Robust nonlinear control via feedback linearization and Lyapunov theory for permanent magnet synchronous generator-based wind energy conversion system

Abstract

In this paper, the method for the nonlinear control design of a permanent magnet synchronous generator based-wind energy conversion system (WECS) is proposed in order to obtain robustness against disturbances and harvest a maximum power from a typical stochastic wind environment. The technique overcomes both the problem of nonlinearity and the uncertainty of the parameter compared to such classical control designs based on traditional control techniques. The method is based on the differential geometric feedback linearization technique (DGT) and the Lyapunov theory. The results obtained show the effectiveness and performance of the proposed approach.

Keywords

permanent magnet synchronous generator / wind energy conversion system / stochastic / differential geometric / feedback linearization / maximum power point tracking / Lyapunov / robust control

Introduction

During the last decades, wind energy has grown rapidly and become the most competitive form of renewable energy. Wind energy conversion system (WECS)-based permanent magnet synchronous generator (PMSG) is found to be very attractive for many applications [¹]. The main problem regarding wind power systems is the irregular character of the primary source which lies in the wind speed stochastic nature. Whatever, any wind energy conversion project within the parameters imposed by the energy market and by technical standards is not possible without the essential contribution of advanced control techniques [²,³]. Recently, due to the contribution of many areas such as mechanics, aerodynamics, power electronics and control, wind industry has grown to become the most widespread renewable energy source, namely in China, North America, and Europe. However, a full WECS is considered as a very complicated system due to the high nonlinearities which is inherent by nature, and the dynamic behavior of the system should be studied by using nonlinear model as the basis [⁴,⁵]. Therefore, in order to control the WECS using an available control technique, it is a prime requirement to linearize the system by transforming the nonlinear models into linear ones, so as to enable them to be applicable for such a control technique [³,⁶]. To overcome the nonlinearity issue, researchers have paid a great deal of attention and made significant contributions [⁷–⁹]. Actually, the popular linearization techniques are mainly categorized into the more standard form of linearization technique (SLT) using Taylor’s series expansion, the direct feedback linearization (DFL), and the differential geometric linearization technique (DGT) [¹⁰,¹¹]. SLT involves linearization of the nonlinear model in the neighborhood of equilibrium, it has been more conventional and known to be widely employed [¹²,¹³]. The DFL involves feedback linearization and is the way to transform an original system models into equivalent models of a simpler form. It is completely different from the SLT linearization, because feedback linearization is achieved by exact state transformation and feedback rather than by linear approximations of the dynamics [⁴,¹⁴]. The DGT is another popular technique for control design [¹⁵,¹⁶], which has been proposed by Alberto Isidori to linearize the highly nonlinear dynamical systems [¹⁷]. It involves a static state feedback and a special nonlinear coordinate change. However, both DFL and DGT techniques are commonly used to handle globally the nonlinearity issues for smooth functions. The major drawback of those two techniques is their direct dependency of the parameters of the plant. In the last two decades, thanks to the developments in the nonlinear control theory, researchers have come up with several robust control laws; the association of those robust laws with feedback linearization techniques has given an admirable success in regards to the robustness of the control scheme. In the literature those control designs can be found in many industrial applications. For example, Ref. [¹⁸] proposes a feedback linearization technique associated with a generalized predictive control strategy to achieve a robust control of a solar furnace. Reference [¹⁹] has also proposed a combination between the feedback linearization technique and sliding mode for the tracking and robustness concerns of a hydraulic generator regulating system (HGRS), which is almost the same for doing so in Refs. [²⁰–²²], in which a feedback linearization is used to drive an electrical motor.

However, for the application of WECS, it is necessary to take into account the extraction of the maximum power from a stochastic wind environment and the robustness against the uncertainty of parameters. In this paper, a combination between the DGT and a robust law is proposed to achieve a maximum wind power capture. The robust law is derived from the Lyapunov theory that minimizes a quadratic cost of the control error. Indeed, the application of the Lyapunov theory for the control and stability concerns can be found in Refs. [²³–²⁸].

System modeling

The schema of the PMSG-based WECS studied is presented in Fig. 1. Indeed, to simplify control, power electronics and local grid are not considered because their dynamics are much faster than other dynamics [³]. However, the maximum power capture is achieved by adjusting the generator speed to its optimum value. This is obtained by changing the chopper equivalent resistance (load) R_L at the generator terminals. This resistance is indeed considered as the system control signal [²⁹,³⁰].

