Hybrid Genetic Algorithm-Particle Swarm Optimization Based Approach for Optimal Sizing and Placement of a Passive Filter

Hybrid Genetic Algorithm-Particle Swarm Optimization Based Approach for Optimal Sizing and Placement of a Passive Filter | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Hybrid Genetic Algorithm-Particle Swarm Optimization Based Approach for Optimal Sizing and Placement of a Passive Filter Ahlam AbuZahew, Sanaa Salama, Ibrahim Mahariq, Muhammet Server Firat This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4447556/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This work presents a hybrid Genetic Algorithm-particle swarm optimization based technique for optimal placement and sizing of a passive power filters to reduce harmonics distortion and mitigate the total active power losses in distribution lines with linear and nonlinear loads. The problem was formulated as a nonlinear multi-objective optimization problem with equality and inequality constraints using hybrid Genetic Algorithm with practical swarm optimization. Recommendations by IEEE standards 519–1992 and 18-2002 are considered as the main constraints in this study. Based on the minimum total harmonic distortion and active power losses as the main optimization criteria the optimal solutions were selected. The proposed approach is validated on the IEEE 14 bus system and on a real power system. The results confirmed that proposed method provides robust, effective and high-quality solutions for large power systems. Passive filter harmonic mitigation Genetic algorithm Particle swarm optimization optimal harmonic power flow Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 1. Introduction The expansion of harmonic currents in electrical systems had risen recently due to various factors, including solid-state power conversion devices, traction drives, industrial variable speed drive systems, A/D converters, and increased utilization of arc welders. Additionally, important sources of harmonic currents in power systems include electrical equipment like fluorescent lamps, large induction motors, near-saturated power transformers, and computer system installations. Presently, nonlinear devices constitute a growing portion of the electrical load in industrial and commercial power systems. When a sinusoidal supply feeds a combination of linear and nonlinear loads, the total supply current will inevitably contain harmonics. The introduction of harmonic currents and the resulting harmonic voltages can lead to power quality issues, impacting the performance of consumers connected to the electric power network, [ 1 ]. Nonlinear devices produce harmonics in line currents, leading to increased conductor heating. This elevated temperature can exceed the rated capacity of cables not designed for harmonic currents, causing potential problems. The effects of harmonics in power distribution systems include aging, reduced capacity of components, malfunctioning protection and measurement devices, a lower power factor, and a consequent decrease in power system efficiency due to increased losses. To mitigate these issues, industrial standards and recommended practices for power systems have been established. However, power systems must still endure harmonic flow as complete elimination is not possible. Consequently, as non-linear loads in power systems continue to grow, there is a growing need for more effective methods to counteract the harmonic effects both among sources and on the power system itself. Extensive efforts have been made in analyzing, simulating, and developing passive and active filters, hardware for identification, classification, and mitigation of power system harmonics. Consequently, there is a growing interest for the application of passive and active filters to mitigate the harmonic effects generated by these loads. Proper consideration of the sizing and placement of passive filters is crucial to prevent the amplification of harmonic currents and voltages, which can occur due to potential resonance at specific harmonic frequencies. Various research studies, such as [ 1 – 5 ], utilize genetic algorithms to determine the optimal size and location of shunt capacitors in radial distribution systems, taking into account harmonic distortion limits imposed by nonlinear power electronic devices. In [ 6 ], the non-dominated sorting genetic algorithm (NSGA-II) is employed to identify the optimal placement of passive filters. Another innovative approach proposed in [ 7 ] involves the use of fuzzy dynamic programming to decide the optimal size and location of compensation shunt capacitors for distribution systems with harmonic distortion. A method for determining optimal or near-optimal locations and sizes of single-tuned passive harmonic filters within existing capacitor buses in a power system is introduced in [ 8 ]. Additionally, [ 9 ] presents a novel method towards passive filter placement, and [ 10 ] utilizes a whale optimization algorithm for filter placement. The methodology of the present study is based on harmonic flow simulation in the Alternative Transient Program (ATP/EMTP) and employs the NSGA-II optimization algorithm simulated on MATLAB® to identify the optimal placement of passive filters. Although the method presented in [ 11 ] accounts for both harmonic current sources from nonlinear loads and the presence of background voltage harmonics, the objective functions are notably complex. This complexity, particularly at lower levels, leads to challenges in filtering due to conflicting requirements. In [ 12 ], a C-type filter is used to reduce harmonic distortion, improve system performance and compensate reactive power. In order to improve the load power factor, also considering economic factors. In [ 13 ], a method for designing an L compensator in time-variant, non-sinusoidal systems, is proposed. It examines the realistic representation of implementing the equivalent system impedances, source voltage harmonics, and the equivalent load impedance, the proposed method shows a high level of complexity. The work in [ 14 ] introduces a method that combines sequential neural-network approximation with orthogonal arrays (SNAOA) to reduce harmonic distortion using passive harmonic filters. While the method yields satisfactory results, it requires a substantial amount of real data to effectively train the neural network. In [ 15 ], the proposed technique aimed to allocate and sizing single tuned passive filters in power distribution systems. The objective is to minimize total harmonic distortion. The method employs the standard genetic algorithm (GA) without any modifications. In the time where most reviewed research emphasizes using a single optimization technique with one or two objective functions, this often doesn't result in finding the global minimum or optimal solution due to the common problem of getting stuck in local minima. The proposed method suggests a hybrid approach. It involves selecting a wide-ranging population for one optimization technique while keeping the population relatively close and locally separated for the other optimization technique. The proposed method can continually search for a global minimum while also exploring local minima solutions. This work aims to introduce a novel approach to passive filter sizing and allocation, utilizing optimization techniques to enhance the filter's impact on reducing harmonics distortion and active power losses. As the filter comprises multiple branches in shunt, the distribution of reactive power among these branches will be optimized. The proposed passive filter design incorporates a hybrid GA-PSO algorithm. The choice of this algorithm, among others, is based on its combination of the unique individual generation functions of both GA and PSO, mimicking social behaviors such as animal breeding and survival of the fittest. This integration enhances the overall efficiency of the algorithm's results. While the PSO algorithm excels in efficient local searches, the GA serves as a potent tool for global searches, albeit with less power in local search. Hence, a hybrid algorithm is employed to capitalize on the strengths of both approaches. 