High-performance MPPT control of DFIG with optimized flux reference in presence of nonlinear magnetic characteristic

Received Oct 29, 2021 Revised April 20, 2022 Accepted May 2, 2022 In recent years, an expansive research exertion has been devoted to the control for aeorogenerators conforming of a Doubly Fed Induction Generator (DFIG) coupled to a wind turbine. Most preceding controllers based on standard models (whose magnetic induction characteristic is considered linear and iron losses have not been taken into account). Since such a magnetic characteristic is nonlinear in practice, this work develops a variable speed control strategy on the basis of a DFIG model in presence of nonlinear magnetic characteristic and iron losses. The main contribution of this paper is to present a highperformance maximum power point (MPPT) control of DFIG based on rotor flux optimization, designed to reach an optimal operational conditions. Of the foregoing, the control goal is twofold: i) minimization of the stator and rotor currents circulating by controlling the rotor flux for a maximum electromagnetic torque; ii) captured the maximum wind power to insure the attainment of MPPT. The performances of the proposed command has been studied theoretically and validated by simulation studies with Matlab/Simulink.


INTRODUCTION
Global warming and lowering fossil reserves have pushed many countries to move towards the use of renewable energies, which are considered clean energy [1]- [4]. In this context, the major areas of research in the world have been focused on the problem of optimizing wind energy captured by the turbine. Admittedly, the control of a wind turbine equipped with a fixed-speed alternator is very simple but can no longer ensure optimization of the aerodynamic energy captured. [5]- [8]. Because of this, variable speed operation was introduced. In this context, the MPPT control is called to maximize the wind energy extraction. Various MPPT control models were based on the standard model [9], [10] which neglects the nonlinearity of the magnetic characteristic as well as the iron losses [11]- [14].
For these reasons, this article develops the principles of the high performance MPPT control for DFIG based on the optimization of the currents passed through the stator and rotor. Optimality is grasped in the context of maximizing torque and decreasing circulating currents [15]- [19]. The theoretical results are justified by the simulation which confirms that the developed controller has a certain robustness to disturbances. For some authors, the reference rotor flux is given the nominal value (or by defluxing). In such a case, the electromagnetic torque is close to its nominal value and the efficiency of the machine is maximum. But, in reality, the electromagnetic torque is not always fixed in a precise value and can have strong variations (depending on the mechanical torque applied) [20]- [23]. So, if the aerodynamic torque is small, there is  [26]. The article is structured as follows: the model of the wind turbine taking into account the nonlinear magnetic characteristic as well as the iron losses are described in section 2. The formulation of the problem, the design of the reference signals, and the development of an improved MPPT controller based on backstepping technique which exploited the magnetic characteristic nonlinearity are established in section 3. The robustness of the control law is presented by simulation in section 4, a conclusion is illustrated in section 5. At the end of the article, the list of references and the bibliography are presented.

WIND TURBINE MODEL IN PRESENCE OF NONLINEAR MAGNETIC CHARACTERISTIC AND IRON LOOSES 2.1. Turbine model
The aerodynamic wind power available at the blades is given by [19]: with β is a blade pitch angle which will remain constant for our case study, C P (λ) is the coefficient of performance (C P (λ, β) ≝ C P (λ)),R is the radius of the wind turbine rotor, ρ is the air density, v w is the wind velocity and λ is the tip speed ratio, its expression is:

Nonlinear dynamic DFIG model 2.1.1. Iron losses
Generally, the iron losses of DFIG are most frequently modeled by of the equivalent iron loss resistance R m associated in parallel with the magnetizing inductance Ӽ. although, in this paper, the proposed equivalent circuit of DFIG is inspired from [19], where the iron losses of DFIG are modeled by an equivalent resistance R m associated at the same time in parallel with the stator cyclic leakage inductance L σs and mutual cyclic inductance Ӽ. In fact, the simplicity and precision of the calculations of this technique justifies this choice [13], because, if we use the theorem of thevenin, we will obtain an equivalent circuit like that of the standard model (without iron losses). Figures 1(a) and (b) show the proposed equivalent circuit of the DFIG in the reference frame α and β respectively using concordia transformation. So, the iron resistance is a variable parameter which was identified experimentally on a 4 kW DFIG driven by a DC motor. The procedure described in [19] was used to determine the values of the iron loss resistance as well as the magnetizing reactance [22]. The iron resistance characteristic is shown in Figure 2.
From the Figure 1, the parameters Rs, Rm and V sαβ will be replaced by an equivalent voltage source V sTαβ in series with an equivalent resistance RsT, from this, we are calculated in the stationary α-β reference frame the thevenin's equivalents parameters as [23]: Therefore, the stator and rotor flux can be rewritten as: On the other hand, the stator and rotor currents contribute to the generation of the magnetizing flux along with the magnetizing current, denoted by: and we have μd = Ӽi μd , μq = Ӽi μq (10) Where φ μ is the magnetizing air-gap fluxes. In that case, the flux in (5), (6), (7) and (8) are given in the dq reference as [13]: = 0 = σ + φ μq (12) = σ + φ μd (13) The leakage inductances are grouped in the stator side, we pose is the equivalent inductance to the stator side of both the stator and the rotor leakage brought, that gives σ = 0 and σ = . Next, the inductances and can be rewritten like that: Figure 3(a) give the variations of mutual inductance as a function of the corresponding magnetizing current. We suppose that the magnetizing flux direction remains unchanged and that it saturates only in amplitude, which allowed us to admit that φ μ and i μ are in phase. Figure 3(b) shows the magnetic characteristic Φ μ = λ(I μ ) where Φ μ and I μ are the norms values. Of the above, the instantaneous values φ μ and i μ are given by: In the following, we pronounce the static magnetization parameter Ӽ, defined as [23]: First, we note some experimental points of the DFIG magnetic characteristic to make the approximation of the mutual inductance Ӽ(I μ ). Then, we can easily deduce the magnetizing flux norm Φ μ (with Φ μ = Φ r ) in function of the magnetizing current I μ .
(a) (b) Where: = = 1 − In the above model, the parameter Γ is the unique parameter that varies in function of magnetic state of the DFIG, this relation has been given a polynomial approximation [24]- [26], i.e.: The ai's are constant coefficients, identified with polyfit matlab function. So, the magnetic characteristic of the DFIG is denoted Φ r = γ(I μ ) where I μ denotes the norm of the magnetizing current and Φ r = Φ μ . From the experimental points (I μ , Φ μ ), we deduce a polynomial function approximation of the characteristic (Φ r , Γ) as depicted Figure 5. The parameter values of DFIG used in experimental tests are given in Table 1.

REFERENCE ROTOR FLUX CONSTRUCTION AND THE ENHANCED OPTIMAL CONTROL
The present section presents a new and optimal approach that properly accounts for the nonlinear magnetic characteristic to control DFIG. Notice that in previous works, the standard model can be referred to linear model (LM), from the nonlinear model (NLM), and this while keeping the parameter Γ constant. Thus, we describe a new control strategy based on a method for optimizing the rotor flux reference to minimize stator  [26]. the performance of this optimization law is validated by means of simulation.

Problem formulation
The problem of controlling variable speed wind turbines is addressed. The majority of previous works considered that the reference of the reactive power of the stator was null and the magnetic characteristic was linear. However, the machine operation mode is not optimal of the large electromagnetic torque with Qs=0.In the present section, we will develop a new optimal MPPT control strategy taking into account the nonlinearities of the machine (in order to lower the total currents consumption in both the stator and rotor).
The control objective is to follow the maximum of the wind power captured and to minimize the currents transited by generating an optimal rotor flux in line. In this context, the rotor flux norm Φ r must track perfectly the flux reference Φ r−ref = H (I r , I s ) , where (I r , I s ) are the rotor and stator currents norms and the function H will be resolute in order that Φ r = Φ r−ref involve the minimization of the stator and rotor currents circulated in the induction machine for getting a maximum electromagnetic torque [23]- [26].

Optimal rotor flux reference
Firstly, we find a relation between the rotor flux, stator and rotor currents norms. In this respect, one considers the DFIG model developed in (19); thus, the stator and rotor currents can be rewritten by: On the other hand, the rotor flux can be rewritten as: Then, using equations (20) and (21), we'll have: Accordingly, one has: Φ r 2 = Ӽ 2 (I s 2 + I r 2 + 2(i sTd i rd + i sTq i rq ) We replace i sd i rd , and i sq i rq by their expressions according to (23) The electromagnetic torque T em can be written by: Then, (26) can be simplified to:  (I s  2 + I r  2 ), there are several different operating points by the value of rotor flux norm Φ r and T em . For example, a torque T em = 5Nm can be produced, for the generator 4kw (Table 1), with: − A current I s 2 + I r 2 = 490 A and a flux Φ r = 0.2 Wb; − A current I s 2 + I r 2 = 20 A and a flux Φ r = 0.40 Wb; The best operation point from an energetic viewpoint, is the one which corresponds to the minimum of currents circulations. From Figure 6, it is usually seen that there is a unique couple (Φ r ,(I s 2 + I r 2 )) for any T em , which engages the least possible values of the stator and rotor currents. For our case study Φ r−ref = 0.4Wb.
The stator active and reactive powers are given by: Introducing (20) and (21) into (31)  The stator flux φ s is assumed constant (the grid is considered stable).

