Robust direct power control based on the Lyapunov theory of a grid-connected brushless doubly fed induction generator

Abstract

This paper deals with robust direct power control of a grid-connected brushless doubly-fed induction generator(BDFIG). Using a nonlinear feedback linearization strategy, an attempt is made to improve the desired performances by controlling the generated stator active and reactive power in a linear and decoupled manner. Therefore, to achieve this objective, the Lyapunov approach is used associated with a sliding mode control to guarantee the global asymptotical stability. Thus, an optimal operation of the BDFIG in sub-synchronous operation is obtained as well as the stator power flows with the possibility of keeping stator power factor at a unity. The proposed method is tested with the Matlab/Simulink software. Simulation results illustrate the performances and the feasibility of the designed control.

Keywords

brushless doubly fed induction generator (BDFIG) / vector control / Lyapunov theory / power factor unity / active and reactive power

Introduction

During the last two decades, the wind energy generation industry has been growing rapidly (at 40% per year), moving fast to sustain more than 20% of the electricity in some countries. Recently, the doubly fed induction generators (DFIG) have become the most used configuration in variable speed wind energy applications because of their advantages [ 1]. The cost of maintenance for traditional DFIG based wind generators increases the pressure to seek other alternative generator systems [ 2].

Brushless doubly fed induction generators (BDFIGs) promise significant advantages for wind-power generation. The absence of a brush gear increases the reliability of the device and requires less maintenance cost [ 3– 5]. This configuration is of great importance for offshore and difficult-to-reach installations. Moreover, the BDFIG manufacturing cost is very less compared to the DFIG [ 4]. This fact results from the absence of the slip-ring system and the simple structure of the rotor winding. Recently, research efforts have been directed towards eliminating the slip rings and brushes while maintaining the benefits of DFIG.

In literature, several BDFIG scalar control algorithms are proposed. For instance, it is shown that the open-loop control, the closed-loop frequency control and the phase-angle control can stabilize the machine over a wide speed range. However, the vector control (VC) methods, also known as field-oriented control, give better dynamic performance [ 3, 4]. This last can be implemented with a conventional proportional plus integral (PI) controller [ 2, 6, 7]. In general these control proposals are intrinsically based on Taylor’s linearization of the system dynamic model around a particular operational point. Therefore, the tuning of the controller is only valid in a restricted operational area. On the other hand, these controllers are not robust against parameter variations, model uncertainties and external perturbations, and can present an asymptotic convergence. Their main advantage lies in their relatively simple implementation with a rather low computational cost.

More accurate methods are developed based on different nonlinear control techniques. For instance, neural networks [ 8], feedback linearization [ 9], variable structure control [ 10], fuzzy logic control [ 1, 11], combinations of some previous techniques [ 12], and other nonlinear approaches [ 13] are possible design alternatives. Despite the recognized advantage of using nonlinear controllers to cope with nonlinear systems, many of these techniques produce control laws with a rather high computational burden. These calculi commonly depend on the system states and on several model parameters having the secondary effect of reducing the control robustness. Finally, sliding mode techniques [ 14] can produce high frequency discontinuous control actions that is very difficult to be implemented on the inverter controller.

The main contribution of this paper consists on developing an alternative control strategy based on the Lyapunov theory in order to handle the active and reactive powers. This technique is robust against parameter variations, model uncertainties and exogenous perturbations.

Theory of operation

As shown in Fig. 1, the stator of this machine incorporates two sets of three phase windings with different number of poles. The first one, called power winding (PW), is connected directly to the grid and handles most of the machine power. The other one, called control winding (CW), is connected via a bi-directional converter to the grid and handles a small percentage of machine power. The rotor of the BDFIG carries a special design cage [ 15, 16].

The BDFG’s electromagnetic torque is achieved throughout an interaction and synchronization between the two BDFG’s stator magnetic fields and the rotor magnetic field [ 15]. This can occur if the frequency and distribution of the current induced in the rotor by the PW field matches those induced by the CW field [ 16].

The BDFIG can be operated in several modes, including the synchronous (doubly fed) mode, cascade mode, and induction mode [ 3]. The synchronous mode is the most desirable one in which the shaft speed is independent of the torque exerted on the machine. This is achieved when Eqs. (1) and (2) are satisfied [ 3, 16].