Stochastic wind model

A typical wind speed can be considered as a high non-stationary random process that can be modeled as a superposition of two components as seen in Eq. (1).

(1)v\left(t\right)=\overline{v}+\mathrm{\Delta }v\left(t\right),

where \overline{v} is the seasonal, slowly variable component (low frequency long-term variations) and Δv(t) stands for the rapidly variable (high frequency) turbulence. The low frequency component is obtained from the data measured, which characterizes the site from the viewpoint of energy. The turbulence component has a stochastic nature and can be described by power spectrums (see Appendix) [³,³⁰].

WECS modeling

For a fixed pitch variable speed wind turbine, there is

(2)\{\begin{array}{l}{P}_{\mathrm{m}}={\mathrm{\Gamma }}_{\mathrm{m}}{\omega }_{\mathrm{r}},\hfill \\ {\mathrm{\Gamma }}_{\mathrm{m}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mathrm{\pi }\rho R^{\mathrm{3}}{v}^{\mathrm{2}}{C}_{\mathrm{\Gamma }}\left(\lambda \right),\hfill \\ C_{\mathrm{\Gamma }}\left(\lambda \right)=q_{\mathrm{2}}{\lambda }^{\mathrm{2}}+q_{\mathrm{1}}\lambda +q_{\mathrm{0}},\hfill \\ \lambda ={\displaystyle \frac{{\omega }_{\mathrm{r}}R}{v}},\hfill \\ C_{\mathrm{P}}=\lambda C_{\mathrm{\Gamma }},\hfill \end{array}

where P_m and G_m are respectively the wind turbine power and torque, r is the air density, R is the wind turbine radius, C_Γ, C_P, l, and ω_r are respectively the torque coefficient, the power coefficient, the tip speed ratio (TSR), and the wind turbine speed (low speed shaft).

For a rigid drive train, the dynamic equation of the mechanical part is already rendered at the high speed shaft, which can be expressed as [³¹,³²]

(3)\{\begin{array}{l}{J}_{\mathrm{h}}{\displaystyle \frac{\mathrm{d}{\omega }_{\mathrm{h}}}{\mathrm{d}t}}={\displaystyle \frac{\eta }{i}}{\mathrm{\Gamma }}_{\mathrm{m}}-{\mathrm{\Gamma }}_{\mathrm{g}},\hfill \\ {\omega }_{\mathrm{h}}={\omega }_{\mathrm{r}}i,\hfill \end{array}

where J_h is the equivalent inertia rendered at high speed shaft, ω_h, ɳ, G_g, and i are respectively the generator speed, the efficiency, the electromagnetic torque, and the gearbox ratio.

The PMSG model with the equivalent circuit in the (d, q) reference frame is presented in Fig. 2.

(4)\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_d=-{\displaystyle \frac{R_{\mathrm{s}}+R_{\mathrm{L}}}{L_d+L_{\mathrm{L}}}}{i}_d+{\displaystyle \frac{p(L_q-L_{\mathrm{L}})}{L_d+L_{\mathrm{L}}}}{i}_q{\omega }_{\mathrm{h}},\hfill \\ {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_q=-{\displaystyle \frac{R_{\mathrm{s}}+R_{\mathrm{L}}}{L_q+L_{\mathrm{L}}}}{i}_q-{\displaystyle \frac{p(L_d+L_{\mathrm{L}})}{L_q+L_{\mathrm{L}}}}{i}_d{\omega }_{\mathrm{h}}+{\displaystyle \frac{p{\mathrm{\Phi }}_{\mathrm{m}}}{L_q+L_{\mathrm{L}}}}{\omega }_{\mathrm{h}}.\hfill \end{array}

The generator electromagnetic torque is expressed as

(5){\mathrm{\Gamma }}_{\mathrm{g}}=p{\mathrm{\Phi }}_{\mathrm{m}}{i}_q,

where i_d, i_q, L_d, and L_q are, respectively the d/q axis currents and inductances; R_s is the stator resistance, p is the pair poles, and R_L and L_L are respectively the equivalent chopper resistance and inductance.

The state vector is defined as

x={[x_{\mathrm{1}},x_{\mathrm{2}},x_{\mathrm{3}}]}^{\mathrm{T}}={[i_{\mathit{d}},i_{\mathit{q}},{\omega }_{\mathrm{h}}]}^{\mathrm{T}},u==[u_{\mathrm{1}},u_{\mathrm{2}}]=[R_{\mathrm{L}},v]

as a control input signal, and the measured output is y=ω_h.

