Investigation of chaotic behavior and adaptive type-2 fuzzy controller approach for Permanent Magnet Synchronous Generator (PMSG) wind turbine system

1.   Introduction

Recently, the amount of wind energy produced has increased at a quicker rate in order to safeguard the environment from pollution by reducing the reliance on fossil fuel ^[1,2]. Different categories of generators such as PMSG ^[3,4], Squirrel Cage Induction generators (SCIG) ^[5,6], multi-phase induction generator (MPIG) ^[7,8], doubly salient electromagnetic generator (DSEG) ^[9,10] and double feed induction generator (DFIG) ^[11,12], can be employed by the wind turbine to produce electrical energy.

The effective utilization of wind turbines, which transforms wind mechanical energy into useful electrical energy has led to significant application of physics in this area ^[13]. Particularly, the use of PMSG in wind turbine exhibit significant advantages in terms of high efficiency and low maintenance ^[14,15]. Based on this, many studies have been directed towards the control of PMSG ^{[16,17,18,19,20]}.

Several investigations have shown that PMSG produces chaotic behavior that can be well controlled. For instance, Ren et al. ^[21] investigated the dynamic motion of the wind turbine PMSG using optimal control theory. The authors demonstrated the efficiency of the control system. Messadi et al. ^[22] addressed the problem of optimality of angle and speed of the turbine blade in PMSG using predictive control approach with the purpose of generating the maximum power. Si et al. ^[23] proposed a delayed feedback control PMSG model of a wind turbine in fractional-order and an extensive study in terms of non-linear dynamics were conducted. The study demonstrated how the parameters of different system can influence the control result of the system. A sliding mode control approach using state feedback on a PMSG was presented by Alamdar and Balochian ^[24]. The authors discovered that the controller can monitor any desired point and can further converge a chaotic system to zero. Hu et al. ^[25] investigated an adaptive control approach integrating the type-2 sequential fuzzy neural network (T2SFNN) and chaotic motion for the PMSG. The authors illustrated how the control strategy can tame chaos by demonstrating the effectiveness of the strategy. For the PMSG-based wind energy conversion system, Kahla et al. ^[26] proposed the maximum power extraction frameworks operating the cutting-edge optimization approaches using the feedback linearization control strategy and the fractional control theory. They demonstrated the efficacy of the suggested 3-kW PMSG control system, based on a wind energy conversion technology. An event-triggered neural adaptative backstepping control approach for the K chaotic PMSG coupled system was presented in Luo et al. ^[27] The study discovered that the Lyapunov function proves the stability of the suggested scheme, and simulation results demonstrated the efficiency of the method.

Recently, the topic of chaotic Permanent Magnet Synchronous Generator (PMSG) based the conversion system of wind energy has been studied intensively. Takhi et al. ^[28] studied the problem of control and synchronization of the chaotic PMSG system which contains perturbations and uncertainties using predictive control. The authors further applied Arduino boards to verify the approach via microcontroller implementation. Dursun et al. ^[29] presented maximum power extraction frameworks operating the state-of-the-art optimization methods for permanent magnet synchronous generator (PMSG). This was achieved using the wind energy conversion system (WECS). Shanmugam and Joo ^[30] investigated the stabilization analysis for nonlinear permanent magnet synchronous generator (PMSG)-based wind turbine system under fuzzy-based memory sampled-data (FBMSD) control scheme.

To the best of our knowledge, the adaptive type-2 fuzzy controller in the chaotic PMSG model has not been studied. Therefore, due to its effectiveness and performance, this study tends to investigate the Adaptive Type-2 Fuzzy Controller method for controlling the chaotic PMSG based on the conversion system of wind energy.

With regard to the aforementioned discussions, we highlight the main contribution and the novelty of this study are as follows:

a) Investigating chaotic behavior in the PMSG based on the wind turbine system using Lyapunov spectrum, bifurcation model, Poincare map, and complexity theory.

b) This work further studied the mathematical model and numerical simulation of the adaptive type-2 fuzzy controller for chaotic PMSG a wind turbine system.

