Harmonic mitigation using meta-heuristic optimization for shunt adaptive power filters: A review

Citation: Duc, M.L.; Hlavaty, L.; Bilik, P.; Martinek, R. Harmonic Mitigation Using Meta-Heuristic Optimization for Shunt Adaptive Power Filters: A Review. Energies 2023,16, 3998. https://doi.org/10.3390/en16103998 Academic Editors: Mojtaba Ahmadieh Khanesar and Abu-Siada Ahmed Received: 16 March 2023 Revised: 25 April 2023 Accepted: 4 May 2023 Published: 9 May 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). energies Review Harmonic Mitigation Using Meta-Heuristic Optimization for Shunt Adaptive Power Filters: A Review Minh Ly Duc 1,2,* , Lukas Hlavaty 2, Petr Bilik 2and Radek Martinek 2 1Faculty of Commerce, Van Lang University, 69/68 Dang Thuy Tram, Ward 13, Binh Thanh District, Ho Chi Minh City 70000, Vietnam 2Department of Cybernetics and Biomedical Engineering, Faculty of Electrical Engineering and Computer Science, VSB–Technical University of Ostrava, 17. Listopadu 2172/15, 708 00 Ostrava, Czech Republic; [email protected] (L.H.); petr[email protected] (P.B.); [email protected] (R.M.) *Correspondence: [email protected] Abstract: Shunt Adaptive Power Filter (SAPF) is widely used in the performance of power quality improvement activities in the power supply industry for processing industries or civil power sources in the world today based on its simplicity, transparency, high reliability, efficiency, and reliability, and their powerful compensating current-providing nature. The PI controller integrated into the SAPF operation mechanism works with extra high efficiency in selecting the current to compensate for the lost current generated in the power supply due to harmonics generated by the Kp , Ki parameter values. The system operates by the PWM method for bridge rectifier circuits that perform the function of selecting the appropriate compensating current, providing correct compensation for the amount of current loss in the power supply. Adjusting the Kp , Ki parameter to reach the optimal value by different methods is a promising and popular research direction at present. The Kp , Ki parameter serves the right purpose for the PI controller to generate enough PWM pulses to excite the bridge rectifiers to generate just the right amount of compensating current and enough current to be compensated on the power supply. The commonly used Kp , Ki parameter adjustment methods include the Ziegler Nichols closed-loop vibration method, the P-Q theoretical method, and several other methods. This study conducts a comprehensive review of the literature on modern strategies for adjusting the Kp , Ki parameters in the PI controller in the SAPF suite by using the meta-heuristic optimization method. This study performs classification according to the operation mode of metaheuristic optimization methods to adjust the Kp , Ki parameter to control the PI to select the correct PWM frequency to activate bridge rectifiers to select the most optimal compensation current to compensate for the loss of current on the power supply to meet the goal of improving power quality in accordance with IEEE 519-2022 standard, leading to the total harmonic distortion (THD) value is below 5%. The study presents in detail some meta-heuristic optimization algorithms, including applications, mathematical equations, and implementation of flow charts for SAPF and provides some open problems for future research. The main objective of this study is to provide an overview of applying meta-heuristic optimization algorithms to the Kp , Ki parameter tuning of PI controllers. Keywords: shunt adaptive power filter; SAPF; harmonic mitigation; IEEE 519-2022; meta-heuristics; swarm optimization 1. Introduction Power quality and, in particular, reducing power loss during transmission or distribution caused by harmonic components and methods need to be taken urgently [ 1 , 2 ]. There are many methods, as well as models, for reducing and eliminating harmonics and improving power quality in the transmission and distribution process [3,4] (Figure 1). Energies 2023,16, 3998. https://doi.org/10.3390/en16103998 https://www.mdpi.com/journal/energies Energies 2023,16, 3998 2 of 55 Figure 1. Methods to reduce and eliminate harmonics. The power system has the problem of generating harmonics, causing a loss of productivity, and techniques to control and minimize harmonics are proposed [ 5 ]. To understand these techniques, it is necessary to analyze the advantages and disadvantages of each technique and analyze the technical conclusions and their performance. To so harmonic-related problems, there are different techniques like Line reactor [ 6 ], Isolation transformer [ 7 ], K-factor transformer [ 8 ], tuned harmonic filter [ 9 ], IGBT-based fast switched harmonic filter [ 10 ], Low pass harmonic filter [ 11 ], 12 and 18 pulse rectifier [ 12 ], Phase-shifting transformer [ 13 ], and active harmonic filters [ 5 ]. The current reactor implements a series connection with an individual nonlinear current and is the simplest means of harmonic reduction [ 3 , 4 ]. The isolation transformer is known as an electrostatic shield between the primary and secondary coil; they couple capacitance between each coil and shield together, then a low impedance is created to reduce noise, transient current, and zero sequences current [ 14 ]. The shielding helps to reduce harmonic interference in normal mode for the initial side of the transformers [ 15 ]. The K-factor transformer is designed as a constant that determines the transformer’s ability to handle transformer warming caused by generated harmonics [ 16 ]. Usually made by coupling multiple insulated and interchanged conductors to reduce phase effect, magnetic errors are designed with lower flux density [ 8 ]. Factor K has two variables with harmonic current magnitude and harmonic order [15]. The turn harmonic filter is a device that is connected in a series of inductive and capacitive reactance forming a tuned LC circuit, shaped like a shunt device, which is a frequency-modulated resonant circuit that provides impedance short helps to reduce harmonic distortion [ 17 , 18 ]. Insulated Gate Bipolar Transistor (IGBT) has a very fast circuit switching function, about 60 times per second, meeting the requirements of reactive power and ensuring harmonic distortion within the specified standard. A low-pass harmonic filter is to connect multiple string elements into a set of tuning elements, increases input impedance, effectively controls harmonics, and attenuates all harmonic frequencies in the circuit [ 19 ]. A pulse rectifier is a device made up of many rectifiers and connected to a special type of transformer and guarantees a displacement of each secondary phase of 360 divided by the number of rectifier pulses [ 20 ]. The Phase Shifting Transformer is made up of two nonlinear loads fed by the two-phase shifting of the transformer windings and acts as 12 pulses, canceling the fifth and seventh harmonics on the primary side of the transformer [ 21 , 22 ]. Active filters are considered independent harmonic filters or combined with technological techniques in the rectification stage of other power electronic devices. It can analyze the frequency content and the magnitude of the current or filter out the fundamental frequency of the current. It provides suitable inverting currents to eliminate individual harmonics through Insulated Gate Bipolar Transistors (IGBTs) [ 3 – 5 ]. Considering Energies 2023,16, 3998 3 of 55 the approximate cost (USD) and performance of the above harmonic reduction techniques for 3-phase harmonics, the following table gives the comparison results (Table 1). Table 1. Considering the approximate cost (USD) and performance of the above harmonic reduction techniques for three-phase harmonics. Harmonic Mitigation Techniques 15 kW (Price) 75 kW (Price) 300 kW (Price) THD-I (%) (Non-Linear Loads) THD-I (%) (Mixed (50–50) Loads) Reactor (5%) 520 1100 3800 35 17.5 Isolation Transformer 2650 6340 18,000 35 17.5 K-factor (13) Transformer 5300 11,000 48,000 35 17.5 Tuned Filter 2800 3900 7000 12–20 3–12 Low Pass Filter 2400 5600 13,000 8–15 N/A Active Filter N/A 27,000 65,000 5 5 In this study, the focus is on understanding meta-heuristic algorithm methods and AI engineering models, combining the above models to improve power loss compensation through a shunt adaptive power filter (Figure 2). Figure 2. Block diagram of the system for compensation of higher harmonic components using Shunt Active Power Filter (SAPF). Combining AI engineering modeling with meta-heuristic algorithm models improves the model’s prediction accuracy and improves the model’s convergence speed [ 23 ]. Along with today’s trend, the amount of electricity is increasing, specifically in addition to the fact that countries around the world want to gradually reduce their dependence on energy and gas sources, improve the use of renewable energy sources and save electric energy, reduce power loss caused by harmonics by applying [ 1 ] meta-heuristic algorithm techniques and technical models. AI techniques to control active filter circuits, such as shunt adaptive power filters [23,24]. This study has several implications as follows: 1. AI engineering models and meta-heuristic algorithm models are applied to SAPF to perform the extraction of the harmonic component from the measurement signals of the sensors and, at the same time, perform the selection of the optimal compensating current value providing compensation to the power supply; Energies 2023,16, 3998 4 of 55 2. Models that combine meta-heuristic algorithm techniques with AI engineering models in shunt adaptive power filter to increase convergence speed into selecting current compensation and improve the quality of the sine wave shape of the power signal. 