Classical vector,first-order sliding-mode and high-order sliding-mode control for a grid-connected variable-speed wind energy conversion system: A comparative study

Abstract

This article proposes a comparative study between different control strategies of a wind energy conversion system. It aims to guarantee a robust control strategy which gives a good performance despite the external disturbances. Studied system comprises a wind turbine, a permanent magnet synchronous generator, and two converters linked by a DC bus. The whole is connected to the grid through a resistor–inductor filter. A classical vector control based on proportional–integral controller is applied to our system. Owing to the sensitivity of this control against external disturbances, a control strategy using first-order sliding mode has been proposed. This strategy provides good performance, such as insensitivity to non-linearity of system. Yet, the theory of first sliding mode has faced the problem of chattering, which proved to be a major drawback. To overcome this problem, a control strategy using sliding mode of higher order was implemented on the basis of the super-twisting algorithm.

Keywords

Permanent magnet synchronous generator first-order sliding mode second-order sliding mode wind energy conversion system super twisting

Introduction

Recently, integration of renewable energies such as photovoltaic systems and wind energy conversion systems (WECS) in the new electrical networks has increased considerably because of the massive industrialization of some countries (Lin and Chen, 2013; Stigka et al., 2014). In fact, this type of renewable sources is clean, free, inexhaustible, and environmentally friendly compared to other traditional energy production centralized sources (Chou et al., 2014; Tseng and Huang, 2014).

Today, the production of electricity from wind energy has become an alternative to traditional sources in terms of cost of production with a very high growth rate. This energy has been used for centuries and led to numerous investigations to improve the technology of wind generators in the electromechanical energy conversion (Rajaei et al., 2013).

Depending on the type of generator, two main configurations of wind turbines are being answered: fixed speed and variable speed. For fixed-speed turbines, the generator is directly coupled to the grid; the rotational speed of this machine is adapted to the desired frequency of the grid via a speed multiplier (Wessels et al., 2013). Concerning variable speed wind turbines configuration, the most usable power generators are doubly fed induction generator, synchronous machine with a wound rotor and permanent magnet synchronous generators (PMSG) (Boukettaya and Krichen, 2014). Variable speed wind turbines can be coupled or decoupled from the electrical network thanks to power converters interfaces. The main privileges of this type of wind turbines compared to fixed speed are as follows: pitch angle system is simpler, and maximum power available even at low wind speeds and wind turbines become more flexible through the use of power electronics (power converter) which increases their integration into the electrical network (Errami et al., 2015). In recent years, PMSGs are increasingly being used for variable speed wind turbines for several reasons such as variable speed operation, noise is lower in case of low power operation, a very high torque, and no significant losses are generated in the rotor system (Xia et al., 2013). The synchronous machine may be coupled directly to the turbine if the number of its poles is large enough (Orlando et al., 2013). Besides, the performance of PMSG equipment has been improving and the price has been decreasing recently (Yaramasu et al., 2014). Therefore, it has been considered a promising candidate for new designs in WECS. With those advantages, PMSGs are attracting great attention and interests all over the world. So, some of them have become commercially accessible, for example, Enercon E70 (2.5 MW), Vestas V112 (3.0 MW), and Gold wind (1.5 MW) series products (Cardenas et al., 2013; Nasiri et al., 2015).

In order to improve the quality of electric power produced by wind generators, a robust control insensitive to external perturbation is needed to control the production system (Dansoko et al., 2014). In Alepuz et al. (2013) and Li et al. (2012), a control strategy on the basis of proportional–integral (PI) controllers have been proposed to address the problem of maximum power extraction and to adjust the frequency and amplitude of voltage and current at the point of connection to the grid. The performances of PI controllers are limited. This limitation is due to nonlinearity of the system (Leonhard, 1990).

In fact, wind energy system has high nonlinearity obtained from uncertainties, wind speed turbulence, and the changes in wind system parameters. To deal with the problem of nonlinearity, nonlinear control strategies have been implemented. Therefore, there are a lot of intelligent nonlinear techniques used to overcome this difficulty, such as fuzzy logic (Muyeen, 2013) and neural network (Cadenas and Rivera, 2009). In addition, sliding mode control (SMC) approach can be used. It has a good performance because of the insensitivity to external disturbance and to the nonlinearity of system (Evangelista et al., 2013). Amimeur et al. (2012) developed a control strategy based on a controller by first-order sliding mode (FOSM). The chattering phenomenon is the major drawback of practical implementation of this strategy. To address this problem, a control strategy by sliding mode of higher order was implemented on the basis of the super-twisting algorithm (Saad et al., 2015). Therefore, this document focuses on the comparative study of the performances of classical vector control, FOSM control, and second-order sliding mode (SOSM) control.

Within this framework, the main interest of this control approach is to insert a control system for pitch angle in order to control the power output of the wind turbine and thus to limit it to its nominal value when the wind speed is too high, to ensure the extraction of maximum power even for low wind speed, to regulate both the reactive and active power independently, to maintain frequency and voltage at the connection point to the grid in qualifying beaches, and to ensure the operation of system at a unity power factor.

This article is organized as follows. In section “Description and modeling of the power generation system,” we modeled the various system elements. In section “Control of the energy conversion system,” a conventional vector control, FOSM control, and SOSM control are proposed to ensure robust control to our system. Finally, in section “Simulation results,” we validated a control algorithms implemented by simulation results, and a dynamic model of the system is modeled and simulated in MATLAB/Simulink.

Description and modeling of the power generation system

Description of the power generation system

The system studied in this article is presented in Figure 1. It consists of a wind turbine with three blades, PMSG, and a three-phase rectifier (converter 1 AC/DC). This system forms a decentralized generator connected to the grid through a common DC bus, a three-phase inverter (converter 2 DC/AC) that transfers the powers provided by this decentralized generator to the grid to participate in the system services, and resistor–inductor (RL) filter that minimizes the harmonics generated by this inverter.

Figure 1.

Structure of the conversion system.

Wind generator modeling

The power captured by the wind turbine also called aerodynamic power is only part of the wind power P_w and described by the following equation (Rashid and Ali, 2017)

P_{a e r} = C_{p} P_{w} = \frac{1}{2} C_{p} (λ, β) ρ S V_{v}^{3}

(1)

where P_aer is the power extracted from the wind (W), ρ is the air density (kg/m³), R is the radius of the wind turbine (m), C_p is the power coefficient, λ is the speed ratio, V_v is the wind speed (m/s), and β is the pitch angle of the blades (°).