(1) {\omega }_{\mathrm{r}} ={\displaystyle \frac{{\omega }_{\mathrm{sc}}\pm {\omega }_{\mathrm{sp}}}{P_{\mathrm{c}}+P_{\mathrm{p}}}},

(2) N_{\mathrm{r}}=P_{\mathrm{c}}+P_{\mathrm{p}},

where {\omega }_{\mathrm{sp}} and {\omega }_{\mathrm{sc}} are the angular frequency of PW and CW, respectively; {\omega }_r is the rotor angular speed; P_{\mathrm{p}} and P_{\mathrm{c}} are the number of pole pairs of PW and CW, respectively, in the case in this paper, (P_{\mathrm{p}}=\mathrm{3}) and (P_{\mathrm{c}}=\mathrm{1}) ; N_{\mathrm{r}} is the number of rotor bars; and±accounts for the case which CW is excited in the positive or negative phase sequence.

If the conditions stated in Eqs. (1) and (2) are satisfied, a cross coupling between the two stator fields occurs via the rotor, and hence a nonzero average torque is produced. This mode of operation of the BDFIG is called the “synchronous mode”. To avoid the direct mutual coupling between the two stator field windings, the number of poles should be different [ 5].

Modeling of the system

Model of BDFIG

The BDFIG equations obtained in the (d-q) reference frame depicted in Fig .2, that rotates synchronously with the power winding stator flux by angular speed of w_{\mathrm{sp}} [ 2, 5] can be expressed as

(3) v_{\mathrm{sp}}=R_{\mathrm{sp}}{i}_{\mathrm{sp}}+{\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{sp}}}{\mathrm{d}t}}+\mathrm{j}{w}_{\mathrm{sp}}{\psi }_{\mathrm{sp}},

(4) v_{\mathrm{sc}}=R_{\mathrm{sc}}{i}_{\mathrm{sc}}+{\displaystyle \frac{d{\psi }_{\mathrm{sc}}}{\mathrm{dt}}}+\mathrm{j}{w}_{\mathrm{sc}}{\psi }_{\mathrm{sc}},

(5) v_{\mathrm{r}}=R_{\mathrm{r}}{i}_{\mathrm{r}}+{\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{r}}}{\mathrm{d}t}}\mathrm{+j}{w}_{\mathrm{r}}{\psi }_{\mathrm{r}},

(6) {\psi }_{\mathrm{sp}}=L_{\mathrm{sp}}{i}_{\mathrm{sp}}+L_{\mathrm{mp}}{i}_r,

(7) {\psi }_{\mathrm{sc}}=L_{\mathrm{sc}}{i}_{\mathrm{sc}}+L_{\mathrm{mc}}{i}_r,

(8) {\psi }_r=L_r{i}_r+L_{\mathrm{mc}}{i}_{\mathrm{sc}}+L_{\mathrm{mp}}{i}_{\mathrm{sp}},

(9) T_{em}={\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{P}_p\mathrm{Im}\{{\psi }_{\mathrm{sp}}^{*}{i}_{\mathrm{sp}}\}+{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{P}_c\mathrm{Im}\{{\psi }_{sc}{i}_{\mathrm{sc}}^{*}\}.

The stator power (PW) expressions are

(10) P_{\mathrm{s}\mathrm{p}}=\mathrm{Re}\{v_{\mathrm{s}\mathrm{p}}{i}_{\mathrm{s}\mathrm{p}}^{*}\},

(11) Q_{\mathrm{s}\mathrm{p}}=\mathrm{Im}\{v_{\mathrm{s}\mathrm{p}}{i}_{\mathrm{s}\mathrm{p}}^{*}\}.

The current-flux equations are

(12) i_{\mathrm{sp}}={\displaystyle \frac{{\psi }_{\mathrm{sp}}-M_p{i}_r^d}{L_{\mathrm{sp}}}},

(13) i_r={\displaystyle \frac{{\psi }_r-M_p{i}_{\mathrm{sp}}-M_c{i}_{\mathrm{sc}}}{L_r}}.

Equation (14) can be obtained by combining Eq. (12) with Eq. (13).

