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

Abstract

This paper discusses the robust control of a grid-connected doubly-fed induction generator (DFIG) controlled by vector control using a nonlinear feedback linearization strategy in order to ameliorate the performances of the control and to govern the developed stator active and reactive power in a linear and decoupled manner, in which an optimal operation of the DFIG in sub-synchronous operation is given, as well as the control stator power flow with the possibility of keeping stator power factor at a unity. The use of the state-all-flux induction machine model gives place to a simpler control model. So, to achieve this objective, the Lyapunov approach is used associated with a sliding mode control to guarantee the global asymptotical stability and the robustness of the parametric variations.

Keywords

doubly fed induction generator (DFIG) / vector control / Lyapunov function / power factor unity / active power / reactive power

Introduction

The doubly fed induction machine (DFIM) is a very attractive solution for variable-speed applications such as electric vehicles and electrical energy production, where it is considered as one of the most important machines in the generation of electrical energy, especially in renewable energy. At the same time, it allows vehicles to work at a variable speed between \pm \mathrm{30}\%around the synchronous speed [1-7]. However, to exploit these advantages in the production of energy, strategy of control should be achieved taking into a count its complex structure and the quality of energy to be generated. Because of the lack or scarcity of control on the produced active and reactive powers, many problems may arise, when the generator connected to the grid or feeding an isolated loads by stand alone operation, such as the lowing power factor and harmonic pollutions. Several designs and arrangements have been implemented to cope with this difficulty [3]. However, an alternative approach consists in using a wound-rotor induction generator fed with variable frequency rotor voltage. This allows fixed-frequency electric power to be extracted from the generator stator. So in this kind of machines, if the rotor current is governed by applying stator-flux-oriented vector control using commercial machine side rotor PWM converter (Fig. 1), certainly a decoupled control of the stator side active and reactive power is resulted [1,2]. The ability to generate electricity with unity power factor would reduce the costs of introducing additional capacitors for reactive power regulation. To achieve this, the transfer of energy between different parts of the global structure (turbine, stator, rotor and grid) should be supervised [1,4,5].

A lot of research has been conducted over the past few years using the Lyapunov theory to controlled and optimized the control structures such the maximization of the power capture given time varying wind conditions [8]. In the present paper, a robust control based on the Lyapunov theory is presented in order to regulate the stator active and reactive power flow.

Model of the DFIM

The model of the DFIM is expressed, in the synchronous reference frame, by the following equations:

Voltage equations:

(1)\{\begin{array}{l}{\overline{u}}_s=R_s.{\overline{i}}_s+{\displaystyle \frac{d{\overline{\mathrm{\Psi }}}_s}{dt}}+j{\omega }_s{\overline{\psi }}_s,\hfill \\ {\overline{u}}_r=R_r{\overline{i}}_r+{\displaystyle \frac{d{\overline{\mathrm{\Psi }}}_r}{dt}}+j{\omega }_r{\overline{\psi }}_r.\hfill \end{array}

Current-flux equations:

(2)\{\begin{array}{l}{\overline{i}}_s=\gamma {\overline{\psi }}_s+\lambda {\overline{\psi }}_r,\hfill \\ {\overline{i}}_r=\chi {\overline{\psi }}_r+\lambda {\overline{\psi }}_s,\hfill \end{array}

with

\gamma ={\displaystyle \frac{\mathrm{1}}{\sigma L_s}},\lambda =-{\displaystyle \frac{\mathrm{1}}{\sigma L_s{L}_r}},\chi ={\displaystyle \frac{\mathrm{1}}{\sigma L_s}}.

From Eqs. (1) and (2), the all flux state model is done as

(3)\{\begin{array}{l}{\overline{u}}_s={\displaystyle \frac{1}{\sigma T_s}}{\overline{\phi }}_s-{\displaystyle \frac{M}{\sigma T_s{L}_r}}{\overline{\phi }}_r+{\displaystyle \frac{d}{dt}}{\overline{\phi }}_s+j{\omega }_s{\overline{\phi }}_s,\hfill \\ {\overline{u}}_r=-{\displaystyle \frac{M}{\sigma T_r{L}_s}}{\overline{\phi }}_s+{\displaystyle \frac{1}{\sigma T_r}}{\overline{\phi }}_r+{\displaystyle \frac{d}{dt}}{\overline{\phi }}_r+j{\omega }_r{\overline{\phi }}_r.\hfill \end{array}

The stator power expressions are

(4)\{\begin{array}{l}{P}_s=\mathrm{Re}\left\{{\overline{u}}_s\cdot {\overline{i}}_s^{*}\right\},\hfill \\ Q_s=\mathrm{Im}\left\{{\overline{u}}_s\cdot {\overline{i}}_s^{*}\right\}.\hfill \end{array}

Replacing Eq. (2) in Eq. (4) with equalization of the real and imaginary parts, Eq. (5) can be obtained.