The other parts of this study are structured as follows. Section 2 present the mathematical model of the PMSG based on the wind turbine system. Section 3 investigate the nonlinear dynamics of PMSG based on different system orders. Section 4 present an adaptive type-2 fuzzy controller of the PMSG model. Finally, Section 5 concludes the findings of this paper.

2.   Modeling and formulation of the PMSG systems

This section will discuss about the mathematical model of PMSG based on selected wind conversion system (see Figure 1). The back-to-back converter employed for transporting power generated by the generator to the grid is where the adaptive type-2 fuzzy controller is primarily focused.

Figure 1.  The configuration of a PMSG wind turbine.

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The parameters description will be presented as follows ^[22]:

V : Wind;

MSC : Machine-side converter;

V_dc : Dc-link voltage;

i_sabc : Stator three-phase current;

u_sabc : Stator three-phase voltage;

β : Pitch angle;

GSC : Grid side converter;

Pe : The active output power;

P_w : The power extracted from the wind;

i_labs : Load current;

Qe : The reactive output power.

According to ^[22], the mathematical model for PMSG following rotor coordinate system can be given as in (1):

The variables description will be presented as follows:

{${i}_{q}$, ${i}_{d}$}, {${u}_{q}$, ${u}_{d}$}, {${L}_{q}$, ${L}_{d}$}: currents, voltages and inductance of quadrature and direct axis stator in pairs;

$t{'}$ : Time;

$f$ : Viscous friction coefficient;

$J$ : Rotor moment of inertia;

${T}_{1}$ : Load torque;

${\phi }_{f}$ : Rotor magnet flux linking the stator;

${R}_{s}$ : Stator resistance;

$p$ : Number of poles pairs.

The simplified form of Eq (1) can be obtained by transforming the affine and time scaling as follows ^[22]:

with $\sigma = \frac{f{L}_{q}}{{R}_{s}J}$, $\epsilon = \frac{pb{L}_{q}^{2}{k}^{2}\left({L}_{q}-{L}_{d}\right)}{J{R}_{s}^{2}}$, $\gamma = -\frac{{\phi }_{f}}{k{L}_{q}}$, $b = \frac{{L}_{q}}{{L}_{d}}$, $k = \frac{fR}{{L}_{q}p{\phi }_{f}}$, ${\stackrel{~}{i}}_{d} = \frac{{L}_{d}p{\phi }_{f}}{f{R}_{s}}{i}_{d}$, ${\stackrel{~}{i}}_{q} = \frac{{L}_{q}p{\phi }_{f}}{f{R}_{s}}{i}_{q}$, $\stackrel{~}{w} = \frac{{L}_{q}}{{R}_{s}}w$, ${\stackrel{~}{u}}_{d} = \frac{1}{{R}_{s}k}{u}_{d}$, ${\stackrel{~}{u}}_{d} = \frac{1}{{R}_{s}k}{u}_{d}$, ${\stackrel{~}{u}}_{q} = \frac{1}{{R}_{s}k}{u}_{q}$, $\stackrel{~}{T} = \frac{{L}_{q}^{2}}{J{R}_{s}^{2}}{T}_{1}$ and $t = \frac{{R}_{s}t{'}}{{L}_{q}}$.

The state vector $y$ assumes the forms $y = {\left[w, {i}_{q}, {i}_{d}\right]}^{T}$ and the PMSG model in Eq (2) is transformable to the normalized PMSG model ^[22]:

where a and b denote the operating parameters. The PMSG exhibits chaotic behavior by limiting the values of a and b into certain boundary and setting the external parameters to zero, that is, $\stackrel{~}{T} = {\stackrel{~}{u}}_{q} = {\stackrel{~}{u}}_{d} = 0$. Then Eq (3) become

where a = 5.45 and b = 20 with initial condition (0.2, 0.2, 0.2). Figure 2 displays the chaotic attractor of PMSG system.

