Sliding Mode Control of a Variable- Speed Wind Energy Conversion System Using a Squirrel Cage Induction Generator

Abstract

This paper deals with the control of a variable-speed wind energy conversion (WEC) system using a squirrel cage induction generator (SCIG) connected to the grid through a back-to-back three phase (AC-DC-AC) power converter. The sliding mode control technique is used to control the WEC system. The objective of the controllers is to force the states of the system to track their desired states. One controller is used to regulate the generator speed and the flux so that maximum power is extracted from the wind. Another controller is used to control the grid side converter, which controls the DC bus voltage and the active and reactive powers injected into the grid. The performance of the controlled wind energy conversion system is verified through MATLAB simulations, which show that the controlled system performs well.

1. Introduction

Wind energy has been used to generate electricity for a long time. However, it is more prevalent nowadays because the cost of wind energy has continuously dropped, and it is approaching the competitive level of conventional energy [1]. Moreover, wind energy generation does not contribute to the pollution of the environment. The generators that are used to convert the mechanical power obtained from the wind turbine into electric power are generally either doubly-fed induction generators (DFIG), or squirrel cage induction generators (SCIG), or permanent magnet synchronous generators (PMSG). Therefore, variable-speed wind energy conversion systems can be broadly classified into three types: (i) DFIG-based wind energy conversion (WEC) systems; (ii) SCIG-based WEC systems; and (iii) PMSG-based WEC systems. For variable-speed operations, all three above-mentioned WEC systems need power electronic converters. The PMSG- and the SCIG-based systems need full-scale power electronic converters, whereas the DFIG-based systems need partial-scale power converters. Compared to the DFIG-based systems, the PMSG- and the SCIG-based systems could be more attractive due to the dropping cost of power electronics over time and due to the absence of brushes [2]. Hence, the main advantages of SCIG-based systems are their reliability, their ruggedness in design and their low operation and maintenance costs.

In an SCIG-based WEC system, the generator is coupled to the grid through back-to-back three-phase (AC-DC-AC) power converters. The AC/DC converter or the stator side converter controls the generator; the DC/AC converter or the grid side converter controls the DC bus voltage and the active and reactive powers injected into the grid. The controllers are designed so that maximum power is extracted from the wind at all wind speeds; thus, maximum system efficiency is achieved; the generated active power is transmitted through the DC-bus to the grid while ensuring the unity power factor. This paper proposes the use of the sliding mode control technique to control both the stator side and the grid side converters of an SCIG-based wind energy conversion system.

Extensive work was done on wind energy conversion systems. Mahela and Shaik [3] presented an overview of wind energy generation systems. Cheng and Zhu [4] surveyed the state of the art of wind energy conversion systems and their technologies. Chen et al. [5] reviewed the power electronics used for wind turbines systems. Papers by [1,6,7,8] discussed the state of the art of the control techniques applied to the different types of wind energy conversion systems.

Several control techniques were used to control the SCIG-based wind energy conversion system. Benchagra and his colleagues [9,10,11,12,13] published several papers on the control of SCIG-based WEC systems. Wang et al. [14] proposed a passivity-based robust controller for an SCIG-based WEC system. Hassan [15] used the backstepping control technique to control an SCIG-based WEC system. Zhao et al. [16] used an adaptive controller for maximum power point tracking (MPPT) for an SCIG-based WEC system. Heydari-doostabad et al. [17] proposed an optimal linear quadratic regulator (LQR) controller for a SCIG-based WEC system using a static compensator (STATCOM). Baloch et al. [18] employed the feedback linearization technique through field control concepts to control an SCIG-based WEC system. Domínguez-García [19] used indirect vector control for the squirrel cage induction generator WEC system. The sliding mode control (SMC) technique was used for induction generator-based WEC systems in [20,21,22,23,24]. Moreover, other types of controllers were also designed for squirrel cage-based WEC systems; for example, see [25,26,27,28,29,30,31].

The sliding mode technique is a very suitable approach to deal with the control of wind energy conversion systems. This is the case because this technique is robust with respect to variations in the parameters of the system and to bounded external disturbances. Therefore, the SMC technique is selected to control the squirrel cage induction generator-based WEC system.

The contribution of this paper involves the design of two controllers, which force the states of a squirrel cage induction generator-based wind energy conversion system to track the desired states of the system and the transmission of the the generated power to the grid. The first sliding mode controller enables maximum extraction of power from the wind at different speeds of the wind; the second sliding mode controller ensures that the generated active power is injected through the DC-bus to the grid with the unity power factor. The proposed controllers are validated through MATLAB simulation.

The rest of the paper is organized as follows. The model of an SCIG-based wind energy conversion system is presented in Section 2. The model of the system consists of two parts. The first part is composed of the model of the wind turbine and the induction generator; the second part consists of the model of the grid side converter. The computation of the desired states of the WEC system and the computation of the reference inputs are presented in Section 3. The design of sliding mode controllers for the WEC system is proposed in Section 4. Simulation results for the controlled system are presented and discussed in Section 5. Finally, the conclusion is given in Section 6.

2. Model of the Wind Energy Conversion System

This section presents the model of the wind energy conversion system, which is depicted in Figure 1. The model of the system consists of two parts. One part deals with the model of the turbine and the squirrel cage induction generator (SCIG). This part of the system will be controlled through the stator side AC/DC Converter. The second part of the model deals with the grid side DC/AC converter sub-system.

2.1. The Model of the Turbine and the Induction Generator

The mechanical power extracted from the wind can be expressed as follows:

$$\begin{array}{ccc}\hfill P_m& =& \frac{1}{2}{C}_p(\lambda ,\beta )\rho A{V}^3\hfill \end{array}$$

(1)

where $P_m$ is the mechanical output power of the wind turbine; the power coefficient is represented by $C_p(\lambda ,\beta )$; $\lambda$ is the tip speed ratio $(TSR)$ of the turbine blade; and $\beta$ is the pitch angle. The constant $\rho$ represents the air density; A is the area swept by the rotor blades; and V is the speed of the wind.