The aerodynamic torque C_aer developed on the wind turbine rotor is deduced by the following relationship

C_{a e r} = \frac{P_{a e r}}{Ω_{m e c}}

(2)

The power coefficient (C_p) represents the ratio between the aerodynamic power and wind power. It is linked to the tip speed ratio λ and the pitch angle β. The expression of the tip speed ratio is given by the following equation

λ = \frac{Ω_{m e c} R}{V_{v}}

(3)

with Ω_mec is the rotational speed of the wind turbine (rad/s).

The wind turbine is controlled to extract the maximum power available. According to Betz theory, the power factor cannot exceed 0.593. Its expression is a function of the geometrical shape of the blade. In our study, we assumed that the blades are characterized by a coefficient C_p described by the following equation (Rashid and Ali, 2017)

\begin{array}{l} C_{p} = 0.5 (\frac{151}{λ_{i}} - 0.58 β - 0.002 β^{2.14} - 10) e^{- \frac{18.4}{λ_{i}}} \\ λ_{i} = \frac{1}{\frac{1}{λ - 0.02 β} - \frac{0.003}{β^{3} + 1}} \end{array}

(4)

The variation of C_p with λ and β is given by Figure 2(a), which shows an optimum value of the speed ratio (λ_opt = 8.15) corresponding to a maximum value of the power coefficient (C_p = 0.4794) with β = 0.

Figure 2.

(a) C_p versus λ, (b) aerodynamic power in function of the mechanical speed for different values of the wind speed, (c) evolution of different characteristics of the wind turbine, and (d) PLL algorithm.

Equation (5) gives the expression of the maximum power obtained by using the maximum power point tracking (MPPT) strategy

P_{M P P T} = \frac{1}{2} \frac{ρ C_{p \max} R^{5} Ω_{mec}^{3}}{λ_{opt}^{3}}

(5)

Then

C_{e m_{M P P T}} = \frac{1}{2} \frac{ρ C_{p \max} R^{5} Ω_{mec}^{2}}{λ_{opt}^{3}}

(6)

With the aim of generating the most possible power, the blade pitch angle should be zero. It is the operating region in MPPT. However, when the wind speed reaches the nominal speed, it is necessary to limit the rotational speed of the turbine to protect our system against mechanical events.

In our study, a control system called “Pitch control” has been inserted in order to control the power and speed of the wind turbine and thus to limit it when the wind speed is too high. The operation of this system is described in Figure 2(c).

In order to convert mechanical energy into electrical energy, the studied wind turbine is coupled to a PMSG. Equations of this machine in the d–q frame are as follows (Rekik and Krichen, 2012)

{\begin{cases} \frac{d}{d t} i_{s d} = \frac{1}{L_{s}} (V_{s d} - R_{s} i_{s d} + ω L_{s} i_{s q}) \\ \frac{d}{d t} i_{s q} = \frac{1}{L_{s}} (V_{s q} - R_{s} i_{s q} - ω L_{s} i_{s d} - ω ϕ) \end{cases}

(7)

The electromagnetic torque C_em is given by the following equation (Rekik and Krichen, 2012)

C_{e m} = p ϕ i_{s q}

(8)

The evolution of the mechanical speed Ω_mec depends on the mechanical torque applied to the generator rotor C_m which is the result of an electromagnetic torque produced by the generator C_em, a couple of C_vis viscous friction and a couple of speed multiplier C_aer. The torque from the friction is modeled by the viscous friction coefficient f.

Taking into account the friction losses, the evolution of the mechanical speed of the PMSG is determined by the fundamental equation of the following dynamics

j \frac{d}{d t} Ω_{m e c} = C_{a e r} - C_{e m} - C_{v i s} = C_{a e r} - C_{e m} - f Ω_{m e c}

(9)

where j is the total inertia of the generator and turbine.

Figure 1 shows that the binding to the grid is carried out via a RL filter. In this part, we will model the input filter in the d–q reference frame as follows (Errami et al., 2015)

{\begin{cases} L_{r} \frac{d}{d t} i_{d} = V_{o d} - R_{r} i_{d} + L_{r} ω_{R} i_{q} - V_{r d} \\ L_{r} \frac{d}{d t} i_{q} = V_{o q} - R_{r} i_{q} - L_{r} ω_{R} i_{d} - V_{r q} \end{cases}

(10)

Phase locked loop

To ensure proper connection of renewable energy generator to the grid, the converter output voltage should have the same characteristic parameters for each of the three phases. This result is obtained if the phase angle of the voltage is properly controlled. Then, the most common technique used to synchronize the outputs of the converter 2 with the mains electricity supply is the phase locked loop (PLL). Thus, we were interested in the study and design of a PLL (Figure 2(d)) capable of evaluating a correct manner the voltage phase angle of an ideal electrical network. This phase angle can synchronize the renewable energy source with respect to the evolution of the latter.

Interest of variable speed

The main interest of variable speed operation is that wind turbine provides the maximum power to each wind speed. The evolution of the power converted by the wind turbine depending on its rotation speed and the wind speed is given in Figure 2(b). It is observable that the peak position of the aerodynamic power varies according to wind speed. In fact, for a wind speed V₁, the point A corresponds to the peak of the aerodynamic power of coordinates (P₁, Ω₁). According to Figure 2b, if the speed of the turbine remains unchanged and the wind speed goes from V₁ to V₂, it is remarkable that the power produced becomes in another point B other than point A, but at this wind speed V₂ the point B does not represent the maximum power point. If it is desired to extract the maximum power, it is necessary to vary the speed of the generator to another rotational speed Ω₂ higher than the previous Ω₁, hence the need to make the variable rotation speed depending on the wind to achieve the maximum power generated.

Control of the energy conversion system

The purpose of this article is to present a control strategy using second-order sliding modes for a grid-connected variable-speed WECS. The control design is based on the super-twisting algorithm as a solution of chattering reduction caused by the FOSM controller. In this work, the control approach is introduced in order to ensure the following services: maximize the energy conversion (operation in MPPT), inserting a control system for pitch angle in order to control the speed of the wind turbine when the wind speed is too high, study and design of a PLL able to evaluating a correct synchronization between the outputs of the converter 2 and the electrical network, the regulation both the reactive and active power independently, achieve the operation of the studied system at a unit power factor and the regulation of the DC bus voltage. SOSM is capable to enhance system robustness to parameter variations and fluctuation of wind speed when compared with classical vector control and FOSM control.

