Optimal tuning of proportional–integral controller using particle swarm optimization algorithm for control of permanent magnet synchronous generator based wind turbine with tip speed ratio for maximum power point tracking

Abstract

This article deals with the study of the particle swarm optimization algorithm and its variants. After modeling the global system, a comparative study is carried out about the algorithms described in order to choose the best of those to be used thereafter. Then, the perturbed particle swarm optimization is presented to determine the optimal parameters of the proportional–integral controller for speed control to certify the tip speed ratio for maximum power point tracking of a wind energy conversion system. A numerical simulation is used in conjunction with the particle swarm optimization algorithm to determine the proportional–integral controller optimal parameters. From the simulations results, we observe that the proportional–integral controller designed with particle swarm optimization gives better results compared to the traditional method (proportional–integral manually) in terms of the performance index.

Keywords

Wind turbine permanent magnet synchronous generator field-oriented control controller proportional–integral tip speed ratio particle swarm optimization

Introduction

In recent years, electrical energy and the constraints related to its production are increasing, such as the effects of pollution. So, the research studies lead to the development of renewable energy sources. In this case, wind energy conversion systems (WECSs) are considered as a very successful solution. Furthermore, it is necessary to optimize the design of WECS parts in order to overcome the efficiency problem, in the case of maximum performance (Bašić et al., 2019).

Over the last decade, several new control techniques such as the variable structure control, adaptive control, and intelligent control have been intensively studied to control non-linear components in electrical machines. Numerous works of literature on wind turbine control performance problems have been discussed such as Jha (2017) has implementing a sliding mode observer to estimate the rotor position of the generator based on the stator voltages controlled and the measured stator currents. Rajan (2016) describes two different methods of permanent magnet synchronous generator based wind turbine (PMSG WT) control. The author uses, on the one hand, the field-oriented control (FOC) in which it is necessary to know the position of the rotor and the speed. On the other hand, the non-linear sliding control mode (SCM) is presented in order to improve the efficiency of the control. A genetic algorithm was applied by Hassanein and Muyeen (2012) to regulate the controller parameters of a PMSG WT. Kim et al. (2013) used a torque compensation strategy for PMSG WT to improve system stability. Also, a method to search for the proportional–integral (PI) controller parameters of a PMSG wind turbine to develop control performance is studied by Kim et al. (2015). An implementation of the direct torque and field control strategy was developed by Busca et al. (2010). In spite of the good results given by these control techniques, they have few real applications. On the one hand, their stability has a lack of confidence. On the other hand, their structures are complicated. In contrast, conventional PI controllers are still the most commonly used control techniques in power systems because of their simple structures. Unfortunately, setting up traditional PI controllers can be difficult to properly adjust PI gains due to non-linearity.

The metaheuristic particle swarm optimization (PSO) is a calculation algorithm based on the intelligence of the swarm originally proposed by J. Kennedy and R. Eberhart (Hajihassani et al., 2018). This stochastic and global optimization technique is stimulated by the collaborative behavior of populations and biological organisms such as schools of birds and schools of fish (Singh et al., 2016). The exchange of information among individuals in the population solves various complex optimization problems in various engineering domains (Nath et al., 2018).

In this article, types of the PSO algorithm such as the case of the canonical PSO algorithm (Liu et al., 2012), PSO with decreasing linear inertia (Raska and Ulrych, 2015), and the disturbed PSO variants (Jensi and Jiji, 2016) particularly used for the optimization of the PI controller parameters. The speed regulation problem has behaved as a constrained optimization problem that is effectively solved by proposed PSO algorithms whose error performance criteria will be minimized as an objective function.

This article is organized as follows. Section “Model of wind turbine” presents the modeling of the wind turbine. A mathematical model of permanent magnet synchronous generator (PMSG) is then established. In section “PMSG model,” the decoupling problem, specifically the determination and adjustment of parameters of the PI controller, is formulated as an optimization problem. Section “Field-oriented control of PMSG” is devoted to the presentation of the proposed PSO algorithms, namely canonical PSO, PSO with a linear decrease in its inertia factor, and the disturbed PSO, used to solve the PI controller agreement problem formulated. In the section “Tuning PI using PSO,” all the simulation results, obtained with the proposed metaheuristic approaches, are shown and compared with each other. The document is finished with a conclusion.