The gearbox transfers power from the rotating wind turbine shaft to the rotating generator shaft. The angular speed of the generator ${\omega }_m$ is related to the angular speed of the wind turbine ${\omega }_t$ through the following equation:

$$\begin{array}{ccc}\hfill {\omega }_m& =& G{\omega }_t\hfill \end{array}$$

(2)

where G is the gear ratio.

The tip speed ratio is related to the angular speed of the wind turbine ${\omega }_t$ and the angular speed of the generator ${\omega }_m$ through the following equation:

$$\begin{array}{ccc}\hfill \lambda & =& \frac{R{\omega }_t}{V}=\frac{R{\omega }_m}{GV}\hfill \end{array}$$

(3)

where R is the radius of the turbine blades.

The power coefficient $C_p$ is a nonlinear function of $\lambda$; several nonlinear models of $C_p$ can be found in the literature. The model of the power coefficient adopted in this paper is as follows [32]:

$$\begin{array}{ccc}\hfill C_p(\lambda ,\beta )& =& a_1\left(\frac{a_2}{\overline{\lambda }}-a_3\beta -a_4\right)exp(-\frac{a_5}{\overline{\lambda }})+a_6\lambda \hfill \end{array}$$

(4)

where $\overline{\lambda }=1/[1/(\lambda +a_7\beta )-a_8/({\beta }^3+1)]$ with $a_1=0.5109$, $a_2=116$, $a_3=0.4$, $a_4=5$, $a_5=21$, $a_6=0.0068$, $a_7=0.08$ and $a_8=0.035$. Figure 2 depicts the power coefficient $C_p$ versus the tip speed ratio $\lambda$ for different pitch angles. The figure shows that the maximum value of the power coefficient $C_p$ is $C_{p_{max}}=0.47$; this value is obtained when the pitch angle $\beta$ equals zero degrees and the corresponding optimum tip speed ratio is ${\lambda }_{opt}=8.1$.

Note that the maximum mechanical power extracted from the wind is such that:

$$\begin{array}{ccc}\hfill P_{m_{max}}& =& \frac{1}{2}{C}_{p_{max}}{\lambda }_{opt}\rho A{V}^3\hfill \end{array}$$

(5)

Figure 3 shows the wind turbine mechanical (aerodynamic) power $P_m$ versus the generator speed ${\omega }_m$ for different values of the wind velocity V. Furthermore, the graph of the maximum mechanical power $P_{m_{max}}$ is shown in black in Figure 3.

The mechanical (or aerodynamic) torque is related to the the power $P_m$ captured by the wind turbine through the equation:

$$\begin{array}{ccc}\hfill T_m& =& \frac{P_m}{{\omega }_t}\hfill \end{array}$$

(6)

The mechanical equation of the system is such that:

$$\begin{array}{c}\hfill J\frac{d{\omega }_m}{dt}=T_{em}-T_m-B{\omega }_m\end{array}$$

(7)

In the above equation, $T_{em}$ is the electromagnetic torque of the induction generator, and the inertia J is the total inertia of the turbine and the induction generator. This inertia is calculated such that $J=\frac{J_t}{G^2}+J_g$ where $J_t$ and $J_g$ represent the inertia of the turbine and the inertia of the generator respectively, and G is the gearbox ratio. Furthermore, B is the total external damping of the system.

In the direct-quadrature frame, the differential equations governing the flux linkages of the stator and the rotor of the generator are as follows [9]:

$$\begin{array}{ccc}\hfill \frac{d{\psi }_{ds}}{dt}& =& -R_s{i}_{ds}+{\omega }_s{\psi }_{qs}+v_{ds}\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill \frac{d{\psi }_{qs}}{dt}& =& -R_s{i}_{qs}-{\omega }_s{\psi }_{ds}+v_{qs}\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill \frac{d{\psi }_{dr}}{dt}& =& -R_r{i}_{dr}+{\omega }_r{\psi }_{qr}\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill \frac{d{\psi }_{qr}}{dt}& =& -R_r{i}_{qr}-{\omega }_r{\psi }_{dr}\hfill \end{array}$$

(11)

where ${\psi }_{ds}$ is the stator direct axis flux linkage and ${\psi }_{qs}$ is the stator quadrature axis flux linkage. The flux ${\psi }_{dr}$ is the rotor direct axis flux linkage, and ${\psi }_{qr}$ is the rotor quadrature axis flux linkage. The currents $i_{ds}$ and $i_{qs}$ are the stator currents in the direct-quadrature frame; and $i_{dr}$ and $i_{qr}$ are the rotor currents in the direct-quadrature frame. The voltages $v_{ds}$ and $v_{qs}$ represent the voltages of the windings of the stator in the direct-quadrature frame. The resistances $R_s$ and $R_r$ are the stator and rotor phase resistances. Furthermore, ${\omega }_s$ is the synchronous speed, and ${\omega }_r$ is the angular speed of the rotor.

The relationship between the stator and rotor flux linkages and the currents can be expressed as:

$$\begin{array}{ccc}\hfill {\psi }_{ds}& =& L_s{i}_{ds}+L_m{i}_{dr}\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill {\psi }_{qs}& =& L_s{i}_{qs}+L_m{i}_{qr}\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill {\psi }_{dr}& =& L_r{i}_{dr}+L_m{i}_{ds}\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill {\psi }_{qr}& =& L_r{i}_{qr}+L_m{i}_{qs}\hfill \end{array}$$

(15)

where $L_s$ is the stator inductance and $L_r$ is the rotor inductance; $L_m$ is the mutual inductance between the stator and the rotor.

The equation of the electromagnetic torque $T_{em}$ in the d-q frame can be written as follows:

$$\begin{array}{c}\hfill T_{em}=\frac{p{L}_m}{L_r}({\psi }_{dr}{i}_{qs}-{\psi }_{qr}{i}_{ds})\end{array}$$

(16)

where p is the number of pole pairs.