3. The equation relationship between the meta-heuristic algorithm models is also compared via the pseudo-code algorithm; 4. Overview of applying shunt adaptive power filter to compensate for power loss for power sources that have been connected to the national power grid such as PV Solar, wind power, and combined AI techniques models with meta-heuristic algorithm models into the above power system; 5. Overview of current control circuits that compensate for power loss caused by harmonics and harmonic analysis circuits generated in power systems are also described in general. Power quality problems are phenomena that arise in the power supply [ 1 ]. The causes that give rise to the above problems are harmonic distortion [ 5 ] and the consequences for the power system and electrical equipment when there is a voltage variation problem as above [ 3 , 4 ]. The waveform of the voltage source or the current source of the power source is distorted, and harmonics are measured as integer multiples of the fundamental supply frequency or the waveform of the voltage or the waveform of the current source, which has a non-sine shape [ 4 , 5 ]. Sources of classical equipment causing harmonics such as arc furnaces, fluorescent lamps), welding machines, rectifiers (Microprocessors, motor drives, any electronic loads), and DC brush motors. Modern sources of equipment cause harmonics such as all non-linear loads such as power electronics equipment, including ASDs, switched-mode power supplies, data processing equipment, and highly efficient lighting [6]. Devices such as rectifiers, ASDs, soft starters, electronic ballasts for discharge lamps, switched-mode power suppliers, and HVAC using ASDs use power and generate harmonics. Harmonic is a form of noise signal that has a direct negative impact on power quality. Harmonics are noticed when the sum of harmonic currents is above the allowable limit. The frequency of the harmonic current is a set of times higher than the fundamental signal frequency. Characteristic oscillations of complete harmonics are in the frequency spectrum. The harmonic component in an AC source is the sine component of a wave period whose frequency is integer times the fundamental frequency of the system [ 25 ]. Harmonics is the main cause of power quality loss and affects other electrical equipment such as transformers, motors, cables, interrupters, capacitors, and protective switching devices [ 3 ]. Bad switching will affect the performance of electrical appliances or electronic control devices, and neutral current is also generated when the electronic devices perform switching modes, devices such as PCs, printers, photocopies, and any triplets generators. The temperature generated in the conductor is caused by the neutral current acting and generated. In addition, the neutral current also adversely affects the performance of the transformer continuously. Harmonics are generated from static frequency converters, cycle converters, induction motors, and arcing devices [ 5 ]. Power quality issues affect devices differently, just as electrical equipment responds to the impact of power quality problems differently, the presence of power electronics is also a factor related to power quality issues, and harmonics management standards are regulated based on the IEEE 519-2022 standard [ 26 ] (Tables 2and 3). Since 1980, harmonics have been considered an essential element that needs to be controlled in electrical systems and electrical equipment [ 1 ]. Harmonics are the cause of voltage source waveforms being distorted; they are causing wires to overheat, which is a serious problem in power transmission and distribution systems [ 6 ]. Harmonics cause transformers to generate heat and heat up transformers and are the cause of failures in electrical equipment [ 26 ]. Harmonic control, eliminating or limiting the generation of harmonics in the power supply, is an urgent issue; currently, solving problems related to harmonics is done by shunt adaptive power filters (SAPF) [27,28]. Energies 2023,16, 3998 5 of 55 Table 2. Current distortion limits for systems rated 120 V–69 Kv (IEEE 519-2022, pg. 19). Isc IL Harmonic Limits a,b TDD Required 2≤h<11 11≤h<17 17≤h<23 23≤h<35 35≤h≤50 <20 4.0 2.0 1.5 0.6 0.3 5.0 20 < 50 7.0 3.5 2.5 1.0 0.5 8.0 50 < 100 10.0 4.5 4.0 1.5 0.7 12.0 100 < 1000 12.0 5.5 5.0 2.0 1.0 15.0 >1000 15.0 7.0 6.0 2.5 1.4 20.0 a: For h≤ 6, even harmonics are below 50% of the harmonic limit; b: Current distortion has resulted in a dc offset; Where I sc : Maximum short circuit current is current that flows through a conductor with very low resistance, almost zero at the point of common coupling (PCC). I L : Maximum demand load current at PCC under normal operating conditions, a function of many factors over time P(t), so they do not obey a certain law. Therefore, it is very difficult to identify them. The electrical load is an important parameter in selecting the equipment for the power system at PCC.PCC (Point of Common Coupling): In many cases, when there is enough source reactance calculated at the point we consider to reduce harmonics, a filter placed at this point can absorb harmonics from many different harmonic sources flowing to them. Even harmonics are limited to 25% of the odd harmonic limits above. Current distortions that result in a dc offset. I sc /I L : All power generation equipment is limited to these values of current distortion. Table 3. Voltage distortion limits (IEEE 519-2022, pp. 17). Bus Voltage (V) at PCC Total Voltage Distortion THD (%) Individual Voltage Distortion (%) V≤1.0 kV ≤8.0% ≤5.0% 1.0 kV ≤V≤69 kV ≤5.0% ≤3.0% 69 kV <V≤161 kV ≤2.5% ≤1.5% 69 kV <V≤1.5% ≤1.0% Previous studies presented algorithms applied to SAPF to perform the harmonic compensation task in the power source. However, many limitations still arise when applying algorithms (Table 4). Table 4. Brief summary of harmonic mitigation methods in SAPF. Ref. Years Methodology Feature Result and Advantage Disadvantage [29] 2019 p-q theory Power 3 phase THDi= 8.2%, Generate reference currents for modern power systems based on the steady-state variation of current and voltage vectors. Unsatisfactory harmonic compensation efficiency less than 5% according to IEEE 519-2022 standard. [29] 2019 DCAP method THDi= 3.5%, Divide the sinusoidal current into n parts and balance the source side. Satisfactory harmonic compensation efficiency is less than 5% according to IEEE 519-2022 standard. [30] 2019 Predictive Direct Power Control (P-DPC) THDi = 1.2%, Maintain the DC bus offset voltage to a specified value and the anti-reverse compensated PI controller to regulate the DC bus voltage. Effect of the sampling period and parameter error on power quality of distribution system. [31] 2020 LCL Filter THDi= 4.56%, The design is higher than the harmonic frequency compensation that the SAPF has to compensate for the higher order harmonics of the grid. The control algorithm is complex. Resonance generation. The parameters of the LCL Filter are very complicated. Energies 2023,16, 3998 6 of 55 Table 4. Cont. Ref. Years Methodology Feature Result and Advantage Disadvantage [32] 2020 SiC-MOSFET THDi= 4.15%. Using the L-locator to suppress the switch sub-harmonics to a smaller level simplifies circuit design and control algorithms. The switching frequency is increased to 50 kHz. Increases the second harmonic. [33,34] 2020 An ADALINE-based Neural Network (ANN) THDi= 2.39%, The current is measured using the Least Mean Square (LMS) algorithm; the weights are obtained with the help of online calculations. Analysis under severe abnormal conditions is the direction of future research. [35] 2021 Space Vector Pulse Width Modulation (SVPWM) THDi= 3.73%, Trace and identify the reference voltage in a static coordinate system through coordinate transformation and determine the reference voltage. The reference structure has only 4 transformation modes and no vector 0. This reduces the freedom of the composite vector and is difficult to control. [36] 2021 Triangle Orthogonal Principle (TOP) THDi= 4.98%, Using the phase signal from the phase-locked loop is synchronized with the grid signal based on the principle of triangle orthogonality. Lack of selective harmonic compensation. [37] 2021 Computation Fluid Dynamics (CFD) THDi= 4.25%, Simulation of a heat transfer coupling under forced cooling conditions. Designing power electronic components requires high precision. [38] 2022 Least Mean Square (LMS) THDi = 3.7%, Separation of the elementary active, reactive, and harmonic components of the distorted current. Performance is low when using the same speed for components when estimating the feedback operation. [39] 2022 Modified Symmetrical Sinusoidal Integrator (MSSI) THDi = 3.94%, Extract the basic components of the corresponding forward sequence and use instantaneous reactive power theory to process the reference flow. Look up the parameters of the transfer function. [40] 2020 Adaptive Backstepping Fuzzy Neural Controller based on Fuzzy Sliding Mode (FNN-based FSM) Power 1 phase THDi= 4.48%, Establish a subsystem and use virtual controls to simplify controller design. Satisfactory harmonic compensation efficiency is less than 5% according to IEEE 519-2022 standard. [41] 2021 Long and Short Term Memory Fuzzy Neural Network (LSTMFNN) THDi= 4.67%, Combine fuzzy neural network and long and short-term memory mechanism to enhance self-learning ability and high performance. Improve control effect, new neural network learning strategies, finite time control and reduction of system chattering are future research directions. [42] 2022 Modified Multiport Interleaved Flyback Convertor (MMPIFC) Photovoltaic (PV) three-phase power THDi= 2.61%, Multi-port interlaced flyback conversion to connect n number of input sources to DC bus to overcome partial shadow problem. Replacing fuzzy controls with advanced artificial intelligence algorithms like bio-inspired optimization is the direction of future research. Energies 2023,16, 3998 7 of 55 The parts of the research paper are organized as follows: Section 2shows the details of random models and optimization models. Section 3provides an overview of harmonic mitigation using meta-heuristic algorithms and artificial intelligence. Section 4presents a discussion and future research problems, and Section 5presents conclusions. 