Microsoft Word - ETASR_V13_N2_pp10588-10594 Engineering, Technology & Applied Science Research Vol. 13, No. 2, 2023, 10588-10594 10588 www.etasr.com Van et al.: Chaos Control of Doubly-Fed Induction Generator via Delayed Feedback Control Chaos Control of Doubly-Fed Induction Generator via Delayed Feedback Control Co Nhu Van Department of Cybernetics, University of Transport and Communications, Vietnam | Institute for Control Engineering and Automation, Hanoi University of Science and Technology, Vietnam vancn@utc.edu.vn Nguyen Thanh Hai Department of Electronic Engineering, University of Transport and Communications, Vietnam nguyenthanhhai@utc.edu.vn (corresponding author) Nguyen Phung Quang Institute for Control Engineering and Automation, Hanoi University of Science and Technology, Vietnam quang.nguyenphung@hust.edu.vn Received: 24 February 2023 | Revised: 10 March 2023 | Accepted: 11 March 2023 ABSTRACT With the increasing wind power penetration, wind farms are directly influencing the power systems, so the need to improve the quality of the system is an open research topic. A Doubly-Fed Induction Generator (DFIG) is often used in wind power systems. However, DFIG has a complex structure and often works in harsh environments, so potential faults may occur. Faults can cause the system to fall into a chaotic working state, which is a harmful phenomenon for DFIG, since it makes the operating quality of the system worse, even leading to system destruction if not fixed on time. This study presents simulations of the chaotic phenomenon that occurs for a DFIG under specific working conditions based on Lyapunov’s exponents. The delay feedback controller is designed, and along with the selection of the appropriate controller parameters, the chaotic phenomenon is quickly eliminated, bringing the system back to stable operation. Keywords-chaos; chaos control; DFIG; DFC; Lyapunov’s exponents I. INTRODUCTION Chaos can be beneficial as it may speed up chemical reactions, enhance mixing, provide a powerful mechanism for heat and mass transfer, etc. However, in many situations, chaos is an undesirable phenomenon, it can lead to increased mechanical fatigue for irregular oscillations, and non- convection energy absorption in a turbulent regime can lead to system parameters exceeding the safe level. Chaos is a phenomenon that only occurs in nonlinear systems, sensitive to initial conditions, and non-periodic but obeying certain laws. A dynamic system with chaotic motion needs to meet the following conditions [1-2]:  The system has at least three independent variables.  The equation of motion must have nonlinear terms. In addition, the phase space of the system must have a dimension of not less than 3 to ensure the existence of divergent trajectories, be confined to a finite domain of the dynamics space, and ensure uniqueness of the orbit. First of all, we need to note that in a linear dynamic system, chaos never occurs. So, when we talk about chaos, we mean nonlinear systems. However, not all nonlinear systems have chaotic motion. There are different methods of detecting chaos such as the Fourier analysis method, time responses, phase portraits, bifurcation diagram of the maximum value of the state variable over time, and calculations of Lyapunov exponents used to illustrate the chaotic behavior when changing the characteristic value. The main interest of this paper is focused on the Lyapunov exponent because it can be calculated with relative ease and it provides confirmation of the presence of chaos in the observed data [3]. When the average Lyapunov exponent is calculated as positive, it indicates chaotic behavior occurring in the system. From that, the chaotic working area of the object is drawn, and a control method is proposed to bring the system to a stable working state, Engineering, Technology & Applied Science Research Vol. 13, No. 2, 2023, 10588-10594 10589 www.etasr.com Van et al.: Chaos Control of Doubly-Fed Induction Generator via Delayed Feedback Control extinguishing the self-sustaining oscillations with high amplitude and abnormal changes. Due to the harsh working environment of wind