Model of wind turbine

The wind turbine captures the wind kinetic energy and converts it into a torque that turns the rotor blades. The air density, the area swept by the rotor, and the wind speed are the factors that determine the relationship between the wind energy and the mechanical energy recovered by the rotor given by the following expression

p_{v} = \frac{1}{2} ρ S V^{3}

(1)

$ρ$ is the density of air $(ρ = 1.22 kg \cdot m^{- 3})$ , S is the circular surface swept by the turbine, the radius of the circle is determined by the length of the blade R such as $S = π R^{2}$ , and V is the wind speed.

Thus, the aero generator is able to convert the wind energy to mechanical energy. The mechanical torque $C_{m}$ is defined as follows

C_{m} = \frac{P_{mec}}{Ω}

(2)

$Ω$ is the turbine rotor speed.

The mechanical power $P_{mec}$ , also called aerodynamic power, depends on the coefficient of power $C_{p}$ . It is given by

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

(3)

$C_{p} (λ, β)$ is the power factor that depends on the characteristics of the turbine. It characterizes the aerodynamic efficiency of the turbine. Theoretically, $C_{p}$ is limited by the Betz limit (0.593). In practice, this limit is never reached (Smaili, 2013). The expression used is the one used by Alizadeh (2014) which is written in this form

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

(4)

$λ$ is given by

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

(5)

The coefficient of power conversion $C_{p}$ depends on the pitch angle of the blades $β$ and the speed ratio of the turbine $λ$ which is defined by this equation

λ = \frac{Ω R}{V}

(6)

PMSG model

The equations of the synchronous generator with permanent magnets in the reference $d - q$ are defined by

[\begin{matrix} u_{d} \\ u_{q} \end{matrix}] = - R_{s} [\begin{matrix} i_{d} \\ i_{q} \end{matrix}] + \frac{d}{dt} [\begin{matrix} Ψ_{d} \\ Ψ_{q} \end{matrix}] + ω_{e} [\begin{matrix} Ψ_{d} \\ Ψ_{q} \end{matrix}]

(7)

$Ψ_{d}$ and $Ψ_{q}$ are expressed in terms of stator current as follows

{\begin{matrix} Ψ_{d} = - L_{d} i_{d} + Ψ_{f} \\ Ψ_{q} = - L_{q} i_{q} \end{matrix}

(8)

By replacing $Ψ_{d}$ and $Ψ_{q}$ with their expressions in the system of voltage equation, we obtain the following equations (Al-Toma et al., 2015)

{\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}

(9)

in which $ω$ is defined with

ω = p Ω

(10)

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 electrical power can be expressed in the DQ-axis reference frame as follows

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

(11)

The expression of the electromagnetic torque is given by

C_{em} = \frac{P_{em}}{ω / p} = \frac{3}{2} p ((L_{d} - L_{q}) i_{q} i_{d} - Ψ_{f} i_{q})

(12)

About this study, the generator used is smooth poles, so the expression of the couple will be in this form

C_{em} = - \frac{3}{2} p Ψ_{f} i_{q}

(13)

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

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

(14)

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

Field-oriented control of PMSG

The principle of vector control with voltage supply and current control makes it possible to impose the torque. However, regardless of the purpose of the control (torque, speed, or position control), current monitoring is still necessary (Padmanathan et al., 2019).

To simplify the control, the proposed strategy is applied to the generator by optimizing the speed with tip speed ratio for maximum power point tracking (TSR-MPPT) to maximize the power produced by the wind turbine. The d-axes stator current of PMSG can be set to zero during the operation to achieve a linear relationship between the stator current and the electromagnetic torque. Then again, the generator can be controlled to produce maximum torque with minimum direct current. In this case, the torque will be controlled by the quadrature component. This command consists of controlling the two components of the stator current $i_{d}$ and $i_{q}$ by imposing the voltages $U_{d}$ and $U_{q}$ . Consequently, in order to require these voltages and to solve the coupling problem between the axes d and q, it suffices to impose the references voltages $U_{dref}$ and $U_{qref}$ at the input of the generator which are written in this form