The active and reactive powers transmitted by the stator $P_s$ and $Q_s$ can be expressed as:

$$\begin{array}{ccc}\hfill P_s& =& \frac{3}{2}(v_{ds}{i}_{ds}+v_{qs}{i}_{qs})\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill Q_s& =& \frac{3}{2}(v_{ds}{i}_{qs}-v_{qs}{i}_{ds})\hfill \end{array}$$

(18)

Combining Equations (7)–(16), we can write the following system of ordinary differential equations (o.d.e.’s):

$$\begin{array}{ccc}\hfill \frac{d{i}_{ds}}{dt}& =& -c_1{i}_{ds}+{\omega }_s{i}_{qs}+c_2{\psi }_{dr}+c_3{\omega }_m{\psi }_{qr}+c_4{v}_{ds}\hfill \\ \hfill \frac{d{i}_{qs}}{dt}& =& -c_1{i}_{qs}-{\omega }_s{i}_{ds}+c_2{\psi }_{qr}-c_3{\omega }_m{\psi }_{dr}+c_4{v}_{qs}\hfill \\ \hfill \frac{d{\psi }_{dr}}{dt}& =& c_5{i}_{ds}-c_6{\psi }_{dr}+({\omega }_s-p{\omega }_m){\psi }_{qr}\hfill \\ \hfill \frac{d{\psi }_{qr}}{dt}& =& c_5{i}_{qs}-c_6{\psi }_{qr}-({\omega }_s-p{\omega }_m){\psi }_{dr}\hfill \\ \hfill \frac{d{\omega }_m}{dt}& =& c_7({\psi }_{dr}{i}_{qs}-{\psi }_{qr}{i}_{ds})-c_8{\omega }_m-c_9{T}_m\hfill \end{array}$$

(19)

where, $c_1=\frac{L_r^2{R}_s+L_m^2{R}_r}{\sigma L_s{L}_r^2}$, $c_2=\frac{L_m{R}_r}{\sigma L_s{L}_r^2}$, $c_3=\frac{p{R}_r}{\sigma L_s{L}_r}$, $c_4=\frac{1}{\sigma L_s}$, $c_5=\frac{L_m{R}_r}{L_r}$, $c_6=\frac{R_r}{L_r}$, $c_7=\frac{p{L}_m}{J{L}_r}$, $c_8=\frac{B}{J}$, $c_9=\frac{1}{J}$.

By fixing the direct axis of the rotating d-q reference frame on the rotating flux vector, we obtain ${\psi }_{qr}=0$. Therefore, the corresponding equation is removed from the model of the system, and the system of o.d.e’s is reduced accordingly. Furthermore, we will define the state variable vector $x_T$ comprised of the states $x_1$–$x_4$, such that:

$$\begin{array}{ccc}\hfill x_T& =& {\left[x_1{x}_2{x}_3{x}_4\right]}^T={\left[i_{ds}{i}_{qs}{\psi }_{dr}{\omega }_m\right]}^T\hfill \end{array}$$

(20)

Therefore, the model of the wind turbine and the induction generator in (20) can be written in compact form as follows:

$$\begin{array}{ccc}{\dot{x}}_1\hfill & =\hfill & -c_1{x}_1+{\omega }_s{x}_2+c_2{x}_3+c_4{v}_{ds}\hfill \\ {\dot{x}}_2\hfill & =\hfill & -c_1{x}_2-{\omega }_s{x}_1-c_3{x}_3{x}_4+c_4{v}_{qs}\hfill \\ {\dot{x}}_3\hfill & =\hfill & c_5{x}_1-c_6{x}_3\hfill \\ {\dot{x}}_4\hfill & =\hfill & c_7{x}_2{x}_3-c_8{x}_4-c_9{T}_m\hfill \end{array}$$

(21)

We will define the desired state vector $x_d$ such that, $x_d={\left[x_{1d}{x}_{2d}{x}_{3d}{x}_{4d}\right]}^T$ = ${\left[i_{ds}^{*}{i}_{qs}^{*}{\psi }_{dr}^{*}{\omega }_m^{*}\right]}^T$ where $x_{id}(i=1,2,\dots ,4)$ are the desired values of the states $x_i(i=1,2,\dots ,4)$ of the WEC system in (21). Since the desired states have to be an operating point of the WEC system, then these states must satisfy the model of the WEC system in (21). Therefore, the desired states of the model of the wind turbine and the induction generator system are governed by the following set of differential equations:

$$\begin{array}{ccc}\hfill {\dot{x}}_{1d}& =& -c_1{x}_{1d}+{\omega }_s{x}_{2d}+c_2{x}_{3d}+c_4{v}_{{ds}_r}\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill {\dot{x}}_{2d}& =& -c_1{x}_{2d}-{\omega }_s{x}_{1d}-c_3{x}_{3d}{x}_{4d}+c_4{v}_{{qs}_r}\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\dot{x}}_{3d}& =& c_5{x}_{1d}-c_6{x}_{3d}\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill {\dot{x}}_{4d}& =& c_7{x}_{2d}{x}_{3d}-c_8{x}_{4d}-c_9{T}_{mr}\hfill \end{array}$$

(25)

Note that $v_{{ds}_r}$, $v_{{qs}_r}$ and $T_{mr}$ in (22), (23) and (25) are the reference input voltages and the reference mechanical torque of the WEC system. The computation of desired states and the reference inputs will be discussed in Section 3.

Define the errors $e_i(i=1,2,.....,4)$ such that:

$$\begin{array}{ccc}\hfill e_i& =& x_i-x_{id}\hfill \end{array}$$

(26)

Using Equations (21)–(26), the dynamic model of the error of the wind turbine and the induction generator system can be written such that:

$$\begin{array}{ccc}{\dot{e}}_1\hfill & =\hfill & -c_1{e}_1+{\omega }_s{e}_2+c_2{e}_3+c_4{u}_1\hfill \\ {\dot{e}}_2\hfill & =\hfill & -c_1{e}_2-{\omega }_s{e}_1-c_3(e_3{e}_4+x_{4d}{e}_3+x_{3d}{e}_4)+c_4{u}_2\hfill \\ {\dot{e}}_3\hfill & =\hfill & c_5{e}_1-c_6{e}_3\hfill \\ {\dot{e}}_4\hfill & =\hfill & c_7(e_2{e}_3+x_{3d}{e}_2+x_{2d}{e}_3)-c_8{e}_4-c_9(T_m-T_{mr})\hfill \end{array}$$

(27)

where the inputs $u_1$ and $u_2$ are defined such that:

$$\begin{array}{ccc}\hfill u_1& =& v_{ds}-v_{{ds}_r}\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill u_2& =& v_{qs}-v_{{qs}_r}\hfill \end{array}$$

(29)

In Section 4, we will use the sliding mode control technique to design the controllers $u_1$ and $u_2$ to force the errors $e_1$–$e_4$ to converge to zero as t tends to infinity.

