Neural network adaptive backstepping control via uncertainty compensation for PMSG-based variable-speed wind turbine: Controller design and stability analysis

Abstract

This paper proposes a design scheme along with stability analysis of a new adaptive backstepping controller designed for permanent magnet synchronous generator-based wind turbine, by using artificial neural network-based uncertainty compensation. The idea is to control the rotor speed and the mechanical power generated under internal and external nonlinear parametric uncertainties. An uncertain model of permanent magnet synchronous generator is designed. Then, two artificial neural network compensators are built to compensate such uncertainties in the current loops. The stability of the closed-loop system is studied according to the Lyapunov function. Simulations of the dynamic model are performed under both variable step and random wind speeds by using the DEV-C++ software, and the results are plotted with MATLAB. Compared to the classical direct torque control technique, the obtained results show the robustness of the proposed controller despite the presence of different uncertainties.

Keywords

Wind turbine permanent magnet synchronous generator parametric uncertainties artificial neural network uncertainty compensation adaptive backstepping control direct torque control

Introduction

Nowadays, energy production and consumption are one of the key factors that simulate the development of societies, and that because of modern life sophistication has become very dependent on electricity. Over the last two centuries, the main source of energy has been fossil fuels (oil, gas, coal) and their derivatives (Elbeji et al., 2021). However, due to the exponential growth of industries and technology, a tremendous demand on these energy resources is growing more and more. As a result, economic and environmental concerns over energy security have been raised, one of them is the threat of global worming caused by greenhouse gas emissions, which is directly related to the massive use of these fossil fuels (Beniysa et al., 2019; Malik et al., 2020). To overcome these concerns, considerable research has been devoted toward developing a more sustainable energy infrastructure, which is based on technologies that extracts energy from renewable resources.

Among all forms of renewable energy sources (e.g. wind turbines, biomass, photovoltaic panels, ocean currents, etc.), wind energy systems are very reliable, thanks to the recent advanced techniques in the wind turbine (WT) aerodynamics and electronic power interfaces. Variable-speed WTs based on permanent magnet synchronous generator (PMSG) are increasingly used by virtue of their high efficiency, high speed, high power density, low cost, and large torque to inertia ratio (Sumathi et al., 2015). Besides, the turbine gearbox can be eliminated since it is characterized by a large number of poles.

To fulfill the increasing demand for electrical power, WT system components must be designed for optimal and safe performance. Such components are supposed to meet some criteria in the market, such as cost optimization, weight reduction, higher performances, and lower non-conformance cost. Considering the fault characteristics of the wind turbines subsystems, studies have shown that among the main challenges to overcome is overheating, which can be caused by running the WT at higher load points. Such overheating can accelerate, for its part, the bearings’ failures through grease lubrication deterioration, and can lead to stator windings’ failures as well (Singh et al., 2019a, 2020; Singh and Sundaram, 2021). To tackle this issue, several solutions were presented in the literature. For instance, a study (Singh et al., 2020) has proposed a unique design solution to mitigate the bearing and the winding overheating challenges, by choosing an air-to-air cooled squirrel cage induction generator (SCIG) as the test subject and later modify it with a new design solution. In another paper (Singh et al., 2018), two air-to-air cooling configurations have been presented for the SCIG-based wind turbine, the totally enclosed air-to-air cooled (IC6A1A6) generator, and the open ventilated (IC3A1) generator. After a detailed theoretical analysis and experimental testing, the concluding remarks of a comparison between the two configurations have given a big motivation to implement the IC3A1 cooling in real practical production levels, since it showed experimentally a remarkable decrease of around 40°C in winding temperature, and about 30°C in the bearing temperature. Furthermore, to better secure the IC3A1 generator, a proper filter housing selection was proposed (Singh et al., 2019c). As for the inadequate and imbalanced bearing cooling for the IC6A1A6 generator, and in order to optimize its bearing life and reliability, a study (Singh et al., 2019b) has proposed two solutions, either a significant reducing in the allowed range of the bearing’s operating temperature, or the standard for rotating electrical machines (IEC60034/IEEE) should come up with different intervals of lubrication for the two bearing ends. The winding’s temperature rise can be caused also by the voltage peaks, which are created by the converter that feeds the WT’s generator, and hence a proper winding insulation system is recommended (Singh et al., 2019a).

On the other hand, due to the fluctuating nature of the wind speed, control strategies must be implemented to extract the maximum of available wind energy. Generally, a PMSG-based WT is connected through a full-scale voltage source converter, which is used to control the torque and the speed of the rotor. In this regard, control theory provides a clear insight into developing the convenient controller that involves the non-linear characteristics and results in maximum energy production with fewer leakages (Vinodkumar et al., 2019).

However, a PMSG-based WT is a highly complex dynamical system containing multiple structures connected all together, which introduces multi-variabilities, nonlinearities, parametric uncertainties, high degree of freedom, and uncertain external disturbances. These complexities can cause undesired behavior of PMSG and hence destroy its stability. That is why the control of these types of generators is still a challenging domain of research, since classical controllers are not very effective with the presence of such complexities (Geng et al., 2011a, 2011b; Kim et al., 2010; Li et al., 2010; Quang and Dittrich, 2015; Wu et al., 2009; Zhang et al., 2012) . Alternatively, other adaptive techniques have been used to control the PMSG with its uncertainties and disturbances, and we cite here some examples: adaptive nonlinear control of PMSG-based wind energy conversion system (WECS) (Magri et al., 2013), adaptive backstepping feedback control implemented and validated under dSPACE board (Youness et al., 2019), passivity-based sliding-mode control design (Yang et al., 2018), and robust control under uncertainty of stator parameters (Zu et al., 2011). However, these control strategies require an accurate mathematical model of the dynamical system. To achieve this, other modern optimized and intelligent techniques are proposed in the literature (Alizadeh and Kojori, 2015; Beddar et al., 2016; Hong et al., 2013; Jan et al., 2008; Lin et al., 2011; Mousa et al., 2019; Rayd et al., 2014).

Recently, thanks to the remarkable development in power electronics and microprocessor systems (microcontrollers, digital signal processing calculators, etc.), adaptive controllers based on fuzzy logic, artificial neural network (ANN), and bio-inspired algorithms are very popular for the control of PMSG-based WECSs. ANN-based optimization algorithm has become an effective method for the control of linear and nonlinear systems, as it shows a high accuracy and robustness against internal and external disturbances (Aguilar-Mejía et al., 2015; Poznyak et al., 2001; Rivals and Personnaz, 1998). Extended versions of ANN algorithms have been found among the most conventional tools in control engineering, identification, and functional approximations (de Jesús Rubio and Yu, 2007; Igelnik and Pao, 1995; Ren and Chen, 2006). The backstepping control optimized by ANN algorithms has a strong ability of handling the uncertain information and can be easily used for the control of systems that are too complex to have an accurate mathematical model. Actually, such controllers are also widely used for the estimation and the prediction of dynamical systems in both continuous and discrete-time. The techniques implemented in discrete-time have the advantage of being able to be directly implemented in digital hardware (Lewis et al., 2002; Sanchez and Ornelas-Tellez, 2013; Sarangapani, 2006).

Unfortunately, one of the most challenges that must be considered when opting for the backstepping control optimized by ANN-based uncertainty compensation is the stability, and especially when the system is in discrete-time (Lewis et al., 2002; Sarangapani, 2006). Over the past decade, the stability analysis of such controllers has been considered as an essential and necessary area of research (Arik, 2014; Lewis et al., 2002; Luo et al., 2014; Mohamad, 2008; Sarangapani, 2006; Zhang and Han, 2014), and was discussed mainly in Lyapunov stability theory and in its other extended versions (Dai et al., 2009; Eghbal et al., 2013; Griggs et al., 2010; Zhu et al., 2007).

The rest of the paper is structured as follows. In Section “Mathematical model description and problem formulation,” an aerodynamic-based model of WT is presented along with a simulation of the output curves related to the power coefficient and the mechanical power. Besides, a detailed mathematical description of PMSG’s electromechanical behavior is introduced. Section “Strategy of the proposed control design” is devoted to presenting the scheme of the control strategy, and an in-depth mathematical formulation in discrete-time for both the proportional integral (PI) speed controller, and the proposed adaptive backstepping controller based on ANN compensation. Furthermore, in section “Stability analysis of the proposed controller,” discrete Lyapunov function is considered to evaluate the PMSG stability toward perturbations. Simulations are implemented through using several reference data inputs for the wind speed in order to verify the robustness of the proposed control technique, and are shown in section “Simulation results.” The results are shown and discussed through tracking performance-based comparison between the proposed control and the classical direct torque control (DTC). The last section presents the concluding remarks and future research directions for the paper.

Mathematical model description and problem formulation

WECS consists of two essential parts, machine side system (MSS) and grid side system (GSS). The MSS that is the studied part in this paper, is composed of WT, PMSG, sensors, AC-DC machine side converter (MSC), and digital signal processing calculator.

