Maximum power extraction framework using robust fractional-order feedback linearization control and GM-CPSO for PMSG-based WECS

Abstract

The most important issue in the use of wind energy conversion systems is to ensure maximum power extraction in terms of efficiency. Therefore, maximum power point tracking algorithms are as important as the maximum power point tracking controller. In this study, maximum power extraction frameworks operating the state-of-the-art optimization methods are presented for permanent magnet synchronous generator–based wind energy conversion system. These frameworks consist of a Gauss map–based chaotic particle swarm optimization and a hybrid maximum power point tracking approach that combines feedback linearization technique with fractional-order calculus. The feedback linearization control strategy can fully decouple and linearize the original state variables of the nonlinear system and thus provide an optimal controller crossing wide-range operating conditions. The objective is to maintain the tip speed ratio at its optimal value, which implies the use of a rotational speed loop. The method is based on the feedback linearization technique and the fractional control theory. Gauss map–based chaotic particle swarm optimization, which is a remarkable and recent optimization technique, is utilized to achieve optimum coefficients to efficiently ensure the maximum power point tracking operation in here. A simulation study is carried out on a 3-kW wind energy conversion system to show the effectiveness of the proposed control scheme.

Keywords

Permanent magnet synchronous generator maximum power point tracking feedback linearization control fractional-order theory Gauss map–based chaotic particle swarm optimization

Introduction

Energy consumption around the world continues to grow during the current century, declining sources of fossil fuels and concern over environmental pollution levels are the main motivators for electricity generation from sources of energy. Renewable energies, such as solar, wind and tidal power, are clean, inexhaustible and environmentally friendly energies. Because of all these factors, the generation of wind energy has attracted great interest in recent years due to the development of the wind turbine (WT) industry and the development of new WT control methods (El Yaakoubi et al., 2017; Mérida et al., 2014). As part of this change, the world market share of wind energy will provide 12% of the electricity needed by 2020, 29.1% by 2030 and 34.2% by 2050 (Abdeddaim el al., 2014; Fateh, 2015).

Wind energy can be used in different applications, including the pumping water, as well as the both grid-connected and stand-alone WT. WT technology aims to capture part of the kinetic energy contained in the wind and convert it into available mechanical power using rotor blades; then, this energy is transformed into electrical energy by a generator. Among the electrical generators, wind energy conversion system (WECS)-based permanent magnet synchronous generator (PMSG) are widely used in medium- and high-power WTs due to its robustness, its mechanical simplicity and except initial installation costs (Bonfiglio et al., 2017). They are the object of particular attention because of their high efficiency. However, their analysis and control is a difficult task because of nonlinearities and load torque. To overcome the difficulties associated with the design of a regulator for the PMSG, several control approaches have been proposed such as conventional proportional–integral (PI) control (Li et al., 2010), fuzzy logic PI control (Muyeen and Al-Durra, 2013), adaptive control (Boudries and Tamalouzt, 2019), neuronal control (Elbeji et al., 2021) and sliding mode control (Ahriche et al., 2016).

Recently, many design methods based on the principles of feedback linearization control have been proposed (Chen et al., 2013; Munteanu et al., 2008). Actually, the popular linearization techniques can be basically categorized into three groups: standard form of linearization technique (SLT) using Taylor’s series expansion, the direct feedback linearization (DFL), and the differential geometric technique (DGT; Bodson and Chiasson, 1998; Shi et al., 2012; Tailor and Bhathawala, 2011).

Feedback linearization techniques are proposed to linearize highly nonlinear dynamic systems (Cheikh et al., 2018). This involves a static state feedback and a special nonlinear coordinate change. The major disadvantage of these techniques is their direct dependence on the parameters of the installation. In recent years, thanks to developments in the theory of nonlinear control, researchers have developed several robust control laws. The combination of these robust laws with feedback linearization techniques has been an admirable success in the robustness of the control scheme. In the literature, these control designs can be found in many industrial applications. For example, Yuan et al. (2016a) have proposed an input/output feedback linearization method–based sliding mode control strategy for the hydraulic generator regulating system (HGRS) with external disturbance and system uncertainties. Soufi et al. (2016) have also proposed a feedback linearization controller–based particle swarm optimization (PSO) for maximum power point tracking (MPPT) of WT equipped by PMSG connected to the grid.