Figure 2.  The chaotic attractors PSMG for (a) y₁-y₂, (b) y₂-y₃, and (c) y₁-y₃ planes using MATLAB.

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Using MATLAB, the following is how we calculate the Lyapunov exponents of model (4) ^[31]:

The Kaplan-Yorke dimension of system (2.4) is computed as follows:

which points to the high level of complexity of the new system (4).

3.   Dynamical analysis

3.1.   Bifurcation diagram and Lyapunov exponent

The Lyapunov spectrum and bifurcation diagrams are respectively presented in Figures 3(a) and 3(b) by varying parameter a such that a ϵ ^[2,18]. They indicate that the PMSG system (4) can generate periodic behavior and exhibit chaotic behavior. The system has periodic behavior in b ϵ ([2, 3.7], [4.35, 4.62], [12.2, 12.72], ^[15,18]), and chaotic behavior in b ϵ ([3.7, 4.35], [4.62, 12.2], [12.72, 15]).

Figure 3.  Dynamic analysis of the PMSG system (4) with parameter s varying and b = 20; (a) Lyapunov exponents spectrum and (b) Bifurcation diagram.

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The Lyapunov exponent spectrum and bifurcation diagram of (4) are displayed in Figures 4(a) and 4(b), respectively. They indicate that the PMSG system (4) can generate stable behavior, periodic behavior and exhibit chaotic behavior. The system has stable behavior in b ϵ ([0, 5.2]), periodic behavior in b ϵ ([24.1, 25.6], [31.45, 50]), and chaotic behavior in b ϵ ([5.2, 24.1], [25.6, 31.45]).

Figure 4.  Dynamic analysis of the PMSG system (4) with parameter b varying and a = 5.45; (a) Lyapunov exponents spectrum and (b) Bifurcation diagram.

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3.2.   Poincare map

Poincaré map is a fundamental and useful tool for learning about periodic state and its associated properties ^[32,33]. It transforms a dynamical system with continuous time into one with discrete time, and converts the study of the flow in the vicinity of a closed orbit or a periodic solution into the study of a map ^[34,35,36]. The Poincaré map of (4) is presented in Figure 5. The diagram show that the Poincaré map exhibits chaotic behavior with densely packed trajectories.

Figure 5.  Poincaré map analysis of the PMSG system (4) with parameter a = 5.45 and b = 20.

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4.   Adaptive type-2 fuzzy controller

To design the controller, the dynamics of (4) are rewritten as:

where,

with ${y}^{T} = \left[{y}_{1}, {y}_{2}, {y}_{3}\right]$, and ${F}_{i}\left(y\right), i = \mathrm{1, 2}, 3$ representing unknown functions approximated by suggested type-2 fuzzy logic systems (T2-FLSs) as ${\widehat{F}}_{i}\left(y, {\psi }_{i}\right), i = \mathrm{1, 2}, 3$, where ${\psi }_{i}$ denotes the rule parameters. The proposed controllers are:

where, ${r}_{i}$ represents the reference, ${\chi }_{i}$ is defined as ${\chi }_{i} = {y}_{i}-{r}_{i}$, and ${\kappa }_{i}$ is a constant. By applying the controllers (9) on the system (7), we have

By considering the optimal ${\widehat{F}}_{i}^{*}\left(y, {\psi }_{1}^{*}\right)$, we get:

also, by considering the following definition:

then from (10), we obtain:

where term ${\widehat{F}}_{i}^{*}\left(y, {\psi }_{i}^{*}\right)-{\widehat{F}}_{i}\left(y, {\psi }_{i}\right)$ is rewritten as:

with, ${\phi }_{i}\left(y\right)$ defining the vector of normalized rule firing, and ${\stackrel{~}{\psi }}_{i}$ is defined as

From (14), the Eq (13) becomes:

To extract the learning rules of T2-FLS, and ensure stability, consider Lyapunov as:

where, $\lambda$ is a constant. From (17), we have

By substituting form (16), we have

From (19), we have

From (20), the learning rules are considered as:

Then from (21), the Lyapunov function in (20), becomes

From Eq (22), and considering the fact that

we can write:

Then by considering the feedback gains ${\kappa }_{i}$, the stability is ensured. The main point is that, by the use of T2-FLSs, the errors ${\vartheta }_{i}, i = \mathrm{1, 2}, 3$ are much small in comparison with type-1 counterparts. It should be noted that ${\kappa }_{i}, i = \mathrm{1, 2}, 3$ are positive constants. The larger values of ${\kappa }_{i}, i = \mathrm{1, 2}, 3$ increase the speed of convergence but leads to a bigger control signal. Then is choosing of ${\kappa }_{i}, i = \mathrm{1, 2}, 3$ a trade-off should be made between speed of convergence and magnitude of control signal. Also, the big ${\kappa }_{i}, i = \mathrm{1, 2}, 3$ help for stability. The values of ${\kappa }_{i}, i = \mathrm{1, 2}, 3$ should be bigger than $\left|{\vartheta }_{i}\right|$.

5.   Simulation

The simulations in this study were conducted on Matlab programming software so as to assess the efficacy of the suggested approach. The designed controller is implemented in Matlab as shown in Figures 6–8. By considering the references as $\mathit{sin}\left(t\right)+\mathit{cos}\left(t\right)$, $\mathit{cos}\left(t\right)$, and $\mathit{sin}\left(t\right)$. Figures 9 and 10 gives the tracking response and tracking error. It can be seen that tracking error converted to small neighborhood around zero, expeditiously. Simulation results confirm that our proposed controller can guarantee the stability and robustness of the close loop system. Figure 11 show that the control signal for the chaotic PMSG based on the conversion system of wind energy. Finally, Figure 12 present the phase portrait after designed controller to remove the chaotic PMSG wind turbine model.

Figure 6.  The general scheme of implemented controller.

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Figure 7.  The MATLAB scheme of controller.

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Figure 8.  The implemented scheme of T2-FLS.

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Figure 9.  Numerical simulation using Matlab (a) Output trajectory y₁, (b) Output trajectory y₂, and (c) Output trajectory y₃.

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Figure 10.  Tracking error.

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Figure 11.  Control signals.

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Figure 12.  Phase Potrait using Matlab.

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The performance of the proposed control method is compared with the Fuzzy PID control and Adaptive fuzzy control in terms of the root mean square (RMS) as presented in Table 1. On these bases, it is concluded that the proposed controller outperforms the benchmarking techniques. In order to determine the robustness of the proposed control method, the simulations are further carried out under different longitudinal velocities, and matched uncertainties.

Table 1.  Root mean square error comparison.

Methods	Error signals
	${\chi }_{1}$	${\chi }_{2}$	${\chi }_{3}$
Proposed	0.0124	0.2511	1.1204
Fuzzy PID [37]	0.0132	0.3612	3.0172
Adaptive fuzzy controller [38]	0.0114	0.5417	6.4120

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6.   Conclusions

In this article, we investigated the chaotic behavior of Permanent Magnet Synchronous Generator (PMSG) in a wind turbine system with quadratic nonlinearities. The bifurcation diagrams, Lyapunov exponents, Poincare maps, and Phase portraits, have all been used to illustrate the intricate dynamical characteristics of PMSG chaotic system. Finally, an adaptive type-2 fuzzy controller is designed for different flow configurations of the PMSG. To further demonstrate the effectiveness of the proposed approach, we presented analysis of the performance by simulating the PMSG. This simulation study of the PMSG guarantees the efficiency and performance of the new procedure.

Acknowledgments

Aceng Sambas appreciates the Postdoctoral Fellowship from Universiti Malaysia Terengganu (UMT), Mohamad Afendee Mohamed was partially funded by CREIM, Universiti Sultan Zainal Abidin (UniSZA) and Mohamad Arif Awang Nawi was partially funded with grant Number: 304/PPSG/6315410.

Conflict of interest

The authors declare no conflict of interest.