1.1 Passive Filters Passive filters come in two primary configurations: Shunt passive filters and Series passive filters. They are employed to either divert harmonic currents away from the line or impede their transmission between different parts of the system by tuning the elements to resonate at specific harmonic frequencies. Figure 1 depicts various common filter configurations. Comprising inductance, capacitance, and resistance elements, passive filters typically utilize a single-tuned "notch" filter, which is both commonly used and cost-effective for many applications. Besides mitigating harmonics, notch filters also contribute to Power Factor (PF) correction [ 2 ]. The application of a series filter is constrained when it comes to blocking multiple harmonic currents since each harmonic requires a dedicated series filter tuned to that specific harmonic. This setup can lead to considerable losses at the fundamental frequency. Therefore, in centralized filter design, the use of shunt filters is more suitable than series filters. This preference arises from the expected outcome of multiple harmonics due to the diverse nature of nonlinear elements in the system. Shunt passive filters rely on three key design parameters: the tuning harmonic orders (h), the quality factor ( \(\text{Q})\) , and the impedance of the filter at the fundamental frequency. The tuning of harmonic orders is contingent upon the nonlinear behavior of the system. The quality factor, denoting the ratio of reactance to resistance at the fundamental frequency, typically ranges between 50 and 150 for a single-tuned filter, [ 5 ]. The filter reactance, \({X}_{L}\) , and resistance, R , can be calculated by \({X}_{L} = \frac{{X}_{c}}{{h}^{2}}\) and \(R = \frac{{{h}_{ }X}_{L}}{\text{Q}}\) , respectively. Where, \({X}_{C}\) is the filter capacitance, h is frequency order, and Q is the filter quality factor. The fundamental steps in the design process of passive filters can be condensed into a few key stages. Initially, the tuned frequency for the filter is chosen. Once this frequency is established, the capacitor bank size and resonant frequency can be calculated. The maximum value for the filter reactive power Q f is determined by the total fundamental reactive power. The filter reactance X f , representing the difference between the capacitive reactance, X C , and the inductive reactance, X L , can be determined by \({X}_{f} = {X}_{C} - {X}_{L} = V /{Q}_{f}.\) Where V is the line to line voltage. So, the filter capacitive reactance, capacitor and inductor values can be calculated by, [ 9 ], \({X}_{C}= {h}^{2}{X}_{L}= \frac{{X}_{f} {h}^{2}}{{h}^{2}-1}\) , C \(= \frac{1}{{X}_{c}{\omega }_{\text{o}}}\) , and L \(= \frac{{X}_{L}}{{\omega }_{o}}\) . 1.2 Objective Function The objective function selected to be minimized for this optimization problem is the branches active power losses (P loss ) and the total harmonic distortion (THD) that can be calculated according to the following: $${P}_{loss}= \sum _{h=1}^{max}\sum _{i=1}^{N-1}{{{P}_{loss(i,i+1)}}^{h}}_{ }^{ }$$ 1 $${{P}_{loss(i,i+1)}}^{h}=\left({{R}_{i,i+1}}^{h}\right({\left({{V}_{i}}^{h}- {{V}_{i+1}}^{h} \right)\text{*}{{ Y}_{i,i+1}}^{h})}^{2})$$ 2 $$THD= \frac{\surd \left(\sum _{h=2}^{max}{{{|V}_{i }}^{h}|}^{2}\right)}{\left|{{V}_{i}}^{1}\right|}$$ 3 These two optimization objectives were employed in the optimization problem. THD served as the primary objective function in the section executed by PSO. Conversely, THD was transformed into a constraint in the section where GA was employed, with the minimum P loss being the objective function. Throughout this problem, the aim was to determine the optimal configuration of certain control variables (e.g., single-tuned filter location, size, and tuned frequency) while adhering to specified equality and inequality constraints. The Newton-Raphson method was utilized in this optimization problem to compute the total fundamental reactive power and to uphold the equality constraints associated with the fundamental active and reactive power at bus i within its specified tolerance. At harmonic frequencies, the power system was reconfigured, taking into account the passive filter, the nonlinear load, and the frequency effect at all the buses, as follows: $${Y}_{l,i}^{h}= \frac{{P}_{l,i}}{{\left|{{V}_{i}}^{1}\right|}^{2}}-j\frac{{\text{Q}}_{l,i}}{{h\left|{{V}_{i}}^{1}\right|}^{2}}$$ 4 $${Y}_{c,i}^{h}=h{Y}_{c,i}^{1}$$ 5 $${Y}_{f,i}^{h}= \frac{1}{{{{Z}_{f,i}}^{h}}^{ }}$$ 6 $${Y}_{i,i+1}^{h}=\frac{1}{{R}_{i,i+1}+jh{X}_{i,i+1}}$$ 7 The impedance for single tuned filter is $${Z}_{f}^{h}=R+j\omega L+ \frac{1}{j\omega C}$$ 8 Efficient simulation is essential for large systems, prompting the recommendation to employ decoupled harmonics power flow for system parameter calculations. In the case of the i th bus, where P nl,i and Q nl,i denote the nonlinear active and reactive power at the fundamental frequency, the nonlinear current can be computed using the provided equations. $${I}_{nl,i}^{1}= {\left[\frac{{P}_{nl,i}+j{\text{Q}}_{i,nl}}{{V}_{i}^{1}}\right]}^{\text{*}}$$ 9 $${I}_{nl,i}^{h}=c\left(h\right){I}_{nl,i}^{1}$$ 10 1.3 Constraints The fundamental active and reactive power at bus (i) is considered as constraints during this optimization method and it can be calculated by: $${P}_{G,i}-{P}_{l,i}=\left|{V}_{i}^{\left(1\right)}\right|\sum _{j=1}^{N}\left|{V}_{i}^{\left(1\right)}\right|\left|{Y}_{i,j}^{\left(1\right)}\right|\text{cos}\left({{\varnothing}}_{i,j}^{\left(1\right)}- {\delta }_{i}^{\left(1\right)}+{\delta }_{j}^{\left(1\right)}\right)$$ 11 $${Q}_{G,i}-{Q}_{l,i}=\left|{V}_{i}^{\left(1\right)}\right|\sum _{j=1}^{N}\left|{V}_{i}^{\left(1\right)}\right|\left|{Y}_{i,j}^{\left(1\right)}\right|sin({{\varnothing}}_{i,j}^{\left(1\right)}- {\delta }_{i}^{\left(1\right)}+{ \delta }_{j}^{\left(1\right)})$$ 12 Where: i = 1…N is the bus number, \({P}_{G,i},{P}_{l,i}{,Q}_{G,i}and{Q}_{l,i}\) are the generated and load active and reactive power respectively. \({Y}_{i,j}^{\left(1\right)}\) is the fundamental admittance between bus i and j \({ {\varnothing}}_{i,j}^{\left(1\right)}, {\delta }_{i}^{\left(1\right)}and {\delta }_{j}^{\left(1\right)}\) are the admittance phase angle and the voltage angle at bus i and j at the fundamental frequency. 1.4 Optimization Techniques This study proposes a combination of two global optimization algorithms, GA and PSO, to solve nonlinear optimization problems. The rationale behind this lies in both algorithms working with an initial population of solutions, making the integration of their search capabilities a sensible strategy. The subsequent sections outline the steps and flowchart of the GA, PSO and the hybrid GA-PSO optimization techniques. 1.4.1 Particle Swarm Optimization Swarm behavior can be replicated through a set of straightforward rules, allowing for the modeling of schools of fish and flocks of birds. Even with individual agents following simple behavior rules, the collective behavior of the entire swarm can become intricate. Agents make decisions based on both their own experiences and the experiences of others, contributing to the fundamental elements that underlie PSO. Each agent is aware of its personal best value (p best ) and its corresponding x, y position, akin to individual experiences. Additionally, each agent is informed about the best value within the group (g best ) among the p best , resembling knowledge about how neighboring agents have performed. Agents seek to modify their positions based on its current position, velocity, p best and g best . This modification is conceptually represented by the notion of velocity, where the velocity of each agent can be adjusted using a specific equation. $${V}_{i}^{k+1}={wv}_{i}^{k}+{c}_{1}{rand}_{1}*\left({{g}_{best}}_{i}- {S}_{i}^{k}\right)+{c}_{2}{rand}_{2}*({{g}_{best}}_{ }- {S}_{i}^{k})$$ 13 where, \({V}_{i}^{K}\) is velocity of agent i at iteration k, \(w\) is weighting function, \({c}_{j}\) is weighting coefficients, rand is a random number between 0 and 1, \({S}_{i}^{K}\) is current position of agent i at iteration k, p besti is p best of agent i, and g best is g best of the group. Namely, velocity of an agent can be changed using three vectors. The velocity is usually limited to a certain maximum value [ 16 – 23 ]. The following weighting function is usually utilized: $$w= {w}_{max}- \frac{{w}_{max}- {w}_{min}}{{iter}_{max}}\text{*}iter$$ 14 where, w max is initial weight, w min is the final weight, iter max is maximum iteration number, and iter is the current iteration number. The three velocity vectors convey the following implications: The initial term represents the agent's previous velocity. The second and third terms are employed to adjust the agent's velocity. Without these latter terms, the agent would persist in moving in the same direction until reaching a boundary. In essence, this implies an exploration of new areas, making the first term synonymous with diversification in the search process. The current position (searching point in the solution space) can be modified by the following equation: $${S}_{i}^{K+1}= {{S}_{i}^{k}}_{ }+ {V}_{i}^{k+1}$$ 15 The general process flow of PSO can be outlined as follows: Step 1 Initialization of each agent's initial conditions. Initial search points (s i 0 ) and velocities (v i 0 ) for each agent are typically randomly generated within the allowable range. The current search point is set as pbest for each agent. The best-evaluated value of pbest becomes gbest, and the corresponding agent number is stored. Step 2 Evaluation of each agent's search point. The objective function value is computed for each agent. If the value surpasses the current pbest of the agent, the pbest value is updated with the current value. If the best value of pbest is superior to the current gbest, gbest is updated with the best value, and the corresponding agent number is stored. Step 3 Modification of each search point. Step 4 Checking the exit condition. If the current iteration number reaches the predetermined maximum iteration number, the process exits; otherwise, it returns to Step 2. Figure 2 provides a visual representation of the overall flowchart of PSO [ 21 , 22 ]. Based on, [ 21 – 25 ], the following parameters were deemed suitable, and their values are found to be independent of specific problem characteristics: ci = 2:0 w max =0,9 w min =0,4 The values are also proved appropriate for power system problems. 1.4.2 Genetic Algorithm Unlike traditional optimization techniques that start with a single candidate and iteratively seek the optimal solution using static heuristics, the Genetic Algorithm (GA) approach employs a population of candidates to simultaneously and adaptively explore various areas within a solution space, [ 3 , 4 ]. While some heuristic search methods rely on local search, such as hill climbing, others opt for a nonconvex optimization approach that accepts cost-deteriorating neighbors. GA stands out as one of the most popular methods extending beyond simple local search. Genetic algorithms operate on a population of individuals, where each individual represents a potential solution. Following the generation of an initial population, either randomly or heuristically, the algorithm evolves the population through sequential and iterative application of three operators, culminating in the formation of a new generation at the end of each iteration, [ 4 ]: Selection. Crossover. Mutation. 2. Material and Method In the realm of heuristic optimization, a robust method for tackling complex nonlinear equations involves the combination of particle swarm optimization (PSO) and genetic algorithms (GA). This hybrid approach, known as GA-PSO, synergizes the strengths of PSO and GA by incorporating PSO mechanisms for generating individuals in a new generation, alongside the traditional crossover and mutation procedures observed in GA, [ 8 ]. In the following section the proposed hybrid GA-PSO optimization technique will be defined in steps as well as in a flowchart shown in Fig. 3 . 2.1 GA-PSO Optimization Technique Steps: Step 1 : Input line data at the fundamental harmonic (i.e: branches impedance). Input the fundamental active ve reactive bus power bus voltages amplitude and phase bus code ( slack, load or generating bus). Insert Sbase, maximum number of iteration = 1000 and the accuracy = 0.001. Generate Ybus, Jacobian Matrix. Use Newton Raphson method to calculate the power solution at the fundamental frequency. Calculate active ve reactive power at the slack bus for the fundamental frequency. Calculate the reactive power required by the filter that is the total fundamental reactive power. Step 2 Input harmonic current spectrum and its frequency order. Enter active and reactive power for the non linear load and its position. Calculate I1 for the non linear load. Calculate Ih using C(h) and I1 for the non linear load. Calculate Ih and Vh at all buses. Step 3 1. Give a first assumption for the filter size and tuned frequency. Step 4 Input PSO parameters (c i =2:0 w max =0,9 w min =0,4). Define objective functions (minimum P loss and THD). Define constrains (fundamental power equation). Generate initial filter position and tuned harmonic order for each agents. Evaluate the searching point for each agents. Modification for each searching agents. Maximum iteration number reached. Step 5 Input GA parameters. Use the PSO output and generate a random value for the filter size and set the frequency order for each chromosome in the population. Calculate the power equations in at the fundamental frequency then recalculate V h Evaluate the objective functions of minimum active power losses with the constraints of not exceeding THD limits for each chromosome in the population. Maximum Number of Iterations. If not, selection, crossover and mutation. New Population. Print filter size, position and set frequency. 