )I_{rq} k_5 I_{rd} +k_6 \omega _w I_{rd} +k_8 \omega _w k_9 } \right) \end{align} $  (17) 
$ \begin{align} \label{eq18} G_2 =&\ (k_4 +\Delta k_4 )I_{rd} \dot {I}_{rdr} +k_5 I_{rq} k_6 \omega _w I_{rq}. \end{align} $  (18) 
In (13) and (15), since
Using the standard SMC with the approaching law method [18], the socalled exponential approaching law is selected as
$ \begin{align} \label{eq19} \dot {\sigma }=\varepsilon {\rm sgn}(\sigma )\delta \sigma \end{align} $  (19) 
where
Then the
$ \begin{align} \label{eq20} &U_{rq} =\frac{1}{k_3 k_7 }(\varepsilon _1 {\rm sgn}(\sigma _1 )\delta _1 \cdot \sigma _1 G_1 ) \end{align} $  (20) 
$ \begin{align} \label{eq21} &U_{rd} =\frac{1}{k_7 }(\varepsilon _2 {\rm sgn}(\sigma _2 )\delta _2 \cdot \sigma _2 G_2 ). \end{align} $  (21) 
Since the switching term
Using the supertwisting algorithm [19], the control inputs
$ \begin{align} \label{eq22} \begin{cases} U_{rq} =u_1 +u_2 \\ u_1 =\gamma _1 \vert \sigma _1 \vert ^\frac{1}{2}{\rm sgn}(\sigma _1 ) \\ \dot {u}_2 =\phi _1 {\rm sgn}(\sigma _1 ) \end{cases} \end{align} $  (22) 
$ \begin{align} \begin{cases} U_{rd} =u_3 +u_4 \\ u_3 =\gamma _2 \vert \sigma _2 \vert ^\frac{1}{2}{\rm sgn}(\sigma _2 ) \\ \dot {u}_4 =\phi _2 {\rm sgn}(\sigma _2 ) \end{cases} \end{align} $  (23) 
where
Notice that the discontinuity of supertwisting SOSM control appears only in its derivative term, such that the control inputs
By using this SOSM scheme, it is very important to choose suitable control parameters for stable operation of WT system. Here a Lyapunov method is employed to determine the range of control parameters, and then guarantee stable operation of the WT system.
In order to ensure
$ \begin{align} \label{eq24} y=H_1 k_3 k_7 \phi _1 \int_0^t {{\rm sgn}(\sigma _1 )} d\tau \end{align} $  (24) 
and taking into consideration (22), then, (15) is deduced as
$ \begin{align} \label{eq25} \begin{cases} \dot {\sigma }_1 =k_3 k_7 \gamma _1 \vert \sigma _1 \vert ^\frac{1}{2}{\rm sgn}(\sigma _1 )+y \\[1mm] \dot {y}=k_3 k_7 \phi _1 {\rm sgn}(\sigma _1 )+\dot {G}_1. \end{cases} \end{align} $  (25) 
As shown in (17), the perturbation
$ \begin{align} \label{eq26} \left {\dot {G}_1 } \right\le \Pi _1 {\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\forall t>0 \end{align} $  (26) 
where
Now, choose the quadratic Lyapunov function
$ \begin{align} \label{eq27} V(\sigma, y)=\zeta ^TP\zeta \end{align} $  (27) 
where
$ P=\left[ {\begin{array}{l} 2k_3 k_7 \phi _1 +\frac{1}{2}k_3 ^2k_7 ^2\gamma _1 ^2{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\frac{1}{2}k_3 k_7 \gamma _1 \\[1mm] {\kern 1pt}\frac{1}{2}k_3 k_7 \gamma {\kern 1pt}_1 {\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}1 \\ \end{array}} \right].{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt} $ 
From
$ \begin{align} \label{eq28} \dot {\zeta }=\frac{1}{\left {\zeta _1 } \right}\left(A\zeta +B\hat {\dot {G}}_1 \right) \end{align} $  (28) 
where the matrix
Setting the matrix
$ \begin{align} \dot {V}(\sigma, y)=&\ 2\dot {\zeta }^TP\zeta \notag\\ =&\ \frac{1}{\left {\zeta _1 } \right}(2\zeta ^TA^T+2\hat {\dot {G}}_1 B^T)P\zeta\notag \\ \le&\ \frac{1}{\left {\zeta _1 } \right}(2\zeta ^TA^TP\zeta +2\hat {\dot {G}}_1 B^TP\zeta +\Pi _1 ^2\left {\zeta _1 } \right\hat {\dot {G}}_1 ^2)\notag \\ =&\ \frac{1}{\left {\zeta _1 } \right}(2\zeta ^TA^TP\zeta +2\hat {\dot {G}}_1 B^TP\zeta +\Pi _1 ^2\zeta ^TC^TC\zeta\notag\\ & \hat {\dot {G}}_1 ^2)\notag \\ \le&\ \frac{1}{\left {\zeta _1 } \right}(\zeta ^TA^TP\zeta +\zeta ^TPA\zeta +\Pi _1 ^2\zeta ^TC^TC\zeta\notag \\ &+\zeta ^TPBB^TP\zeta )\notag \\ =&\ \frac{1}{\left {\zeta _1 } \right}\zeta ^T(A^TP+PA+\Pi _1 ^2C^TC+PBB^TP)\zeta. \end{align} $  (29) 
Choose
$ \begin{align} \label{eq30} \dot {V}\le \frac{1}{\left {\zeta _1 } \right}\zeta ^TQ\zeta. \end{align} $  (30) 
$ \begin{align} Q&=(A^TP+PA+\Pi _1 ^2C^TC+PBB^TP)\notag \\ &=\left[{\begin{array}{l} k_3 ^2k_7 ^2\Big(\dfrac{1}{2}k_3 k_7 \gamma _1 ^3+\gamma _1 \phi _1 \dfrac{1}{4}\gamma _1 ^2\Big)\Pi _1 ^2{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\dfrac{1}{2}k_3 k_7 \Big(\gamma _1k_3 k_7 \gamma _1 ^2\Big) \\[2mm] {\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\dfrac{1}{2}k_3 k_7 \Big(\gamma _1 k_3 k_7 \gamma _1 ^2\Big){\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\dfrac{1}{2}{\kern 1pt}k_3 k_7 \gamma _1 {\kern 1pt}1{\kern 1pt}{\kern 1pt}{\kern 1pt} \\ \end{array}} \right] \end{align} $  (31) 
When
The matrix
Set