farms, the DFIG parameters can change with temperature, time, load conditions, etc. DFIG is prone to faults such as gearbox faults, power converter faults, stator winding faults, rotor winding faults, velocity sensor faults, etc. [4-7], from which the system may fall into a state of chaotic operation, leading to poor working quality, which can be the cause of problems and failures. Among electrical failures, insulation failures in the stator windings cause the majority (30%-40%) of induction machine failures, including DFIGs [8-10]. Although a DFIG has a complex structure, prone to failure, it is still the most used structure in wind power systems due to its outstanding advantages. Therefore, there have been many research works put forth to improve the quality and performance of the system [6, 11-16]. The research on bifurcation and chaos for the DFIG system is very limited [17-23] and does not cover all the potential risks. The above studies have shown that, under certain operating conditions, a DFIG can be chaotic. Authors in [18, 20] evaluate that phenomenon based on computer simulations but have not evaluated it on a solid theoretical basis, or do not give specific conditions on which parameter changes in the system lead to chaos [20]. Authors in [21] analyzed the phenomenon of chaos and presented several risks that make the DFIG system chaotic. On the basis of theoretical analysis (algorithm to test 0-1) and through simulation, authors in [23] evaluated the stability and chaos of the system. The study also gives specific conditions of the parameter set causing chaos for the DFIG, however the condition of 3 simultaneous faults (stator winding fault, rotor winding fault, and rotor speed sensor fault) rarely occurs in practice. This study demonstrates that DFIG faces a chaotic phenomenon when the system parameters change in case of stator winding failure, based on theoretical analysis and simulations. From there, the delay response controller will be designed to eliminate chaos, return the system to stability and avoid possible risks. II. MATHEMATICAL MODEL OF DFIG, CHAOS CONTROL, AND DELAY FEEDBACK CONTROL METHOD A. Mathematical Model of DFIG One of the main control objectives for the DFIG is the independent control of active and reactive power through decoupled control of rotor currents ��� and ��� [24]. The starting point for deducing the state-space model of the DFIG is the voltage equations for the stator and rotor windings [24-25]: Stator voltage in stator winding system: ��� = � �� + �� �� (1) Rotor voltage in rotor winding system: ��� = � �� + ������ (2) Rotor and stator flux: ��� = �� � + �� � �� = �� � + �� � (3) where: ��� = �� + ��� �� = �� + ��� Converting (1) and (2) to the dq coordinate system, for any internal rotation with angular speed �� we get: � �� = � � + �� �� + ���ψ��� = � � + ����� + ���ψ� (4) where �� = � + �� The generator stator is connected to the grid with constant voltage and frequency. Since the stator frequency is always identical to the grid frequency, the voltage drop across the stator resistor can be neglected in comparison with the voltage drop across the mutual inductance �� and the dissipation inductance ��� stator, which can be written as follows: �� = � � + �� �� => �� ≈ �� �� or �� ≈ �ω��� (5) From (3), we obtain: � = ! (Ψ� − �� � ) &� = �� � + �� ' After suppressing the stator current � and the rotor flux &� , from system (4) we get: ⎩⎪ ⎨ ⎪⎧ � ,�� = − � - .� + /�. 0 1 − ��� 1+ /�� - . + ��0 �23 + �!� �1 + /��!4 �2 �567�� = . 1 − - . + ���0 523 + !4 �2 (6) where 523 = &� /�� , 9 = 1 − ��; /(���� ) , <� = �� / � , <� = �� / � . After decoupling both equations into real and imaginary components, we obtain the complete system of electrical equations of DFIG (7). ⎩⎪ ⎪⎪ ⎪⎨ ⎪⎪ ⎪⎪ ⎧ �>�?�� = − � - .� + /�. 0 ��� + �� ��� + /�� - . &��3 − �&��3 0 + �!� ��� − /��!4 ��� �>�@�� = − � - .� + /�. 0 ��� − �� ��� + /�� - . &��3 + �&��3 0 + �!