Optimal torque maximum power point technique for wind turbine: Proportional–integral controller tuning based on particle swarm optimization

Abstract

This article presents the problem of the energy system optimization for wind generators. The goal of this work is to maximize power extraction for a permanent magnet synchronous generator–based wind turbine with maximum power point technique. This goal is achieved using a proportional–integral controller for optimal torque tuning with the particle swarm optimization algorithm. In order to indicate the effectiveness and superiority of the particle swarm optimization algorithm–based proposal, a comparison with the genetic algorithm and the artificial bee colony algorithm is studied. The system is modeled and tested under MATLAB/Simulink environment. Simulation results validate the advantages of the designed particle swarm optimization–tuned proportional–integral controller compared to P&O and the proportional–integral controller manually in terms of performance index.

Keywords

Wind turbine genetic algorithm artificial bee colony algorithm particle swarm optimization algorithm optimum torque maximum power point technique proportional–integral controller

Introduction

Among the most important development preferences in the world, energy is considered as the first area to be developed. On one hand, with the shortage of fossil fuels, international electricity needs are increasing rapidly which make costs rise as rapidly. On the other hand, the burning of fossil fuels such as oil, natural gas, and coal causes a lot of emissions of carbon dioxide (CO₂) into the atmosphere. This effect results in the absorption of heat and global warming. So, we are forced to look for another solution to save the lives of people and the earth. In order to avoid all these problems, renewable energies are considered as one of the effective sources capable of generating better energy than conventional fuels and especially without the disadvantages of carbon dioxide emissions or to produce radioactive waste (Storey et al., 2013). Thus, renewable energies are usually renewed and abundant such as the case of the wind, which is considered as a more profitable source of energy currents.

At the performance problem level of the wind turbine (WT) drive, many literatures have been presented. For example, the research by Sarvi et al. (2013) describes two different permanent magnet synchronous generator (PMSG) WT control modes. The field control (FOC) is the first control strategy studied which requires knowing the speed and position of the rotor. In addition, in order to improve control efficiency, the non-linear sliding mode control (SCM) strategy is applied to the second control method. Qiao et al. (2012) have implemented a sliding mode observer in order to estimate the position of the generator rotor based on the stator voltages controlled and the stator currents measured. The optimization method based on the genetic algorithm (GA) was applied in Hasanien and Muyeen (2012) to regulate the control parameters of a PMSG WT. An implementation of the direct control strategy of the couple and the field was developed around the study by Kim et al. (2015). Kim et al. (2013) discuss a torque compensation strategy for PMSG WT to ensure system stability. Still, a method of finding proportional–integral (PI) controller parameters of a PMSG WT to enhance control performance is presented in Busca et al. (2010).

Today, the proportional–integral–derivative (PID) controller is one of the most used controllers for several reasons. First, it is very simple to set up and is effective for most real systems. Moreover, the calculation of coefficients allows the choice between several methods of increasing difficulty. Thus, an experimental method that is very easy to set up makes it possible to quickly obtain correct coefficients for systems that do not require very great precision. However, it is important to note that this type of conventional PID controllers have a certain limitation, in the case where rather large variations of the perturbation factors act on the system to be adjusted. The classic regulator PID does not always react optimally.

An optimization problem can be defined as a search, in a solution space, of an optimal solution quantified by an objective function. This quantification leads to maximize or minimize the problem. For the case of a difficult problem which cannot be solved in a polynomial time or by an exact method, the metaheuristics are considered as optimizers of this type. These methods are iterative algorithms, possessing a random component and traversing the research space by different techniques for generating solutions. Metaheuristics is often inspired by physical, biological, or ethological systems. Among the most popular optimization metaheuristics is the GA, the artificial bee colony (ABC) algorithm, and the particle swarm optimization (PSO) algorithm.

The PSO algorithm does not need many of the numbers of parameters for it to be executed. The PSO technique is an easy and simple method to implement compared to other metaheuristic algorithms (Bechouat et al., 2015).