Classical vector control of WECS

In this part, a classical vector control based on PI controllers will be proposed to control our conversion system. The purpose of this command is to show the sensitivity of the PI controllers against the high large range of wind speed variations and the nonlinearity of the studied system. The structure of the classical vector control applied to our system is detailed in Figure 3.

Figure 3.

Schematic of the vector control strategy of WECS.

The first part of Figure 3 shows the control strategy of the wind generator. This strategy is based on the vector control applied to PMSG to extract maximum wind power (MPPT). The principle of this control is to impose a reference direct current i_sd-ref equal to zero and a quadratic current reference i_sq-ref proportional to the reference torque given by the MPPT algorithm as follows

{\begin{cases} i_{s d - r e f} = 0 \\ i_{s q - r e f} = \frac{C_{e m_{M P P T}}}{ω ϕ} \end{cases}

(11)

A controller at the DC bus is mastered in Part 2 of Figure 3 to maintain its voltage to a constant value regardless of the wind speed variations.

Using a decentralized renewable generator to participate in the system services requires a control strategy to ensure a smooth and fast connection between the WECS and the grid (Part 3 of Figure 3). Consequently, a control vector is applied to the converter 2 based on the management of active and reactive power exchanged between the main grid and our production system. This control is similar to that of generator. Two control loops are established to control the two components of current i_rd and i_rq. Therefore, it can regulate, respectively, active and reactive power of the grid connection. The active and reactive powers of references $P_{r}^{*}$ and $Q_{r}^{*}$ provide the currents i_rd-ref and i_rq-ref as follows

{\begin{cases} i_{r d - r e f} = \frac{P_{r}^{*} V_{r d} + Q_{r}^{*} V_{r q}}{V_{r d}^{2} + V_{r q}^{2}} \\ i_{r q - r e f} = \frac{P_{r}^{*} V_{r q} - Q_{r}^{*} V_{r d}}{V_{r d}^{2} + V_{r q}^{2}} \end{cases}

(12)

We choose to work with a unity power factor. In this case, the reactive power reference is zero $(Q_{r}^{*} = 0)$ .

It is found that the active power P_r and reactive power Q_r are a function of the direct component and the quadratic component of the current grid. The problem here is to independently control the active power and reactive power. To do this, it is sufficient to orient the reference (d, q) so to cancel the component of the quadratic voltage (V_rq = 0). Consequently V_r = V_rd.

As a result, the dynamic model of the grid connection given by equation (10) becomes

{\begin{cases} L_{r} \frac{d}{d t} i_{r d} = V_{o d} - R_{r} i_{r d} + L_{r} ω_{r} i_{r q} - V_{r} \\ L_{r} \frac{d}{d t} i_{r q} = V_{o q} - R_{r} i_{r q} - L_{r} ω_{r} i_{r d} \end{cases}

(13)

And the reference currents become

{\begin{cases} i_{r d - r e f} = \frac{P_{r}^{*}}{V_{r d}} \\ i_{r q - r e f} = 0 \end{cases}

(14)

The classical vector control laws of the PI type give good results in the case of linear systems with constant parameters. For nonlinear systems such as wind systems, these laws may be insufficient because they are not robust, especially when requirements on accuracy and other dynamic characteristics of the system are severe during the wind speed fluctuation (Errami et al., 2015). To work around this obstacle, first-order SMC approach is proposed.

SMC

The study of non-linear control is of great interest since the majority of real systems are essentially non-linear. Conventional linear methods are satisfactory only for restricted operating ranges. For this, as soon as the system leaves its operating range, the controller is no longer able to guarantee the stability of the closed loop, hence the interest of studying more deeply nonlinear controls methods. A nonlinear system is described by the following equation system (Levant and Alelishvili, 2007)

{\begin{cases} \overset{•}{x} = f (x, t) x + g (x, t) u \\ y = h (x, t) x \end{cases}

(15)

where f(x, t), g(x, t), and h(x, t) are two nonlinear continuous and uncertain bounded functions; u and y are the input and output of the system, respectively; and x = [i_sd, i_sq, i_rd, i_rq].

The objective of our study is to ensure the control and continuation of the trajectory of reference y_r by the output y of the system, thus making the error e = y_r–y tends toward zero in the presence of uncertainties and disturbances. For this, the sliding surface is chosen as follows (Manceur et al., 2012)

\begin{matrix} S (x, t) = {(\frac{\partial}{\partial t} + σ)}^{(n - 1)} e \\ = {\sum_{k = 0}^{n - 1} \frac{(n - 1)!}{k! (n - k - 1)!} (\frac{\partial}{\partial t})}^{(n - k - 1)} σ^{k} e \end{matrix}

(16)

where n is the system order and σ is a positive constant. The choice σ > 0 guarantees a polynomial of Hurwitz.

The objective of the control law is to constrain the trajectories of state of the system to be reached and then to remain on the sliding surface in spite of the presence of uncertainties on the system. Thus, the control law must be calculated by checking a condition ensuring the stability of S(x, t) = 0. One of the methods for testing the stability of the SMC is based on the second Lyapunov theorem. Suppose that the state of equilibrium is zero, let “ $ϒ = (1 / 2) S^{2}$ ” be the Lyapunov function, the sign of the derivative of this function gives information about the stability of the system. A necessary and sufficient condition that the slip variable S(x, t) tends to zero is that the derivative of $ϒ$ is negative definite (Ouassaid et al., 2014)

\overset{•}{ϒ} = \frac{1}{2} \frac{d}{d t} S^{2} (x, t) = S (x, t) \overset{•}{S} (x, t) < 0

(17)

where

\begin{matrix} \overset{•}{S} (x, t) = {\sum_{k = 0}^{n} \frac{(n - 1)!}{k! (n - k - 1)!} (\frac{\partial}{\partial t})}^{(n - k - 1)} σ^{k} \overset{•}{e} \\ = {(y_{r} - y)}^{n} + δ_{s} \end{matrix}

(18)

and

δ_{s} = {\sum_{k = 1}^{n} \frac{(n - 1)!}{k! (n - k - 1)!} (\frac{\partial}{\partial t})}^{(n - k - 1)} σ^{k} \overset{•}{e}

(19)

The studied system in this article is of a first order “n = 1.” So the sliding surface becomes

S = e = y_{r} - y

(20)

FOSM control of WECS

The performance of the conventional vector control is limited, in particular its sensitivity against strong wind speed variation, which cannot follow the changes in WECS parameters. To overcome this problem, an SMC approach is proposed. It offers many advantages such as insensitivity against strong wind speed fluctuation and system nonlinearity. The structure of this control strategy is detailed in Figure 4.