Recently, fractional calculus (FC)-based fractional-order PID (FOPID) control found its way into complex mathematical and physical problems to achieve improved control performance over the traditional PID controller (Ramadan, 2017).

Stochastic behaviour is utilized in various themes like numerical function optimization, hybrid classifier design and electrical generator applications (Hannachi et al., 2021). The convergence of methods is proposed to be efficient for the handled fitness function, and this process should be realized with influential methods to achieve a remarkable performance. In the literature, most studies consider different derivatives of PSO so as to provide better convergence than other techniques. In Chen et al. (2018), two efficient derivatives of PSO that are sine map–based chaotic PSO (SM-CPSO) and dynamic weight PSO (DW-PSO) are suggested. By combining SM-CPSO and DW-PSO algorithms, chaotic dynamic weight PSO (CDW-PSO) was generated to design a robust structure. On numerical function optimization, CDW-PSO surpassed 20 optimization algorithms including basic methods, chaotic approaches, PSO-based derivatives and the state-of-the-art studies. Koyuncu (2020) examined 10 chaotic maps in PSO to reveal whether chaotic maps are essential in the update of inertia weight. At the first part of experiments, the CPSOs including 10 chaotic maps are compared with each other besides general PSO. Thus, it was apparent that Gauss map–based CPSO (GM-CPSO) was the only one outperforming general PSO in every condition and was also the best one among all CPSOs. At the second part, GM-CPSO was compared with SM-CPSO, DW-PSO and CDW-PSO methods in different dimensional conditions.

In this article, GM-CPSO algorithm is handled to design an efficient framework operating hybrid optimization and robust fractional MPPT method to extract maximum power from WECS via feedback linearization strategy. The major contributions of this study made to the literature can be seen as follows:

GM-CPSO algorithm has been diversely hybridized with the fractional MPPT method to create an effective maximum power extraction framework that has not been used in WECS before.

The proposed MPPT framework is mechanically sensorless, has a very fast convergence without the need for any system prior knowledge and ensures maximum power extraction by switching between the best coefficients and by following the generator speed changes during operation.

At the two wind speed profiles, the proposed method is analysed in detail by comparing with the fractional MPPT method, and also contributes to MPPT efficiency by providing higher power extraction than the others.

The rest of this study is presented as follows: In section ‘WECS configuration’, the designed WECS configuration is given in detail. Section ‘Feedback linearization strategy of PMSG-based WECS’ presents the MPPT method and feedback linearization control in a comprehensive manner. GM-CPSO optimization of fractional control–based feedback linearization strategy is presented in section ‘GM-CPSO optimization of fractional control design–based feedback linearization strategy’. Simulation results and interpretations are evaluated in section ‘Simulation results and discussions’, and concluding remark is given in section ‘Conclusion’.

WECS configuration

The block diagram of the PMSG-based WECS which is connected to the power grid bus via a back-to-back voltage source converter (VSC) is shown in Figure 1. The produced active power and reactive power of PMSG is regulated through the generator-side VSC, while the grid-side VSC attempts to deliver active power to the power grid through the DC link and to maintain the DC link voltage at the rated value.

Figure 1.

Configuration of a PMSG-based WECS.

WT model

The aerodynamic part transforms the kinetic energy of the wind into a mechanical energy and torque translated by the rotation of the rotor of the generator (Kahla et al., 2015)

P_{wt} = \frac{1}{2} ρ π R^{2} v^{3} C_{p} (λ)

(1)

Γ_{wt} = \frac{1}{2} ρ π R^{3} v^{2} C_{Γ} (λ)

(2)

where $P_{wt}$ is the mechanical power, $ρ$ is the air density, $R$ is the blade radius of WT, $v$ is the wind speed, $C_{p} (λ)$ is the power coefficient, $C_{Γ} (λ)$ is the torque coefficient and $λ$ is the tip speed ratio given by