2.2 The Hybrid GA‑PSO Test on the IEEE 14 Bus System and on a Real Electrical Network To explore the optimal placement and dimensions of a passive filter within a power system, aiming to reduce total harmonic distortion to the IEEE standard-defined legal limits, minimize active power losses in branches, and improve the system power factor using a hybrid optimization technique integrating PSO and GA; two tests had been done. In the following sections the details of the tested systems were identified. 2.2.1 The IEEE 14 Bus System The 14-bus IEEE standard system was selected as the basis for the optimization study. The optimization task involved addressing the passive filter design and allocation problem. The GA-PSO hybrid optimization technique previously defined was applied to assess solutions for the IEEE 14-bus system. This undertaking aimed to evaluate the effectiveness, robustness, and quality of the proposed technique. The IEEE 14-bus system comprises four generation buses, one slack bus, and nine load buses, all operating at the same voltage level (1kV). The characteristics of each bus, including its type, rated voltage amplitude and angle, active and reactive power for loads and generators, as well as reactive power limits, are detailed in Table 1 . The single line diagram for the IEEE 14 bus is represented using ETAP program as shown in Fig. 4 . In the assumption that the system is initially free from harmonics, a nonlinear load was introduced to serve as the harmonic source in the system. Specifically, a nonlinear load in the form of an IEEE typical 6-pulse rectifier was added to bus number 10. This load contributes 18.75 MW and 16.53 Mvar of active and reactive power, respectively. The harmonic spectrum associated with this load is depicted in Fig. 5 . Table 1 Generation Bus Information for IEEE 14 Bus System. Generation Bus Voltage Generation Mvar Limits ID kV Type % Mag. Angle MW Mvar Max Min Bus1 1.000 Voltage Control 109.0 0.0 0.000 1.000 -1.000 Bus 1 H_1 1.000 Swing 106.0 0.0 Bus 2 H_2 1.000 Voltage Control 104.5 0.0 29.145 29.145 -29.145 Bus 3 H_3 1.000 Voltage Control 101.0 0.0 0.000 23.400 0.000 Bus 6 L_6 1.000 Voltage Control 107.0 0.0 0.000 12.200 -6.000 The harmonics orders are [5 7 13 15 17 19 23 25 29 32 35 37 42 45 47 49]. The harmonic current can be determined using Eq. 10 where C(h) represents a fraction of the fundamental current components or it is defined based on real values from its power spectrum. The coefficients C(h) for different harmonic orders (h) are specified as follows: C(h)=[ 0.2 0.1429 0.0769 0.0667 0.0588 0.0526 0.0435 0.04 0.0345 0.0313 0.0286 0.0270 0.0238 0.0222 0.0213 0.0204] In this work, optimization algorithms were employed to determine the optimal locations and sizes for one and two single-tuned filter. The results obtained were then compared with those obtained without any filters. Prior to the initiation of the optimization algorithm, the total reactive power for the filters was calculated using the Newton-Raphson power flow analysis method, resulting in a value of 73.299 Mvar, representing the total fundamental reactive power. The THD at each bus of the system before the filter installation is illustrated in Fig. 6 . It is evident that the THD has surpassed IEEE limits at buses 10 and 9, and that the THD at buses 1 and 14 is very close to the IEEE limit. Bus 9 exhibits the highest THD value at 5.61%. 2.2.2 The Second Case Study a Real Electrical Network that is Tubas District Electricity Company: The Tubas District Electricity Company (TDECO), a publicly limited company, was established on 2006 with a capital of $ 3,491,320. It operates in Tubas city, a small governorate covering an area of 402 km 2 in the Middle East of Palestine. In 2002, the company commenced its operations by providing electricity to its initial 23 member shareholders, comprising municipal and village councils. Over the years, the company expanded its services to include electricity departments in municipalities and local authorities, serving a total of 15,000 customers by 2011. With the goal of supplying electricity to 20,000 customers with a daily consumption rate of 25 mega-volt amperes, the company relies on two connection points receiving electricity from the Israeli company. The primary feeding point is located in Tayasir, delivering more than 20 MW, while the secondary point is in Al-Jalamah. The electrical network in Tubas comprises 674 buses, 267 transformers, 408 cables and transmission wires, including 33 underground cables. To visualize and analyze this network, a single-line diagram was created using ETAP software, capturing the network's structure and components. The single-line diagram is illustrated in Fig. 7 . For the TUBAS network, the total reactive power used to determine the filter sizes was 6 Mvar, calculated through the Newton-Raphson power flow analysis method before initiating the optimization algorithm. This value represents the total fundamental reactive power. To assess the impact of passive filters on the network's behavior, a nonlinear load was introduced to one of the buses. Given the absence of specific harmonic distortion data, except for its exceedance of legal limits in multiple buses and the main feeding point, a nonlinear load was added to the bus with the maximum PV system, considering photovoltaic systems as potential sources of harmonics. At Bus 5, where the largest PV system with 5 MWp is installed (the biggest PV system in the network exceeding 19 MWp), a nonlinear load in the form of an IEEE typical 6-pulse rectifier was added. This load contributes 3.3 MW and 3.7 Mvar of active and reactive power, respectively. The Total Harmonic Distortion (THD) at each bus of the system before installing the filters is illustrated in Fig. 8 . It is evident that the THD has exceeded IEEE limits at buses 459, 550, 565, and 717, and that the THD at buses 163, 164, 168, and 716 is very close to the IEEE limit. Bus 549 exhibits the highest THD value at 6.34%. 3. Results and Discussion 3.1 IEEE 14 Bus System Test Results The effect of installating a single passive filter with a capacity of 21.6 Mvar at bus number 9, on the 14 bus IEEE system, could be notice over the THD as shown in Fig. 9 as well as over the power factor improvment and active power loss reduction. The maximum THD has seen a reduction of 27.6%, decreasing from 5.61–4.06%. Additionally, the total active power losses at branches have experienced a 2.4% reduction, decreasing from 46.3 MW to 45.2 MW. It is noteworthy that these reductions could further enhance if more than filter is installed. The results of installing one and two single tunde filter is summrized in Table 2 . Table 2 IEEE 14 Bus System Original Case and after Installing One and Two Single Tuned Passive Filters. Objective Function values with the Design parameters Original case (without Filters) One filter solution Two filter solution Filter 1 Filter 2 Filter size Mvar - 21.6 37 32 Active power losses MW 46.3 45.2 42.6 THD% 5.61 4.06 4.4 Filter Location - 9 10 13 Tuned Frequency - 7.04 7.12 7.14 Filter Inductance (mH) - 2.9 3.96 4.56 Filter capacitance ( \(\mu F)\) - 65529 47509 41089 Filter Resistance (mΩ) - 97.4 