Enhanced Control of Doubly Fed Induction Generator Based Wind Turbine System Using Fractional-Order Fuzzy PD+I Regulator

The intricate setup employed in this article is shown in Figure 1. Here, the stator windings are directly interconnected with the three-phase power grid, while the rotor windings are linked to the prime mover via the AC-DC-AC converter [5]. This converter furnishes three-phase rotor excitation power, offering adjustability in phase, frequency, and amplitude, and ensures bidirectional flow of slip power.

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Figure 1. Configuration of DFIG-based WTS

2.1 DFIG model and control theory

The comprehensive mathematical representation of the DFIG within the rotating coordinate system of Park (d-q) encompasses four state equations. These equations encapsulate the mechanical dynamics, rotor and stator voltages, rotor and stator flux, as well as rotor and stator powers, denoted by Eqs. (1) to (4) respectively [47]:

$\left\{\begin{array}{c}J \frac{d}{d t} \Omega_r+f_r \Omega_r=T_e-T_r \\ T_e=\frac{3}{2} P \frac{M}{L_r}\left(\psi_{q s} I_{d r}-\psi_{d s} I_{q r}\right)\end{array}\right.$  (1)

$\left\{\begin{array}{l}V_{d r}=R_r I_{d r}+\frac{d}{d t} \psi_{d r}-\left(\omega_{o b s}-\omega_m\right) \psi_{q r} \\ V_{q r}=R_r I_{q r}+\frac{d}{d t} \psi_{q r}+\left(\omega_{o b s}-\omega_m\right) \psi_{d r} \\ V_{d s}=R_s I_{d s}+\frac{d}{d t} \psi_{d s}-\omega_{o b s} \psi_{q s} \\ V_{q s}=R_s I_{q s}+\frac{d}{d t} \psi_{q s}+\omega_{o b s} \psi_{d s}\end{array}\right.$         (2)

$\left\{\begin{array}{l}\psi_{d r}=L_r I_{d r}+M I_{d s} \\ \psi_{q r}=L_r I_{q r}+M I_{q s} \\ \psi_{d s}=L_s I_{d s}+M I_{d r} \\ \psi_{q s}=L_s I_{q s}+M I_{q r}\end{array}\right.$  (3)

$\left\{\begin{array}{l}P_s=\frac{3}{2}\left(V_{d s} I_{d s}+V_{q s} I_{q s}\right) \\ Q_r=\frac{3}{2}\left(V_{q s} I_{d s}-V_{d s} I_{q s}\right)\end{array}\right.$        (4)

Table 1. The symbols in the DFIG mathematical model

Table 1 provides a comprehensive list of symbols used in the mathematical model of DFIG, along with their corresponding descriptions.

This section elucidates how the previously derived equations can be effectively applied to autonomously regulate the reactive and active powers of the DFIG, without impacting the control of the RSC current. Two prominent vector control techniques employed for the RSC, as discussed in the literature [5, 47] are the Stator Flux Oriented (SFO) and Stator Voltage Oriented (SVO) methods. In the SFO approach, control actions occur within a rotating reference frame dq, with the d-axis aligned with the stator flux. Conversely, in SVO, the d-axis aligns with the stator voltage. This study primarily concentrates on SFO, owing to its widespread adoption in DFIG control strategies. It involves the conversion of three-phase voltages, currents, and fluxes into a rotating reference frame, followed by the implementation of cascaded control techniques to track stator active and reactive power. This alignment decision results in the direct rotor current being proportionate to the stator reactive power, while the quadrature rotor current is proportional to the active stator power, as elucidated in subsequent discussions. In the SFO reference frame theory, the stator flux $\Psi_s$ is associated with the d-axis of the synchronous frame, which gives us the advantage that:

$\psi_{d s}=\psi_s$   (5)

$\psi_{q s}=0$  (6)

Assuming that the stator flux stays constant because of the AC voltage from the grid provides a constant voltage to the stator [5], and by neglecting the stator resistance drop. We can simplify the stator voltage amplitude as:

$V_{d s}=0$     (7)

$V_{q s}=V_s=\omega_s \psi_s=$ constant         (8)

$\frac{d\left|\psi_s\right|}{d t}=0$    (9)

The following equations are obtained when applied to Eqs. (1)-(9):

$\left\{\begin{array}{c}V_{d r}=R_r I_{d r}+\sigma L_r \frac{d I_{d r}}{d t}-g \omega_s \sigma L_r I_{q r} \\ V_{q r}=R_r I_{q r}+\sigma L_r \frac{d I_{q r}}{d t}-\sigma g \omega_s I_{d r}+g \frac{M V_s}{L_s}\end{array}\right.$       (10)

where:

$\sigma=1-\frac{M}{L_r L_s}, g=\frac{\omega_s-\omega_r}{\omega_s}$

The stator power equations may be defined as:

$\left\{\begin{array}{c}P_s=-\frac{3}{2} p \frac{M V_s}{L_s} I_{q r}=K_p I_{q r} \\ Q_s=\frac{3}{2} \frac{V_s^2}{\omega_s L_s}-\frac{3}{2} \frac{M V_s}{L_s} I_{d r}=K_Q\left(I_{d r}-\frac{V_s}{\omega_s L_s}\right)\end{array}\right.$       (11)

where, g indicates the generator slip.

The following equation illustrates the correlation between the rotor current and generator torque.

$T_e=\frac{P_s}{\omega_s}=-\frac{M V_s}{\omega_s L_s} I_{q r}$     (12)

According to Eq. (11), the stator reactive and active powers can be regulated separately by controlling the quadrature and the direct rotor currents $I_{q r}$ and $I_{d r}$, respectively. A PI regulator can be used to control the rotor current for each rotor current component $(d-q)$. The reference values of the direct and quadrature rotor currents $I_{d r}{ }^*$ and $I_{q r}{ }^*$ can be determined through the outer loops. Additionally, cross terms of (10) can be added to the output of each current regulator as an aid.

Within this configuration, as illustrated in Figure 2, it results in a cascade structure of two equivalent closed-loop systems. The inner loop controls the rotor currents, while the outer loop exclusively manages the stator's active and reactive power.

The PI regulators are employed to control the rotor currents ($I_{d r}-I_{q r}$), as demonstrated in (13):

$\left\{\begin{array}{l}V_{q r}^*=k_{p_{-} I_{d r}} \cdot\left(I_{d r}^*-I_{d r}\right)+k_{i_{-} I_{d r}} \cdot \int\left(I_{d r}^*-I_{d r}\right) d t \\ V_{d r}^*=k_{p_{-} I_{q r}} \cdot\left(I_{q r}^*-I_{q r}\right)+k_{i_{-} I_{q r}} \cdot \int\left(I_{q r}^*-I_{q r}\right) d t\end{array}\right.$        (13)

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Figure 2. Direct vector control of the DFIG-based WTS

Figure 3 depicts the convergence of both axes towards identical characteristics, with their plant being simplified to a first-order transfer function. Additionally, the equivalent closed-loop systems of both current loops are portrayed as second-order systems with two poles and a zero, which can be manipulated using classical control theory by selecting suitable gains for the PI regulators. Various techniques for tuning PI controllers are extensively documented in the literature, such as the Ziegler-Nichols method, the pole placement method, and the Bode plot method [5].

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Figure 3. The equivalent closed-loop systems of current loops

Again, a simple PI regulator can control the stator active and reactive power as described by (14):

$\left\{\begin{array}{l}I_{q r}^*=k_{p_{-} P_s} \cdot\left(P_s^*-P_s\right)+k_{i_{-} P_s} \cdot \int\left(P_s^*-P_s\right) d t \\ I_{d r}^*=k_{p_{-} Q_s} \cdot\left(Q_s^*-Q_s\right)+k_{i_{-} P_s} \cdot \int\left(Q_s^*-Q_s\right) d t\end{array}\right.$      (14)

As it is presented in (14), the $d$-axis rotor current reference $I_{r d}{ }^*$ is obtained by the output of the stator active power regulator, and the $q$-axis rotor current reference $I_{r q}{ }^*$ is generated by the output of stator reactive power regulator.

