Robust control of wind turbine based on doubly-fed induction generator optimized by genetic algorithm

Received Jul 7, 2021 Revised Feb 28, 2022 Accepted Mar 16, 2022 In this research paper we will cover the study of the modelling and the control methods of the variable speed wind turbine based on doubly-fed induction generator (DFIG). It represents the most stressed structure given its distinctive characteristics. To control the electrical powers generated by this system independently, the vector control with the stator flux orientation is founded according to two techniques: i) the control of the powers by the backstepping technique and ii) the robust control based on the active disturbances rejection control. After the synthesis of the controllers of those two methods, their performances will be tested and compared to evaluate their effectiveness. We are mainly interested in the robustness test of the two control strategies with respect to the internal parameters’ fluctuation of the generator. The computing of the different parameters regulators of these two strategies is carried out using a genetic algorithm. This computing method makes it possible to arrive at an optimal solution of the DFIG power control. The different parts are simulated using MATLAB/Simulink environment.


INTRODUCTION
Faced with the ever-increasing demand for energy, industrialized countries have made massive use of nuclear power plants. This type of energy has the advantage of not causing atmospheric pollution, but the risk of nuclear accident [1], [2]. Nowadays countries are increasingly turning to the use of renewable energy sources as a solution to that problem. Wind power in term of example, which represent an attractive alternative that may in the future replace traditional power plants [1], [3], [4].
The variable speed wind turbine based on a doubly-fed induction generator (DFIG), driven by a turbine and controlled by the rotor is the most used system in wind energy conversion considering the advantages it offers [3], [5]- [7]. In this structure, the stator of the generator is connected directly to the grid, on the other hand the rotor is linked to the grid via a transformer and link filter with two bidirectional converters [3], [4], [8], [9]. Figure 1 shows the functional scheme of this structure based on the DFIG and two power converters.
For this system command, various approaches are available in the literature. A study by Nadour and Essadki [10] shows that the backstepping approach provides good results compared to the proportional integral (PI) control especially with regard to the hounding the references values and robustness against external and internal disturbances. However, this approach does not present more significant results faced with the change of internal DFIG parameters with higher variation rates (100% for example). Especially since it does not present an analytical method allowing the computing of the regulator parameters. Bossoufi and Karim [11] have presented a control technique based on a control approach of pole placement incorporated with a backstepping control system. The experimental results obtained based on a prototyping platform realized with DFIG-generator, field programmable gate array (FPGA) and wind turbine [11] show the efficiency of the technique for the system stability. The problem of this solution is that it does not present any detail about the robustness in the case of internal parameters' variation. Indeed, these parameters may undergo changes caused by many physical phenomena (temperature, inductors saturation) [12].
Rouabhi and Rachid [13] presented a combination of vector and backstepping control. Simulation results show the decoupled control of both electrical powers and a good performance likened to the results acquired in the case of the compensator use. However, we do not find in this approach a method for the synthesis of the controllers' parameters, on which the performances of this approach are based.
To ensure system stability and rapid dynamic response, the control parameters must be well chosen. However, in all the previous works and in the majority of the documents which treat the same subject, one does not find a study which presents a procedure for the determination of these control strategy parameters. The calculation is often done by a random adjustment during the simulation.
In recent years, several works have dealt with the DFIG command used in wind energy conversion system (WECS) by active disturbance rejection control (ADRC) [14]- [16]. With this control method, the system is able to compensate external disturbances and those due to parameter changes. The work of this research paper is focused on the study and comparison of this methods robustness with that based on Backstepping. We also propose a method based on genetic algorithms, for the calculation of the different controllers' parameters, which allows to arrive at an optimal solution of DFIG power control. We can show that replacing backstepping controllers with ADRC controllers optimized by genetic algorithm (GA) allows to improve significantly the system performances. This paper is structured as shown in: the first part is devoted to the studied system modeling with its different components. In the next part, we treat the DFIG control principle by the backstepping technique. The third part is entirely devoted to the presentation of the DFIG robust control founded on the ADRC. The next part is devoted to the presentation of the procedure used for the optimization of the two control approaches using the genetic algorithm. The last section of this paper is devoted to the presentation and analysis of the simulation results, and the robustness test using the MATLAB/Simulink environment.

Wind turbine model
The wind turbine converts the kinetic wind energy and drives the variable speed generator. The maximum power which can be obtained according to Betz's law is [17]- [19]: where Cp is the power coefficient, which is defined by the ratio between the total extracted power and the incident one [17]. This coefficient depends on λ, the velocity ratio, and β the pich angle, it is formulated by the following function [16], [20]: ( , ) = the aerodynamic torque Taer is given by:

DFIG modelling
The equations which describe the electric part of the generator in the two-phase reference are given by the following relations [12], [21]: where p=2 is the number of pole pairs. We have considered the stator flux control oriented along the d axis of the two-phase reference. It is also considered that the stator resistance Rs is negligible during the model computing [12], [17], [21], [22]. So, we deduce: Finally, we can deduce the block diagram of the DFIG model ( Figure 3).

