Multi-objective optimization strategy for parallel tuning of multiple fractional order controllers in a PMSG based WECS

Abstract

In this paper, a novel multi-objective optimization strategy is proposed for the parallel tuning of six fractional order controllers used in regulation loops of a PMSG based wind energy conversion system connected to the electric grid. The nonlinear nature of the WECS components has made controllers design challenging. To enhance the transient response of system variables, Fractional Order Proportional Integral (FOPI) controllers are considered as they are more suitable for such nonlinear physical systems. The additional parameters introduced by FOPI are normally tuned using a single objective, multi-dimensional particle swarm optimization algorithm. However, the inter-dependence of regulation loops introduces additional complexity that is solved in this work using a multi-objective optimization strategy based on a succession of Single-Objective PSO and Multi-Objective PSO. A simulation study has been conducted in order to demonstrate the higher performance and superior tracking accuracy of the proposed multi-objective optimization strategy in variable wind speed operating conditions.

Keywords

Multi-objective optimization particle swarm optimization (PSO)fractional order proportional integral controllers (FOPI)permanent magnet synchronous generator (PMSG)wind energy conversion systems (WECS)three-level NPC

Introduction

Several control techniques have been proposed in the literature to improve the performance of the different components of Wind Energy Conversion Systems (WECS) (Chatri et al., 2022), (Mondal and Dey, 2022) in order to enhance their efficiency in the presence of various disturbances and operating conditions such as varying atmospheric conditions which cause fluctuations in the wind speed. It has been established that Proportional Integral (PI) controllers are widely used in industrial applications in addition to their simple implementation and good transient and steady state response with good performances in terms of overshoot and settling-time (Sung and Lee, 1996). However, these controllers may not be very effective for complex nonlinear systems that may also include parameter variations (Noureddine et al., 2021).

On the other hand, the doubly-fed induction generator (DFIG) has been used for a long period of time in wind energy generation systems, mainly because of its variable speed operation and overall good performance. However, additional investigations have led to the implementation of Permanent Magnet Synchronous Generator (PMSG) in wind turbines combined with equivalent rated power converters. The additional losses in power converters when compared with DFIG based WECS that uses only 30% of the nominal power are covered by the fact that PMSG based WECS are not equipped with a gearbox which extends its operating speed range. This results in higher power efficiency at variable wind speeds, and better fulfillment of grid requirements. However, additional problems associated with their complex control loops, higher installation cost, and interconnection with grid are major concerns yet to be solved, as this kind of wind energy system requires fully rated power conversion stages with associated control of different parameters (active/reactive power, voltage, current, frequency, harmonics, … etc).

Various studies have been conducted to deal with complex control loops of PMSG-based WECS in order to develop control and regulation systems based on modern and available technology (Hachana et al., 2022). Recently, there are many topologies of the electronic power converters that have been proposed to control wind energy conversion systems based on the PMSG. The most important of them, the three-level NPC, which has significant advantages compared to the two-level converter. The three-level NPC improves the energy extracted from the turbines as well as control units that allow controlling the electrical power supplied to the grid (Vancini et al., 2022). The main advantages of the three-level NPC are higher power output, lower output voltage, lower DC-link voltage, and lower switching stress (Blaabjerg et al., 2012). Generally, power electronics are combined with aerodynamic controls to regulate physical quantities such as power, current, speed, and torque. In Maaruf et al. (2021), a fast terminal sliding mode control (FTSMC) was proposed as a hybrid approach combining the particle swarm optimization (PSO) in WECS. In Barros and Barros (2017), an internal model state feedback control (IMSFC) based current control strategy is proposed in order to prove the dq current steady-state error elimination achieved through a linear quadratic regulator (LQR) compared to a conventional PI. In Pan and Wang (2020), a hybrid control technique which combines a PI controller for the current and speed loops regulation and a T-S fuzzy PID control scheme, is proposed to stabilize the output power. An enhanced design of classical PID controllers has been investigated and Fractional Order PID (FOPID) has been proposed by Podlybny (Podlubny, 1994) where the order of integral and derivative terms of the classical PID have been made of fractional values so that this will allow additional degrees of freedom to the controller (Yang et al., 2019). Fractional operators can be derived from various definitions, but the most used are those of Riemann-Liouville, Crone, Carlson, high frequency, continued fraction and low frequency continued fraction. Due to their design complexity, Fergani (2022) carried out a synthesis of the controller parameters using optimization techniques that are used in order to find the best controller parameters. Such algorithms include: Genetic Algorithm (GA) (Ozturk et al., 2015), Particle Swarm Optimization (PSO) (Zamani et al., 2009), Gray Wolf (Komathi and Umamaheswari, 2020), Chaotic Algorithm (Hekimoglu, 2019) and Bat Optimization Algorithm (Gandomi and Yang, 2014).