2. Random Models and Optimization Models The input parameters of the optimal models are usually partially known, or they are not defined to be known; these parameters can also be called uncertain parameters. They are implemented through probabilistic statistical models or experimental design [ 6 ]. The model used to implement the above parameters is called the stochastic programming model and is expressed through Formula (1) as follows: min x∈X{g(x)=f(x)+E[Q(x,ε)]}(1) With: X is a nonempty closed subset of Rn , ε is a random vector whose probability distribution P is supported on a set [I]⊂Rd and Q:X×[I]→R . In the framework of twostage stochastic programming, Q(x,ε) is given by the optimal value of the corresponding second-stage problem. g(x) is well-defined and finite valued for all x∈X . This implies that for every x∈Xthe value Q(x,ε)is almost surely finite. The key to making the model change is the input parameters, and in particular, the objective functions that are set up containing random parameters whose values are unknown or known. However, the input variables of the objective function obey the distribution law of a given probability previously [ 6 ]. There are many related studies applying models using unknown, unspecified random input parameters and following probability distributions, such as the meta-heuristic algorithm (Figure 3). Figure 3. Classification of meta-heuristic algorithms. Evolution-based algorithms are models that form algorithms inspired by natural evolution to generate populations for algorithmic solutions [ 43 , 44 ]. Individuals are created from the best solution of the mathematical model, mutation, or crossover, or select the best solution in the mathematical model to create new individuals [ 45 ]. The genetic Algorithm (GA) is a figure point. This mathematical modeling technique is based on Darwin’s evolutionary technique. In addition, there are other techniques that have been developed, such as evolution strategy, genetic programming, Backtracking Search Algorithm (BSA), and Differential Evolution (DE). Swarm Intelligence based Algorithms are social behavior from insects, animals such as fish, birds, and so on while they are foraging or hunting, specifically their behavior of moving to find the best location and space best for the process of social behavior. Mathematical models are built from those social behaviors [ 46 , 47 ]. The most popular is particle swarm optimization (PSO),developed by Kennedy and Eberhart. There are also many other algorithm models, such as Ant Colony Optimization, Honeybee colony optimization algorithm, and Cat Swarm Optimization (CSO). Physics-based algorithms are based on the laws of physics in the universe around us, re-modeled [ 48 ] into algorithms like Simulated Annealing (SA) and Gravitational Search Algorithm (GSA). Human behavior relation algorithms are based on human behavior modeled into mathematical models. The performance of a mathematical model is directly related to Energies 2023,16, 3998 8 of 55 human behavior [ 49 , 50 ]. The algorithms were conceived as a teaching-learning-based optimization (TLBO) and a League Championship algorithm. The above plans meet the requirements of high equivalence search criteria and have fast convergence when using stochastic methods with unknown input parameter variables or undefined according to the distribution law of probability statement. However, because the input factor is a random variable, the meta-heuristic optimization methods can loop around to find the approximate value of the criterion function over a long time or possibly indefinitely [ 6 ]. This is a limitation of the above optimization models; the variables can be used to optimize the randomness of the input parameter variables of the optimal model, to reduce the randomness, reduce the size of the random data or eliminate the finite difference, as well as remove the confounding factors, to bring the optimal results for the model [ 44 , 46 ]. Each meta-heuristic optimization model has its own characteristics, its own mainstream, and at the same time, its own limitations [ 48 , 49 ]. Meta-heuristics-based optimization is considered for use on the following grounds: 1. Meta-heuristic optimization is applied by many researchers to research many aspects of optimization and is widely used, which means that there are many recent research publications in many prestigious journals around the world catalog ISI/SCOPUS and is used in almost every field from engineering to economics and other sciences; 2. Artificial intelligence uses meta-heuristic optimization models in training activities and as well as improves the ability to predict results of artificial intelligence (AI) technical models such as artificial neural network (ANN), fuzzy logic, and adaptive neural fuzzy system (ANFIS); 3. Meta-heuristic optimization is done very simply with not too complicated mathematical models, with no need for additional training data or initial implementation solutions, just building suitable mathematical models and precise distribution functions’ respective performance to improve the optimization level for the operations; 4. Researchers only need to use the population size and number of iterations to build an optimal research model using meta-heuristic optimization without the need to delve into the knowledge of complex mathematical models; 5. Researchers only need to build fitness functions and constraints to freely choose meta-heuristic models and modify them to perform optimal problem-solving; 6. Meta-heuristic research models are integrated into the test models and validated based on simulation models with various tools available; 7. Meta-heuristic optimization gives good processing results for multi-objective processing models and, with many decision variables and constraints, does not restrict solutions and is not dominated; 8. Meta-heuristic optimization is used to solve multi-disciplinary problems, and along with many publications in prestigious journals in the world at the present time, it is useful for analysis, comparison, and analysis activities to compare the research results of the proposed work of the authors; 9. Compared with the training and learning requirements with complex mathematical models of artificial intelligence (AI) techniques, meta-heuristic optimization shows that the computation process is much simpler with the use of algorithms. Math models are much simpler than those applied in AI techniques; 10. Nowadays, the development of computer technology needs to use optimization models more and more to optimize the processing time of real-time problems. 3. Harmonic Mitigation Using Meta-Heuristic Algorithms and Artificial Intelligence Models of harmonic control and reduction are often related to (1) analyzing and detecting harmonic components. (2) Control and generate a suitable compensating current into the power supply to compensate for the electrical loss caused by the harmonic component (Figure 2). In some cases, researchers create analyzers that identify harmonics used in single-phase [ 51 ] or three-phase power networks [ 52 ]. In many cases, the problem to be optimized is necessary; the reason is that the problems are very complex. Nowadays, Energies 2023,16, 3998 9 of 55 modern optimization methods implemented into SAPF filters are a promising research direction. Meta-heuristic optimization and artificial intelligence techniques have been applied by many researchers to SAPF control to generate a compensating current that provides compensation to the power supply [ 46 ]. Finding the optimal power supply parameter to compensate for the harmonic component is a very complex problem. Applying a mathematical model or more to solve a problem is necessary. Choose one or more available information about the problem, or interactions between them, to apply to the optimization algorithm, which produces an optimal result better than the individual algorithms [ 3 , 6 ]. Meta-heuristic optimization models using combined with shunt adaptive power filters to control harmonics is a research direction that is interesting in scientists and managers at the current time, as well as a development orientation for the application of advanced signal processing control by computer in power quality improvement activities [26–28]. 3.1. Analyze and Detect Harmonic Components Detecting and extracting harmonic components of voltage and current sources is essential for power quality improvement [ 3 ]. The purpose of this work is to find a suitable method to select the compensating current to compensate for the current, or power voltage, loss caused by harmonic components. Components such as amplitude and phase of harmonics require a reasonable technique for extraction, detection, and classification at the input source. Harmonics cause distortion of the voltage waveform or input voltage current, and a suitable compensating current is required to compensate for the loss caused by harmonic distortion to correct the waveform distortion of the voltage source or current source [ 3 ]. To do this, a suitable method is needed (Figure 4). The methods for extracting, detecting, and classifying harmonics are divided into two groups. Group 1 is a type of frequency harmonic component analysis technique, and group 2 is a time domain harmonic component analysis technique. Figure 4. Harmonic detection methods. Group 1 is a group of frequency harmonic component analysis techniques performed by Fourier series analysis to extract the harmonic components of the input source. The methods of analyzing and extracting harmonic components in the power source include Discrete Fourier Transform (DFT) [ 53 ], Fast Fourier Transform (FFT) [ 54 ], and Sliding Discrete Fourier Transform (SDFT) [ 55 ].The disadvantages to note when using the above methods are that it takes a certain period of time to solve the problem, requires a large Energies 2023,16, 3998 16 of 55 Table 6. Cont. Step Step-By-Step Explanation of the ABC Algorithm Method Step 5: bees observe, receive information, evaluate the information received and choose a quality food source. They then pass the information back to the swarm, and together they rate the quality of the nectar and compare it to the quality of the previous nectar. Where the quality of the nectar is better than the quality of the previous nectar, they switch to a source with better quality nectar. At the same time, they also change the memory of the old nectar information (Equation (10)). P(i)=α×Fitness(i) max(Fitness)+b (10) With, i: is the i-th food source. P(i): Probability that the observed bee chooses a