Combining Eqs. (2) and (5) gives the full nonlinear state space model of the PMSG-based WECS:

(6)\{\begin{array}{l}\dot{\mathit{x}}=\mathit{f}\left(\mathit{x}\right)+\mathit{g}\left(\mathit{x}\right)\mathit{u},\hfill \\ y=h\left(\mathit{x}\right),\hfill \end{array}

where

\mathit{f}\mathit{\left(}\mathit{x}\mathit{\right)}=\left[\begin{array}{l}{\displaystyle \frac{\mathrm{1}}{L_d+L_{\mathrm{L}}}}\left(-R_{\mathrm{s}}{x}_{\mathrm{1}}+p(L_q-L_{\mathrm{L}})x_{\mathrm{2}}{x}_{\mathrm{3}}\right)\hfill \\ {\displaystyle \frac{\mathrm{1}}{L_q+L_{\mathrm{L}}}}\left(-R_{\mathrm{s}}{x}_{\mathrm{2}}-p(L_d+L_{\mathrm{L}})x_{\mathrm{1}}{x}_{\mathrm{3}}+p{\mathrm{\Phi }}_{\mathrm{m}}{x}_{\mathrm{3}}\right)\hfill \\ {\displaystyle \frac{\mathrm{1}}{J_{\mathrm{h}}}}\left(d_{\mathrm{3}}{x}_{\mathrm{3}}^{\mathrm{2}}-p{\mathrm{\Phi }}_{\mathrm{m}}{x}_{\mathrm{2}}\right)\hfill \end{array}\right],

\mathit{g}\left(\mathit{x}\right)\mathit{u}=

{\left[-{\displaystyle \frac{\mathrm{1}}{L_d+L_{\mathrm{L}}}}{x}_{\mathrm{1}}{u}_{\mathrm{1}},-{\displaystyle \frac{\mathrm{1}}{L_q+L_{\mathrm{L}}}}{x}_{\mathrm{2}}{u}_{\mathrm{2}},{\displaystyle \frac{\mathrm{1}}{J_{\mathrm{h}}}}\left(d_{\mathrm{1}}{u}_{\mathrm{2}}^{\mathrm{2}}+d_{\mathrm{2}}{u}_{\mathrm{2}}{x}_{\mathrm{3}}\right)\right]}^{\mathrm{T}},

\mathit{h}\mathit{\left(}\mathit{x}\mathit{\right)}=\left[\mathrm{0}{}^{}\mathrm{0}{}^{}\mathrm{1}\right]x.

The parameters d_{\mathrm{1}}={\displaystyle \frac{\eta \rho \mathrm{\pi }{R}^{\mathrm{3}}}{\mathrm{2}i}}{q}_{\mathrm{0}},d_{\mathrm{2}}={\displaystyle \frac{\eta \rho \mathrm{\pi }{R}^{\mathrm{4}}}{\mathrm{2}{i}^{\mathrm{2}}}}{q}_{\mathrm{1}},d_{\mathrm{3}}={\displaystyle \frac{\eta \rho \mathrm{\pi }{R}^{\mathrm{5}}}{\mathrm{2}{i}^{\mathrm{3}}}}{q}_{\mathrm{2}}.

Differential geometric feedback linearization (DGT)

To simplify the complexity of the full nonlinear system of Eq. (6), let’s consider that the wind speed changes slightly from an operation point to another. The simplified state space model of the PMSG-based WECS can be rewritten with a unique control input u=R_L and output y=ω_h as

(7)\{\begin{array}{l}\dot{\mathit{x}}=\mathit{f}\left(\mathit{x}\right)+\mathit{g}\left(\mathit{x}\right)\mathit{u},\hfill \\ y=h\left(\mathit{x}\right).\hfill \end{array}