In this work, a multi-objective optimization strategy is adopted for the optimal tuning of fractional order controllers used in regulation loops of the WECS. A single-objective PSO is firstly used in order to tune individual regulation loops, where inner current loops are tuned first. The resulting optimal controllers’ parameters are implemented in order to construct narrow search spaces for the multi-objective PSO optimization algorithm. The objective functions of the different controllers are normalized by the values of the optimal fitness obtained from the single-objective optimization, and then added in order to construct a single objective function for the multi-objective optimization. The inter-dependence of the WECS regulation loops is then solved using the multi-objective optimization strategy based on the execution of the single-objective PSO that reduces the search space for the multi-objective PSO algorithm and then using a normalized single objective function for the multi-objective optimization.

This paper is organized as follows: in the next section, the WECS description is presented, where the model of the system is developed. In section 3, the fractional order control approach is presented. Section 4 gives the details of the multi-objective optimization strategy. Simulation results and discussion are given in Section 5. Conclusions are given at the end.

Description and modeling of the wind energy conversion system

Description of the WECS

In wind turbines based on PMSG systems, power converters are generally classified as direct and indirect: In the direct conversion, there is an AC/AC power supply with one converter stage. In the indirect conversion, on the other hand, there are two stages (AC/DC and DC/AC) or three-stages (AC/DC, DC/DC and DC/AC) (Yaramasu et al., 2017).

The structure of wind turbine-PMSG proposed in this work is mainly composed of a wind turbine which converts kinetic energy of the wind to mechanical energy to rotate a shaft coupled directly to the PMSG, whose output produces alternating current to the electrical grid through two stages of 3L-NPC back-to-back power converters as shown in Figure 1.

Wind turbine model

For variable speed wind turbines, the output torque and mechanical power are given as follows (Putri et al., 2016):

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

(1)

T_{m} = \frac{P_{m}}{ω_{m}} = \frac{1}{2} \frac{ρ A C_{p} (λ, β) {v_{w}}^{2}}{ω_{m}}

(2)

Where A is the area swept by the turbine blades, ρ the density of air, $ω_{m}$ the mechanical angular speed of the wind turbine and $v_{w}$ the wind velocity. The turbine power coefficient $C_{p} (λ, β)$ describes the performance coefficient of the turbine which depends on the rotor blade pitch angle $β$ and the Tip Speed Ratio $λ$ (TSR).

$λ$ is given by the following equation:

λ = \frac{R_{t} ω_{m}}{v_{w}}

(3)

where R_t is the wind turbine blade radius.

The coefficient $C_{p}$ is a nonlinear function described by a generic function (Lertnuwat and Oonsivilai, 2017):

C_{p} (λ, β) = 0.5176 [\frac{116}{λ_{i}} - 0.4 β - 5] e (\frac{- 21}{λ_{i}}) + 0.0068 λ

(4)

where

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

(5)

Figure 2 shows the power coefficient C_p for a typical WT. For each ß, there is only one λ corresponding to the maximum power coefficient C_pmax, which is λ_opt optimal. The maximum value of C_pmax = 0.48 is reached for β = 0 and λ_opt = 8.1 which results of maximum wind power being captured by the wind turbine.