λ = \frac{Ω_{l} v}{R}

(3)

where $Ω_{l}$ is the low-speed shaft rotational speed. $C_{p} (λ)$ and $C_{Γ} (λ)$ are related by the following equation

C_{p} (λ) = λ C_{Γ} (λ)

(4)

The theoretical upper limit of the power coefficient $C_{p max}$ is given by Betz’s limit (Kahla et al., 2017)

C_{p max} = \frac{16}{27} \approx 0.59

(5)

To obtain the maximum power at different wind speeds, the MPPT algorithm is implemented to keep WT operating at MPP below the rated wind speed. An efficient MPPT algorithm should regulate the rotor speed to maximize the output power. The MPP is obtained during operation at optimal values of $λ_{opt}$ and $C_{p max}$ which equal to 7 and 0.475, respectively. Figure 2 describes the relationship of the optimal power generation at different wind speeds while operating at MPP (Dahbi et al., 2016).

Figure 2.

Wind turbine power characteristics.

Mechanical shaft system modelling

The dynamics of the mechanical shaft system and mechanical torque of PMSG is given by (Yang et al., 2018)

J_{t} \frac{d Ω_{h}}{dt} = Γ_{wt} - Γ_{em} - F Ω_{h}

(6)

where $Ω_{h}$ is the high-speed shaft rotational speed. $J_{t}$ is the total inertia of the drive train which equals to the summation of WT inertia constant and generator inertia constant and $F$ is the viscous friction coefficient. In addition, active power is calculated as

P_{a} = Γ_{em} Ω_{h}

(7)

with $Γ_{em}$ being the electromagnetic torque.

PMSG model

The dynamics of PMSG in the d-q reference frames is written as (Yang et al., 2018)

{\begin{matrix} V_{d} = R_{s} i_{d} + L_{d} \frac{{di}_{d}}{dt} - φ_{q} ω_{s} \\ V_{q} = R_{s} i_{q} + L_{q} \frac{{di}_{q}}{dt} + φ_{d} ω_{s} \end{matrix}

(8)

where $R_{s}$ is the stator resistance, $V_{d}$ and $V_{q}$ are d and q stator voltages, $L_{d}$ and $L_{q}$ are d and q inductances, $i_{d}$ and $i_{q}$ are the currents in the d-q axis and $ω_{s}$ is the stator pulsation, respectively

φ_{d} = L_{d} i_{d} + φ_{m}

(9)

φ_{q} = L_{q} i_{q}

(10)

are fluxes in d-q axis and $φ_{m}$ is the flux that is constant due to permanent magnets. Thus, the model at equation (8) becomes

{\begin{matrix} V_{d} = R_{s} i_{d} + L_{d} \frac{{di}_{d}}{dt} - L_{q} i_{q} ω_{s} \\ V_{q} = R_{s} i_{q} + L_{q} \frac{{di}_{q}}{dt} + (L_{d} i_{d} + φ_{m}) ω_{s} \end{matrix}

(11)

The PMSG model with the equivalent circuit in the (d, q) reference can be expressed as

{\begin{matrix} \frac{d}{dt} i_{d} = - \frac{R_{s} + R_{L}}{L_{d} + L_{L}} i_{d} + \frac{p (L_{q} - L_{L})}{L_{d} + L_{L}} i_{q} Ω_{h} \\ \frac{d}{dt} i_{q} = - \frac{R_{s} + R_{L}}{L_{q} + L_{L}} i_{q} - \frac{p (L_{d} + L_{L})}{L_{q} + L_{L}} i_{d} Ω_{h} + \frac{p φ_{m}}{L_{q} + L_{L}} \end{matrix}

(12)

The electromagnetic torque is obtained as

Γ_{em} = p [φ_{m} i_{q} + (L_{d} - L_{q}) i_{d} i_{q}]

(13)

where $p$ is the number of pole is pairs and $R_{L}$ and $L_{L}$ are the equivalent chopper resistance and inductance, respectively.

Feedback linearization strategy of PMSG-based WECS

The idea of feedback linearization technique is to use a transformation, z=T (x), and to apply a feedback control law, u, which transforms the nonlinear system into a linear system. Then, after obtaining a linear system, the usual linear control design methods can be applied for stabilization.

