Abstract
Many studies have been made on the double-fed induction generator wind turbine system (DFIG-WTS) in recent decades due to its power management capability, speed control operation, low converter cost, and minimized energy losses. In contrast, induction machine control is a more complex task because of its multivariable and nonlinear nature. In this work, a new robust nonlinear generalized predictive control (RNGPC) is developed to maximize the extracted energy from the wind without the use of aerodynamic torque measurements or an observer. The aim of the predictive control is to produce an anticipated impact by employing explicit knowledge of the present condition. By revisiting the cost function of the conventional nonlinear generalized predictive control (NGPC), which is based on Taylor series expansion, in that way, the resilience of the system is improved. An integral action is included in the nonlinear predictive controller. As a result, if the closed loop system is stable, the suggested controller totally eliminates the steady state error, even if unknown perturbations and mismatched parameters are present. The output locating error’s convergence to the source is utilized to show the locked system’s stability. Simulation results demonstrate and verify the efficiency, the good performance, and robustness of this proposed control technique.
Introduction
Due to its multiple advantages over other types of wind turbines, the doubly fed induction generator (DFIG) is becoming more common in wind generation. In general, the system is designed to operate in a 33% slip range. In this situation, the power requirement of the rotor converter is 25% of the maximum power of the wind turbine (). The supply of the rotor by a converter, with variable frequency, makes it possible to separate the mechanical speed of the rotor from the frequency of the electrical grid. The control system for an induction machine is more complicated because it is a multivariable, nonlinear, and highly linked system. Techniques like voltage-field-oriented control and power-torque direct control, which use a linear conventional controller like the proportional-integral-predictive (PID) controller, have not always achieved flawless results ().
An important number of nonlinear controls have appeared in the literature to ensure better performances control of a DFIG-based wind turbine. In , an algorithm is proposed for the improvement of the DFIG by sharing between the converters (machine-side and grid-side) an optimal reactive power. An active disturbance rejection control with important robustness properties to design the current control law is developed in to enhance the ride-through low voltage. A coordinated super-twisting sliding mode control is investigated in for grid synchronization and power optimization of the conversion system. A new exponential reaching law-based sliding mode control is proposed in for hastening the approaching process and lowering the system chattering phenomena. A multi-objective particle swarm optimization algorithm in studied in , improving dynamic stability under electrical disturbances and thereby assisting the ride-through low voltage.
However, to ensure higher performances of the DFIG-based wind energy converter, it requires the utilization of advanced nonlinear controllers. Indeed, model predictive control (MPC) emerged in the late 70s (), and since then, it has evolved into a variety of forms such as model algorithmic control, predictive functional control (), generalized predictive control (), extended horizon adaptive control (), and robust model predictive control ().
The MPC is a plant model-based feedback control approach that forecasts future plant outputs over a specific time horizon. These predictions are then utilized to choose the optimum control by solving an optimization problem while adhering to a set of constraints (). To deal with the features and control needs of specific types of variable-speed wind turbines, a revisiting variant of a recent adaptive second-order sliding algorithm is presented in . Robust adaptive MPC constructed through linear matrix inequality and obtained estimation with a nonlinear observer is designed in for the optimal, effective, and robust operation of the conversion system. Based on an analysis of the effectiveness of nonlinear MPC with a fixed weight coefficient under various wind conditions, a fuzzy regulator is created to update the weight coefficient of the cost function, and an improved multi-objective marine predator algorithm is proposed in for optimization on energy capture and generator torque fluctuation. A DFIG wind system based on balanced and unbalanced network conditions is proposed using a universal and low-complexity model predictive direct power control (). To handle high-voltage ride through, a P-Q coordination-based MPC is developed in . On the basis of intelligent scenario generation (ISG), a stochastic model predictive yaw control (SMPYC) technique is suggested in . Here, the ISG approach is suggested to develop scenarios that define the uncertainty of wind direction prediction. Then, the yaw action is improved using the suggested scenario-based SMPYC to increase the energy capture effectiveness of WTs.