(14) i_{\mathrm{sp}}={\displaystyle \frac{L_r}{L_{\mathrm{sp}}{L}_r-M_p^2}}{\psi }_{\mathrm{sp}}-{\displaystyle \frac{M_p}{L_{\mathrm{sp}}{L}_r-M_p^2}}{\psi }_r+{\displaystyle \frac{M_c{M}_p}{L_{\mathrm{sp}}{L}_r-M_p^{\mathrm{2}}}}{i}_{\mathrm{sc}}.

Replacing Eq. (14) in Eqs. (10) and (11) with equalization of the real and imaginary parts, Eq. (15) can be obtained.

(15) \{\begin{array}{l}{P}_{\mathrm{sp}}={\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{v}_{\mathrm{sp}}({\delta }_{\mathrm{5}}{\psi }_{\mathrm{sp}}^q-{\delta }_{\mathrm{4}}{\psi }_r^q+{\delta }_{\mathrm{3}}{i}_{\mathrm{sc}}^q),\\ Q_{\mathrm{sp}}={\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{v}_{\mathrm{sp}}({\delta }_{\mathrm{5}}{\psi }_{\mathrm{sp}}^d-{\delta }_{\mathrm{4}}{\psi }_r^d+{\delta }_{\mathrm{3}}{i}_{\mathrm{sc}}^d),\end{array}

where

\begin{array}{l}{\delta }_{\mathrm{1}}={\displaystyle \frac{M_c{L}_{\mathrm{sp}}}{L_{\mathrm{sp}}{L}_r-M_p^{\mathrm{2}}}},{\delta }_{\mathrm{2}}=L_{\mathrm{sc}}-{\displaystyle \frac{M_c^{\mathrm{2}}{L}_{\mathrm{sp}}}{L_{\mathrm{sp}}{L}_r-M_p^{\mathrm{2}}}},{\delta }_{\mathrm{3}}={\displaystyle \frac{M_c{M}_p}{L_{\mathrm{sp}}{L}_r-M_p^{\mathrm{2}}}},\\ {\delta }_{\mathrm{4}}={\displaystyle \frac{M_p}{L_{\mathrm{sp}}{L}_r-M_p^{\mathrm{2}}}},{\delta }_{\mathrm{5}}={\displaystyle \frac{L_r}{L_{\mathrm{sp}}{L}_r-M_p^{\mathrm{2}}}}.\end{array}

From Eqs. (4) to (11), the dynamic relation between the CW current and the voltage in the d-q axis ( v_{\mathrm{sc}} and i_{\mathrm{sc}} ) can be obtained as

(16) \{\begin{array}{l}{v}_{\mathrm{sc}}^q=R_s{i}_{\mathrm{sc}}^q+{\displaystyle \frac{d}{dt}}({\delta }_{\mathrm{1}}{\psi }_r^q+{\delta }_{\mathrm{2}}{i}_{\mathrm{sc}}^q)+{\omega }_{\mathrm{sc}}({\delta }_{\mathrm{1}}{\psi }_r^d+{\delta }_{\mathrm{2}}{i}_{\mathrm{sc}}^d-{\delta }_{\mathrm{3}}{\psi }_{\mathrm{sp}}^d),\\ v_{\mathrm{sc}}^d=R_s{i}_{\mathrm{sc}}^d+{\displaystyle \frac{d}{dt}}({\delta }_{\mathrm{1}}{\psi }_r^d+{\delta }_{\mathrm{2}}{i}_{\mathrm{sc}}^d)-{\omega }_{\mathrm{sc}}({\delta }_{\mathrm{1}}{\psi }_r^q+{\delta }_{\mathrm{2}}{i}_{\mathrm{sc}}^q-{\delta }_{\mathrm{3}}{\psi }_{\mathrm{sp}}^q).\end{array}

Modeling of grid side converter

The purpose of the grid side power converter is to establish a fixed DC-link voltage regardless of the operational condition of the BDFIG. The AC source model and the rectifier model are given by the state space representation as