The maximum power point technique (MPPT) is of paramount importance in renewable energy conversion systems. Due to the great inertia of a WT system, the MPPT methods studied for Wind Energy System (WES) are more difficult compared to those of photovoltaic system. Among the most used indirect methods, we quote the Tip Speed Ratio (TSR) (Cheng and Zhu, 2014) method which is working with the measurement of the wind speed. Other methods are based on the power curve of the WT such as the case of the Optimal Torque Control (OTC) (Johnson et al., 2006) and Power Signal Feedback (PSF) (Pagnini et al., 2015) method. These last algorithms have almost the same strategy of which PSF is based on the generator’s speed-power interrogation method in order to calculate the optimal power while OT produces the optimal torque by means of a conventional control. By comparison in the search (Song et al., 2017), the PSF algorithm may be slightly lower than the Optimal Torque (OT) algorithm. To further improve OT performance, work was discussed such as the example of adding an anticipation term to facilitate acceleration or deceleration (Johnson et al., 2004). Estimation methods are presented to readjust the parameter of the gain of torque (Hand et al., 2004). In this article, the PSO will be chosen to adjust and improve the torque gain for the MPPT OT. This optimization algorithm looks for the optimal parameters of the PI controller in order to extract the maximum power from the WT equipped with a synchronous generator with permanent magnets. In the simulation part, the proposed OT technique optimized by the PSO will be compared with the conventional MPPT OT and the MPPT Perturb and Observe (P&O) for the improvement to be illustrated.

This article is organized as follows. Section “WT” presents detailed WT modeling with its own equations. In section “Comparative analysis,” the study of some optimization algorithms such as GA, ABC, and PSO are treated so an example of application is applied which contains a comparison between proposed metaheuristic approaches. Section “MPPT strategy optimal torque control” is devoted to the presentation of MPPT with optimal torque associated with a PI controller optimized by the PSO algorithm. In section “Simulation results,” all numerical simulation results, obtained with the PSO metaheuristic, are presented and compared with the PI manual and P&O. Section “Conclusion” concludes with a conclusion and future work.

WT

The WT has the role of capturing the kinetic power of the wind $P_{v}$ and transforming it into a mechanical power $P_{mec}$ whose expressions of these two powers, respectively, are written in the following form (Soufi et al., 2016)

P_{v} = \frac{1}{2} ρ π R^{2} V^{3}

(1)

P_{mec} = P_{v} C_{p} = \frac{1}{2} ρ π R^{2} V^{3} C_{p}

(2)

where $V$ is the wind speed, $ρ$ is the air density, $R$ is the radius of WT, and $C_{p}$ is the coefficient of power conversion defined as a function of the blade tip speed ratio $λ$ and blade pitch angle $β$ , and this dependence is clearly shown in Figure 1 (Elbeji et al., 2014)

C_{p} (λ, β) = 0.5176 (\frac{116}{λ_{i}} - 0.4 β - 5) e^{(\frac{- 21}{λ_{i}})} + 0.006795 λ

(3)

$λ_{i}$ is given by

λ_{i} = \frac{1}{\frac{1}{(λ + 0.08 β)} - \frac{0.035}{β^{3} + 1}}

(4)

$λ$ is the tip-speed ratio which is obtained by the following expression (Elbeji et al., 2014)

λ = \frac{Ω R}{V}

(5)

Figure 1.

Power coefficient curve according to $β$ and $λ$ .

The power coefficient is maximum around the value 0.48 when $β = 0$ .

Figure 2 shows the mechanical power curves of the WT as a function of the variation of the rotor speed with different values of the wind speed.

Figure 2.

The mechanical power curves with variation of $Ω$ and $V$ .

Comparative analysis

In this section, we present some metaheuristic algorithms which are the most widely used optimization algorithms.

