Optimized fuzzy fractional PI-based MPPT controllers for a variable-speed wind turbine

Abstract

In recent years, wind power has become one of the most popular sources of renewable electricity generation. However, the wind is, by its nature, a highly intermittent source of energy. To capture the maximum power, wind turbines are generally equipped with a Maximum Power Point Tracking (MPPT) Controller. This paper proposes effective and robust MPPT control strategies based on Fuzzy Logic controller PI (FPI) and Fuzzy logic Fractional Order controller PI (FFOPI). Particle Swarm Optimization (PSO) is used to optimize the membership functions of FPI and FFOPI. The proposed MPPT strategies are validated on a Permanent Magnet Synchronous Generator (PMSG)-variable-speed wind energy conversion system. The overall model of the wind turbine-PMSG and control scheme is developed in MATLAB/Simulink and SimPower Systems toolbox. The results show that the MPPT based on FFOPI control optimized by PSO leads to the best transient response performance and robustness.

Keywords

Wind turbine MPPT FOPI PSO FPI and FFOPI controllers

Introduction

Wind power is becoming an important alternative source of electricity supply and has gained a great popularity in recent years (Chavero-Navarrete et al., 2019). The main drivers are the escalation of electricity demand globally, the need to meet the decarbonization targets of the energy sector and the cost decline of wind turbine technology. However, wind energy depends on several factors such as the atmospheric conditions and geographic locations which may not always provide a strong enough or constant wind speed to generate electricity (Maroufi et al., 2020).

Consequently, with these intermittent characteristics, wind power cannot be fully extracted to meet electrical demands. In addition, wind turbines are not capable of producing as much electricity as conventional energy sources such as coal, gas or nuclear power plants (Parvin et al., 2019).

There are two types of wind turbines: fixed speed turbine and variable speed turbine (Salih et al., 2016). A variable-speed wind turbine allows the rotor speed of the turbine to follow the changes in wind speed. To maintain a maximum power output and maximize the efficiency of a variable speed wind turbine, a Maximum Power Point Tracking (MPPT) controller is used.

In recent years, Fuzzy Logic Controller (FLC) has been widely used for MPPT of WECS due to fact that it is able to take care of the nonlinearities and uncertainties (Parvin et al., 2019). optimization of fuzzy fractional controller using PSO algorithm improved the results for many kinds of systems, since it gives additional flexibility to the design. In this line of thought many applications of this type of controller were developed in the last few years Jesus and Barbosa, 2014).

Many researchers have devoted their studies to developing intelligent controllers, among these controllers we find the fuzzy FOPI optimized by the PSO, (PSO-fuzzyFOPI) (Elyaalaoui et al., 2021; Truong and Ngo, 2019). Our new contribution is to build a fuzzy program in the Matlab script, and introduce the values obtained from the PSO algorithm such as the error and error variation to obtain the results of $K_{p}$ , $K_{i}$ , and $λ$ . the PSO is we help also to determine the universe of discourse of different parameters.

Compared to other evolutionary computation techniques such as Genetic Algorithms (GAs); the advantages are that PSO is easy to implement and there are few parameters to adjust. PSO has successfully applied in many areas such as function optimization, artificial neural network training, fuzzy system control, and other areas where GA can be applied. PSO has already been a new and fast developing research topic (Dorrah et al., 2012).

This article presents a new approach to robust and optimal MPPT controller design for a wind power system. Two controllers techniques for tuning the MPPT are proposed based on fuzzy PI (FPI) and fuzzy FOPI (FFOPI). The membership functions of these fuzzy controllers are tuned using an evolutionary algorithm based on Particle Swarm Optimization (PSO).

The remaining of the paper is organized as follows: Section 1 describes the modeling of the wind energy conversion system. Section 2 overviews the PSO algorithm and presents the fuzzy logic and fractional PI (FOPI); Section 3 describes the combination between the two tuning techniques (FLC and PSO), as well as the objective function for optimization, section 4 discusses the simulation result, and section 5, finally concludes the article.

Wind turbine model

Figure 1 shows the schematic of a small wind turbine system based on a permanent magnet synchronous generator (PMSG). Wind turbines convert wind energy into mechanical energy. This energy drives a generator, which produces electrical energy.

Figure 1.

Wind power system with MPPT controller scheme.