with

\mathit{f}\mathit{\left(}\mathit{x}\mathit{\right)}=\left[\begin{array}{l}-{\displaystyle \frac{R_{\mathrm{s}}}{L_d+L_{\mathrm{L}}}}{x}_{\mathrm{1}}+{\displaystyle \frac{p(L_q-L_{\mathrm{L}})}{L_d+L_{\mathrm{L}}}}{x}_{\mathrm{2}}{x}_{\mathrm{3}}\hfill \\ -{\displaystyle \frac{R_{\mathrm{s}}}{L_q+L_{\mathrm{L}}}}{x}_{\mathrm{2}}-{\displaystyle \frac{p(L_d+L_{\mathrm{L}})}{L_q+L_{\mathrm{L}}}}{x}_{\mathrm{1}}{x}_{\mathrm{3}}+{\displaystyle \frac{p{\mathrm{\Phi }}_{\mathrm{m}}}{L_q+L_{\mathrm{L}}}}{x}_{\mathrm{3}}\hfill \\ {\displaystyle \frac{\mathrm{1}}{J_{\mathrm{h}}}}\left(d_{\mathrm{1}}{v}^{\mathrm{2}}+d_{\mathrm{2}}v{x}_{\mathrm{3}}+d_{\mathrm{3}}{x}_{\mathrm{3}}^{\mathrm{2}}-p{\mathrm{\Phi }}_{\mathrm{m}}{x}_{\mathrm{2}}\right)\hfill \end{array}\right],

\mathit{g}\left(\mathit{x}\right)={\left[-{\displaystyle \frac{\mathrm{1}}{L_d+L_{\mathrm{L}}}}{x}_{\mathrm{1}},-{\displaystyle \frac{\mathrm{1}}{L_q+L_{\mathrm{L}}}}{x}_{\mathrm{2}},\mathrm{0}\right]}^{\mathrm{T}},

\mathit{h}\left(\mathit{x}\right)=x_{\mathrm{3}}.

Equation (7) is considered as a nonlinear system, but with smooth nonlinearities, so a possible control design solution can be the DGT technique [¹⁷,³³].

In general, x∈Rⁿ is the state vector, u∈R is the control input, y∈R is the system output, f(x)∈Rⁿ and g(x)∈Rⁿ are smooth vector fields, and h(x)∈Rⁿ is a scalar smooth function. It is assumed that the nonlinear system of Eq. (7) has a relative degree r (r<n) in a neighborhood D_x of a point x₀, and has a stable internal dynamics [¹⁶], taking derivatives of the output y with respect to time up to r times gives

(8)y^{\left(r\right)}=L_f^{r}h\left(x\right)+\underset{\ne \mathrm{0}}{\underbrace{L_g{L}_f^{r-\mathrm{1}}h\left(x\right)u}},

where L_f^{r}h\left(x\right) is the Lie derivative of h(x) along the direction of the vector field f(x) up to r times and L_g{L}_f^{r-\mathrm{1}}h\left(x\right) is the Lie derivative of h(x) along the direction of the vector field g(x).

Defining: \alpha \left(x\right)=L_f^{r}h\left(x\right), \beta \left(x\right)=L_g{L}_f^{r-\mathrm{1}}h\left(x\right),

rewriting Eq. (8) yields

(9)y^{\left(r\right)}=\alpha \left(\mathit{x}\right)+\beta \left(\mathit{x}\right)u

choosing the state feedback linearization control

(10)u^{*}\left(x\right)={\displaystyle \frac{\mathrm{1}}{\beta \left(x\right)}}(-\alpha (x)+\upsilon ).

Then the input-output nonlinear relation (Eq. (9)) becomes linear as

(11)y^{\left(r\right)}=\upsilon .

Equation (11) has a relative degree r = 2 (double-integrator) and any linear control technique can be designed for a stabilizing or tracking objective. It is apparent that the ideal feedback linearization control Eq. (10) is only applicable if the field vectors α(x) and β(x) are completely known [³,¹⁷].

Hence, after computation using the WECS data, there is

(12)\{\begin{array}{l}\alpha \left(x\right)=-{\displaystyle \frac{p{\mathrm{\Phi }}_{\mathrm{m}}}{J_{\mathrm{h}}}}\left[b_{\mathrm{2}}{x}_{\mathrm{2}}+b_{\mathrm{1}}{x}_{\mathrm{1}}{x}_{\mathrm{3}}+b_{\mathrm{4}}{x}_{\mathrm{3}}\right]+{\displaystyle \frac{\mathrm{1}}{J_{\mathrm{h}}}}\left[{\displaystyle \frac{d_{\mathrm{2}}}{i}}v+\mathrm{2}{d}_{\mathrm{3}}{x}_{\mathrm{3}}\right]{\dot{x}}_{\mathrm{3}},\hfill \\ \beta \left(x\right)={\displaystyle \frac{b_{\mathrm{4}}}{J_{\mathrm{h}}}}{x}_{\mathrm{2}}.\hfill \end{array}

From Eq. (12), it can be seen clearly that α(x) and β(x)are strongly dependent on the parameters of the system.