134 113.5 Q Factor 66 66 90 Power factor at the main connection point 96.4 97.6 99.06 3.2 The Second Case Study Tubas District Electricity Company Test Results The Total Harmonic Distortion (THD) after installing a single passive filter with a capacity of 5 Mvar at bus number 165 in the TUBAS network is illustrated in Fig. 10 . Notably, the maximum THD has undergone a reduction of 48% decreasing from 6.34–3.0%. Additionally, the active power losses at branches have experienced a 8% reduction, decreasing from 1.63 MW to 1.4 MW. Moreover, the power factor at the main connection point has increased by 11% rising from 61.7 to 68.5. The impact of installing two passive filters, with capacities of 1.62 Mvar and 2.8 Mvar, at bus numbers 163 and 717 in the TUBAS network is illustrated in Fig. 11 . The results indicate a significant improvement in system performance: The maximum THD has been reduced to 4.4%, representing a reduction of 30.6% compared to the original THD of 6.34%. Active power losses at branches have been reduced by 5.5%, decreasing from 1.63 MW to 1.54 MW. The power factor at the main connection point has increased by 14.3%, rising from 61.7 to 72%. These improvements highlight the effectiveness of strategically placing passive filters in the network to mitigate harmonic distortions, decrease active power losses, and enhance power factor. The result of installing one and two single tuned passive filters is summarized in Table 3 . Table 3 TDECO Original Case and after Installing One and Two Single Tuned Passive Filters. Objective Function values with the Design parameters Original case (without Filters) One filter solution Two filter solution Filter 1 Filter 2 Filter size Mvar - 5 1.62 2.8 Active power losses MW 1.63 1.4 1.54 THD% 6.34 3.0 4.4 Filter Location - 165 163 717 Tuned Frequency - 5.125 5.14 5.26 Filter Inductance (mH) - 2.9 7.1 µ 0.016 Filter capacitance ( \(\mu F)\) - 65529 53978 23077 Filter Resistance (mΩ) - 97.4 0.124 0.26 Q Factor 66 99 102 Power factor at the main connection point 61.7 68.5 72 GA and PSO optimization methods were used to select the position of two single tuned passive filter for this system. Using PSO decreasing the THD by 30.6% and reducing the active power losses by 4.3% while enhancing the power factor by 17%, while using GA decreasing the THD by 27.4% and reducing the active power losses by 8.6% while enhancing the power factor by 17.7% as summarized in Table 4 . In one hand, the heuristic optimization methods used on this research shows satisfied results, on the other hand the hybrid GA-PSO methods shows a robust effective and high-quality solutions specially for large systems. Table 4 TDECO Original Case and after Installing Two Single Tuned Passive Filters Using PSO and GA. Objective Function values with the Design parameters Original case (without Filters) PSO Filter 1 Filter 2 GA Filter 1 Filter 2 Filter size Mvar - 1.6 2.4 1.2 2.4 Active power losses MW 1.63 1.56 1.49 THD% 6.34 4.41 4.6 Filter Location - 163 722 163 716 Tuned Frequency - 5.13 5.29 5.39 5.26 Q Factor 100 99 99 101 Power factor at the main connection point 61.7 72.29 72.6 4. Conclusion The results achieved through the hybrid GA-PSO approach align with the predefined objective functions, focusing on minimizing active power losses, and comply with the IEEE standard for Total Harmonic Distortion limits, serving as specified constraints met within this proposed methodology. The results shows that the THD could be reduced to a proximately 50% of its initial value. The reduction on the active power loss had a lower reduction percentage and that’s due to the large number of buses in the time where the algorithm had restricted by allocating two filters as a max. Comparative analysis between the outcomes of the hybrid GA-PSO approach and those of GA or PSO reveals that the hybrid GA-PSO method offers efficient, and high-quality solutions for extensive systems, whereas PSO demonstrates exceptional solutions for smaller systems. The accuracy and efficiency of the hybrid GA-PSO approach are successfully validated through the utilization of ETAP and MATLAB programs, illustrated effectively by both the standard IEEE 14 and the real system of Tubas Electric Company. Irrespective of the objective function's structure, the suggested hybrid GA-PSO approach can be easily and promptly applied to any other distribution system featuring linear and/or nonlinear loads. Declarations Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this paper. Funding Statement The authors declare that there was no funding for this work. References T.S. Chung, H.C. Leung, "A genetic algorithm approach in optimal capacitor selection with harmonic distortion considerations ", International Journal of Electrical Power & Energy Systems , vol. 21, no. 8, pp. 561–569, 1999. https://doi.org/10.1016/S0142-0615(99)00025-3 . M. Ghiasi, V. Rashtchi, and S.H. Hoseini, "Optimum location and sizing of passive filters in distribution networks using genetic algorithm", 4th International Conference on Emerging Technologies , Rawalpindi, Pakistan, Oct. 2008. DOI: 10.1109/ICET.2008.4777493 . M. Milovanović, J. Radosavljević, D. Klimenta, P. Perović, “GA–based approach for optimal placement and sizing of passive power filters to reduce harmonics in distorted radial distribution systems”. Electrical Engineering 101,787–803, 2019. DOI: 10.1007/s00202-019-00805-w . A. Berizzi, C.Bovo, "The use of genetic algorithms for the localization and the sizing of passive filters", Ninth International Conference on Harmonics and Quality of Power. Proceedings (Cat. No.00EX441), Orlando, FL, USA, Oct. 2000. DOI: 10.1109/ICHQP.2000.896992 . Y.Ming Chen," Passive Filter Design Using Genetic Algorithms", IEEE Transactions on Industrial Electronics, vol. 50, no.1, pp. 202–207, 2003. DOI: 10.1109/TIE.2002.807664 . M. Pomalis C.S.a, R. Chouhy Leborgnea, A. Ricardo Herrera-Orozcob, A. Suman Bretasc," NSGAII optimization for single phase passive filter allocation in distribution systems", Electric Power Systems Research, vol. 176, 2019. https://doi.org/10.1016/j.epsr.2019.105923 . H. Chan Chin, "Optimal shunt capacitor allocation by fuzzy dynamic programming", Electric Power Systems Research, vol. 35, no. 2, pp. 133–139, 1995. https://doi.org/10.1016/0378-7796(95)00999-X . G. W. Chang, H. Wang, and S. Yung Chu," Strategic Placement and Sizing of Passive Filters in a Power System for Controlling Voltage Distortion", IEEE Transactions on Power Delivery, vol. 19, no. 3, pp. 1204–1211, 2004. DOI: 10.1109/TPWRD.2003.822954 . F.N. Belchior, et. al., " A Novel Approach towards Passive Filter Placement", 2015 IEEE Power & Energy Society General Meeting , Denver, CO, USA, July 2015. DOI: 10.1109/PESGM.2015.7286399 . A. Rosyadi, O. Penangsang, A. Soeprijanto "Optimal Filter Placement and Sizing in Radial Distribution System Using Whale Optimization Algorithm", 2017 International Seminar on Intelligent Technology and Its Applications (ISITIA) . Surabaya, Indonesia, August 2017. DOI: 10.1109/ISITIA.2017.8124060 . A. Menti, T. Zacharias, J. Milias-Argitis “Optimal sizing and limitations of passive filters in the presence of background harmonic distortion”, Electrical Engineering, 91(2), pp.89–100, 2009. DOI 10.1007/s00202-009-0120-3 . I. F. Mohamed, S. H. E. Abdel Aleem, A. M. Ibrahim & A. F. Zobaa. “Optimal Sizing of C -Type Passive Filters under Non-Sinusoidal Conditions”, Energy Technology & Policy, vol. 1, PP. 35–44, 2014. DOI: 10.1080/23317000.2014.969453 A. F. Zobaa, A. Vaccaro, H. H. Zeineldin, A. Lecci,, and A. M. Abdel Monem, “Sizing of Passive Filters in Time-Varying on sinusoidal Environments,” Proceedings of 14th International Conference on Harmonics and Quality of Power - ICHQP , Bergamo, Italy Sep. 2010. DOI: 10.1109/ICHQP.2010.5625346 . C. Low, Y. Chang, S. Hung.” An application of sequential neural-network approximation for sitting and sizing passive harmonic filters”, Expert Systems with Applications , vol. 36, no. 2, part 2, pp. 2910–2920, 2009. https://doi.org/10.1016/j.eswa.2008.01.004 . I. D. Meloa, J. L.R. Pereiraa, A. M. Variza, P. F. Ribeirob,” Allocation and sizing of single tuned passive filters in three-phase distribution systems for power quality improvement,” Electric Power Systems Research, vol.180, 2020. https://doi.org/10.1016/j.epsr.2019.106128 . H. Yoshida, K. Kawata, Y. Fukuyama, S. Takayama, and Y. Nakanishi, “A particle swarm optimization for reactive power and voltage control considering voltage security assessment,” IEEE Transactions on Power Systems, vol. 15, no. 4, pp. 1232–1239, 2000. DOI: 10.1109/59.898095 . Y. Fukuyama and H. Yoshida, “A particle swarm optimization for reactive power and voltage control in electric power systems,” Proceedings of the 2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546), Seoul, Korea, May 2001. DOI: 10.1109/CEC.2001.934375 . A. H. Mantawy and M. S. Al-Ghamdi, “A new reactive power optimization algorithm,” 2003 IEEE Bologna Power Tech Conference Proceedings , Bologna, Italy, June 2003. DOI: 10.1109/PTC.2003.1304768 . B. Zhao, C. X. Guo, and Y. J. Cao, “A multiagent-based particle swarm optimization approach for optimal reactive power dispatch, ” IEEE Transactions on Power Systems , vol. 20, no. 2, pp. 1070–1078, 2005. DOI: 10.1109/TPWRS.2005.846064 . A. A. A. Esmin, G. Lambert-Torres, and A. C. Zambroni de Souza, “A hybrid particle swarm optimization applied to loss power minimization,” IEEE Transactions on Power Systems, vol. 20, no. 2, pp. 859–866, 2005. DOI: 10.1109/TPWRS.2005.846049 . J. Chuanwen and E. Bompard, “A hybrid method of chaotic particle swarm optimization and linear interior for reactive power optimisation,” Mathematics and Computers in Simulation , vol. 68, no. 1, pp. 57–65, 2005. https://doi.org/10.1016/j.matcom.2004.10.003 . W. Zhang; Y. Liu, “Reactive power optimization based on PSO in a practical power system,” IEEE Power Engineering Society General Meeting , Denver, CO, USA, June 2004. DOI: 10.1109/PES.2004.1372792 . G. Coath, M. Al-Dabbagh, and S. K. Halgamuge, “Particle swarm optimisation for reactive power and voltage control with grid-integrated wind farms,” IEEE Power Engineering Society General Meeting , Denver, CO, USA, June 2004. DOI: 10.1109/PES.2004.1372803 . J. Cabral Leite, I. Pérez Abril · M. Emilia de Lima, R. Celio Limão de Oliveira, “Multi-objective optimization of passive filters in industrial power systems", Electrical Engineering , vol. 99, pp.387–395, 2017. Doi: 10.1007/s00202-016-0420-3 . V. Miranda and N. Fonseca, “EPSO-evolutionary particle swarm optimization, a new algorithm with applications in power systems,” IEEE/PES Transmission and Distribution Conference and Exhibition , Yokohama, Japan, Oct. 2002. DOI: 10.1109/TDC.2002.1177567 . Additional Declarations No competing interests reported. Supplementary Files HybridGAPSOCircuitTheoryJournalV2.pdf Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4447556","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":305785549,"identity":"9ff049c1-65d1-407e-bcd9-99ffd0b10e2d","order_by":0,"name":"Ahlam 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19:45:16","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":85280,"visible":true,"origin":"","legend":"\u003cp\u003ePSO Searching Flowchart\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-4447556/v1/768fb12cc24a7b29e49148ad.png"},{"id":58232341,"identity":"dad7b544-5b4a-451c-b10b-e243ee5e4944","added_by":"auto","created_at":"2024-06-12 19:45:16","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":331745,"visible":true,"origin":"","legend":"\u003cp\u003eHybrid GA-PSO Flowchart\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4447556/v1/274f7e8528aaa4682a79ff86.png"},{"id":58232166,"identity":"ab7fc643-9310-4a92-9240-bb258b638a6c","added_by":"auto","created_at":"2024-06-12 19:37:16","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":122672,"visible":true,"origin":"","legend":"\u003cp\u003eIEEE 14 Bus System Single Line Diagram\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4447556/v1/7abef26b414a674a836f3412.png"},{"id":58232169,"identity":"daf5137a-9b61-4a9a-a69f-7193f5999c50","added_by":"auto","created_at":"2024-06-12 19:37:16","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":18517,"visible":true,"origin":"","legend":"\u003cp\u003eNonlinear Load Harmonic Spectrum\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-4447556/v1/b8393df73848b83cc52f66a6.png"},{"id":58232643,"identity":"6249b7d5-1ee1-441e-baca-dda8a8bd2f97","added_by":"auto","created_at":"2024-06-12 19:53:16","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":15229,"visible":true,"origin":"","legend":"\u003cp\u003eTHD for the IEEE 14 Bus System without Installing Passive Filter.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-4447556/v1/3c6fc7b8ce52bf3161b6d71a.png"},{"id":58232343,"identity":"95870684-b6f6-4478-99fb-48453c6f301a","added_by":"auto","created_at":"2024-06-12 19:45:16","extension":"jpeg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":156108,"visible":true,"origin":"","legend":"\u003cp\u003eTDECO Single Line Diagram\u003c/p\u003e","description":"","filename":"floatimage6.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-4447556/v1/63ca3c456ab8b36737d3b3b2.jpeg"},{"id":58232342,"identity":"5f145cb7-8ba1-4400-9f12-f11bcf38ada6","added_by":"auto","created_at":"2024-06-12 19:45:16","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":21186,"visible":true,"origin":"","legend":"\u003cp\u003eTHD for the TDECO Buses before Installing Passive Filters.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-4447556/v1/bed3da605e68c3236d7fa32b.png"},{"id":58232172,"identity":"a24b7895-b758-4be8-a6d2-aaab719fcb9c","added_by":"auto","created_at":"2024-06-12 19:37:16","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":13635,"visible":true,"origin":"","legend":"\u003cp\u003eTHD for the IEEE 14 Bus System after Installing Passive Filter with 21Mvar Capacity\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-4447556/v1/bf70bc47c0be24931d0f3e04.png"},{"id":58232168,"identity":"484b29b5-2031-4b37-b2a4-8942ea662085","added_by":"auto","created_at":"2024-06-12 19:37:16","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":25321,"visible":true,"origin":"","legend":"\u003cp\u003eTHD for the TDECO Buses after Installing One Passive Filters\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-4447556/v1/d85a13fba49b06881a9e0d6a.png"},{"id":58232344,"identity":"d7ed9c98-1bcd-4533-b04a-d256b50962f0","added_by":"auto","created_at":"2024-06-12 19:45:16","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":445053,"visible":true,"origin":"","legend":"\u003cp\u003eTHD for the TDECO Buses before and after Installing Two Passive Filters\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-4447556/v1/92702a0546abece0d0567bc1.png"},{"id":65713385,"identity":"4b2a9139-44c8-45ce-ae36-f013fa2f911a","added_by":"auto","created_at":"2024-10-01 15:16:14","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1902469,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4447556/v1/73315c3b-85d6-49fb-8a64-0830b32fe56f.pdf"},{"id":58232174,"identity":"17219158-d23a-405c-b6af-7bcc83cabed0","added_by":"auto","created_at":"2024-06-12 19:37:16","extension":"pdf","order_by":2,"title":"","display":"","copyAsset":false,"role":"supplement","size":1072869,"visible":true,"origin":"","legend":"","description":"","filename":"HybridGAPSOCircuitTheoryJournalV2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4447556/v1/1775f698c942d5aa039393a7.