Indirect power control of DFIG based on wind turbine operating in MPPT using backstepping approach

Received Jul 15, 2020 Revised Dec 12, 2020 Accepted Dec 22, 2020 This paper describes a MPPT control of the stator powers of a DFIG operating within a wind energy system using the backstepping control technique. The objective of this work consists of providing a robust control to the rotor-side converter allowing the stator active power to be regulated at the maximum power extracted from the wind turbine, as well as maintaining the stator reactive power at zero to maintain the power factor at unity, under various conditions. We have used the Matlab/Simulink platform to model the wind system based on a 7.5 kW DFIG and to implement the MPPT control algorithm in a first step, then we have implemented the field-oriented control and the backstepping controller in a second step. The simulation results obtained were very satisfactory with a fast transient response and neglected power ripples. They furthermore confirmed the high robustness of the approach used in dealing with the variation of the internal parameters of the machine.


INTRODUCTION
The investigated wind power system consists of a wind turbine driven by the DFIG shaft via a gearbox as shown in Figure 1. The generator's stator is wired straight into the electrical grid while its rotor dissipates or absorbs its power into the electrical grid through a back-to-back converter, which is composed of an AC/DC converter on the rotor side, a DC link capacitor on the DC bus side, and a DC/AC converter on the grid side. This configuration allows the rotor speed to be adjustable according to the wind speed [1][2][3].
Variable speed allows the wind turbine to operate over a wider range of wind speeds and to extract the maximum possible power for each wind speed. Compared to some synchronous machines, which use converters traversed by the entire rated power, DFIG's converters are sized to allow only 25% of the rated power to pass through [4]. They are therefore less cumbersome and less expensive [5,6].
The research work on the control of the DFIG is almost all based on the field oriented control. This technique which makes the DFIG similar to the DC machine through the separation of the rotor and stator fluxes provides the basis for the implementation of the control methods [7]. In the recent years, the shift towards non-linear control methods by researchers has reached an interesting place in the control scope of converters in systems with sustainable energy, precisely in photovoltaics and wind turbines. Their robustness against disturbances and their ability to withstand non-linear systems renders these control methods an  [8,9].
Although various control methods have recently been proposed in the literature, the linear PI with vector control is still the most famous one, but its drawbacks are numerous, namely its lack of robustness in rejecting internal and external disturbances. Nowadays, researchers have switched to intelligent control strategies such as fuzzy logic, neural networks, and genetic algorithms. While these methods are very robust and efficient, their implementation is not obvious, it requires a high level of expertise on the one hand and they require very fast and powerful calculation processors on the other hand. Lyapunov's stability based nonlinear control methods such as SMC and backstepping remain a good compromise that is easily implementable, robust and offers high satisfaction. These methods have already been proposed by researchers as an application in the control of DFIG-based wind energy systems. In [10] Zamzoum et al. have proposed the technical adaptive sliding mode as a control solution, its results showed that the use of high order sliding mode control offers robustness against variations of internal machine parameters with fast transient response. The backstepping control of the DFIG which was treated by Mensou et al. in [11] also has showed a great robustness against external disturbances as well as offering a very fast response time.
In this paper a backstepping control has been proposed in order to control the rotor-side converter to ensure a smooth regulation of the DFIG stator active power to follow the variation of the maximum power extracted by the turbine. And with the aim of validating the robustness of this control technique we have tested its behaviour under the variation of the internal parameters of the DFIG. This paper is organized as follows: Section 2 deals with wind turbine modeling and the MPPT algorithm. The modeling of the DFIG with field oriented control is presented in section 3. Section 4 describes the design of the backstepping controller. Section 5 is reserved for the presentation and discussion of the simulation results, and the last section gives a conclusion of this work.

MODELING OF WIND TURBINE AND MPPT ALGORITHM
The power generated by the wind turbine is defined by the formula (1) [12]: where ρ is the air density, S is the surface swept by the turbine, V is the wind speed, β is the angle of the orientation of the blades. λ is the speed ratio defined by (2) [13] : where Ω t is the angular velocity of the turbine shaft. Cp(λ, β) is the power coefficient and refers to the efficiency of the wind turbine in terms of energy conversion, which is expressed (3) [14] : The power coefficient evolutions as a function of the speed ratio λ and the beta β are illustrated in Figure 2. It is shown on this figure that for operation in MPPT mode with β=0°, the optimum speed ratio is λopt=8.1 for Cpmax=0.48.

Figure 2. Power coefficient Cp
The expression of the aerodynamic torque is (4): The rotary speed and torque at the generator shaft are defined as (5) [15]: where Ωmec: mechanical speed of the generator. Tg: The generator torque.
Applying Newton's second law of motion to the generator shaft gives: with: J, Jg,Jt : the total inertia, the generator inertia, and the turbine inertia. f: Damping coefficient. Tem: the electromagnetic torque of the generator. We can eventually express the optimum electromagnetic torque corresponding to the maximum power as shown in (7) The block diagram of the wind turbine with the MPPT algorithm is shown in Figure 3.