PMSG modeling

The PMSG stator voltage equations in d-q reference frame may be expressed in terms of instantaneous currents and stator flux linkages as:

{\begin{matrix} Vsd = - R_{s} i_{s d} - L_{d} \frac{d}{d t} i_{s d} + ω_{e} L_{q} i_{s q} \\ Vsq = - R_{s} i_{s q} - L_{q} \frac{d}{d t} i_{s q} - L_{d} ω_{e} i_{i_{s q}} + ω_{e} Φ_{m} \end{matrix}

(6)

where $i_{s d}$ , $i_{s q}$ and V_sd, V_sq, represent the stator currents and voltages in the (d, q) axis. R_s, L_d and L_q are the resistance and the inductances of the stator windings in (d, q) axis, respectively. Φ_m is permanent flux linkage and ω_e is the electrical speed which is given by:

ω_{e} = p ω_{m}

(7)

where p is the number of pairs of poles.

The electromagnetic torque developed by the machine is given by:

T_{e} = \frac{3}{2} p (i_{s q} . Φ_{m} + (L_{q} - L_{d}) i_{s d} i_{s q})

(8)

The design of the rotor in our case is considered almost smooth and thus offers equal reluctance for the two d-q axis inductances (L_d = L_q). The equation of electromagnetic torque may be modified as:

T_{e} = \frac{3}{2} p i_{s q} . Φ_{m}

(9)

The mechanical equation of the PMSG is given by:

J \frac{d ω_{m}}{d t} = T_{m} - T_{e} - f ω_{m}

(10)

where J is the total moment of inertia, f the friction coefficient, T_m and T_e the mechanical torque produced by a WT and the electromagnetic torque of PMSG respectively.

Topology and modeling of the back-to-back power converter

Figure 3 illustrates the equivalent circuit model of the back-to-back power converter where the transistors are considered as ideal switches. The fundamental topology of the back-to-back converter is composed of two neutral-point three-level power converters (3L-NPC). The machine-side converter (MSC) works as a rectifier, while the grid-side one (GSC) works as a rectifier. As an inverter, the two converters are separated by two capacitors (C₁ and C₂) forming a DC-link. Each converter is made up of three identical arms, each of which contains four switches and two diodes. It also includes a DC bus having two capacitors C₁ and C₂ in series whose voltages are V_dc1 and V_dc2, respectively. i_af, i_bf and i_cf are the three-phase currents of the filter. As the arms of these converters are identical, the study of a single arm allows the explanation of the operation of the structure. In order to avoid short circuits of voltages V_dc1 and V_dc2, the switches of the same arm are complementary: if one is closed, the other must be open and vice versa.

In order to obtain the mathematical model, the switching states of phase A, Sa is broken down into:

{\begin{matrix} S_{a} = 2 \Rightarrow S_{1 a} = 1, S_{2 a} = 0, S_{3 a} = 0 \\ S_{a} = 1 \Rightarrow S_{1 a} = 0, S_{2 a} = 0, S_{3 a} = 1 \\ S_{a} = 0 \Rightarrow S_{1 a} = 0, S_{2 a} = 1, S_{3 a} = 0 \end{matrix}

(11)

Where S_1a, S_2a, S_3a represent the connection functions which describe the state of the first arm (1a represents the upper half of the arm, 2a the lower half of the arm and 3a the neutral point of the arm). Using the same method, the switching states of phases B and C, Sb and Sc are also broken down. The three-phase voltages (V_A, V_B and V_C) of the three-level NPC converter are assumed balanced, using the DC voltage (V_dc1 and V_dc2), the relations between the voltages (V_A, V_B, and V_C) with connection functions (S_1a, S_2a, S_3a) are defined as follows:

[\begin{matrix} V_{A} \\ V_{B} \\ V_{C} \end{matrix}] = \frac{1}{3} [\begin{matrix} 2 & - 1 & - 1 \\ - 1 & 2 & - 1 \\ - 1 & - 1 & 2 \end{matrix}] ([\begin{matrix} S_{1 a} \\ S_{1 b} \\ S_{1 c} \end{matrix}] V_{d c 1} - [\begin{matrix} S_{2 a} \\ S_{2 b} \\ S_{2 c} \end{matrix}] V_{d c 2})

(12)

Assuming that C₁ = C₂ = C, the DC bus voltage is therefore given by:

C \frac{d V_{d c}}{d t} = C \frac{d V_{d c 1}}{d t} + C \frac{d V_{d c 2}}{d t} = i_{c 1} + i_{c 2}

(13)