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eHybrid Genetic Algorithm-Particle Swarm Optimization Based Approach for Optimal Sizing and Placement of a Passive Filter \u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eThe expansion of harmonic currents in electrical systems had risen recently due to various factors, including solid-state power conversion devices, traction drives, industrial variable speed drive systems, A/D converters, and increased utilization of arc welders. Additionally, important sources of harmonic currents in power systems include electrical equipment like fluorescent lamps, large induction motors, near-saturated power transformers, and computer system installations. Presently, nonlinear devices constitute a growing portion of the electrical load in industrial and commercial power systems. When a sinusoidal supply feeds a combination of linear and nonlinear loads, the total supply current will inevitably contain harmonics. The introduction of harmonic currents and the resulting harmonic voltages can lead to power quality issues, impacting the performance of consumers connected to the electric power network, [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. Nonlinear devices produce harmonics in line currents, leading to increased conductor heating. This elevated temperature can exceed the rated capacity of cables not designed for harmonic currents, causing potential problems. The effects of harmonics in power distribution systems include aging, reduced capacity of components, malfunctioning protection and measurement devices, a lower power factor, and a consequent decrease in power system efficiency due to increased losses.\u003c/p\u003e \u003cp\u003eTo mitigate these issues, industrial standards and recommended practices for power systems have been established. However, power systems must still endure harmonic flow as complete elimination is not possible. Consequently, as non-linear loads in power systems continue to grow, there is a growing need for more effective methods to counteract the harmonic effects both among sources and on the power system itself. Extensive efforts have been made in analyzing, simulating, and developing passive and active filters, hardware for identification, classification, and mitigation of power system harmonics. Consequently, there is a growing interest for the application of passive and active filters to mitigate the harmonic effects generated by these loads. Proper consideration of the sizing and placement of passive filters is crucial to prevent the amplification of harmonic currents and voltages, which can occur due to potential resonance at specific harmonic frequencies. Various research studies, such as [\u003cspan additionalcitationids=\"CR2 CR3 CR4\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e], utilize genetic algorithms to determine the optimal size and location of shunt capacitors in radial distribution systems, taking into account harmonic distortion limits imposed by nonlinear power electronic devices. In [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], the non-dominated sorting genetic algorithm (NSGA-II) is employed to identify the optimal placement of passive filters. Another innovative approach proposed in [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] involves the use of fuzzy dynamic programming to decide the optimal size and location of compensation shunt capacitors for distribution systems with harmonic distortion. A method for determining optimal or near-optimal locations and sizes of single-tuned passive harmonic filters within existing capacitor buses in a power system is introduced in [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. Additionally, [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] presents a novel method towards passive filter placement, and [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e] utilizes a whale optimization algorithm for filter placement. The methodology of the present study is based on harmonic flow simulation in the Alternative Transient Program (ATP/EMTP) and employs the NSGA-II optimization algorithm simulated on MATLAB\u0026reg; to identify the optimal placement of passive filters. Although the method presented in [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] accounts for both harmonic current sources from nonlinear loads and the presence of background voltage harmonics, the objective functions are notably complex. This complexity, particularly at lower levels, leads to challenges in filtering due to conflicting requirements. In [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e], a C-type filter is used to reduce harmonic distortion, improve system performance and compensate reactive power. In order to improve the load power factor, also considering economic factors. In [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e], a method for designing an L compensator in time-variant, non-sinusoidal systems, is proposed. It examines the realistic representation of implementing the equivalent system impedances, source voltage harmonics, and the equivalent load impedance, the proposed method shows a high level of complexity. The work in [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] introduces a method that combines sequential neural-network approximation with orthogonal arrays (SNAOA) to reduce harmonic distortion using passive harmonic filters. While the method yields satisfactory results, it requires a substantial amount of real data to effectively train the neural network. In [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], the proposed technique aimed to allocate and sizing single tuned passive filters in power distribution systems. The objective is to minimize total harmonic distortion. The method employs the standard genetic algorithm (GA) without any modifications.\u003c/p\u003e \u003cp\u003eIn the time where most reviewed research emphasizes using a single optimization technique with one or two objective functions, this often doesn't result in finding the global minimum or optimal solution due to the common problem of getting stuck in local minima. The proposed method suggests a hybrid approach. It involves selecting a wide-ranging population for one optimization technique while keeping the population relatively close and locally separated for the other optimization technique. The proposed method can continually search for a global minimum while also exploring local minima solutions.