Figure 4.

Schematic of the FOSM control strategy of WECS.

The first part of Figure 4 shows the control strategy of the wind generator. The first-order SMC consists of two terms: a discontinuous control as a function of the sign of the sliding surface and a so-called equivalent control characterizing the dynamics of the system on the sliding surface

{\begin{cases} V_{s d r} = V_{s d - e q} + V_{s d n} \\ V_{s q r} = V_{s q - e q} + V_{s q n} \end{cases}

(21)

Therefore, there are two controllers by sliding mode, which are used to regulate the currents i_sd and i_sq, so it is necessary to define the sliding surfaces as follows

{\begin{cases} S_{d} = i_{s d - r e f} - i_{s d} \\ S_{q} = i_{s q - r e f} - i_{s q} \end{cases}

(22)

The equivalent part of the V_sdq-eq command describes an ideal sliding movement, that is to say, without taking into account the uncertainties and disturbances of the system. Physically, it can be seen as the average value of the actual command. It is obtained by the invariance conditions of the sliding surface

{\begin{cases} S_{d} = \frac{d}{d t} S_{d} = 0 \\ S_{q} = \frac{d}{d t} S_{q} = 0 \end{cases}

(23)

Then, we obtain the equivalent control law as follows

{\begin{cases} V_{s d - e q} = R_{s} i_{s d} - L_{s} ω i_{s q} \\ V_{s q - e q} = L_{s} \frac{d}{d t} i_{s q - r e f} + R_{s} i_{s q} + L_{s} ω i_{s d} + ω ϕ \end{cases}

(24)

In order to satisfy the condition of attractiveness equation (17), it is sufficient to choose the discontinuous part of the order as follows

{\begin{cases} V_{s d n} = k_{d} s i g n (S_{d}) \\ V_{s q n} = k_{q} s i g n (S_{q}) \end{cases}

(25)

where k_d > 0, k_q > 0 are the command gain and sign(S_dq) is the sign function defined by

s i g n (S (x)) {\begin{cases} 1 i f S (x) > ε; x = d, q \\ \frac{S (x)}{ε} i f - ε < S (x) < ε \\ - 1 f S (x) < - ε \end{cases}

(26)

As a result

{\begin{cases} V_{s d r} = R_{s} i_{s d} - ω L_{s} i_{s q} + L_{s} k_{d} s i g n (S_{d}) \\ V_{s q r} = R_{s} i_{s q} + ω L_{s} i_{s d} + ω ϕ + L_{s} \frac{d}{d t} i_{s q - r e f} + L_{s} k_{q} s i g n (S_{q}) \end{cases}

(27)

To make the surface attractive, SMCler must be selected such that the Lyapunov function satisfies the criterion of stability Lyapunov. This function is defined as follows

ϒ_{1} = \frac{1}{2} S_{d}^{2} + \frac{1}{2} S_{q}^{2}

(28)

For the Lyapunov function to decrease, simply ensure that its derivative is negative, and to guarantee the attraction of the system throughout the surface, this is verified if

\frac{d}{d t} ϒ_{1} = S_{d} \frac{d}{d t} S_{d} + S_{q} \frac{d}{d t} S_{q} < 0

(29)

On the other hand

\begin{matrix} S_{d} \frac{d}{d t} S_{d} = S_{d} (- \frac{1}{L_{s}} (V_{s d} - R_{s} i_{s d} + ω L_{s} i_{s q})) \\ = S_{d} (- \frac{1}{L_{s}} ((R_{s} i_{s d} - ω L_{s} i_{s q} + L_{s} k_{d} s i g n (S_{d})) - R_{s} i_{s d} + ω L_{s} i_{s q})) \\ = - S_{d} k_{d} s i g n (S_{d}) \end{matrix}

(30)

\begin{matrix} S_{q} \frac{d}{d t} S_{q} = S_{q} (\frac{d}{d t} i_{s q - r e f} - \frac{1}{L_{s}} (V_{s q} - R_{s} i_{s q} - ω L_{s} i_{s d} - ω ϕ)) \\ = S_{q} (\frac{d}{d t} i_{s q - r e f} - \frac{1}{L_{s}} (\begin{array}{l} (R_{s} i_{s q} + ω L_{s} i_{s d} + ω ϕ + L_{s} \frac{d}{d t} i_{s q - r e f} + L_{s} k_{q} s i g n (S_{q})) \\ - R_{s} i_{s q} - ω L_{s} i_{s d} - ω ϕ \end{array})) \\ = - S_{q} k_{q} s i g n (S_{q}) \end{matrix}

(31)

Finally, equation (22) becomes

\frac{d}{d t} ϒ_{1} = S_{d} \frac{d}{d t} S_{d} + S_{q} \frac{d}{d t} S_{q} = - k_{d} S_{d} s i g n (S_{d}) - k_{q} S_{q} s i g n (S_{q}) < 0

(32)

Therefore, the system stability is achieved.

Part 2 of Figure 4 represents the control loop of the DC bus. A PI controller is applied to maintain the voltage measured U_bus equivalent to that of reference U_bus-ref.

The role of the command applied to the converter 2 presented in Part 3 of the Figure 4 is to maintain the frequency of the currents injected into the grid within the permissible range and improve the power factor. The vector control of grid side converter control consists of two terms: a discontinuous control as a function of the sign of the sliding surface and a so-called equivalent control characterizing the dynamics of the system on the sliding surface

{\begin{cases} V_{o d r} = V_{o d - e q} + V_{o d n} \\ V_{o q r} = V_{o q - e q} + V_{o q n} \end{cases}

(33)

The sliding surfaces for i_rd and i_rq are given by the following equation

{\begin{cases} S_{d - g} = i_{r d - r e f} - i_{r d} \\ S_{q - g} = i_{r q - r e f} - i_{r q} \end{cases}

(34)

When the system is restricted to the sliding surface, that is, S_dq-g = 0, it will be governed by an equivalent command V_odq-eq that is obtained using the invariance conditions of the surface, that is, S_dq-g = 0 and ${\overset{•}{S}}_{d q - g} = 0$ . We obtain the equivalent control law as follows

{\begin{cases} V_{o d - e q} = L_{r} \frac{d}{d t} i_{r d - r e f} + R_{r} i_{r d} - L_{r} ω_{r} i_{r q} + V_{r} \\ V_{o q - e q} = R_{r} i_{r q} + L_{r} ω_{r} i_{r d} \end{cases}

(35)

In order to satisfy the condition of attractiveness in equation (17), it is sufficient to choose the discontinuous part of the order as follows

{\begin{cases} V_{o d n} = k_{d - g} s i g n (S_{d - g}) \\ V_{o q n} = k_{q - g} s i g n (S_{q - g}) \end{cases}

(36)

where K_d-g > 0, K_q-g > 0.

