Robust nonlinear control of wind turbine driven doubly fed induction generators

*Corresponding author: Hamzamesai2@gmail.com © The Author(s) 2020. Published by ARDA. Abstract This paper presents the active and reactive powers control of a doubly fed induction generator (DFIG) connected to the grid utility and driven by a wind turbine, this machine allowing a large speed variation and so a large range of wind is achieved. Traditionally vector control is introduced to the DFIG control strategies, which decouples DFIG active and reactive powers, and reaches good performances in the wind energy conversion systems (WECS). However, this decoupling is lost if the parameters of the DFIG change. In this direction, a robust control scheme based on the nonlinear input-output linearizing and decoupling control strategy for the rotor side converter (RSC) of the WECS is presented. Simulation results show that the proposed control strategy provides a robust decoupled control and perfect tracking of the generated active and reactive powers of the wind turbine driven DFIG with a low THD rate of the generated currents.


Introduction
Due to the economic impact of wind power, efficient control techniques are becoming a focus of the world's renewable energy challenges, making it a promising field of study. With better controllers and the constant development of new technologies, wind power will maintain its rapidly growing market ratio. In recent years, the concept of the variable speed wind turbine (VSWT) equipped with a doubly fed induction generator (DFIG) has received increasing attention due to its noticeable advantages over other wind turbine concepts [1]. In the DFIG concept, the stator is usually connected directly to the three-phase grid; the rotor is also connected to the grid but via a transformer and two back-to-back converters ( Figure 1).

RSC GSC
This arrangement provides a flexibility of operation in sub and super-synchronous speeds in both generating and motoring modes (±30% around the synchronous speed). The power inverter needs to handle a fraction (20-30%) of the total power to achieve full control of the generator. Recently, several nonlinear controls laws based a DFIG are presented in literatures such as sliding mode control (SMC) technique [2], [3]. However, the major problem of this strategy of control is the chattering phenomenon caused by the control switching, which may have undesirable effects on the machine, such as overheating of the windings, torque pulsation, current harmonics, acoustic noise, etc. [4]. This paper discusses the synthesis of a multivariable nonlinear controller by input-output linearizing technique for the control of a doubly fed induction generator dedicated to a wind turbine system. The objective is to the modeling and control of a wind conversion system based on DFIG, associated with a control of active and reactive power based on stator flux orientation. This technique is applied to the rotor side converter (RSC) in both sub and super-synchronous mode operation of DFIG. Finally, the simulation results will be analyzed in terms of robustness against parameters variations of the DFIG and the quality of energy supplied to the grid.

Mathematical model of DFIG
The equivalent circuit of the DFIG in the synchronous reference frame rotating at the angular synchronous speed ωs is shown in Figure 2.
The rotor-side converter is controlled in a synchronously rotating d-q axis frame, with the d-axis oriented along the stator flux vector position ( Figure 3).
The influence of the stator resistance can be neglected and the stator flux can be held constant as the stator is connected to the grid. Consequently: Since the stator is directly connected to the grid and the stator flux can be considered constant, and if the voltage dropped in the stator resistance has been neglected [5], the stator voltage equations, flux equations, currents equations and stator active and reactive powers equations can be respectively simplified in study state as: Replacing the stator currents by their expressions given in (11), the equations below are expressed by:

Lie derivative
Given the system [4]: Where "x" is the state vector; "u"is the input; "y" is the output; "f"and "g" are smooth vector fields; "h" is smooth scalar function. The Lie derivative of the function h(x) along a vector field f T (x) = (f1 (x), f2 (x), ..., fn (x)) is given by [7]: Where Lf h(x) is a derivative of Lie its means: the derivative of h(x) in the direction of f. The purpose of the linearization application is to find a nonlinear feedback control as the following form: This brings the nonlinear system in closed-loop back to a linear system. This makes it possible to obtain a linear behavior of the system over the entire operating range, unlike the vector control that linearizes the system around an operating point.