GA

GAs were invented by John Holland in the 1960s in his work (Holland, 1975) whose original purpose was to study the mechanisms of adaptation in nature to develop methods for to import these mechanisms into computer systems.

These algorithms have been further developed in Goldberg and Holland (1988) in order to have an ambiguity allowing the origin of solving problems with discrete variables.

The principle of this algorithm is to search for potential solutions in a given population in order to find those that approach certain specifications or criteria. At each generation, the principle of survival is applied as a new set of approximations is created by the process of selecting individual potential solutions (individuals) basing at their level of form in the problem domain and their reproduction moderately operators borrowed from natural genetics. Just as in natural adaptation, this process leads to the evolution of the population of individuals better adapted to their environment than the individuals from whom they were created.

A basic GA comprises three genetic operators’ selection, mutation, and crossover.

Selection. It has the role of examining a set of individuals of the population based on their performances. A selection preferably retains the best individuals, but must also give the less fortunate a chance to avoid the problems of premature convergence.

Mutation. This operation reflects the operating character of the algorithm. The mutation introduces a small perturbation on the chromosome of an individual.

Crossing. It is responsible for exploration in the research space by diversifying the population. This operation usually manipulates the chromosomes of two parents to generate two children.

These operations are applied iteratively in the GA defined as follows in the following flow chart (Figure 3).

Figure 3.

Flow chart of GA algorithm.

ABC algorithm

The ABC algorithm, introduced by Dervis Karaboga (2005), is a population-based, naturalist-inspired algorithm based on the foraging of bees.

The operating principle of the optimization algorithm of the ABC is formulated according to the natural feeding behavior of bees which is a candidate solution to the problem of optimization and is represented by a food source (Karaboga and Basturk, 2008). The ABC algorithm begins its search with a random set of solutions (colony size) unlike other optimization methods such as the case in the GA that is based on a single solution. Thus, each member of the population is evaluated for the given objective function and is assigned a physical form. The best adjustments are planned for the next generation, while the others are rejected and offset by a new set of random solutions at each generation. At the end of the cycles, the solution of the best fit is the desired solution whose only stopping criterion is the completion of the maximum number of cycles or generations.

The operating steps of this optimization algorithm are well detailed in the flow chart (Figure 4).

Figure 4.

Flow chart of ABC algorithm.

PSO algorithm

The PSO algorithm is a population algorithm, proposed in 1995 by Russel Eberhart and James Kennedy (1995). This algorithm considered as a global and stochastic optimization technique is inspired by the collaborative behavior of biological populations and organisms such as schools of fish and flocks of birds. In order to solve various complex optimization problems in various fields, the principle of information change between individuals of the population found in this algorithm is used in several resolutions.

A swarm is a set of particles positioned in the search space of the objective function. The principle of the algorithm is to move these particles in the search space to find the optimal solution. Each particle represents a solution of the problem of which it possesses a position and a speed. The speed is actually a vector of displacement, of modifying its current position, defining his future probable position. Each particle remembers its best personal position at all times, as well as the best position of its group of informants (or neighborhood).

The PSO search principle is described in the following flow chart (Figure 5).

Figure 5.

Flow chart of PSO algorithm.

Comparison between GA, ABC, and PSO

This part is dedicated to a comparative study made between the algorithms described in the last part in order to choose the algorithm that will be used later. The example used is the search for the minimum of the “my Rastrigin” test function in order to examine the performance of each algorithm.

The size of the population is $n = 30$ and the number of generations is $I_{max} = 100$ . These control parameters are chosen to be identical for different algorithms.

For PSO algorithm, the cognitive and social coefficients are $c_{1} = 0.5, c_{2} = 0.3$ and the possibility models are given as $0.001 < σ < 0.15$ and $0.4 < ω < 0.9$ .

For GA algorithm, the crossover percentage is $P_{cro} = 0.7$ and the mutation percentage is $P_{mut} = 0.3$ .