(5)\{\begin{array}{l}{P}_s={\gamma }_{u_{sd}{\phi }_{sd}}+{\lambda }_{u_{sd}{\phi }_{rd}}+{\gamma }_{u_{sq}{\phi }_{sq}}+{\lambda }_{u_{sq}{\phi }_{rq}},\hfill \\ Q_s={\gamma }_{u_{sd}{\phi }_{sd}}+{\lambda }_{u_{sd}{\phi }_{rd}}-{\gamma }_{u_{sq}{\phi }_{sq}}-{\lambda }_{u_{sq}{\phi }_{rq}}.\hfill \end{array}

The equalization of the real parts and imaginary parts of Eq. (3) give the following equations [6]:

(6)\{\begin{array}{l}{u}_{sd}={\gamma }_1{\phi }_{sd}-{\gamma }_2{\phi }_{rd}+{\displaystyle \frac{d}{dt}}{\phi }_{sd}-{\omega }_s{\phi }_{sq}=-f_1+{\displaystyle \frac{d}{dt}}{\phi }_{sd},\hfill \\ u_{sq}={\gamma }_1{\phi }_{sq}-{\gamma }_2{\phi }_{rq}+{\displaystyle \frac{d}{dt}}{\phi }_{sq}+{\omega }_s{\phi }_{sd}=-f_2+{\displaystyle \frac{d}{dt}}{\phi }_{sq},\hfill \\ u_{rd}=-{\gamma }_3{\phi }_{sd}+{\gamma }_4{\phi }_{rd}+{\displaystyle \frac{d}{dt}}{\phi }_{rd}-{\omega }_r{\phi }_{rq}=-f_3+{\displaystyle \frac{d}{dt}}{\phi }_{rd},\hfill \\ u_{rq}=-{\gamma }_3{\phi }_{sq}+{\gamma }_4{\phi }_{rq}+{\displaystyle \frac{d}{dt}}{\phi }_{rq}+{\omega }_r{\phi }_{rd}=-f_4+{\displaystyle \frac{d}{dt}}{\phi }_{rq},\hfill \end{array}

with

{\gamma }_{\mathrm{1}}={\displaystyle \frac{\mathrm{1}}{\sigma T_s}},{\gamma }_{\mathrm{2}}={\displaystyle \frac{M}{\sigma T_s{L}_r}},{\gamma }_{\mathrm{3}}={\displaystyle \frac{\mathrm{1}}{\sigma T_r{L}_s}},{\gamma }_{\mathrm{4}}={\displaystyle \frac{\mathrm{1}}{\sigma T_r}}

and

(7)\{\begin{array}{l}-f_1={\gamma }_1{\phi }_{sd}-{\gamma }_2{\phi }_{rd}-{\omega }_s{\phi }_{sq},\hfill \\ -f_2={\gamma }_1{\phi }_{sq}-{\gamma }_2{\phi }_{rq}+{\omega }_s{\phi }_{sd},\hfill \\ -f_3=-{\gamma }_3{\phi }_{sd}+{\gamma }_4{\phi }_{rd}-{\omega }_r{\phi }_{rq},\hfill \\ -f_4=-{\gamma }_3{\phi }_{sq}+{\gamma }_4{\phi }_{rq}+{\omega }_r{\phi }_{rd}.\hfill \end{array}

The rewrite of (6) gives:

(8)\{\begin{array}{l}{\displaystyle \frac{d}{dt}}{\phi }_{sd}=f_1+u_{sd},\hfill \\ {\displaystyle \frac{d}{dt}}{\phi }_{sq}=f_2+u_{sq},\hfill \\ {\displaystyle \frac{d}{dt}}{\phi }_{rd}=f_3+u_{rd},\hfill \\ {\displaystyle \frac{d}{dt}}{\phi }_{rq}=f_4+u_{rq}.\hfill \end{array}