(17) {\displaystyle \frac{d}{dt}}\left[\begin{array}{l}{i}_{\mathrm{1}}\\ i_{\mathrm{2}}\\ i_{\mathrm{3}}\end{array}\right]=\left[\begin{array}{lll}-{\displaystyle \frac{R}{L}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& -{\displaystyle \frac{R}{L}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& -{\displaystyle \frac{R}{L}}\end{array}\right]\left[\begin{array}{l}{i}_{\mathrm{1}}\\ i_{\mathrm{2}}\\ i_{\mathrm{3}}\end{array}\right]+{\displaystyle \frac{\mathrm{1}}{L}}\left[\begin{array}{l}{V}_{\mathrm{1}}-V_{an}\\ V_{\mathrm{2}}-V_{bn}\\ V_{\mathrm{3}}-V_{cn}\end{array}\right].

The converter

(18) \left[\begin{array}{l}{V}_A\\ V_B\\ V_C\end{array}\right]={\displaystyle \frac{U_C}{\mathrm{3}}}\left[\begin{array}{lll}\mathrm{2}& -\mathrm{1}& -\mathrm{1}\\ -\mathrm{1}& \mathrm{2}& -\mathrm{1}\\ -\mathrm{1}& -\mathrm{1}& \mathrm{2}\end{array}\right]\left[\begin{array}{l}{S}_{\mathrm{1}}\\ S_{\mathrm{2}}\\ S_{\mathrm{3}}\end{array}\right].

In addition, the rectified current is given by

(19) i_s=[S_{\mathrm{1}}{S}_{\mathrm{2}}{S}_{\mathrm{3}}]\left[\begin{array}{l}{i}_{\mathrm{1}}\\ i_{\mathrm{2}}\\ i_{\mathrm{3}}\end{array}\right].

The output voltage is governed by

(20) {\displaystyle \frac{d{U}_C}{dt}}={\displaystyle \frac{\mathrm{1}}{C}}(i_s-i_L).

Modeling of stator (CW) side converter

The stator (CW) side converter is used to control the active and reactive powers injected by the stator (PW) of the BDFIG to the grid. The used converter is a simple two levels three-phase inverter (Fig. 3).

The mathematical model of the rotor side converter is given by

(21) \left[\begin{array}{l}{V}_A\\ V_B\\ V_C\end{array}\right]={\displaystyle \frac{E}{\mathrm{6}}}\left[\begin{array}{lll}\mathrm{2}& -\mathrm{1}& -\mathrm{1}\\ -\mathrm{1}& \mathrm{2}& -\mathrm{1}\\ -\mathrm{1}& -\mathrm{1}& \mathrm{2}\end{array}\right]\left[\begin{array}{l}{S}_{\mathrm{1}}\\ S_{\mathrm{2}}\\ S_{\mathrm{3}}\end{array}\right].

Vector control strategy of BDFIG

The model of the BDFG is derived in the PW synchronously rotating d-q reference frame with the d-axis aligned with the PW flux [ 5, 16]. Accordingly

{\psi }_{\mathrm{sp}}^d=\left|{\psi }_{\mathrm{sp}}\right|,

but there is no component in the q-axis

(22) \{\begin{array}{l}{\psi }_{\mathrm{sp}}^d=\left|{\psi }_{\mathrm{sp}}\right|\\ {\psi }_{\mathrm{sp}}^q=\mathrm{0.}\end{array}

To simplify calculations, the stator voltage constraint given can be considered as Eq. (23) in the dq-axis.

(23) \{\begin{array}{l}{v}_{\mathrm{sp}}^d=\mathrm{0},\\ v_{\mathrm{sp}}^q=v_{\mathrm{sp}}.\end{array}

Replacing Eqs. (22) and (23) in Eq. (20), the power expressions become

(24) \{\begin{array}{l}{P}_{\mathrm{sp}}={\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{v}_{\mathrm{sp}}^q(-{\delta }_{\mathrm{4}}{\psi }_r^q+{\delta }_{\mathrm{3}}{i}_{\mathrm{sc}}^q),\\ Q_{\mathrm{sp}}={\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{v}_{\mathrm{sp}}^q({\delta }_{\mathrm{5}}{\psi }_{\mathrm{sp}}^d-{\delta }_{\mathrm{4}}{\psi }_r^d+{\delta }_{\mathrm{3}}{i}_{\mathrm{sc}}^d).\end{array}

Then, a Lyapunov function can be defined as

(25) V_{\mathrm{1}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{(Q_{\mathrm{sp}}-Q_{\mathrm{sp}}^{\mathrm{ref}})}^{\mathrm{2}}+ {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}( P_{\mathrm{sp}}-P_{\mathrm{sp}}^{\mathrm{ref}}){ }^{\mathrm{2}}>\mathrm{0.}

The derivate of the function is

(26) {\dot{V}}_{\mathrm{1}}=(P_{\mathrm{sp}}-P_{\mathrm{sp}}^{\mathrm{ref}})\left({\dot{P}}_{\mathrm{sp}}-{\dot{P}}_{\mathrm{sp}}^{\mathrm{ref}}\right)+(Q_{\mathrm{sp}}-Q_{\mathrm{sp}}^{\mathrm{ref}})\left({\dot{Q}}_{\mathrm{sp}}-{\dot{Q}}_{\mathrm{sp}}^{\mathrm{ref}}\right).

Substituting Eq. (19) in Eq. (21), it results in

(27) \begin{array}{l}{\dot{V}}_{\mathrm{1}}=(P_{\mathrm{sp}}-P_{\mathrm{sp}}^{\mathrm{ref}})\left[{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{v}_{\mathrm{sp}}^q\left(-{\delta }_{\mathrm{4}}{\dot{\psi }}_r^q+{\delta }_{\mathrm{3}}\dot{i_{\mathrm{sc}}^q}\right)-{\dot{P}}_{\mathrm{sp}}^{\mathrm{ref}}\right]\\ +(Q_{\mathrm{sp}}-Q_{\mathrm{sp}}^{\mathrm{ref}})\left[{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{v}_{\mathrm{sp}}^q\left(-{\delta }_{\mathrm{4}}{\dot{\psi }}_r^d+{\delta }_{\mathrm{3}}\dot{i_{\mathrm{sc}}^d}\right)-{\dot{Q}}_{\mathrm{sp}}^{\mathrm{ref}}\right].\end{array}

From Eq. (16), the derivate of control current ( \stackrel{.}{i_{\mathrm{sc}}^d},\stackrel{.}{i_{\mathrm{sc}}^q} ) can be expressed as

(28) \{\begin{array}{l}{\displaystyle \frac{d}{dt}}(i_{\mathrm{sc}}^q)={\displaystyle \frac{V_{\mathrm{sc}}^q}{{\delta }_{\mathrm{2}}}}-{\displaystyle \frac{R_s}{{\delta }_{\mathrm{2}}}}{i}_{\mathrm{sc}}^q-{\displaystyle \frac{w_{\mathrm{sc}}}{{\delta }_{\mathrm{2}}}}({\delta }_{\mathrm{1}}{\psi }_r^d+{\delta }_{\mathrm{2}}{i}_{\mathrm{sc}}^d-{\delta }_{\mathrm{3}}{\psi }_{\mathrm{sp}}^d)-{\displaystyle \frac{{\delta }_{\mathrm{1}}}{{\delta }_{\mathrm{2}}}}\stackrel{.}{{\psi }_r^q,}\\ {\displaystyle \frac{d}{dt}}(i_{\mathrm{sc}}^d)={\displaystyle \frac{V_{\mathrm{sc}}^d}{{\delta }_{\mathrm{2}}}}-{\displaystyle \frac{R_s}{{\delta }_{\mathrm{2}}}}{i}_{\mathrm{sc}}^d+{\displaystyle \frac{w_{\mathrm{sc}}}{{\delta }_{\mathrm{2}}}}({\delta }_{\mathrm{1}}{\psi }_r^q+{\delta }_{\mathrm{2}}{i}_{\mathrm{sc}}^q)-{\displaystyle \frac{{\delta }_{\mathrm{1}}}{{\delta }_{\mathrm{2}}}}\stackrel{.}{{\psi }_r^d.}\end{array}