The equations of the currents ic₁ and ic₂ can be deduced as:

i_{c 1} = i_{inv 1} - i_{inv 2}

(14)

i_{c 2} = i_{c 1} + i_{M 1} - i_{M 2} = i_{inv 1} - i_{inv 2} + i_{M 1} - i_{M 2}

(15)

The simplified model of the DC bus voltage is as follows:

C \frac{d V_{d c}}{d t} = 2 (i_{inv 1} - i_{inv 2}) + i_{M 1} - i_{M 2}

(16)

The neutral point current is shown by:

i_{M} = C \frac{d (V_{d c 1} - V_{d c 2})}{d t} = i_{M 2} - i_{M 1}

(17)

In this work, the Pulse Width Modulation (PWM) technique is used for the control of the three-level NPC back-to-back converter.

Modeling of the L_fR_ffilter

An (L_f-R_f) filter is used for the connection between the wind system and the electrical network, the purpose of which is to minimize the harmonic frequencies resulting from the switching operation of the three-level NPC converter.

The application of the mesh law will allow the calculation of the voltages appearing at the terminals of the (L_f-R_f) filter which are function of the currents between the converter and the network (i_af, i_bf, i_cf) and also the electrical network voltages (V_as, V_bs, V_cs) according to:

{\begin{matrix} V_{a f} = - R_{f} i_{a f} - L_{f} \frac{d i_{a f}}{d t} + V_{a s} \\ V_{b f} = - R_{f} i_{b f} - L_{f} \frac{d i_{b f}}{d t} + V_{b s} \\ V_{c f} = - R_{f} i_{c f} - L_{f} \frac{d i_{c f}}{d t} + V_{c s} \end{matrix}

(18)

The filter voltages can be represented in Park’s rotating reference as follows:

\begin{matrix} V_{d f} = - R_{f} i_{d f} - L_{f} \frac{d i_{d f}}{d t} + ω_{s} L_{f} i_{qf} + V_{ds} \\ V_{qf} = - R_{f} i_{qf} - L_{f} \frac{d i_{qf}}{d t} - ω_{s} L_{f} i_{d f} + V_{q s} \end{matrix}

(19)

where V_df, and V_qf denote the voltages emerging at the terminals of the (L_f-R_f) filter along the two axes: direct and quadrature respectively., i_df, i_qf are the currents between the converter and the network along the direct and quadrature axes respectively.

Fractional order PID controllers

Fractional Order (FO) controllers are based on the theory of fractional calculus, which is considered as a generalization of ordinary differentiation and integration from integer to arbitrary real values (Tapadar et al., 2022). FOPID controllers is more robust and offers more flexibility compared to the conventional PID controller. Figure 4 shows a graphical representation of the FOPID controller.

The fractional order differentiator may be denoted by a general operator $a^{D_{t}^{μ}}$ (Sutha et al., 2015), given by:

a^{D_{t}^{μ}} = {\begin{matrix} \frac{d^{μ}}{d t^{μ}} ℜ (μ) > 0 \\ 1 ℜ (μ) = 0 \\ \int_{a}^{t} {(d τ)}^{- μ} ℜ (μ) < 0 \end{matrix}

(20)

Different descriptions define mathematically Fractional and integral derivatives. Among them we note that of Riemann-Liouville whose integral of fractional order is given by:

a^{D_{t}^{- μ}} f (t) = \frac{1}{Γ (μ)} \int_{a}^{t} {(t - τ)}^{μ - 1} f (τ) d τ

(21)

The fractional order derivative definition is given by:

a^{D_{t}^{μ}} f (t) = \frac{1}{Γ (n - μ)} \frac{d^{n}}{d t^{n}} [\int_{a}^{t} \frac{f (τ)}{{(t - τ)}^{μ - n + 1}} d τ]

(22)

or :

Γ (x) = \int_{0}^{\infty} y^{x - 1} e^{- y} dy

(23)

$Γ (x)$ represents the Gamma function, a and t are the limits of the operation, and μ is the number which identifies the fractional order.

In order to implement the fractional controller in the industrial process and digital simulations, several analog approximations are used. The fractional order controller called FOPID proposed by Podlubny (Kumar et al., 2017), has the general form given in (24). This controller has two additional parameters compared to the classic PID, so five parameters in all: the proportional gain (K_P), integral gain (K_I), derivative gain (K_D), integral order (λ) and the derivative order (µ) (Podlubny, 1994).