� ��� − /��!4 ����A ?7�� = . ��� − . &��3 + ��&��3 + !4 ��� �A @7�� = . ��� − . &��3 − ��&��3 + !4 ��� (7) In grid voltage, the orientated coordinates are: ��� = �� , &��3 = 0, &��3 = &�3. Engineering, Technology & Applied Science Research Vol. 13, No. 2, 2023, 10588-10594 10590 www.etasr.com Van et al.: Chaos Control of Doubly-Fed Induction Generator via Delayed Feedback Control Summarizing the equation system (7) yields the following state space model for the DFIG in the grid voltage orientated reference frame (8): �D�� = ED + F� �� + F� �� (8) with: State vector D. = G��� , ��� , &��3 , &��3 I , stator voltage vector ��. = G��� , ��� I, as the input vector on the stator side, and stator voltage vector ��. = G��� , ��� I as the input vector on the rotor side. The system matrix A, the rotor input matrix F� and the stator input matrix F� may be written as: (9) B. Chaos Control and Delayed Feedback Control Method Every parameter of the system has the potential to cause bifurcation and chaos, the idea of control is to bring these two properties towards its purpose, by adjusting the system parameters, or to hit the target by methods such system parameters transformation, changes in the structure of the system, input stimulus from outside, change the controller, or a control method based on the system's strange attractor set when a chaotic phenomenon occurs, bringing the trajectory to the target orbit with the new set parameter combination compatible with the old parameter. To do so, it is necessary to know the rate at which chaos or exponential growth increases and the oscillation amplitude of the system state variable. From there, a way to approach is by using accurate measurement equipment of system state variables and thus determining the stable and unstable working parameter areas, thereby determining the appropriate control method and the way to change the structure of the system if necessary. There are many methods of chaos control applied to electric drive systems and they have certain effects. The chaotic control method based on the delay feedback controller has been studied extensively [1, 26-30]. The delayed feedback control method, described in [31], is: J(K) = LMN(K − O) − N(K)P = LQ(K) where τ is the delay time. If this time coincides with the period of i th periodic orbit O = <> , i.e. N(K) = N> (K), it does not change the periodic orbits in the system. Choosing the appropriate weight K of the feedback, we can achieve the stabilization of the system. The delayed feedback control method is less complex and more responsive than other control methods. It allows for non- invasive stabilization in the sense that the control force disappears when the state has reached the target. Authors in [32] showed that the simple self-feedback delay controller gives better results than the sliding mode self-feedback delay controller when applied to control chaotic behavior in the IFOC of a 3-phase induction motor. Today, the DFC has become one of the most popular methods in chaos control research [33]. So far the number of published works on chaos control for DFIG is very limited and has not covered all the problems. To the best of our knowledge there are no works that apply DFC method to control the DFIG. This study aims to fill that gap. III. CHAOTIC PHENOMENA AND CHAOS CONTROL FOR DFIG A. Chaotic Behavior Analysis of DFIG It can be seen from (7) that DFIG is a multivariable and nonlinear system (which is a necessary condition for a chaotic motion system). DFIG contains many parameters and the system works in harsh environments, therefore, the parameters can vary according to temperature, time, and system operating conditions. When the parameters of the system change, the system may appear to be chaotic. To clarify this issue, this study presents the chaotic phenomenon that occurs in DFIGs in cases of stator winding failure (this is the problem that accounts for the majority of electrical problems for DFIGs [8-10]). From the equation of motion of the rotor [6, 17], we combine with the first two equations of (7) to obtain a system of equations representing the rotor currents (ird, irq) and the rotor speed of DFIG as follows: ⎩⎪ ⎪⎪ ⎨ ⎪⎪⎪ ⎧ �>�?�� = − � - .� + /�. 0 ��� + (�� − �)��� − /�� � A !4 � �!� ��� $ /��!4 �� �>�@�� � $ � - .� � /�. 0 ��� $ "�� $ �%��� � /�� . A !4 � �!� ��� �R�� � S; TUV !4A W! ��� $ XW � $ TUW