The mechanical power produced by the wind turbine is expressed as follows (Sabzevari et al., 2017):

P_{m} = \frac{1}{2} ρ {AV}_{w}^{3} C_{p} (λ, β)

(1)

Where $ρ$ is the air density (kg/m³), $C_{p}$ is the coefficient of power, which ranges from 0.4 to 0.48. However, its theoretical maximum is 0.59 according to Betz’s law, $A (m^{2})$ is the swept area by the turbine blades and $V_{w}$ (m/seconds) is the velocity of available wind (Das and Buragohain, 2015).

The power captured by the wind turbine depends highly on $C_{p}$ for a given wind speed and the relationship of $C_{p}$ with $λ$ represents the output characteristics of the wind turbine:

C_{p} = 0.5176 (\frac{116}{λ} - 0.4 β - 5) e^{(\frac{- 21}{λ})} + 0.0068 λ

(2)

Where $β$ is the blade pitch-angle, $λ$ is tip speed ratio (TSR). The TSR can be expressed in terms of the wind speed and rotor speed as follows:

λ = \frac{Ω_{m} R}{V_{w}}

(3)

The mechanical equation of the PMSG-based wind turbine is given by:

J \frac{d Ω_{m}}{dt} = C_{m} - C_{em} - f Ω_{m}

(4)

Where $J$ is the inertia and $f$ is turbine total external damping coefficient. The electromagnetic torque of the PMSG is given by:

C_{em} = \frac{3}{2} P i_{q} ϕ_{f}

(5)

The adjustment (or variation) of the torque is obtained by acting on the quadrature current $iq$ .

The speed regulation of the turbine with the maximization of extracted power is given in the following Figure 2:

Figure 2.

Block diagram of the turbine with maximization of the extracted power.

Remarks

(i) For a given WT with fixed pitch angle, $C_{p}$ presents a unique maximum for $C_{p - \max}$ for $λ$ = $λ_{\max}$ .

(ii) In our project, $C_{p - \max}$ =0.48 is achieved for $λ$ =8.1.

PI and FOPI controllers

PI controller

PI controller has wide applicability in the industry, because of its simplicity in implementation, most industrial controllers belong to the PID family Farahani and Rahmani (2019). Two parameters of the PI controller are defined as proportional (P) which is equal to the difference between the input signal and the feedback signal; integral (I) which is proportional to the integral error of the exciting signal. The transfer function of the PI regulator G(s) is expressed in equation (6).

G (s) = K_{p} + \frac{K_{i}}{s}

(6)

FOPI controller

Fractional calculus is an extension of the $n^{th}$ order successive differentiation and integration of an arbitrary function having the order as any real value. There are three main definitions of fractional calculus, the Grünwald-Letnikov (GL), Riemann-Liouville (RL) and Caputo definitions. The Caputo definition is widely used in fractional order control system design problems. According to Caputo’s definition, the $α^{th}$ order differ-integral of a function $f (t)$ with respect to time $t$ is given by equation (7) (Pan and Das, 2016).

D^{α} f (t) = \frac{1}{Γ (m - α)} \int_{0}^{t} \frac{D^{m} f (τ)}{{(t - τ)}^{α + 1 - m}} d τ

(7)

α \in R^{+}, m \in Z^{+}, m - 1 \leq α \leq m

(7)

The generalized FOPI controller G(s), has a transfer function of the form (Jesus and Barbosa, 2014):

G (s) = K_{p} + \frac{K_{i}}{s^{λ}}

(8)

Where $λ$ is order of fractional integrator, the parameters $K_{p}$ , $K_{i}$ are correspondingly the proportional and integral gains of the controller. Clearly, taking (1,0), we get the classical (P, PI). The fractional-order controller is more flexible and gives the possibility of adjusting more carefully the closed-loop system characteristic.

Particle swarm optimization (PSO)

Particle swarm optimization (PSO) is a heuristic algorithm that finds the best solution by simulating the movement and clustering of birds. It can also be said to be a social psychology-based algorithm. According to Kennedy and Eberhart, the main idea of PSO is to assume that there is a flock of birds in a particular space.