The most generalized form of the fractional order controller is given by (Mondal and Dey, 2022):

y (t) = K_{P} e (t) + K_{I} {D_{t}}^{- μ} e (t) + K_{D} {D_{t}}^{μ} e (t)

(24)

where e (t) is the error and u (t) represents the response action of the controller based on the error. By applying the Laplace transform to equation (24), the transfer function of the FOPID controller is described as:

U (S) = (K_{P} + K_{I} S^{- λ} + K_{D} S^{μ}) E (S)

(25)

Multi-objective optimization of fractional order controllers for WECS

The WECS is considered as a complex system that is composed of interconnected components which interact with each other and need to be globally optimized in order to achieve an overall good performance. Such task is challenging because of the multi-dimensional search space of the fractional order controllers parameters, in addition to the inter-dependent nature of its variables which affect each other. In order to remedy to such issue, a single objective optimization is considered first in order to limit the search space for the multi-objective optimization algorithm.

Single-objective optimization

Multi-dimensional Particle Swarm Optimization

In a D-dimensional search space of the Particle Swarm Optimization (PSO) algorithm, the particle i of the swarm is modeled by its position vector ${\vec{x}}_{i j} = {(x_{i 1} x_{i 2} \dots x_{iD})}^{T}$ and by its velocity vector ${\vec{v}}_{i j} = {(v_{i 1} v_{i 2} \dots v_{iD})}^{T}$ . The evaluation of the particles position is determined by the value of the objective function at this point. The particle stores the best position through which it has already passed which is noted ${\vec{p}}_{i j} = (p_{i 1} p_{i 2} \dots . p_{iD})^{T}$ . The best position reached by its neighboring particles is defined by ${\vec{g}}_{i j} = (g_{i 1} g_{i 2} \dots . g_{iD})^{T}$ .

The velocity vector is calculated using the following update equation:

υ_{i j} (k) = w . υ_{i j} (k - 1) + c_{1} . r_{1} (p_{i j} (k - 1) - x_{i j} (k - 1)) + c_{2} . r_{2} (g_{j} (k - 1) - x_{i j} (k - 1))

The position at iteration k of the ith particle is then defined by the equation:

x_{i j} (k) = x_{i j} (k - 1) + υ_{i j} (k), w h e r e i = 1, 2, 3 \dots N_{p}, j = 1, 2, 3 \dots N_{d}, k = 1, 2, 3 \dots k_{\max}

where N_P is the number of particles in the swarm, N_d is the number of problem variables (i.e. dimensions of a particle), k_max is the maximum number of iterations. υ_ij (k) is the velocity of the jth component of the ith particle of the swarm, at the kth iteration. p_ij is the jth component of the best position occupied by the ith particle of the swarm recorded in previous iterations (local best). g_ij is the jth component of the best position occupied by the ith global particle of the swarm (global best). $x_{i j} (k)$ is the jth coordinate of the current position of the i particle at the kth iteration. r₁ and r₂ are random numbers in [0,1], w is a constant called inertia coefficient (learning factor), c₁ and c₂ are two acceleration coefficients.

The objective function of each component in the position update of the particle in the PSO algorithm is detailed as follows:

$w . υ_{i j} (k - 1)$ corresponds to the rate of displacement. The parameter w controls the weight of the previous rate of displacement on the future move. The inertia coefficient (or learning factor) w plays an important role in the search procedure. It guarantees a balance between local search and global search, a good choice of this function increases the efficiency of the method to have a global solution. Experience has shown that linearly decreasing the value of w from 0.9 to 0.1 during the search procedure gives best results.

$c_{1} . r_{1} (p_{ijbest} (k - 1) - x_{i j} (k - 1))$ corresponds to the cognitive component of the displacement where c₁ controls the particle’s cognitive behavior.

$c_{2} . r_{2} (g_{jbest} (k - 1) - x_{i j} (k - 1))$ corresponds to the social component of movement, where c₂ controls the particle’s social ability.

The appropriate ranges of values for c₁ and c₂, are in [1, 2], but 2 is the most appropriate in most cases.

