Sensorless Nonlinear Control of Wind Energy Systems with Doubly Fed Induction Generator

Abstract

The problem of controlling doubly fed induction generators (DFIG) associated with wind turbines is addressed. The control objective is twofold: maximum power point tracking and reactive power regulation in the DFIG. Unlike previous works, we seek the achievement of this control objective without resorting to physical sensors of mechanical variables (e.g., wind turbine velocity and DFIG rotor speed). Interestingly, wind velocity is also not assumed to be accessible to measurements. The control problem is dealt with using an output feedback controller designed on the basis of the nonlinear state-space representation of the controlled system. The controller is constituted of a high-gain nonlinear state observer and a nonlinear sliding state feedback mode. Using tools from Lyapunov’s stability, it is formally shown that the closed-loop control system, expressed in terms of the state estimation errors and the output-reference tracking errors, enjoys a semi-global practical stability. Accordingly, it is possible to tune the controller design parameters so that it meets its objectives with an arbitrarily high accuracy, whatever the initial conditions are. These theoretical results are confirmed by simulations involving wide range variation of the wind speed.

Appendix: Development and Bounding of the Disturbance Terms in (74)

Bounding Term 1:

The first term of (74) satisfies the following inequality

$$\begin{aligned} \Vert 2 e_o ^TP\dot{\varGamma }\left( s \right) {\varGamma ^{ - 1}} \left( s \right) {e_o} \Vert \le 2 \lambda _{\mathrm{max}}\Vert \dot{\Gamma }\left( \mathrm{s}\right) \Vert \Vert \Gamma ^{-1}\left( s \right) \Vert \Vert e_o^2\Vert \nonumber \\ \end{aligned}$$

(86)

Then with assumptions A1–A3 and remark 2, using (93), inequality (86) can be rewritten as

$$\begin{aligned}&\Vert 2 e_o^TP\dot{\varGamma } \left( s \right) \varGamma ^{-1}\left( s \right) {e_o}\Vert \nonumber \\&\quad \le 2{\lambda _{\mathrm{max}}}{L_{\Gamma }^{-1}} \Vert e_o^2\Vert \left( \alpha _{10} + \alpha _{11}\left| e_i \right| + \alpha _{12}\left| e_2 \right| \right) \end{aligned}$$

(87)

where

$$\begin{aligned}&\bullet \quad \alpha _{10} =\left\| \left[ \begin{array}{cc} {0_2 }&{} {O_2 } \\ {O_2 }&{} \rho \\ \end{array} \right] \right\| \hbox { with } \rho =\left[ \begin{array}{cc} {\alpha \rho _{11} }&{} {\frac{\alpha }{\beta }\rho _{21} } \\ {-\alpha \rho _{21} }&{} {\frac{\alpha }{\beta }\rho _{11} } \\ \end{array} \right] \end{aligned}$$

(88)

$$\begin{aligned}&\left\{ \begin{array}{l} \rho _{11} = \left( {\frac{{{L_s}{\omega _s}}}{{m{V_s}}}{k_1}{{\ddot{\varOmega } }_{ref\_M}} + {g_{3\_M}}{{\varOmega } _{\mathrm{max}}}} \right) \\ \rho _{21} = - \frac{{{L_s}}}{{{V_s}{M_{\mathrm{sr}}}{k_2}}}{d_2}\hbox {sign}\left( {{S_2}} \right) \\ \quad \qquad - {{\varOmega } _{\mathrm{max}}} \left( {\frac{{a{M_{\mathrm{sr}}}}}{\sigma } + \frac{1}{\sigma }} \right) p I_{rqref\_M} - \frac{{{L_s}}}{{{V_s}{M_{\mathrm{sr}}}}}{\dot{Q}_{ref\_M}} \end{array} \right. \end{aligned}$$

(89)

$$\begin{aligned}&\hbox {with }g_{3\_M} \nonumber \\&\quad =\frac{V_s a}{\omega _s \sigma }p+\left( {\frac{1}{\sigma }p-\frac{aM_{\mathrm{sr}} }{\sigma }} \right) \nonumber \\&\qquad \frac{V_s }{\omega _s M_{\mathrm{sr}} }+g_4 Q_{ref\_M} \end{aligned}$$