$ \begin{align} & {\begin{cases} \gamma _1 >\dfrac{2}{k_3 k_7 }\\ \phi _1 >\dfrac{k_3 k_7 \gamma _1 ^2}{4(k_3 k_7 \gamma _1 2)}+\dfrac{\Pi _1 ^2}{k_3 k_7 \gamma _1 } \end{cases}} {\kern 1pt} \notag\\[2mm] &\quad\ \ \Rightarrow \begin{cases} \gamma _1 >\dfrac{4J(L_r L_s L_m ^2)}{3n_g n_p L_m \phi _s } \\ \phi _1 >\dfrac{3L_m \phi _s n_p n_g \gamma _1 ^2}{12L_m \phi _s n_p n_g \gamma _1 16J(L_r L_s L_m ^2)}\\[4mm] \qquad\ \ +\dfrac{2J\Pi _1 ^2(L_r L_s L_m ^2)}{3L_m \phi _s n_p n_g \gamma _1 }. \end{cases} \end{align} $  (32) 
Similarly, the perturbation
$ \begin{align} \label{eq31} \left {\dot {G}_2 } \right\le \Pi _2 {\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\forall t>0. \end{align} $  (33) 
The ranges of
$ \begin{align} \label{eq32}&\begin{cases} \gamma _2 >\dfrac{2}{k_7 } \\ \phi _2 >\dfrac{k_7 \gamma _2 ^2}{4(k_7 \gamma _2 2)}+\dfrac{\Pi _2 ^2}{k_7 \gamma _2 } \end{cases} {\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt} \notag\\[2mm] &\quad\ \ \Rightarrow \begin{cases} \gamma _2 >\dfrac{2(L_r L_s L_m ^2)}{L_s } \\ \phi _2 >\dfrac{L_s \gamma _2 ^2}{4L_s \gamma _2 8(L_r L_s L_m ^2)}+\dfrac{\Pi _2 ^2(L_r L_s L_m ^2)}{L_s \gamma _2 }. \end{cases} \end{align} $  (34) 
As long as the control parameters
The proposed SOSM was simulated on a 1.5 MW WT system. The system parameters based on a 1.5 MW wind turbine are shown in Table Ⅰ [14]. The parameter perturbations of resistances
In order to verify the control effect under different operating environment, two case studies are considered in this section. Case 1 uses a stepwisevarying wind speed to test the validity of the proposed SOSM control strategy under a sudden wind change. A PID control and a standard first order sliding mode (FOSM) control are also constituted in this case, to show the robustness and chatteringfree behavior of the SOSM controller. The effectiveness has been further demonstrated using the turbulent wind speed in Case 2, to test validity of the proposed SOSM control strategy under realistic VSWT conditions.
Generally, the cutin wind speed of 1.5 MW WT is 3 m/s, the cutout wind speed is 25 m/s, and the rated wind speed is in the range of 11 m/s to 13 m/s. Therefore, in these simulations, the wind speed varies within the range between 3 m/s and 12 m/s, making the WT system operates in the partial load region. Two case studies are described as follows.
4.1 Case 1: Stepwise Wind SpeedIn order to show the controller performances under a sudden wind change, the fast step variations of the wind speed are used in this simulation, as shown in Fig. 5. For comparing purpose, a PID control and a FOSM control are also constituted. The PID control parameters are chosen as
Fig. 6 shows the regulation performances of the wind turbine rotor speed
The actual controller output
In order to evaluate the tracking performance, robustness and chatteringfree behavior of the proposed SOSM control strategy under realistic VSWT conditions, a 10min randomly varying wind speed is adopted in the simulation. As shown in Fig. 8, the wind speed is ranging between 3 m/s and 10 m/s. The evolution of the rotor speed and the optimal speed is depicted in Fig. 9 (a). Compared with the PID controller, the SOSM controller features more accurate and faster response in rotor speed tracking.
The evolution of the SOSM controller output
In this paper, a new SOSM control approach for DFIGbased VSWT system is proposed to achieve the objectives of maximum power point tracking and minimum stator reactive power. The two SOSM controllers are designed based on the supertwisting algorithm, which only require measurement of the sliding variables without using information about the time derivatives of the sliding constraint. Quadratic form Lyapunov function method is employed to choose controller parameters, and guarantee the finite time stabilization of closedloop system. The controllers are simulated based on a complete model of the DFIGbased VSWT, which includes both the mechanical and the electric dynamics, together with parameter uncertainties and perturbations. A PID control method and a standard SMC are also carried out for comparison. The simulation results indicate that the proposed control strategy is rather suitable for controlling the DFIGbased VSWT system, and the control objectives are successfully achieved in both step wisevarying wind speed and a randomly varying turbulent wind speed. Compared with the PID controller and conventional firstorder SMC, the proposed control strategy shows a higher robustness, and the chattering phenomenon of rotor control voltage is almost eliminated.
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