Wind turbine model

WTs convert the kinetic energy present in the wind into mechanical energy by producing a mechanical torque $T_{m}$ . The power coefficient $C_{p}$ gives the fraction of the kinetic energy that is converted into mechanical energy by the wind turbine. It depends on the tip-speed ratio $λ$ and the adjustable blade pitch angle $β$ (Hore and Sarma, 2018).

The two equations below, show the kinetic power $P_{w}$ captured by the turbine, and the resulted mechanical one $P_{m}$ transmitted to the PMSG, respectively (Hong et al., 2013).

\begin{matrix} P_{w} = 0.5 ρ {AV}_{w}^{3} \\ P_{m} = 0.5 ρ C_{p} (λ, β) {AV}_{w}^{3} \end{matrix}

(1)

Where $C_{p}$ is the power coefficient, $ρ$ is the air density $(kg . m^{3})$ , $A$ is the area swept by the rotor blades, and $V_{w}$ is the wind speed $(m . s^{- 1})$ . $C_{p}$ is calculated by:

\begin{matrix} C_{p} = 0.73 (\frac{151}{λ_{i}} - 0.58 β - 0.002 β^{2.14} - 13.2) \exp (\frac{- 18.4}{λ_{i}}) \\ λ_{i} = \frac{1}{\frac{1}{λ - 0.02 β} - \frac{0.003}{β^{3} + 1}} \end{matrix}

(2)

Where $λ = \frac{ω_{r} r}{V_{w}}$ is the tip-speed ratio, $r$ is the wind turbine blade radius, $β$ is the adjustable pitch angle, and $ω_{r}$ is the turbine speed.

In Figure 1, a family of power coefficient curves versus the tip-speed ratio $λ$ for different values of blade pitch angle $β$ are shown. From such figure, it is clear that the power coefficient is remarkably influenced by the pitch angle. For each value of $β$ , there is an optimum value $C_{p - opt}$ of the power coefficient at which the turbine operates at the best performances through producing maximum power, and the optimum value $β_{opt}$ that results in maximum power depends on the turbine and also on the wind speed. The rest of the simulations presented in this work is performed at $β = 0$ , then $C_{p - opt} = 0.4412$ and the optimal tip-speed ratio is $λ_{opt} = 6.9444$ .

Figure 1.

Power coefficient versus tip-speed ratio for different values of wind blade pitch angle.

Furthermore, the curves that describe the wind turbine mechanical power $P_{m}$ versus rotor speed $ω_{r}$ are obtained for different values of wind speed $V_{m}$ and shown in Figure 2. At a fixed wind speed, the mechanical power reaches its highest level at a certain optimum rotor mechanical speed. So, in order to ensure the extraction of maximum power during the wind speed variation, the WT should be controlled to operate at its optimum rotational speed $ω_{r}$ (Elbeji et al., 2021).

Figure 2.

WT mechanical power versus rotor speed for different values of wind speed.

PMSG model

The mechanical torque $T_{m}$ is expressed as the quotient of the power transmitted to PMSG by its rotor mechanical speed, as follows (Hong et al., 2013).

\begin{array}{l} T_{m} = \frac{P_{m}}{ω_{r}} \end{array}

(3)

By considering the simplifying conditions, assumptions, and physical laws, the three-phase mathematical model of PMSG can be expressed easily in the three-phase ( $a, b, c$ ) frame (Bose, 2002; Quang and Dittrich, 2015). Then, the ( $d, q$ ) axis of current, voltage, and flux are obtained from two transformations, the first one converts the three stationary phases ( $a, b, c$ ) model to two stationary phases model in ( $α, β$ ) frame, and the second converts the obtained model in ( $α, β$ ) frame to rotational model in ( $d, q$ ) frame. So, according to the above steps, the electromechanical behavior of PMSG system is described by the following discrete equations (Hong et al., 2013):

{\begin{matrix} L_{d} \frac{d i_{d}}{dt} = u_{d} - R i_{d} + p L_{q} ω_{r} i_{q} \\ L_{q} \frac{d i_{q}}{dt} = u_{q} - R i_{q} - p L_{d} ω_{r} i_{d} - p φ_{v} ω_{r} \\ J \frac{d ω_{r}}{dt} = T_{e} - T_{m} - B ω_{r} \\ \frac{d θ_{r}}{dt} = ω_{r} \\ ω_{e} = p ω_{r} \\ T_{e} = \frac{3 p}{2} (φ_{v} i_{q} + (L_{d} - L_{q}) i_{d} i_{q}) \end{matrix}

(4)

Where $R$ is the stator resistance, $θ_{r}$ is the rotor position, $u_{d}$ and $u_{q}$ are the stator voltages in the rotational reference frame, $i_{d}$ and $i_{q}$ are the direct and quadrature stator currents, $ω_{e}$ and $ω_{r}$ are the electrical and mechanical rotor speeds, $φ_{v}$ is the permanent magnetic flux generated by the magnets, $T_{e}$ is the electrical torque, $p$ is the number of pole pairs, $J$ is the rotor inertia, $B$ is the friction, $L_{d}$ and $L_{q}$ are the direct and quadrature stator inductances, and $T_{r}$ is the mechanical torque.

Furthermore, by using the following notations: $i_{d} (k) = x_{1} (k)$ , $i_{d} (k + 1) = x_{2} (k)$ , $i_{q} (k) = x_{3} (k)$ , $i_{q} (k + 1) = x_{4} (k)$ , $ω_{r} (k) = x_{5} (k)$ , $ω_{r} (k + 1) = x_{6} (k)$ , $θ_{r} (k) = x_{7} (k)$ , $θ_{r} (k + 1) = x_{8} (k)$ , $T_{e} (k) = x_{9} (k)$ , and $T_{e} (k + 1) = x_{10} (k)$ . Then, the PMSG dynamics can be rewritten according to Newton discretization method in Brunovsky discrete-time domain form as:

{\begin{matrix} x_{1} (k + 1) = x_{2} (k) \\ x_{2} (k + 1) = x_{2} (k) + (\begin{matrix} u_{d} (k) - R x_{2} (k) + p L_{q} x_{4} (k) x_{6} (k) \end{matrix}) (\begin{matrix} \frac{Ts}{Ld} \end{matrix}) \\ x_{3} (k + 1) = x_{4} (k) \\ x_{4} (k + 1) = x_{4} (k) + (\begin{matrix} u_{q} (k) - R x_{4} (k) - p L_{d} x_{2} (k) x_{6} (k) - p φ_{v} x_{6} (k) \end{matrix}) (\begin{matrix} \frac{Ts}{Lq} \end{matrix}) \\ x_{5} (k + 1) = x_{6} (k) \\ x_{6} (k + 1) = x_{6} (k) + (\begin{matrix} x_{10} (k) - T_{m} - B x_{6} (k) \end{matrix}) (\begin{matrix} \frac{Ts}{J} \end{matrix}) \\ x_{7} (k + 1) = x_{8} (k) \\ x_{8} (k + 1) = x_{8} (k) + x_{6} (k) T_{s} \\ x_{9} (k + 1) = x_{10} (k) \\ x_{10} (k + 1) = 1.5 p (φ_{v} x_{4} (k + 1) + (L_{d} - L_{q}) x_{2} (k + 1) x_{4} (k + 1)) \end{matrix}

(5)

Where $T_{s}$ is the sampling time. In fact, the parameters $φ_{v}$ , $R$ , $L_{d}$ , $L_{q}$ , $B$ , $J$ and mechanical torque $T_{m}$ are not constant during the operation, then we put:

\begin{matrix} \hat{B} = B + \tilde{B}; \hat{J} = J + \tilde{J}; {\hat{T}}_{m} = T_{m} + {\tilde{T}}_{m}; {\hat{L}}_{d} = L_{d} + {\tilde{L}}_{d} \\ {\hat{L}}_{q} = L_{q} + {\tilde{L}}_{q}; \hat{R} = R + \tilde{R}; {\hat{φ}}_{v} = φ_{v} + {\tilde{φ}}_{v} \end{matrix}

(6)

The parameters defined by cap depict parameter actual values, the tiled parameters depict the individual errors and the parameters without accent depict constant (nominal) values. Substituting nominal parameters in equation (5) by those caped in equation (6), the uncertain PMSG model described by the system of equation (7) is obtained.