The control objective first is the MPPT which should drive the generator speed ω_h to track asymptotically an optimal reference y_r\left(t\right)={\omega }_{\mathrm{h}}^{*}.

The generator speed reference can be expressed as

(13)y_r\left(t\right)={\omega }_{\mathrm{h}}^{*}={\displaystyle \frac{{\lambda }^{*}iv\left(t\right)}{R}}

with λ* being the optimal TSR.

Define the output tracking error e(t)=y_r(t)−y(t), and introduce the following error vector \xi \left(t\right)={\left[e\left(t\right)\dot{e}\left(t\right)\right]}^{\mathrm{T}}.

Hence, the control objective can be achieved using the ideal control law

(14)u^{*}\left(x\right)={\displaystyle \frac{\mathrm{1}}{\beta \left(x\right)}}\left(-\alpha \left(x\right)+{\ddot{y}}_r\left(t\right)-{\mathit{k}}^{\mathrm{T}}\xi \right),

where the feedback gain matrix k=[k₂ k₁]^T is chosen such that the polynomial s²+k₁s+k₂ has all its roots in the left-half complex plane. That is leading to \ddot{e}\left(t\right)+k_{\mathrm{1}}\dot{e}\left(t\right)+k_{\mathrm{2}}e\left(t\right)=\mathrm{0}, which implies the tracking error asymptotically converges to zero. Besides, the eigenvalues of the error vector matrix are placed following to specifications concerning overshoot and settling time. The values of the selected gains k₁ and k₂ are listed in Appendix.

Robust control law based on Lyapunov theory

A novel robust control scheme based on Lyapunov theory is proposed to handle the large model uncertainty caused by the variation of parameters.

In the case in this paper, the nonlinear field vectors (α(x), β(x)) of Eq. (12) is directly affected by the uncertainty of the parameter.

Globally to model uncertainty, Eq. (7) is rewritten as

(15)\{\begin{array}{l}\dot{\mathit{x}}=\mathit{f}(\mathit{x})+\mathrm{\Delta }\mathit{f}(\mathit{x})+\left(\mathit{g}\left(\mathit{x}\right)+\mathrm{\Delta }\mathit{g}\left(\mathit{x}\right)\right)u,\hfill \\ y=h\left(\mathit{x}\right),\hfill \end{array}

where the exact part of the plant is represented by the functions (f, g, h), while (Δf, Δg) present the uncertain part of the plant.

From Eq. (15), the uncertain WECS model input-output relationship is obtained as

\ddot{y}=\upsilon +\Delta \upsilon ,

where the effect of uncertainty is also represented by \Delta \upsilon =\Delta \alpha \left(x\right)+\Delta \beta \left(x\right)u.

However, as \Delta \upsilonis an unknown term, a robust control of the uncertain WECS is not guaranteed by the linear feedback control law of Eq. (11). Thus, the following robust law based on Lyapunov theory is proposed to achieve an input-output stability and a robust tracking performance of the uncertain WECS [²⁰,²⁵–²⁷].

First, a state-space form of the error dynamics is written as

(16)\dot{\xi }=\left[\begin{array}{ll}\mathrm{0}\hfill & \mathrm{1}\hfill \\ -k_{\mathrm{2}}\hfill & -k_{\mathrm{1}}\hfill \end{array}\right]\xi +\left[\begin{array}{l}\mathrm{0}\hfill \\ \mathrm{1}\hfill \end{array}\right](\upsilon +\Delta \upsilon ).

Taking matrices

{\mathit{A}}_{\mathrm{c}}=\left[\begin{array}{ll}\mathrm{0}\hfill & \mathrm{1}\hfill \\ -k_{\mathrm{2}}\hfill & -k_{\mathrm{1}}\hfill \end{array}\right],\mathit{B}=\left[\begin{array}{l}\mathrm{0}\hfill \\ \mathrm{1}\hfill \end{array}\right],

so that

\dot{\xi }={\mathit{A}}_{\mathrm{c}}\xi +\mathit{B}(\upsilon +\Delta \upsilon ).

A Lyapunov function V={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\xi }^{\mathrm{T}}\mathit{P}\xi is chosen, in which P is chosen as a positive semi-definite matrix satisfying the Lyapunov equation

(17){\mathit{A}}^{\mathrm{T}}\mathit{P}+\mathit{P}\mathit{A}=–\mathit{Q},

where Q is an identity matrix.

The Lyapunov function time derivative is

\dot{V}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\left[\dot{\xi }\mathit{P}\xi +{\xi }^{\mathrm{T}}\mathit{P}\dot{\xi }\right].