For mechanical sensorless control technology of induction machines, several solutions have been developed. A direct torque control associated with an artificial neural network was applied to a dual power induction motor (). This strategy is carried out without a mechanical sensor using the extended Kalman filter as an observer. In order to ensure the estimation of the five-phase induction motor rotor speed, the authors, in , performed two approaches: the first is based on the adaptive flux and speed observer, the second, on the other hand, is based on a model reference adaptive system estimator. A disturbance observer is constructed to estimate the aerodynamic torque in , and a nonlinear generalized predictive control for wind energy conversion systems is proposed to achieve the controller’s robustness. For induction motor drives, a load torque observer based on a second-order sliding mode technique is suggested ().
The present study is aimed to design a robust controller in order to maximize the amount of energy taken from the wind without the use of an observer or aerodynamic torque measurements. The system’s robustness is increased by revising the cost function of the traditional nonlinear generalized predictive control (NGPC), which is based on Taylor series expansion. The nonlinear predictive controller includes an integral action. As a result, assuming that the closed loop system is stable, the proposed controller completely eliminates the steady state error, even in the presence of unknown disturbances and mismatched parameters. The convergence of the output locating error to the source is used to demonstrate the stability of the locked system. The validation of the trajectory following and perturbation rejection of the applied control using simulation results of the dynamic behavior of the researched system was performed under Matlab/Simulink.
The paper is organized as follows: in the Introduction section, the DFIG wind turbine system is described, and in the DFIG wind turbine system description and energy conversion section, the DFIG modeling is examined. The Proposed robust nonlinear generalized predictive control section develops the proposed structure controller. Finally, in the Simulation results Section, the simulation results are presented.
DFIG wind turbine system description and energy conversion
The DFIG wind turbine system is depicted schematically in Figure 1. The stator is directly linked to the three-phase power grid, while the rotor is fed at a variable frequency through the AC-DC-AC converter ().
FIGURE 1
Modeling of the wind turbine
Only a small portion of the acquired wind energy can be converted by a wind turbine; thus, the wind’s mechanical power is calculated as follows (; ):where is the wind energy extracted (power), is the air density (kg/m3), R is the blade radius (m), is the wind speed (m/s), and is the turbine power coefficient as a function of speed ratio and blade pitch angle β given as
The blade’s tip-speed ratio is defined as follows:
It is preferable for the generator to have a power characteristic that follows the maximum line (Figure 2) in order to convert the maximum amount of wind power. For a speed ratio , the power coefficient Cp achieves a maximum of 0.5506 for a blade pitch angle of β = 0°.
FIGURE 2
The control block’s rotor speed and active power references were provided by maximum power point tracking (MPPT), while the reactive power was set to zero to obtain a unity power factor ().where Pgrid−refis the electrical grid's active power reference, Ωr−ref is the rotor speed reference, λopt is the optimal speed ratio, and Cp−max is the maximal power coefficient.
Model of the DFIG
The DFIG model is commonly characterized by the following state equations in the synchronous orientation frame when the d-axis is aligned with the stator flow vector (Figure 3) and by disregarding the stator resistance ():where are the dq rotor voltage and current components, respectively. Rr is the resistance of the rotor winding, and Lr is the rotor winding inductance. ωsand ωr are, respectively, the stator and the rotor angular velocities. and are, respectively, the generator slip and dispersion ratio.
FIGURE 3
In the Park transformation of the rotor quantities, the relative angle between the rotor and the Park reference (dq-axis) is used, which is determined as , , is the grid frequency.
The generator torque can then be expressed by
The following formulas give active and reactive stator powers:
The mechanical equation iswhere is the aerodynamic torque after the gearbox.
Proposed robust nonlinear generalized predictive control
The predictive control idea is to create an anticipatory effect using the explicit knowledge of the current state (; ). The classical cost function is given as
In this study, a robust nonlinear generalized predictive control (RNGPC) is suggested to solve the robust control problem by revising the classical cost function (9). Consider the nonlinear multivariable system:
where x and are the state vector and control, respectively. The function f(x) and h(x)are assumed to be continuously differentiable a sufficient number of times, and B(x)is a continuous function of x. The predictive command will be optimal if the novel cost function I(x,u) is minimized.with
To solve the nonlinear optimization problem (11), the predicted term )th order Taylor series expansion.
