Design and Experimental Validation of a Hybrid GWO-PSO Tuned PI Controller for Predictive Direct Power Control in SAPF

Modern power systems use SAPFs extensively to reduce harmonic distortions, offset reactive power, and improve general power quality. These filters operate by injecting compensating currents into the grid to offset nonlinear loads. The control algorithms used determine how successful SAPFs are in suppressing harmonics and effectively controlling reactive power [10]. Several control strategies have been presented to maximize SAPF performance.

Using mathematical transformations, conventional methods such as the synchronous reference frame (SRF) technique and the instantaneous reactive power theory (p–q theory) separate harmonic components and provide suitable compensation signals [10, 11]. More complex methods, such as artificial neural networks (ANNs) and fuzzy logic control (FLC) [12, 13], have been investigated to increase adaptation and dynamic responsiveness. Moreover, model predictive control (MPC) has been popular since its switching operation optimization and real-time forecasting powers [6].

Usually including multiple key components, the SAPF system architecture consists of the control algorithm, power supply unit, nonlinear load, and voltage source inverter (VSI). Usually, a setup provides sufficient current compensation utilizing a VSI with pulse-width modulation (PWM). Maintaining the required running power to store VSI energy, a DC-link capacitor, and precision sensing and signal processing circuits track important system variables like voltage and current in real time [12].

This section details the design of each component within the integrated SAPF system. Figure 1 depicts the overall configuration.

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Figure 1. Overall system diagram of the studied SAPF configuration

DC bus voltage management is critical to the correct operation of SAPF. The DC voltage control loop provides accurate reference for active power in PDPC and keeps the bus voltage steady at a preset reference value. This study compares the performance of a proposed PI-GWO-PSO controller to existing PI controllers. The investigation focuses on assessing the PI-GWO-PSO controller's efficacy and performance, specifically its impact on the stability and regulation of the DC bus voltage in SAPF.

4.1 Design of standard PI controller

Figure 2 shows the closed-loop representation of the SAPF's DC bus regulation. A reference value ($V_{D C}^*$) is used to compare the measured DC bus voltage ($V_{D C}$). The PI controller receives this error as an input [18]. The instantaneous active power $P^*$ is then produced by $V_{D C}$ multiplying the output of this controller ($I_{s \max }$) by ($V_{D C}$). By ensuring that the DC bus voltage stays within the specified reference value, this control mechanism helps the SAPF to operate steadily and effectively [19,20].

Eq. (1) provides the system's transfer function (TF), and Eq. (2) provides the TF of the PI controller:

$P(s)=\frac{1}{C_{D C} \cdot s}$                      （1）

$C_c(s)=K_{p c}+\frac{K_{i c}}{s}$                      （2）

Figure 3 presents the voltage regulation loop and is given by Eq. (3):

$G(s)=\frac{V_{D C}}{V_{D C}{ }^*}=\frac{\frac{K_{p c}}{C_{D C}}+\frac{K_{i c}}{C_{D C}}}{s^2 \frac{K_{p c}}{C_{D C}} s+\frac{K_{i c}}{C_{D C}}}$                           （3）

When simplifying Eq. (2), Eq. (3) has been gotten which is a second degree:

$G(s)=\frac{V_{D C}}{V_{D C}{ }^*}=\frac{2 \xi \omega_n s+\omega_n^2}{s^2+2 \xi \omega_n s+\omega_n^2}$                 （4）

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Figure 2. PI controller-based DC bus architecture

Figure 3 shows the conventional PI controller to maintain the DC bus of the SAPF at a reference value.