The rewrite of Eq. (28) gives

(29) \{\begin{array}{l}{\displaystyle \frac{d}{dt}}(i_{\mathrm{sc}}^q)={\displaystyle \frac{\mathrm{1}}{{\delta }_{\mathrm{2}}}}{V}_{\mathrm{sc}}^q+f_{\mathrm{1}},\\ {\displaystyle \frac{d}{dt}}(i_{\mathrm{sc}}^d)={\displaystyle \frac{\mathrm{1}}{{\delta }_{\mathrm{2}}}}{V}_{\mathrm{sc}}^d+f_{\mathrm{2}},\end{array}

where

f_{\mathrm{1}}=-{\displaystyle \frac{R_s}{{\delta }_{\mathrm{2}}}}{i}_{\mathrm{sc}}^q-{\displaystyle \frac{w_{\mathrm{sc}}}{{\delta }_{\mathrm{2}}}}({\delta }_{\mathrm{1}}{\psi }_r^d+{\delta }_{\mathrm{2}}{i}_{\mathrm{sc}}^d-{\delta }_{\mathrm{3}}{\psi }_{\mathrm{sp}}^d)-{\displaystyle \frac{{\delta }_{\mathrm{1}}}{{\delta }_{\mathrm{2}}}}\stackrel{.}{{\psi }_r^q},

f_{\mathrm{2}}=-{\displaystyle \frac{R_s}{{\delta }_{\mathrm{2}}}}{i}_{\mathrm{sc}}^d+{\displaystyle \frac{w_{\mathrm{sc}}}{{\delta }_{\mathrm{2}}}}({\delta }_{\mathrm{1}}{\psi }_r^q+{\delta }_{\mathrm{2}}{i}_{\mathrm{sc}}^q)-{\displaystyle \frac{{\delta }_{\mathrm{1}}}{{\delta }_{\mathrm{2}}}}\stackrel{.}{{\psi }_r^d}.

Substituting Eq. (29) in Eq. (27), it results in

(30) \begin{array}{l}{\dot{V}}_{\mathrm{1}}=(P_{\mathrm{sp}}-P_{\mathrm{sp}}^{\mathrm{ref}})\left([\gamma v_{\mathrm{sp}}^q{V}_{\mathrm{sc}}^q+{\alpha }_{\mathrm{1}}]-{\dot{P}}_{\mathrm{sp}}^{\mathrm{ref}}\right)\\ +(Q_{\mathrm{sp}}-Q_{\mathrm{sp}}^{\mathrm{ref}})\left([\gamma v_{\mathrm{sp}}^q{V}_{\mathrm{sc}}^d+{\alpha }_{\mathrm{2}}]-{\dot{Q}}_{\mathrm{sp}}^{\mathrm{ref}}\right),\end{array}

where

a_{\mathrm{1}}=\gamma v_{\mathrm{sp}}^q\left[-R_s{i}_{\mathrm{sc}}^q-{\omega }_{\mathrm{sc}}({\delta }_{\mathrm{1}}{\psi }_r^d+{\delta }_{\mathrm{2}}{i}_{\mathrm{sc}}^d-{\delta }_{\mathrm{3}}{\psi }_{\mathrm{sp}}^d)-{\delta }_{\mathrm{1}}\dot{{\dot{\psi }}_r^q}\right]-{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{v}_{\mathrm{sp}}^q{\delta }_{\mathrm{4}}{\dot{\psi }}_r^q,

{\alpha }_{\mathrm{2}}=\gamma v_{\mathrm{sp}}^q\left[-R_s{i}_{\mathrm{sc}}^d-{\omega }_{\mathrm{sc}}({\delta }_{\mathrm{1}}{\psi }_r^q+{\delta }_{\mathrm{2}}{i}_{\mathrm{sc}}^q)-{\delta }_{\mathrm{1}}\dot{{\dot{\psi }}_r^d}\right]-{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}{v}_{\mathrm{sp}}^q{\delta }_{\mathrm{4}}{\dot{\psi }}_r^d.

\gamma ={\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}\cdot {\displaystyle \frac{{\delta }_{\mathrm{3}}}{{\delta }_{\mathrm{2}}}}.