To determine the required condition for the existence of the sliding mode, it is fundamental to design the Lyapunov function. So, the Lyapunov function can be chosen as follows

ϒ_{2} = \frac{1}{2} S_{d - g}^{2} + \frac{1}{2} S_{q - g}^{2}

(37)

Following the Lyapunov stability criterion, to guarantee the attraction of the system throughout the surface, the sliding manifold is reached after a limited time; this is verified if

\frac{d}{d t} ϒ_{2} = S_{d - g} \frac{d}{d t} S_{d - g} + S_{q - g} \frac{d}{d t} S_{q - g} < 0

(38)

On the other hand

\begin{matrix} S_{d - g} \frac{d}{d t} S_{d - g} = S_{d - g} (\frac{d}{d t} i_{r d - r e f} - \frac{1}{L_{r}} (V_{o d} - R_{r} i_{r d} + L_{r} ω_{r} i_{r q} - V_{r})) \\ = S_{d} (\frac{d}{d t} i_{r d - r e f} - \frac{1}{L_{r}} (\begin{array}{l} L_{r} \frac{d}{d t} i_{r d - r e f} + R_{r} i_{r d} - L_{r} ω_{r} i_{r q} + V_{r} + \\ L_{r} k_{d - g} s i g n (S_{d - g}) - R_{r} i_{r d} + L_{r} ω_{r} i_{r q} - V_{r} \end{array})) \\ = - S_{d - g} k_{d - g} s i g n (S_{d - g}) \end{matrix}

(39)

\begin{matrix} S_{q - g} \frac{d}{d t} S_{q - g} = S_{q - g} (- \frac{1}{L_{r}} (V_{o q} - R_{r} i_{r q} - L_{r} ω_{r} i_{r d})) \\ = S_{d} (- \frac{1}{L_{r}} (R_{r} i_{r q} + L_{r} ω_{r} i_{r d} + k_{q - g} s i g n (S_{q - g}) - R_{r} i_{r q} - L_{r} ω_{R} i_{r d})) \\ = - S_{q - g} k_{q - g} s i g n (S_{q - g}) \end{matrix}

(40)

Finally, equation (38) becomes

\frac{d}{d t} ϒ_{2} = S_{d - g} \frac{d}{d t} S_{d - g} + S_{q - g} \frac{d}{d t} S_{q - g} = - k_{d - g} S_{d - g} s i g n (S_{d - g}) - k_{q - g} S_{q - g} s i g n (S_{q - g}) < 0

(41)

As a result, the loop currents are asymptotically stable.

During the sliding regime, the discontinuities applied to the control may cause high-frequency oscillations of the system trajectory around the sliding surface. This phenomenon is called chattering. This phenomenon can lead to premature wear of the actuators or of certain parts of the system due to strong solicitations. In order to reduce or eliminate the phenomenon of chattering, many techniques have been proposed. The most commonly used techniques are boundary layer, observer, adaptive system fuzzy, and high-order sliding-mode control.

In this article, a second-order sliding mode command is used to solve the chattering problem. They represent an extension of the first-order sliding mode to a higher degree. This generalization retains the main characteristic in terms of robustness than that of conventional sliding modes. They also reduce their main disadvantage: the effect of chattering in the vicinity of the sliding surface. The extension of the first-order sliding modes to the higher order sliding modes is characterized by the choice of discontinuous control acting not on the sliding surface but on its upper derivatives.

SOSM control of WECS

The major drawback of the SMC is the phenomenon of chattering caused by the use of first sliding mode. An effective method to deal with this problem is to use a command by higher order sliding mode. This method helps reduce the effect of chattering, keeping the main properties of the original approach rugged. This strategy, illustrated in Figure 5, is based on the algorithm super twisting.

Figure 5.

Schematic of the SOSM control strategy of WECS.

Stability proprieties

The global control U* is composed of two super-twisting terms U₁ and U₂ in addition to the equivalent control Ueq. Originally, this algorithm has been developed to control systems with relative degree 1 in order to avoid chattering. The two terms are given by the following equation (Derbeli et al., 2016).

{\begin{cases} U^{*} = U_{1} + U_{2} + U_{e q} \\ U_{1} = \int λ_{1} s i g n (S (x, t)) \\ U_{2} = λ_{2} {| S (x, t) |}^{\frac{1}{2}} s i g n (S (x, t)) \end{cases}

(42)

In this study, control of the power produced by the WECS shown in Part 1 of Figure 5 is provided by the applied control strategy to PMSG. The sliding surfaces are determined by the relationship provided in equation (22).

The primary derivative of sliding surfaces is given by the following equation

{\begin{cases} \frac{d}{d t} S_{d} = \frac{d}{d t} i_{s d - r e f} - \frac{d}{d t} i_{s d} = - \frac{1}{L_{s}} (V_{s d} - R_{s} i_{s d} + ω L_{s} i_{s q}) \\ \frac{d}{d t} S_{q} = \frac{d}{d t} i_{s q - r e f} = \frac{d}{d t} i_{s q} = \frac{d}{d t} i_{s q - r e f} - \frac{1}{L_{s}} (V_{s q} - R_{s} i_{s q} - ω L_{s} i_{s d} - ω ϕ) \end{cases}

(43)

In order to satisfy the transition condition in equation (17) and test the stability and the robustness in closed loop, we consider a second derivative of the sliding surface (Benelghali et al., 2011)

{\begin{cases} \overset{• •}{S_{d}} = φ_{d} (x, t) + γ_{d} (x, t) {\overset{•}{V}}_{s d} \\ \overset{• •}{S_{q}} = φ_{q} (x, t) + γ_{q} (x, t) \overset{•}{V_{s q}} \end{cases}

(44)

So we guarantee the convergence of S_d and S_q to zero in the presence of uncertainties and disturbances, and the functions φ_d(x, t), φ_q(x, t), γ_d(x, t), and γ_d(x, t) must satisfy the following conditions: φ_i > 0, 0 < Г_mi < γ_i < Г_Mi, |φ_i| > Φ_i where i = d or q.