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Figure 3. Structure of the PI controller integrated into the PDPC algorithm

The gains $k_{p c}$ and $k_{i c}$ are given by the identification of Eqs. (5)-(6). They are given by the following expression:

$k_{p c}=2 . \xi . \omega_n . C_{D C}$                        （5）

$k_{i c}=C_{D C} . \omega^2$                         （6）

4.2 Proposed GWO-PSO

MOAs are intended to search for the best solutions within a specific range. The GWO-PSO hybrid method, a new development, combines the strengths of GWO and PSO to increase exploitation and exploration while searching for optimum solutions. The fundamental goal of this hybrid GWO-PSO approach is to avoid premature convergence and improve accuracy in discovering the global optimal solution, especially in complicated, multi-modal search spaces. The method seeks to balance local refinement and global research using the complementing properties of GWO and PSO, resulting in improved performance in a wide range of optimization tasks.

4.2.1 Grey wolf optimization

Grey wolf optimization algorithm draws inspiration from the hunting strategies exhibited by grey wolves [21]. Grey wolves hold a significant position as apex predators, occupying the uppermost tier of the food chain. Their collective hunting prowess is particularly noteworthy; they are renowned for their adeptness at cooperative group hunting. These groups generally consist of 5 to 12 members on average and can be categorized into four types: alpha $\left(\alpha_{G W O}\right)$, beta $\left(\beta_{G W O}\right)$, delta $\left(\delta_{G W O}\right)$, and omega $\left(\omega_{G W O}\right)$. A distinctive aspect of their behavior is the strict adherence to a well-defined social hierarchy, visually represented in Figure 4.

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Figure 4. Hierarchy of GWO

The alpha wolf is pivotal in decision-making concerning hunting, resting locations, wake-up schedules, and other pack-related affairs in the grey wolf hierarchy. The alpha $(\alpha)$ is commanding, dictating the pack's actions and, therefore, earns the "alpha wolf" [6]. Following the alpha is the beta, constituting the next tier in the grey wolf hierarchy. Beta wolves collaborate with the alpha in decision-making and active participation in pack endeavors. The beta $(\beta)$ wolf assumes a dual role as the alpha's advisor and enforcer of pack discipline. They must respect the alpha while leading the lower-ranked members, namely the delta $(\delta)$ and omega $(w)$. Sitting at the lowest rung of the hierarchy, the omega wolf is tasked with a critical function as a scapegoat. While the omega might appear less influential, its absence has been observed to trigger internal conflicts and turmoil within the entire pack. Figure 5 shows the hunting techniques of grey wolves.

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Figure 5. Hunting techniques employed by grey wolves [21]

If a wolf does not fall into the alpha, beta, or omega categories, it is referred to as a delta. Delta wolves submit to the authority of the alpha and beta members but wield dominance over omega wolves. The fundamental stages of grey wolf hunting behavior are depicted in Figure 5; Stage (a) encompasses tracking, approach, and pursuit of prey. Subsequent stages (b) through (d) correspond to pursuit, harassment, and the strategic encircling of the prey. The static encounter and attack on the prey are depicted in phase (E) [21].

(1) GWO algorithm's mathematical formulas

The social hierarchy, tracking, encircling, and attacking aspects of the GWO algorithm can be captured through the following mathematical equations [6]:

$\vec{D}=\left|\vec{C} . \vec{X}_p(l)-\vec{X}(l)\right|$                       （7）

$\vec{X}(l+1)=\left|\vec{X}_p(l)-\vec{A} \cdot \vec{D}\right|$                        （8）

where:

$l$:Iteration number.

$\vec{A}$ and $\vec{C}$: Coefficients vectors.

$\vec{X}_p$: The prey's position vector and $\vec{X}$ refers to a position vector for a grey wolf.

$\vec{D}$: The distance between wolves and prey.

$\vec{C}$ and $\vec{A}$ vectors can be expressed by:

$\vec{C}=2 . \vec{r}_1$                        (9)

$\vec{A}=2 . \vec{a} . \vec{r}_2-\vec{a}$                        (10)

where, $\vec{r}_1, \vec{r}_2$ are two randomly generated numbers falling within the range $[0,1]$. These parameters are subject to variation within each iteration. And, $\vec{a}$ commences from 2 and gradually decreases to 0 across the iterations.