Vector control strategy of the DFIG

To simplify calculations, let us consider the stator voltage constraint given as follows in dq-axis [3]:

(9)\{\begin{array}{l}{u}_{sd}=0,\hfill \\ u_{sq}=u_s.\hfill \end{array}

Replacing Eq. (9) in Eq. (5), the power expressions become

(10)\{\begin{array}{l}{P}_s=u_s\left(\lambda {\phi }_{rq}+\gamma {\phi }_{sq}\right),\hfill \\ Q_s=u_s\left(\lambda {\phi }_{rd}+\gamma {\phi }_{sd}\right).\hfill \end{array}

Then a Lyapunov function can be defined as

(11)V_{\mathrm{1}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{(P_s-P_{sref})}^{\mathrm{2}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{(Q_s-Q_{sref})}^{\mathrm{2}}>\mathrm{0.}

The derivate of the function is

(12)\stackrel{.}{V_{\mathrm{1}}}=(P_s-P_{sref})\left({\stackrel{.}{P}}_s-{\stackrel{.}{P}}_{sref}\right)+(Q_s-Q_{sref})\left({\stackrel{.}{Q}}_s-{\stackrel{.}{Q}}_{sref}\right).

Substituting Eqs. (8) and (10) in Eq. (12), it results in

(13)\stackrel{.}{V_{\mathrm{1}}}=(P_s-P_{sref})({\alpha }_{\mathrm{1}}+\lambda u_s{u}_{rq}-P_{sref})+\stackrel{.}{(Q_s-Q_{sref})\left({\alpha }_{\mathrm{2}}+\lambda u_s{u}_{rd}-{\stackrel{.}{Q}}_{sref}\right)},

with

{\alpha }_{\mathrm{1}}=\lambda u_s{f}_{\mathrm{4}}+\gamma (f_{\mathrm{2}}+u_s),{\alpha }_{\mathrm{2}}=\lambda u_s{f}_{\mathrm{3}}+\gamma f_{\mathrm{1}}.

Then, Eq. (13) can be definitely negative, if the following control law is defined:

(14)\{\begin{array}{l}\stackrel{.}{u_{rd}={\displaystyle \frac{1}{\lambda u_s}}\left[-{\alpha }_2+{\stackrel{.}{Q}}_{sref}-K_2\left(Q_s-Q_{sref}\right)\right]},\hfill \\ u_{rq}={\displaystyle \frac{1}{\lambda u_s}}\left[-{\alpha }_1+{\stackrel{.}{P}}_{sref}-K_1\left(P_s-P_{sref}\right)\right].\hfill \end{array}

Replacing Eq. (14) in Eq. (13), Eq. (15) can be obtained.

(15)

So Eq. (15) is stable if K_i (i = 1,2) were, of course, all positives [6], in other words,

(16)\{\begin{array}{l}\underset{t\to +\infty }{\lim }{Q}_s-Q_{sref}=0,\hfill \\ \underset{t\to +\infty }{\lim }{P}_s-P_{sref}=0.\hfill \end{array}

Robust nonlinear feedback control

An attempt is made to design a robust control in order to solve the large model uncertainties due to parameter variations, errors measurement and noises. 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. In practice, these nonlinear feedback functions can be strongly affected by the conventional effect of induction machine (IM) such as temperature, saturation and skin associated to the different no linearly caused by harmonic pollution and the noise measurements. Globally, it can be written:

(17)\{\begin{array}{l}{f}_i=\widehat{f_i}+\mathrm{\Delta }{f}_i,\hfill \\ {\alpha }_i=\widehat{{\alpha }_i}+\mathrm{\Delta }{\alpha }_i,\hfill \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}; {\widehat{f}}_i, the true nonlinear feedback function.

The \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 are bounded as , where {\beta }_i are known bounds. The knowledge of {\beta }_i is not difficult to obtain, since a sufficiently large number can be used to satisfy the constraint.

Replacing Eq. (17) in Eq. (8), Eq. (18) can be obtained.