There are two additional estimators with a similar structure in the proposed DVC method. The first is used to predict Ps value, while the second predicts the $Q_s$. The following are the predicted stator powers:

$\left\{\begin{array}{c}P_s=-\frac{3}{2} \frac{M}{\sigma L_r L_s} \cdot\left(\psi_{r \alpha} V_s\right) \\ Q_s=-\frac{3}{2}\left(\frac{V_s}{\sigma L_s} \cdot \psi_{r \beta}-\frac{V_s M}{\sigma L_r L_s} \cdot \psi_{r \alpha}\right)\end{array}\right.$      (15)

Concisely, Figure 4 depicts the overall control architecture of a DFIG-based WTS. The composite control system consists of the stator power regulators, the direct and quadrature rotor current regulators, and the MPPT controller.

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Figure 4. Description of the DFIG control system

The SSO is a completely new swarm optimization method introduced by Cuevas et al. [42]. He studied the nature of social-spider colonies and their cooperative behavior when developing this intelligent algorithm. Furthermore, the SSO scheme underwent experimental evaluation across three distinct classes of systems: higher-order plants, non-minimum systems, and dynamic fractional systems. It was juxtaposed against other akin evolutionary approaches including GA, HS, PSO and GSA) These experiments conclusively showcased that The proposed SSO method demonstrates superior performance in terms of both solution accuracy and convergence compared to other techniques. Nowadays, there are many publications dedicated to exploring modifications, improvements, or applications of SSO for solving various intricate optimization problems. [42-46]. At the heart of the SSO algorithm lies a sophisticated framework comprising two fundamental search agents: the social spiders and the communal web. These spiders are intricately subdivided into distinct genders, delineating a structured hierarchy within the algorithm. Within the SSO methodology, the placement of each solution within the vast search space mirrors the spatial arrangement of spiders within the communal web. Furthermore, each spider is imbued with a weighted significance, meticulously determined by the fitness value of the corresponding solution, as meticulously evaluated by the discerning social spider. The initialization phase of the SSO algorithm commences with the establishment of a population, denoted as S, comprising an ensemble of N spider locations representing diverse solutions. This diverse assembly encompasses both female (fi) and male (mi) spiders, thereby encapsulating the comprehensive breadth of the search space.

Given that female spiders typically constitute a larger proportion than males in authentic spider societies (approximately $70-90 \%$ ), we opted to randomly designate the number of female spiders, denoted as $N_f$, to comprise approximately $85 \%$ of the total population, $N$ [42]. As such, Eqs. (22) and (23) are used to determine $N_f$ and $N_m$:

$N_f=$ floor $[N \cdot(0.9-0.25 . \operatorname{rand}(0.1))]$     (22)

$N_m=N-N_f$         (23)

In this context, "rand" denotes a randomly generated value ranging from 0 to 1 , while "floor(.)" function converts a real number to the nearest integer value Consequently, $N$ elements compose the entire population $S=\{s 1, s 2 \ldots, s N\}$, which is then divided into the sub-groups male spiders $M=$ $\{m 1, m 2, \ldots, m N m\}$ and female spider $F=$ $\{f 1, f 2, \ldots, f N f\}$, such that: $S=\{s 1=f 1, s 2=$ $f 2, \ldots, s N f=f N f, s N f+1=m 1, s N f+2=$ $m 2, s N f+2=m 2, \ldots ., s N=m N m\}$.