Then, Eq. (25) can be definitely negative, if the control law is defined as

(31) \{\begin{array}{l}{V}_{\mathrm{sc}}^q={\displaystyle \frac{\mathrm{1}}{\gamma v_{\mathrm{sp}}^q}}\left[-{\alpha }_{\mathrm{1}}+{\dot{P}}_{\mathrm{sp}}^{\mathrm{ref}}-K_{\mathrm{1}}(P_{\mathrm{sp}}-P_{\mathrm{sp}}^{\mathrm{ref}})\right],\\ V_{\mathrm{sc}}^d={\displaystyle \frac{\mathrm{1}}{\gamma v_{\mathrm{sp}}^q}}\left[-{\alpha }_{\mathrm{2}}+{\dot{Q}}_{\mathrm{sp}}^{\mathrm{ref}}-K_{\mathrm{2}}(Q_{\mathrm{sp}}-Q_{\mathrm{sp}}^{\mathrm{ref}})\right].\end{array}

Replacing Eq. (31) in Eq. (30), Eq. (32) can be obtained.

(32)

So Eq, (32) is stable if K_i ( i=\mathrm{1},\mathrm{2} ) were all positives [ 9].

Then, Eq. (30) is asymptotically stable. Hence, using the Lyapunov theorem [ 17], it can be concluded that

(33) \{\begin{array}{l}\underset{t\to +\infty }{\lim }(P_{\mathrm{sp}}-P_{\mathrm{sp}}^{\mathrm{ref}})=\mathrm{0},\\ \underset{t\to +\infty }{\lim }(Q_{\mathrm{sp}}-Q_{\mathrm{sp}}^{\mathrm{ref}})=\mathrm{0.}\end{array}

Robust nonlinear feedback control

An attempt is made to design a robust control in order to solve the large model uncertainties resulted from parameter variations, errors measurement and noises [ 18– 20]. In the kind of feedback control, the model uncertainties are more globally related to the nonlinear function, f_i (i =1, 2, 3, 4), than to the parameter drifts. As it is known, in practice, many factors such as temperature, saturation, skin effect, as well as different non linearities caused by pollution and noise, can affect the induction machine’s parameters and hence affect these nonlinear feedback functions [ 18– 20]. Globally, it can be written as

(34) \{\begin{array}{l}{f}_i=\widehat{f_i}+\mathrm{\Delta }{f}_i,\\ {\alpha }_i=\widehat{{\alpha }_i}+\mathrm{\Delta }{\alpha }_i,\end{array}

where NLFF is the nonlinear feedback function; f_i , the NLFF effective; \mathrm{\Delta }{f}_i , the NLFF variation around of \widehat{f_i} ; and {\widehat{f}}_i , the true nonlinear feedback function. \mathrm{\Delta }{f}_i can be generated from all of the parameters and variables as indicated above. It is assumed that all of the \mathrm{\Delta }{f}_i is bounded as , where {\beta }_i are known bounds [ 18– 20]. The knowledge of {\beta }_i is not difficult to obtain, since a sufficiently large number can be used to satisfy the constraint

Replacing Eq. (34) in Eq. (29), Eq. (35) can be obtained.

(35) \{\begin{array}{l}{\displaystyle \frac{d}{dt}}(i_{\mathrm{sc}}^q)={\displaystyle \frac{\mathrm{1}}{{\delta }_{\mathrm{2}}}}{V}_{\mathrm{sc}}^q+f_{\mathrm{1}}+\mathrm{\Delta }{f}_{\mathrm{1}},\\ {\displaystyle \frac{d}{dt}}(i_{\mathrm{sc}}^d)={\displaystyle \frac{\mathrm{1}}{{\delta }_{\mathrm{2}}}}{V}_{\mathrm{sc}}^d+f_{\mathrm{2}}+\mathrm{\Delta }{f}_{\mathrm{2}}.\end{array}

Taking into account of \mathrm{\Delta }{f}_i , the new law control can be chosen as

