Evaluation of the possibility of chaos for doubly-fed induction generator in wind power generation system

ABSTRACT


INTRODUCTION
DFIG in the power generation system has the outstanding characteristics that the stator side is directly connected to the grid and the rotor side is connected to two back-to-back voltage source converters to the grid.Because the control device is located in the rotor, the power of the control device is much smaller than the generator power (in the limited speed range, the power of the converter is only 30% of the power transmitted to the grid [1], [2]), which is economically attractive, especially when the generator power is large.The system is capable of operating in a slip coefficient in a fairly wide range (±33% compared with the synchronous  [3], [4] allowing to make good use of the power source hybridized by the main machine, which is working in either over synchronous or sub synchronous mode.In both modes, the machine supplies active power P to the grid on the stator side, and on the rotor side, the machine draws reactive power Q from the grid in the sub synchronous mode and returns the reactive power Q to the grid in the over synchronous mode.
Due to the great benefits of DFIG, this configuration has become now very popular for variable-speed wind turbines, so there are many control methods for DFIG to perfect the system, typically studied [5]- [10].However, in industrial practice, reliability is extremely important, therefore, it is necessary to study more deeply and widely to help the system achieve the best efficiency and quality.The parameters of DFIG can change with temperature, life, and load conditions.Due to the challenging working environment of wind farms.DFIG is vulnerable to faults like gearbox faults, power converter faults, stator winding faults, rotor winding faults and velocity sensor faults.The word [11]- [13], from which the system may fall into a state of chaotic work, which results in poor system working quality and is the reason for issues and failures.
Recently, the research investigation into the bifurcations and chaos of motor drives and permanent magnet synchronous generators has received much attention [14]- [19], enabling a better comprehension of the system's dynamical activities, which may provide some information that is helpful for actual control and design work.However, research on chaos for DFIG is only a few scattered studies, that do not cover all issues of chaos in DFIG.While DFIG is considered complex, the parameters of DFIG can vary with temperature, life, environment, load conditions, etc., so the system may fall into a chaotic behavior, which makes the system's working quality poor, which is the main cause of system failure.Thus, it is necessary to have an overview and study more fully, applying chaos to control DFIG for wind power generators will lay the foundation for new research, thereby providing appropriate control structures to complete the system.
The paper is organized as follows: i) The overview of the chaos phenomenon and the identification to control chaos will be presented in section 2; ii) Section 3 will present some mathematical models of DFIG as a basis for implementing section 4chaos in DFIG, to perfect the system, a controller is introduced to eliminate or avoid the phenomenon of chaos; iii) The discussion will be done in section 5; and iv) Finally, section 6 presents the conclusions the of study.

CHAOS AND APPLICATIONS IN CONTROL
In the real world, any system is disturbed by external noise, the oscillations of systems are always nonlinear, the linearization of oscillation processes is nothing more than a simplification of the real oscillation problem, and in computer simulations, a small perturbation appears due to numerical round-off.A little perturbation that occurs in a chaotic system accumulates exponentially over time, dramatically altering the system's behavior [20], [21].There are many chaotic regimes in nature and in human-made devices, chaotic dynamics is one of the most general methods of the evolution of nonlinear systems.When chaos intensifies mixing and speeds up chemical reactions, it is advantageous because it creates a strong mechanism for the transfer of mass and heat.Chaos is a common undesired phenomenon that can, for instance, lead to additional mechanical fatigue of the elements of construction due to their irregular vibrations.In a chaotic regime, there is a chance that no resonant energy absorption will push the system parameters over safe limits.
Chaos brings undesired results and can be self-maintaining, affecting the quality of the system.This phenomenon only manifests in nonlinear systems, is nonperiodic, sensitive to initial conditions, and follows certain laws that are not the same as noise.It was not until the last decades of the 20th century that chaos theory began to be deeply explored for powertrain systems.Because the input and output variables of the powertrain vary from the time, most of the calculations are instantaneous, to simplify the control problem we have to accept the idealized input variable and treat the drivetrain parameter as constant.However, that turns them into noise, which causes chaos in the system for a while.With the control requirements set out in the drive array, the accuracy of response, and stable operation, the application of chaos theory about the control object to increase reliability is very feasible.
The most crucial aspect of chaos motion is that it is highly sensitive to the initial conditions of the system.That is, a very small difference in the input that is amplified will make a very large difference in the output.This is shown very clearly by the simple system of (1) of Lorenz-1963 [22] when the initial conditions are different (Figures 1).
The parameters have the values σ = 24.1,ρ = 10, ꞵ = 8/3, Figure 1 shows that, with very little different initial conditions, from the value [0 1.79980000000000000002 0] to the value [0 1.7998000000000001 0], but the state of the system has changed from a steady state (Figure 1(a)) to a chaotic state (Figure 1(b)).Basically, a chaotic system has the following properties [23]- [25]: − Nonlinearity: Chaos only occurs in nonlinear dynamical systems or systems that exhibit a certain degree of nonlinearity.
Int J Pow Elec & Dri Syst ISSN: 2088-8694  − Determinism: chaos can be predicted by simple deterministic equations, determining the domain of parameters corresponding to the chaotic solution.That is, chaos obeys one or more deterministic equations, without an element of randomness, or probability.− Sensitive to initial conditions: A small change in the system's initial state can result in drastically different final-state behavior.As a result, even though the behavior of the system is determined by deterministic fundamental laws, it is impossible to anticipate it over the long term.− Aperiodicity: Chaotic orbits are aperiodic, but follow a certain law or principle.People are often interested in a special form of this family called the attractor.The orbits wander forever in an internal domain, without reaching a fixed point or a closed orbit, they are attracted to a complex geometrical body called the strange attractive, which is a chaotic phenomenon [20].In addition to being a strange mesomorphic structure, the chaotic attractor is also dynamically strange.The orbits are attracted to the attractive, but they are unstable on that set and are sensitive to initial conditionsa characteristic of chaos motion.Once inside the strange attractor, the phase portraits are confined to it and can approach any point arbitrarily attractor to the suction set but never repeat the same at a later.The phase trajectory is not stable anywhere on the strange attractor, but overall, the attractor is very stable as shown in Figure 2. We must first recognize that chaos never exists in a linear dynamic system.Thus, when we refer to chaos, we actually mean nonlinear systems.Chaotic motion is not present in all nonlinear systems, though.The Fourier analysis method, time responses, phase portraits, bifurcation diagrams of the maximum value of the state variable over time, and calculations of Lyapunov exponents are a few of the techniques used to identify chaos.These techniques are used to show the chaotic behavior when changing the characteristic value, to be able to identify the chaotic working area of the object, that propose controls method to eliminate the self-sustaining oscillations with high amplitude and irregular variation, return the system to a stable working state.As analyzed, the DFIG system has a complex structure, especially DFIG in wind powers working in unstable conditions from wind energy, which easily leads to a chaotic state.Therefore, identification and elimination from the "third state" in wind power generation systems using DFIG to increase reliability are important goals of this research.