$\vec{D}_\alpha, \vec{D}_\beta$ and $\vec{D}_\delta$ representing the separations between wolves' distances of different categories, and the prey, are defined as follows:

$\begin{aligned} & \vec{D}_\alpha=\left|\vec{C}_1 . \vec{X}_\alpha-\vec{X}\right| \\ & \vec{D}_\beta=\left|\vec{C}_2 . \vec{X}_\beta-\vec{X}\right| \\ & \vec{D}_\delta=\left|\vec{C}_3 . \vec{X}_\delta-\vec{X}\right|\end{aligned}$                    (11)

where:

$\vec{D}_\alpha$: Distance between wolves in this category ($\alpha$) and their prey.

$\vec{D}_\beta$: Distance between wolves in this category $(\beta)$ and their prey.

$\vec{D}_\delta$: Distance between wolves in this category ($\delta$)and their prey.

$\vec{C}_1, \vec{C}_2$ and $\vec{C}_3$: Represent coefficient vectors corresponding to the first three best positions $\vec{X}_1, \vec{X}_2$ and $\vec{X}_3$.

$\vec{X}_\alpha, \vec{X}_\beta$ and $\vec{X}_\delta$: Refer to the best, 2nd best, and 3rd best search agents, respectively, within the algorithm.

The first three best positions (solutions) of the grey wolves, denoted as $\vec{X}_1, \vec{X}_2$ and $\vec{X}_3$ are:

$\begin{aligned} & \vec{X}_1=\vec{X}_\alpha-\vec{A}_1 .\left(\vec{D}_\alpha\right) \\ & \vec{X}_2=\vec{X}_\beta-\vec{A}_2 .\left(\vec{D}_\beta\right) \\ & \vec{X}_3=\vec{X}_\delta-\vec{A}_3 .\left(\vec{D}_\delta\right)\end{aligned}$                      (12)

Eq. (13) that updates the positions of the prey for the best search agent's position is expressed as follows:

$\vec{X}(l+1)=\frac{\vec{X}_1+\vec{X}_2+\vec{X}_3}{3}$                         (13)

4.2.2 PSO technique

One of metaheuristic optimization is PSO that uses swarms to find the global optimum. A population-based exploration strategy drives the search. What sets PSO apart is its unique inspiration from bird social behavior. It portrays candidate solutions as particles, each with its position in the search space represented by $x$. Every particle has a velocity $v$, which determines its movement in terms of distance, direction, and step size. In addition to its position and velocity, each particle remembers its best position $\left(P_{\text {best}}\right)$. In contrast, the best position any particle in the swarm reaches is called the global best $\left(G_{\text {best}}\right)$ [22]. PSO’s simplicity stems from its ease of implementation and minimum requirement for parameter adjustment. The particle’s velocity is changed according to specific rules, allowing for fine-tuning its path toward better solutions. This velocity adjustment is crucial in optimization because it helps balance exploration (finding new regions of the solution space) with exploitation (fine-tuning existing potential solutions). PSO is a metaheuristic optimization approach that uses swarms to find the global optimum [7].

PSO’s simplicity stems from its ease of implementation and minimum requirement for parameter adjustment. The particle’s velocity is changed according to specific rules, allowing for fine-tuning its path toward better solutions. This velocity adjustment is crucial in optimization because it helps balance exploration (finding new regions of the solution space) with exploitation (fine-tuning existing potential solutions). The rules for velocity updates are expressed as follows:

$\begin{aligned} & v_i(k+1)=r_1 c_1\left(P_{\text {best }(i)}(k)-x_i(k)\right) \\ & +w v_i(k)++r_2 c_2\left(G_{\text {best }(i)}(k)-x_i(k)\right)\end{aligned}$                       (14)

$x_i(k+1)=x_i(k)+v_i(k+1)$                         (15)

where:

$\omega$: The inertial weight constant regulates the influence of the particle's previous velocity on its current velocity.