In some cases, the particle tends to leave the search space during its movement, the algorithm then executes a confinement mechanism in order to manage the displacement of the particle and brings it back to a new point in the boundaries of the search space.

{\begin{matrix} i f x_{i j} > x_{j m a x} t h e n x_{i j} = x_{j m a x} \\ i f x_{i j} < x_{j m i n} t h e n x_{i j} = x_{j m i n} \\ v_{i j} = 0 \end{matrix}

(26)

where $x_{j m i n} and x_{j m a x}$ : are the limit values of the parameter $x_{i j} .$

Single-objective optimization of fractional order controllers in WECS

Fractional order proportional integral (FOPI) controllers are used in WECS in order to track the reference of the following variables: i_sq, i_sd, ω_m, i_dg, i_qg and v_dc. As shown in Figure 5, the controllers’ parameters (K_p, K_i, λ) are tuned using the single-objective multi-dimensional particle swarm optimization algorithm, where performance indices are used as objective functions. The performance of the integral indices are defined using the system’s error e(t) may be considered as a quantitative measure of the feedback control system performance. A system is said to be an optimal control system if its parameters are varied until the performance index reaches a minimum value. The following performance indices are used in this work: integral of the absolute error (IAE), the integral of the square of the error (ISE), the integral of the absolute error multiplied by time (ITAE), the integral squared error multiplied by time (ITSE).

IAE = \int_{0}^{τ} | e (t) | dt, ISE = \int_{0}^{τ} e (t)^{2} dt, ITAE = \int_{0}^{τ} t | e (t) | d t, ITSE = \int_{0}^{τ} te (t)^{2} d t

(26)

The flowchart shown in Figure 6 gives execution sequence of the SOPSO optimization algorithm, where the first step considers the initialization of the controllers parameters with values having acceptable tracking performance. The next step takes into consideration the optimization of the first controller parameters, which is in this case the stator current i_sq. The performance indices (ISE, IAE, ITSE, ITAE) are considered once at a time, which results in a set of controller parameters for each performance index. The remaining five controllers for the variables i_sd, ω_m, i_dg, i_qg, v_dc are tuned using the same procedure.

The order of controllers optimization execution has been chosen so that inner reference tracking loops are optimized first and then the outer loop is considered. Such choice will allow better performance optimization as generally, inner regulation loops are faster than the outer loops.

The optimization results of the SOPSO algorithm are used in the following section in order to further improve the reference tracking of FOPI controllers.

Multi-objective optimization of FOPI controllers in WECS

The single-objective optimization is used mainly to tune independently the FOPI controller of each WECS variable in order to limit the search space for the multi-objective optimization algorithm. The reduced search space is used in addition to a multi-objective evaluation function in order to globally optimize the WECS performance for a variable wind speed profile.

The flowchart shown in Figure 7 shows the different steps of the proposed multi-objective optimization procedure of the fractional order controllers’ parameters. A random initialization of the parameters is carried out first, then the single objective PSO is used to tune the parameters until an acceptable error evaluation is obtained. The resulting optimal controllers’ parameters are used to construct a narrow search space for the multi-objective optimization algorithm. The multi-dimensional multi-objective optimization is then executed in order to minimize the following objective function:

J = \frac{J_{s d}}{J_{s d_o p t}} + \frac{J_{s q}}{J_{s q_o p t}} + \frac{J_{ω}}{J_{ω_o p t}} + \frac{J_{q g}}{J_{q g_o p t}} + \frac{J_{d g}}{J_{d g_o p t}} + \frac{J_{d c}}{J_{d c_o p t}}

(27)

Where $J_{s d}, J_{s q}, J_{ω}, J_{q g}, J_{d g}, J_{d c}$ are evaluation indices for i_sd, i_sq, ω_m, i_qg, i_dg, v_dc respectively. $J_{s d_o p t}, J_{s q_o p t}, J_{ω_o p t}, J_{q g_o p t}, J_{d g_o p t}, J_{d c_o p t}$ are optimal evaluation indices obtained from the SOPSO optimization procedure, which are used in the MOPSO in order to normalize the global evaluation index. The aforementioned evaluation indices are calculated for the different performance metrics, namely: IAE, ISE, ITAE, ITSE.