STATE EQUATION OF DFIG
Decoupled regulation of active and reactive current components is one of the key control goals.This advises using a reference frame for the control architecture that is stator voltage orientated.The real axis of the grid voltage vector uN might be chosen as the d axis in generating systems like wind power plants where the stator is directly connected to the grid.The following relations of the DFIM are obtained [26]: where Where  = 1 -  2 /(    ),   =   /  ,   =   /  .In grid voltage orientated coordinates:   = 0;   =   ;   ′ = 0;   ′ =   ′ and: The rotor current component   acts as a control variable for the generator torque mG and respectively for the active power P, because the stator flux   is governed by the grid voltage and can be seen as constant.The power factor cosφ or the reactive power Q can be controlled by the control variable irq.The system of (3) shows very clearly the multivariable and nonlinear of the system, which is a necessary condition for the system to be chaotic.Thus, when the DFIG system operates under certain conditions, it can lead to chaotic behavior.This issue will be considered in detail in content 4 and content 5 below.

CHAOTICS PHENOMENA IN DFIG 4.1. Some causes of DFIG working unstable, bifurcation, and chaotic behavior − When the controller parameter changes
From the classical control structure PI for DFIG in [27], based on theoretical analysis, from the Jacobian matrix, it is possible to determine the conjugate complex solutions whose real parts correspond from negative to less negative and become positive, as shown in the Table 1.One pair of complex eigenvalues' real parts become less negative as   increases, and at a critical value of   , the real parts switch from negative to positive, leading to Hopf bifurcation.

− When the mechanical torque changes
The research of Yang et al. [28] has determined that the DFIG wind generator is capable of bifurcation to instability when the mechanical torque Tm increases to characteristic values through simulation and theoretical analysis.On the basis of theoretical analysis, the loci of the Jacobian's eigenvalues are computed and the analysis shows that the system loses stability via a Hopf bifurcation, there exist four pairs of complex conjugating eigenvalues as Tm is varied.From Table 2 we see, for a small Tm value, all four pairs have a negative real part; as Tm increased, the real part of the eigenvalues pairs became less negative and at the critical value of Tm the real part changes from negative to 0; As Tm continued to increase, the real part became positive and the system became unstable.This suggests that the instability is caused by Hopf bifurcation.− When DFIG is connected to a "weak" grid with unbalanced loads and parasitic At low wind speeds, bifurcation has been identified in the DFIG wind turbine, which manifests as a low-frequency oscillation of the DC link capacitor voltage of the system when the DFIG wind turbine is coupled to a "weak" grid that has parasitic and unbalanced loads, and if the wind speed further increases, chaotic behavior may emerge [29].− When the parameters of DFIG vary This is a common phenomenon for DFIG during operation, and there have been studies demonstrating that chaos occurs when the parameters change to a certain value [30]- [32].In [31], from system (5).