The flock exchanges and communicates information through each other’s messages, follow the bird closest to the food, and change its search direction, gradually approach the only piece of food in the space. The theoretical basis of PSO is to find the optimal solution ( $P_{best}$ ) of the particle itself by a single particle, and to find the best solution for the group ( $G_{best}$ ) through the interaction with other particles in the particle group. In each iteration, each particle updates the search direction and speed, and adjusts its velocity based on the particle’s own momentum $P_{best}$ and $G_{best}$ (Huang and Chou, 2019).

In a $d$ -dimensional search space, the position $X_{i}$ and velocity $V_{i}$ for particle $i$ are defined as follows:

X_{i} = (X_{i}^{1}, X_{i}^{2}, X_{i}^{3}, . . ., X_{i}^{d}) for i = 1, 2, 3, . . ., N

(9)

V_{i} = (V_{i}^{1}, V_{i}^{2}, V_{i}^{3}, . . ., V_{i}^{d}) for i = 1, 2, 3, . . ., N

(10)

where $X_{i}^{d}$ and $V_{i}^{d}$ are the position and velocity of particle $i$ in the $d$ -dimensional space. The particle updates velocity and position according to equations ?? (11).

\begin{matrix} V_{i}^{d} (t + 1) = {wV}_{i}^{d} (t) + C 1^{*} rand (p_{i}^{d} - X_{i}^{d} (t)) \\ + C 2^{*} rand (p_{g}^{d} - X_{g}^{d} (t)) \end{matrix}

(11)

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

(12)

Where $t$ number of iterations, $V_{i}^{d} (t + 1)$ is the velocity of particle at time $t + 1$ , w is the inertia weight, $C 1$ and $C 2$ are positive constants, rand is a uniform random variable in the interval [0,1], $X_{i}^{d}$ presents the position of particle $i$ in the d-dimension, $p_{i}^{d}$ is the optimal position of particle $i$ at iteration $t$ , and $p_{g}^{d}$ is the current group optimal solution $G_{best}$ . the Flowchart of the PSO algorithm is shown in Figure 3.

Figure 3.

PSO process flowchart.

Fuzzy logic controller (FLC)

A fuzzy logic system (FLC) is a rule-base system that implements a nonlinear mapping between its inputs and outputs (Dorrah et al., 2012). Mamdani fuzzy system consists of four mains parts, namely Fuzzy, Fuzzy Inference, fuzzy rule base and defuzzification. A typical fuzzy system takes a state value and passes it through a fuzzification process and, then, it is processed by an inference engine. Finally, it goes through a defuzzification process (Safari et al., 2013). Figure 4 below shows the basic block diagram of an FLC (Parvin et al., 2019). There are two inputs in this Fuzzy algorithm and two outputs for PI controller which are $K_{p}$ and $K_{i}$ and three outputs for FOPI controller, which are $K_{p}$ , $K_{i}$ , and $λ$ .

Figure 4.

FLC basic block diagram.

Tuning through fuzzy controller

The structure of a tuned FPI and FFOPI controllers with two inputs ( $e$ and $de / dt$ ), two outputs for PI controller ( $K_{p}$ , $K_{i}$ ), and three outputs ( $K_{p}$ , $K_{i}$ , $λ$ ) for FOPI are shown in Figure 5.

Figure 5.

Mamdani fuzzy system for proposed FPI and FFOPI controllers.

Three membership functions for inputs: error ( $e$ ) and error variation ( $de / dt$ ), and five membership function for outputs ( $K_{p}$ , $K_{i}$ , $λ$ ). The linguistic levels are allocated as: PS small positive, PMS- Medium small positive, PM-Medium positive, PL- Large positive, PB-Big positive.

Suspension fuzzy control tuning by PSO

This section presents the proposal for an active suspension fuzzy controller with PSO tuning, which is outlined in Figure 6. The purpose of the PSO is to adjust the scaling factors of a fuzzy controller which are A and B for inputs and $K_{p}$ , $K_{i}$ , $λ$ for outputs, so that it minimizes the objective function and therefore minimizes the error between the desired speed wind turbine and the speed at the exit of the turbine. The components of the system are described below (Hurel et al., 2012). The range of these outputs ( $K_{p}$ , $K_{i}$ , $λ$ ) is from 0 to 300, 0 to 300, and 0 to 1, respectively.

Figure 6.

Block diagram for the PSO-Fuzzy control system.