In addition to the equivalent command V_sdq-eq given in equation (24), the global command consists of the two terms of super-twisting U_1i and U_2i. After grouping the various components, the global command will be given by the following equation

{\begin{cases} V_{s d r} = U_{1 d} + U_{2 d} + V_{s d - e q} \\ V_{s q r} = U_{1 q} + U_{2 q} + V_{s q - e q} \end{cases}

(45)

where

{\begin{cases} U_{1 d} = \int λ_{1 d} s i g n (S_{d}); U_{2 d} = λ_{2 d} {| S_{d} |}^{\frac{1}{2}} s i g n (S_{d}) \\ U_{1 q} = \int λ_{1 q} s i g n (S_{q}); U_{2 q} = λ_{2 q} {| S_{q} |}^{\frac{1}{2}} s i g n (S_{q}) \end{cases}

(46)

Sufficient conditions for convergence in finite time toward the sliding surface are as follows (Benbouzid et al., 2014)

{\begin{cases} λ_{1 i} > \frac{Φ_{i}}{Γ_{m i}} \\ λ_{2 i} \geq \sqrt{\frac{4 Φ_{i} Γ_{M i} (λ_{1 i} + Φ_{i})}{Γ_{m i} (λ_{1 i} - Φ_{i})}} \end{cases}

(47)

The controlling of DC bus voltage is similar to that discussed in the previous part. A PI controller is applied to maintain the voltage measured U_bus equivalent to that of reference U_bus-ref.

The role of the command applied to the converter 2 presented in Part 3 of the Figure 5 is to maintain the frequency of the currents injected into the grid within the permissible range and ensure the operation at unity power factor.

The i_rd-ref and i_rq-ref current references are linked to the active and reactive power of grid, as it is shown by equation (14). The sliding surfaces are determined by the relationship provided in equation (34).

After the first derivation of both the surfaces, we have

{\begin{cases} \frac{d}{d t} S_{d g} = \frac{d}{d t} i_{r d - r e f} - \frac{d}{d t} i_{r d} = \frac{d}{d t} i_{r d - r e f} - \frac{1}{L_{r}} (V_{o d} - R_{r} i_{r d} + L_{r} ω_{r} i_{r q} - V_{r}) \\ \frac{d}{d t} S_{q g} = \frac{d}{d t} i_{r q - r e f} - \frac{d}{d t} i_{r q} = - \frac{1}{L_{r}} (V_{o q} - R_{r} i_{r q} - L_{r} ω_{r} i_{r d}) \end{cases}

(48)

The second derivative of sliding surfaces is given by the following equation (Benelghali et al., 2011)

{\begin{cases} {\overset{• •}{S}}_{d g} = φ_{d g} (x, t) + γ_{d g} (x, t) {\overset{•}{V}}_{o d} \\ {\overset{• •}{S}}_{q g} = φ_{q g} (x, t) + γ_{q g} (x, t) {\overset{•}{V}}_{o q} \end{cases}

(49)

So we guarantee the convergence of S_dg and S_qg to zero in the presence of uncertainties and disturbances, and the functions φ_dg(x, t), φ_qg(x, t), γ_dg(x, t), and γ_dg(x, t) must satisfy the following conditions: φ_j > 0, 0 < Г_mj < γ_j < Г_Mj, |φ_j| > Φ_j where j = dg or qg.

In addition to the equivalent command V_odq-eq given by equation (35), the global command consists of the two terms of super-twisting U_1j and U_2j. After grouping the various components, the global command will be given by the following equation

{\begin{cases} V_{o d r} = U_{1 d g} + U_{2 d g} + V_{o d - e q} \\ V_{o q r} = U_{1 q g} + U_{2 q g} + V_{o q - e q} \end{cases}

(50)

where

{\begin{cases} U_{1 d g} = \int λ_{1 d g} s i g n (S_{d g}); U_{2 d g} = λ_{2 d g} {| S_{d g} |}^{\frac{1}{2}} s i g n (S_{d g}) \\ U_{1 q g} = \int λ_{1 q g} s i g n (S_{q g}); U_{2 q g} = λ_{2 q g} {| S_{q g} |}^{\frac{1}{2}} s i g n (S_{q g}) \end{cases}

(51)

Sufficient conditions for convergence in finite time toward the sliding surface are as follows (Benbouzid et al., 2014)

{\begin{cases} λ_{1 j} > \frac{Φ_{j}}{Γ_{m j}} \\ λ_{2 j} \geq \sqrt{\frac{4 Φ_{j} Γ_{M j} (λ_{1 j} + Φ_{j})}{Γ_{m j} (λ_{1 j} - Φ_{j})}} \end{cases}

(52)

Stability study

Using the same Lyapunov function in SMC, it is clear that ϒ₃ is positive definite. The command law is expressed by the following equation

{\overset{•}{ϒ}}_{3} = S_{d} {\overset{•}{S}}_{d} + S_{q} {\overset{•}{S}}_{q} + S_{d g} {\overset{•}{S}}_{d g} + S_{q g} {\overset{•}{S}}_{q g}

(53)

By the use of equations (43), (44), (46), (47), (49), (51), and (52), equation (54) becomes

{\overset{•}{ϒ}}_{3} = - S_{d} \frac{1}{L_{s}} (U_{1 d} + U_{2 d}) - S_{q} \frac{1}{L_{s}} (U_{1 q} + U_{2 q}) - S_{d g} \frac{1}{L_{r}} (U_{1 d g} + U_{2 d g}) - S_{q g} \frac{1}{L_{r}} (U_{1 q g} + U_{2 q g}) < 0

(54)

Accordingly, the global asymptotic stability is ensured.

SOSM strategy developed meets the objectives outlined in the introduction with robustness. It therefore increases reliability, improves energy efficiency and with little chatter it causes, it reduces the mechanical stress on the entire transmission of the wind turbine.

Simulation results

To validate control approaches proposed in this study, a dynamic model of system is developed and simulated under MATLAB /Simulink using the system parameters given in Tables 1 to 4. For 10 s, we have applied to our system a wind profile that is shown in Figure 6(a).

Table 1.

Parameters of utility grid.

Parameters	Value
Grid-side filter resistance, R_r	0.2 Ω
Grid-side filter inductance, L_ra	25 mH
Grid frequency, F	50 Hz
Grid voltage, V_r	230 V
DC capacitance, C_bus	2200 µF
U_bus-ref reference DC	400 V

Table 2.

Parameters of controllers.

Parameters	Value
k_d	10
k_q	10
k_d-g	15
k_q-g	15

Table 3.

Parameters of wind turbine.

