Super-Twisting Sliding Mode Control Design for Cascaded Control System of PMSG Wind Turbine

Abstract

This study focuses on an advanced second-order sliding mode control strategy for a variable speed wind turbine based on a permanent magnet synchronous generator to maximize wind power extraction while simultaneously reducing the mechanical stress effect. The control design based on a modified version of the super-twisting algorithm with variable gains can be applied to the cascaded system scheme comprising the current control loop and speed control loop. The proposed control inheriting the well-known robustness of the sliding technique successfully deals with the problems of essential nonlinearity of wind turbine systems, the effects of disturbance regarding variation on the parameters, and the random nature of wind speed. In addition, the advantages of the adaptive gains and the smoothness of the control action strongly reduce the chatter signals of wind turbine systems. Finally, with comparison with the traditional super-twisting algorithm, the performance of the system is verified through simulation results under wind speed turbulence and parameter variations.

I. INTRODUCTION

The wind turbine is one of the fastest growing renewable energy technologies available today. The total globally installed wind capacity has been growing significantly during the past decade, reaching 318GW at the end of 2013. This is an increase of 12.5% compared with the 2012 market [1]. Among currently available wind energy conversion systems (WECSs), variable speed wind turbines are preferable to fixed speed ones because of their ability to maximize power extraction energy and increase efficiency [2]. In variable speed wind turbines, wind turbine based on permanent magnet synchronous generator (PMSG) offers several benefits in the wind power applications, such as their high power density, high efficiency, more reliability, and easy maintenance. These features make it an attractive choice in the wind turbine systems [3], [4]. It is widely known that the main control objective for the wind turbines in the low speed region is power efficiency maximization.

To achieve this goal, the turbine tip speed ratio should be maintained at its optimum value despite wind speed variation. For such a purpose, many different control strategies can be used. First, proportional-integral (PI) based control schemes were proposed to deal with the problem of maximum power point tracking [5], [6]. The authors presented control methods using the conventional and direct-current vector configuration to implement the maximum power extraction. It should be realized that PI control is a linear regulator, which is suitable for linear time invariant systems. However, the electrical and mechanical components in wind turbines work globally as nonlinear systems where electromechanical parameters vary considerably. Additionally, wind turbines are expected to operate effectively under a wide range of wind speeds; therefore, control designs become more complicated. To overcome this drawback, there are many different intelligent and nonlinear control strategies, such as gain scheduling [7], fuzzy logic control [8], neural network [9], feedback linearization [10], and backstepping [11]. In this context, methods based on sliding mode control (SMC) seem to be an interesting approach because they make plants more robust and invariant with respect to matched uncertainties and disturbance. Concerning these methods, SMC schemes have been proposed in many studies [12]-[15]. The authors proposed one of these control strategies applied to the field-oriented control (FOC) to relatively overcome the effect of model uncertainties and wind speed turbulence [15]. However, its discontinuity of control variables results in variable switching frequency of the generator-side converter; thus, it may lead to a high chattering phenomenon and mechanical stress effect [13]. To keep a constant switching frequency without sacrificing robustness and tracking ability, second-order sliding mode control strategies were adopted [16]-[18]. Among them, the super-twisting algorithm (STA) has a simple law and allows synthesizing a continuous control action with discontinuous time derivative, which makes the trajectories reach the 2-sliding manifold in finite time and significantly reduces the chattering phenomenon. The disadvantage of the traditional STA is that it does not allow compensating uncertainties and turbulence growing together with the state variable; therefore, the sliding motion cannot be assured. To overcome this problem, the STA with variable gains (VGSTA) was proposed [19]. The proposed algorithm, whose adaptive gains are determined by the known functions, allows the exact compensation of smooth uncertainties and turbulence bounded together with their derivative by the known function. The new advanced algorithm was presented in literature [20], [21] where a VGSTA controller was developed for a WECS double output induction generator (WECS-DOIG), and this controller was applied to the WECS doubly fed induction generator (WECS - DFIG). To develop the new advanced algorithm for the multiple input multiple output (MIMO) of the WECS–PMSG, a VGSTA controller was designed in this paper. Using the Lyapunov function, the convergence of state variables was proved and the convergence time was estimated. In this paper, the results of the simulation demonstrated a better VGSTA control quality compared with the fixed-gain variable STA. This controller is not difficult to implement; thus, the computational burden is low during the online operation of the controller.