food source. Step 6: The old food source is also removed and improved with a new food source after each establishment, and this work is performed by scout bees. Step 7: Loop when reaching the maximum value, the algorithm terminates. Otherwise, the loop is updated next with the formula iter = iter + 1 and goes back to step 4 to continue executing the program. The ABC method solves the nonlinear equation of the Harmonic and Selective rejection sample considering unequal DC current sources. The ABC method classifies the fundamental components that respond to the rejection of low-order harmonics. ABC method is one of the powerful and new evolutionary optimization patterns, finding optimized transformation angles with higher accuracy and higher convergence than others. The ABC algorithm determines the optimal switching angles and finds the optimal switching angles to generate the desired voltage. The ABC algorithm outperformed algorithms such as GA, PSO and BA in 30 runs with the same initial values according to the criteria of convergence and accuracy. ABC algorithm finds the optimal gain value of controller PI for SAPF for different errors as functional fitness variables. The ABC algorithm gives good results for monitoring the reference voltage and adjusting the PI for THD minimization. Stable dc-link voltage is in less than 1 cycle during transient load. ITSE is a performance indicator that shows better dynamic response harmonic compensation in a current source. ABC algorithm is a good tool to find the optimized gain of PI controller with ITSE as a fitness function. Ant Colony Optimization (ACO) for SAPF Ant Colony Optimization (ACO) is applied to Shunt Adaptive Power Filter (SAPF) with the goal of optimizing the gain Kp , Ki of the PI controller [ 82 , 83 ], and optimize the indicators of Integral Square Error (ISE), Integral Time Square Error (ITSE), Integral Absolute Error (IAE) and Integral Time Absolute Error (IATE) [ 84 ]. Dorigo (1992) introduced the ACO method based on the foraging behavior of ants used when foraging, such as (1) positive feedback, (2) distributed computation, and (3) constructive greedy heuristics. The goal of the ant colony’s foraging behavior is to find the best solutions to discover the fastest food source, evaluate and select the best food source for the ant colony, prevent mistakes in choosing food sources, and choose the best method for the ants to find the best food source [84,85]. Artificial ants mimic the behavior of biological ants and find the best way to find food sources. The leader ant emits hormones; the ant follows the shortest distance, with many hormones remembering positive feedback. All the ants in the ant colony move at the same speed and send the same proportions of hormones [ 78 ]. The ACO modulating the Kp , Ki gain in the PI controller is depicted in the block diagram (Figure 9). Energies 2023,16, 3998 17 of 55 Figure 9. ACO tuning approach for SAPF PI controller. The cost function e(t) provides a mathematical model for ACO’s optimal search. Each ant moves through the Kp , Ki nodes, and the Ki search space for 100 nodes. Each node Kp in the range (0.1~1.0) and Ki in the range of 1–300 and is condensed in two different vectors. The goal of ACO is to find the path with the smallest cost function, that is, to find the most suitable Kp , Ki parameters. Each ant is made to move according to the probability function. When the Kth ant moves to the ith position and builds each part Sp , according to the distribution law of the Kth ant selected according to the jnut from the inut. The distribution function is: ρk ij =  hτα ij (n)ihµβ ij(n)i ∑N(sP)τα il (n)hµβ il(n)ii f CijeN(Sp)   (11) where τij and µij : pheromone and metaheuristic intensity value index information between nodes i and j . Indexes N(Sp) and l are the set values of nodes and paths that may not have been visited by ant k. The ant deposit updates hormone value (Local hormone) according to Formula (12). τk ij(n)=τk ij(n−1)+0.2 ∗α C(12) The global hormone is updated according to Formula (13). τij(n)best =τij(n)best +α Cbest (13) The negative hormone is updated according to Formula (14). τij(n)worst =τij(n)worst −0.3 ∗a Cworst (14) Global hormones are updated after each hour according to Formula (15). τij(n)=τij(n)∂+∆(15) where τij(n)best and τij(n)worst is the hormone of the paths for the ant to follow in its search for a food source with the lowest cost ( Cbest ) and maximum cost ( Cworst ) values. λ is the evaporation constant, and ∆is the sum of Formulas (13) and (14). The ACO algorithm for PI tuning is shown in Table 7. Energies 2023,16, 3998 18 of 55 Table 7. The ACO algorithm code. Step 1: (Parameter Initiation) SettherangeforKp,Ki, tour = 0, m = 20, maximumtour = 100, α=0.06, l=0.95 For every combination (i,j) Set an initial value τij(0)=1, ∆τijlocal(0)=0and ∆τijglobal(0)=0 End Step 2: (Local Update Rules) For k = 1 to m and choose Kp , Ki with a transition probability given in Equation (12). Calculate cost k End For ∀(i,j) Fork=1tom For ∀(i,j) Fork=1tom Update the pheromone using Equation (11) End End Step 3: (Global update rules) Update pheromone for best and worst tours of ant using Equations (11) and (12). Globally update pheromone using Equation (16) Tour = tour + 1 If (tour < maximum tour) Go to step 2 Else Print the best node values for the minimum cost function End Details of the steps to implement the ACO algorithm in SAPF are described in Figure 10 in the flow chart of ACO for SAPF. The ACO method performs optimal tuning of the membership functions and normalized gain in SAPF. The ACO method is a choice for an effective DC link voltage to compensate for harmonic currents in the power supply [ 86 ]. The ACO algorithm is the best for the controller, and satisfactory performance is an effective solution for the growing energy demands now and in the future. Ant colony optimization (ACO) is a technique that optimizes the gain values of the PI controller used in the Shunt Active Power Filter (SAPF), improving its dynamic performance [ 87 ]. The minimization of Integral Square Error (ISE), Integral Time Squared Error (ITSE), Integral Absolute Error (IAE), and Integrated Time Absolute Error (ITAE) are considered functional costs of the ACO-SAPF system and the ACO method improve the resolution time (Ts) with ISE as the cost function. Ant Lion Optimizer (ALO) for SAPF ALO is implemented in SAPF to adjust the control parameters and K P K I in the PI controller to adjust the compensatory current for the power supply in order to reduce harmonic according to IEEE 519-2022 standards for both power and current on the load [ 88 ]. ALO implements the gain and loss adjustment method of the PI controller to adjust the required DC current at the output and meet the required voltage compensation on the power supply [ 89 ]. The synchronous and theoretical P-Q reference method is implemented in the circuit to generate a suitable compensating current [90,91]. Energies 2023,16, 3998 19 of 55 Figure 10. Flow chart of ACO for SAPF. ALO uses two search agents, including ant lions and ants, in a swarm. Ant lions are always looking for the best food element, and their location is always known to be fixed. Ants in the swarm are always free to move in space foraging and are likely to be caught when trapped in a wormhole [ 90 ]. The position of Ants in the swarm is done according to Formula (16). Antk j=PK A−PK E 2(16) where PK A represents the nearest random path of the Ant lion. PK E represents the position of the ith ant closest to the ant lion in the E group of the ant. The distance between ant j and ant lion g after ant j moves in loop K is determined by Formula (17). MK g=wj−rj∗dj−Cj bK j−rj(17) where rj , bj determines the largest and smallest steps of ants in the foraging zone of size K.C,dthe random walk region of ant j and the maximum and minimum limits of the threshold are in the range [0.5]. In the natural environment, ants roam randomly around in search of food sources. This behavior is expressed by Formula (18). X(it)=[0, cusu(2r(it1)),cus(2r(it2)), . . . , cusu(2r(itn)−1)] (18) Energies 2023,16, 3998 20 of 55 where cusu: cumulative sum spending time r(it) and is valued according to Formula (19) r(it)=(1, i f rand >0.5 0, i f rand ≤0.5 (19) The antlion senses that prey has entered the hole. The antlion throws sand to reassemble the prey and captures the prey into the hole. This action is described by Formulas (20) and (21). Cit =Cit 10w.it itmax (20) dit =dit 10w.it itmax (21) where Cit , dit : the low and high variables are converged. W: the constant is fixed at the current loop and is determined by Formula (22). w=                2, i f it >10%. itmax 3, i f it >50%. itmax 4, i f it >75%. itmax 5, i f it >90%. itmax 6, i f it >95%. itmax (22) The final stage is ALO traps the prey and stops the trap, and this action is built according to Formula (23) Antlionit j=Antit i;i f f Antit i>fAntlionit j(23) where Antlionit j: the jth chosen location of the antlion at ith. Antit i:the ant’s i-th position. The pseudo-code (Algorithm 1) and flow chart of ALO for SAPF are shown in Figure 11 . Algorithm 1: The pseudo-code of the ALO Algorithm 1 Input (Set input data of SAPF. Set parameters of ALO) 2 K = 1 3 Create 3 initial sizes of ant and ant lion are Kp,Ki,Gα 4 Run SAPF with Kp,Ki,Gαand evaluate the fitness function value of ants and ant lions 5 Identify the best ant lion 6 While K≤Kmax do 7 For i = 1 to the number of agents, do: 8 Choose the antlion based on the movement circle 9 Update the position of ants according to Formulas (18) and (19) 10 Update the location of the antlion (update the Kp,Ki,Gαvalue) according to Formula (17). 