$c_1$ and $c_2$: These are the cognitive and social coefficients, respectively. They control the weight given to the particle's personal best $\left(P_{\text {best}}, i\right)$ and the best position among the group ($G_{\text {best}}$) when updating the velocity.

$r_1$ and $r_2$: These are uniformly distributed random variables within the range $[0,1]$, contributing to the stochastic nature of the algorithm.

$P_{\text {best}}, i$: Represents the personal best position of particle $i$.

$G_{\text {best}}$: Represents the best particle position among all members of the swarm.

4.2.3 GWO-PSO

GWO component features three essential agents: Alpha $(\alpha)$, Beta $(\beta)$, and Delta $(\delta)$, which represent the most effective solutions found thus far. These wolves, known as Omega wolves, coordinate the movements of the remaining population and lead the search [7]. The GWO mathematical model updates the locations of all search agents (wolves) based on their relative attraction to $\alpha, \beta$, and $\delta$. This technique helps the GWO strike a balance between exploration and exploitation. The agent’s updated locations are determined using the following Eqs. (11) and (12).

The PSO component updates each agent's velocity depending on its previous velocity and attraction to the newly calculated coordinates $X_1, X_2$ and $X_3$. The formula for updating velocity in PSO is as follows:

• Velocity update

$\begin{aligned} & v_i(k+1)=w \cdot v_i(k)+C_1 \cdot r_1 \cdot\left(X_1-X_i\right)+ \\ & +C_2 \cdot r_2 \cdot\left(X_2-X_i\right)+C_3 \cdot r_3 \cdot\left(X_3-X_i\right)\end{aligned}$                       (16)

• Position update

$X_i(k+1)=X_i(k)+v_i(k+1)$                           (17)

This equation adjusts each agent’s position based on the newly estimated velocity. The hybrid method achieves the best balance of exploration and exploitation by combining GWO’s exploitation-driven technique with PSO’s velocity-based exploration.

4.2.4 Anti-windup PI controller tuned by GWO-PSO

The PI-GWO-PSO controller efficiently manages the DC link in the SAPF. However, the anti-windup PI controller's dependence on an accurate mathematical model presents challenges when dealing with system nonlinearities and parameter variations. Consequently, the PI anti-windup controller, designed using conventional methods, struggles to perform optimally across different operating conditions. To address this, the GWO-PSO algorithm is proposed to enhance the anti-windup PI controller's performance. The optimization process focuses on three parameters of the PI-GWO-PSO controller $\left(K_{p c}, K_{i c}\right.$ and $\left.G_a\right)$. The GWO-PSO algorithm aims to solve an objective function that minimizes overall costs, with the study targeting three key objectives: reducing DC link voltage overshoot, minimizing the instantaneous $\left(T H D_i\right)$ of supply current, and controlling the instantaneous error in the DC link voltage loop $(\varepsilon(t))$.

A fitness function evaluates the robustness of the PDPC-SAPF model during its iterative execution, determining the relevance of each search agent until the algorithm reaches the maximum specified iterations. As depicted in Figure 6, the PDPC-SAPF model operates using the closed-loop transfer function of the DC link voltage and the anti-windup PI controller. The number of problem variables corresponds to the initial set of search agents, randomly generated by the GWO-PSO algorithm. In this case, the search agent operates in a three-dimensional space $\left(K_{p c}, K_{i c}\right.$ and $\left.G_a\right)$. With each iteration, the GWO-PSO algorithm adjusts these parameters. After running the PDPC-SAPF model, the fitness function evaluates the parameter values, and the optimal parameters are those that minimize the fitness function.

The study aims to reduce power fluctuations, mitigate over and under-voltages in the DC link, and reduction of $(T H D)$. The ideal PI-GWO-PSO controller gains $\left(K_{p c}, K_{i c}\right.$ and $\left.G_a\right)$ are determined by solving the following equation, which minimizes the fitness function.