 

II. WIND TURBINE SYSTEM DESCRIPTION

Wind energy first transformed into mechanical energy through the wind turbine blades and then into electrical energy through the PMSG. The PMSG is connected into the grid through a two-level back-to-back converter. The electrical power is completely delivered to the grid by regulating the controller of the two-level back-to-back converter, which consists of a machine side converter (MSC), a grid side converter (GSC), and a DC link. The generator side control comprising the current control loop and speed control loop is presented in Fig. 1.

Fig. 1.Control block diagram of the PMSG.

A. Wind Turbine Modeling

Aerodynamic power extracted from the wind is expressed by the following equation [8]:

where ρ is the air density, γ is the wind turbine rotor radius, ν is the wind speed, and Cp(λ,β) is the turbine power coefficient. This coefficient can be calculated as follows:

where β is the blade pitch angle, and λ is the tip speed ratio, which is defined as [10]:

where wr is the angular speed of the rotor.

To effectively extract wind power while maintaining safe operation, the wind turbine should operate in two possible regions, namely, high and low speed regions [16]. In the high-speed region, the wind turbine must limit the captured wind power so that it does not exceed the rated power of thePMSG to maintain the safety of the electrical and mechanical loads. Control of the blade pitch is typically used to limit the power and speed wind turbine in this region. Meanwhile, control of speed, torque, and power is frequently implemented in the low speed region to extract the maximum power from the wind. In this paper, speed control is used in the low speed region to reach maximum power extraction without controlling pitch blade angle (β = const) .

From Equ. (2), the Cp - λ characteristics, for different values of the pitch angle β, are shown in Fig. 2. From Fig. 2, the wind turbine operates with a constant pitch angle β = 0° and the maximum power coefficient 0.4412 Cpmax is achieved for an optimum tip speed ratio value λopt = 6.91.

Fig. 2.Power coefficient versus tip speed ratio. Optimum value Cpmax = 0.4412 ; λopt = 6.91

Thus, the reference rotor speed of wind turbine is given by:

The Pm - wr characteristics with maximum power point tracking for variable wind speed are presented in Figure 3, which shows that for any wind speed value, there is a unique rotor speed that generates the maximum power Pmax.

Fig. 3.Turbine characteristic with maximum power point tracking.

From Equ. (1), Equ. (3), and Equ. (4), the maximum mechanical power is written as follows:

Aerodynamic torque can be expressed as [15]:

B. PMSG Modeling

Dynamic modeling of the PMSG can be described in (d,q) synchronous rotating reference frame, and is given by the following equations [22]:

where id, iq are the d-axis and q-axis stator current, respectively, Ud and Uq are the d-axis and q-axis stator voltage, respectively, Rs is the resistance of stator windings, Ls is the inductance of stator windings, ψr is the permanent magnetic flux, and p is the number of pole pairs of the PMSG.

Moreover, the electromagnetic torque of the PMSG is computed using the following equation:

with . Finally, the dynamic equation of the wind turbine is expressed as follows:

where J represents the inertia of the rotating parts, κ is the viscous friction coefficient, and Tm denotes the aerodynamic torque.

From Equs. (7)–(10), the dynamic characteristics of the PMSG modeling with variation of parameters can be expressed as follows [15], [23]:

where Lsn, Rsn, ψrn, Jn, κn, Ktn are the nominal values; ΔRs, ΔLs, Δψr, Δκ, ΔKt represent the variations of parameters. These variations of parameters are assumed less than 50% of the nominal parameters. Then, the PMSG modeling can be rewritten as:

where the functions f1, f2, and f3 represent the nominal model. Meanwhile, the functions Δf1, Δf2, and Δf3 denote the modeling errors and parameter uncertainties, and the signals u1, u2, u3 are the control input variables.