11 Run SAPF updates the Kp , Ki , Gα value and evaluates the fitness function value of the ant lion 12 Substitute the antlion with ants according to Formula (24) 13 Update elite position 14 K=K+1 15 End for 16 End while 17 Return optimization elite 18 Output: Print optimization gains Kp,Ki,Gαof the PI controller in SAPF according to the optimal elite value Energies 2023,16, 3998 21 of 55 The objective function f(g) performs error minimization of the two gains, K P and K I , of the PI controller according to Formula (24). f(g)=ITSE =f(q,t)=Z∞ 0e2(q,t)tdt (24) where ITSE = Integral Time Square Error. ITSE extracts and provides data for ALO to optimize the parameters of gain, K P and K I , of the PI controller. The error(e) of the index showing the agreement between the reference voltage V dc,ref and the voltage of the capacitor Vdc is shown by Formula (25). Error(e)=Vdc,re f −Vdc (25) The gain controller K P ,K I is determined according to Formula (26). Gain Control (GC) is a closed-loop feedback regulating circuit in an amplifier or chain of amplifiers to maintain a suitable signal amplitude. The average or peak output signal level is used to dynamically adjust the gain of the amplifiers, allowing the circuit to work satisfactorily with a greater range of input signal levels. q=Kp,KiT∈M(26) where Mis a positive real value index. The foraging space is expressed by Formula (27). S={q∈M,qmin ≤x≤qmax}(27) Figure 11. Flow chart of ALO for SAPF. Energies 2023,16, 3998 22 of 55 The ALO method searches for the parameter values of K p and K i that achieve the minimum value of the fit function, then is provided as the optimal parameter at the output of ALO-SAPF. The goal is to reduce the maximum overshoot and lower the DC-link voltage with reduced power ripple and as low THD as possible. The ALO method properly adjusts the circuit to reduce harmonics in the source current and load voltage, adjusting the gain of the controller to adjust the required DC output voltage. The ALO method is used to extract the optimal values of frequency and increase the PI voltage of the PI controller in SAPF. The reception ALO algorithm maintains sinusoidal patterns for the source current waveform with a good terminal voltage and frequency within a limited range under unbalanced and variable load conditions. THD is less than 5%. The reception control algorithm with ALO integrated into SAPF is very efficient in terms of power quality. Bat Algorithm (BA) for SAPF BA is implemented into SAPF, which performs DC voltage rectifier controller optimization and reactive power theory and P-Q theory used to extract the reference current of the power supply [92]. BA builds on the bat’s perceptual behavior by using echolocation to recognize and classify food sources and barriers. Bat’s velocity speed (V i ) and Bat’s position (X i ), Bat’s broadcast frequency (F min ), the wavelength of echo ( ∂ ) and reverberation (A 0 ) during the search for food sources. The magnitude of A 0 is calculated according to the A min constant. BA adjusts the DC index value to optimize the PI controller. However, BA is limited by the foraging region (in this case, the tuning parameters KPand Kiin the PI controller) [93]. BA follows two main activities, including exploration and exploitation. Exploration to find new solutions is given by Formula (28), and exploitation to search for food sources in the vicinity is given by Formula (29). Xi(t+1)=Xi(t)+Vi(t)(28) Xnew =Xold +e×A0(t)(29) BA pseudo codes are done in the following steps (Table 8). Table 8. BA Algorithm code. Step Step-By-Step Explanation of the ABC Algorithm Method Step 1: Establish the function of the bat according to the formula F. Step 2: Initialize functional variables, including upper bound information and lower bound information of each bat, number of bats, maximum number of repetitions, and number of variables looking for food sources. Each bat has different upper and lower bound parameters in the foraging zone. Step 3: Call and Find the initial value of the objective function Step 4: The maximum number of repetitions is to be performed from the start of the main loop, and the frequency is randomly chosen according to Formula (30). F(i)=Fmin +(Fmax −Fmin)×rand (30) Step 5: Update the speed and position of the bat. After each update of the upper and lower bound values, the new position of the bat is updated according to Formula (31). Xnew =Xold +X+−X−+ub ×X++lb ×X− (31) Step 6: Check the pulse rate of each bat. The random step size limiting factor is 0.001. Step 7: Recalculate the fitness value after optimization according to Formula (32). Plot the convergence curve for the best fit and repeat. The best position is also called the optimized value. Fmin =fitnessnew (32) Energies 2023,16, 3998 23 of 55 The results of applying BA to SAPF (Figure 6) and THD coefficient = 0.7% meet the requirements of IEEE 519-2022 standard. The objective function is for the optimization controller parameter, according to Formula (33). F=minmin(ITAE)+min(Tr)+min(Ts)+minMp+min(ess)(33) where ITAE: Integral Time Absolute Error, Tr : rise time, Ts : setting time, Mp : peak overshoot, and ess : steady state error.The flow chart of BA for SAPF is shown in Figure 12. Figure 12. Flow chart of BA for SAPF. The BA method performs dc-link voltage regulator optimization. The stability of the current controller with the SAPF system is a mathematical model evaluated in terms of time and frequency area. The BA method implements a PI controller to adjust the harmonic current in SAPF theoretically analyzed for stability and suitability; dc-link optimization performs well harmonic harmonization and reactive power consumption of the load. The harmonics in the current are effectively suppressed by the SAPF, the reactive power required by the load is compensated by the SAPF, and the power supply operates the power required by the load and the inverter losses. A faster SAPF dynamic response is achieved to sudden load changes of nonlinear loads. Energies 2023,16, 3998 24 of 55 Bacterial Foraging Algorithm (BFA) for SAPF BFO deployed in SAPF adjusts the control coefficients KP and KI of the PI controller (Figure 13) to provide compensatory power for the power system to improve the quality of power supply for balanced loads and unbalanced [ 94 , 95 ]. The results show that the THD = 1.37% value is within the IEEE 519-2022 standard. Figure 13. Block diagram of BFO for PI controller. BFO was built inspired by microbial foraging with the goal of optimizing bacterial energy consumption per unit of time (T). BFO works by four observed mechanisms of micro-emergent, including chemotaxis, swarming, reproduction, and elimination or dispersal [96,97] . The four mechanisms of action of BFO are explained as follows (Table 9). Table 9. BFO Algorithm code. Step Step-By-Step Explanation of the BFO Algorithm Method Step 1: (Chemotaxis): bacteria move to find a source of more nutrients in the intestines thanks to the mechanism of bladder action in directions such as swimming or somersaults. Assume θi(j,k,l)is the ith bacterium in the jth trophic zone, the kth spawning zone, and the lth elimination dispersal step. The bacteria in motion were calculated according to Formula (34). θi(j+1, k.l)=θi(j,k,l)+C(i)∆(i) √∆T(i)×∆(i) (34) where C(i) is the size of a single step and movement in a random direction, and ∆ (i) is the vector in an arbitrary direction of the elements in the range [−1, 1]. Step 2: (Swarming): bacteria move in swarms with high density in the activity of sourcing nutrients through mechanisms of attracting and repelling substances given by Formula (35). Jcc(θ(i,j,k,l)) =S ∑ i=1Jccθ,θi(j,k,l)= =S ∑ i=1−dattractantexp−wattractant p ∑ m=1θm−θi m2+ (35) +S ∑ i=1hrepellantexp−wrepellant p ∑ m=1θm−θi m2 where Jcc(θ(i,j,k,l)) is the objective function used to optimize goals over time. S: Totalnumber of bacteria in the population. pis the optimization variable and θ=θ1,θ2, . . . , θpTis a point in the p−dimension in the search for nutrients. dattractant,wattractant,hrepellant,wrepellant are measures of the number, rate of diffusion, and strength of the forward and backward effects of bacteria, respectively. Step 3: (Reproduction): the acclimatization value of bacteria iin NC migration and calculated according to Formula (36). Ji health =Nc+1 ∑ j=1Ji(j.k.l) (36) where Ji health is the health of the representative ith bacterium. The healthy bacteria eventually eliminate other healthy bacteria, and the population stays the same in the end. Energies 2023,16, 3998 25 of 55 Table 9. Cont. Step Step-By-Step Explanation of the BFO Algorithm Method Step 4: (Elimination or Dispersal) : the bacteria are removed and dispersed with probability ped after the Nre spawning event with the goal that the bacteria are not trapped and ensure that the local optimum replaces the global optimal. The objective function is optimized following Formula (37). J=Rt 0(∆Vdc)2dt =β∗∆Vdcmax +(1−β)(ts−t0)+α∗|Ess| (37) where α: steady − state voltage error correction index Ess.βis the decisive index of the value of voltage (∆Vdcmax ) . tsis the maximum value of βwithout overshoot , and t0is the start time, tsis the steady-state time of the transition period. The BFO method gives optimal results that outperform traditional methods by ensuring excellent SAPF functionality and rapidly overpowering harmonics in the current source, even when the power supply is unbalanced [ 98 ]. The BFO method is implemented to adjust the coefficients of the PI controller in SAPF to improve the performance of the power system under balanced and unbalanced supply voltage conditions. The dc link voltage is stable for about one cycle, and also the voltage variation is less than that of conventional PI controllers. The BFO-SAPF method performs harmonic rejection and function superior to the PSO-SAPF method and has excellent functional confirmation of its superior and powerful harmonic compensation. Firefly Algorithm (FA) for SAPF Predator-Prey-based firefly optimization (PPFO) is implemented into SAPF to select the appropriate compensation current to provide compensation for the loss of mains current to improve power quality [ 99 , 100 ]. Shape the sine wave shape of the power supply for balanced and unbalanced loads. The results for the THD = 1.909% index belong to the IEEE 519-2022 standard [99]. PPFO is inspired by the flickering light of fireflies to explore and exploit food sources in the search for food sources [ 101 ]. The proposed problem variables form fitness functions, and these variables formed in SAPF include Cdc , Vdc,re f , Lf , Rf , Kp , and Ki and randomly generate a swarm of fireflies from initialization. Each firefly represents an optimal solution in the foraging zone and has as many dimensions as the number of designed variables. Each firefly is parameterized according to Formula (38). f=hCdc,Vdc,re f ,Lf,Rf,Kp,Kii(38) The final search area space is limited according to Formula (39) fK(min)≤fK≤fK(max),K=1, 2, . . . , n(39) A mathematical model is established from bioluminescence communication to change into the motion of fireflies in the foraging space. Every firefly is mesmerized by other fireflies’brightness, and they try to fly toward where the light is. Firefly’s brightness has an impact on the efficiency of the designed problem point [ 99 , 101 ]. In repeating theprocess, the algorithm model is evaluated by each firefly’s brightness and attractiveness, and the position value of the firefly is updated based on these values. The brightness function (BFun) is made to reduce total harmonic distortion (THD) and is calculated by Formula (40) . MaximizeBFun =1 1+THD (40) Attractiveness of the ith and jth fireflies are shown by Formula (41). βij =β0exp−γrij2(41) Energies 2023,16, 3998 32 of 55 Table 13. WOA code. Step Step-By-Step Explanation of ATS Algorithms Step 1: At first, the whale acquaints itself with the prey, then surrounds the prey. The whale predicts the best solution and calls it objective prey and is substituted when there is another better solution. Variables are updated according to the Formulas (47) and (48). → D= → C . → X∗(t)−→ X(t) (47) → X(t+1)=→ X∗(t)−→ A.