Optimization problem

The goal of all optimization algorithms is to minimize system error performance. Therefore, the objective function must take into account the reduction in error. The characteristics of the response are implemented as an objective function. In this work, we use the integral absolute error (ITAE) criterion in goal to minimize the error signal in other words (Solihin et al., 2020); overshoots and setting time. The investigated objective function is given below:

ITAE = \int_{0}^{tss} t | e (t) | dt

(13)

where $t_{ss}$ is total simulation time and we his tacked 01 second with a step $t_{ss}$ = 0.01 second.

Results of simulation and discussion

Results of simulation

We have noted that figures 7, 8, 9, 10, 11, 12, 13 represent the memberships functions for the inputs: error and error variation and the outputs: Kp, Ki, and λ; the wind profile represents the image of the turbine speed, it is illustrated in figure 14.

Figure 7.

Memberships functions inputs.

Figure 8.

Memberships functions output 1.

Figure 9.

Memberships functions outputs 2.

Figure 10.

Memberships functions inputs.

Figure 11.

memberships functions outputs 1.

Figure 12.

memberships functions outputs 2.

Figure 13.

memberships functions outputs 3.

Figure 14.

Wind speed profile.

Discussion

We can clearly see in Figure 15 that the FPI controller optimized by PSO gives good follow-up in terms of robustness, speed and precision compared to other controllers.

Figure 15.

Performance of controllers: PI, FPI, and PSO-FPI.

It is also clearly seen in Figure 16 that the FFOPI controller optimized by PSO gives good performance compared to other controllers.

Figure 16.

Performance of controllers: FOPI, FFOPI, and PSO-FFOPI.

We can see in Figure 17, that the PSO-FFOPI controller gives a good flexibility compared to PSO-FPI, causing the impact of $λ$ .

Figure 17.

The Performance comparison between two controllers: PSO-FPI and PSO-FFOPI.

The final figure shows the robustness of two controllers PSO-FPI, and PSO-FFOPI by the parametric variation with $Δ J$ = +0.0007 and $Δ f$ = +0.002.

From Figure 18, the controllers PSO-FPI, and PSO-FFOPI present good robustness despite the parametric variation of the system.

Figure 18.

The Performance of PSO-FPI, and PSO-FFOPI controllers with parametric variation.

Results of calculation

This part of the work gives the results obtained from the fuzzyPI and fuzzyFOPI programs using a PSO code.

The parameters of the simulated PMGS-based WT system and PSO algorithm are given in the Tables 1 and 2 (Asgharnia et al., 2018; Dorrah et al., 2012; Iqbal et al., 2020).

Table 1.

The parameters of the PMGS.

Parameters	Values
Pole pairs $P$	3
Grid frequency $fn$	50 hz
Total damping $f$	0.001 (N.m.s)/rad
Total inertia $J$	0.0008 kg.m²
Number of blades	3
Blades length	01 m
Electromagnetic flux	0.175 Weber

Table 2.

The parameters of PSO algorithm.

Parameters	Values
Population size	49
Number of iteration	50
Number of variables	2 for PI and 3 for FOPI
Inertia weight $w$	0.9
Acceleration constant $C 1$ , $C 2$	2

In this work, we have optimized the interval of the universe of discourse for two inputs $e$ , $de / dt$ and three outputs $K_{p}$ , $K_{i}$ , and $λ$ and the fuzzy rules. The typical ranges of the optimized parameters had given by:

(i) for FPI: $A_{\min} < A < A_{\max}$

$B_{\min} < B < B_{\max}$

$K_{p - \min} < K_{p} < K_{p - \max}$

$K_{i - \min} < K_{i} < K_{i - \max}$

(ii) for FFOPI: $A_{\min} < A < A_{\max}$

$B_{\min} < B < B_{\max}$

$K_{p - \min} < K_{p} < K_{p - \max}$

$K_{i - \min} < K_{i} < K_{i - \max}$

$λ_{\min} < λ < λ_{\max}$

The results obtained by the PSO and on the interval [0 0] a [300 300] for FPI controller, and [0 0 0] a [300 300 1] for FFOPI controller are showing in the following Table 3:

Table 3.

Results parameters of PI and FOPI obtained by PSO algorithm.