U_{dref} = K_{p} e_{d} + K_{i} \int e_{d} dt - Ω L_{s} i_{q}

(15)

U_{qref} = K_{p} e_{q} + K_{i} \int e_{q} dt + Ω L_{s} i_{d} + Ω φ

(16)

$K_{p}$ and $K_{i}$ are the parameters (proportional and integral) of the current controller PI, $e_{d}$ and $e_{q}$ are the errors of the current given by

e_{d} = i_{dref} - i_{d}

(17)

e_{q} = i_{qref} - i_{q}

(18)

where

i_{dref} = 0

(19)

and

i_{qref} = \frac{C_{emref}}{K_{t}}

(20)

The reference torque $C_{emref}$ is obtained after the loop of the speed control that can be written in the form

C_{emref} = K_{p} e_{w} + K_{i} \int e_{w} dt

(21)

with

K_{t} = \frac{3}{2} p φ ()

(22)

e_{w} = Ω_{ref} - Ω

(23)

In order to extract the maximum power generator, the turbine speed should be changed with wind speed so that the optimum tip speed $λ_{opt}$ is maintained which ensures the optimization of the power coefficient. For the speed control of the PMSG, the reference speed of the wind turbine is obtained by (Munteanu et al., 2013)

Ω_{ref} = \frac{λ_{opt} V_{v}}{R}

(24)

Tuning PI using PSO

The coefficients of the conventional controllers PI used within the vector control are directly calculated from the parameters of the machine when the drifts of the latter cause an alteration of the control of the machine. In order to obtain better performances, the optimization of these controllers is used. As an optimization technique, the optimization method called “particle swarm optimization” will be used in this part including the operating principle of the PSO technique as well as these vary by highlighting their similarities and differences in the application at the speed setting through the gains of the PI controller.

Designing of PI controller using PSO

Regarding the field of machine control, the PI controller is widely used. It is a good controller except that the problem of the mathematical model of the installation must be known thus the presence of the error between the desired value and the mustered one. As a solution to this disadvantage, several methods have been introduced to adjust the PI controller. In this work, the proposed method is to use the PSO algorithm to optimize PI controller parameters based on the error of stator current $i_{d}$ and $i_{q}$ also that of speed. The performance of the PMSG varies according to the gains of the PI controller and is judged by the value of the integral time absolute error (ITAE) of which the sum of the performance indices (ITAE) is chosen as the objective function. Stochastic algorithms are to minimize the objective function so that all particles in the population are decoded for $K_{p}$ and $K_{i}$ (see Figure 1).

Figure 1.

The global control system: PMSG variable speed wind energy conversion MPPT control.

The ITAE is the one used in this search, which is described by

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

(25)

The diagram in Figure 2 shows the PSO steps followed to find the controller parameter.

Figure 2.

The flowchart of the PSO–PI control system.

Canonical PSO algorithm

In order to reach an optimal solution of a potential generic optimization solution, the canonical PSO algorithm uses a swarm of randomly distributed particles in the initial search space considered which is characterized by a position $x_{k}^{i}$ and a speed $v_{k}^{i}$ given by

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

(26)

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})

(27)

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]; $p_{k}^{i}$ is the top of position attained previously from the particle; and $p_{k}^{g}$ has the best position succeeded in the entire swarm at the present iteration k.

PSO with inertia weight decreasing

The w parameter is the inertial weight that increases the overall performance of PSO. It is reported that greater values of greater capacity for global research while the lower value of greater local search capability. To achieve better performance in improving the exploration and exploitation abilities of the PSO canonical algorithm, we linearly reduced the value of inertial weights in generative search and search generation which is calculated by the following formula (Singh et al., 2016)

w_{k + 1} = w_{max} - (\frac{w_{max} - w_{min}}{k_{max}}) k

(28)

$w_{max} = 0.9$ and $w_{min} = 0.4$ represent the maximum and minimum inertia factor values, respectively; $k_{max}$ is the maximum iteration number.

PSO with perturbed particle updating strategy

In order to correct the very high PSO convergence speed defect, which often results in a rapid loss of diversity during the optimization process and inevitably leads to undesired premature convergence, optimization of the oscillation of disturbed particles is treated by Jensi and Jiji (2016) which has almost the same code as canonical but reformulated in order to escape the optimal local trap.