Parameters	Value
Number of blades	3
Density of air (kg/m), ρ	1.225 kg/m³
Blade radius (m), R	2
Rated power, P_eol	3.5 kW
Rated wind speed, V_vn	13 m/s

Table 4.

Parameters of PMSG.

Parameters	Value
Resistance of the stator, R_s	0.82 Ω
Inductance of the stator, L_s	15.1 mH
Pole pair number, p	2
Flux of the permanent magnet (Wb), φ	0.4832 Wb

PMSG: permanent magnet synchronous generators.

Figure 6.

(a) Wind speed, (b) mechanical speed, (c) pitch angle β, and (d) power coefficient.

A wind turbine is designed to rotate at a nominal speed and produce a nominal power. For this, a control system called “pitch angle” was used in order to control the power output of the wind turbine and thus to limit it when the wind speed is high. The mechanical speed of the machine is given by Figure 6(b), and Figure 6(c) gives the variation of the pitch angle β. The operation of this control system is summarized as follows: when the wind speed exceeds its nominal value, the angle β increases, subsequently a power coefficient C_p decreases (Figure 6(d)). Therefore, a limitation of the power produced is applied (Figure 8).

A performance comparison of a PI and FOSM control

This test is done to show the robustness of the control applied to our system according to wind speed variation. The responses obtained with the two types of control clearly show that the system operated with the FOSM is more robust compared to the PI structure. For the PI control, it is observed that the error on the quadratic component of stator current (Figure 7) and electromagnetic torque (Figure 8) caused by the change in speed is very important, so the couple and the current do not respond instantly. For this, a control by FOSM which gives good performance was implanted.

Figure 7.

Quadratic component of stator current.

Figure 8.

Electromagnetic torque.

Figure 9 shows the evolution of direct current component injected into the grid, and the response of these curve shows that the FOSM is better than the PI structure from the standpoint of response time and disturbances rejection. The strong wind speed change affects classical vector control compared to FOSM control. Despite the robustness and insensitivity against the high wind speed, the performance of FOSM is limited by the presence of the phenomenon of chattering.

Figure 9.

Direct current component injected into the grid.

The FOSM has several advantages such as robustness, high accuracy, stability, simplicity, and very low response time. Since its appearance, the theory of first sliding mode has faced the problem of chattering which proved to be a major drawback. In particular, it is difficult in such conditions to consider developments for practical applications when their implementation involves a relatively rapid wear of the process control devices. To work around this obstacle, a control by higher order sliding mode has been proposed. The proposed SOSM control has been designed using the super twisting algorithm; it is possible to reduce or even to exclude any phenomenon of chattering while maintaining the robustness and convergence properties in finite time.

A performance comparison of a FOSM and SOSM control

To validate the performance of the implemented control strategies based on SMC, a comparative study between SOSM and FOSM was presented.

As shown in Figure 10, the response of electromagnetic torque by the methodology of SOSM is more robust, while the major problem of the response by the methodology of FOSM is the phenomenon of chattering. This phenomenon results in several undesirable effects on the quality of electrical energy produced and moreover on the whole system. The use of control algorithms for SOSM is the most effective solution to fix this problem.

Figure 10.

Electromagnetic torque.

Figures 11 and 12 show, respectively, the evolution of quadratic component of stator current and direct current component injected into the grid. These figures show that the system response in the two control types has an almost linear characteristic and reached its reference without overshoot in a very small response time. The SOSM allows excluding any phenomenon of chattering caused by FOSM while maintaining the robustness properties.

Figure 11.

Quadratic component of stator current.

Figure 12.

Direct current component injected into the grid.

Figure 13 shows the simulation result of reactive power injected into grid. As can be seen, the system operates at unity power factor. It is clear that the SOSM has allowed the reduction or even the mitigation of chattering phenomenon while keeping the robustness properties and convergence in finite time.

Figure 13.

Reactive power injected into the grid.

Note that the system operates in sliding exhibiting practically no chattering and proving the robustness of the controller in the presence of the aforementioned disturbances. Illustratively, the trajectory of the controlled system in the state space $(S - \overset{•}{S})$ plane is shown in Figure 14.

Figure 14.

$S - S - \overset{•}{S}$ plane.

The different characteristics of comparison between the different control techniques are given in Table 5.

Table 5.

Comparison of the different techniques against the high variation in wind speed.

Comparison criteria	PI	FOSM	SOSM
Stability proprieties	Low	Good	Excellent
Dynamic	Limited	Good	Very good
Complexity of implementation and tuning	Low	High	Very high
Response time	High	Low	Low
Perturbations	High	Chattering	Low

PI: proportional–integral; FOSM: first-order sliding mode; SOSM: second-order sliding mode.

Response time

Good tracking responses of electromagnetic torque and currents are obtained for using a nonlinear sliding mode controller compared to the PI controller.

Stability proprieties

The stability analysis of these three methods is done by using pole compensation for classical vector control method and Lyapunov stability for SMC. It can be seen that both methods have stable performances, but faster convergence for SOMC.

Complexity of implementation and tuning

The PI controller consists only of two control parameters contrary to the SMC method, which made the PI control easier to implement and design. Thus, complexity of implementation and tuning of the SMC more specifically the SOSM controller with super-twisting algorithm are much higher.

Perturbations

The SOSM control with super-twisting algorithm offers high performance, as there is weak perturbation, on the contrary, for the chattering problem of the FOSM control and PI regulator disturbance due to high variation in wind speed.

Dynamic

Good tracking responses of electromagnetic torque, stator current and the injected grid current are obtained for a WECS using a non-linear super-twisting and second order sliding-mode control-based approach compared with the first order sliding mode control and conventional PI control.

Conclusion

In this article, a non-linear control for a grid-connected wind energy conversion chain called “higher order sliding mode control” based on super-twisting algorithm and the stability of Lyapunov is developed. The control has two main objectives. The first one is the control of converter 1 allowing maximum power extraction from the wind turbine. The second is the control of active and reactive power exchange, the regulation of the DC bus voltage, the regulation of the frequency, and voltage at the connection point to the grid to ensure the operation at unity power factor. This nonlinear approach is compared with a classical vector control and also with a first-order SMC. The study of this strategy confirms the high performance of the SOSM control compared to the PI control and FOSM control in terms of rise time and faster transient response. The FOSM has several advantages compared with the conventional vector control such as fast dynamic response, insensitivity against disturbances, and wind velocity fluctuations. However, the major drawback of FOSM is the chattering problem. To deal with this problem, a SOSM is investigated. SOSM control offers better response compared with other non-linear control of WECS, such that the system becomes stable in all operating regimes. Indeed, SOSM gives a good tracking response, an excellent stability properties, and low perturbations.