2.2. The Model of the Grid Side Converter

The following equations relate the grid, converter voltages and the line currents [10]:

$$\begin{array}{ccc}\hfill v_{g_1}& =& R_t{i}_{g_1}+L_t\frac{d{i}_{g_1}}{dt}+v_{i_1}\hfill \\ \hfill v_{g_2}& =& R_t{i}_{g_2}+L_t\frac{d{i}_{g_2}}{dt}+v_{i_2}\hfill \\ \hfill v_{g_3}& =& R_t{i}_{g_3}+L_t\frac{d{i}_{g_3}}{dt}+v_{i_3}\hfill \end{array}$$

(30)

where, $v_{g_1}$, $v_{g_2}$ and $v_{g_3}$ are the three phase grid voltages; and $i_{g_1}$, $i_{g_2}$ and $i_{g_3}$ are the line currents. The voltages, $v_{i_1}$, $v_{i_2}$ and $v_{i_3}$ are the voltages of the voltage source inverter (VSI). The resistance and the inductance of the grid-side transmission line are $R_t$ and $L_t$, respectively.

Using the d-q transformation, the equations in (31) can be written as:

$$\begin{array}{ccc}{v}_{dg}\hfill & =\hfill & v_{d_i}-R_t{i}_{dg}-L_t\frac{d{i}_{dg}}{dt}+\omega L_t{i}_{qg}\hfill \\ v_{qg}\hfill & =\hfill & v_{q_i}-R_t{i}_{qg}-L_t\frac{d{i}_{qg}}{dt}-\omega L_t{i}_{dg}\hfill \end{array}$$

(31)

where $v_{dg}$ is the direct component of the grid voltage, and $v_{qg}$ is the quadrature component of the grid voltage. The currents $i_{dg}$ and $i_{qg}$ are the direct and quadrature output line currents. The voltages $v_{d_i}$ and $v_{q_i}$ are the direct and quadrature VSI input voltages. Furthermore, $\omega =2\pi f$, where f is the frequency of the grid.

Let the constants $c_{10}$ and $c_{11}$ be such that $c_{10}=\frac{R_t}{L_t}$ and $c_{11}=\frac{1}{L_t}$. Therefore, the above equations can be written as:

$$\begin{array}{ccc}\frac{d{i}_{dg}}{dt}\hfill & =\hfill & -c_{10}{i}_{dg}+\omega i_{qg}+c_{11}(v_{d_i}-v_{dg})\hfill \\ \frac{d{i}_{qg}}{dt}\hfill & =\hfill & -c_{10}{i}_{qg}-\omega i_{dg}+c_{11}{v}_{q_i}\hfill \end{array}$$

(32)

The output active power $P_g$ and the output reactive power $Q_g$ of the wind energy conversion system can be written as:

$$\begin{array}{ccc}{P}_g\hfill & =\hfill & \frac{3}{2}(v_{dg}{i}_{dg}+v_{qg}{i}_{qg})\hfill \\ Q_g\hfill & =\hfill & \frac{3}{2}(v_{qg}{i}_{dg}-v_{dg}{i}_{qg})\hfill \end{array}$$

(33)

Setting the initial angle of the direct and quadrature reference frame to $\frac{\pi }{2}$ and setting the initial angle of Phase 1 to 0^o yield the voltage $v_{qg}=0$ and $v_{dg}=V_{grid}$, where $V_{grid}$ is the grid voltage. Hence, the output active and reactive powers of the system can be written as:

$$\begin{array}{ccc}\hfill P_g& =& \frac{3}{2}{v}_{dg}{i}_{dg}\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill Q_g& =& -\frac{3}{2}{v}_{dg}{i}_{qg}\hfill \end{array}$$

(35)

The voltage equation for the DC-bus is such that:

$$\begin{array}{ccc}\hfill C\frac{d{U}_{dc}}{dt}& =& i_{dc}=i_s-i_g\hfill \end{array}$$

(36)

where $U_{dc}$ is the voltage of the DC-link, $i_s$ is the current from the induction generator side and $i_g$ is the current from the grid side; refer to Figure 4. The capacitor of the DC-link is denoted as C.

Remark 1.

The DC-bus voltage and the output active and reactive powers $P_g$ and $Q_g$ of the wind energy conversion system are controlled by the direct and quadrature components of the line current $i_g$. Hence, the term $i_s$ in Equation (36) can be assumed as a disturbance in the control [15].

Motivated by the work in [9], we multiply both sides of (36) by $U_{dc}$ and use the fact that the SCIG active power can be expressed as $P_s=U_{dc}{i}_s$ and that $P_g=U_{dc}{i}_g$, to obtain the following equation:

$$\begin{array}{ccc}\hfill \frac{1}{2}C\frac{d{U}_{dc}^2}{dt}& =& P_s-P_g\hfill \end{array}$$

(37)

Using the expression $P_g=\frac{3}{2}{v}_{dg}{i}_{dg}$, the above equation can be written as:

$$\begin{array}{ccc}\hfill \frac{d{U}_{dc}^2}{dt}& =& -\frac{3}{C}{v}_{dg}{i}_{dg}+\frac{2}{C}{P}_s\hfill \end{array}$$

(38)

We will define the state variable vector $x_C$, which contains the states of the converter $x_5$–$x_7$, such that:

$$\begin{array}{ccc}\hfill x_C& =& {\left[x_5{x}_6{x}_7\right]}^T={\left[i_{dg}{i}_{qg}{U}_{dc}^2\right]}^T\hfill \end{array}$$

(39)

Using Equations (32) and (38), the dynamic model of the grid side converter can be written as follows:

$$\begin{array}{ccc}\hfill {\dot{x}}_5& =& -c_{10}{x}_5+\omega x_6+c_{11}(v_{d_i}-v_{dg})\hfill \\ \hfill {\dot{x}}_6& =& -c_{10}{x}_6-\omega x_5+c_{11}{v}_{q_i}\hfill \\ \hfill {\dot{x}}_7& =& -\frac{3}{C}{x}_5{v}_{dg}+\frac{2}{C}{P}_s\hfill \end{array}$$