Powers control using the backstepping technique 2.2.1. Backstepping principle
The principle of backstepping relies on its ability to dissociate a complex nonlinear problem control for a system into simpler ones. Basically, this technique synthesis is separated into different design steps. At each one, we're virtually dealing with a simpler system with single-input and single-output problem, and each sub system supply a baseline for the next synthesis step. The global stability and overall performance are obtained by a Lyapunov function for the global system [4], [6], [11], [13].

DFIG control based on the backstepping method
The backstepping method that we are going to apply for controlling the double-fed asynchronous machine is based on the vector control. In this approach, the command will be established in the reference ( and ) according the condition in (11). The control of the rotor side converter (RSC) allows the regulation of the exchanged powers with the grid. the synthesis of the control blocks with the backstepping method is divided into two subsystems: the first one is to generate the current reference Ird and Irq and the second one is to generate the voltage reference Vrd and Vrq [4], [10], [23]. the relations linking the stator powers and the rotor currents are expressed by (14) and (15): the rotor currents derivatives can be mathematically determined and given by the following: where = (1 − 2 ) represent the dispersion coefficient. The computing of the reference quantities of the currents and voltages control is based on the Lyapunov functions according to two steps [4], [11]- [13].
-Step 1: the errors E1 and E2 between the currents Irq, Ird and their reference Irqref and Irdref is given by (18).
The derivative of this error is given by (19).
We select the Lyapunov function so such that: The errors are chosen so as to have a zero derivative of the Lyapunov function: -Step 2: From the previous equations, the voltages reference can be mathematically determined: K1 and K2 are the backstepping controller parameters, they are strictly positive, and ensure the system stability as well as a fast dynamic response [14]. These performances are ensured with a zero derivative of Lyaponov function. The functional scheme of the backstepping command is given in the Figure 4.

. Active disturbances rejection control strategy
The ADRC is a robust control strategy which is based on the model system extension using a state observer extended state observer (E.S.O) to have an estimation and to cancels, in real time, all disturbances affecting the system whether internal or external [15], [16], [18], [24]- [26]. To explain the principle of the ADRC, we treat the case of a first order system: where b is a parameter to be estimated, b0 represents the known part of b and Δb represents the modeling error due to the system parameters changes. u and y are input and output variables, K is the static gain and T is a system constant.
f(y,d,t) represents the effect resulting from internal parameters variations and the external disturbances impact. The basic idea is the estimation and compensation of the disturbance. The (31) can be written in an augmented state space form as [27]: where: = ( 0 0 0 0 ), = ( 0 0 ), = (10) and = ( 0 1 ). The observer will be able to estimate the derivatives of y and f and it is developed as shown in: In Figure 5(a), we represent the structure of the ADRC controller previously studied [14], [15], and the structure of the extended state observer is given by the Figure 5(b) [28].
(a) (b) Figure 5. Structure of (a) the ADRC controller and (b) ESO

Control of the rotor side converter
By referring to the assumptions relating to the stator flux orientation along the d-axis, and neglecting the stator resistance, the equations of the mathematical model of DFIG can be simplified [16]. So, the rotor currents are expressed by the following relations: we can also put these relations in the (37) fd and fq represent the disturbances impacting the rotor currents Ird and Irq. Vrd and Vrq are respectively the command variables of the current's loops Ird and Irq. br0 represent the known part of the process parameters. The schematic diagram of the rotor side control is given by Figure 6. The parameters Kp, β1 and β2 are determined to have a perfect follow-up of Idr and Iqr references values. These parameters are obtained in our case using GA.

Control of the GSC
The currents iqf and idf are given by (39). The computing of the parameters used in the regulation algorithm, is done to have a perfect tracking of the reference values by the currents iqf and idf. This control algorithm by ADRC can be summarized according to Figure 7. Figure 6. The GSC control diagram using ADRC Figure 7. The GSC control diagram using ADRC

Genetic algorithm
The genetic algorithm is an optimization method for finding optimal solutions to research problems. It is based on the natural selection methods. This approach was proposed by John Holland in 1975, and then adopted as a very efficient method to find the solutions to an optimization problem. It makes it possible to avoid the local minima constituting a major problem in the case of nonlinear systems [29], [30].
The basic principle is based on the repeated modification of the population so as to evolve towards an optimal solution, and this starting from an initial population consisting of a certain number of elements appointed chromosomes. The relevance of each solution is measured by a fitness function. The algorithm is based primarily on three stages: selection, crossing, and mutation. These three operations must be applied to create new individuals to be evaluated with respect to their parents. The algorithm stops when we obtain individuals with optimal performances [29], [31], [32].
In the case of DFIG powers' control, the objective function can be computed and evaluated for each individual belonging to the actual population. Therefore, the parameters of the ADRC or Backstepping controller can be decoded for each individual, then the DFIG model is simulated to obtain the objective function value which consists in following the reference power required by the turbine. The following population is generated based on the genetic algorithm operators (selection, crossing and mutation) [30], [33]. These two steps are repeated from generation to generation until a final population is obtained. In Figure 8, we present the optimization diagram to be carried out to reach the optimal parameters values of the backstepping and ADRC regulators. The types of the GA operations used and the parameters' initialization are given in Tables 1 and 2. In this work, the power of the machine used is 1.5 MW. The algorithm stops when the best fitness value is less than or equal to the limit fitness value which has been set here at 5.10 -4 MW.