We consider a class of nonlinear systems of the form

{\begin{matrix} x^{\cdot} = f (x) + g (x) u \\ y = h (x) \end{matrix}

(14)

with

f (x) = [\begin{matrix} f_{1} \\ f_{2} \\ f_{3} \end{matrix}] = [\begin{matrix} \frac{1}{L_{d} + L_{q}} (- R x_{1} + p (L_{q} - L_{s}) x_{2} x_{3}) \\ \frac{1}{L_{q} + L_{s}} (- R x_{2} - p (L_{d} + L_{s}) x_{1} x_{3} + p φ_{m} x_{3}) \\ \frac{1}{J_{t}} (d_{1} v^{2} + d_{2} v x_{3} + d_{3} x_{3}^{2} - p φ_{m} x_{2}) \end{matrix}]

g (x) = {[\begin{matrix} g_{1} & g_{2} & g_{3} \end{matrix}]}^{T} = {[\begin{matrix} - \frac{1}{L_{d} + L_{L}} x_{1} & - \frac{1}{L_{q} + L_{s}} x_{2} & 0 \end{matrix}]}^{T} h (x) = x_{3} = Ω_{h}

The state vector is defined as

x = {[\begin{matrix} x_{1} & x_{2} & x_{3} \end{matrix}]}^{T} = {[\begin{matrix} i_{d} & i_{q} & Ω_{h} \end{matrix}]}^{T}

u = [\begin{matrix} u_{1} & u_{2} \end{matrix}] = [\begin{matrix} R_{L} & v \end{matrix}]

with the parameters

{\begin{matrix} d_{1} = \frac{1}{2} ρ π R^{3} a_{0}; d_{2} = \frac{1}{2} ρ π R^{4} a_{1}; d_{3} = \frac{1}{2} ρ π R^{5} a_{2} \\ a_{0} = 0.1253; a_{1} = - 0.0047; a_{2} = - 0.0005 \end{matrix}

The system given by equation (14) is considered as nonlinear with smooth functions whose synthesis of feedback linearization control is possible to make the system linear (Cheikh et al., 2018). The nonlinear system equation (14) is said to be feedback linearizable if there exists a diffeomorphism $T : D \to R^{n}$ such that $D_{z} = T (D)$ contains the origin and the change of variables $z = T (x)$ transforms the system equation (14) into the form

\overset{\cdot}{z} = Az + B γ (x) [u - δ (x)]

(15)

The coordinate transform that leads to a partial linearization of the system is

z = Φ (x_{1}, x_{2}, x_{3}) = [\begin{matrix} x_{3} \\ d_{1} v^{2} + d_{2} v x_{3} + d_{3} x_{3}^{2} - d_{4} x_{2} \\ a_{3} \frac{x_{1}}{x_{2}} \end{matrix}]

(16)

To be able to perform the inverse transform, $Φ (x_{1}, x_{2}, x_{3})$ should not be singular. The direct coordinate transform is

{\begin{matrix} z_{1} = h (x) \\ z_{2} = L_{f} h (x) = d_{1} v^{2} + d_{2} v x_{3} + d_{3} x_{3}^{2} - d_{4} x_{2} \\ z_{3} = a_{3} \frac{x_{1}}{x_{2}} \end{matrix}

(17)

And the inverse coordinate transform is

{\begin{matrix} x_{1} = a_{3} z_{3} \frac{d_{1} v^{2} + d_{2} v z_{1} + d_{3} {xz}_{1}^{2} - z_{2}}{d_{4}} \\ x_{2} = \frac{d_{1} v^{2} + d_{2} v z_{1} + d_{3} z_{1}^{2} - z_{2}}{d_{4}} \\ x_{3} = z_{1} \end{matrix}

(18)

The advantage of the representation in equation (15) is the ability to choose a feedback control law that cancels out the nonlinearities of the system. The control objective can be achieved using the ideal control law

u^{*} = \frac{1}{β (x)} (- δ (x) + υ)

(19)

Hence, we need to calculate the functions $δ (x)$ and $β (x)$ . These functions are given by

{\begin{matrix} δ (x) = L_{f}^{2} h (x) \\ β (x) = L_{g} L_{f} h (x) \end{matrix}