A simplified form of \dot{V} can be obtained as

(18)\dot{V}=-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\xi }^{\mathrm{T}}\mathit{Q}\xi +{\xi }^{\mathrm{T}}\mathit{P}\mathit{B}(\upsilon +\mathrm{\Delta }\upsilon ).

If the condition is satisfied along the solution trajectory of the system, it implies that

(19)

The linear control \upsilon is chosen as

(20)\upsilon =-F\mathrm{sgn}\left({\xi }^{\mathrm{T}}\mathit{P}\mathit{B}\right)

with the condition F>\left|\Delta \upsilon \right|.

The substitution of Eq. (20) in Eq. (19) gives

(21)

This equation is valid for any value of {\xi }^{\mathrm{T}}PB, provided F>\left|\Delta \upsilon \right|, where

After computation, the P matrix is found as

\mathit{P}=\left[\begin{array}{cc}\hfill {\displaystyle \frac{k_{\mathrm{1}}}{\mathrm{2}{k}_{\mathrm{2}}}}+{\displaystyle \frac{\mathrm{1}+k_{\mathrm{2}}}{\mathrm{2}{k}_{\mathrm{1}}}}\hfill & \hfill {\displaystyle \frac{\mathrm{1}}{\mathrm{2}{k}_{\mathrm{2}}}}\hfill \\ \hfill {\displaystyle \frac{\mathrm{1}}{\mathrm{2}{k}_{\mathrm{2}}}}\hfill & \hfill {\displaystyle \frac{\mathrm{1}+k_{\mathrm{2}}}{\mathrm{2}{k}_{\mathrm{1}}{k}_{\mathrm{2}}}}\hfill \end{array}\right].

The final expression of the robust linear control \upsiloncan be written as

(22)\upsilon =y_{\mathrm{r}}^{\left(\mathrm{2}\right)}\left(t\right)-{\mathit{k}}^{\mathrm{T}}\mathit{\xi }-\mathit{F}\mathrm{sgn}\left(\left[\begin{array}{ll}{\displaystyle \frac{\mathrm{1}}{\mathrm{2}{k}_{\mathrm{2}}}}\hfill & {\displaystyle \frac{\mathrm{1}+k_{\mathrm{2}}}{\mathrm{2}{k}_{\mathrm{1}}{k}_{\mathrm{2}}}}\hfill \end{array}\right]\xi \right).

Simulation results

The simulation is conducted within a Matlab/Simulink environment, in which the studied 3 kW PMSG-based WECS (Fig. 1) has a maximum power coefficient of C_pmax≈ 0.476, which corresponds to an optimal tip-speed ratio λ*≈ 7. The simulation control scheme is indeed depicted in Fig. 3 and the system data are given in Appendix. The states variables of the WECS are considered available for feedback. Hence, to ensure the efficiency of the proposed control scheme, the simulation steps are organized as following:

First, a maximum power point tracking check of the proposed controller under a realistic wind profile (Fig. 4 (a)), namely for a wind sequence having an average speed of about 7 m/s and a medium turbulence intensity (σ_v = 0.15) using the von Karman spectrum. Secondly, robustness checks of the proposed controller have been done against both sharp variation of the wind speed and the uncertainty of the parameter. Finally, a comparison with another robust nonlinear controller (First order sliding mode control) is offered to ensure the advantage of the proposed control scheme.

However, for the tracking objective, under wind speed fluctuations, the optimal power extraction of the WECS is guaranteed by keeping both the power coefficient value around its maximum value (C_pmax) and the tip-speed ratio value around its optimal value (λ*). Hence, it is obviously observed from Fig. 4 that under a stochastic wind environment, the tip speed ratio (Fig. 4(b)) and the power coefficient C_p (Fig. 4(c)) are easily maintained to their corresponding optimal values. Indeed, the generator rotates at an optimal speed as illustrated in (Fig. 4(d)), and the WECS extracts a maximum power from the wind (Fig. 5(b)).

Figure 5(a) demonstrates the control signal (R_L) (equivalent chopper resistance), which has a chattering phenomenon owing to the discontinuous control effect (see Eq. (20)). Through Figs. 5(c) and 5(d), it can be seen that for each wind speed value, it is possible to drive any operating point into the neighborhood of optimal operating points by adjusting the control signal.