(36) \{\begin{array}{l}{V}_{\mathrm{sc}}^q={\displaystyle \frac{\mathrm{1}}{\gamma v_{\mathrm{sp}}^q}}\left[-{\alpha }_{\mathrm{1}}+{\dot{P}}_{\mathrm{sp}}^{\mathrm{ref}}-K_{\mathrm{1}}(P_{\mathrm{sp}}-P_{\mathrm{sp}}^{\mathrm{ref}})-K_{\mathrm{11}}\mathrm{sgn}(P_{\mathrm{sp}}-P_{\mathrm{sp}}^{\mathrm{ref}})\right],\\ V_{\mathrm{sc}}^d={\displaystyle \frac{\mathrm{1}}{\gamma v_{\mathrm{sp}}^q}}\left[-{\alpha }_{\mathrm{2}}+{\dot{Q}}_{\mathrm{sp}}^{\mathrm{ref}}-K_{\mathrm{2}}(Q_{\mathrm{sp}}-Q_{\mathrm{sp}}^{\mathrm{ref}})-K_{\mathrm{22}}\mathrm{sgn}(Q_{\mathrm{sp}}-Q_{\mathrm{sp}}^{\mathrm{ref}})\right].\end{array}

where K_{ii}\ge {\beta }_i , K_i>\mathrm{0} and i=1, 2.

Then the analogue derivate Lyapunov function, established from Eq. (30) using Eqs. (35) and (36), becomes

(37)

Hence, the variations can be absorbed when system stability is increased if chosen

(38) \{\begin{array}{l}{K}_{\mathrm{11}}=\left|\Delta {\alpha }_{\mathrm{1}}\right|,\\ K_{\mathrm{22}}=\left|\mathrm{\Delta }{\alpha }_{\mathrm{2}}\right|.\end{array}

Finely, it can be written [ 10] as

(39)

It can be concluded that the control law given by Eq. (36) to end at the convergent processes stability for any α_i. Figure 4 illustrates a general block diagram of the suggested BDFIG control scheme.

Simulation results

The machine data are given in the Table 1. In order to validate the approach proposed in this paper, digital simulation has been conducted using the Matlab/Simulink software. Hence, the results obtained are organized respectively according to the following specifications where the speed is fixed at 62.83 rad/s. Figure 5 illustrates the response of the stator active and reactive powers. The active power is varied between –1200 W, –1800 W and –1600 W respectively at 0.5 s and 1.5 s while the reactive power is fixed at 0 var, from where the good tracking of the proposed control can be observed clearly. So in this case, the stator power factor versus time which is easily maintained to unity is demonstrated in Fig. 6.

It can be noted that the stator voltage and current are in opposite phases (Fig. 7), which shows that the machine is operating in a generation mode ( \cos \phi =-\mathrm{1} ).

After seeing the good tracking of the proposed set-points illustrated in Fig. 8, the robustness of the structure of control against the incertitude of parameters should be checked. So the following step is taken to check the robustness against the resistance stator (PW and CW) changing.

It is well-known that the problem in electrical machines is the changing of their electrical parameters due to the changing of the temperature, which may cause some problems. Therefore, and in order to know the control structure behaviour under these critical conditions, an attempt is made to check it by test of robustness, when the stator (PW and CW) resistances are changed as shown in Fig. 9..

Figure 10 displays both a set-point tracking response of the stator active and reactive powers and a robustness test against a 100% PW and CW stator resistance increase at 0.5 s and 1 s, respectively. Figure 10(a) depicts the results of a traditional vector control (PI controller) and Fig.10(b) exhibits those of the proposed Lyapunov control. However, remarkable improvements of dynamic and static performances are achieved by the proposed Lyapunov control approach in comparison with those obtained by the traditional vector control. Furthermore, the proposed approach leads to a perfect decoupling between the two components of the generated powers of the stator. It is clearly observed in Fig. 10(b) that there a dramatic oscillation canceling and almost a zero error set-point tracking.

Conclusions

In this paper, a robust vector control intended for the BDFIG was investigated. A robust control was designed and the stability of the control is proven using the Lyapunov theory. To check the reliability of the control structure, a test of robustness was conducted in two ways. First, an abrupt change was made to the stator active power set-points aiming to see how the controllers can enforce the responses to follow their set-points. Then a test of robustness against parameters variation was made, in which the stator resistances were chosen in the test due to their sensitivity toward the temperature. Besides, the results obtained exhibit a good set-point tracking without any recorded effect, and a good robustness against the parameters variation. As a conclusion, the proposed BDFIG system control can be considered as an interested solution in wind energy conversion system area.

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