(20)

After computation using the WECS data, there is

{\begin{matrix} L_{f}^{2} h (x) = - d_{4} f_{2} + (d_{2} v + + 2 d_{3} x_{3}) f_{3} \\ L_{g} L_{f} h (x) = - d_{4} a_{3} . x_{2} \end{matrix}

(21)

with $f_{2}$ and $f_{3}$ being components of function $f$ from equation (14). The control input has a state feedback component – the Lie derivates $L_{f}^{2} h (x)$ and $L_{g} L_{f} h (x)$ and a component $υ$ , which imposes the dynamic of the linear input–output mapping.

GM-CPSO optimization of fractional control design–based feedback linearization strategy

Fractional control design–based feedback linearization strategy

FC is an extension of regular integral calculus (IC) to noninteger case. In comparison with IC, FC is adequate and natural to fully characterize many physical phenomena. In general, the extra degrees of freedom from the use of fractional-order integrator and differentiator could further enhance the control performance compared with that of traditional integer-order controller.

Fractional-order calculus generalizes the integer-order integration and differentiation into the noninteger-order domain. The fundamental operator $_{a} D_{t}^{α}$ is defined as (Podlubny, 1999)

_{a} D_{t}^{α} = {\begin{matrix} \frac{d^{α}}{{dt}^{α}} & α > 0 \\ 1 & α = 0 \\ \int_{a}^{t} {(d τ)}^{- α} & α < 0 \end{matrix}

(22)

where a and t denote the lower and upper limits, respectively, while $α \in ℜ$ means the operation order.

Here, Riemann–Liouville (RL) definition for fractional-order derivative is adopted with gamma function $Γ (•)$ , as follows

{}_{a}D_{t}^{α} = \frac{1}{Γ (n - α)} \frac{d^{n}}{d t^{n}} \int_{a}^{t} \frac{f (τ)}{{(t - τ)}^{α - n + 1}} d τ

(23)

where n is the first integer which is not less than $α$ , for example, $n - 1 \leq α < n$ . The Laplace transformation of the RL-based fractional-order derivative equation (23) can be obtained as

\int_{0}^{\infty} {_{0} D_{t}^{α} f (t) e^{- st} dt = s^{α} ℑ {f (t)} - \sum_{k = 0}^{n - 1} s^{k} D_{t}^{α - k - 1} f (t) |}_{t = 0}

(24)

where $ℑ {•}$ represents the Laplace operator.

In addition, the Oustaloup approximation (Yuan et al., 2016b) is used for a recursive distribution of zeros and poles, which gives

s^{α} \approx K Π_{n = - N}^{N} \frac{1 + (\frac{s}{w_{z, n}})}{1 + (\frac{s}{w_{p, n}})}, α > 0

(25)

where $2 N + 1$ denotes the number of zeros and poles; K is the gain which causes both sides of equation. $w_{z, n}$ and $w_{p, n}$ are given as

w_{z, n} = w_{b} {(\frac{w_{h}}{w_{b}})}^{(n + N + 1 + (1 - α) / 2) / (2 N + 1)}

(26)

w_{p, n} = w_{b} {(\frac{w_{h}}{w_{b}})}^{(n + N + 1 + (1 + α) / 2) / (2 N + 1)}

(27)

In equations (26) and (27), lower limit $w_{b}$ and upper limit $w_{h}$ normally satisfy $w_{b} w_{h} = 1$ and $k = w_{h}^{α}$ .

The case $α < 0$ can be resolved by inverting equation (25). Besides, for $| α | > 0$ , the approximation becomes unsatisfactory. To handle such case, the fractional powers of $s$ is usually split, as follows

s^{α} = s^{n} s^{σ}, α = n + σ, σ \in [0.1]

(28)

Hence, only the latter term $σ$ needs to be approximated.

The following Lemma 1 states the stability of the fractional-order system.

Lemma 1 (Matignon, 1998). Consider the following autonomous system

_{0} D_{t}^{α} = C ℓ, ℓ (0) = ℓ_{0}

(29)

where $ℓ \in ℜ^{n}$ and $C \in ℜ^{nxn}$ are asymptotically stable if $| \arg (eig (C)) | > α π / 2$ , in which each component of the states decays towards 0 like $t^{- α}$ .