→ D (48) Vector of coefficients → Aand → Care updated by Formulas (49) and (50) → A=2→ a.→ r−→ a (49) → C=2. → r (50) wheret =indicates the current iteration. → X∗=position vector of the current best solution obtained. → X=position vector should be updated whenever there is a better solution. → A=coefficient vector. → C=coefficient vector. → a=linearly decreased vector from 2 to 0. → r= random vector between [0, 1]. Step 2: Exploitation phase, whales will attack their prey with a bubble net strategy and do so with twomethods, including shrinking, encircling, and spiral updating. Shrinking encircling performs a new search defined between the current best range and the updated → avalue search range with Formula (11)and the→ avalue is assigned from 1 to −1. Sprial updating performs a calculation of the distance between whale X and Y from the prey X∗and Y∗. Spiral is shown according to Formulas (51) and (52). → X(t+1)=→ D0.eb1.cos(2πL)+→ X∗(t) (51) → D0= → X∗(t)−→ X(t) (52) Formula (13)calculates the distance from the ith whale compared to the best updated solution. L = random number in [ − 1,1], b = fixed number for the spiral algorithm. The algorithm model is built as Formula (53). → X(t+1)=   → X(t)−→ A.→ Di f p ≤0.5 → D0.tt +→ X∗(t)i f p ≥0.5 (53) where the random value of pis selected in [0, 1] and H =eb1.cos(2πL). Step 3: The search for prey or the exploration phase and the→ avector number is selected randomly to update the search location and perform according to Formulas (54) and (55). → D= → C.→ Xrand −→ X (54) → X(t+1)=→ Xrand −→ A.→ D (55) where → Xrand = random vector chosen from whales’ location from the current population. After applying WOA and SAPF, the THD = 1.49%, within the IEEE 519-2022 standard, where the objective function is according to Formula (56). PLoss =Kp.Error +KiRt 0(Error)dt (56) With : Error =Vdc re f −Vdc actual The WOA algorithm implemented in SAPF performs optimization of gain parameters in the PI controller to select the current to compensate for the disturbance current in the power supply and compare the results of optimal performance parameters with other parameters. For other algorithms, the WOA algorithm gives the best results [ 115 , 116 ]. The WOA algorithm used in SAPF shows that the signal processing by Width Modulation (PWM) is very simple and uses the Technical Width Modulation (PWM) parameter to tune the controller in SAPF [ 117 ]. The WOA method addresses power quality problems caused by interruptions caused by electrical equipment using electricity, such as nonlinear loads Energies 2023,16, 3998 33 of 55 or renewable energy sources. The WOA method performs direct tuning of the relevant parameters to facilitate power quality improvement. Figure 17. Flow chart of WOA for SAPF. Swarm Particle Swarm Optimization (PSO) for SAPF PSO is applied in compensating current control for the shunt adaptive power filter (SAPF). The goal is to ensure the quality of the power supply to the load [118–120]. PSO is inspired by the swarm, and PSO’s mechanism generates particles randomly and is assigned an arbitrary parameter. The velocities of particles in space group together to form a global convergence value [ 98 , 121 – 129 ]. The flight movements of the particles in the respective search area of each individual and their particles in the swarm population, the position of the ith particle in the swarm xid(t) moving with speed Vid(t) , the positions and the velocities of the particles repeated successive times, xid(t+1) and Vid(t+1) , respectively, are updated as Formulas (57) and (58): Vid(t+1)=n.Vid(t)+C1.r1[Pid(t)−xid(t)] +C2.r2[gid(t)−xid(t)] (57) xid(t+1)=xid(t).Vid(t+1)(58) where w isthe inertia constant which maintains the balance between the neighborhood and global search regions. C1 , C2 = accelerator constant. r1 , r2 = two random constants are generated independently and evenly distributed in the interval [ − 1, 1]. Pid(t) = coordinatesof the best position detected at the ith particle. gid(t) = coordinates of the best-detected location for the entire swarm or global optimal. The value of the inertia constant w specifies the search space operation and is performed according to Formula (59). w=wmax −(wmax −wmin)g G(59) Energies 2023,16, 3998 34 of 55 where g = the current number of evolutionary generations. Wmax , Wmin = maximum and minimum weight. The initial value w= 0.9 allows the fastest global optimal value search. W = 0.4 isoptimal for search switching from exploratory mode to exploitative mode. The search process ends when the global optimal value is defined to be the best. PSO algorithms are explained step-by-step in Table 14. Table 14. PSO algorithm’s code. Step Step-By-Step Explanation of PSO Algorithms Step 1: Initialization particle size, search space size, maximum number of iterations and constant values of the PSO included w,C1,C2and determine the random number r1,r2, find the current fitness of each particle in the population. Step 2: Assign the particles a random initial position (x)and velocity (v). Set initial counter value = 0. Initial population value , current best Fitness value of each county with its own matching value and global best position Pid of each county at their respective current position according to Formula (60). Pid =current position of ith particle (60) Step 3: The global best fitness value is calculated according to Formula (61) Global best fitness =min(local best fitness) (61) The position that meets global best fitness is the position that meets global best gid. Step 4: Update the position and velocity of the particles according to Formulas (62) and (63). Step 5: Increase the number of iterations of K=K+1 and find the current fitness of each particle. If current fitness < local best fitness, set. Local best fitness =current fitness, (62) Pid =current fitness (63) Step 6: After calculating the local best fitness of each particle, the current global best fitness of the each kth loop is calculated as follows: Current global best fitness =min(local best fitness) (64) If current global best fitness < global best fitness, then. Global best fitness =current global best fitness (65) Position that meets global best fitness value, assigned for gid. Step 7: Repeat steps 5 and 6 until k is equal to the maximum value of the loop defined in step 1 or there is no global best fitness improved. Step 8: End the algorithm loop or until no more loops are executed The flow chart of the PSO algorithmis shown in Figure 18. The PSO algorithm is applied in SAPF to adjust the gain parameter of the PI controller in order to improve the performance of SAPF in the process of selecting suitable and accurate compensating current to provide current compensation. The interference in the distribution system is generated by harmonics and reactive power compensation to improve the power factor of the power supply. The PSO method implements DC link voltage regulation to adjust the offset current. The PSO method adjusts the gain of the PI controller and calculates the parameters according to IEEE 519-2022 conventions. The PSO algorithm applied in SAPF helps the system operate with little overshoot, providing the correct amount of compensation current to compensate for the noise current, helping to minimize the sine wave of the power supply and the compensation implementation time to the power supply with the least amount of time compared to other algorithms. PSO + ANN PSO and ANN hybrid method control parameter Kp , Ki of PI controller of SAPF filter to reduce THD value in power supply meeting IEEE 519-2022 standard. The PSO performs the optimization of the supply voltage and DC voltage of the SAPF filter operating under different load conditions. Optimal data set to improve the optimization prediction with the lowest error of SAPF [130,131]. Energies 2023,16, 3998 35 of 55 Figure 18. Flow chart of PSO algorithm. PI amplification parameters are optimized by PSO and ANN (Figure 19), in which the optimal solutions are performed by the PSO algorithm after many iterations. The output of the PSO optimization serves as the input of the ANN for accurate PWM around prediction for increased minimal error tolerance. The result of this combined method is that the THD value reaches 2.22 to meet the IEEE 519-2022 standard [132]. Energies 2023,16, 3998 36 of 55 Figure 19. Schematic diagram of PSO ANN. PSO is used for dataset generation. PSO starts with a group of random variables, then finds optimal solutions according to Formula (66). Is1 Is2 Is3 =   sin(wst)cos(wst) sinwst−2π 3coswst−2π 3 sinwst+2π 3coswst+2π 3  Id∗ Iq∗(66) PSO updates the twobest values after each iteration; the first best solution is Pbest , and the second best value solution is called the global best value Gbest . The optimization process is done as follows: Pbest =Pbestk1,Pbestk2, . . . Pbestkd (67) The best global particle Gbest is defined, and the velocity of the kth particle is calculated by Formula (68). Vk=Vk1,Vk2, . . . Vkd (68) The current velocity is recalculated according to the newly calculated position and velocity. Then the distance is calculated from Pbest kd to Gbest kd using Formulas (69) and (70) . xt+1 k1m=wV(t) k1m+C1rand()Pbestk1m−xt k1m+C2rand()Pbestk1m−xt k1m(69) xt+1 k1m=xt k1m+V(t+1) k1m(70) The optimal solution is calculated so that the PSO reaches the minimum error value, and the system calculates those parameters by Formula (71). xi=   K11 pK11 iK12 pK12 i. . . K1n pK1n i K21 pK21 iK22 pK22 i. . . K2n pK2n i Km1 pKm1 iKm2 pKm2 i. . . Kmn pKmn i   (71) In the control scheme of ANN, the proposed parameters are implemented by PSO. ANN is implemented as a three-layer network, including threenodes in the input layer, 20 nodes in the hidden layer, and one node in the output layer. The optimized performance of core functions and training time is done by the hidden layers. Selected hidden layers are validated by cross-validation. Sigmoid functions are used as the hidden layers and show all Energies 2023,16, 3998 37 of 55 effects obtained from a random mapping of standard sigmoidal functional variables in the range [0, 1]. The weights of the neural network are updated by the Levenberg–Marquardt back-propagation algorithm (LMBP) [ 133 ].The output of the ANN is used for a three-phase reference current. The LMBP algorithm is a combination of Gauss–Newton and Gradient using good responses for local or global transport. The 2D recursive neural network used restricts overtraining of the whole process. The ANN network is performed according to the following steps (Table 15). Table 15. The ANN network’s code. Step Step-By-Step Explanation of ANN Network Algorithms Step 1: The training network generates a control pulse (z) with a time interval (t) input Step 2: The error target of x(1),x(2), . . . , x(n)is made using Formula (72) LMBP1 error =Z(1)NN(target)−Z(1)NN(out) LMBP2 error =Z(2)NN(target)−Z(2)NN(out) LMBPn error =Z(n)NN(target)−Z(n)NN(out) (72) Step 3: The above equation is the output of the network. Z(1)NN(out)=a1+N ∑ n=1w1n.z(1)NN(k) Z(2)NN(out)=a2+N ∑ n=1w1n.z(2)NN(k) Z(n)NN(out)=an+N ∑ n=1w1n.z(1n)NN(k) (73) where a is a function node deviation of one or two and n Step 4: The weight of each neuron is calculated using Formula (74). z(1)NN(k)=1 1+exp(−h1n.z(1)−h2n.z(2)) z(2)NN(k)=1 1+exp(−h2n.z(2)−hnn.z(n)) z(n)NN(k)=1 1+exp(−hnn.z(n)−h1n.z(1)) (74) Step 5: Weight adjustment is calculated as follows: ∆h1=Lr.z(1).LMBP1 error ∆h2=Lr.z(2).LMBP1 error ∆hn=Lr.z(n).LMBP1 error (75) Step 6: All above steps repeat until LMBP min (LMBP < 1) The desired control signal is generated from the SAPF after the ANN is successfully trained. ANN training performance was assessed using Root Mean Square Error (RMSE), coefficient of determination ( R2) and Mean Absolute Error (MAE). The Artificial Neural Adaptive Linear Neural Network (ADALINE) (ANN) acts as the reference flow selector of the PSO and ANN application system in the SAPF. Meanwhile, the PSO performs the role of the gain parameter adjustment controller in the SAPF PI controller and controls the DC voltage to select the correct compensating current for the system with a noisy power source. The PSO algorithm has strengths in accurate estimation in terms of adjusting the gain parameters of the PI controller and is superior in performance compared to traditional methods. The application system that combines the ANN algorithm and the PSO algorithm into the SAPF shows high efficiency in providing compensating current for the power supply and improving the quality of the power supply. Flower Pollination Algorithm (FPA) FPA is used to maintain a constant DC voltage by controlling the PI ratio integrator of the SAPF unit between voltage reference V∗ dc and the actual DC voltage value Vdc in order to reduce harmonics in the power system.FPA is used to select the best value of Kp , KI in the PI controller system [134–138]. Energies 2023,16, 3998 38 of 55 FPA works based on flower pollination or the process of transferring pollen from one species to another, including two main activities: self-pollination/biological and crosspollination/abiotic. The self-pollination process is the movement of pollen of the same species by wind. The cross-pollination process is the movement of pollen by honey bees, birds or bats. In fact, 90% is cross-pollination, and the remaining 10% is self-pollination. FPA performs self-pollination of flowers according to the following rules (Table 16) step-by-step to implement the FPA algorithm (Table 17). Table 16. FPA algorithm’s rule. Rule Explanation of the FPA Algorithm’s Rules Rule 1: Pollen and the best global solutions are defined by Formula (76) xk+1 i=xk i+LGbest −x−k i (76) where Gbest is the most recent best pollen with oneset of pollen. L= represents theLevy factor that is responsible for the movement of the pollen group, and this factor follows the Levy distribution and is calculated using Formula (77) L=λΓ(λ)sin(πλ 2) π1 s1+λSS0>0 (77) where Γ(λ)=standard gamma value for the biggest move SS0>0 Rule 2: The equation for local pollination or self-pollination, following Formula (78) xk+1 i=xk i+εxk m−xk i (78) where xk mand xk iis a random number in the range 0–1. Rule 3: Set the probability switch value in the range pe[0,1], make the transition from local to global search, and a p-value = 0.8 often gives the optimal result. Table 17. FPA algorithm’s code. Step Step-By-Step Explanation of FPA Algorithms Step 1: Set the initial parameters. The first step is setting the initial parameters consisting of population size (N), probability switch (p), the max number of iterations (itermax), decision variable size (d), and scaling factor (λ) Step 1: Main FPA algorithm First of all, the first decision variable is chosen randomly in the lower and upper bounds, as shown in the flowchart below. For i = 1:n; x(i)=Lbi+(Ubi−Lbi).rand(d,1); End Next, identify the fitness or error of the first population and do the following flowchart. For i = 1:n; CF(i)=PICx(i)) End Where CF = current fitness and PIC is a function that combines the Matlab and Simulink models of SAPF. Normally, CF is in 50 ×1 size Following that , find the pollen variable/ best decision variable . Pollen has aminimum fitness value of Kp,KI In the next step, pollen is updated according to rules 1 and 2, and the probability p-value is randomly selected in the range 0–1. If the random number is greater than p, then the pollen value is calculated according to Formula (79) vα vβ=q2 3"1−1 2−1 2 0√3 2√3 2#  va vb vc  (79) Energies 2023,16, 3998 39 of 55 Table 17. Cont. Step Step-By-Step Explanation of FPA Algorithms Provide by rule 1. On the other hand, if the random number is less than p, then the pollen obeys rule 2 Evaluate the fitness value after updating the pollen value according to the following equation. For I = 1:n CFUi= PIC(x.u(i)); where CFUi: updated value of fitness and x.u: updated pollen value End Updating the current global best fitness value from local best fitness is described in detail by the following equation If CFU < CF BESTP = PIC(x()i); CF = CFU End These steps are repeated until the value of the mathematical equation reaches convergence and the iteration becomes more than the maximum number of iterations initially set; then, the program is stopped. Find the Kp,KIflower pollination value achieved with the minimum error value The FPA algorithm applied in SAPF performs the function of stabilizing the DC link value in the SAPF filter to improve the efficiency of current compensation for noisy power sources. The issue of power quality improvement is important, and the FPA algorithm applied in SAPF has fulfilled the role of controlling the gain values in the PI controller to help SAPF select the correct compensating current to compensate for the current noise caused by the PI controller as a result of harmonics. The FPA algorithm takes care of the gain parameter adjustment to help minimize the error between the reference voltage and the actual DC link voltage. The FPA algorithm applied in SAPF optimizes the gain values to help the system reduce harmonic distortion with high efficiency and compensate current compensation time to reach the system setting in a short time with 0.01 s. Grey Wolf Optimization (GWO) Algorithm for SAPF GWO applied to SAPF optimizes the THD value in the power supply to meet IEEE 519-2022 standards and achieve THD = 3.815%, and the configuration applied by GWO to the SAPF is shown in detail in Figure 20. GWO is built on action inspired by the hunting behavior of gray wolves. Gray wolves have a herd behavior of 5–12 animals and organize the herd according to four levels, including Alpha (aGWO), Beta (bGWO), Delta (dGWO), and Omega (xGWO). In it, the aGWO-level gray wolf performs hunting, arranging sleep and wake times for the whole pack and the gray wolf aGWO is the leader of the pack. The bGWO-level gray wolf is the second tier in the pack that does the job of helping the aGWO-grade gray wolf make other decisions in the pack. Gray wolves of rank xGWO are the lowest level in the pack and always perform tasks under the direction of gray wolves of other ranks, namely aGWO, bGWO and Dgwo [ 139 – 144 ]. The mathematical equation of GWO in the process of tracking, encircling and attacking slugs is described by Formulas (80) and (81). → DGWO = → CGWO·→ Xp(it)−→ X(it) (80) Energies 2023,16, 3998 40 of 55 → X(it +1)= → Xp(it)−→ AGWO.→ DGWO (81) where it: Current iteration; → AGWO·→ CGWO .: Coefficient vector; → Xp : Position vector of sardines; → X : Grey wolf position vector; → DGWO : Distance between gray wolves and sardines and → CGWO = 2 ·→ r1GWO ; → AGWO = 2 ·→ aGWO·→ r2GWO −→ aGWO where → r1GWO ; → r2GWO : Random parameter with a value in the range 0–1, and these two parameters are loop variables. → aGWO : Starting from value two, this runs to zero until the end of the loop. The distance → Dα,→ → Dβ,→ Dδbetween gray wolves and sardines is determined by Formula (82).                  → Dα= → C1·→ Xα−→ X → Dβ= → C2·→ Xβ−→ X → Dδ= → C3·→ Xδ−→ X (82) where → Dα , → Dβ , → Dδ : the distance between αGWO , βGWO , δGWO gray wolves and sardines; → C1 , → C2 , → C3 : the vector coefficients of the three best positions → X1 , → X2 , → X3 ; → Xα , → Xβ , → Xδ : the first-, second-, and third-best search areas. The three best positions of gray wolves are updated according to Formula (83).          → X1=→ Xα−→ A1·→ Dα → X2=→ Xβ−→ A2·→ Dβ → X3=→ Xδ−→ A3·→ Dδ (83) Update gray wolf position in best location search area by Formula (84). → X(it+1)= → X1+→ X2+→ X3 3(84) Figure 20. Configuration of GWO in SAPF. Energies 2023,16, 3998 41 of 55 The Pseudocode of GWO used for PI controller in SAPF by the algorithm below (Algorithm 2): Algorithm 2: Pseudocode of GWO Algorithms 1 Input (Set input data of SAPF. Set initialize parameters of GWO) 2 K=1 3 Create an initial population of search agent (Xiwith i =1, 2, 3, . . . , N)with 3 dimension Kp,Ki,Gα 4 Run SAPF using Kp,Ki,Gαand evaluate the fitness function value in the search area 5 Sort the Xα,Xβ,Xγpositions in the order of first-, second-, and third-best in the search area. 6 While K≤Kmax do 7 For i = 1 to the number of search agents, do 8 Update the position → X(it+1)and update the value of Kp,Ki,Gαfollowing Equation (84) 9 Update α 10 Update → CGWO and → AGWO 11 Run SAPF using updated values of Kp , Ki , Gα and evaluate the fitness function value of the search area. 12 Update Xα,Xβ,Xγ 13 K=K+1 14 End for 15 End while 16 Return Xα(best solution) 17 Output: Print the optimum again Kp,Ki,Gαof the PI controller in SAPF in terms of Xα The GWO algorithm applied in SAPF brings many benefits, such as simple calculation because the algorithm requires few control parameters, and the algorithm is flexible and easy to optimize globally. The gain parameters of the PI controller in SAPF are optimized by the GWO algorithm to help the SAPF unit select the correct amount of offset current to compensate for the power supply. The GWO algorithm applied in SAPF shows outstanding feedback architecture and optimization of high-performance parameters. Interference in the system is responded to quickly; the SAPF unit responds to interferences highly efficiently and provides a timely compensating current to improve power quality. The GWO-SAPF system helps the power system to measure the voltage and frequency of the power supply, helping the system to control overshoot and quickly stabilize the power system, improving the quality of electricity in operation and electrical system safety. The distribution power network currently has the participation from many renewable energy power sources, and these are also considered sources of harmonics generation and also an opportunity for researchers to apply the technique to calculate GWO in SAPF into activities to improve the quality of distribution power in the future. 3.2.4. Physics-Based Algorithms Calculated in the period from 1966 to 2021, there are 21 methods to the advantage of Physics based Algorithms. However, the study authors have applied two methods to harmonics mitigation in shunt adaptive power filters, which is Gravitational Search Algorithm (GSA). This demonstrates that there is a large scaling problem for researchers using the remaining methods in SAPF in the future, including the direct implementation of each individual method and the possible implementation of a single method or a hybrid method in corporating individual methods. Gravitational Search Algorithm (GSA) for SAPF GSA applied to SAPF performs optimal compensating current selection to compensate for the loss of current on the power source and minimizes the THD value of the power supply [ 145 , 146 ] that meets IEEE 519-2022 standard (Figure 21), and the THD = 4.0% value meets the THD standard less than 5%. Energies 2023,16, 3998 48 of 55 Table 19. Summary application of meta-heuristic in SAPF. Ref. Method Results and Benefits of Applying Meta-Heuristic Optimization to SAPF Limitation or Future Research [67–70] DE Improve turning of the proportional-integral control loop of SAPF. The THD value reaches 3.42% to meet the IEEE 519-2022 standard. The meta-heuristic hybrid method is different from DE; the aim is to reduce the THD value to meet the IEEE 519-2022 standard. [71–77] GA Controller turning to obtain optimum gain values to switch SAPF and THD in the supply current present in the hardware is 1.4%, more than the simulation results of 1.24%. Control technique for the SAPF system with time-varying parametric uncertainties. [78–82] ABC To solve the nonlinear equation of selective harmonic elimination patterns considering unequal direct current sources, satisfying fundamental components, and eliminating low-order harmonics. The THD of the hardware is 11.78%, more than the simulation results of 10.46%. Propose a hybrid method that combines meta-heuristics and ABC to reduce THD and meet the IEEE 519-2022 standard. [83–87] ACO Optimize the gain values of the PI controller used in SAPF. The setting time (Ts) is 28 ms, and the THD of the supply current is 3.85%, 2.92%, and 3.49% for phase a, phase b, and phase c, respectively. Consider the proposed systems to be an efficient solution to the growing demand forpower at the present and in the future. [88–91] ALO To properly tune the circuit in order to reduce the harmonics in the source current and load voltages, the THD of the supply current with RL load is 3.73%, and the RLC load is 4.03%. The THD of the supply voltage with RL load is 4.2%, and the RLC load is 4.44%. The technique works for different load variations in the system. [92–94] BA Proportional resonant controller-based pulse width modulation. Current control for three-phase, three-leg SAPF with the optimized DC-link controller. The THD value reaches 0.7% to meet the IEEE 519-2022 standard. BA is very promising for solving other multi-objective optimization problems. [95–97] BFO To optimize the parameters of the PI controller through an online self-adaptive self-turning algorithm. The THD value reaches 1.9% to meet the IEEE 519-2022 standard. BFO-based SAPF proves to be a significant approach to reducing the ripple current harmonics. [98–106] FA Optimization problems with the objective of minimizing the THD and solving it using predator-prey-based firefly optimization. The THD is 1.9092%. The proposed method can be extended to designing hybrid active power filters in future works. [107,108] ASNS The optimization of conventional control scheme used in SAPF. THD of supply current is 1.21%, 1.14%, and 1.11% for balanced, unbalanced, and distorted loads, respectively. Compensation time (Ts) is 0.055 (s), 0.003 (s), and 0.001 (s) for balanced, unbalanced, and distorted loads, respectively. Design for all different types of HAPF. [109–112] TS The instantaneous power theory with Fourier and the optimal design of the current predictive controller. The THD of the supply current is 0.96%. The proposed novel active filter can be applied to higher-frequency systems. [113–119] WOA To control the DC − link voltage to a constant value, a PI controller is used by the gains of the controller Kp,Ki). The THD of the supply current is 3.07% A tuned PI controller can be used in hardware for real-time implementation. The proposed modern industrial optimization should be tested under various range constraints by using new techniques to handle the constraints. Energies 2023,16, 3998 49 of 55 Table 19. Cont. Ref. Method Results and Benefits of Applying Meta-Heuristic Optimization to SAPF Limitation or Future Research [120–133] PSO The selection of a proper reference compensation current extraction scheme plays the most crucial role in the performance of SAPF and includes conventional instantaneous active and reactive power (p-q),modified p-q, and instantaneous active and reactive current component (id-iq) schemes.THD of supply current is 3.45%, 2.97%, and 3.07%, based on phase a, phase b, and phase c, respectively. A hybrid method that combines other meta-heuristic methods into the search area of PSO to help limit the fast convergence error of PSO, such as DE-PSO, GA-PSO, and Levy-flight-PSO. [133–138] FPA To maintain the DC link voltage constant, the proportional-integral (PI) controller being employed on the DC side of SAPF is used to minimize the error between voltage and actual value. The THD of the supply current is 3.13%, and Tsis 0.001 s. Application of some hybrid optimization algorithm for the determination of optimal controller parameters. [139–144] GWO To reduce the maximum overshoot and undershoot of the DC-link voltage variation and minimize power ripples with current distortion in IEEE 519-2022. Improve the predictive direct power control of three-phase SAPF. The THD of the supply current is 3.8% and 57%, based on simulation and experimental data, respectively. Propose a hybrid method that combines meta-heuristics and GWO to enhance work efficiency. [145,146] GSA The harmonic content reduction in the source current is carried out with optimal turning of the PI controller. The THD of the supply current is 1.76%. Propose a hybrid method that combines meta-heuristics and GSA with the aim of maximizing work efficiency. [147–150] TLBD The reference current signals are generated by sensing the source voltage load current and DC bus voltage; with these signals, the gate driving pulses are generated by a band current controller. THD of the supply current is 1.06%. Propose a hybrid method that combines meta-heuristics and TLBO to maximize productivity. Author Contributions: Conceptualization, M.L.D. and P.B.; methodology, M.L.D.; software, M.L.D.; validation, M.L.D. and P.B.; formal analysis, M.L.D. and L.H.; investigation, M.L.D.; resources, M.L.D.; data curation, M.L.D. and L.H.; writing—original draft preparation, M.L.D.; writing—review and editing, M.L.D. and L.H.; visualization, M.L.D. and L.H.; supervision, P.B. and R.M.; project administration, P.B. and R.M.; funding acquisition, P.B. and R.M. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported in part by the Ministry of Education of the Czech Republic (Project No. SP2023/090). Data Availability Statement: Not applicable. Acknowledgments: The authors are extremely grateful to VSB Technical University of Ostrava, Czechia, for financial support. They would also like to express their gratitude to Van Lang University, Vietnam, for supporting this research. Conflicts of Interest: The authors declare no conflict of interest. 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