Relative degree
The relative degree "r" of the output "y", is the number of times it is necessary to derive "y" to make the input "u" explicitly appear in the derivatives of "y". According to this definition we can say that: The system in (23) is said to have a relative degree "r" if: Indeed, if we apply this principle to the system in (15), we obtain: If Lf h(x)=0, we continue the derivation of the output "y" until the appearance of "u" for the first time, for a relative degree equal to "r", we obtain: The input-output relation of the system in (15) is therefore the following [8]:

Multivariable input-output linearization
Considering now a system with "p" inputs and "p" outputs given by: : is the state vector where n is the order of the system; : is the vector of control input signals; : is the vector of the output signals; If the system to be linearized admits the vector of relative degrees , then the input-output differential equations of the system are given by: With at least one of the We can write the input-output equations of the system in matrix form as follows: (  : is called decoupling matrix, it must be non-singular (D -1 exists).
To linearize the system, we define the linearizing control law: : is the new control input vector of the system to be linearized.

Input-output linearizing control of the DFIG
The active and reactive powers of the DFIG are controlled by the rotor current that is controlled through the output voltage of the rotor side converter. Therefore, the applied voltage of the rotor is the direct control variable.
According to the (11) the direct and quadrature components of stator and rotor currents are linear dependent respectively, thus we choose the state vector of the DFIG as follows: We replace (39) in (34), and then we find: We can rewrite (41) in the form: Since the decoupling matrix D(x) is nonsingular, the control law is given as: The Proportional-Integral (PI) controller achieves the tracking of the stator powers. Hence, the new input "v" is given by [6,9]: Where e1 is the error between the desired and the measured active power, and e2 is that relates to the reactive power:

Simulation results and discussions
In this section, the input-output linearizing control of 1.5MW DFIG (see appendix) is tested by simulation under Matlab/ Simulink software. Two types of tests were applied to the wind energy conversion system to observe the behavior of its regulation: 1-References tracking test at variable wind speed. 2-Robustness test against the parameters variations of the DFIG at fixed wind speed.

References tracking test
In this test, the wind turbine is driven by a variable wind speed with an average value of 8.2 m/sec ( Figure 5.a). The reference active power is generated by the MPPT strategy and the reactive power reference is kept at zero in order to guarantee a unit power factor on the stator side of the DFIG (see Figure 5).  Figure 5.c), in order to achieve a unit power factor (FP = 1) on the stator side of the DFIG (Figure 5.c). The spectral analysis of the stator current shows that the input-output linearizing control of DFIG guarantees a better quality of the stator current waveform injected to the grid, where the total harmonic distortion (THD) rate is only 1.93%.

Robustness test
In order to test the robustness of the proposed control strategy of the DFIG, we also studied the influence of parameters variations of the generator on the performances of this last one.
To realize this test we increase the rotor resistance Rrby 100% of its nominal value (case of warming-up of rotor windings) and decrease the mutual inductance Lmby 50% of its nominal value (case of inductances saturation). The Figure 6 shows the active and reactive powers responses of DFIG for the input-output linearization control, whose the wind turbine is driven at fixed wind speed of 12 m/sec. Simulation results in Figure 6 show the robustness of the proposed control strategy against parameters variations of the DFIG, contrary to the vector control strategy (FOC) based PI (Proportional-Integral) controllers [10].
(a) active power response (b) reactive power response Figure 6. Simulation results of the robustness test against parameters variations of DFIG for the input-output feedback linearization control

Conclusion
In this paper it has been presented a robust nonlinear control strategy based on input-output linearization control, allowing an independent control of the active and reactive stator powers of the DFIG, driven by a variable speed wind turbine. The performances obtained by this control strategy are very satisfactory even in the presence of parameters variations as shown by the trajectory tracking and the fast convergence of the measured variables towards their desired references. Moreover, this control strategy has a very low harmonic distortion rate compared to other nonlinear control strategies such as sliding mode control technique. Finally, the proposed