Moreover, system equation (29) is stable if $| \arg (eig (C)) | \geq α π / 2$ with those critical eigenvalues satisfying $| \arg (eig (C)) | = α π / 2$ have geometric multiplicity one.

Besides, Figure 3 briefly shows the stability region when $0 < α < 2$ . It demonstrates that the stability region of fractional-order system with $0 < α < 1$ is the largest than that of the other two scenarios.

Figure 3.

Stability region of fractional-order system determined with varying operation order.

The control input is calculated using pole placement technique and fractional theory

u^{*} = - [\begin{matrix} k_{1} & k_{2} \end{matrix}] [\begin{matrix} z_{1} \\ z_{2} \end{matrix}] + k_{I} ε

(30)

with

ε = \frac{1}{s^{μ}} (y_{ref} - y)

(31)

Since the control of the WECS aims at the extraction of the maximum wind power (MPPT) that will be obtained by the drive of the generator at a speed $Ω_{h}$ which must continue asymptotically, the optimal reference speed

y_{ref} (t) = Ω_{h}^{*} = \frac{λ_{opt} . i . v (t)}{R}

(32)

with $λ_{opt}$ being the optimal tip speed ratio.

GM-CPSO

GM-CPSO involves nearly all phenomena in the PSO method, except stable or linear updated inertia weight. As in the PSO algorithm, GM-CPSO utilizes velocity and position concepts to provide the convergence towards the global point(s), and the concepts are shown in equations (33) and (34), respectively (Chen et al., 2018)

\begin{matrix} V_{i} (t + 1) = ω V_{i} (t) + c_{1} r_{1} (t) (X_{pbest (i)} (t) - X_{i} (t)) \\ + c_{2} r_{2} (t) (X_{gbest} (t) - X_{i} (t)) \end{matrix}

(33)

X_{i} (t + 1) = X_{i} (t) + V_{i} (t + 1)

(34)

In equations (33) and (34), V_i(t) and X_i(t), respectively, symbolize the current velocity and position vectors of ith particle, while V_i(t+ 1) and X_i(t+ 1) add up to the new ones. X_pbest_(i)(t) and X_gbest(t) specify the individual best position of ith particle and global best solution of the whole swarm. c₁ and c₂ are acceleration constants adjusting the movements to the individual and global best positions and are formulated according to equation (35) (Koyuncu, 2020). In the literature, these constants are generally chosen as equal to ‘2’ to acquire better fitness (Chen et al., 2018). r₁(t) and r₂(t) are generated in the range of (0, 1) and provide diversity to better explore the search space. ω connotes the inertia weight limiting the effect of the current velocity to the step size (new velocity). In GM-CPSO, the arrangement of ω is performed using Gauss (Mouse) map presented in equation (36) (Chen et al., 2018; Koyuncu, 2020)

c_{1} + c_{2} \leq 4

(35)

ω (i + 1) = {\begin{matrix} 1 ω (i) = 0 \\ \frac{1}{mod (ω (i), 1)} otherwise \end{matrix} \begin{matrix}  \end{matrix}, ω (0) = 0.7

(36)

In equation (35), the first assignment of inertia weight is advised as ‘0.7’ to obtain better chaotic behaviour . In terms of the presented chaotic behaviour, Gauss map can change or eliminate the effect of the current (Koyuncu, 2020) velocity on step size. Even if this process indirectly affects the new position, it improves the searching capability by diversifying the solutions (positions) in search space. Herein, Gauss map induces more variety to guarantee the better convergence for optimal solutions (Chen et al., 2018; Koyuncu, 2020).