(40)

Let $x_{5d}=i_{dg}^{*},x_{6d}=i_{qg}^{*}$ and $x_{7d}={U_{dc}^2}^{*}$ represent the desired values of $i_{dg}$, $i_{qg}$ and $U_{dc}^2$ respectively. Furthermore, let $c_{12}$ be such that $c_{12}=\frac{3}{C}$. Define the errors $e_5,e_6$ and $e_7$ such that:

$$\begin{array}{ccc}\hfill e_i& =& x_i-x_{id}(i=5,6,7)\hfill \end{array}$$

(41)

The model of the errors of the grid side converter can be written as:

$$\begin{array}{ccc}\hfill {\dot{e}}_5& =& -c_{10}{e}_5+\omega e_6+u_3\hfill \\ \hfill {\dot{e}}_6& =& -c_{10}{e}_6-\omega e_5+u_4\hfill \\ \hfill {\dot{e}}_7& =& -c_{12}{e}_5{v}_{dg}+d_p\hfill \end{array}$$

(42)

where:

$$\begin{array}{ccc}\hfill u_3& =& -c_{10}{x}_{5d}+\omega x_{6d}-{\dot{x}}_{5d}+c_{11}(v_{d_i}-v_{dg})\hfill \end{array}$$

(43)

$$\begin{array}{ccc}\hfill u_4& =& -c_{10}{x}_{6d}-\omega x_{5d}-{\dot{x}}_{6d}+c_{11}{v}_{q_i}\hfill \end{array}$$

(44)

$$\begin{array}{ccc}\hfill d_p& =& -c_{12}{x}_{5d}{v}_{dg}-{\dot{x}}_{7d}+\frac{2}{C}{P}_s\hfill \end{array}$$

(45)

The objective of controlling the grid side converter is to design the controllers $u_3$ and $u_4$ so that the errors $e_5$, $e_6$ and $e_7$ converge to zero even with the existence of the bounded “disturbance” $d_p$. We will use the sliding mode control technique to accomplish this task.

3. Computation of the Desired States and the Reference Inputs of the System

This section deals with the computation of the desired states and the reference inputs of the wind energy conversion system. Assuming the d-axis of the rotating frame is aligned with the rotor flux vector, then the quadrature component of the flux of the rotor is equal to zero (${\psi }_{qr}=0$), and the direct component of the flux of the rotor is always at a maximum constant value. Therefore, we can write the following:

$$\begin{array}{ccc}\hfill x_{3d}& =& {\psi }_{dr}^{*}={\psi }_{r_{max}}\hfill \end{array}$$

(46)

where ${\psi }_{dr}^{*}$ is the desired value of ${\psi }_{dr}$, and:

$$\begin{array}{ccc}\hfill {\psi }_{r_{max}}& =& L_m{i}_{ds}\hfill \end{array}$$

(47)

Furthermore, the desired value of the current $i_{ds}$ can be calculated from (47), such that:

$$\begin{array}{ccc}\hfill x_{1d}& =& i_{ds}^{*}=\frac{1}{L_m}{\psi }_{dr}^{*}=\frac{1}{L_m}{x}_{3d}\hfill \end{array}$$

(48)

The wind turbine changes its speed by following the maximum of the power coefficient $C_{p_{max}}$, so that maximum power is extracted from the wind. Using Equation (3), the desired value of the rotor speed ${\omega }_m$ is such that:

$$\begin{array}{ccc}\hfill x_{4d}& =& {\omega }_m^{*}=\frac{G{\lambda }_{opt}V}{R}\hfill \end{array}$$

(49)

where ${\lambda }_{opt}$ is the optimum value of tip speed ratio of the wind turbine.

The desired value of the current $i_{qs}$ can be calculated from the electromagnetic torque given in (16), such that:

$$\begin{array}{ccc}\hfill x_{2d}& =& i_{qs}^{*}=\frac{2}{3}\frac{L_r}{p{L}_m}\frac{T_{mr}}{x_{3d}}\hfill \end{array}$$

(50)

Note that $T_{mr}$ is the input reference mechanical torque, and it is set to the maximum mechanical (aerodynamic) torque such that $T_{mr}=\frac{P_{m_{max}}}{x_{4d}}$.

The desired values of the direct and quadrature line currents $i_{dg}$ and $i_{qg}$ can be found from Equations (17), (34) and (35) after setting $P_g=P_s$ to transfer all active power from the side of the stator to the grid side while maintaining $Q_g=0$ to ensure the unity power factor. Hence:

$$\begin{array}{ccc}\hfill x_{5d}& =& i_{dg}^{*}=\frac{2}{3}\frac{P_s}{V_{grid}}\hfill \end{array}$$

(51)

and the desired value of the quadrature line current $i_{qg}$ is such that:

$$\begin{array}{ccc}\hfill x_{6d}& =& i_{qg}^{*}=0\hfill \end{array}$$

(52)

The DC-link voltage $U_{dc}$ must be maintained constant at a preset value. The desired value of the voltage of the DC-link $U_{dc}$ is such that:

$$\begin{array}{c}\hfill U_{dc}^{*}=U_N\end{array}$$

(53)

where $U_N$ is the rated DC-bus voltage.

On the other hand, the input reference voltages $v_{{ds}_r}$ and $v_{{qs}_r}$ are computed using Equations (22) and (23) such that:

$$\begin{array}{ccc}\hfill v_{{ds}_r}& =& \frac{1}{c_4}\left(c_1{x}_{1d}-{\omega }_s{x}_{2d}-c_2{x}_{3d}\right)\hfill \\ \hfill \end{array}$$

(54)

$$\begin{array}{ccc}\hfill v_{{qs}_r}& =& \frac{1}{c_4}\left(c_1{x}_{2d}+{\omega }_s{x}_{1d}+c_3{x}_{3d}{x}_{4d}+\frac{2}{3}\frac{L_r}{p{L}_m}\frac{{\dot{T}}_{mr}}{x_{3d}}\right)\hfill \end{array}$$

(55)

4. Control of the Wind Energy Power System

In this section, we will use the sliding mode control technique to control the wind energy conversion system. Figure 4 depicts the block diagram representation of the controlled SCIG-based WEC system.