All the algorithms are simulated and its results are illustrated in Figure 6.

Figure 6.

Convergence properties of the proposed algorithms.

It is very clear that the PSO is the first algorithm that holds the object sought before the other algorithms. At iteration 25, the objective function executed by the PSO algorithm is equal to zero on the other hand GA arrived at this value at iteration 41. the last algorithm ABC attacked the position sought only at iteration 61.

Based on the curve obtained after the execution of the proposed algorithms, we note that the best cost function is that given by PSO, of which it produces a result close by a total convergence toward the same interesting region (Appendix 1). This indicates the success of this algorithm to explore and find the overall optimum in the search space.

After the good result obtained by PSO in the example treated in this part, this optimization algorithm will be used later to determine the optimal parameters of a PI regulator.

MPPT strategy optimal torque control

In this section, the optimal torque control technique will be discussed in which the purpose of this MPPT is to control the electromagnetic torque to facilitate the control of the mechanical speed to maximize the generated electrical power.

Optimal torque technique formulation

The system of voltage equation of the synchronous generator with permanent magnets in the reference $d - q$ is defined in the following equations

{\begin{matrix} u_{d} = - R_{s} i_{d} -_{s} L_{d} \frac{d i_{d}}{dt} + ω L_{q} i_{q} \\ u_{q} = - R_{s} i_{q} -_{s} L_{q} \frac{d i_{q}}{dt} - ω L_{d} i_{d} + ω Ψ_{f} \end{matrix}

(6)

in which $ω$ is defined with

ω = p Ω

(7)

where p is the number of pairs of poles, $L_{d}, L_{q}$ are, respectively, the inductances of the direct and quadrature axis, $Ψ_{f}$ is the inductive flux, and $ω$ is the electrical pulsation of the PMSG voltages.

The expression of the electromagnetic torque is given by

T_{em} = \frac{3}{2} p ((L_{d} - L_{q}) i_{q} i_{d} - Ψ_{f} i_{q})

(8)

The dynamic of the machine is given by the following mechanical equation

J \frac{d Ω}{dt} + f Ω = T_{m} + T_{em}

(9)

where $T_{m}$ is the mechanical torque, $J$ presents the total moment of inertia of the machine, and $f$ is the viscosity coefficient of friction.

In order to capture and keep the optimal position in which the power started remains at its maximum, a hypothesis on the variation of the wind speed must be applied. This condition is that this change in steady state is negligible. If we neglect the coefficient of friction, the following equality is extracted from the mechanical equation such that

T_{m} = T_{em}

(10)

T_{m} = \frac{P_{mec}}{Ω} = \frac{0.5 ρ π R^{3} V^{2} C_{p}}{λ}

(11)

The estimation of the wind speed is done in order to solve the problems due to the use of the speed sensor. It can be determined from the expression of the gear ratio as a result

V = \frac{Ω R}{λ}

(12)

The new expression of the torque as a function of the mechanical speed, after the insertion of the estimated wind equation, is written in the following form

T_{em} (Ω) = \frac{0.5 ρ π R^{5} Ω^{2} C_{p}}{λ^{3}}

(13)

In order to extract the maximum of the power, the reference electromagnetic torque must be calculated at the optimum operating point with the pair $(C_{p max}, λ_{op})$ which is given by

T_{em - ref} (Ω) = \frac{0.5 ρ π R^{5} Ω^{2} C_{p max}}{{λ_{op}}^{3}}

(14)

The reference torque is proportional to the square of the speed of the generator in this form

T_{em - ref} (Ω) = K_{opt} Ω^{2}

(15)

With

K_{opt} = \frac{0.5 ρ π R^{5} C_{p max}}{{λ_{op}}^{3}}

(16)

Designing of PI controller using PSO

During the search for the optimal position (state) of such a problem, the PSO optimization algorithm is based on the change of position (state) of particles over time. These latter move in a multidimensional search space. During the flight, each particle adapts its position according to its own experience which is called $p_{k}^{i}$ and according to the experience of a neighboring particle named $p_{k}^{g}$ . In other words, each particle takes the best position discovered by itself and its neighbor.