The boundaries of velocity values are regulated according to equation (37). X_min and X_max are defined as the minimum and maximum values that a position can become, while V_min and V_max are specified for the velocity limitations. The parameter ‘τ’ is used to connect the velocity boundaries with the position limitations, and it is preferred to be assigned as ‘0.2’ as advised in most of the PSO-based derivatives (Koyuncu, 2020)

[V_{min}, V_{max}] = τ * [X_{min}, X_{max}]

(37)

Define the tracking error $e = Ω_{h} - Ω_{h}^{ref}$ , with $Ω_{h}^{ref}$ being the high-speed shaft’s rotational speed reference. The proposed control for tracking error dynamics for the low-frequency loop can be designed by

u^{*} = - [\begin{matrix} k_{1} & k_{2} \end{matrix}] [\begin{matrix} z_{1} \\ z_{2} \end{matrix}] + \frac{k}{s^{μ}} (Ω_{h}^{ref} - Ω_{h})

(38)

where fractional-order feedback linearization (FOFL) control gains $k_{1}$ , $k_{2}$ and $k$ and fractional integrator order $μ$ are chosen to realize a satisfactory convergence of tracking error dynamics in equation (38). The FOFL control parameters in equation (38) are optimally tuned through GM-CPSO under two cases, for example, (a) low-turbulence stochastic wind speed and (b) high-turbulence stochastic wind speed. The optimization goal is to minimize the tracking error of high-speed shaft rotational speed, which model is given as follows

Minimize F (x) = \sum_{Two cases} \int_{o}^{T} (| Ω_{h}^{ref} - Ω_{h} |) d t

(39)

Subject to {\begin{matrix} k_{1}^{min} \leq k_{1} \leq k_{1}^{max} \\ k_{2}^{min} \leq k_{2} \leq k_{2}^{max} \\ k^{min} \leq k \leq k^{max} \\ μ^{min} \leq μ \leq μ^{max} \end{matrix}

(40)

The GM-CPSO parameters are chosen as the total number of iterations $N_{iter} = 100$ , minimum velocity $v_{min} = 0.1$ , maximum velocity $v_{max} = 1$ , weight coefficients $c_{1} / c_{2} = 2 / 2$ , respectively. In addition, the convergence criteria is chosen as

| F_{j} - F_{j - 1} | \leq ε

(41)

where $ε$ is the tolerance of convergence error, whose value is chosen to be 10⁻⁴ in this article and $F_{j}$ and $F_{j - 1}$ represent the fitness function value calculated at the $j th$ iteration and the $(j - 1) th$ iteration, respectively. To this end, Figure 4 describes the overall control framework of FOFL for PMSG-based WECS.

Figure 4.

Overall control framework of FOFL-based GM-CPSO for MPPT.

Simulation results and discussions

In this section, all designs, validation tests and comparisons for proposed MPPT frameworks and FOFL method are performed in a simulation environment. Parameters of the low-power (3 kW) rigid drive train PMSG-based WECS are presented in Table 1.

Table 1.

Parameters of the low-power (3 kW) rigid drive train PMSG-based WECS.

Turbine rotor	Drive train	PMSG
Blade length: R= 2.5 m	Multiplier ratio: i = 7 Jh= 0.5042 kg m² Efficiency: η = 1	p = 3, Rs = 3.3 Ω, Φm= 0.4382 Wb, Ld = 41.56 mH, Lq = 41.56 mH, Rln = 80 Ω, Vs = 380

PMSG: permanent magnet synchronous generator; WECS: wind energy conversion system.

Figure 5 shows the wind speed models of higher and lower turbulence intensity characteristics. The simulations have been done for 300 s for a wind sequence having the average speed of about 7 m/s via two-class studies of the turbulence categories: class A with 10% turbulence intensity; class B with 18% turbulence intensity, obtained using the von Karman spectrum in the International Electrotechnical Commission (IEC) standard.

Figure 5.

Wind speed profile: (a) 10% turbulence intensity, (b) 18% turbulence intensity.

Figures 6(a), 7(a), 8(a) and 9(a) show the mechanical power obtained by the two control methods under different turbulence levels. These results show the predominance of energy production by using the proposed FOFL-based GM-CPSO method over the FOFL method.

Figure 6.

PMSG responses obtained under a low-turbulence stochastic wind speed with FOFL: (a) mechanical power, (b) generator speed, (c) power coefficient, (d) tip speed ratio.

Figure 7.