4.1. Design of a Sliding Mode Controller for the Turbine and the Induction Generator

To facilitate the design of a controller for the wind turbine and induction generator system, we define the following nonlinear transformation:

$$\begin{array}{ccc}{z}_1\hfill & =\hfill & e_3\hfill \\ z_2\hfill & =\hfill & c_5{e}_1-c_6{e}_3\hfill \\ z_3\hfill & =\hfill & e_4\hfill \\ z_4\hfill & =\hfill & c_7(e_2{e}_3+x_{3d}{e}_2+x_{2d}{e}_3)-c_8{e}_4-c_9(T_m-T_{mr})\hfill \end{array}$$

(56)

Using the above transformation, the dynamic model of wind turbine and induction generator error system in (27) can be written as follows:

$$\begin{array}{ccc}{\dot{z}}_1\hfill & =\hfill & z_2\hfill \\ {\dot{z}}_2\hfill & =\hfill & f_1+g_1{u}_1\hfill \\ {\dot{z}}_3\hfill & =\hfill & z_4\hfill \\ {\dot{z}}_4\hfill & =\hfill & f_2+g_2{u}_2\hfill \end{array}$$

(57)

where $f_1,f_2$, $g_1$ and $g_2$ are such that:

$$\begin{array}{ccc}\hfill f_1& =& c_5\left(-c_1{e}_1+{\omega }_s{e}_2+c_2{e}_3\right)-c_6\left(c_5{e}_1-c_6{e}_3\right)\hfill \end{array}$$

(58)

$$\begin{array}{ccc}\hfill f_2& =& c_7(e_3+x_{3d})\left(-c_1{e}_2-{\omega }_s{e}_1-c_3(e_3{e}_4+x_{4d}{e}_3+x_{3d}{e}_4)\right)\hfill \\ & & +c_7(e_2+x_{2d})\left(c_5{e}_1-c_6{e}_3\right)+c_7{e}_2{\dot{x}}_{3d}+c_7{e}_3{\dot{x}}_{2d}-c_9({\dot{T}}_m-{\dot{T}}_{mr})\hfill \\ & & -c_8\left(c_7(e_2{e}_3+x_{3d}{e}_2+x_{2d}{e}_3)-c_8{e}_4-c_9(T_m-T_{mr})\right)\hfill \end{array}$$

(59)

$$\begin{array}{ccc}\hfill g_1& =& c_4{c}_5\hfill \end{array}$$

(60)

$$\begin{array}{ccc}\hfill g_2& =& c_4{c}_7(e_3+x_{3d}).\hfill \end{array}$$

(61)

Let the design constants ${\beta }_1,{\beta }_2,K_1,K_2,W_1$ and $W_2$ be positive scalars. Define the sliding surfaces $s_1$ and $s_2$ such that:

$$\begin{array}{ccc}\hfill s_1& =& z_2+{\beta }_1{z}_1\hfill \end{array}$$

(62)

$$\begin{array}{ccc}\hfill s_2& =& z_4+{\beta }_2{z}_3\hfill \end{array}$$

(63)

Proposition 1.

The following sliding mode control law:

$$\begin{array}{ccc}\hfill u_1& =& \frac{1}{g_1}\left(-f_1-{\beta }_1{z}_2-K_1{s}_1-W_{1}sign(s_1)\right)\hfill \end{array}$$

(64)

$$\begin{array}{ccc}\hfill u_2& =& \frac{1}{g_2}\left(-f_2-{\beta }_2{z}_4-K_2{s}_2-W_{2}sign(s_2)\right)\hfill \end{array}$$

(65)

when applied to the error system given by (27) guarantees that the errors $e_i$$(i=1,\dots ,4)$ converge to zero as t tends to infinity. Therefore, the states of the wind turbine and the induction generator given by (21) converge to their desired values as t tends to infinity.

Proof. 

Taking the time derivatives of $s_1$ and $s_2$ in (62) and (63), and using the dynamic model of the errors in (27) and the control laws given by (64) and (65), we obtain:

$$\begin{array}{ccc}{\dot{s}}_1\hfill & =\hfill & {\dot{z}}_2+{\beta }_1{\dot{z}}_1=f_1+g_1{u}_1+{\beta }_1{z}_2\hfill \\ & =\hfill & -K_1{s}_1-W_{1}sign(s_1)\hfill \end{array}$$

(66)

and:

$$\begin{array}{ccc}-K_2{s}_2-W_{2}sign(s_2)\hfill & =\hfill & {\dot{z}}_4+{\beta }_2{\dot{z}}_3=f_2+g_2{u}_2+{\beta }_2{z}_4\hfill \\ & =\hfill & -K_2{s}_2-W_{2}sign(s_2)\hfill \end{array}$$

(67)

From (66) and (67), we conclude that ${\dot{s}}_i=-K_i{s}_i-W_{i}sign(s_i)$ for $i=1,2$. It is clear that $s_i{\dot{s}}_i=-K_i{s}_i^2-W_i\left|s_i\right|$ for $i=1,2$. Hence, the equations given by (66) and (67) guarantee that $s_i{\dot{s}}_i<0$ when $s_i\ne$ for $i=1,2$. Therefore, the trajectories of these discontinuous dynamic equations exhibit a finite time reachability to zero from any given initial conditions provided that the gains $K_1$, $K_2$, $W_1$ and $W_2$ are chosen to be sufficiently large, strictly positive scalars.

Since $s_1$ and $s_2$ are driven to zero in finite time, then after such a finite time, we have $z_2+{\beta }_1{z}_1=0$ and $z_4+{\beta }_2{z}_3=0$. Thus, $z_2=-{\beta }_1{z}_1$ and $z_4=-{\beta }_2{z}_3$. Using the equations in (57), we can write: ${\dot{z}}_1=z_2=-{\beta }_1{z}_1$, and ${\dot{z}}_3=z_4=-{\beta }_2{z}_3$. The solutions of these two o.d.e.’s are, $z_1(t)=exp(-{\beta }_{1}t)z_1(0)$ and $z_3(t)=exp(-{\beta }_{2}t)z_3(0)$. Because ${\beta }_1$ and ${\beta }_2$ are positive scalars, it can be concluded that the states $z_1(t)$ and $z_3(t)$ tend to zero as t goes to infinity. Furthermore, since $s_1=0$ and $s_2=0$ on the sliding surfaces, it can be concluded that the variables $z_2(t)$ and $z_4(t)$ are driven to zero as t tends to infinity.