{\begin{matrix} x_{1} (k + 1) = x_{2} (k) \\ x_{2} (k + 1) = x_{2} (k) + (\begin{matrix} u_{d} (k) - R x_{2} (k) + p L_{q} x_{4} (k) x_{6} (k) \end{matrix}) (\begin{matrix} \frac{T_{s}}{L_{d}} \end{matrix}) + [- \tilde{R} x_{2} (k) + p {\tilde{L}}_{q} x_{4} (k) \\ x_{6} (k) - (\begin{matrix} u_{d} (k) - R x_{2} (k) + p L_{q} x_{4} (k) x_{6} (k) \end{matrix}) (\begin{matrix} \frac{{\tilde{L}}_{d}}{L_{d}} \end{matrix})] (\begin{matrix} \frac{T_{s}}{L_{d} + {\tilde{L}}_{d}} \end{matrix}) \\ = f_{d} (x (k)) + τ_{d} (k) + d_{d} (k) \\ x_{3} (k + 1) = x_{4} (k) \\ x_{4} (k + 1) = x_{4} (k) + (\begin{matrix} u_{q} (k) - R x_{4} (k) - p L_{d} x_{2} (k) x_{6} (k) - p φ_{v} x_{6} (k) \end{matrix}) (\begin{matrix} \frac{T_{s}}{Lq} \end{matrix}) - [\tilde{R} x_{4} (k) \\ + p {\tilde{L}}_{d} x_{2} (k) x_{6} (k) + p {\tilde{φ}}_{v} x_{6} (k) + (\begin{matrix} u_{q} (k) - R x_{4} (k) - p L_{d} x_{2} (k) x_{6} (k) - p φ_{v} x_{6} (k) \end{matrix}) \\ (\begin{matrix} \frac{{\tilde{L}}_{q}}{L_{q}} \end{matrix})] (\begin{matrix} \frac{T_{s}}{L_{q} + {\tilde{L}}_{q}} \end{matrix}) \\ = f_{q} (x (k)) + τ_{q} (k) + d_{q} (k) \\ x_{5} (k + 1) = x_{6} (k) \\ x_{6} (k + 1) = x_{6} (k) + (x_{10} (k) - T_{m} - B x_{6} (k)) (\begin{matrix} \frac{T_{s}}{J} \end{matrix}) + [- {\tilde{T}}_{m} - \tilde{B} x_{6} (k) - (x_{10} (k) - T_{m} \\ - B x_{6} (k)) (\begin{matrix} \frac{\tilde{J}}{J} \end{matrix})] (\begin{matrix} \frac{T_{s}}{J + \tilde{J}} \end{matrix}) \\ = x_{6} (k) + (x_{10} (k) - T_{m} - B x_{6} (k)) (\begin{matrix} \frac{T_{s}}{J} \end{matrix}) + d_{ω} (k) \\ x_{7} (k + 1) = x_{8} (k) \\ x_{8} (k + 1) = x_{8} (k) + x_{6} (k) T_{s} \\ x_{9} (k + 1) = x_{10} (k) \\ x_{10} (k + 1) = 1.5 p (\begin{matrix} φ_{v} + (L_{d} - L_{q}) x_{2} (k + 1) \end{matrix}) x_{4} (k + 1) + 1.5 p ({\tilde{φ}}_{v} x_{4} (k + 1) + ({\tilde{L}}_{d} - {\tilde{L}}_{q}) \\ x_{2} (k + 1) x_{4} (k + 1)) \\ = 1.5 p (\begin{matrix} φ_{v} x_{4} (k + 1) + (L_{d} - L_{q}) x_{2} (k + 1) x_{4} (k + 1) \end{matrix}) + d_{T} (k) \end{matrix}

(7)

Where $x_{i} (k) \in ℜ^{10}$ , $i = {1, 2 . . ., 10}$ , and $τ_{i} (k) \in ℜ^{2}$ , $i = {d, q}$ . The vector $d_{i} (k) \in ℜ^{4}$ , $i = {d, q, ω, T}$ denotes an unknown disturbance acting on the system at the $k^{th}$ time with a known constant.

From the equation (7), we conclude that the actuator outputs $τ_{i} (k)$ are related to the uncertain control inputs $u_{i} (k)$ through the following nonlinear expression, $i = {d, q}$ :

{\begin{matrix} τ_{d} (k) = u_{d} (k) (\begin{matrix} \frac{T_{s}}{L_{d}} \end{matrix}) + [- \tilde{R} x_{2} (k) + p {\tilde{L}}_{q} x_{4} (k) x_{6} (k) - (u_{d} (k) - R x_{2} (k) + p L_{q} x_{4} (k) \\ x_{6} (k)) (\begin{matrix} \frac{{\tilde{L}}_{d}}{L_{d}} \end{matrix})] (\begin{matrix} \frac{T_{s}}{L_{d} + {\tilde{L}}_{d}} \end{matrix}) \\ τ_{q} (k) = u_{q} (k) (\begin{matrix} \frac{T_{s}}{L_{q}} \end{matrix}) + [- \tilde{R} x_{4} (k) - p {\tilde{L}}_{d} x_{2} (k) x_{6} (k) - p {\tilde{φ}}_{v} x_{6} (k) - (u_{q} (k) - R x_{4} (k) \\ - p L_{d} x_{2} (k) x_{6} (k) - p φ_{v} x_{6} (k)) (\begin{matrix} \frac{{\tilde{L}}_{q}}{L_{q}} \end{matrix})] (\begin{matrix} \frac{T_{s}}{L_{q} + {\tilde{L}}_{q}} \end{matrix}) \end{matrix}

(8)

Since a backlash is a dynamic nonlinearity, the equation (8) can be modeled by the following backlash in discrete time domain (Lewis et al., 2002; Sarangapani, 2006):

\begin{matrix} τ_{i} (k) = Backlash (τ_{i} (k - 1), u_{i} (k - 1), u_{i} (k)) \\ = {\begin{matrix} m_{i} u_{i} (k) if ((u_{i} (k) > 0 and (u_{i} (k) = m_{i} . τ_{i} (k) - m_{i} . d_{i +} (k))) \\ or (u_{i} (k) < 0 and (u (k) = m_{i} . τ_{i} (k) - m_{i} . d_{i -} (k))) \\ 0 Otherwise \end{matrix} \end{matrix}

(9)

Where $u_{i} (k - 1)$ and $u_{i} (k)$ are the inputs of backlash function, $τ_{i} (k - 1)$ is the state, and $m_{i}$ , $d_{i +}$ , $d_{i -}$ are unknown values for $i = {d, q}$ .

The control law of this nonlinear uncertain PMSG system in discrete-time is based on the following assumptions (Lewis et al., 2002; Sarangapani, 2006):

Assumption 1. The desired trajectory is assumed to be bounded in the sense that

$‖ \begin{matrix} x_{(i, 1)}^{*} (k) \end{matrix} ‖^{T} \leq X_{(i, 1)}$ and $‖ \begin{matrix} x_{(i, 2)}^{*} (k) \end{matrix} ‖^{T} \leq X_{(i, 2)}$ for a known bound $X_{(i, 1)}$ and $X_{(i, 2)}$ , where $i = {d, q}$ and $x_{(i, 1)}^{*} (k)$ is the delayed value of $x_{(i, 2)}^{*} (k)$ .

Assumption 2. The nonlinear functions $f_{i} (x (k))$ are assumed to be unknown, but a fixed estimate ${\hat{f}}_{i} (x (k))$ is assumed known such that the functional estimation error ${\tilde{ε}}_{i} (x (k)) = {\hat{f}}_{i} (x (k)) - f_{i} (x (k))$ satisfies $∥ {\tilde{ε}}_{i} (x (k)) ∥ \leq f_{iM} (x (k))$ for some known bounding function $f_{iM} (x (k))$ , $i = {d, q}$ .

Assumption 3. The unknown disturbance is also assumed to satisfy: $∥ τ_{d} ∥ \leq τ_{M}$ , with $τ_{M} (k)$ a known positive constant.

Assumption 4. The unknown values $m_{i}$ , $d_{i +}$ , and $d_{i -}$ are bounded by unknown constants $M_{i}$ , $D_{i +}$ , and $D_{i -}$ as: $∥ m_{i} ∥ \leq M_{i}$ , $∥ d_{i +} ∥ \leq D_{i +}$ , and $∥ d_{i -} ∥ \leq D_{i -}$ .

Assumption 5. The dynamical backlash is invertible. Its inverse is indicated by the following equation:

u_{i} (k) = Backlas h^{- 1} (u_{i} (k - 1), w_{i} (k - 1), w_{i} (k))

(10)

Where $w_{i} (k - 1)$ and $w_{i} (k)$ are the input signals of the backlash inverse that generates the signal $u_{i} (k)$ , this latter is subsequently sent to the backlash to produce $τ_{i} (k)$ .

Assumption 6. The backlash parameters $m_{i}$ , $d_{i +}$ , and $d_{i -}$ used in the backlash inverse are exactly correct.

Our objective is to control the speed and currents of PMSG such that the tracking errors of the uncertain PMSG converge closely to zero in the presence of parameter uncertainties and external disturbances. To achieve this objective, two ANN compensators (ANNCs) are developed and implemented in discrete-time. Besides, a rigorous stability analysis via Lyapunov function is applied considering the uncertainties explicitly.

Strategy of the proposed control design

Main design scheme of the proposed adaptive controller

The tracking control system is presented schematically in Figure 3. The overall system consists of the WT, uncertain PMSG fed by variable mechanical torque, space vector pulse width modulation (SVPWM), voltage-source rectifier (VSR), two backlash actuators, two estimated functions, two filters, two proportional-plus-derivative (PD) controllers, two ANNCs based on the backstepping technique, and three essential loops.

Figure 3.

The adaptive controller’s overall structure for the uncertain PMSG-based WT.

The controllers employ a structure of cascade control loop including a speed loop and two current loops, and are used to stabilize the $d$ -axis and $q$ -axis currents errors according to $λ_{i}$ Hurwitz gains, which are followed by two inner PD controllers with gains ( $K_{i}$ and $κ_{i}$ , $i = {d, q}$ ). The PI controller is also used to track the desired speed. As shown in Figure 3, the angular speed $ω_{r}$ can be obtained from the position sensor. The currents $i_{d}$ and $i_{q}$ are calculated from the three-phase currents $i_{a}$ and $i_{b}$ , which are obtained from measurement by Clarke and Park transformations.

According to the actual value of the angle $θ_{r}$ , the backlash actuator outputs can be transformed from the rotating ( $d, q$ ) frame to the stator-fixed coordinates $(α, β)$ , to be the inputs of SVPWM that activates the transistors’ state of the rectifier. Finally, the suitable stator voltages are transmitted to the rectifier with respect to the desired active power. But the uncertainty of PMSG destroys the tracking, and then the PD controllers are not capable of compensating it. To correct the errors due to this uncertainty, two ANNCs are added to the currents’ loops.

More details about the design procedure for the proposed control strategy (Figure 3) are presented step-by-step in the following subsections.

Discrete-time PI controller for the speed

To handle the speed trajectory, a PI controller is considered. The expressions of its gains $K_{p}$ and $K_{I}$ are determined by the following equations, respectively (Rayd et al., 2014):

{\begin{matrix} K_{I} = \frac{4 J R^{2}}{L_{q}^{2}} \\ K_{p} = \frac{K_{I} L_{q}}{R} \end{matrix}

(11)

The actual speed value is compared to its set reference value. To track the speed reference perfectly, the speed errors defined by the equation (12) are processed by the equation (13).

{\begin{matrix} e_{5} (k) = x_{5}^{*} (k) - x_{5} (k) \\ e_{6} (k) = x_{6}^{*} (k) - x_{6} (k) \\ e_{5} (k + 1) = x_{5}^{*} (k + 1) - x_{5} (k + 1) \\ e_{6} (k + 1) = x_{6}^{*} (k + 1) - x_{6} (k + 1) \end{matrix}

(12)

Where $e_{5} (k)$ , $e_{6} (k)$ , $e_{5} (k + 1)$ , and $e_{6} (k + 1)$ represent the speed errors at discrete-times $k$ and $(k + 1)$ .

{\begin{matrix} x_{9}^{*} (k) = x_{9}^{*} (k - 1) + K_{P} . e_{5} (k) + (K_{I} T_{s} - K_{P}) . e_{5} (k - 1) \\ x_{10}^{*} (k) = x_{10}^{*} (k - 1) + K_{P} . e_{6} (k) + (K_{I} T_{s} - K_{P}) . e_{6} (k - 1) \\ x_{9}^{*} (k + 1) = x_{9}^{*} (k) + K_{P} . e_{5} (k + 1) + (K_{I} T_{s} - K_{P}) . e_{5} (k) \\ x_{10}^{*} (k + 1) = x_{10}^{*} (k) + K_{P} . e_{6} (k + 1) + (K_{I} T_{s} - K_{P}) . e_{6} (k) \end{matrix}

(13)

For $i_{d}^{*} = 0$ strategy, the desired trajectory of $i_{q}^{*}$ is related to the desired trajectory of the torque with the following equations:

{\begin{matrix} x_{3}^{*} (k) = \frac{2}{3 p φ_{v}} x_{9}^{*} (k) \\ x_{3}^{*} (k + 1) = \frac{2}{3 p φ_{v}} x_{9}^{*} (k + 1) \\ x_{4}^{*} (k) = \frac{2}{3 p φ_{v}} x_{10}^{*} (k) \\ x_{4}^{*} (k + 1) = \frac{2}{3 p φ_{v}} x_{10}^{*} (k + 1) \end{matrix}

(14)

Hence, by substituting equation (13) in equation (14), the desired trajectory of $i_{q}^{*}$ is related to the tracking errors of the speed with the following equations:

{\begin{matrix} x_{3}^{*} (k) = x_{3}^{*} (k - 1) + \frac{2}{3 p φ_{v}} (\begin{matrix} K_{P} e_{5} (k) + (K_{I} T_{s} - K_{P}) e_{5} (k - 1) \end{matrix}) \\ x_{3}^{*} (k + 1) = x_{3}^{*} (k) + \frac{2}{3 p φ_{v}} (\begin{matrix} K_{P} e_{5} (k + 1) + (K_{I} T_{s} - K_{P}) e_{5} (k) \end{matrix}) \\ x_{4}^{*} (k) = x_{4}^{*} (k - 1) + \frac{2}{3 p φ_{v}} (\begin{matrix} K_{P} e_{6} (k) + (K_{I} T_{s} - K_{P}) e_{6} (k - 1) \end{matrix}) \\ x_{4}^{*} (k + 1) = x_{4}^{*} (k) + \frac{2}{3 p φ_{v}} (\begin{matrix} K_{P} e_{6} (k + 1) + (K_{I} T_{s} - K_{P}) e_{6} (k) \end{matrix}) \end{matrix}

(15)

Discrete-time ANN backlash compensators

In this work, multi-layer perceptron neural networks (MLPNNs) are used as compensators (Figure 4). These compensators have dynamic weights between hidden and output neurons, and static weights between input and hidden layer neurons. Consequently, they possess lesser number of weights to be updated. As a result, they can be trained quickly and easily. The concept of the adaptation technique is to minimize the tracking errors of the uncertain PMSG. Such errors are related to the desired trajectories $x_{i, 2}^{*} (k)$ and their first delayed values $x_{i, 1}^{*} (k)$ , and are expressed as:

{\begin{matrix} e_{i, 1} (k) = x_{i, 1} (k) - x_{i, 1}^{*} (k) \\ e_{i, 2} (k) = x_{i, 2} (k) - x_{i, 2}^{*} (k) \end{matrix}

(16)

Equation (17) defines a filtered tracking error $r_{i} (k)$ calculated from the tracking errors $e_{i, 2} (k)$ and their delayed values $e_{i, 1} (k)$ .

r_{i} (k) = e_{i, 2} (k) + λ_{i} e_{i, 1} (k); i = {d, q}

(17)

Where $λ_{i}$ are chosen according to Hurwitz stability criterion. Equation (17) is used also to calculate $r_{i} (k + 1)$ for the discrete-time $(k + 1)$ . From equations (7), (16), and (17), we obtain the expression of the filtered tracking errors $r_{i} (k + 1)$ :

r_{i} (k + 1) = f_{i} (x (k)) - x_{i, 2}^{*} (k + 1) + λ_{i} e_{i, 1} (k) + τ_{i} (k) + d_{i} (k); i = {d, q}

(18)

Where the nonlinear functions $f_{d} (x (k))$ and $f_{q} (x (k))$ can be estimated successively by the terms ${\hat{f}}_{d} (x (k))$ and ${\hat{f}}_{q} (x (k))$ as follows:

{\begin{matrix} f_{d} (x (k)) = x_{2} (k) + (- R x_{2} (k) + p L_{q} x_{4} (k) x_{6} (k)) (\frac{T_{s}}{L_{d}}) \\ f_{q} (x (k)) = x_{4} (k) + (- R x_{4} (k) - p L_{d} x_{2} (k) x_{6} (k) - p φ_{v} x_{6} (k)) (\frac{T_{s}}{L_{q}}) \end{matrix}

(19)

A robust compensation scheme for the unknown term $f_{i} (x (k))$ is provided by selecting the following tracking controller:

\begin{matrix} τ_{i}^{*} (k) = K_{i} r_{i} (k + 1) - {\hat{f}}_{i} (x (k)) + x_{i, 2}^{*} (k + 1) - λ_{i} e_{i, 1} (k + 1); i = {d, q} \end{matrix}

(20)

The term $τ_{i}^{*} (k)$ is the actuator output, which is also the desired control signal. The feedback gain matrix $K_{i}$ is often selected diagonal and greater than zero (Elmas et al., 2008).

The complete error of the system dynamics is expressed by the following equations:

{\begin{matrix} {\tilde{τ}}_{d} (k) = τ_{d}^{*} (k) - τ_{d} (k) \\ {\tilde{τ}}_{q} (k) = τ_{q}^{*} (k) - τ_{q} (k) \end{matrix}

(21)

Using the equation (20), the equation (18) can be rewritten as follows:

r_{i} (k + 1) = K_{i} . r_{i} (k) + {\tilde{ε}}_{i} (\begin{matrix} x (k) \end{matrix}) + d_{i} (k) - {\tilde{τ}}_{i} (k); i = {d, q}

(22)

The substitution of $τ_{i} (k)$ by $B_{i} (\begin{matrix} τ_{i} (k), u_{i} (k), u_{i} (k + 1) \end{matrix})$ in equation (21) gives the following equation:

{\tilde{τ}}_{i} (k) = τ_{i}^{*} (k) - τ_{i} (k) = τ_{i}^{*} (k) - B_{i} (\begin{matrix} τ_{i} (k), u_{i} (k), u_{i} (k + 1) \end{matrix}); i = {d, q}

(23)

Furthermore, the ideal backlash inverse and its approximation is given by:

\begin{matrix} u_{i} (k + 1) = B_{i}^{- 1} (\begin{matrix} u_{i} (k), τ_{i} (k), τ_{i} (k + 1) \end{matrix}) \\ {\hat{u}}_{i} (k + 1) = {\hat{B}}_{i}^{- 1} (\begin{matrix} {\hat{u}}_{i} (k), τ_{i} (k), {\hat{τ}}_{i} (k + 1) \end{matrix}) \end{matrix}

(24)

The backlash errors are defined by the following equation:

{\tilde{B}}_{i} (\begin{matrix} τ_{i} (k), u_{i} (k), u_{i} (k + 1) \end{matrix}) = B_{i} (\begin{matrix} τ_{i} (k), u_{i} (k), u_{i} (k + 1) \end{matrix}) - {\hat{B}}_{i} (\begin{matrix} τ_{i} (k), u_{i} (k), u_{i} (k + 1) \end{matrix})

(25)

Thus,

τ_{i} (k + 1) = {\hat{τ}}_{i} (k + 1) + {\tilde{τ}}_{i} (k + 1) = {\hat{B}}_{i} (\begin{matrix} τ_{i} (k), u_{i} (k), u_{i} (k + 1) \end{matrix}) + {\tilde{B}}_{i} (\begin{matrix} τ_{i} (k), u_{i} (k), u_{i} (k + 1) \end{matrix})

(26)

In order to design a stable closed-loop system with backlash compensation, we select a nominal backlash inverse ${\hat{u}}_{i} (k + 1)$ . The system dynamics can be represented as follows:

{\hat{u}}_{i} (k + 1) = - κ_{i} {\tilde{τ}}_{i} (k) + δ_{i} (k) + \hat{W} (k)^{T} σ (V^{T} X_{NN} (k)); i = {d, q}

(27)

The term $\hat{W} (k)^{T} σ (V^{T} X_{NN} (k))$ is used to compensate the backlash inversion and the filtered error by using the ANN approximation property. Where $\hat{W} (k)$ and $V (k)$ are the weights of ANN. The term $δ_{i} (k)$ is the dynamic filter of $τ_{i}^{*} (k)$ processed by a discrete time filter $\frac{az}{z + a}$ . The output $δ_{i} (k)$ is calculated by the following equation:

δ_{i} (k) = a_{i} δ_{i} (k - 1) - a_{i} τ_{i}^{*} (k); i = {d, q}

(28)

Where $a_{i}$ represents the constant value of the discrete-time filter.

A neural network compensator (ANNC) is composed of: Input, hidden, and output layers (Figure 4). The input layer vector of each $ANN C_{i}$ , $i = {d, q}$ is expressed by:

X^{i} (k) = {[\begin{matrix} 1 & r_{i} {(k)}^{T} & {(x_{i}^{*})}^{T} (k) & {\tilde{τ}}_{i} {(k)}^{T} & τ_{i} {(k)}^{T} \end{matrix}]}^{T}; i = {d, q}

(29)

The hidden output $j$ of $ANN C_{i}$ , $i = {d, q}$ is expressed by the following sigmoidal function:

Y_{j}^{i} (k) = \frac{1}{1 + \exp (\begin{matrix} \sum_{ι = 1}^{n_{1}} W_{(ι, j)}^{i} (k) X_{ι}^{i} (k) + {Rand}_{j}^{i} \end{matrix})}; i = {d, q}

(30)

Where ${Rand}_{j}^{i}$ is the $j^{th}$ normalized random bias value of the compensator $ANN C_{i}$ , $i = {d, q}$ , $ι = {1, 2, \dots, n_{1}}$ , $j = {1, 2, \dots, n_{2}}$ , and $n_{1}$ , $n_{2}$ are respectively the numbers of the input layer neurons and the hidden layer neurons.

Figure 4.

The structure of artificial neural network backlash compensators.

The output layer of each $ANN C_{i}$ , $i = {d, q}$ has one output neuron, which is expressed by:

Z^{i} (k) = \sum_{j = 1}^{n_{2}} W_{(j, h)}^{i} (k) Y_{j}^{i} (k)

(31)

Then, the expression of $u_{i} (k + 1)$ at discrete-time $(k + 1)$ is:

u_{i} (k + 1) = - κ_{i} (τ_{i}^{*} (k) - τ_{i} (k)) + δ_{i} (k) + Z_{j}^{i} (k)

(32)

Where $κ_{i}$ is a design parameter which is always selected greater than zero (Elmas et al., 2008). Finally $τ_{i} (k + 1)$ is expressed as:

τ_{i} (k + 1) = Backlash (τ_{i} (k), u_{i} (k), u_{i} (k + 1))

(33)

\begin{matrix} τ_{i} (k + 1) = {\begin{matrix} m_{i} . u_{i} (k + 1) if (((u_{i} (k + 1) > 0) \\ and (u_{i} (k) = m_{i} . τ_{i} (k) - m_{i} . d_{i +})) \\ or ((u_{i} (k + 1) < 0) \\ and (u_{i} (k) = m_{i} . τ_{i} (k) - m_{i} . d_{i -}))) \\ 0 Otherwise \end{matrix} \\ i = {d, q} \end{matrix}

(34)

Where $d_{i +}$ , $d_{i -}$ , and $m_{i}$ are the backlash parameters. The right choice of their values provides a correct backlash.

The weights vector vector $W_{(ι, j)}^{i}$ between neurons of layers $ι$ and $j$ of $ANN C_{i}$ ( $i = {d, q}$ ) are not time-dependent, since they are selected randomly at the initial time to provide a basis (Igelnik and Pao, 1995), and then they are kept constant through the tuning process. For the hidden layer, the ANN weights are adjusted in real time with no preliminary offline learning required. The online tuning law of hidden layer weights are updated by the following expression (Sarangapani, 2006):

\begin{matrix} {\hat{W}}^{i} (k + 1) = {\hat{W}}^{i} (k) + α_{i} Y^{i} (k) [\begin{matrix} r_{i}^{T} (k + 1) + {\tilde{τ}}_{i} (k + 1) \end{matrix}] - γ_{i} ∥ I - α_{i} Y^{i} (k) Y^{iT} (k) ∥ {\hat{W}}^{i} (k); \\ i = {d, q} \end{matrix}

(35)

Where $γ_{i}$ and $α_{i}$ are the ANNs learning rates (Lewis et al., 2002), $I$ is the identity matrix of size $(n_{2} + 1) (n_{2} + 1)$ , and $Y^{i} (k) Y^{iT} (k)$ is expressed by:

Y^{i} (k) Y^{iT} (k) = \sqrt{\sum_{ι = 1}^{n_{2}} \sum_{j = 1}^{n_{2}} {(\begin{matrix} Y_{(i + 1)}^{i} (k) Y_{(j + 1)}^{i} (k) \end{matrix})}^{2}}; i = {d, q}

(36)

The convergence of equation (35) in such a manner that the closed-loop stability is guaranteed, can be ensured by Lyapunov stability theory (Lewis et al., 2002; Sarangapani, 2006).

Stability analysis of the proposed controller

To demonstrate the boundedness of all closed-loop signals, the following positive Lyapunov function in discrete-time is selected:

V (k) = V_{1} (k) + V_{2} (k)

(37)

The positive $V_{1} (k)$ represents the Lyapunov function for the closed-loop speed, which is used to weight the speed error $e_{ω} (k)$ . The positive $V_{2} (k)$ is used to weight the filtered tracking errors $r_{i} (k)$ , the ANN estimation errors ( ${\tilde{W}}_{i} (k)$ ), and the backlash errors ${\tilde{τ}}_{i} (k)$ ; $i = {d, q}$ . These functions are defined by the following equations:

V_{1} (k) = \frac{1}{2} e_{ω}^{2} (k)

(38)

\begin{matrix} V_{2} (k) = 2 r_{d}^{T} (k) r_{d} (k) + 2 r_{q}^{T} (k) r_{q} (k) + 2 r_{d}^{T} (k) {\tilde{τ}}_{d} (k) + 2 r_{q}^{T} (k) {\tilde{τ}}_{q} (k) + {\tilde{τ}}_{d}^{T} (k) {\tilde{τ}}_{d} (k) + {\tilde{τ}}_{q}^{T} (k) {\tilde{τ}}_{q} (k) \\ + \frac{1}{α_{d}} tr ({\tilde{W}}_{d}^{T} (k) {\tilde{W}}_{d} (k)) + \frac{1}{α_{q}} tr ({\tilde{W}}_{q}^{T} (k) {\tilde{W}}_{q} (k)) \end{matrix}

(39)

The variation of the equation (38) is given as:

Δ V_{1} (k) = \frac{1}{2} e_{ω}^{2} (k + 1) - \frac{1}{2} e_{ω}^{2} (k) = \frac{1}{2} {(\begin{matrix} ω_{r}^{*} (k + 1) - ω_{r} (k + 1) \end{matrix})}^{2} - \frac{1}{2} e_{ω}^{2} (k)

(40)

Using the PI controller for the speed via loop design method, and according to Euler integration, we consider the virtual desired $q$ -axis current $i_{q}^{*}$ and the desired $d$ -axis current $i_{d}^{*}$ . Then, the expressions of input currents are:

{\begin{matrix} i_{d}^{*} = 0 \\ i_{q}^{*} = \frac{2}{3 p φ_{v}} (B ω_{r} + T_{m} + J (K_{p} e_{ω} (k) + K_{I} \sum e_{ω} (k)) \end{matrix}

(41)

Where $K_{I}$ and $K_{p}$ are positive constant gains of the PI controller. Substituting equations (5) and (41) in equation (40), $Δ V_{1} (k)$ is expressed by the following equation:

\begin{matrix} Δ V_{1} (k) = \frac{1}{2} [ω_{r}^{*} (k) + (\begin{matrix} (T_{e} (k) - T_{m} - B ω_{r}^{*} (k) \end{matrix}) (\begin{matrix} \frac{T_{s}}{J} \end{matrix}) - ω_{r}^{*} (k) + (\begin{matrix} T_{e}^{*} (k) - T_{m} - B ω_{r}^{*} (k) \end{matrix}) \\ (\begin{matrix} \frac{T_{s}}{J} \end{matrix})]^{2} - \frac{1}{2} e_{ω}^{2} (k) \\ = - \frac{B T_{s}}{J} (\begin{matrix} 2 - \frac{B T_{s}}{J} \end{matrix}) e_{ω}^{2} (k) + 2 (\begin{matrix} 1 - \frac{B T_{s}}{J} \end{matrix}) \frac{T_{s}}{J} [T_{e}^{*} (k - 1) - T_{e} (k - 1) + K_{P} e_{ω} (k) \\ + (K_{I} T_{s} - K_{P}) e_{ω} (k - 1)] e_{ω} (k) + {(\begin{matrix} \frac{T_{s}}{J} \end{matrix})}^{2} [T_{e}^{*} (k - 1) - T_{e} (k - 1) + K_{P} e_{ω} (k) \\ + (K_{I} T_{s} - K_{P}) e_{ω} (k - 1)]^{2} \end{matrix}

(42)

Substituting equation (5) in equation (42), the following equation is obtained:

\begin{matrix} Δ V_{1} (k) = - \frac{B T_{s}}{J} (\begin{matrix} 2 - \frac{B T_{s}}{J} \end{matrix}) e_{ω}^{2} (k) + 2 (\begin{matrix} 1 - \frac{B T_{s}}{J} \end{matrix}) \frac{T_{s}}{J} e_{ω} (k) [(\begin{matrix} \frac{J}{T_{s}} + B + K_{P} \end{matrix}) e_{ω} (k) \\ + (\begin{matrix} \frac{J}{T_{s}} + (K_{I} Ts - K_{P}) \end{matrix}) e_{ω} (k - 1)] + {(\begin{matrix} \frac{T_{s}}{J} \end{matrix})}^{2} [(\begin{matrix} \frac{J}{T_{s}} + B + K_{P} \end{matrix}) e_{ω} (k) \\ + (\begin{matrix} \frac{J}{T_{s}} + (K_{I} T_{s} - K_{P}) \end{matrix}) e_{ω} (k - 1)]^{2} \end{matrix}

(43)

Finally, the expression of Lyapunov function variation $Δ V_{1} (k)$ is:

\begin{matrix} Δ V_{1} (k) = - \frac{B T_{s}}{J} (\begin{matrix} 2 - \frac{B T_{s}}{J} \end{matrix}) e_{ω}^{2} (k) + 2 (\begin{matrix} 1 - \frac{B T_{s}}{J} \end{matrix}) \frac{T_{s}}{J} [e_{ω} (k) + K_{0} e_{ω} (k - 1)] e_{ω} (k) \\ + {(\begin{matrix} \frac{T_{s}}{J} \end{matrix})}^{2} [e_{ω} (k) + K_{0} e_{ω} (k - 1)]^{2} \end{matrix}

(44)

Where, $K_{0} = \frac{(\begin{matrix} \frac{J}{T_{s}} + (K_{I} Ts - K_{P}) \end{matrix})}{(\begin{matrix} \frac{J}{T_{s}} + B + K_{P} \end{matrix})}$ is a positive constant, and $- \frac{B T_{s}}{J} (\begin{matrix} 2 - \frac{B T_{s}}{J} \end{matrix}) < 0$ according to the parameters values, which are declared in the simulation.

Then, according to Slotine and Li (1991), if $K_{0}$ is chosen properly for the controller, $(\begin{matrix} e_{ω} (k) + K_{0} e_{ω} (k - 1) \end{matrix})$ tends to zero. Hence, it is evident that the equation (44) is negative, meaning that the speed tracking errors are guaranteed to converge to zero.

On the other hand, the variation of $V_{2}$ is expressed by:

\begin{matrix} Δ V_{2} (k) = V_{2} (k + 1) - V_{2} (k) \\ = 2 r_{d}^{T} (k + 1) r_{d} (k + 1) + 2 r_{q}^{T} (k + 1) r_{q} (k + 1) + 2 r_{d}^{T} (k + 1) {\tilde{τ}}_{d} (k + 1) + 2 r_{d}^{T} (k + 1) \\ {\tilde{τ}}_{d} (k + 1) + {\tilde{τ}}_{d}^{T} (k + 1) {\tilde{τ}}_{d} (k + 1) {\tilde{τ}}_{q}^{T} (k + 1) {\tilde{τ}}_{q} (k + 1) + \frac{1}{α_{d}} tr ({\tilde{W}}_{d}^{T} (k + 1) {\tilde{W}}_{d} (k + 1)) \\ + \frac{1}{α_{q}} tr ({\tilde{W}}_{q}^{T} (k + 1) {\tilde{W}}_{q} (k + 1)) - 2 r_{d}^{T} (k) r_{d} (k) - 2 r_{q}^{T} (k) r_{q} (k) - 2 r_{d}^{T} (k) {\tilde{τ}}_{d} (k) - 2 r_{d}^{T} (k) \\ {\tilde{τ}}_{d} (k) - {\tilde{τ}}_{d}^{T} (k) {\tilde{τ}}_{d} (k) {\tilde{τ}}_{q}^{T} (k) {\tilde{τ}}_{q} (k) - \frac{1}{α_{d}} tr ({\tilde{W}}_{d}^{T} (k) {\tilde{W}}_{d} (k)) - \frac{1}{α_{q}} tr ({\tilde{W}}_{q}^{T} (k) {\tilde{W}}_{q} (k)) \end{matrix}

(45)

To not burden this paper, we note that the proof of $Δ V_{2} (k) < 0$ is similar to that determined in Lewis et al. (2002) and Sarangapani (2006). So, according to the standard Lyapunov theorem extension, the tracking error $r_{i} (k)$ , the actuator error ${\tilde{τ}}_{i} (k)$ , and the estimated ANN weights ${\tilde{W}}_{i} (k)$ , $i = {d, q}$ are globally, uniformly, and ultimately bounded (GUUB), and hence the proposed PMSG the proposed PMSG control scheme is theoretically stable in the presence of different parametric uncertainties and mechanical torque disturbances.

Simulation results

In this simulation, the rectifier is simulated by an ideal switching frequency $20 KHz$ . The generator is represented by its dynamic model in Park frame. Thus, after several simulation tests we found that the average execution time of our program is around $0.012 s$ for $10000$ simulation cycles, which means that the execution time needed for one cycle is around $1.2 μ s$ . Bearing in mind that practically the sensors, the analog/digital converters, the switching elements of the rectifier, and the digital signal processing (DSP) algorithms are time consuming, and hence it is practically difficult to achieve such system with small sampling period. Thus, in practice, convenient sampling periods that are equal or higher than $100 μ s$ , are normally selected for processing. In our work, to make our the proposed adaptive control feasible for the PMSG with the parameters shown in Table 1 below, it is simulated with the sampling time $T_{s} = 100 μ s$ , meaning that for $10000$ samples the time of simulation is $T = 1 s$ .

Table 1.

Parameters of PMSG-based WT.

Parameters	Symbols	Values (units)
$d$ -Axis magnetizing current	$I_{fd}$	46.75 $A$
$d$ -Axis mutual inductance	$L_{md}$	0.0048 $mH$
Stator resistance	$R$	1.47 $Ω$
$d$ -Axis inductance	$L_{d}$	0.00533 $mH$
$q$ -Axis inductance	$L_{q}$	0.00533 $mH$
Magnetic flux constant	$φ_{v}$	0.2244 $mWb$
Friction coefficient	$B$	0.00578 $N . m . ra d^{- 1} s^{- 1}$
Motor inertia	$J$	0.00132 $Kg . m^{2}$
Air density	$ρ$	1.25 $Kg . m^{3}$
Wind turbine blade radius	$r$	0.5 $m$
Optimum tip-speed ratio	$λ_{opt}$	6.9444 $rad$

In this work, the chosen values of PMSG parameters are given in Table 1 (Hong et al., 2013). The two gains $K_{p} = 0.3669$ and $K_{I} = 88.5612$ of the PI controller were calculated according to their expressions. In order to better show the robustness of the proposed controller, the parameters of the two backlash actuators are considered as follows: $m_{d} = 0.5$ , $m_{q} = 0.55$ , $d_{d +} = 0.2$ , $d_{d -} = 0.2$ , $d_{q +} = 0.2$ , and $d_{q -} = 0.2$ .

After several simulation tests, the following parameters values are chosen: The Hurwitz and PD controller parameters gains are $λ_{d} = 2.5$ , $λ_{q} = 0.23$ , $K_{d} = 0.0001285$ , $K_{q} = 0.1285$ , $κ_{d} = 2.25$ , $κ_{q} = 2.0$ . The number of hidden layer neurons, and their bounded weights values with sigmoidal activation are respectively chosen as: $n_{2} = 10$ , $W_{j, h}^{i} \in [- 0.1, + 0.1]$ . The first layer weights and bias are initialized randomly and uniformly distributed in $[- 0.1, + 0.1]$ and ${Rand}_{j}^{i} \in [- 100, + 100]$ . The filters that generate the signals $δ_{d}$ and $δ_{q}$ are implemented respectively with the gains $a_{d} = 0.05006$ and $a_{q} = 0.133425$ .

In order to demonstrate the high performance of the proposed technique, against parametric uncertainties of the generator, affected by temperature variation, wear-and-tear of the generator, etc., numerous PMSG parameters are set randomly to the following variations: Increasing $300 %$ in the stator resistance; $55 %$ in the rotor moment of inertia and the friction, decreasing $25 %$ in the stator inductances. The perturbed PMSG magnetic flux is expressed by the following function: $φ_{v} = - 0.05 t + 0.1546$ where $t$ is time.

The generator starts running under mechanical torque $T_{m}$ applied by the wind. The desired speed references is calculated by the expression $ω_{r} = \frac{V_{w} λ_{opt}}{r}$ , where $λ_{opt} = 6.9444$ .

Simulation tests were performed at different operating conditions. Such conditions allow the generator to run under disturbances of the mechanical torque, parameters uncertainties indicated above and the different values of wind speed as indicated below.

Operation under a step change in wind speed:

$V_{w} = 10 m . s^{- 1}$ , $V_{w} = 12 m . s^{- 1}$ , $V_{w} = 5 m . s^{- 1}$ , respectively in the following time intervals $[0 s; 0.25 s]$ , $[0.25 s; 0.75 s]$ , $[0.75 s; 1.0 s]$ .

The Figures 6 to 9 show the simulations with the conditions indicated above, where the wind speed is a step (Figure 5).

Figure 5.

Step wind speed reference.

The Figure 6 plots the reference and the actual speed responses. As is shown in the zoomed Figure 6(a), it is clear that the uncertain PMSG tracks precisely the reference speed trajectory with the proposed adaptive controller, whereas with the classical PI controller, it follows the reference quietly well at the beginning, but after the wind speed variation, the value of the mechanical torque changes too, then the classical controller loses its performance. The zoomed Figure 6(b) to 6(d) show that with the proposed method, the speed follows the desired value without oscillations in steady state time, and drops rapidly to the desired reference during mechanical torque application. In addition, the Figure 7 displays the speed tracking errors, which tend to zero in steady state with the proposed controller, but for the classical PI in DTC, the tracking errors rate is higher with large steady states.

Figure 6.

Speed response under mechanical torque disturbances and parameter uncertainties: (a) speed response, (b) zoom around $0 s$ , (c) zoom around $0.25 s$ , and (d) zoom around $0.75 s$ .

Figure 7.

Tracking errors of the rotor speed.

Figures 8 and 9 show respectively the dynamic evolution of mechanical power $P_{m}$ and power coefficient $C_{p}$ . The fluctuations that appear at the beginning and during the speed variation can be eliminated easily by applying the proposed control scheme, while the classical controller is unable to track the desired value perfectly. So, the simulation results related to power tracking control at steady state time show that the ANN backstepping controller has a superior control performance as compared to the conventional PI controller in DTC.

Figure 8.

Mechanical power evolution at step change in wind speed.

Figure 9.

Power coefficient evolution at step change in wind speed.

To maximize the aerodynamic efficiency, the WT should operate at $C_{p - opt}$ . This requires the tip-speed ratio to be kept near the optimal value continuously so as to track the wind speed variation.

In this work, the power coefficient $C_{p}$ is calculated by using the expression 2, and shown for each control strategy in Figure 9. From such figure it is clear that the $C_{p}$ obtained with the adaptive ANN backstepping control drops rapidly to maximum power range as compared to DTC, where the $C_{p}$ curve tracks the maximum power after a large steady state time. The observed shifts in $C_{p}$ curve at $0.25 s$ and $0.75 s$ are due to sudden changes in wind speed. As a result, the overall captured energy is optimized.

Operation under a random change in wind speed (Beddar et al., 2016):

V_{w} = 0.9 (10 + 2 \sin (2 π \frac{t}{T}) + 2 \sin (7 π \frac{t}{T}) + \sin (25 π \frac{t}{T})) + \sin (2 \frac{t}{T}) + 0.2 \sin (70 \frac{t}{T}) m . s^{- 1} .

(45)

Once the speed and current regulation system are designed with a step change in wind speed reference under parameters uncertainties and mechanical torque disturbances, we proceed to prove the effectiveness of the proposed adaptive controller by applying a random wind speed reference under high frequency fluctuations, and with the same conditions indicated above, as shown in Figure 10. The dynamic reference and the actual speed response of the proposed controller are given in Figure 11. We observe that the disturbances rejection capacity leads to a good speed tracking performance. In addition, the Figure 12 presents the evolution of the mechanical power $P_{m}$ , and it shows that the asymptotic speed tracking objective is obtained with a good sensibility and accuracy with the adaptive controller under different uncertainties.

Figure 10.

Random wind speed reference.

Figure 11.

Speed response under mechanical torque disturbances and parameters uncertainty.

The proposed ANN backstepping controller, for its part, maintains its performance due to the adaptive compensation by reacting quickly to clear the errors occurred by parameters uncertainties and the applied mechanical torque. Whereas the PI controller loses the reference almost completely in this case, as is demonstrated in Figures 11 and 12.

Figure 12.

Mechanical power evolution at random change in wind speed.

Figure 13 presents the evolution of power coefficient $C_{p}$ for each control strategy. From such figure, one can notice clearly that the $C_{p}$ obtained with the proposed adaptive control reaches rapidly its maximum power range as compared to DTC. For this latter, the $C_{p}$ curve doesn’t track the optimal value accurately, and presents losses and oscillations resulted from changes in the wind speed.

Figure 13.

Power coefficient evolution at random change in wind speed.

Conclusion

This paper presents a novel and robust adaptive backstepping controller for the uncertain PMSG-based WT via uncertainty compensation, and by using dynamic inversion based on ANN. The compensators are built by online training methodology, and are designed to stabilize the system despite the presence of different uncertainties (e.g. nonlinearities and external disturbances resulting from variations of the mechanical torque). The Hurwitz technique is used to determine the current controllers’ gains, whereas the speed control is processed by a PI controller. The stability of the closed-loop system is proven by applying the Lyapunov function. The simulation results that were conducted have proven that the proposed adaptive controller remarkably outperforms the PI in DTC, through achieving a good tracking performance of the speed regardless of the desired input signal references, and suppressing the uncertain behavior resulting from the uncertainties. In future works, we will be focusing on integrating the grid side system (GSS), along with implementing maximum power point tracking (MPPT) algorithms.

Footnotes

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

Mohsin Beniysa

Aziz El Janati El Idrissi

References

Aguilar-Mejía

Tapia-Olvera

Valderrabano-González

, et al. (2015) Adaptive neural network control of chaos in permanent magnet synchronous motor. Intelligent Automation & Soft Computing 22(3): 499–507.

Alizadeh

Kojori

(2015) Augmenting effectiveness of control loops of a PMSG (permanent magnet synchronous generator) based wind energy conversion system by a virtually adaptive PI (proportional integral) controller. Energy 91: 610–629.

Arik

(2014) An improved robust stability result for uncertain neural networks with multiple time delays. Neural Networks 54: 1–10.

Beddar

Bouzekri

Babes

, et al. (2016) Experimental enhancement of fuzzy fractional order PI+I controller of grid connected variable speed wind energy conversion system. Energy Conversion and Management 123: 569–580.

Beniysa

El Idrissi

AEJ

Bouajaj

, et al. (2019) An iterative approach for modeling a photovoltaic module using the complete single-diode model. In: 2019 international conference on intelligent systems and advanced computing sciences (ISACS), Taza, Morocco, 26–27 December, pp.1–7. New York: IEEE.

Bose

(2002) Modern Power Electronics and AC Drives, 1st edn. Upper Saddle River, NJ: Prentice Hall PTR.

Dai

Teel

, et al. (2009) Piecewise-quadratic Lyapunov functions for systems with deadzones or saturations. Systems & Control Letters 58(5): 365–371.

de Jesús Rubio

(2007) Nonlinear system identification with recurrent neural networks and dead-zone Kalman filter algorithm. Neurocomputing 70(1315): 2460–2466.

Eghbal

Pariz

Karimpour

(2013) Discontinuous piecewise quadratic Lyapunov functions for planar piecewise affine systems. Journal of Mathematical Analysis and Applications 399(2): 586–593.

10.

Elbeji

Hannach

Benhamed

, et al. (2021) Artificial neural network–based sensorless control of wind energy conversion system driving a permanent magnet synchronous generator. Wind Engineering 45(3): 459–476.

11.

Elmas

Ustun

Sayan

(2008) A neuro-fuzzy controller for speed control of a permanent magnet synchronous motor drive. Expert Systems with Applications 34(1): 657–664.

12.

Geng

, et al. (2011a) Unified power control for PMSG-based WECS operating under different grid conditions. IEEE Transactions on Energy Conversion 26(3): 822–830.

13.

Geng

, et al. (2011b) Active damping for PMSG-based WECS with DC-link current estimation. IEEE Transactions on Industrial Electronics 58(4): 1110–1119.

14.

Griggs

King

Shorten

, et al. (2010) Quadratic Lyapunov functions for systems with state-dependent switching. Linear Algebra and its Applications 433(1): 52–63.

15.

Hong

C-M

Chen

C-H

C-S

(2013) Maximum power point tracking-based control algorithm for PMSG wind generation system without mechanical sensors. Energy Conversion and Management 69: 58–67.

16.

Hore

Sarma

(2018) Neural network–based improved active and reactive power control of wind-driven double fed induction generator under varying operating conditions. Wind Engineering 42(5): 381396.

17.

Igelnik

Pao

Y-H

(1995) Stochastic choice of basis functions in adaptive function approximation and the functional-link net. IEEE Transactions on Neural Networks 6(6): 1320–1329.

18.

Jan

R-M

Tseng

C-S

Liu

R-J

(2008) Robust PID control design for permanent magnet synchronous motor: A genetic approach. Electric Power Systems Research 78(7): 1161–1168.

19.

Kim

H-W

Kim

S-S

H-S

(2010) Modeling and control of PMSG-based variable-speed wind turbine. Electric Power Systems Research 80(1): 46–52.

20.

Lewis

Selmic

Campos

(2002) Neuro-Fuzzy Control of Industrial Systems with Actuator Nonlinearities. Philadelphia, PA: Society for Industrial and Applied Mathematics.

21.

Haskew

(2010) Conventional and novel control designs for direct driven PMSG wind turbines. Electric Power Systems Research 80(3): 328–338.

22.

Lin

W-M

Hong

C-M

T-C

, et al. (2011) Hybrid intelligent control of PMSG wind generation system using pitch angle control with RBFN. Energy Conversion and Management 52(2): 1244–1251.

23.

Luo

Zhong

Zhu

, et al. (2014) Further results on robustness analysis of global exponential stability of recurrent neural networks with time delays and random disturbances. Neural Networks 53: 127–133.

24.

Magri

Giri

Fadili

, et al. (2013) Adaptive nonlinear control of wind energy conversion system with PMS generator. IFAC Proceedings Volumes 46(11): 318–325.

25.

Malik

Khan

Aamir

(2020) Modeling and design study of energy generation through high-speed wind at the spillways of hydroelectric power station. Wind Engineering 45(3): 538–546.

26.

Mohamad

(2008) Exponential stability preservation in discrete-time analogues of artificial neural networks with distributed delays. Journal of Computational and Applied Mathematics 215(1): 270–287.

27.

Mousa

HHH

Youssef

A-R

Mohamed

EEM

(2019) Variable step size P&O MPPT algorithm for optimal power extraction of multi-phase PMSG based wind generation system. International Journal of Electrical Power & Energy Systems 108: 218–231.

28.

Poznyak

Sánchez

(2001) Differential Neural Networks for Robust Nonlinear Control: Identification, State Estimation and Trajectory Tracking. Singapore: World Scientific.

29.

Quang

Dittrich

J-A

(2015) Vector Control of Three-Phase AC Machines, 2nd edn. Berlin Heidelberg: Springer.

30.

Rayd

El Idrissi

AEJ

Zahid

, et al. (2014) Robust emerged artificial intelligence speed controller for PMSM drive. In: 2014 second world conference on complex systems (WCCS), Agadir, Morocco, 10–12 November, pp.510–517. New York: IEEE.

31.

Ren

T-J

Chen

T-C

(2006) Robust speed-controlled induction motor drive based on recurrent neural network. Electric Power Systems Research 76(12): 1064–1074.

32.

Rivals

Personnaz

(1998) A recursive algorithm based on the extended Kalman filter for the training of feedforward neural models. Neurocomputing 20(1–3): 279–294.

33.

Sanchez

Ornelas-Tellez

(2013) Discrete-Time Inverse Optimal Control for Nonlinear Systems. Boca Raton, FL: CRC Press.

34.

Sarangapani

(2006) Neural Network Control of Nonlinear Discrete-Time Systems. Boca Raton, FL: CRC Press.

35.

Singh

Lentijo

Sundaram

(2019a) The impact of the converter on the reliability of a wind turbine generator. In: Proceedings of the ASME 2019 power conference, Salt Lake City, UT, 15–18 July, p.V001T06A016. New York: American Society of Mechanical Engineers (ASME).

36.

Singh

Matuonto

Sundaram

(2019b) Impact of imbalanced wind turbine generator cooling on reliability. In: 2019 10th international renewable energy congress (IREC), Sousse, Tunisia, 26–28 March, pp.1–6. New York: IEEE.

37.

Singh

Saleh

Amos

, et al. (2018) IC6A1A6 vs. IC3A1 squirrel cage induction generator cooling configuration challenges and advantages for wind turbine application. In: 2019 ASME 2018 power conference collocated with the ASME 2018 12th international conference on energy sustainability and the ASME 2018 Nuclear Forum, Lake Buena Vista, FL, 24–28 June, p.V001T06A002. New York: American Society of Mechanical Engineers (ASME).

38.

Singh

Sundaram

(2021) Methods to improve wind turbine generator bearing temperature imbalance for onshore wind turbines. Wind Engineering. Epub ahead of print 13 May 2021. DOI: 10.1177/0309524X211015292.

39.

Singh

Sundaram

Matuonto

(2020) A solution to reduce overheating and increase wind turbine systems availability. Wind Engineering 45(3): 491–504.

40.

Singh

Sundaram

Saleh

(2019c) Addressing reduced ingress protection class & proper filter selection for open ventilated (IC3A1) wind turbine generator. In: 2019 10th international renewable energy congress (IREC), Sousse, Tunisia, 26–28 March, pp.1–5. New York: IEEE.

41.

Slotine

JJE

(1991) Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall.

42.

Sumathi

Ashok Kumar

Surekha

(2015) Solar PV and Wind Energy Conversion Systems: An Introduction to Theory, Modeling with MATLAB/SIMULINK, and the Role of Soft Computing Techniques. Switzerland: Springer International Publishing.

43.

Vinodkumar

Prakash

Joo

(2019) Impulsive observer-based output control for PMSG-based Wind Energy Conversion System. IET Control Theory & Applications 13(13): 2056–2064.

44.

Zhang

X-P

(2009) Small signal stability analysis and control of the wind turbine with the direct-drive permanent magnet generator integrated to the grid. Electric Power Systems Research 79(12): 1661–1667.

45.

Yang

Shu

, et al. (2018) Passivity-based sliding-mode control design for optimal power extraction of a PMSG based variable speed wind turbine. Renewable Energy 119: 577–589.

46.

Youness

Aziz

Abdelaziz

, et al. (2019) Implementation and validation of backstepping control for PMSG wind turbine using dSPACE controller board. Energy Reports 5: 807–821.

47.

Zhang

X-M

Han

Q-L

(2014) Global asymptotic stability analysis for delayed neural networks using a matrix-based quadratic convex approach. Neural Networks 54: 57–69.

48.

Zhang

Zhao

Qiao

, et al. (2012) A space-vector modulated sensorless direct-torque control for direct-drive PMSG wind turbines. In: 2012 IEEE industry applications society annual meeting, Las Vegas, NV, 7–11 October, pp.1–7. New York: IEEE.

49.

Zhu

Cheng

Qin

(2007) Constructing common quadratic Lyapunov functions for a class of stable matrices. Acta Automatica Sinica 33(2): 202–204.

50.

Zhang

Fei

, et al. (2011) H robust control of direct-drive permanent magnetic synchronous generator considering the uncertainty of stator parameters. In: Proceedings of the 30th Chinese control conference, Yantai, China, 22–24 July, pp.2368–2373. New York: IEEE.