 

III. CONTROL SYSTEM

A. Control Design

At this point, to achieve the desired control objectives when operating in the sliding mode, the sliding variables can be chosen as:

where idref and iqref are the reference values of the stator current, wref denotes the reference value of rotor speed of the PMSG. Taking the time derivative of the sliding variables Equ. (17), incorporated with Equs. (14)–(16), leads to:

Moreover, each component of the action control variables consist of two terms [21]:

where ueqi (i=1,2,3) are the equivalent control terms for the nominal and unperturbed model, and is the term of control variables designed by using the STA with variable gains [19]. The expression for the equivalent control term ueqi can be determined as [21]:

Substituting Equ. (20) and Equ. (19) into Equ. (18) produces:

Functions Δfi are divided into two terms expressed as:

where the functions g1i(s,t) = 0 when si = 0 , and the functions g2i(t) are the perturbations on the 2-SM manifold. By contrast, the components of the control term can be given as [21]:

where is the partial derivative of ϕ1i(si) with respect to si , and kci >0 is the constant. Values k1i(s,t) and k2i(s,t) are the variable gains which are determined by certain bounding functions of Equ. (26), Equ. (27). Then, the functions g1i(s,t) and the time derivative the functions g12(s,t) are bounded by the following equations:

where denote the known positive functions. Under these conditions, the finite-time convergence to the 2-SM manifold and the robustness against all disturbance and uncertainty considered in functions Δfi are guaranteed if the variable gains of Equ. (23) are calculated as:

where βi, εi, and δi are the arbitrary constants.

A block diagram of the controller is presented in Fig. 4.

Fig. 4.Block diagram of the controller.

B. Stability of Control System

Defining as new states.

The closed loop system for Equ. (21) with the VGSTA control law of Equ. (23) can be written as follows:

The Lyapunov function can be chosen as:

where The time derivative of each term Vi can be calculated as [21]:

where

The values α1i(s,t) and α2i(s,t) are derived from the equations:

From [19], the built matrix [Qi - 2εiI] is positive definite, where I is the identity matrix of size 2. Thus, Equ. (33) can be bounded as follows:

where represents the square of the 2-Euclidean norm determined as:

Additionally, the standard inequality for quadratic forms:

where λm(Pi) , λM(Pi) are the minimum and maximum eigenvalues of matrix Pi , respectively. From Equ. (36) and Equ. (37), the following inequalities can be expressed as:

From Equ. (35), Equ. (38), and Equ. (39), the time derivative of Vi can be bounded as:

where , which shows that the V function is a strong Lyapunov function. Finally, the total time derivative of Vi is bounded as follows:

From Equ. (32), Equ. (41), V is positive definite and is negative definite, by considering the principle [24], the trajectories of Equ. (30), Equ. (31) converge to zero in the finite time.

C. Fixed-Gains – STA

The fixed-gain STA is used to compare with VGSTA. In this case, the term in Equ. (21) is similar to Equ. (23), but using fixed-gains can be expressed as follows:

where k1i, k2i are the fixed-gain. From Equ. (30), Equ. (31), the bounding the components of gi is:

where δ1i, δ2i are the positive constants, and the fixed-gain can be chosen as:

 

IV. SIMULATION RESULT

To verify the validity of the proposed algorithm, Matlab–Simulink was performed for a 10 kW PMSG wind turbine system. The parameters of the wind turbine are listed in Table I. During the simulations, the discrete model was used with a simulation time step of 5 μs . The sample time of the proposed control was 50 μs , and switching frequency converters was set at 2.5 kHz. Of the variable parameters, 50% were considered for wind turbine system. Namely:

TABLE IWIND TURBINE PARAMETER VALUES

ΔRs = 0.225, ΔLs = 0.0004175, Δψr = 0.349165, ΔJ = 0.075, ΔK = 0.005         

Moreover, the following VGSTA controller settings were used:

For the STA controller, the fixed-gains can be chosen as follows:

Fig. 5 shows the ability of tracking their reference value of the control variable in generator side control using the VGSTA controller for current control loop and speed control loop. The wind speed profile is shown in Fig. 5(a). Wind speed consists of the mean speed varying between 11 m/s and 13 m/s with the time interval [0, 4s] and the wind speed turbulence with 15% turbulence intensity. In response to the wind speed profile, the actual rotor speed (ωr) and its reference value (ωref) are shown in Fig. 5(b). The rotor speed with the proposed controller tracks more closely the reference value, which is the optimal rotor speed leading to more power capture from nature wind. In Fig. 5(c), the tip speed ratio equals its optimum value (λopt = 6.91) with a small oscillation ensuring the maximum power extraction.

Fig. 5.(a) Wind velocity profile (m/s). (b) Rotor speed and rotor speed reference. (c) Tip speed ratio. (d) Aerodynamic torque and electromagnetic torque. (e) Mechanical power and generator active power.

Fig. 5(d) and Fig. 5(e) reveal that, despite the wind speed turbulence and variable parameters, the electromagnetic torque and active power of the PMSG accurately follow their reference values. The reference value Pm describes the maximum power extracted from the wind, and the reference value Tm denotes the aerodynamic torque calculated from the measurements. Although the chatter electromagnetic torque and the chatter active power exist because of the speed turbulence and the high converter frequency, it is a small value considered acceptable.

Fig. 6 shows the comparisons of the characteristics of the PMSG controlled by the VGSTA and STA under the aforementioned variable parameters. In these comparisons, the assumption is that the wind speed profile consists of the step value changing from 10 m/s to 13 m/s at 1 s and the turbulence shown in the Fig. 6(a). The rotor wind speed is governed by the VGSTA (see Fig. 6(b)) and the STA illustrated in Fig. 6(c). These figures reveal a smaller oscillation of the rotor wind speed regulated by the VGSTA against the one regulated by the STA. Moreover, the comparisons of the electromagnetic torque are implemented in Fig. 6(d) (VGSTA) and Fig. 6(e) (STA). The chatter electromagnetic torque governed by VGSTA, kept below 2% its rated value, is much smaller than the one controlled by STA. A similar situation also happens in Fig. 6(f) and Fig. 6(g). VGSTA is much better than STA in this control system. The different comparison of the VGSTA and STA is derived from their control gains. The variable gains (VGSTA) allow compensating the uncertainties and the turbulences increasing in the system states, whereas the fixed-gains (STA) cannot do this. Thus, to maintain the robustness of the system under variable parameters and the wind turbulences, the fixed-gains (STA) must be much larger leading to the growth of the chatter signals in the wind system. Meanwhile, the VGSTA whose adaptive gains vary according to the system states completely reduces the chattering phenomenon affecting the system.

Fig. 6.(a) Wind speed profile (m/s). (b) Rotor wind speed (VGSTA). (c) Rotor wind speed (STA). (d) Electromagnetic torque (VGSTA). (e) Electromagnetic torque (STA). (f) Generator active power (VGSTA). (g) Generator active power (STA).

 

V. CONCLUSION

A cascaded control system of a wind turbine based on VGSTA was proposed for a variable speed wind energy conversion system. With controlled speed wind turbine under disturbance conditions, two main goals are achieved. First, generator side control ensures maximum power extraction regardless of wind speed turbulence and parameter variation. Second, it considerably reduces chatter active power and electromagnetic torque, avoiding the damage of excessive mechanical stress wind turbine. The candidate Lyapunov function was chosen for the cascaded control loops of the wind turbine system. By using this function, designing the control laws for cascaded control loops and demonstrating the convergence of the algorithm is easy. The most important contribution of this study is the successful application of this algorithm to control the current control loop and speed control loop, and the demonstration of the advantages of this algorithm over the traditional STA. The problem of VGSTA is that it is not simple to exactly determine the known bounded functions derived from the disturbances and to turn parameters of this controller. However, computational burden is low during the online operation of the controller. Finally, the excellent performances of VGSTA were verified by simulation results under wind speed noise and parameters derivation.