(90)

and

$$\begin{aligned}&\bullet \quad \alpha _{11} \!=\!\left\| \begin{array}{cccc} 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} {-\alpha d_3 }&{} {-\frac{\alpha }{\beta }\left( {\frac{aM_{\mathrm{sr}} }{\sigma }+\frac{1}{\sigma }} \right) p{\varOmega } _{\mathrm{max}} } \\ 0&{} 0&{} {\alpha \left( {\frac{aM_{\mathrm{sr}} }{\sigma }+\frac{1}{\sigma }} \right) p{\varOmega } _{\mathrm{max}} }&{} {-\frac{\alpha }{\beta }d_3 } \end{array}\right\| \nonumber \\ \end{aligned}$$

(91)

$$\begin{aligned}&\bullet \quad \alpha _{12} \!=\!\left\| \begin{array}{cccc} 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} 0&{} 0 \\ 0&{} 0&{} {\alpha \left[ {g_4 {\varOmega } _{\mathrm{max}} } \right] }&{} 0 \\ 0&{} 0&{} 0&{} {\frac{\alpha }{\beta }\left[ {g_4 {\varOmega } _{\mathrm{max}} } \right] } \\ \end{array}\right\| \end{aligned}$$

(92)

$$\begin{aligned}&\hbox {and }\alpha =\frac{M_{\mathrm{sr}} }{\sigma L_s }p\left( {1-aM_{\mathrm{sr}} } \right) \end{aligned}$$

(93)

Using (29), (66), and using Young’s inequality, (87) can be written as follows:

$$\begin{aligned} Term1\le & {} \left( \frac{\lambda _{\mathrm{max}} }{\lambda _{\mathrm{min}} }L_{\Gamma ^{-1}} \alpha _1 \right) ^{2}V_{cT} \nonumber \\&+\,2\frac{\lambda _{\mathrm{max}} }{\lambda _{\mathrm{min}} }L_{\Gamma ^{-1}} \alpha _{10} V_o +V_o \end{aligned}$$

(94)

With

$$\begin{aligned} \alpha _1 =\sqrt{2}\,\hbox {sup}\left( {\alpha _{11} ,\frac{\alpha _{12} }{k_2 }} \right) \end{aligned}$$

(95)

Bounding Term 2:

From (22) and (26), one has:

$$\begin{aligned}&\Vert \Gamma \left( \hbox {s} \right) \Delta _{\theta } \partial \left( {x_4 ,x_3 ,s} \right) \Vert \nonumber \\&\quad \le {\left\| \begin{array}{c} {T_a \varphi _{\mathrm{max}} } \\ {T_a \varphi _{\mathrm{max}} } \\ {\frac{\gamma _{31} }{\theta }+\frac{\gamma _{32} }{\theta }e_1 } \\ {\frac{\gamma _{31} }{\theta }+\frac{\gamma _{32} }{\theta }e_1 } \\ \end{array}\right\| }\end{aligned}$$

(96)

$$\begin{aligned} \hbox {with }\gamma _{31}= & {} \beta \varphi _{\mathrm{max}} \left( {\frac{T_a }{J}+\frac{f}{J}{\varOmega } _{ref\_M} } \right) ,\nonumber \\ \gamma _{32}= & {} \beta \varphi _{\mathrm{max}} \frac{f}{J}\end{aligned}$$

(97)

$$\begin{aligned} \bullet \quad \alpha _{20}= & {} T_{Max} \varphi _{\mathrm{max}} +\left| {\frac{\gamma _{31} }{\theta }} \right| ,\nonumber \\ \alpha _{21}= & {} \gamma _{32} \end{aligned}$$

(98)

Using (29), (66), and using Young’s inequality, inequality (96) can be written in the following form:

$$\begin{aligned} Term2\le & {} 4\frac{\lambda _{\mathrm{max}} }{\sqrt{\lambda _{\mathrm{min}} }}\alpha _{20} \sqrt{V_o }\nonumber \\&+\,\left( {2\frac{\lambda _{\mathrm{max}} }{\sqrt{\lambda _{\mathrm{min}} }}\frac{\sqrt{2}\alpha _{21} }{\theta k_1 }} \right) ^{2}V_o +V_{cT} \end{aligned}$$

(99)

Bounding Term 3:

Using (31) and (36)

$$\begin{aligned}&S^{T}M_r \left( y \right) \tilde{x} \nonumber \\&\quad =S_1 \frac{\theta K_2 }{\beta \varphi _{\mathrm{max}}^2 }\varphi _{\mathrm{rq}} \tilde{x} _1 \nonumber \\&\qquad -S_1 \frac{\theta K_2 }{\beta \varphi _{\mathrm{max}}^2 }\varphi _{\mathrm{rd}} \tilde{x} _2 \end{aligned}$$