The Multi-Objective PSO (MOPSO) has been used as the main multidimensional and multi-objective optimization algorithm. The algorithm manages a swarm of 18 controllers’ parameters which gives 18 dimensions for the optimization problem. Furthermore, the optimization is carried out using six objective functions that are normalized (as in equation (27)) by optimal evaluations from the single-objective optimization, which allows to give equal weights for the different objective functions. Figure 8 shows how the MOPSO is used in order to modify the controllers’ parameters for different regulation loops in the WECS.

Results and discussion

Single-objective optimization for WECS

The WECS is fed with a variable wind speed profile in order to tune the FOPI controller for various operating points. The following parameters are used for the WECS: direct and quadrature stator inductances (L_d = L_q = 8.4 mH), nominal power (p = 6 kw), field flux (Φ_m = 0.4 wb), stator resistance (R_s = 0.4 Ω), number of pole pairs (p = 12), inertia (J = 0.089 kg·m²), friction (f = 0.0016 N/m). For the wind turbine: radius of the turbine (R_t = 2 m), volume density of the air (ρ = 1.225 kg/m³), pitch angle (β = 0), specific optimal speed (λ_opt = 8.1), coefficient of maximum power (C_pmax = 0.48).

Figure 9(a) shows the optimal response of ω_m for tracking a reference speed profile ω_m* using different evaluation metrics (IAE, ISE, ITAE, ITSE). Table 1 shows the overall evaluation metrics along the simulation time (3 seconds), where it is noted the different controllers’ parameters obtained for the different evaluation indices. The responses of stator currents (i_sd, i_sq), DC bus voltage (v_dc) and grid currents (i_qg, i_dg) are given in Figure 9(b) to (f) respectively, where the reference has been omitted from the plot as it has high frequency oscillations. The performance indices for these variables are given in Table 1, where the obtained controller parameters also vary with the chosen evaluation index.

Table 1.

Simulation results for single objective optimization.

		K _p	K _i	λ	Evaluation
ω_m	IAE	60.8074	1202.9557	0.5986	0.0762
	ISE	58.3898	800.0	1.9	0.2074
	ITAE	43.1631	1825.6393	0.5812	0.0736
	ITSE	46.2983	3000	0.6557	0.0715
i_sd	IAE	1.0294	150.4655	0.7729	10.1159
	ISE	4.1862	169.5824	0.7759	11.3752
	ITAE	3.4192	73.9137	0.4338	11.1951
	ITSE	2.1581	80.7694	0.8661	10.7170
i_sq	IAE	3.7343	515.0213	1.3227	8.9817
	ISE	2.0999	381.1036	1.2611	11.9610
	ITAE	2.9273	415.2141	1.216	9.9117
	ITSE	3.1122	846.8179	1.3478	9.0653
v_dc	IAE	3.8957	190.7481	0.8251	1.7215
	ISE	3.5094	68.9178	0.8586	2.2168
	ITAE	3.6914	139.2068	0.863	2.2451
	ITSE	4.2196	116.1885	0.6741	2.3000
i_dg	IAE	1.3018	125.9083	0.7765	15.3039
	ISE	5.182	463.9678	0.9860	15.9304
	ITAE	1.4135	523.3061	1.4484	15.0292
	ITSE	1.5165	514.3395	1.2937	17.1390
i_qg	IAE	3.5432	499.0768	0.8071	10.7645
	ISE	3.3918	787.1605	0.7442	10.6440
	ITAE	2.9688	42.6368	0.7022	9.4515
	ITSE	2.6318	1205.1934	1.5844	9.5677

Multi-objective optimization for WECS

The simulation results obtained from the execution of the MOPSO for the variable wind profile are given in Figure 10. A precise tracking of the reference ω_m* is achieved as shown in Figure 10(a). The different evaluation indices give close responses that have noticeable differences only in transients. The same is noted for the remaining responses in Figure 10(b) to (f).

It is to be noted that the response for a given evaluation index is simultaneously evaluated for the six controllers. Whereas in the case of the SOPSO results shown in Figure 9, the responses for a given evaluation index the evaluations are executed sequentially for each optimized variable. In addition, a large over/under shoot may be noticed at some responses due to the unconstrained control signal. Implementing control signal constraints will further add more complexity to the multi-objective optimization algorithm.