The gbest is denoted as “possibly at gbest” instead of a crisp location and it is noted $p - gbest = (p_{g, 1}, p_{g, 2}, \dots, p_{g, D})$ which is characterized by a normal distribution given as follows (Jensi and Jiji, 2016)

{\hat{p}}_{k}^{q} ~ N (p_{k}^{q}, σ)

(29)

where $σ$ represents the degree of uncertainty about the optimality $g_{jbest}$ of the standard PSO algorithm.

Concerning the strategy for updating disturbed particles, $σ$ it is presented as a non-increasing function of the number of cycles k. Thus, the function $p - gbest$ is to encourage particles to explore a region beyond that defined by the search path.

$p - gbest$ foresaw an effective and simple exploration at an early stage when $σ$ is large and favors local adjustment in the last stage as $σ$ is small. Jensi and Jiji (2016) have proposed three models such as update min–max model, linear model, and random model which are presented as follows.

The disturbed PSO with min–max update mechanism is given by this equation

σ = {\begin{matrix} σ_{max}, k ≺ σ k_{max} \\ σ_{min}, otherwise \end{matrix}

(30)

with $σ_{max}$ , $σ_{min}$ , and $σ$ are manually fixed parameters.

The disturbed PSO with a linear update strategy is based on the following mechanism

σ_{k} = σ_{max} - \frac{k - 1}{k_{max}} (σ_{max} - σ_{min})

(31)

The random model consents to update randomly which can be written in the form

σ_{k} = σ_{min} + U (0, 1) (σ_{max} - σ_{min})

(32)

where $U (0, 1)$ is a random value uniformly distributed in the range $[0, 1]$ .

Comparison between the proposed PSO algorithms

In this section, a comparative study will be made between the algorithms described in the last part in order to choose the algorithm that will be used later. The PSO control parameters are chosen to be identical for different algorithms. The size of the population is $n = 5$ ; the number of generations is $K_{max} = 15$ ; the cognitive and social coefficients are $c_{1} = 2, c_{2} = 2$ ; and the possible models are given as $0.001 < σ < 0.15$ and $0.4 < ω < 0.9$ .

Figure 3 shows the comparison of the convergence dynamics of the objective function with the proposed PSO algorithms. We note the superiority, in terms of convergence of speed and quality of the solutions obtained, of the disturbed variants of the PSO tool to solve the problem to be optimized. The exploration and exploitation capabilities of these variants of the PSO algorithm are clearly improved thanks to this practical optimization problem.

Figure 3.

Convergence properties of the proposed PSO algorithms.

We note that all the reached best cost functions of both standard PSO and its improved variants produced with near results by a total convergence to the same interesting region. This indicates the success of these algorithms to explore and find the global optimum in the search space.

Through all proposed metaheuristic methods, the PSO version with a linear decrease in the inertia weight shows the best performance and superiority while comparing with other proposed algorithms (Table 1).

Table 1.

Optimization results from 15 runs of problem.

Algorithm	K_p	K_i	ISE	ITAE	Cost function
Canonical PSO	104.3883	51,218	1.2163	0.0459	$2.6275 e^{- 4}$
wPSO	617.6684	2584.1	$6.9288 e^{- 4}$	$6.1709 e^{- 4}$	$4.5355 e^{- 4}$
pPSO min–max	72.9949	214.9241	0.0886	0.0052	$1.3198 e^{- 4}$
pPSO random	969.0231	9300	$6.9674 e^{- 5}$	$1.9629 e^{- 4}$	$1.3198 e^{- 4}$
pPSO linear	714.2866	6170	$1.9697 e^{- 4}$	$2.0890 e^{- 4}$	$1.2834 e^{- 4}$

ISE: integral squared error; ITAE: integral time absolute error; PSO: particle swarm optimization; pPSO: perturbed particle swarm optimization; wPSO: weight PSO.