PMSG responses obtained under a low-turbulence stochastic wind speed with FOFL-based GM-CPSO:(a) mechanical power, (b) generator speed, (c) power coefficient, (d) tip speed ratio.

Figure 8.

PMSG responses obtained under a high-turbulence stochastic wind speed with FOFL: (a) mechanical power,(b) generator speed, (c) power coefficient, (d) tip speed ratio.

Figure 9.

PMSG responses obtained under a high-turbulence stochastic wind speed with FOFL-based GM-CPSO:(a) mechanical power, (b) generator speed, (c) power coefficient, (d) tip speed ratio.

Figures 6(b), 7(b), 8(b) and 9(b) show the generator speed obtained by using two different control models. These results confirm that the disturbance of the wind speed variation is strongly deflecting the generator speed to its optimal trajectory. This deviation has obvious consequences for the capture of wind energy, counter to FOFL-based GM-CPSO control, which manages to assure the optimal speed tracking.

The evolution of the power coefficient obtained by the two control methods under different turbulence levels is depicted in Figures 6(c), 7(c), 8(c) and 9(c). It should be noted that $C_{p max} = 0.475$ for the WT in this simulation. It can be seen from the figures that the proposed method of optimization is able to track quickly the maximum power coefficient better than the other method of control.

The main control objective of variable-speed WT is power efficiency maximization. To reach this goal, the turbine tip speed ratio should be maintained at its optimum value, despite wind variations. In Figures 6(d), 7(d), 8(d) and 9(d), one can see the effectiveness of the proposed FOFL control–based GM-CPSO optimization to more accurately maintain the tip speed ratio around the optimal value as compared with the FOFL method.

Figures 10 and 11 show the performance of wind turbine generator systems (WTGSs) obtained by the two control methods under different turbulence levels. One could remark the operating point distribution around optimal regimes characteristic in speed-power plane (Figures 10(c) and 11(c)) and in tip speed ratio-electromagnetic torque plane (Figures 10(d) and 11(d)). Figures show a better performance of the control law, and the variance of the operating point around optimal regimes characteristic (ORC) is satisfactory and the torque follows the reference given by the control and allows us to reduce the torque ripples compared with FOFL control (Figures 10(a) and (b) and 11(a) and (b)).

Figure 10.

WTGS performance based on 10% turbulence intensity wind speed : (a) ORC tracking with FOFL, (b) electromagnetic torque versus tip speed with FOFL, (c) ORC tracking with FOFL-based GM-CPSO, (d) electromagnetic torque versus tip speed with FOFL-based GM-CPSO.

Figure 11.

WTGS performance based on 18% turbulence intensity wind speed : (a) ORC tracking with FOFL, (b) electromagnetic torque versus tip speed with FOFL, (c) ORC tracking with FOFL-based GM-CPSO, (d) electromagnetic torque versus tip speed with FOFL-based GM-CPSO.

Conclusion

This article has presented a newly developed nature-inspired algorithm called GM-CPSO to find the optimal parameters of fractional feedback linearization controller applied to a PMSG used in a WECS connected to the grid. First, an effective framework has been presented to maximize the wind power extraction. This framework consists of GM-PSO and hybrid MPPT algorithm by combining feedback linearization technique with fractional-order calculus that has never been used in a WECS before. GM-CPSO has been used to find the optimum coefficients by searching the peak value of the maximum output power to be extracted. To ensure MPPT efficiency, these coefficients must be determined optimally to achieve the maximum power. After these algorithms generate the optimum coefficients, the framework passes to the second part in which the FOFL MPPT method is operated according to these coefficients. According to the results, it can be seen that the proposed optimization-based FOFL contributes to MPPT efficiency and reaches superior wind power than the other FOFL method. Furthermore, the proposed combined framework has relatively simple design and can be easily implemented in the actual WECS where system prior knowledge and parameters are unknown.

Footnotes

Acknowledgements

The authors would like to thank the Pervasive Artificial Intelligence (PAI) group of the informatics department of Fribourg University – Switzerland – for their valuable suggestions and comments which helped us to improve this article. Special thanks to Prof. Béat Hirsbrunner and Prof. Michèle Courant.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iDs

Sami Kahla

Badreddine Babes

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