After seeing the good tracking performance of the proposed controller, the robustness of the control scheme has been checked under critical conditions. Hence, two attempts have been made to check its behavior. First, the system has been tested under a sharp variation of the wind speed (rise and drop) as shown in Fig. 6(a) and in Fig. 6(b) for the wind turbine electrical power response. Even though the wind speed is abruptly changed, some transitory disturbances can be observed in power coefficient, tip speed ratio, and generator speed as shown in Figs. 7(a)–7(c). In fact, there are small static errors but they do not have notable effects on the control scheme stability and performance.

Secondly, as α(x) and β(x) are directly dependent on the parameters of the system, especially with the high speed shaft inertia J_h (Eq. (12)), an attempt has been made to check the robustness of the control scheme against the uncertainty of parameters.

The robustness test has only taken into account the high speed shaft inertia J_h as exhibited in Fig. 8(a). Although small static errors and very small disturbances can be observed in Fig. 9 (zoom), Fig. 10 (zoom), and Fig. 11(b) (zoom), they do not have any notable effect on the stability of the control scheme. Figure 11(a) shows the equivalent chopper resistance during the disturbance, Fig. 11(b) shows the generator speed, and Figs. 11(c) and 11(d) present respectively the variations of the field vectors α(x) and β(x) versus time. Even though their magnitudes are changed during the disturbance, the system remains stable and robust.

For the comparison study, a robust nonlinear control scheme based on a first order sliding mode control (SMC) is proposed for the tracking objective under stochastic wind speed, in which, from Figs. 12(a)–12(c), the simulation results (index ‘com’ for the SMC) show clearly the advantages of the proposed controller.

The results obtained confirm that the proposed robust controller guarantees an optimal power conversion and a good robustness property to overcome the effect of nonlinearities, as well as the uncertainty of the parameter.

Conclusions

To ensure a maximum power capture for the variable speed permanent magnet-based wind energy conversion system, a robust control design was presented.

Since the WECS model studied is nonlinear in the d-q reference frame, a geometric derivative feedback linearization technique (GDT) was used to handle the nonlinearity issue of the system. However, due to the dependency of the feedback controller on the parameters of the system, such as the high speed shaft inertia, it was strongly affected by any uncertainty of the parameter. To enhance the system robustness, a robust control law based on the Lyapunov theory was proposed in this paper, where the uncertainty and disturbances of the nonlinear model were fully handled.

References

[1] Harrabi N, Souissi M, Aitouche A, Chabaane M. Intelligent control of wind conversion system based on PMSG using T-S Fuzzy Scheme. International Journal of Renewable Energy Research, 2015, 5(4): 52–60

[2] Emna M E, Adel K, Mimouni M F. The wind energy conversion system using PMSG controlled by vector control and SMC strategies. International Journal of Renewable Energy Research, 2013, 3(1): 41–50

[3] Munteanu I, Bratcu A I, Cutululis N A, Ceang Ă E. Optimal Control of Wind Energy Systems: Toward a Global Approach. London: Springer, 2008

[4] Yadaiah N, Ramana N V. Linearization of multi-machine power system: modeling and control—a survey. International Journal of Electrical Power and Energy Systems, 2007, 29(4): 297–311

[5] Ghasemi S, Tabesh A, Askari-Marnani J. Application of fractional calculus theory to robust controller design for wind turbine generators. IEEE Transactions on Energy Conversion, 2014, 29(3): 780–787

[6] Tabesh A, Iravani R. On the application of the complex torque coefficients method to the analysis of torsional dynamics. IEEE Transactions on Energy Conversion, 2005, 20(2): 268–275

[7] Tabesh A, Iravani R. Frequency response analysis of torsional dynamics. IEEE Transactions on Power Systems, 2004, 19(3): 1430–1437

[8] Farmad M, Farhangi S, Gharehpetian G B, Afsharnia S. Nonlinear controller design for IPC using feedback linearization method. International Journal of Electrical Power & Energy Systems, 2013, 44(1): 778–785

[9] Boukhezzar B, Siguerdidjane H. Nonlinear control with wind estimation of a DFIG variable speed wind turbine for power capture optimization. Energy Conversion and Management, 2009, 50(4): 885–892

[10] Shi Z, Li X, Hu S. Direct feedback linearization based control in variable air volume air-conditioning system. Procedia Physics, 2012, 24(Part B): 1248–1254

[11] Bodson M, Chiasson J. Differential-geometric methods for control of electric motors. International Journal of Robust and Nonlinear Control, 1998, 8(11): 923–954