The final parameters after optimization are given in Table 2. The multi-objective algorithm combines the evaluation indices of every regulation loop into a normalized global evaluation index that is used as the main performance evaluation index for the multi-objective PSO algorithm. The individual performances of the MOPSO are characterized by better performance when compared with the SOPSO evaluation indices given in Table 1, which justifies the transition to the proposed MOPSO algorithm in order to enhance the reference tracking performance.

Table 2.

Simulation results for multi-objective optimization.

		IAE	ISE	ITAE	ITSE
ω_m	Evaluation	0.0703	0.1336	0.0718	0.0664
	K _p	55	63.37566	40.036	41.3337
	K _i	1228.1469	769.7112	1863.3	2991.966
	λ	0.55	0.85	0.5338	0.6657
i_sd	Evaluation	10.5047	11.0429	10.6091	10.4606
	K _p	1.2877	3.5	2.0119	2
	K _i	153.9339	170.4550	63.398	81.5846
	λ	0.7610	0.7482	0.7548	0.8281
i_sq	Evaluation	8.2907	10.8453	8.4195	8.0258
	K _p	3.39055	2.4593	3.4048	3.4657
	K _i	494.5334	377.375	487.86	864.1293
	λ	1.1542	1.2437	1.1328	1.4536
v_dc	Evaluation	1.6518	2.0765	2.5618	2.0213
	K _p	3.5408	3.0937	3.0774	3.5259
	K _i	192.3429	72.4287	140.91	109.321
	λ	0.9053	0.801	0.8343	0.5941
i_dg	Evaluation	14.3755	15.4992	15.2469	15.2971
	K _p	1.2531	5.0759	1.3057	1.135
	K _i	126.7581	437.1846	595.18	535.1854
	λ	0.7413	1.2799	1.4865	1.5763
i_qg	Evaluation	9.7474	10.2431	8.5796	8.6417
	K _p	3.1375	3.5079	2.4339	2.3696
	K _i	516.0607	793.012	44.703	1208.005
	λ	0.8305	0.7953	0.6738	1.4431
Global evaluation		5.6890	7.2228	5.6737	5.2709

The controllers’ parameters evolution along the MOPSO optimization iterations are given in Figures 11 to 16. The 100 particles for every controller parameters start from random initial value in the narrow search space obtained from the SOPSO and evolve through the MOPSO optimization position update until they reach the final value after the completion of the 50 iterations of the algorithm.

Conclusion

Wind Energy Conversion Systems (WECS) are characterized by the dependence of their overall conversion efficiency on the proper regulation of the different system variables that are mainly related to the variations in the wind speed profile.

In this paper, a multi-objective optimization strategy has been used as the main tuning technique for simultaneously calculating the appropriate parameters of the six fractional order controllers used in regulation loops of a permanent magnet synchronous generator (PMSG) based wind energy conversion system (WECS) connected to the electric grid. The Single Objective PSO optimization strategy allowed the limitation of the optimal controller search space which allowed controllers parameters fine tuning using the Multi-Objective PSO algorithm to rapidly find newer parameters with better performance.

Simulation results show the high performance, faster time response, and superior tracking accuracy in variable wind speed operating conditions of the WECS controllers using parameters tuned by the proposed multi-objective optimization strategy. Practical implementation of the proposed multi-objective optimization technique may be carried out offline on a model of the physical system then the obtained optimal parameters of the controllers can be implemented online.

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 iD

Nadir Boutasseta

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Multi-objective optimization strategy for parallel tuning of multiple fractional order controllers in a PMSG based WECS

Abstract

Keywords

Introduction

Description and modeling of the wind energy conversion system

Description of the WECS

Wind turbine model

PMSG modeling

Topology and modeling of the back-to-back power converter

Modeling of the LfRffilter

Fractional order PID controllers

Multi-objective optimization of fractional order controllers for WECS

Single-objective optimization

Multi-dimensional Particle Swarm Optimization

Single-objective optimization of fractional order controllers in WECS

Multi-objective optimization of FOPI controllers in WECS

Results and discussion

Single-objective optimization for WECS

Multi-objective optimization for WECS

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Modeling of the L_fR_ffilter