Using the transformation given by (56), it can be deduced that the errors $e_1(t)$, $e_2(t)$, $e_3(t)$ and $e_4(t)$ asymptotically converge to zero as t tends to infinity. ☐

Hence, it can be concluded that the application of the controller (64) and (65) to the model of the wind turbine and the induction generator system (21) guarantees the convergence of the states $i_{ds},i_{qs},{\psi }_{dr}$ and ${\omega }_m$ to their desired values.

Remark 2.

The control laws $v_{ds}$ and $v_{qs}$, which will be used to control the model of the wind turbine and the induction generator given by (21), are as follows:

$$\begin{array}{ccc}\hfill v_{ds}& =& u_1+v_{{ds}_r}\hfill \end{array}$$

(68)

$$\begin{array}{ccc}\hfill v_{qs}& =& u_2+v_{{qs}_r}\hfill \end{array}$$

(69)

where $v_{{ds}_r}$ and $v_{{qs}_r}$ are obtained from (54) and (55).

4.2. Design of a Sliding Mode controller for the Grid Side Converter

This subsection deals with the design of a sliding mode controller for the grid side converter.

Let ${\beta }_3,K_3,K_4,W_3$ and $W_4$ be positive scalars. Furthermore, let $B_d$ be a positive scalar bounding the “disturbance” $d_p$ defined in (45) such that $|{\dot{d}}_p+{\beta }_3{d}_p|\le B_d$.

Define the sliding surfaces $s_3$ and $s_4$ such that:

$$\begin{array}{ccc}\hfill s_3& =& e_6\hfill \end{array}$$

(70)

$$\begin{array}{ccc}\hfill s_4& =& {\dot{e}}_7+{\beta }_3{e}_7\hfill \end{array}$$

(71)

Proposition 2.

The following sliding mode control law:

$$\begin{array}{ccc}\hfill u_3& =& \frac{-1}{c_{12}{v}_{dg}}\left[c_{12}{v}_{dg}(-c_{10}{e}_5+\omega e_6)+c_{12}{\dot{v}}_{dg}{e}_5+c_{12}{\beta }_3{v}_{dg}{e}_5-K_4{s}_4-(B_d+W_4)sign(s_4)\right]\hfill \end{array}$$

(72)

$$\begin{array}{ccc}\hfill u_4& =& c_{10}{e}_6+\omega e_5-K_3{s}_3-W_{3}sign(s_3)\hfill \end{array}$$

(73)

when applied to the model of the errors of the grid side converter given by (43) guarantees that the errors $e_5$, $e_6$ and $e_7$ converge to zero as t tends to infinity. Therefore, the states of the grid side converter converge to their desired values as t tends to infinity.

Proof. 

Let the Lyapunov function candidate $V_x$ be such that:

$$\begin{array}{ccc}\hfill V_x& =& \frac{1}{2}{s}_3^2+\frac{1}{2}{s}_4^2\hfill \end{array}$$

(74)

Taking the derivative of $V_x$ with respect to time along the trajectories of (43) and using the control law given by (72) and (73), we obtain:

$$\begin{array}{ccc}\hfill {\dot{V}}_x& =& s_3{\dot{s}}_3+s_4{\dot{s}}_4=s_3{\dot{e}}_6+s_4({\ddot{e}}_7+{\beta }_3{\dot{e}}_7)\hfill \\ & =& s_3\left(-c_{10}{e}_6-\omega e_5+u_4\right)\hfill \\ & & +s_4\left(-c_{12}{v}_{dg}(-c_{10}{e}_5+\omega e_6+u_3)-c_{12}{e}_5{\dot{v}}_{dg}+{\dot{d}}_p+{\beta }_3(-c_{12}{e}_5{v}_{dg}+d_p)\right)\hfill \\ & =& -K_3{s}_3^2-W_3{s}_{3}sign(s_3)\hfill \\ & & +s_4\left(-c_{12}{v}_{dg}(-c_{10}{e}_5+\omega e_6+u_3)-c_{12}{e}_5{\dot{v}}_{dg}+{\dot{d}}_p+{\beta }_3(-c_{12}{e}_5{v}_{dg}+d_p)\right)\hfill \\ & \le & -K_3{s}_3^2-W_3{s}_{3}sign(s_3)\hfill \\ & & +s_4\left(-c_{12}{v}_{dg}(-c_{10}{e}_5+\omega e_6+u_3)-c_{12}{e}_5{\dot{v}}_{dg}-{\beta }_3{c}_{12}{e}_5{v}_{dg}\right)+\left|s_4\right|B_d\hfill \\ & =& -K_3{s}_3^2-W_3{s}_{3}sign(s_3)\hfill \\ & & +s_4\left(-c_{12}{v}_{dg}(-c_{10}{e}_5+\omega e_6+u_3)-c_{12}{e}_5{\dot{v}}_{dg}-c_{12}{\beta }_3{e}_5{v}_{dg}+B_{d}sign(s_4)\right)\hfill \\ & =& -K_3{s}_3^2-W_3{s}_{3}sign(s_3)-K_4{s}_4^2-W_4{s}_{4}sign(s_4)\hfill \\ & =& -K_3{s}_3^2-W_3|s_3|-K_4{s}_4^2-W_4\left|s_4\right|\hfill \end{array}$$

(75)

Therefore, ${\dot{V}}_x$ is negative definite. Hence, it can be concluded that the sliding surfaces $s_3$ and $s_4$ exhibit a finite time reachability to zero from any given initial conditions provided that the gains $K_3$, $K_4$, $W_3$ and $W_4$ are chosen to be sufficiently large, strictly positive scalars. Therefore, the error $e_6(t)=s_3(t)$ converges to zero in finite time. Furthermore, because ${\beta }_3$ is a positive scalar, we can conclude that the error $e_7(t)$ converge to zero as t tends to infinity.