The female spider location fi is randomly initialized between the lower initial value $p_j^{\text {low }}$ and the upper initial value $p_f^{\text {high }}$ by using the following expression:

$f_{i, j}^0=p_j^{\text {low }}+\left(p_i^{\text {high }}-p_j^{\text {low }}\right) \cdot \operatorname{rand}(0.1)$     (24)

While the male spider location mi is randomly initialized as follows:

$\begin{gathered}m_{k, j}^0=p_j^{\text {low }}+\operatorname{rand}(0.1) \cdot\left(p_j^{\text {high }}-p_j^{\text {low }}\right)  =1,2, \ldots, n, i=1,2, . ., N_f, k =1,2, \ldots, N_m\end{gathered}$         (25)

where, "0" signifies the initial population, while "$j$", "$i$", and "$k$" denote individual indexes. The function rand $(0,1)$ generates a random value within the range of 0 to 1 . " $f i$, jis " represents the $j^{t h}$ variable of the $i^{\text {th }}$ female spider location. In the proposed SSO method, the weight of each spider $\left(\omega_i\right)$ reflects the quality of the corresponding solution within the population (S). $\omega_i$ is determined by the following equation:

$\omega_i=\frac{J\left(s_i\right)-\text { worst }_s}{\text { best }_s-\text { worst }_s}$    (26)

In this context, " $J\left(s_i\right)$ " represents the objective function or fitness value obtained by evaluating the spider location " $s_i$ ".

Eq. (27) is used to get the worst and best values (worsts and bests) by considering the following constrained optimization problem into account:

best$_s=\min _{k \in\{1, \ldots, N\}} J\left(s_k\right)$ and$worst_s$=$\max _{k \in\{1, \ldots, N\}} J\left(s_k\right)$    (27)

Colony members interact through the communal web, exchanging information via small vibrations essential for organizing all spiders collectively within the population. These vibrations' transmission is influenced by both the weight and distance of the spider.

produced them. The vibrations perceived by the individual member i from member j are given by Eq. (28):

$V_i b_{i, j}=\omega_j \cdot e^{-d_{i, j}^2}$      (28)

where, the $d_{i j}$ is the Euclidean distance between the member $i$ and $j$, such that $d_{i, j}=\left\|s_i-s_j\right\|$.

There are three types of vibrations in the SSO approach: Vibrations $V_i b c_i$. It is possible to describe the information (vibration) exchanged between individual $i\left(s_i\right)$ and member $c\left(s_c\right)$, which is the nearest member to individual $i$ and having greatest weight, as follows:

$V_i b c_i=\omega_c \cdot e^{-d_{i, c}^2}$     (29)

Vibrations $V_i b b_i$. The information exchanged between the individual $i\left(s_i\right)$ and the best member $b\left(s_b\right)$ of the total population $S$ may be described as:

$V_i b b_i=\omega_b \cdot e^{-d_{i, b}^2}$      (30)

Vibrations $V_i b f_i$. The transmitted exchanged between the individual $i\left(s_i\right)$ and the nearest female individual $f\left(s_f\right)$ may be described as:

$V_i b f_i=\omega_f \cdot e^{-d_{i, f}^2}$       (31)

At each iteration k, the female members update their position as follows:

$\begin{gathered}f_i^{k+1}=\left\{\begin{array}{c}f_i^k+\rho \cdot V_i b c_i \cdot\left(s_c-f_i^k\right)+\tau . V_i b b_i \cdot\left(s_b-f_i^k\right) \\ +\delta \cdot(\text { rand }-0.5) \text { withprobabilityPF } \\ f_i^k-\rho \cdot V_i b c_i \cdot\left(s_c-f_i^k\right)-\tau . V_i b b_i \cdot\left(s_b-f_i^k\right) \\ +\delta \cdot(\text { rand }-0.5) \text { withprobability } 1-P F\end{array}\right.\end{gathered}$         (32)

where, $\rho, \tau$ and $\delta$ are random numbers in $[0,1]$, whereas $P F$ represents the probability threshold. Within the colony, male spiders are categorized into dominant (D) or nondominant (ND) members, and they are organized in descending order based on their weight values. The male member whose weight $w_{N f+m}$ falls in the middle is identified as the median male member. At each iteration $k$, the male members undergo positional adjustments according to the following protocol:

$m_i^{k+1}\left\{\begin{array}{c}m_i^k+\rho \cdot V_i b f_i \cdot\left(s_f-m_i^k\right)+\delta .(\text { rand }-0.5) ; \\ \text { if } \omega_{N_{f+i}}>\omega_{N_{f+m}} \\ m_i^k+\rho \cdot\left(\frac{\sum_{h=1}^{N_m} m_h^k \cdot \omega_{N_{f+h}}}{\sum_{h=1}^{N_m} \omega_{N_{f+h}}}-m_i^k\right) ; \\ \text { if } \omega_{N_{f+i}} \leq \omega_{N_{f+m}}\end{array}\right.$      (33)

where, the members symbolizes the nearest female member to the male member i.

Once all female and male members are update, the last operator represents the mating process where only dominant male members will collaborate with female members who are within a particular radius named mating radius computed by:

$r=\frac{\sum_{j=1}^n\left(P_j^{\text {high }}-P_j^{l o w}\right)}{2 \cdot n}$       (34)

where $n$ is the dimension of the problem, $P_j^{\text {high }}$ and $P_j^{\text {low }}$ are the upper and lower limits for a given dimension, respectively. It is obvious that the spider with the higher weight has the most influence on the new product. The SSO method determines the influence probability $P_{s i}$ of each member as the follows:

$P s_i=\frac{\omega_j}{\sum_{j \in T^k} \omega_j}$      (35)

where, $T^k$ stands for the group of individuals participating in the mating process and $j \in T^k$. To summarize the previous equations of the SSO approach, The flowchart illustrated in Figure 8 outlines the computational steps required to execute the SSO algorithm.

The five parameters of the FO Fuzzy PD+I regulator in the DFIG-based WTS are considered as the SSO population. The goal is to minimize the current deviations $\left(\Delta I_{d r}, \Delta I_{q r}\right)$ as well as reduce the power fluctuation $\left(\Delta P_s, \Delta Q_s\right.$) upon system uncertainties. This can be achieved by optimizing the parameters of the FO Fuzzy PD+I current regulator. The ITAE is considered in this study as the fitness function that is utilized to evaluate the performance of the FO Fuzzy PD+I current regulator, and can be expressed according to:

${ITEA} =\int_0^{T_{\operatorname{sim}}} t|e(t)| d t$    (36)

where: $e(t)=I^*-I_{\text {real }}$         

In Eq. (36), $e(t)$ represents a close loop error, or a difference between the desired rotor current and real rotor current. $T_{\text {sim }}$ symbolizes the simulation time. Hence, the formulation of the problem is as follows:

$J\left(\alpha, K_p, K_d, K_i, K_u\right)=I T E A$     (37)

Subject to the following constraints:

$\left\{\begin{array}{c}\alpha_{\min } \leq \alpha \leq \alpha_{\max } \\ K_{\text {min }} \leq K_p \leq K_{\text {max }} \\ K_{d \min } \leq K_d \leq K_{\text {max }} \\ K_{\text {imin }} \leq K_i \leq K_{\text {max }} \\ K_{u \min } \leq K_u \leq K_{u \max }\end{array}\right.$

where, max and $\min$ are the limit values of each parameter. In the considered design method, $K_{pmin }=K_{{dmin }}=K_{{imin }}=$ $K_{umin}=0$ and $K_{pmax }=K_{{dmax }}=K_{{imax }}=K_{{max }}=20$. The values of $\alpha_{\min }=0$ and $\alpha_{\max }=1$. The selection of these limits was informed by previous insights and familiarity with the model, driven by two primary considerations. Firstly, these limits ensure that the optimized parameters fall within practical and feasible ranges, avoiding solutions that are unrealistic or impractical. Secondly, they serve to narrow down the search space for optimization algorithms for facilitating quicker convergence and enhancing computational efficiency [24, 25].