Demonstration of DFIG's chaotic phenomenon based on theoretical analysis and simulation 4.2.1. State equation of DFIG
From the equation of motion of the rotor [27]. where: (    -    ).The control method is based on the grid voltage orientated coordinates, we have   = 0,   =   , From (2) we can rewrite in ( 6) as (7).
From the first 2 equations of the system of (3), combined with (7) we get the system of equations representing the rotor current on the rotating dq coordinate and the rotor speed of DFIG as ( 8 where  3 lists the key system parameters that we employed in our simulations.Figure 3 and Figure 4 illustrate when the system is operating under normal conditions.Figure 3

The chaotic phenomenon is determined based on theoretical analysis
In the event of a rotor winding failure, which results in the rotor resistance and inductance changing to a different value (other system parameters remain unchanged): Rr = 15 ; Lr = 38.6 H.The equilibrium points of the system are determined by solving the system of ( 9), getting the following results: Substituting the parameters in (10) and solving, we get an equilibrium point: E(0; -16310; 314,2).Stability evaluation of equilibria through Jacobian matrix: From (9) we get: The eigenvalues of the Jacobian matrix calculated at the equilibrium point are what determine the stability of the equilibrium point.It is customary to solve equation det[λI-  ] = 0 for .
With the equilibrium point E(0; -16310; 314,2) the solution of ( 11) is determined: We see,  1 ,  2 are conjugate complex solutions with positive real part, so the equilibrium point  1 is unstable.Thus, the system has 1 unstable equilibrium point.Next, we determine the Lyapunov exponent to 1972 determine whether the system is chaotic or not.From the system of (9) with the given set of parameters, we can determine that the system has 3 Lyapunov's exponent ( 1 ,  2 ,  3 ) with the same value as Table 4.We see, the system always exists at least 1 Lyapunov's exponent that is always positive, so the system has a chaotic phenomenon.For more clarity, the following simulations are performed in the following content.

The chaotic phenomenon is determined through simulation
From the set of parameters of the system given in section 4.2.3.The simulation results show that the rotor speed fluctuates violently and does not follow any rules as shown in Figure 5(a), component amperage   ,   oscillates around the equilibrium point, the trajectory does not repeat the past trajectory as shown in Figure 5

ELIMINATE CHAOTIC PHENOMENA TO STABILIZE THE SYSTEM
The chaotic phenomenon for DFIG is when the parameters of the system change to a certain value, which has been clearly shown through simulation and theoretical analysis (as shown in section 4.2).Chaos is harmful to the DFIG system, so it is necessary to design the controller to avoid or eliminate it when it occurs.The content below will take care of those issues.
When chaos occurs, the controller's task is to bring the trajectory to the equilibrium point, we use the following control law: where ̅ 1 , ̅ 2 , ̅ 3 are equilibrium points determined from (10).According to the Hurwitz stability criterion, from the conditions for the system to be stable, we can choose the parameters:  1 = 520,000;  2 = 1;  3 = 1: i) Control to eliminate the chaotic phenomenon.When chaos occurs, at 0.4 seconds the controller is inserted.Through simulation, we can see that the controller has eliminated the chaotic phenomenon, bringing the system to stable operation as shown in Figure 7(a); and ii) Control to avoid chaos: Once the boundary has been determined, the DFIG system can fall into a chaotic working state.The chaos avoidance controller will be included in the system from the very beginning.Based on the controller has been defined and the parameters have been selected.The simulation results show that the system operates stably as shown in Figure 7

CONCLUSION
Due to DFIG's many advantages, variable-speed wind turbines are now increasingly common in this design.However, research on chaos for DFIG is only a few scattered studies, that do not cover all issues of chaos in DFIG.therefore, it is necessary to study more deeply and widely to make the system more complete.The paper has summarized the errors that can cause DFIG to fall into the phenomenon of bifurcation and chaos, to get a general picture.At the same time, through analysis on a theoretical basis and through simulation, which has demonstrated the chaos for the system when the rotor winding fails, the working area causing chaos of the system can be determined, thereby facilitating system parameterization to build a controller to avoid or eliminate that chaos, making the system work more reliably and with better quality.

Figure 1 .Figure 2 . 3 −
Figure 1.Simulation results of (1) for (a) system is stable and (b) the system appears to be in a chaotic state

2 .
: x1 =   ; x2 =   ; x3 = ; c1 = − With the normal operating mode of the system Table (a) is time series diagrams of   , Figure 3(b) is phase trajectory diagram between   and .

Figure 4 (Figure 3 .Figure 4 .
Figure 3.The system is stable under normal conditions (a) time series diagrams of   and (b) phase trajectory diagram between   and

Figure 5 .Figure 6 .
Figure 5.The system works chaotically when an error occurs (a) time series diagrams of  and (b) phase trajectory diagram between   and (b).

Figure 7 .
Figure 7. Simulation results when the controller is inserted for (a) time series diagrams of  and (b) time series diagrams of

Table 3 .
Main parameters of the DFIG system