PSO algorithm	$K_{p}$	$K_{i}$
PI Rang [0 0] to [300 300]	37.6938	282.8085
FOPI Rang [0 0 0] to [300 300 1]	300	300
$λ$	Error (output)	Error variation (output)
-	1.0415e-09	1.0415e-11
0.7943	4.5873e-06	4.5873e-08

In this step, we introduce the value of the error and variation of error calculated from the PSO. The values of the outputs $K_{p}$ , $K_{i}$ , and $λ$ obtained by the PSO-fuzzy algorithm are compared with those of the values obtained by the algorithm of PSO. The results obtained by the PSO-fuzzy algorithm have presented by following Table 4:

Table 4.

Results parameters of FPI and FFOPI optimized by PSO algorithm.

Fuzzy algorithm by PSO	$K_{p}$	$K_{i}$
PI Rang [0 0] to [300 300]	37.6949	282.6113
FOPI Rang [0 0 0] to [300 300 1]	297.5455	297.5455
$λ$	Error (input)	Error variation (input)
-	1.0415e-09	1.0415e-11
0.7959	4.5873e-06	4.5873e-08

We have seen from the results in two Tables 3 and 4 that the parameters of $K_{p}$ , $K_{i}$ for PI and $K_{p}$ , $K_{i}$ , and $λ$ for FOPI calculated by the PSO algorithm are almost equaled close to those calculated by fuzzy algorithm, then the fuzzy logic for PI and FOPI are optimized. Therefore, the results rules fuzzy for PI and FOPI controllers are presented In Tables 5, 6:

Table 5.

Results of fuzzy rules for FPI.

Fuzzy rules

ans =

1. If (error is negative) and (error variation is negative) then (Kp is PB)(Ki is PB) (1)

2. If (error is negative) and (error variation is zeros) then (Kp is PB)(Ki is PB) (1)

3. If (error is negative) and (error variation is positive) then (Kp is PB)(Ki is PB) (1)

4. If (error is zeros) and (error variation is negative) then (Kp is PMS)(Ki is PB) (1)

5. If (error is zeros) and (error variation is zeros) then (Kp is PMS)(Ki is PB) (1)

6. If (error is zeros) and (error variation is positive) then (Kp is PMS)(Ki is PB) (1)

7. If (error is positive) and (error variation is negative) then (Kp is PS)(Ki is PB) (1)

8. If (error is positive) and (error variation is zeros) then (Kp is PS)(Ki is PB) (1)

9. If (error is positive) and (error variation is positive) then (Kp is PS)(Ki is PB) (1)

Out =

37.6929 282.6113

Table 6.

Results of fuzzy rules for fuzzy FOPI.

Fuzzy rules

ans =

1. If (error is negative) or (error variation is negative) then (Kp is PB)(Ki is PB)(lambda is PB) (1)

2. If (error is negative) or (error variation is zeros) then (Kp is PB)(Ki is PB)(lambda is PB) (1)

3. If (error is negative) or (error variation is positive) then (Kp is PB)(Ki is PB)(lambda is PB) (1)

4. If (error is zeros) or (error variation is negative) then (Kp is PB)(Ki is PB)(lambda is PB) (1)

5. If (error is zeros) or (error variation is zeros) then (Kp is PB)(Ki is PB)(lambda is PB) (1)

6. If (error is zeros) or (error variation is positive) then (Kp is PB)(Ki is PB)(lambda is PB) (1)

7. If (error is positive) or (error variation is negative) then (Kp is PB)(Ki is PB)(lambda is PB) (1)

8. If (error is positive) or (error variation is zeros) then (Kp is PB)(Ki is PB)(lambda is PB) (1)

9. If (error is positive) or (error variation is positive) then (Kp is PB)(Ki is PB)(lambda is PL) (1)

Out =

299.3666 299.3666 0.7943

Conclusion

We assumed a mixed approach of PSO and FLC to achieve the optimal parameters of PI and FOPI controllers in MPPT for a wind turbine. PSO is used to optimize ranges of membership functions and fuzzy rules. The developed algorithm provided a high quality solution in an efficient manner and offered full control to extract the maximum power provided by the wind. the designed FLC gives better control performance than the status feedback controller. However, the FLC is not robust, but when we use the PSO algorithm to tune the FLC, the FPI and FFOPI controllers become very robust. but according to the studies which already made and the study in this work, the new commands PSO-FFOPI and PSO-FPI are very effective compared to the controllers PI, FOPI, FPI, and FFOPI.

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

Sarir Noureddine

Sebaa Morsli

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