Replacing (12) in (13)
Since , it follows that
The Lie derivation of function h(x) along a vector field f(x) is indicated by
Invoking (15) and (16) with (14) yieldswith
It is possible to write (17) in the following format:where
If all output systems have the same relative degree r,where and , is the diagonal of the matrix identity.
Replacing (19) in (11), the expression of the novel cost function will bewhere
The condition that must be met in order to achieve optimal control is given by
According to (20) and (21), the optimal control is calculated as follows:
If n = m, then the optimal control can be indicated as follows:where
An integral action is included in the nonlinear predictive controller (23). As a result, the suggested controller completely eliminates the steady state error even in the presence of unknown perturbations.
The control system’s proposed topology consists of two loops: an inner loop that controls torque and direct current and an outside loop that controls DFIG speed (Figure 4). The rotor speed reference is specified in (4). To reduce the control effort, the reference speed signal is passed via a second-order linear filter, which is provided bywhere wn= 5 and =0.7.
FIGURE 4
Predictive control of electromagnetic torque and current (inner loop)
The electromagnetic torque reference is calculated from the speed controller (outer loop), and to keep the DFIG reactive power at zero, the d-axis rotor current reference is determined. From (5), the electrical equations are written in the matrix form as follows:with , , and
The following are the outputs that will be regulated in the inner loop:
The relative degrees are equal to 1; it follows thatwith
Thus, as follows, the predicted term is expanded into a second-order Taylor series expansion:wherewhere and is the diagonal of the matrix identity (2*2). Replacing (28) in (23), the optimal nonlinear control law is determined aswhere .
Predictive control of the speed (outer loop)
The DFIG’s mechanical dynamics are given by the equation below:where , and .
We do not consider the aerodynamic torque in the mechanical Equation 8. The relative degree of the outputs is equal to 1, and it follows thatwhere .
A second-order Taylor series expansion is used to expand the expected term :where
Replacing (32) in (23), the optimal control is calculated aswhere
Closed-loop system stability
The convergence of the output tracking error to the origin can be used to demonstrate the closed-loop system’s stability. Substituting control law (33) into Equation 31 yieldswhere Kp is the proportional gain, Ki is the integral gain, and Kd is the derivative gain.
As a result of (34), the closed loop system’s characteristic polynomial equation is given as
Computing the solutions of Equation 35 may be used to determine the stability condition that is given as follows:
The true parts of the roots (36) are positive because the predicted time T is positive can be proven to be negative.
Simulation results
The operation performance of the DFIG wind system and control scheme has been investigated using Matlab/Simulink software. Table 1 lists the system’s parameters. The prediction horizon is set to Tp1 = 0.5 ms (inner loop) and Tp2 = 5 ms (outer loop). Figure 5 depicts the wind speed profile utilized in the simulation.
TABLE 1
| Parameter | Value |
|---|---|
| Generator rated power | 1.5 Mw |
| Nominal voltage | 690v |
| Nominal frequency | 50 hz |
| Poles | 2 |
| Rotor resistance | 0.021 Ω |
| Stator resistance | 0.012 Ω |
| Rotor inductance | 0.0137 H |
| Stator inductance | 0.0137 H |
| Mutual inductance | 0.0135 H |
| Coefficient of friction | 0.0071 |
| Moment of inertia | 50 kg m2 |
| Turbine diameter | 60 m |
| number of blades | 3 |
| hub height | 85 m |
| Rayon | 36.5 m |
| Gearbox | 90 |
| Filter resistance | 1Ω |
| Filter inductance | 30 mH |
| Capacitance | 500 µF |
| Nominal voltage | 500v |
Parameter values.