Every one of the particles is a potential solution of which it is characterized by a position $x_{k}^{i}$ and a speed $v_{k}^{i}$ . At each iteration of the algorithm, the $i th$ particle position $x^{i} \in 1 R^{m}$ evolves according to the following update rules (Xinchao, 2010)

x_{k + 1}^{i} = x_{k}^{i} + v_{k + 1}^{i}

(17)

v_{k + 1}^{i} = {wx}_{k}^{i} + c_{1} r_{1, k}^{i} (p_{k}^{i} - x_{k}^{i}) + c_{2} r_{2, k}^{i} (p_{k}^{g} - x_{k}^{i})

(18)

where $w$ is the inertia factor; $c_{1}$ and $c_{2}$ are, respectively, the cognitive factors and the social scale factors; $r_{1, k}^{i}$ and $r_{2, k}^{i}$ are random numbers evenly distributed in the interval [0, 1].

Control of the electromagnetic torque is necessary in order to guarantee the correct operation of the MPPT of the optimal torque. For this purpose, a PI regulator is introduced in the chain. But the problem is that the mathematical model of the installation must be known. Our proposed method is to use the PSO to adjust and optimize the parameters of the PI controller in order to solve the problems of the system as a whole. All the particles of the population are decoded for KP and Ki. The gain parameters of the PI controller are judged by the value of the integral of time absolute error (ITAE). The sum of the performance index (ITAE) is chosen as an objective function. Stochastic algorithms aim to minimize the objective function. The ITAE index is expressed as follows (Allaoua et al., 2009)

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

(19)

Figure 7 illustrates the systematic design of the studied chain.

Figure 7.

PMSG variable speed wind energy conversion MPPT control.

The diagram in Figure 8 shows the PSO steps followed to find the regulator parameter.

Figure 8.

The flowchart of the PSO–PI control system.

Simulation results

The wind model is considered as a Fourier series representation of the wind. This form is composed of a superposition of many of the harmonics written according to $W_{k}$ and $a_{k}$ which are, respectively, the average value of the wind speed and the harmonic amplitude (Allaoua et al., 2013)

V_{v} = A + \sum_{k = 1}^{i} a_{k} \sin (W_{k} t)

(20)

Due to the absence of a pitch angle control system, the results are discussed in this work with a wind speed lower than nominal $V_{nominal} = 10$ and $β = 0$ . In this article, the wind variation is defined as follows

V (t) = 6 + 0.2 \sin (0.1047 t) + 2 \sin (0.2665 t) + \sin (1.293 t) + 0.2 \sin (3.6645 t)

(21)

In order to apply the MPPT studied according to the optimal torque using the optimized PSO PI controller, the proposed optimization algorithm is executed in the MATLAB/Simulink environment in order to adjust the desired parameters. To demonstrate the efficiency of the optimization carried out, the MPPT P&O is applied in this part in order to compare it with the traditional OT and that opted with PSO. The results of the simulation are presented in Figures 9 to 14.

Figure 9.

Wind speed profile.

Figure 10.

Coefficient of power.

Figure 11.

Tip speed ratio.

Figure 12.

Mechanical power.

Figure 13.

Electromagnetic torque.

Figure 14.

Cp versus lambda.

According to the results, the OTC optimized by the PSO algorithm proved to be the fastest to reach steady state. The recovery time during the wind speed change was also faster for this algorithm. In addition, this method reached the highest value of $C_{p}$ , even with the change in wind speed, almost the optimum value of the turbine equal to 0.48 and kept it (Figure 10). Next, the MPPT OT takes 0.479 as the maximum value for $C_{p}$ with a small variation during the wind change. Finally, the P&O method takes longer to reach the optimum value with a $C_{p}$ value of 0.45.