(18)\{\begin{array}{l}{\displaystyle \frac{d}{dt}}{\phi }_{sd}={\hat{f}}_1+\mathrm{\Delta }{f}_1+u_{sd},\hfill \\ {\displaystyle \frac{d}{dt}}{\phi }_{sq}={\hat{f}}_2+\mathrm{\Delta }{f}_2+u_{sq},\hfill \\ {\displaystyle \frac{d}{dt}}{\phi }_{rd}={\hat{f}}_3+\mathrm{\Delta }{f}_3+u_{rd},\hfill \\ {\displaystyle \frac{d}{dt}}{\phi }_{rq}={\hat{f}}_4+\mathrm{\Delta }{f}_3+u_{rq}.\hfill \end{array}

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

(19)\{\begin{array}{l}{u}_{rd}={\displaystyle \frac{1}{\lambda u_s}}\left[-{\alpha }_2+{\stackrel{.}{Q}}_{sref}-K_2\left(Q_s-Q_{sref}\right)-K_{22}sign\left(Q_s-Q_{sref}\right)\right],\hfill \\ u_{rq}={\displaystyle \frac{1}{\lambda u_s}}\left[-{\alpha }_1+{\stackrel{.}{P}}_{sref}-K_1\left(P_s-P_{sref}\right)-K_{11}sign\left(P_s-P_{sref}\right)\right].\hfill \end{array}

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

Than the analog derivate Lyapunov function, established from Eq. (13) using Eqs. (18) and (19) becomes

(20)

Hence the {\widehat{f}}_i variations can be absorbed when system stability is increased if chosen

(21)$$\begin{gathered} K_{\mathrm{11}}=\left|\mathrm{\Delta }{\alpha }_{\mathrm{1}}\right|, \\ {K}_{\mathrm{22}}=\left|\mathrm{\Delta }{\alpha }_{\mathrm{2}}\right|. \end{gathered}$$

Finely, it can be written

(22)

It can be concluded that the control law given by Eq. (19) to end at the convergent processes stability for any {\widehat{\alpha }}_i.

Figure 2 illustrates a general block diagram of the suggested DFIG control scheme.

Simulation results

The machine data are given in the Appendix. To validate the approach proposed in this paper, digital simulation has been realized using the Matlab/Simulink software. Hence, the obtained results are organized respectively according to the following specifications where the speed is fixed at 151.83 rad/s. Figure 3 illustrates the response of the stator active and reactive powers. The active power is varied between -2000 W, -7500 W and -5500 W respectively at 1 s, 2 s and 4 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 (Fig. 4).

It can be noted that the stator voltage and current are in opposite phases (Fig. 5), which shows that the machine is operating in a generation mode (\cos \phi =-\mathrm{1}). Figure 6 presents the variations of the nonlinear functions versus time, where it can be observed that their magnitudes are varied with the profile variation of stator active power in order to make the system more stable and robust.

After seeing the good tracking of the proposed set-points, the robustness of the structure should be checked against the incertitude of parameters. So the following step is taken to check the robustness against the resistance stator and rotor changing.

It is well-known that the problem in the electrical machines is the changing of their electrical parameters due to the changing of the temperature; these changes may cause some problems. Therefore and in order to know the control structure behavior under these critical conditions, an attempt is made to check it by test of robustness, when the stator and rotor resistances are changed as follows:

After showing the simulation results, it can be noted that these results show the robustness (Figs. 7 and 8) of the structure of control against an 100% changing of the rotor and stator resistances; although a small disturbance can be observed, it does not have a notable effect on the stator active and reactive responses (figures zoom respectively).

Conclusions

The DFIM double accessibility is an important advantage. This induces good control of the power flow between machine and grid permitting to inject the power such that the grid power factor is closed to unity. In this paper, a robust vector control intended for the DFIG has been investigated. The stability of the robust control has been proven using the Lyapunov theory with a sliding mode controller. The robustness test is realized in two ways in order to check the structure of the control used, where instantaneous set-point variations of the active and reactive powers, and a test by changing of the rotor and stator resistances have been taken. So the obtained results have demonstrated the efficiency of the adopted structure of control when the good set-point tracking without any recorded effect, and the good robustness against the parameters changing have been observed. It can be concluded that the proposed DFIG system control is an interesting solution in the wind energy area.

References

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