Parameter's optimisation using GA for backstepping technique
As mentioned before, the control system must be able to guarantee the two main objectives: closed loop stability of the system and zero regulation of the tracking error. The problem which arises at this level is the determination of the backstepping adjustment coefficients K1 and K2. To avoid determining the gains of the regulators with non-analytical methods (random), we propose in this work an optimization by the genetic algorithm. The principle consists in considering an adaptation law established by the Lyapunov function as a minimization problem. This function is given as presented before by (43) [10], [11], [13].
Where 1 = − and 2 = − . In fact, the control of the Irq and Ird currents allows the generation of an active power P in accordance with the reference power required by the mechanical part (turbine). A GA based backstepping control optimization can be represented as shown in Figure 9.

Parameter's optimisation using GA for ADRC
The ADRC allows an estimation, using a state observer, of modeling uncertainties and external disturbances, and a compensation for them in real time [20]. The objective is to have a faster observer dynamic than the closed loop system one, as well as a reduced noise sensitivity [15], [16], [18], [28]. The genetic algorithm is used to determine the Kp, β1 and β2 parameters. These parameters are adjusted so as to obtain the optimum performance in terms of dynamics, robustness and disturbance's rejection. The objective function is therefore chosen to minimize the difference between the measured power and the reference one. A GA based ADRC optimization can be represented in Figure 10 and the parameters values used for genetic algorithm in this work are presented in Table 2. The best score value and mean score versus generation for the two control methods are represented in Figure 11. The parameters' evolution trought the generation is presented in Figure 12. The backstepping gains K1 and K2, and the ARDC controller parameters Kp, β1 and β2 are listed in Table 3.

. Tracking and control test
The parameters of the DFIG are listed in Table 4. In Figure 13 we represent the wind profile used in this work which is modelled by (46): v(t) = 8 + 2sin (2.5 − 5 ) + 2sin (4 − 3 ) + 1.5sin (5.4 − 12 ) + 0.5sin (2.5 − 12 ) (46) Figures 14 and 15 show the simulation results of the produced powers and the rotor currents Ird and Irq with their references, for the Backstepping technique and ADRC. The reference reactive power is chosen zero to guarantee a unit power factor. As can be seen in the previous figures, the produced powers controlled by backstepping and ADRC, optimized by genetic algorithm, follow perfectly their references. We can notice more precision and stability with a response without fluctuations or exhibiting overshoot on that based on the backstepping technique. From these figures, we can see that the stator currents vary according to the wind profile provided for the simulation and the frequency of the rotor voltages follows the DFIG speed variation. In Figure 16, we present the stator currents and rotor voltages of the DFIG.

Test of the robustness
The objective of the robustness test is to verify the control methods performance against the internal parameters variation of the DFIG. In fact, the regulator parameters computing is based on a DFIG model whose parameters are considered to be fixed. We are more particularly interested in highlighting the performance limits of the backstepping method in this case. However, these parameters can change over time or because of external phenomena. In Figures 17 and 18 we represent the produced powers evolution and the rotor currents Irq and Ird after a variation Rr and Lr with a rate of 100%.
From the Figures 17 and 18, the variations of the Rr and Lr values did not have much impact on the performance of the system controlled by the active disturbances rejection control unlike the backstepping technique. Indeed, the ADRC allows the estimation and compensation of disturbances. The tracking of the reference values is always assured as well as the stability which is not affected.

CONCLUSION
In this research article, we have dealt the modeling and control of the wind turbine driven by DFIG. We are interested to highlight and evaluate the dynamic performances of ADRC and backstepping methods and their sensitivity to the DFIG's internal parameters variation. The two control methods studied consist in controlling the rotor currents in order to adapt the DFIG rotation speed to the wind speed, by the control of the electromagnetic torque, and also in controlling the active and reactive powers injected into the grid. The synthesis of the regulation parameters is based on the genetic algorithm which allows to have the optimal values to guarantee the best performances. According to the simulation results, and for normal operation assuming that the DFIG's internal parameters are fixed, we notice that the backstepping control, optimized by GA, allows good performances, in particular a fast response and without overshoot, unlike the ADRC which has some undulations at the start. However, internal parameters are always subject to variations. In that case, we have found that the ADRC allows a good tracking of the desired values as well as a robustness against the DFIG's internal parameters variation. In terms of perspectives, the rate of the internal parameter's variation treated in this work is equal to 100%. However, we do not find any work that studies the order of magnitude of this variation rate, in order to have more precise values. We are therefore considering the development of methods or algorithms making it possible to estimate this variation rate and take it into account in the control laws used.