Comparing Figures 11 and 14, it can be seen that the results given by PSO optimized PI are the same optimum values of the turbine and are more accurate than those obtained by manual PI and P&O. Thus, the peak velocity ratio λ given to the PSO algorithm is optimal with its value equal to 8, as shown in Figure 14, which presents the optimum value of the turbine. However, the value obtained by manual PI is almost equal to 8.2 and for the control P&O is equal to 9.

Figure 12 presents the mechanical power. The power obtained by controller OT was regular while that of P&O has intense variations.

The curves obtained in Figure 13 show the effectiveness of the proposed torque control strategy whose resulting torque of the PSO algorithm is almost identical to the reference torque contrary that of P&O (presence of error) and the normal PI (Tables 1 –3).

Table 1.

Comparison between PI tuned manually and PI tuned by PSO algorithm.

Controller	PI tuned manually	PI tuned by PSO
Generator	$K_{i} = 0.3$	$K_{i} = 1846$
Response	$K_{p} = 3$	$K_{p} = 1812$
ITAE	$0.759$	$0.088$

PI: proportional integral; PSO: particle swarm optimization; ITAE: absolute integral time error.

Table 2.

Parameters of the wind turbine and those of the machine.

Parameter	Values
$ρ$	$1.225 kg m^{- 3}$
$R$	$2 m$
$P_{n}$	$3 KW$
$R_{a}$	$0.82 Ω$
$L_{s}$	$15.1 e^{- 3} H$
$J$	$99 e^{- 4} kg m^{2}$
$f$	$10^{- 3} N m s ra d^{- 1}$
$φ$	$0.5 Wb$
$p$	4

Table 3.

Parameters of the PSO with a linear algorithm.

Parameter	Values
Swarm size	10
The maximum number of iteration	30
$c_{2} = c_{1}$	2
$σ_{max}$	0.15
$σ_{min}$	0.001
$ω$	0.9

PSO: particle swarm optimization.

Conclusion

First of all, in this article, we presented the mathematical modeling of the WT. Then, we studied in detail the existing algorithms such as the optimization of PSO, ABC, and GA. Thus, a comparative study has been carried out granted to a simple example in order to choose the best algorithm among them which will be used thereafter. Second, the MPPT optimum torque is implied in the wind energy conversion chain whose PI controller is optimized by the PSO algorithm. Demonstrative simulation results are performed in order to show the efficiency of PI controllers set by proposing metaheuristic. The effectiveness of the proposed methodology is verified. It is well illustrated at the moment of the stabilization of the torque response of the generator, which has been optimized with the optimal parameters obtained with PSO.

To clearly show the innovation sought, we have provided a comparison with traditional OT and P&O. It can be seen that the PSO-adjusted OTC method is one of the simplest and most effective methods among MPPT methods for strong power. The P&O always remain less efficient compared to OT even if it overcomes the obstacle of the OTC method since it is independent of the characteristics of the turbine. From the results of the comparison of the two controllers, we can prove that the OT process works well in terms of energy production, it gives almost optimal values of the studied turbine, whereas the P&O process produces values close to those optimal values. In spite of the efficiency of the OT found in this work, this control encounters, thanks to the great inertia, a reduction of power due to a delayed response. Also, when the WT parameters are unknown, an OT cannot be used so in this case one needs an adaptive method in order to follow the actual MPP.

Footnotes

Appendix 1

The figure below shows the variation of the objective function (Fitness) during the iterations.

popul1: population N°1; ppo size: size of population; upbnd, lwbnd: up and low bandaged

The initial population is defined by the following random equation

popul 1 = rand (\dim, ppo - size) * (upbnd - lwbnd) + lwbnd

The optimal values are those of the iteration 17 whose minimum value of the objective function remains fixed until the final iteration.

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

Marwa Hannachi

Omessaed Elbeji

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