FIGURE 5
Reference tracking test performance
Figures 6, 7 show that the rotor speed and electromagnetic torque follow their references correctly and with good performance. Figure 8 depicts the DFIG slip’s'. As a consequence, in Figure 9, the DFIG active power flawlessly tracks its reference to enhance the conversion efficiency. From Figure 10, it can be noticed that the DFIG reactive power is kept at zero, which helps to adjust for the grid power factor. The rotor voltage-current at sub-synchronous (s > 0), synchronous (s = 0), and hyper-synchronous modes (s < 0) is plotted in Figure 11. These results clearly show that the proposed RNGPC algorithm provides great performance in both steady state and dynamic modes.
FIGURE 6
FIGURE 7
FIGURE 8
FIGURE 9
FIGURE 10
FIGURE 11
Robustness under DFIG’s parameter and prediction horizon variations
The values for the parameter variations included in the DFIG model are as follows: 10% in the coefficient of friction at t = 7s and 25% in the rotor resistance at t = 10s. This test is also used to assess the controller’s performance when the prediction horizon is changed. Only the outer loop Tp2 prediction horizon is lowered from 15 ms to 1.5 ms. These variations are not taken into account in the controller.
Figures 12, 13 show that despite the DFIG’s parameter fluctuation, the system response converges to the reference values. Indeed, the speed and the electromagnetic torque track perfectly their references. From Figures 14, 15, the control effort is limited by the speed filter; thus, a smaller prediction horizon results in faster disturbance rejection.
FIGURE 12
FIGURE 13
FIGURE 14
FIGURE 15
Conclusion
A robust nonlinear generalized predictive controller for a DFIG wind turbine system has been devised in this study. A novel finite horizon cost function is introduced into the control law to ensure robustness against aerodynamic torque and parameter fluctuations without the necessity for an aerodynamic torque observer. The active power delivered to the grid varies with wind speed and follows the optimal wind power precisely. The simulation results show that the control architecture is resistant to unknown aerodynamic torques. This supports the suggested control’s efficiency and dependability in tracking the projected references. The RNGPC’s efficacy and resilience are confirmed by simulation results under a variety of operating conditions and DFIG parameter variations.
Statements
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.
Nomenclature
- ΩtΩr
Turbine, generator speed (rd/s)
- Tt,Tr
Aerodynamic and generator torque(N.m)
- Ωr-ref
Reference generator speed
- G
Gear ratio
- Pgrid -ref
Reference grid active power (W)
- η
System (wind turbine + DFIG) efficiency
- Pr, Qr
Active and reactive rotor power (W,Var)
- Vs
Stator voltages (v)
- Rr, Lr
Per-phase rotor resistance and self-inductance
- M, P
Mutual inductance, number of pole pairs
- J, fr
Moment of inertia, coefficient of friction
- s,σ
Generator slip and dispersion ratio
- Tp
Predictive time
Relative degree of output i
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Summary
Keywords
DFIG wind turbine systems, robust generalized predictive control, doubly fed induction generator, wind, controller
Citation
Ouari K, Belkhier Y, Djouadi H, Kasri A, Bajaj M, Alsharef M, Elattar EE and Kamel S (2022) Improved nonlinear generalized model predictive control for robustness and power enhancement of a DFIG-based wind energy converter. Front. Energy Res. 10:996206. doi: 10.3389/fenrg.2022.996206
Received
17 July 2022
Accepted
01 August 2022
Published
23 September 2022
Volume
10 - 2022
Edited by
Dongran Song, Central South University, China
Reviewed by
Feihhang Zhou, Xi’an University of Posts and Telecommunications, China
Jian Yang, Central South University, China
Updates
Copyright
© 2022 Ouari, Belkhier, Djouadi, Kasri, Bajaj, Alsharef, Elattar and Kamel.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Kamel Ouari, kamel.ouari@univ-bejaia.dz; Salah Kamel, skamel@aswu.edu.eg
† These authors have contributed equally to this work
This article was submitted to Smart Grids, a section of the journal Frontiers in Energy Research
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.