[12] Tailor M R, Bhathawala P H. Linearization of nonlinear differential equation by Taylor’s series expansion and use of Jacobian linearization process. International Journal of Theoretical and Applied Science, 2011, 4(1): 36–38

[13] Jouybari-Moghaddam H, Hosseinian S H, Vahidi B. Grid reconnection detection for synchronous distributed generators in stand-alone operation. International Transactions on Electrical Energy Systems, 2015, 25(1): 138–154

[14] Akhrif O, Okou F A, Dessaint L A, Champagne R. Application of a multivariable feedback linearization scheme for rotor angle stability and voltage regulation of power systems. IEEE Transactions on Power Systems, 1999, 14(2): 620–628

[15] Lu Q, Sun Y Z. Nonlinear stabilizing control of multi machine systems. IEEE Transactions on Power Systems, 1988, 4(1): 36–38

[16] Hunt L R. Su R, Meyer G. Design for multi-input nonlinear systems in differential geometric control theory. Progress in Mathematics, 1982, 27:268–298

[17] Isodori A. Nonlinear Control Systems. 3rd ed. Berlin: Springer-Verlag, 1995

[18] Beschi M, Berenguel M, Visioli A, Guzmán J L, Yebra L J. Implementation of feedback linearization GPC control for a solar furnace. Journal of Process Control, 2013, 23(10): 1545–1554

[19] Yuan X, Chen Z, Yuan Y, Huang Y, Li X, Li W. Sliding mode controller of hydraulic generator regulating system based on the input/output feedback linearization method. Mathematics and Computers in Simulation, 2016, 119(C): 18–34

[20] Mahboub M A, Drid S, Sid M A, Cheikh R. Robust direct power control based on the Lyapunov theory of a grid-connected brushless doubly fed induction generator. Frontiers in Energy, 2016, 10(3): 298–307

[21] Zarchi H A, Arab Markadeh Gh R, Soltani J. Direct torque and flux regulation of synchronous reluctance motor drives based on input–output feedback linearization. Energy Conversion and Management, 2010, 14(1): 71–80

[22] Bouzidi M, Benaissa A, Barkat S. Hybrid direct power/current control using feedback linearization of three-level four-leg voltage source shunt active power filter. International Journal of Electrical Power and Energy Systems, 2014, 61: 629–646

[23] Mehrasa M, Pouresmaeil E, Akorede M F, Jorgensen B N, Catalão J P S. Multilevel converter control approach of active power filter for harmonics elimination in electric grids. Energy, 2015, 84: 722–731

[24] Alizadeh M, Kojori S S. Augmenting effectiveness of control loops of a PMSG (permanent magnet synchronous generator) based wind energy conversion system by a virtually adaptive PI (proportional integral) controller. Energy, 2015, 91: 610–629

[25] Drid S, Tadjine M, Nait-Said M S. Robust backstepping vector control for the doubly fed induction motor. IET Control Theory & Applications, 2007, 1(4): 861–868

[26] Mehrasa M, Pouresmaeil E, Zabihi S, Rodrigues E M G, Catalão J P S. A control strategy for the stable operation of shunt active power filters in power grids. Energy, 2016, 96: 325–334

[27] Phan D C, Yamamoto S. Rotor speed control of doubly fed induction generator wind turbines using adaptive maximum power point tracking. Energy, 2016, 111: 377–388

[28] Cheikh R, Menacer A, Drid S. Robust control based on the Lyapunov theory of a grid-connected doubly fed induction generator. Frontiers in Energy, 2013, 7(2): 191–196

[29] Bianchi F D. Wind Turbine Control Systems: Principles, Modeling and Gain Scheduling Design. London: Springer-Verlag, 2007

[30] Nguyen H M, Naidu D S. Direct fuzzy adaptive control for standalone wind energy conversion systems. In: Proceedings of the World Congress on Engineering and Computer Science, 2012, San Francisco, USA

[31] Nichita C. Study and development of structures and digital control laws for the realization of 3 kW wind turbine simulator. Dissertation for the Doctoral Degree. France: Université du Havre, 1995 (in French)

[32] Welfonder E, Neifer R, Spanner M. Development and experimental identification of dynamic models for wind turbines. Control Engineering Practice, 1997, 5(1): 63–73

[33] Chapman J W, IIic M D, King C A, Eng L, Kaufman H. Stabilizing a multi machine power system via decentralized feedback linearizing excitation control. IEEE Transactions on Power Systems, 1993, 8(3): 830–839

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