2. Aerodynamic model of wind turbine

installed for a total of approximately 487 GW by the end of 2016. Conservative prognoses forecast an increment of 60 GW in 2017 with a continuous annual grow of about 75 GW by 2021. With the continued improvement in wind turbines technology and ecological concerns, the wind power is now a serious competitor against heavily subsidized energy industries [1, 2]. Doubly-fed induction generator (DFIG) is a wound-rotor induction machine with voltage injection in the rotor winding. This allows a limited speed control of the electric machine, which is sufficient to implement maximum aerodynamic efficiency by maintaining the tip-

The success of DFIG in wind energy applications lies in the reduced power converter, which is typically about 25–30% of the nominal power of the electric machine; furthermore, by current injection in the rotor, it is possible to control the reactive power injected to the electric grid, which is very important for minimizing cooper loses or reactive power compensation [3]. More than 70% of the installed wind turbines use DFIG. However, this machine is very sensitive to voltage variations in the grid because the stator is directly connected to the grid [4], in contrast to other variable-speed machines that are connected through a full-size power converter.

Sliding mode control (SMC) is a nonlinear control technique that ensures finite-time convergence of the sliding surface to zero guarantying robustness against bounded disturbances and parameter variations [9]. The main disadvantage of SMC is the chattering effect (high frequency oscillations with finite amplitude) caused by unmodeled dynamics and discretization [9]. On the other hand, power electronics are controlled by means of the injection of discontinuous signals matching with the discontinuous nature of conventional SMC; therefore, conventional SMC can be used for direct switching of power electronics on DFIG applications, avoiding modulation. SMC has been successfully implemented in DFIG control and tested under unbalanced conditions and harmonics [11, 15–18]. However, the proposals given in [16, 17] require modulation, and the tested faults are moderate since these do not represent a brusque variation in the stator voltage. The SMC presented in [15–18] works under unbalanced conditions but implementation

In this chapter, it is presented a SMC with the following advantages: (a) do not required modulation; (b) do not require modifications of the controller structure to withstand stator voltage perturbances (compared with classical control approach as direct-torque-control [DTC], see [5]); and (c) the controller can regulate torque and reactive power even under unbalanced conditions, which is equivalent to negative current regulation. Therefore, the proposed SMC offers a very simple alternative that requires neither symmetrical decomposition nor pulse width modulation and is not affected by parameter variation. Furthermore, the DFIG system with SMC is characterized in the frequency domain for estimating the commutation frequency of power electronics. A maximum switching frequency value is ensured by means of the addition of hysteresis in the sign function. The hysteresis value is computed applying the Tsypkin's method, a theoretically exact technique to analyze nonlinear systems [12, 14]. Due to the nature of the power converter, the voltage gain seen by the controller is variable in time, which makes difficult to maintain the switching frequency constant. However, it is possible to compute a maximum switching frequency value (minimum hysteresis value), which provides a commutation fre-

speed ratio at the nominal value at most operational conditions of the turbine.

56 Adaptive Robust Control Systems

of the SMC regarding the commutation of the power electronics is not discussed.

quency inside of the acceptable values given in the datasheet of the power electronics.

Horizontal axis wind turbines are used to extract mechanical power from the wind resource based on the lifting force of the rotor blades. The mechanical power is a function of the kinematic energy of the wind and the power coefficient:

$$P\_{wind} = \frac{1}{2} \mathcal{C}\_P(\Lambda, \mathcal{Y}) \rho A V^3 \tag{1}$$

The power coefficient is a function of the tip speed ratio and the pitch angle. Since very complex aerodynamic analyses are required to characterize turbine blades, the power coefficient is usually approximated using mathematical expressions such as [7]:

$$\mathbf{C}\mathbf{p} = \mathbf{c}\_1(\mathbf{c}\_2\mathbf{x} - \mathbf{c}\_3\mathbf{y} - \mathbf{c}\_4\mathbf{y}^{\varepsilon\_5} - \mathbf{c}\_6)\mathbf{e}^{-\mathcal{D}\mathbf{x}} \tag{2}$$

where <sup>κ</sup> <sup>¼</sup> <sup>1</sup> <sup>Λ</sup>þ0:<sup>06</sup> <sup>γ</sup> � <sup>0</sup>:<sup>035</sup> <sup>1</sup>þγ<sup>3</sup> and <sup>c</sup><sup>1</sup> to <sup>c</sup><sup>7</sup> are coefficients dependent on the blades geometry.

The tip speed ratio is the relationship between the tangential speed of the blades tip and the wind speed if it is expressed using the DFIG mechanical speed and the gearbox ratio we obtain:

Figure 1. Power coefficient as a function of tip speed ratio and pitch angle.

$$
\Lambda = \frac{\omega\_m r}{\eta V} \tag{3}
$$

Pitch angle is normally used for aerodynamically reduce power extraction when the wind speed is above the nominal value. For normal operation, it is maintained constant, while the rotor speed is controlled by the DFIG to maintain the tip speed ratio constant, for the blades model shown in Figure 1, the nominal pitch angle is zero and the nominal tip speed ratio is 8 for a maximum power coefficient of approximately 0.41. The parameters used to generate the displayed function are c<sup>1</sup> = 0.5 ; c<sup>2</sup> = 116 ; c<sup>3</sup> = 0.4 ; c<sup>4</sup> =0; c<sup>6</sup> =5; c<sup>7</sup> = 21:
