Maximum Power Point Tracking control of a variable speed wind turbine via a T-S fuzzy model-based approach

Abstract

This paper proposes a multi-objective H₂/H_∞ maximum power tracking control of a variable speed wind turbine to minimize the H₂ tracking error and ensure the H_∞ model reference-tracking performance, simultaneously. The optimal condition is obtained via a boost converter use, which adapts the load impedance to the wind turbine generator. Thus, based on the fuzzy T-S model, a multi-objective Maximum Power Point Tracking (MPPT) controller is developed, ensuring maximum power transfer, despite wind speed variation and system uncertainty. To specify the optimal trajectory to follow, a TS reference model is proposed taking as input the optimal rectified DC current. The conditions of stability and stabilization are expressed in terms of linear matrix inequality (LMI) for uncertain and disturbed T-S models leading to determining the controller gains. Finally, an example of MPP tracking applied to a Wind Energy Conversion System (WECS) illustrates the effectiveness of the proposed fuzzy control law.

Keywords

Multi-objective fuzzy tracking control maximum power point tracking (MPPT)linear matrix inequalities (LMIs)robust control T-S fuzzy model

1 Introduction

Renewable energy recovered from wind turbines is considered the most important energy since is safe, abundant, and clean. Several wind turbine structures equipped with electrical AC generators are used to convert mechanical energy into electrical energy (vertical or horizontal axis wind turbine). The performance of WECS depends essentially on the control strategy used. This control strategy is strongly influenced by the integration of the MPPT algorithm, to capture a maximum of energy over the widest range of wind speed variation. The maximization of the captured energy, i.e., the extraction of the maximum of the available wind power, is a necessary need for better efficiency, mostly in isolated sites. Thus, MPPT technique takes an important consideration, not only to maximize the energy efficiency of the wind system but also to minimize the effects of the disturbance [1, 5, 9, 11].

In the literature, there are several methods known by the acronym MPPT have been proposed to perform the optimal operation of wind energy conversion system. They are classified according to the employed of speed sensors, namely: Tip Speed Ratio (TSR) control [6], Power Signal Feedback (PSF) control [6] and Optimal Torque Control (OTC) [5, 11]. Perturb and Observe (PO) [12] and Incremental Conductance (INC) [1] fall into the category of MPPT algorithms that are devoid of wind speed sensors.

The TSR is considered a simple method commonly used in the industry, it requires an anemometer as well as knowledge of the wind turbine parameters, especially the optimal tip speed ratio, which allows for the determination of the optimal angular rotor speed for each wind speed value. However, an accurate measurement of the wind speed is required in real-time that increases the cost of the overall system. In addition, this control strategy is not suitable for an abrupt change in wind speed [1, 5, 8].

The method based on the PSF control requires knowledge of the maximum power curve of the wind turbine illustrating the relationship between the maximum power and the turbine speed [6]. This curve can be obtained by experiment tests or simulation. The drawback of the PSF method is the use of a speed sensor (tachometer) which is indispensable in the global control scheme of the system. To overcome this problem, we resort to a speed observer use, which is similar, lose its performance when the system is subjected to the variation of a parameter [1, 8].

The Perturb and observe (P&O) method is simple, easy to implement, and does not require prior knowledge of the wind turbine characteristics [12, 13]. However, it is ineffective in a wind turbine system characterized by fast dynamics. Since this, leads to losing its performance under a rapid variation of wind speed and generates oscillations around the maximum power point. Rather, it operates well for photovoltaic systems, characterized by a slower dynamic [4, 8, 10].

Different control schemes based on fuzzy logic approach’s [3 , 20] and artificial neural networks (ANN) [5, 10] are proposed in the literature to improve the energy efficiency of the wind system. The fuzzy logic control is robust to parameter variation because it does not require knowledge of the system model, but it needs a judicious choice of membership functions and fuzzy rules.

Currently, the well-known TS fuzzy models are considered useful tools to approximate nonlinear systems [4, 7, 8, 17, 18]. It permits the representation of the behaviors of the nonlinear system by a combination of linear sub-models with weighting functions. The interest in performing this decomposition is the possibility of applying all theories used in the Linear Time-Invariant system (LTI). This paper proposes a multi-objective H₂/H_∞ controller design for maximum power point tracking of the WECS. It consists to determine an appropriate controller gain, able to minimize the upper bounds (α, β) related to H₂ tracking and H_∞ performance by using a convex optimization tool “LMIs”. Conventional PI controller and (P&O) methods are presented for comparison to illustrate the performance of the proposed TS fuzzy approach.

2 Modelling of wind energy conversion system

The overall structure of the WECS is described in the Fig. 1. The system is composed of a wind turbine coupled to a PMSG generator that supplies a resistive load using an uncontrolled rectifier associated to the boost converter.

Fig. 1

Structural scheme of the wind energy conversion system.

The recovered wind power at the low-speed shaft is proportional to the cube of the wind speed V_w as follows [1, 5, 11]:

$P_{t} = \frac{1}{2} C_{p} (λ, β) ρ π R^{2} V_{w}^{3}$ (1)

Where ρ is the air density, R is the wind turbine rotor radius and C_p (λ, β) is the power coefficient defined as follows [11]:

$\begin{matrix} C_{p} (λ, β) = & 0.5176 [\frac{116}{λ_{i}} - 0.4 β - 5] \\ exp (\frac{- 21}{λ_{i}}) + 0.0068 λ_{i} \end{matrix}$ (2) with $λ_{i} = (\frac{1}{λ + 0.08 β} - \frac{0.055}{β^{3} + 1})$ .

The power coefficient depends on the orientation of the blades defined by the pitch angle β and the tip speed ratio λ expressed by the following relation [1]:

$λ = \frac{ω_{m} R}{V_{w}}$ (3)

Where ω_m is the mechanical angular rotor speed of the wind turbine.

Figure 2 shows the power coefficient versus TSR curve given by (2). It can be seen that there is an optimum TSR λ_opt at which the power coefficient is maximum C_Pmax. Therefore, the mechanical power extracted from the wind is also maximum.

Fig. 2

Power coefficient versus Tip Speed Ratio.

In a variable speed turbine, the goal is to make the turbine operate at the optimal tip speed ratio λ_opt=6.9, which keeps the performance of the turbine at the maximum power coefficient C_pmax = 0.47. This leads to extracting the maximum of the available power on the wind turbine shaft. For a PMSG with constant flux, the rectified DC voltage V_dc is a linear function to the generator rotor speed [6]. Figure 3 illustrates the characteristics of the generator power with respect to the rectified DC voltage V_dc for different wind speed values. It should be noted that the output voltage of the uncontrolled rectifier could be adjusted to the optimum value V_dcop, which results in maximum power extraction.

Fig. 3

Generator power curves for different wind speeds.

By forcing the wind energy conversion system to operate at the optimal conditions, i.e., at λ_opt = 6.9, the torque developed by the turbine leading to the maximum power transfer is given by the relationship [2]:

$C_{t} = K_{opt} ω_{m}^{2}$ (4)

Where

$K_{opt} = 0.5 π ρ C_{p_{max}} (\frac{R^{5}}{λ_{opt}^{3}})$ (5)

Next, consider the dynamic model of the PMSG defined in the d-q frame as follows [3]:

${\begin{matrix} \frac{d}{dt} i_{sd} = - (\frac{R_{s}}{L_{d}}) i_{sd} + n_{p} ω_{m} i_{sq} - \frac{1}{L_{d}} V_{sd} \\ \frac{d}{dt} i_{sq} = - (\frac{R_{s}}{L_{d}}) i_{sq} - n_{p} ω_{m} i_{sd} + (\frac{n_{p} ω_{m} ψ_{f}}{L_{d}}) - \frac{1}{L_{d}} V_{sq} \\ \frac{d}{dt} ω_{m} = \frac{1}{J} (C_{t} - C_{em}) \end{matrix}$ (6)

where R_s is the stator resistance, J is the total moment inertia, L_d is the d-axis stator inductance, ψ_f is the fixed flux, n_p is the number of poles, V_dc is the rectified DC voltage and C_em is the electromagnetic torque developed by the PMSG generator which is proportional to the q-axis stator given as follows [11]:

$C_{em} = \frac{3}{2} n_{p} (i_{sq} ψ_{f})$ (7)

Next, we assume that all losses of the full-bridge diode rectifier are neglected, this leads to the following relation:

$\frac{3}{2} (V_{sd} i_{sd} + V_{sq} i_{sq}) = I_{dc} V_{dc}$ (8)

The wind turbine control system with PMSG uses an energy modulator, i.e a boost converter, whose function is to force the power drawn from the wind turbine to follow the optimum operating point. The dynamic model of the converter can be presented by two operational modes depending on the switch power state. Thereafter, we propose to design the averaged bilinear model of the boost converter, which leads thereafter to the TS representation that serves to the control design step. Thus, by considering the rectified DC voltage V_dc (t), the inductor current I_L (t), and the load voltage V_ch (t) as a state variable, the first operating mode is determined during the “ON” period, when the switch is closed [4].

${\begin{matrix} \frac{{dV}_{dc} (t)}{dt} = - \frac{1}{C_{1}} V_{dc} (t) + \frac{1}{C_{1}} I_{dc} \\ \begin{matrix} \frac{{dI}_{L} (t)}{dt} = \frac{1}{L} V_{dc} (t) - \frac{R_{L}}{L} I_{L} (t) \\ \frac{{dV}_{ch} (t)}{dt} = - \frac{1}{{RC}_{2}} V_{ch} (t) \end{matrix} \end{matrix}$ (9) the second operating mode is defined during the Off period such as the controllable switch is open and this leads to the following differential equation [4]:

${\begin{matrix} \frac{{dV}_{dc} (t)}{dt} = - \frac{1}{C_{1}} V_{dc} (t) + \frac{1}{C_{1}} I_{dc} (t) \\ \frac{{dI}_{L} (t)}{dt} = \frac{1}{L} V_{dc} (t) - \frac{R_{L}}{L} I_{L} (t) - \frac{1}{L} V_{ch} (t) \\ \frac{{dV}_{ch} (t)}{dt} = \frac{1}{C_{2}} I_{L} (t) - \frac{1}{{RC}_{2}} V_{ch} (t) \end{matrix}$ (10)

The reformulation under a state equation leads to:

$\dot{x} (t) = A_{k} x (t) + E w (t) k = 1, 2$ (11)

$A_{1} = [\begin{matrix} 0 & - \frac{1}{C_{1}} & 0 \\ \frac{1}{L} & - \frac{R_{L}}{L} & 0 \\ 0 & 0 & - \frac{1}{R C_{2}} \end{matrix}], A_{2} = [\begin{matrix} 0 & - \frac{1}{C_{1}} & 0 \\ \frac{1}{L} & - \frac{R_{L}}{L} & - \frac{1}{L} \\ 0 & \frac{1}{C_{2}} & - \frac{1}{{RC}_{2}} \end{matrix}],$

$x (t) = [\begin{matrix} V_{dc} (t) \\ \begin{matrix} I_{L} (t) \\ V_{ch} (t) \end{matrix} \end{matrix}], E = [\begin{matrix} \frac{1}{C_{1}} \\ 0 \\ 0 \end{matrix}], w (t) = I_{dc} (t)$

Based on the bilinear models (11), the average model of the boost converter is given as:

$\begin{matrix} \dot{x} (t) = & [A_{1} x (t) + Ew (t)] u (t) + [A_{2} x (t) + Ew (t)] \\ [1 - u (t)] \end{matrix}$ (12)

Or,

$\dot{x} (t) = A_{2} {x (t) + (A}_{1} {- A}_{2}) x (t) u (t) + E w (t)$ (13)

Similarly:

$\dot{x} (t) = A_{2} x (t) + B (x (t)) u (t) + E w (t)$ (14) where

$B (x (t)) = {[\begin{matrix} 0 & \frac{V_{ch} (t)}{L} & - \frac{I_{L} (t)}{C_{2}} \end{matrix}]}^{T}$ and u(t) represent the duty ratio.

3 Multiobjective fuzzy MPPT controller design

Figure 4 illustrates the MPPT control scheme structure for the goal of maintaining the WECS to operate at the optimal condition regardless of the wind speed variation. The power transferred to the load is controlled by the duty cycle of the boost in order to track the optimum operation point.

Fig. 4

Structure of the wind energy control system.

The fuzzy control strategy includes the following steps.

Measure of wind speed V_w.

Calculate the optimal speed ω_mopt by knowing the optimal tip speed ratio λ_opt.

The optimal speed ω_mopt is then used to determine the optimal DC current I_dcopt based on the MPP searching bloc.

The DC current I_dcopt is considered as a control input to the reference model, which generate the reference state x_r.

3.1 MPP searching algorithm

The MPP searching algorithm is used to generate the optimal rectified current Idc_opt. If the steady state is established, the torque developed by the wind turbine is equal to the electromagnetic torque of the generator, this leads to determining the optimal q-axis stator current as follows:

$i_{sqopt} = \frac{2 K_{opt} ω_{mopt}^{2}}{3 n_{p} ψ_{f}}$ (15)

Under optimal operating conditions and using the vector control strategy, the d-axis stator current i_sdopt must be maintain to zero. So, the optimal rectified DC current $I_{dcopt} = \frac{π}{2 \sqrt{3}} \sqrt{i_{sdopt}^{2} + i_{sqopt}^{2}}$ , can be written as [15, 19]:

$I_{dcopt} = \frac{π K_{opt}}{3 \sqrt{3} n_{p} ψ_{f}} ω_{mopt}^{2}$ (16)

Thus, knowing the optimal speed of the wind turbine $ω_{mopt} = \frac{V_{w} λ_{opt}}{R}$ and considering the optimum rectifier DC current as a control input, the TS fuzzy reference model can provide the reference state x_r (t) allowing for the optimum operation. Here, the objective of the proposed fuzzy controller is to make the tracking error (x_r (t) - x (t)) converge to zero for all wind speed variations. The MPPT controller design needs a T-S fuzzy modeling step of the WECS.

3.2 Uncertain T-S fuzzy model

The objective is to obtain a T-S fuzzy representation of the non-linear WECS model (14) using the sector nonlinearity approach. This modeling tool leads to an exact representation of the system because it’s considered a an universal approximator of any smooth nonlinear system [7 , 18]. According to the LPV model (14), the premise variables are chosen as:

${\begin{matrix} z_{1} (t) = V_{ch} (t) \\ z_{2} (t) = I_{L} (t) \end{matrix}$ (17)

Therefore z₁(t) and z₂(t) can be expressed versus four membership functions F_1,min, F_1,max, F_2,min and F_2,min defined as follows:

${\begin{matrix} F_{k, min} (z (t)) = \frac{z_{k} (t) - min (z_{k} (t))}{max (z_{k} (t)) - min (z_{k} (t))} k = {1, 2} \\ F_{k, max} (z (t)) = \frac{max (z_{k} (t)) - z_{k} (t)}{max (z_{k} (t)) - min (z_{k} (t))} \end{matrix}$ (18)

Here we generalized that the ith rule of the nonlinear model (14) are of the following forms:

RuleR_i: If (z₁ (t) is F_1,i) and (z₂ (t) is F_2,i)

Then $\begin{matrix} \dot{x} (t) = & (A_{2} + Δ A_{2}) x (t) + (B_{i} + Δ B_{i}) u (t) \\ + Ew (t), i = {1, 2, 3, 4} \end{matrix}$

$\begin{matrix} B_{1} = [\begin{matrix} \begin{matrix} 0 \\ \frac{V_{C 2 min}}{L} \end{matrix} \\ - \frac{I_{L min}}{C_{2}} \end{matrix}], B_{2} = [\begin{matrix} \begin{matrix} 0 \\ \frac{V_{C 2 max}}{L} \end{matrix} \\ - \frac{I_{L min}}{C_{2}} \end{matrix}], \\ B_{3} = [\begin{matrix} \begin{matrix} 0 \\ \frac{V_{C 2 min}}{L} \end{matrix} \\ - \frac{I_{L max}}{C_{2}} \end{matrix}], B_{4} = [\begin{matrix} \begin{matrix} 0 \\ \frac{V_{C 2 max}}{L} \end{matrix} \\ - \frac{I_{L max}}{C_{2}} \end{matrix}] \end{matrix}$

Thus, the aggregation of four local models by means of weighting functions h_i (z (t)) leads to obtaining the uncertain T-S multimodel given by the following nonlinear form:

$\begin{matrix} \dot{x} (t) = & \sum_{i = 1}^{4} h_{i} (z (t)) ((A_{2} + Δ A_{2}) x (t) \\ + (B_{i} + Δ B_{i}) u (t) + Ew (t)) \end{matrix}$ (19)

the matrix ΔA₂ and ΔB_i represent the parametric uncertainties, such as ΔA₂ = D_AiF_i (t) E_Ai and ΔB_i = D_BiF_i (t) E_Bi. D_(A,B)i and E_(A,B)i are a known constant matrices of appropriate dimensions. F_i (t) is an unknown function satisfying $F_{i} (t) F_{i}^{T} (t) < I$ .

3.3 TS reference model design

To force the WECS to follow the maximum power trajectory, we consider a nonlinear reference model in which the optimal rectified current I_dcopt is taken as a control input. The reference model providing the maximum power trajectory is defined by the following equation:

${\dot{x}}_{r} (t) = A_{r} x_{r} (t) + E I_{dcopt}$ (20)

or equivalently:

${\dot{x}}_{r} (t) = A_{r} x_{r} (t) + r (t)$ (21)

$A_{r} = [\begin{matrix} 0 & - \frac{1}{C_{1}} & 0 \\ \frac{1}{L} & - \frac{R_{L}}{L} & - \frac{1}{L} (1 - u_{opt}) \\ 0 & \frac{1}{C_{2}} (1 - u_{opt}) & - \frac{1}{{RC}_{2}} \end{matrix}],$ $r (t) = [\begin{matrix} \frac{I_{dcopt}}{C_{1}} \\ 0 \\ 0 \end{matrix}]$

Where $u_{opt} = 1 - \sqrt{\frac{V_{dcopt}}{{RI}_{dcopt}}}$

the model (21) is nonlinear and can be represented by a convex combination of two linear submodels originates of the premise variable z_r = (1 - u_opt). It can be defined by two fuzzy rules [2]:

Rule 1: If z_r (t) is N_min Then ${\dot{x}}_{r} (t) = A_{r 1} x (t) + r (t)$

Rule 2: If z_r (t) is N_max Then ${\dot{x}}_{r} (t) = A_{r 2} x (t) + r (t)$

where the membership functions are given as:

{\begin{matrix} N_{min} (z_{r} (t)) = \frac{z_{r} (t) - z_{r, min}}{z_{r, max} - z_{r, min}} \\ N_{max} (z_{r} (t)) = 1 - N_{min} (z_{r} (t)) \end{matrix}

and the reference state matrices are defined as follow:

A_{r 1} = [\begin{matrix} 0 & - \frac{1}{C_{1}} & 0 \\ \frac{1}{L} & - \frac{R_{L}}{L} & - \frac{1}{L} z_{r, min} \\ 0 & \frac{1}{C_{2}} z_{r, min} & - \frac{1}{R C_{2}} \end{matrix}],

A_{r 2} = [\begin{matrix} 0 & - \frac{1}{C_{1}} & 0 \\ \frac{1}{L} & - \frac{R_{L}}{L} & - \frac{1}{L} z_{r, max} \\ 0 & \frac{1}{C_{2}} z_{r, max} & - \frac{1}{R C_{2}} \end{matrix}]

Finally, the nonlinear model (21) can be derived out of defuzzification process as:

${\dot{x}}_{r} (t) = \sum_{k = 1}^{2} h_{k} (z_{r} (t)) (A_{rk} x_{r} (t) + r (t))$ (23)

3.4 TS fuzzy controller design

To extract maximum power from the wind turbine shaft, we force the state x (t) to follow the reference state x_r(t) despite the wind speed change and system parameter variation. As a result, the tracking problem is converted into a tracking error feedback control defined as follows [2, 6, 18]:

$u (t) = \sum_{j = 1}^{4} h_{j} (z (t)) K_{j} (x (t) - x_{r} (t))$ (24)

By substituting the control law (24) in the system (19) and taking into account the reference model (24), the dynamics of the tracking error is defined as follows:

$\begin{matrix} {\dot{e}}_{r} (t) = \sum_{i = 1}^{4} \sum_{j = 1}^{4} \sum_{k = 1}^{2} h_{i} (z (t)) h_{j} (z (t)) h_{k} (z (t)) \\ (\begin{matrix} ((A_{2} + Δ A_{2}) + (B_{i} + Δ B_{i}) K_{j}) e_{r} (t) \\ + (A_{2} - A_{rk} + Δ A_{2}) x_{r} (t) \end{matrix}) \\ + E w (t) - r (t) \end{matrix}$ (25)

Then, by considering the augmented state vector $\bar{x} (t) = {(\begin{matrix} e_{r} (t) & x_{r} (t) \end{matrix})}^{T}$ , the closed-loop augmented system is given by the following equation:

$\begin{matrix} \dot{-} x (t) = & \sum_{i = 1}^{4} \sum_{j = 1}^{4} \sum_{k = 1}^{2} h_{i} (z (t)) h_{j} (z (t)) h_{k} (z_{r} (t)) \\ ({\bar{G}}_{ijk} \bar{x} (t) + \bar{E} \bar{W} (t)) \end{matrix}$ (26) where

${\bar{G}}_{ijk} = {\bar{A}}_{ijk} + Δ {\bar{A}}_{ij} = (\begin{matrix} (A_{2} + B_{i} K_{j}) & (A_{2} - A_{rk}) \\ 0 & A_{ri} \end{matrix}) + (\begin{matrix} Δ A_{2} + Δ B_{i} K_{j} & Δ A_{2} \\ 0 & 0 \end{matrix})$ , $\bar{E} = (\begin{matrix} E & - I \\ 0 & I \end{matrix})$ , $\bar{W} (t) = (\begin{matrix} w (t) \\ r (t) \end{matrix})$

By neglecting the general disturbance effect $\bar{W} (t)$ , the H₂ tracking performance can be defined as follows [4, 8]:

$J = \int_{0}^{\infty} (e_{r} (t)^{T} Q_{1} e_{r} (t) + u (t)^{T} R u (t)) ⩽ α$ (27) where Q₁ and R are the weighting matrices for e_r (t) and u (t) respectively.

The tracking error e_r(t) can be expressed with $\bar{x} (t)$ as $e_{r} (t) = (\begin{matrix} I & 0 \end{matrix}) \bar{x} (t)$ , so, equation (27) can be written: as:

$J = \int_{0}^{\infty} (\bar{x} (t)^{T} {\bar{Q}}_{1} \bar{x} (t) + u (t)^{T} R u (t)) ⩽ α$ (28)

Furthermore, the robust H_∞ tracking performance which serve to attenuate the general disturbance effect $\bar{W} (t)$ , is given by the following equation [4, 8]:

$J_{\infty} = \frac{\int_{0}^{t_{f}} e_{r}^{T} (t) Q_{2} e_{r} (t) dt}{\int_{0}^{t_{f}} {\bar{W}}^{T} (t) \bar{W} (t) dt} = \frac{\int_{0}^{t_{f}} {\bar{x}}^{T} (t) {\bar{Q}}_{2} \bar{x} (t) dt}{\int_{0}^{t_{f}} {\bar{W}}^{T} (t) \bar{W} (t) dt} ⩽ β$ (29)

α et β are the upper bounds of the H₂ tracking and robust H_∞ tracking performance, respectively. ${\bar{Q}}_{i}$ are defined as follows:

${\bar{Q}}_{i} = (\begin{matrix} Q_{i} & 0 \\ 0 & 0 \end{matrix}), for i = 1, 2 .$ (30)

Thus, the objective of the design of the robust multi-objective control law is to determine the fuzzy controller (24) such as the optimal H ₂ performance (28) and the tracking performance H_∞ (29) are all minimized simultaneously. Based on the above analysis, we propose the following theorem.

Theorem 1. The closed-loop wind energy conversion system (26) is globally asymptotically stable and minimizes simultaneously the cost functions (27) and (29), if there a symmetric definite positive matrices X₁, P₂, matrices Y_i, and the scalars ɛ_Ai1, ɛ_Ai2, ɛ_Bi1 satisfying the following optimization problem:

$\underset{(X_{1}, P_{2})}{min (α, β)}$ $\begin{matrix} Subject \\ [\begin{matrix} Π_{11} & X_{1} E_{Ai 1}^{T} & Y_{j}^{T} E_{Bi}^{T} & (A_{2} - A_{rk}) & - Y_{i} & X_{1} \\ * & - ɛ_{Ai 1}^{- 1} & 0 & 0 & 0 & 0 \\ * & * & - ɛ_{Bi 1}^{- 1} & 0 & 0 & 0 \\ * & * & * & Π_{44} & 0 & 0 \\ * & * & * & * & - R^{- 1} & 0 \\ * & * & * & * & * & - Q_{1}^{- 1} \end{matrix}] < 0 \end{matrix}$ (31)

$[\begin{matrix} x_{r} (0)^{T} P_{2} x_{r} (0) - α I & e_{r}^{T} (0) \\ e_{r} (0) & - X_{1} \end{matrix}] < 0$ (32)

$\begin{matrix} [\begin{matrix} Π_{11} & X_{1} E_{Ai}^{T} & Y_{j}^{T} E_{Bi}^{T} & (A_{2} - A_{rk}) & E & - I & X_{1} \\ * & - ɛ_{Ai 1}^{- 1} & 0 & 0 & 0 & P_{2} & 0 \\ * & * & - ɛ_{Bi 1}^{- 1} & 0 & 0 & 0 & 0 \\ * & * & * & Π_{44} & 0 & 0 & 0 \\ * & * & * & * & - β . I & 0 & 0 \\ * & * & * & * & * & - β . I & 0 \\ * & * & * & * & * & * & - Q_{2}^{- 1} \end{matrix}] \\ < 0 \end{matrix}$ (33)

where

$Π_{11} = X_{1} A_{2}^{T} + A_{2} X_{1} + Y_{j}^{T} B_{i}^{T} + B_{i} Y_{j} + (ɛ_{Ai 1} + ɛ_{Ai 2}^{- 1}) D_{Ai} D_{Ai}^{T} + ɛ_{Bi 1} D_{Bi} D_{Bi}^{T}$

$Π_{44} = A_{ri}^{T} P_{2} + P_{2} A_{ri} + ɛ_{Ai 2} E_{Ai} T E_{Ai}$ , and $K_{i} = Y_{i} X_{1}^{- 1}$

Proof. The proof of the theorem is partitioned into two steps; the first step concerns the H₂ tracking performance while the second step is dedicated to the robust H_∞ tracking performance. Beginning by considering the following quadratic Lyapunov function:

$\begin{matrix} V (\bar{x} (t)) = {\bar{x}}^{T} (t) \bar{P} \bar{x} (t) \\ = e_{r}^{T} (t) P_{1} e_{r} (t) + x_{r}^{T} (t) P_{2} x_{r} (t) \end{matrix}$ (34)

Where P₁ > 0, P₂ > 0 and $\bar{P} = diag {P_{1}, P_{2}}$

For the H₂ tracking performance part with $\bar{W} = 0$ , we have:

$\begin{array}{l} J_{2} = \int_{0}^{t_{f}} ({\bar{x}}^{T} (t) {\bar{Q}}_{1} \bar{x} (t) + u^{T} (t) R u (t)) d t \\ \leq {\bar{x}}^{T} (0) \bar{P} \bar{x} (0) + \int_{0}^{t_{f}} ({\bar{x}}^{T} (t) {\bar{Q}}_{1} \bar{x} (t) + u^{T} (t) R u (t) \\ + {\dot{\bar{x}}}^{T} (t) \bar{P} \bar{x} (t) + {\bar{x}}^{T} (t) \bar{P} \dot{\bar{x}} (t)) d t \end{array}$ (35)

Due to ${\bar{x}}^{T} (t_{f}) \bar{P} \bar{x} (t_{f}) ⩾ 0$ . Next, by replacing equation (26) in equation (35), we obtain:

$J_{2} ⩽ {\bar{x}}^{T} (0) \bar{P} \bar{x} (0) + \int_{0}^{t_{f}} (\begin{matrix} {\bar{x}}^{T} (t) {\bar{Q}}_{1} \bar{x} (t) + \\ \sum_{i = 1}^{4} \sum_{j = 1}^{4} \sum_{k = 1}^{2} h_{i} h_{j} h_{k} {\bar{x}}^{T} (t) (\begin{matrix} {\bar{G}}_{ijk}^{T} \bar{P} + \bar{P} {\bar{G}}_{ijk} + {\bar{Q}}_{1} + \\ {[\begin{matrix} I & 0 \end{matrix}]}^{T} K_{j}^{T} {RK}_{j} [\begin{matrix} I & 0 \end{matrix}] \end{matrix}) \bar{x} (t) \end{matrix}) dt$ (36)

If the LMIs (31) have a feasible solution, then condition (37) is satisfied as follows:

$\begin{matrix} {\bar{G}}_{ijk}^{T} \bar{P} + \bar{P} {\bar{G}}_{ijk} \\ + {\bar{Q}}_{1} + {[\begin{matrix} I & 0 \end{matrix}]}^{T} K_{j}^{T} {RK}_{j} [\begin{matrix} I & 0 \end{matrix}] ⩽ 0 \end{matrix}$ (37)

So, take account that ${\bar{x}}^{T} (0) \bar{P} \bar{x} (0) ⩽ 0$ , we can deduce that J₂ ⩽ α.

For the second part, we consider the robust H_∞ tracking performance with $\bar{x} (0) = 0$ . Similar to the previous analysis with ${\bar{x}}^{T} (t_{f}) \bar{P} \bar{x} (t_{f}) ⩾ 0$ , and ${\bar{x}}^{T} (0) \bar{P} \bar{x} (0) = 0$ , we obtain:

$\begin{array}{l} \int_{0}^{t_{f}} {\bar{x}}^{T} (t) {\bar{Q}}_{2} \bar{x} (t) d t \\ \leq \int_{0}^{t_{f}} ({\bar{x}}^{T} (t) {\bar{Q}}_{2} \bar{x} (t) + {\dot{\bar{x}}}^{T} (t) \bar{P} \bar{x} (t) + {\bar{x}}^{T} (t) \bar{P} \dot{\bar{x}} (t)) d t \end{array}$ (38)

According to equation (26), inequality (38) become:

$\begin{matrix} \int_{0}^{t_{f}} {\bar{x}}^{T} (t) {\bar{Q}}_{2} \bar{x} (t) dt ⩽ \int_{0}^{t_{f}} \sum_{i = 1}^{4} \sum_{j = 1}^{4} \sum_{k = 1}^{2} h_{i} h_{j} h_{k} \\ (\begin{matrix} {\bar{x}}^{T} (t) {\bar{Q}}_{2} \bar{x} (t) + {\bar{x}}^{T} (t) {\bar{G}}_{ijk} \bar{P} \bar{x} (t) + \bar{x} (t) \bar{P} {\bar{G}}_{ijk}^{T} {\bar{x}}^{T} (t) \\ + \bar{W} (t) {\bar{E}}^{T} \bar{P} \bar{x} (t) + {\bar{x}}^{T} (t) \bar{P} \bar{E} {\bar{W}}^{T} (t) \end{matrix}) dt \end{matrix}$ (39)

By considering $N^{T} M + M^{T} N ⩽ β N^{T} N + \frac{1}{β} M^{T} M$ for a positive scale value β and any vector or matrix M and N, equation (39) become:

$\begin{matrix} \int_{0}^{t_{f}} {\bar{x}}^{T} (t) {\bar{Q}}_{2} \bar{x} (t) dt ⩽ \int_{0}^{t_{f}} \sum_{i = 1}^{4} \sum_{j = 1}^{4} \sum_{k = 1}^{2} h_{i} h_{j} h_{k} \\ (\begin{matrix} {\bar{x}}^{T} (t) {\bar{Q}}_{2} \bar{x} (t) + {\bar{x}}^{T} (t) {\bar{G}}_{ij} \bar{P} \bar{x} (t) \\ + \bar{x} (t) \bar{P} {\bar{G}}_{ij}^{T} {\bar{x}}^{T} (t) + \\ β {\bar{W}}^{T} (t) \bar{W} (t) + \frac{1}{β} {\bar{x}}^{T} (t) \bar{P} \bar{E} {\bar{E}}^{T} \bar{P} \bar{x} (t) \end{matrix}) dt \end{matrix}$ (40)

Similarly, if the LMIs (33) have a feasible solution, then the following matrix inequalities are satisfied:

${\bar{G}}_{ij}^{T} \bar{P} + \bar{P} {\bar{G}}_{ij} + {\bar{Q}}_{2} + \frac{1}{β} \bar{P} \bar{E} {\bar{E}}^{T} \bar{P} ⩽ 0$ (41) which is equivalent to:

$[\begin{matrix} {\bar{G}}_{ij}^{T} \bar{P} + \bar{P} {\bar{G}}_{ij} + {\bar{Q}}_{2} & \bar{P} \bar{E} \\ {\bar{E}}^{T} \bar{P} & - β . I \end{matrix}] < 0$ (42)

If we take into account the condition (41), Eq. (40) becomes: $\int_{0}^{t_{f}} (\bar{x} (t)^{T} {\bar{Q}}_{2} \bar{x} (t)) ⩽ β \int_{0}^{t_{f}} (\bar{W} (t)^{T} \bar{W} (t))$

To transform the BMI (42) into LMI (33), we must rewrite it in a developed form which can appear the uncertain term as follows:

${\bar{ϒ}}_{ijk} + Δ {\bar{ϒ}}_{ij} < 0$ (43) where ${\bar{ϒ}}_{ijk} = (\begin{matrix} (P_{1} (A_{i} + B_{i} K_{j}) + {(P_{1} (A_{i} + B_{i} K_{j})}^{T} + Q_{2} & P_{1} (A_{i} - A_{rk}) & P_{1} E & - P_{1} \\ * & A_{ri}^{T} P_{2} + P_{2} A_{ri} & 0 & P_{2} \\ * & * & - β I & 0 \\ * & * & * & - β I \end{matrix})$ $Δ {\bar{ϒ}}_{ij} = (\begin{matrix} P_{1} (Δ A_{i} + Δ B_{i} K_{j}) + (Δ A_{i} + Δ B_{i} K_{j})^{T} P_{1} & P_{1} Δ A_{i} & 0 & 0 \\ * & 0 & 0 & 0 \\ * & * & 0 & 0 \\ * & * & * & 0 \end{matrix})$

Pre and post-multiply the BMI (43) by $diag (X_{1} = P_{1}^{- 1}, I, I, I)$ , consider the variable change Y_j = K_jX and using the uncertainty parametric (ΔA₂ = D_AiF_i (t) E_Ai, ΔB_i = D_BiF_i (t) E_Bi) defined in section 3.2, $Δ {\bar{ϒ}}_{ij}$ can be bounded as follows:

$Δ {\bar{ϒ}}_{ij} ⩽ (\begin{matrix} (\begin{matrix} (ɛ_{Ai 1} + ɛ_{Ai 2}^{- 1}) D_{Ai} D_{Ai}^{T} + ɛ_{Bi 1} D_{Bi} D_{Bi}^{T} \\ + ɛ_{Ai 1} X_{1} E_{Ai}^{T} E_{Ai} X_{1} + ɛ_{Bi 1} Y_{j}^{T} E_{Bi}^{T} E_{Bi} Y_{j} \end{matrix}) & 0 & 0 & 0 \\ * & ɛ_{Ai 2} E_{Ai}^{T} E_{Ai} & 0 & 0 \\ * & * & 0 & 0 \\ * & * & * & 0 \end{matrix})$ (44)

Finally, using the condition (44) and applying the Schur complement, Equation (43) is transformed into LMI (33).

4 Simulation results

The performance of the proposed MPPT fuzzy controller is evaluated for two wind speed profiles. The specification of the wind power generator is stated in Table 1. The proposed simulation tests include the analysis of tracking accuracy degree of MPP as well as the convergence speed to the optimal operating point, during the application of a variable wind speed step.

Table 1
Specifications of the wind power generator

Parameters Values

R 1 m

ρ 1.205 kg/m3

R_s 0.57Ω

L_d 0.0055 H

J 0.01645 Kgm2

n_p 4

P_n 6400 W

Parameters	Values
R	1 m
ρ	1.205 kg/m3
R_s	0.57Ω
L_d	0.0055 H
J	0.01645 Kgm2
n_p	4
P_n	6400 W

The TS model exactly represents the dynamic behaviors of the system (14) under the bounded interval of the premise variables z_k (t) as follows: V_C2min =-20 V, V_C2max= 400 V, I_Lmin= -20A and I_Lmax= 20A.

A suitable selecting of the weighting matrices: Q₁= 0.005×I₃, Q₂= 0.02×I₃ and R = 0.01, leading to the controller gains obtained via LMI (31), (32) and (33) as follows: $K_{1} = [\begin{matrix} 0 . 0038 & 0 . 2226 & - 0 . 0086 \end{matrix}]$ $K_{2} = [\begin{matrix} - 0 . 0037 & - 0 . 2112 & 0 . 0079 \end{matrix}]$ $K_{3} = [\begin{matrix} 0 . 0074 & 0 . 3917 & - 0 . 0145 \end{matrix}]$ $K_{4} = [\begin{matrix} - 0 . 1819 & 0 . 2289 & 0 . 4788 \end{matrix}]$

To test the performance of the developed fuzzy controller, we propose a comparison study with conventional methods like the PO and the Proportional-Integral (PI) controller. The PO is considered as a direct method, which requires only rectified DC current Idc and rectified DC voltage V_dc measurements. The PI controller widely used in literature, classified as an indirect method, it is consists to follow the reference voltage V_dcref to ensure an optimal operation of the WECS. The goal is to realize a more subjective comparison with the developed T-S fuzzy controller. Thus, the structure of the PI controller is given

$\begin{matrix} u (t) = & K_{p} (V_{dcopt} (t) - V_{dc} (t)) \\ + K_{I} \int_{0}^{t} (V_{dcopt} (t) - V_{dc} (t)) dt \end{matrix}$ (45) with well-tuned parameters: K_p= 0.5 and K_I= 0.1.

The proposed TS control strategy is simulated for a two-wind speed profile. First, we consider a step range of wind speed from 10 m/s to 8 m/s and then back to 9 m/s as given in Fig. 5-a. By using the proposed MPPT controller, the turbine power response and power coefficient are illustrated in Figs. 5-b and 5-c, respectively

Fig. 5

a) Profile of wind speed, (b) turbine power and (c) power coefficient.

By adopting the MPPT fuzzy controller the power generated by the PMSG generator is maximized, and it follows the optimal power trajectory as illustrated in Fig. 4-b. Similarly, by comparing the developed fuzzy approach to the PO and PI control methods, Fig. 4-c shows that the power coefficient Cp is kept constant around Cp_max with fewer fluctuations despite the wind profile variation.

The rectified DC voltage, the generator rotor speed, and the rectified DC current are illustrated in Fig. 6. We can observe that by adopting the TS fuzzy controller the rectified DC current follows the optimal trajectory I_dcopt, and this leads to extract the maximum power from the wind turbine to the load.

Fig. 6

a) Rectified DC voltage, b) generator rotor speed and c) rectified DC current.

Using the PI controller, the rotor speed and the rectified DC voltage do not closely track the reference signal in the steady state. However, the PO method presents oscillations around MPP, which becomes intense with the increase of the wind speed value. This decrease the extracted turbine power and then reduces the efficiency of the wind system. The duty cycle and the load voltage are illustrated in Fig. 7.

Fig. 7

a) Duty cycle and b) load voltage.

Compared to the conventional control methods, we can deduce that the proposed fuzzy controller has a good transition response, a low tracking error, and a better disturbance attenuation against abrupt wind speed variations.

Next, to check the robustness of the proposed fuzzy controller toward the varying wind speed profile and parameter uncertainties, we consider a sum of several sinusoidal signals, which leads to appear the fluctuating nature of the wind speed: $\begin{matrix} V_{w} (t) = 8 + 0 . 4 \sin (3 . 664 t) + 0 . 7 \sin (1 . 293 t) \\ - 1 . 8 \sin (0 . 265 t) + 0 . 4 \sin (0 . 104 t) \end{matrix}$

We assume that the global fuzzy model (19) has undergone a parametric variation of 20% compared to the nominal model at t = 8 sec. In this case, we consider D_Ai=I₃, E_Ai= 0.2A₂ and E_Bi= 0.2B_i. Figure 7 illustrates the responses of the power generator and the power tracking error.

Compared to the PO and the PI methods, we can notice that the proposed TS fuzzy controller preserved its tracking performance regardless of wind speed variation (11 m/s and 6 m/s) and insensitivity to uncertainties. It converges towards the maximum power trajectory with less tracking error and robust behaviour versus parameter variations. However, the PO and PI methods have a slow dynamic to track the MPP, characterized by oscillations in a steady state which becomes significant with the system parameters variation, thus resulting in a loss of global efficiency.

Moreover, compared to the PO and PI methods, Fig. 8 shows that the power coefficient curve C_p get from the TS controller presents a small fluctuation approximately its maximum value C_pmax . This keeps energy production of the PMSG generator at a maximum level.

Fig. 8

a) Generator power, b) power tracking error, c) Power coefficient.

5 Conclusion

In this paper, a robust fuzzy controller for maximum power tracking of the WECS, under different wind speed profiles was proposed. A comparison study with conventional algorithms based on PO and PI controllers is carried out. A TS reference model is added to the global control scheme to set the maximum power trajectory for optimal operation. LMI conditions for an optimal controller gains synthesis are solved, by minimizing a quadratic cost function related to the H₂ tracking performance and H_∞ disturbance attenuation level. Based on simulation results, the proposed MPPT fuzzy controller is highly efficient over classical controllers regards to:

i) accuracy, especially during unexpected changes in wind speed, ii) robustness to uncertainty and disturbance, iii) a smooth response without oscillation around the MPP.

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