• Minimize F

$J=J_1+J_2+J_3$                      (18)

To reduce the error related to the DC-link voltage in both transient and steady states, the Integral Time Square Error (ITSE) index is utilized to select $J_1$ [6]. The ITSE can be defined as follows:

$J_1=w_1 \int_0^t t .(\varepsilon(t))^2 d t$                         (19)

$J_2$ can be expressed as follows and represents the overshoot in the DC-link voltage that was noticed:

$J_2=w_2 \frac{D_{\max }-D_{\infty}}{D_{\infty}}$                          (20)

where:

$D_{\text {max}}$: Maximum value.

$D_{\infty}$: Final value.

To enhance power quality, The THD of the source current is represented by the formula for $J_3$, which is as follows:

$J_3=w_3 \frac{\sqrt{\sum_{n=2}^{\infty} I_n^2}}{I_f} \times 100$                       (21)

$J_3$ involving the Root Mean Square (RMS) values of $I_s$ for the nth harmonic ($I_n$) and the fundamental ($I_f$).

Different impacts are assigned coefficients, which are the weighting factors $w_1, w_2$ and $w_3$. The relative significance of each impact in the overall assessment is established by these weights. The absolute values of these weights sum together are [6]:

$\left|w_1\right|+\left|w_2\right|+\left|w_3\right|=1$                       (22)

Several investigations revealed that raising the weight allocated to f1 (ITSE) increased overshoot and undershoot. In contrast, increasing the weight provided to f3 (THD) resulted in a significant steady-state inaccuracy and increased overshoot and undershoot.

The proposed GWO-PSO based PI controller is illustrated in Figure 6.

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Figure 6. Block diagram of proposed PI-GWO-PSO

The pseudo code of GWO-PSO used to tune the proposed controller of SAPF is listed in Algorithm of Figure 7.

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Figure 7. GWO-PSO pseudo-code

Table 1. Key parameters used in the PSO component of the GWO–PSO algorithm

The development of the control parameter $a$ across iterations is analyzed to show how the GWO-PSO algorithm transitions from global exploration to local exploitation. Figure 8 illustrates that the parameter $a$ drops linearly from 2 to 0, promoting exploration in the early phases and gradually favoring exploitation. This behavior is crucial for avoiding premature convergence and ensuring $a$ strong global search capacity during first iterations, which progressively improves solution refinement in later stages.

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Figure 8. Evolution of the control parameter $a$ across Iterations in the GWO-PSO algorithm

The parameters for the PSO component of the hybrid GWO-PSO algorithm are carefully chosen to strike a compromise between convergence speed and solution quality. The cognitive and social coefficients ($c_1$ and $c_2$) are both set to 2 , encouraging both individual and group learning. The inertia weight $\omega$ decreases linearly from 0.9 to 0.4 throughout the shift from exploration to exploitation. Table 1 contains details on this and other parameter settings.

Figure 9 depicts the whole workflow for the proposed hybrid GWO-PSO-based PI controller tuning method. The method starts with the initialization phase, which generates a set of candidate solutions at random inside the chosen search space, each representing a combination of and gains. Next, each candidate's performance is assessed using a composite fitness function that incorporates primary potwer quality objectives: overshoot (ITSE), DC-link voltage variation, and THD, with weights $w_2=0.1, w_1=0.6$, and $w_3=0.3$.

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Figure 9. Flow diagram of the hybrid GWO–PSO-based PI tuning process

The hybrid GWO-PSO method is then used to update the search agents' locations and velocities based on a combination of exploitation (by the GWO leadership hierarchy) and exploration (via PSO velocity updates). Figure 8 illustrates how the control parameter $a$ declines linearly over repetitions, ensuring a gentle shift from exploration to exploitation.

The iterative procedure continues until a convergence condition is fulfilled. Finally, the best-performing PI parameter set is retrieved and used for real-time control in the SAPF system. This hierarchical tuning method assures that the controller performs reliably and adaptively under dynamic load and nonlinear situations.