Moreover, since $s_3$ and $s_4$ are driven to zero in finite time, then the controller $u_3$ after such time is: $u_3=c_{10}{e}_5-\frac{{\dot{v}}_{dg}}{v_{dg}}{e}_5-{\beta }_3{e}_5$. Hence, after such time, the dynamics of the error $e_5$ is such that ${\dot{e}}_5=-c_{10}{e}_5+u_3=-(\frac{{\dot{v}}_{dg}}{v_{dg}}+{\beta }_3)e_5$. Therefore, the error $e_5(t)$ asymptotically converges to zero provided that the control gain ${\beta }_3$ is chosen to be sufficiently large, strictly positive scalar. ☐

Hence, it can be concluded that the application of the controller (72) and (73) to the model of the grid side converter (41) guarantees the convergence of the states $i_{dg}$, $i_{qg}$ and $U_{dc}$ to their desired values.

Remark 3.

The control laws $v_{d_i}$ and $v_{q_i}$, which will be used to control the model of the grid side converter given by (41), are as follows:

$$\begin{array}{ccc}\hfill v_{d_i}& =& \frac{1}{c_{11}}\left(u_3+c_{10}{x}_{5d}-\omega x_{6d}+{\dot{x}}_{5d}\right)+v_{dg}\hfill \end{array}$$

(76)

$$\begin{array}{ccc}\hfill v_{q_i}& =& \frac{1}{c_{11}}\left(u_4+c_{10}{x}_{6d}+\omega x_{5d}+{\dot{x}}_{6d}\right).\hfill \end{array}$$

(77)

5. Simulation Results of the Proposed Sliding Mode Control Scheme

The performance of the SCIG-based wind energy conversion system given by (21) and (41) using the sliding mode controllers (68), (69), (76) and (77) was simulated using the MATLAB software. The parameters of the system used for the simulation studies are given in

Table 1. The parameters of the SCIG and the wind turbine.

Parameter	Value
The rated power	300 KW
The rated apparent power	300 KVA
The rated voltage (line to line)	575 v
The frequency/angular speed	$2\pi 50$ rad/s
The nominal frequency of the system	50 Hz
The resistance of the stator	$0.0063\mathsf{\Omega }$
The leakage inductance of the stator	11.8 mH
The resistance of the rotor	$0.0048\mathsf{\Omega }$
The leakage inductance of the rotor	11.6 mH
The magnetizing inductance	11.6 mH
The capacitor of the DC bus	20 mF
The rated voltage of the DC bus	760 v
The grid resistance	$0.1\mathsf{\Omega }$
The grid leakage inductance	0.6 mH
The inertia of the generator	10.0 kg m${}^2$
The number of pole pairs	2
The rated generator speed	158.7 rad/s
The rotor diameter of the wind turbine	14 m
The density of air	1.22 kg/m${}^3$
The rated wind speed	12 m/s
The rated rotor speed	19.7 rpm
The gearbox ratio	23
The inertia of the turbine	50.0 kg m${}^2$

.

The model of the wind speed used for simulation purposes is given by the following formula [33],

$$\begin{array}{ccc}V\hfill & =\hfill & 10+0.55[sin(0.0625w)-0.875sin(0.1875w)+0.75sin(0.3125w)\hfill \\ & -\hfill & 0.625sin(0.625w)+0.5sin(1.875w)+0.25sin(3.125w)+0.125sin(6.25w)]\hfill \end{array}$$

(78)

with $w=\frac{2\pi }{10}t$. The profile of the wind speed versus time is depicted in Figure 5.

The simulation results are presented in Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17. The currents $i_{ds}$ and $i_{ds}^{*}$ versus time and the currents $i_{qs}$ and $i_{qs}^{*}$ versus time are depicted in Figure 6 and Figure 7, respectively. It can be seen that the direct and quadrature currents are tracking their desired values. The fluxes ${\psi }_{dr}$ and ${\psi }_{dr}^{*}$ versus time are depicted in Figure 8. Figure 9 shows the angular speeds ${\omega }_m$ and ${\omega }_m^{*}$ of the generator versus time. It can be noticed that ${\omega }_m$ is tracking the desired speed and following the maximum power coefficient $C_{p_{max}}=0.47$, as shown in Figure 10. The electromagnetic torque $T_{em}$ versus time and the stator active power $P_s$ versus time are shown in Figure 11 and Figure 12. For the grid side, the direct currents $i_{dg}$ and $i_{dg}^{*}$ versus time and the quadrature currents $i_{qg}$ and $i_{qg}^{*}$ versus time are presented in Figure 13 and Figure 14, respectively. It is clear from these figures that the direct and quadrature currents track their desired values. The voltages $U_{dc}$ and $U_{dc}^{*}$ versus time are depicted in Figure 15. It is clear that the DC link voltage is almost constant at all times. Finally, the output active power of the grid $P_g$ versus time and the output reactive power of the grid $Q_g$ versus time are shown in Figure 16 and Figure 17, respectively. Figure 16 shows that approximately all active power from the stator side has been transferred to the grid side. Furthermore, Figure 17 shows that the output reactive power of the grid $Q_g$ is approximately zero, which ensures that the power factor is approximately one.

The simulation results clearly indicate that the proposed sliding mode control scheme when applied to the a squirrel cage induction generator-based wind energy conversion system is able to force the states of the system to converge to their desired values. Hence, the controller enables maximum extraction of power from the wind at different speeds of the wind, and it ensures that the generated active power is transmitted through the DC-bus to the grid while maintaining the unity power factor.

6. Conclusions

This paper dealt with the control of a squirrel cage induction generator-based wind energy conversion system. Sliding mode controllers are proposed for the WEC system so that the actual states of the system track the desired states of the system. The control laws are designed to control the stator side converter, as well as the grid side converter. The stability of the controlled system is analyzed. Furthermore, simulation results are presented to verify the validity of the proposed control scheme. The simulation results indicate that the proposed sliding mode control scheme works well.

Future work will address the design of observer-based controllers for SCIG-based wind energy conversion systems.