Simulation results

The wind model is obtained by a Fourier series representation of the wind whose signal consists of a superposition of several harmonics which is written in the function of and which are, respectively, the average value of the wind speed, the harmonic amplitude (Porate et al., 2017)

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

(33)

In this article, the equation describing the wind variation is defined as follows

V (t) = 10 + 0.2 \sin (0.1047 t) + 2 \sin (0.2665 t) + \sin (1.293 t) + 0.2 \sin (3.6645 t)

(34)

The simulation results were carried out with MATLAB/Simulink in order to verify the effectiveness of the considered system using PSO to find the optimal parameter values of the PI controller. The results of the simulation of reference speed and real speed, without PSO and with PSO, are respectively shown in figures 4 and 5. It is very remarkable that the curves obtained with the use of PSO show the efficiency of the speed control strategy proposed since the measured speed is identical to the reference speed, on the other hand, the existence of the error with the manual regulation.

Figure 4.

Wind turbine rotational speed without PSO and the reference speed.

Figure 5.

Wind turbine rotational speed with PSO and the reference speed.

The error between the reference speed value and those of regulated speed without PSO and with PSO is illustrated, respectively, in Figures 6 and 7. This difference is varied between $\pm 5$ for manual regulation; however, it does not exceed $\pm 0.05$ in regulation with PSO which guarantees the right choice and presents the effectiveness of this method.

Figure 6.

Error without PSO.

Figure 7.

Error with PSO.

Figure 8 presents the appearance of the coefficient of power $C_{p}$ which directly reaches its optimal value (0.48) with the PSO regulation and remains fixed. By counting, it arrives at this value with delay by means of manual regulation (without PSO) with the presence of variability.

Figure 8.

Coefficient of power.

The tip speed ratio $λ$ is presented in Figure 9 in which it is clear that the curve obtained by PSO takes the optimal value (8.1) while that given without PSO varies around this value.

Figure 9.

Tip speed ratio.

After having obtained the results obtained with the regulation of the manual PI controller parameters and that with the PSO algorithm, we can see clearly that the PSO gives good curves better than the manual regulation thus with a minimum of the error as it is indicated in Table 2 (see figures 4 to 15; Tables 3 and 4).

Table 2.

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

Controller	PI tuned manually	PI tuned by PSO
Generator response	$K_{i} = 3.2939$	$K_{i} = 6.170 e^{3}$
	$K_{p} = 0.3484$	$K_{p} = 714.288$
ISE	$456.6246$	$1.9697 e^{- 4}$
ITAE	$1.0456$	$2.0890 e^{- 4}$

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

Figure 10.

Mechanical torque.

Figure 11.

Electromagnetic torque.

Figure 12.

Quadrature current.

Figure 13.

Direct current.

Figure 14.

Quadrature voltage.

Figure 15.

Direct voltage.

Table 3.

Parameters of the wind turbine and those of the machine.

Parameter	Values
$ρ$	$1.225 kg \cdot m^{- 3}$
R	$2 m$
$C p_{opt}$	$0.48$
$λ_{opt}$	$8.1$
$P_{n}$	$3.5 kW$
$R_{a}$	$0.82 Ω$
$L_{s}$	$15.1 e^{- 3} H$
J	$99 e^{- 4} kg \cdot m^{2}$
f	$10^{- 3} N \cdot m \cdot s \cdot ra d^{- 1}$
$φ$	$0.5 Wb$
p	4

Table 4.

Parameters of the PSO with a linear algorithm.

Parameter	Values
Swarm size	5
The maximum number of iteration	15
$c_{2} = c_{1}$	2
$σ_{max}$	0.15
$σ_{min}$	0.001
$ω$	0.9

PSO: particle swarm optimization.

Conclusion

In this study, the complete model of the variable speed wind turbine with permanent magnets synchronous generator in the company of its command has been presented. The purpose of the control is to extract maximum wind power using FOC and an optimal speed reference that is estimated from the wind speed.

The PSO algorithm and these variants have been presented, which are then compared with each other at their convergences to choose the best optimal gains of the PI controllers. The performance index for various error criteria for the proposed controller using the PSO algorithm is proven to be lower than the manually set controller.

The effectiveness of the proposed methodology is verified. It is well illustrated at the time of stabilization of the intermediate circuit voltage response and the response of the generator speed which was reduced with the optimal parameters obtained using PSO. In order to further improve controller parameters, biogeography-based optimization (BBO) technology could be used.

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

Marwa Hannachi

Omessaad Elbeji

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