Improved MPPT control
The main objective of the control strategy is to develop an optimal control laws in order to optimize the different operations points of the wind turbine [13]. For this, an enhanced backstepping controller has been designed to force the system to track their references where Ω ref is the optimal rotor speed reference illustrated in Figure 7 and Φ r−ref is the rotor flux reference (35).  Figure 7. Optimal rotor speed reference in function the speed wind [13] For reasons of simplification (backstepping control of the rotor flux in only one step), the control of the rotor flux will be assured by controlling the stator reactive power described by (34). Firstly, we consider the following errors: Step 1. It follows from (19) Where k 1 and k 3 are positive real parameters to be chosen as described in [13]. On the other hand, in (38), is considered as a virtual control signal which will temporarily be like the real control signal, the error in (38) assures the global asymptotic stability, letting: As the quantity can't be equal to µ 1 because isn't actual control signal. Indeed, we take the stabilizing function µ 1 in the first. Step 2. Now introduce the new error: Then, the dynamic of the error 1 are given as: From (3) and (4), we can write: Introducing (25) and (26) in (27) The control signals and can be deduced from (51) and (52) satisfying ̇2 <0: Where k 2 is a new positive constant. In fact by substituting (54) and (55) in (53), the time derivative of (53): Which ensures the global asymptotic stability of the system.

SIMULATIONS RESULTS
In this section, we will present the importance of the new control strategy (non-linear model with optimal flux NLMOF), introducing the optimal rotor flux reference illustrated by Figure 8

Hegemony of the NLMOF strategy over the LMCF strategy
The simulation protocol is designed in order to test the performance of the HP-MPPT controller (used the NLMOF) operating under different conditions by considering two steps of the wind speed ( Figure 9) and load torque ( Figure 10). Recall that the NLMOF regulator involves the optimal flux reference depicted in Figure 8. The HP-MPPT control laws defined by (54) and (55) receives the following design parameter values, which have proven to be practical: k 1 = 1000, k 2 = 300 and k 3 = 50.
The optimal specific speed 8,1 with MPPT control are presented in Figure 11. The dynamic responses of the optimal flux NLMOF and constant flux LMCF regulator are presented in Figure 12. The control of rotor flux with the NLMOF and LMCF strategies are presented in Figures 13 and 14 respectively. According to the results obtained in Figure 15, it is confirmed that under all operating conditions, the NLMOF regulator requires smaller current than the constant flux reference regulator LMCF. Figure 16 confirms the performance of our regulator by the pace of joule effect losses for LMCF and NLMOF strategies. The active stator and rotor power are presented in Figures 17 and 18 respectively for LMCF and NLMOF regulators. The dynamic and the performance under different operating parameter changes of the proposed regulator have been affirmed further by Figures 19 and 20 which presented the reactive stator and rotor power respectively for LMCF and NLMOF controllers.

Hegemony of the NLMOF strategy over the LMOF strategy
The simulation protocol is similar to part 4.1 (Figures 9 and 10). Recall that the NLMOF regulator is defined for (19) in combination with the optimal flux generator (shown in Figure 8). So to have the LMOF regulator, it is sufficient to make the following changes in (19): − The parameter Γ in equation (19) is held constant, equal to à ≈ 460 Ω −1 . − The rotor flux reference is given by the signal depicted in Figure 8. − The controller parameters are set to the following values: k 1 = 1000, k 2 = 300 and k 3 = 50 . Figure 21 shows that the Sum of squares norm (I s 2 + I r 2 ) is lower with the NLMOF regulator. Figure 22 shows the quality of flux tracking obtained with the NLMOF and LMOF control strategies. So, Figure 23 presents the active stator power for LMOF and NLMOF regulators.Also, from the energy point of view, is also illustrated by Figure 24 when the rotor power produced by NLMOF regulator is more than LMOF regulator.

CONCLUSION
The problem of controlling the DFIG in the presence of nonlinear saturation magnetic state are evoked. A new speed and rotor flux regulator has been developed using the optimal backstepping approach to optimize the currents circulated in the machine. Also, the proposed regulator comprises a reference rotor flux generator dependent on the nonlinear magnetic state. In addition, it is demonstrated by simulation that, under all operating conditions, that the current transited is really lower with the regulator NLMOF. The performance of the proposed control laws have been demonstrated by simulation studies performed in the Matlab/Simulink® environment.