(100)

introducing (27) in (100), one has:

$$\begin{aligned}&S^{T}M_r \left( y \right) \tilde{x} \nonumber \\&\quad =k_1 \frac{\theta K_2 }{\beta \varphi _{\mathrm{max}}^2 }\varphi _{\mathrm{rq}} N_1 \varDelta _{\theta } ^{-1} \varGamma \left( s \right) ^{-1}e_o e_1 \nonumber \\&\qquad -k_1 \frac{\theta K_2 }{\beta \varphi _{\mathrm{max}}^2 }\varphi _{\mathrm{rd}} N_2 \varDelta _{\theta } ^{-1} \varGamma \left( s \right) ^{-1}e_o e_1\end{aligned}$$

(101)

$$\begin{aligned}&\hbox {where }N_1 =\left[ \begin{array}{cccc} 1&{} 0&{} 0&{} 0 \\ \end{array} \right] ; \hbox {N}_2 =\left[ \begin{array}{cccc} 0&{} 1&{} 0&{} 0 \\ \end{array} \right] \end{aligned}$$

(102)

On the other hand, using Eqs. (22) and (34)–(35) one has:

$$\begin{aligned} N_1 \Delta _{\theta }^{-1} \Gamma \left( \hbox {s} \right) ^{-1}= & {} \left[ \begin{array}{cccc} 1&{} 0&{} 0&{} 0 \\ \end{array} \right] \end{aligned}$$

(103)

$$\begin{aligned} N_2 \Delta _{\theta }^{-1} \Gamma \left( \hbox {s} \right) ^{-1}= & {} \left[ \begin{array}{cccc} 0&{} 1&{} 0&{} 0 \\ \end{array} \right] \end{aligned}$$

(104)

To avoid that the observer gain $\theta$ boosts this disturbing term, one can choose the observer design parameter $K_2 =\frac{1}{\theta ^{2}}$

Then term 3 becomes:

$$\begin{aligned} Term\,\,3\le \alpha _{30} \Vert e_o\Vert \left| {e_1 } \right| \le \left( {\frac{\alpha _{30} }{2\theta \sqrt{\lambda _{\mathrm{min}} }}\frac{\sqrt{2}}{k_1 }} \right) ^{2}V_o +V_{cT}\nonumber \\ \end{aligned}$$

(105)

with

$$\begin{aligned} \bullet \quad \alpha _{30} =2\,\,Sup\left( {\frac{k_1 }{\beta \varphi _{\mathrm{max}}^2 }\varphi _{\mathrm{max}} \Vert N_1\Vert ,\frac{k_1 }{\beta \varphi _{\mathrm{max}}^2 }\varphi _{\mathrm{max}} \Vert N_2\Vert } \right) \nonumber \\ \end{aligned}$$

(106)

Bounding Term 4:

Using remark 3, (57) and (63)–(64), the four terms of inequality (74) will be discussed separately.

$$\begin{aligned}&\left| {g_3 \left( s \right) x_3 e_i } \right| \le g_{3\_M} {\varOmega } _{\mathrm{max}} \left| {e_i } \right| \end{aligned}$$

(107)

$$\begin{aligned}&\left| {g_4 x_3 e_{2} e_i } \right| \le g_4 {\varOmega } _{\mathrm{max}} \left| {e_i } \right| \left| {e_2 } \right| \end{aligned}$$

(108)

$$\begin{aligned}&\left| {g_1 x_3 e_{2} e_i } \right| \le g_1 {\varOmega } _{\mathrm{max}} \left| {e_i } \right| \left| {e_2 } \right| \end{aligned}$$

(109)

$$\begin{aligned}&\left| {g_1 I_{\mathrm{rqref}} x_3 e_2 } \right| \le g_1 I_{rqref\_M} {\varOmega } _{\mathrm{max}} \left| {e_i } \right| \end{aligned}$$

(110)

Using (29), (66), inequalities (107–110) can take the form:

$$\begin{aligned} Term\,\,4\le \sqrt{2}\alpha _{40} \sqrt{V_{cT} }+\frac{2\alpha _{41} }{k_2 }V_{cT} \end{aligned}$$

(111)

where

$$\begin{aligned}&\bullet \quad \alpha _{40} =Sup\left( {g_{3\_M} \Omega _{\mathrm{max}} ,g_1 I_{rqref\_M} {\varOmega } _{\mathrm{max}} } \right) \end{aligned}$$

(112)

$$\begin{aligned}&\bullet \quad \alpha _{41} =Sup\left( {g_4 {\varOmega } _{\mathrm{max}} ,g_1 \Omega _{\mathrm{max}} } \right) \end{aligned}$$

(113)