MODELING AND VECTOR CONTROL OF THE DFIG
DFIG's rotor and stator voltages equations in the park frame are expressed as (8) where Rr and Rs are respectively the rotor and stator resistances per phase. Rotor and stator flux are defined as (9) [18]: { φ ds = L s I ds + MI dr φ qs = L s I qs + MI qr φ dr = L r I dr + MI ds φ qr = L r I qr + MI qs (9) where Lr, Ls are respectively the rotor and stator inductances. M is the machine mutual inductance. Idr, Iqr, Ids, Iqs, are respectively the (d, q) rotor and stator currents. The stator active and reactive powers are defined by (10): { P s = V ds I ds + V qs I qs Q s = V qs I ds − V ds I qs Electromagnetic torque is expressed as [19]: T em = pM L s (I dr φ qs − I qr φ ds ) where p is the pair pole number:

1955
By aligning the d-axis in the direction of the stator flux [20], we obtain: φ ds = φ s , φ qs = 0 As a consequence, the stator flux expressions become: { φ s = L s I ds + MI dr 0 = L s I qs + MI qr The stator resistance is negligible because of its small value. In this sense, the stator voltages can be described as (14): From (13) and (14) the stator currents can be expressed as: By combining (10), (14) and (15) In which the coupling voltages ed and eq are expressed as: where g is the slip.

BACKSTEPPING CONTROL IN THE LOOP OF INDIRECT FIELD ORIENTED CONTROL
The fundamental idea of backstepping is to synthesize the control law in a recursive way, at each step a system state is stabilized, assuming the next state as a virtual control. The objective of the method is to design a control law that brings the system to a desired state, which is generally a stable closed-loop steady state [21]. These can be achieved by using Lyapunov's functions which ensure stepwise stability [22]. -Step 1 Active and reactive power control The active and reactive power errors are defined as: By deriving the error we obtain: The first Lyapunov candidate function is: The derivative of the function V1 is expressed as: Therefore V1 can be expressed as: C1 and C3 are positive constants ensuring the Lyapunov stability condition [23][24]. The derivative of the power errors becomes: The second Lyapunov function chosen is defined as: The derivative of V2 is defined as (30): V̇2 = z 1 ż1 + z 2 ż2 + z 3 ż3 + z 4 ż4 (30) We can rewrite (30) is being as: The Figure 4 illustrates the indirect flux oriented control of DFIG using the backstepping method. The diagram contains the calculation blocks for the desired rotor currents and voltages, the PARK transformation blocks, a PWM inverter on the rotor side, and the model of the DFIG in the PARK frame dq.

Normal test simulation
This part presents the results of the simulation of the wind turbine assembly with DFIG. We have applied a wind profile to the turbine illustrated by the Figure 5. Figure 6 presents the mechanical speed response of the DFIG. It can be clearly seen on the figure that the DFIG angular velocity follows perfectly its reference speed image of the MPPT, as validated by maintaining of the power coefficient Cp around its optimal value of 0.48 as shown in Figure 7. The maximum power produced by the wind turbine is therefore the reference applied to the active stator power control loop of the DFIG, the simulation results as presented in Figure 8 show a perfect tracking of the MPPT power by the DFIG while the response time is tr5%=0.4 (ms) and some neglected power oscillations, on the same figure one can see the variation of the rotor active power which varies according to the operating mode of the machine (hypo-synchronous or hyper-synchronous). Figure 9 also shows a very satisfactory regulation of the reactive stator power at a zero value, this being confirmed by Figure 10 which shows that as a result the power factor is maintained at unity. The coupling terms presented in (17) are perfectly compensated and this is of obvious benefit by the use of indirect flux control. In Figure 11 the variations of the stator current injected into the electrical grid are shown, they are smooth and present a very low THD%=0.75 according to the FFT analysis Figure 12.

Robustness test simulation
In order to test the robustness of the Backstepping controller against unexpected variations in the internal parameters of the DFIG, caused by an increase in temperature or other events. We have elaborated a robustness test in which the rotor resistance and inductance values are increased by 50%: { L′ r = L r × 1.5 R′ r = R r × 1.5 (34) Figures 13-15 illustrate the simulation results. It is immediately observed that the regulations have been fully ensured both for the active and reactive power but there is some slight degradation of the performances obtained previously. At the level of active power we have an increase in response time to tr5%=1.7 (ms) with an overshoot of 100 (W) which remains very negligible for a few (ms). For the reactive power we also have an overshoot of -100 (var) during the first 40ms. These results confirm that the backstepping control is highly robust through indirect field oriented control. It offers a perfect decoupling between the d and q axes, a much reduced response time and minimal power oscillations in the face of disturbances resulting from variations in the internal DFIG's parameters.

CONCLUSION
This work presents the modeling and the control of a DFIG-based wind energy system using backstepping technique under variable wind speed. The maximum power point tracking was achieved by using the tip speed ratio algorithm. The results of the simulation of the indirect field oriented control with backstepping controller showed a high stability of the system as well as a very fast response, with a very low total harmonic distortion of the stator currents injected into the power grid, even with the variation of the internal parameters of the DFIG. However, these results confirm the efficiency of this non-linear backstepping control which is easy to implement and gives sometimes better results than the use of the new control based on artificial intelligence such as fuzzy logic or neural network control.

APPENDIX
The following Table 1 presents the nominal values of the different parameters of the studied wind turbine system.