Footnotes

Appendix 1 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.

References

Alepuz

Calle

Busquets-Monge

et al . (2013) Use of stored energy in PMSG rotor inertia for low-voltage ride-through in back-to-back NPC converter-based wind power systems. IEEE Transactions on Industrial Electronics 60(5): 1787–1796.

Amimeur

Aouzellag

Abdessemed

et al . (2012) Sliding mode control of a dual-stator induction generator for wind energy conversion systems. International Journal of Electrical Power & Energy Systems 42: 60–70.

Benbouzid

Beltran

Amirat

et al . (2014) Second-order sliding mode control for DFIG-based wind turbines fault ride-through capability enhancement. ISA Transactions 53(3): 827–833.

Benelghali

Benbouzid

MEH

Charpentier

et al . (2011) Experimental validation of a marine current turbine simulator: Application to a permanent magnet synchronous generator-based system second-order sliding mode control. IEEE Transactions on Industrial Electronics 58(1): 118–126.

Boukettaya

Krichen

(2014) A dynamic power management strategy of a grid connected hybrid generation system using wind, photovoltaic and flywheel energy storage system in residential applications. Energy 71: 148–159.

Cadenas

Rivera

(2009) Short term wind speed forecasting in La Venta, Oaxaca, Mexico, using artificial neural networks. Renewable Energy 34(1): 274–278.

Cardenas

Pena

Alepuz

et al . (2013) Overview of control systems for the operation of DFIGs in wind energy applications. IEEE Transactions on Industrial Electronics 60(7): 2776–2798.

Chou

S-F

Lee

C-T

H-C

et al . (2014) A low-voltage ride-through method with transformer flux compensation capability of renewable power grid-side converters. IEEE Transactions on Power Electronics 29(4): 1710–1719.

Dansoko

Nkwawo

Diourté

et al . (2014) Robust multivariable sliding mode control design for generator excitation of marine turbine in multimachine configuration. International Journal of Electrical Power & Energy Systems 63: 423–428.

10.

Derbeli

Farhat

Barambones

et al . (2016) Control of PEM fuel cell power system using sliding mode and super-twisting algorithms. International Journal of Hydrogen Energy 42: 8833–8844.

11.

Errami

Ouassaid

Maaroufi

(2015) A performance comparison of a nonlinear and a linear control for grid connected PMSG wind energy conversion system. International Journal of Electrical Power & Energy Systems 68: 180–194.

12.

Evangelista

Puleston

Valenciaga

et al . (2013) Lyapunov-designed super-twisting sliding mode control for wind energy conversion optimization. IEEE Transactions on Industrial Electronics 60(2): 538–545.

13.

Leonhard

(1990) Control of Electrical Drives. London: Springer-Verlag.

14.

Levant

Alelishvili

(2007) Integral high-order sliding modes. IEEE Transactions on Automatic Control 52(7): 1278–1282.

15.

Haskew

Swatloski

et al . (2012) Optimal and direct-current vector control of direct-driven PMSG wind turbines. IEEE Transactions on Power Electronics 27(5): 2325–2337.

16.

Lin

Chen

(2013) Distributed optimal power flow for smart grid transmission system with renewable energy sources. Energy 56: 184–192.

17.

Manceur

Essounbouli

Hamzaoui

(2012) Second-order sliding fuzzy interval type-2 control for an uncertain system with real application. IEEE Transactions on Fuzzy Systems 20(2): 262–275.

18.

Muyeen

Al-Durra

(2013) Modeling and control strategies of fuzzy logic controlled inverter system for grid interconnected variable speed wind generator. IEEE Systems Journal 7(4): 817–824.

19.

Nasiri

Milimonfared

Fathi

(2015) A review of low-voltage ride-through enhancement methods for permanent magnet synchronous generator based wind turbines. Renewable & Sustainable Energy Reviews 47: 399–415.

20.

Orlando

Liserre

Mastromauro

et al . (2013) A survey of control issues in PMSG-based small wind-turbine systems. IEEE Transactions on Industrial Informatics 9(3): 1211–1221.

21.

Ouassaid

Elyaaloui

Cherkaoui

(2014) Nonlinear control for a grid-connected wind turbine with induction generator. In: Proceedings of the 2014 international conference on electrical sciences and technologies in Maghreb (CISTEM), Tunis, Tunisia, 3–6 November. New York: IEEE.

22.

Rajaei

Mohamadian

Varjani

(2013) Vienna-rectifier-based direct torque control of PMSG for wind energy application. IEEE Transactions on Industrial Electronics 60(7): 2919–2929.

23.

Rashid

Ali

(2017) Fault ride through capability improvement of DFIG based wind farm by fuzzy logic controlled parallel resonance fault current limiter. Electric Power Systems Research 146: 1–8.

24.

Rekik

Krichen

(2012) Control of a wind turbine for grid fault detection in stand-alone or grid-connected modes. In: Proceedings of the 2012 1st international conference on renewable energies and vehicular technology (REVET), Hammamet, Tunisia, 26–28 March, pp. 244–250. New York: IEEE.

25.

Saad

Sattar

Mansour

(2015) Low voltage ride through of doubly-fed induction generator connected to the grid using sliding mode control strategy. Renewable Energy 80: 583–594.

26.

Stigka

Paravantis

Mihalakakou

(2014) Social acceptance of renewable energy sources: A review of contingent valuation applications. Renewable & Sustainable Energy Reviews 32: 100–106.

27.

Tseng

K-C

Huang

C-C

(2014) High step-up high-efficiency interleaved converter with voltage multiplier module for renewable energy system. IEEE Transactions on Industrial Electronics 61(3): 1311–1319.

28.

Wessels

Hoffmann

Molinas

et al . (2013) StatCom control at wind farms with fixed-speed induction generators under asymmetrical grid faults. IEEE Transactions on Industrial Electronics 60: 2864–2873.

29.

Xia

Wang

Shi

et al . (2013) A novel cascaded boost chopper for the wind energy conversion system based on the permanent magnet synchronous generator. IEEE Transactions on Energy Conversion 28(3): 512–522.

30.

Yaramasu

Rivera

et al . (2014) A new power conversion system for megawatt PMSG wind turbines using four-level converters and a simple control scheme based on two-step model predictive strategy—Part II: Simulation and experimental analysis. IEEE Journal of Emerging and Selected Topics in Power Electronics 2(1): 14–25.