\u003c/p\u003e \u003cp\u003eThis work aims to introduce a novel approach to passive filter sizing and allocation, utilizing optimization techniques to enhance the filter's impact on reducing harmonics distortion and active power losses. As the filter comprises multiple branches in shunt, the distribution of reactive power among these branches will be optimized. The proposed passive filter design incorporates a hybrid GA-PSO algorithm.\u003c/p\u003e \u003cp\u003eThe choice of this algorithm, among others, is based on its combination of the unique individual generation functions of both GA and PSO, mimicking social behaviors such as animal breeding and survival of the fittest. This integration enhances the overall efficiency of the algorithm's results. While the PSO algorithm excels in efficient local searches, the GA serves as a potent tool for global searches, albeit with less power in local search. Hence, a hybrid algorithm is employed to capitalize on the strengths of both approaches.\u003c/p\u003e \u003cdiv id=\"Sec2\" class=\"Section2\"\u003e \u003ch2\u003e1.1 Passive Filters\u003c/h2\u003e \u003cp\u003ePassive filters come in two primary configurations: Shunt passive filters and Series passive filters. They are employed to either divert harmonic currents away from the line or impede their transmission between different parts of the system by tuning the elements to resonate at specific harmonic frequencies. Figure\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e depicts various common filter configurations. Comprising inductance, capacitance, and resistance elements, passive filters typically utilize a single-tuned \"notch\" filter, which is both commonly used and cost-effective for many applications. Besides mitigating harmonics, notch filters also contribute to Power Factor (PF) correction [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e].\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe application of a series filter is constrained when it comes to blocking multiple harmonic currents since each harmonic requires a dedicated series filter tuned to that specific harmonic. This setup can lead to considerable losses at the fundamental frequency. Therefore, in centralized filter design, the use of shunt filters is more suitable than series filters. This preference arises from the expected outcome of multiple harmonics due to the diverse nature of nonlinear elements in the system.\u003c/p\u003e \u003cp\u003eShunt passive filters rely on three key design parameters: the tuning harmonic orders (h), the quality factor (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{Q})\\)\u003c/span\u003e\u003c/span\u003e, and the impedance of the filter at the fundamental frequency. The tuning of harmonic orders is contingent upon the nonlinear behavior of the system. The quality factor, denoting the ratio of reactance to resistance at the fundamental frequency, typically ranges between 50 and 150 for a single-tuned filter, [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. The filter reactance, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{L}\\)\u003c/span\u003e\u003c/span\u003e, and resistance, \u003cem\u003eR\u003c/em\u003e, can be calculated by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{L} = \\frac{{X}_{c}}{{h}^{2}}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(R = \\frac{{{h}_{ }X}_{L}}{\\text{Q}}\\)\u003c/span\u003e\u003c/span\u003e, respectively. Where, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{C}\\)\u003c/span\u003e\u003c/span\u003e is the filter capacitance, \u003cem\u003eh\u003c/em\u003e is frequency order, and \u003cem\u003eQ\u003c/em\u003e is the filter quality factor.\u003c/p\u003e \u003cp\u003eThe fundamental steps in the design process of passive filters can be condensed into a few key stages. Initially, the tuned frequency for the filter is chosen. Once this frequency is established, the capacitor bank size and resonant frequency can be calculated. The maximum value for the filter reactive power \u003cem\u003eQ\u003c/em\u003e\u003csub\u003e\u003cem\u003ef\u003c/em\u003e\u003c/sub\u003e is determined by the total fundamental reactive power. The filter reactance X\u003csub\u003ef\u003c/sub\u003e, representing the difference between the capacitive reactance, \u003cem\u003eX\u003c/em\u003e\u003csub\u003e\u003cem\u003eC\u003c/em\u003e\u003c/sub\u003e, and the inductive reactance, \u003cem\u003eX\u003c/em\u003e\u003csub\u003e\u003cem\u003eL\u003c/em\u003e\u003c/sub\u003e, can be determined by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{f} = {X}_{C} - {X}_{L} = V /{Q}_{f}.\\)\u003c/span\u003e\u003c/span\u003e Where \u003cem\u003eV\u003c/em\u003e is the line to line voltage. So, the filter capacitive reactance, capacitor and inductor values can be calculated by, [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e], \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({X}_{C}= {h}^{2}{X}_{L}= \\frac{{X}_{f} {h}^{2}}{{h}^{2}-1}\\)\u003c/span\u003e\u003c/span\u003e, \u003cem\u003eC\u003c/em\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(= \\frac{1}{{X}_{c}{\\omega }_{\\text{o}}}\\)\u003c/span\u003e\u003c/span\u003e, and \u003cem\u003eL\u003c/em\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(= \\frac{{X}_{L}}{{\\omega }_{o}}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e1.2 Objective Function\u003c/h2\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThe objective function selected to be minimized for this optimization problem is the branches active power losses (P\u003csub\u003eloss\u003c/sub\u003e) and the total harmonic distortion (THD) that can be calculated according to the following:\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Equ1\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${P}_{loss}= \\sum _{h=1}^{max}\\sum _{i=1}^{N-1}{{{P}_{loss(i,i+1)}}^{h}}_{ }^{ }$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e \u003cdiv id=\"Equ2\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${{P}_{loss(i,i+1)}}^{h}=\\left({{R}_{i,i+1}}^{h}\\right({\\left({{V}_{i}}^{h}- {{V}_{i+1}}^{h} \\right)\\text{*}{{ Y}_{i,i+1}}^{h})}^{2})$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e \u003cdiv id=\"Equ3\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$THD= \\frac{\\surd \\left(\\sum _{h=2}^{max}{{{|V}_{i }}^{h}|}^{2}\\right)}{\\left|{{V}_{i}}^{1}\\right|}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThese two optimization objectives were employed in the optimization problem. THD served as the primary objective function in the section executed by PSO. Conversely, THD was transformed into a constraint in the section where GA was employed, with the minimum P\u003csub\u003eloss\u003c/sub\u003e being the objective function. Throughout this problem, the aim was to determine the optimal configuration of certain control variables (e.g., single-tuned filter location, size, and tuned frequency) while adhering to specified equality and inequality constraints.\u003c/p\u003e \u003cp\u003eThe Newton-Raphson method was utilized in this optimization problem to compute the total fundamental reactive power and to uphold the equality constraints associated with the fundamental active and reactive power at bus i within its specified tolerance. At harmonic frequencies, the power system was reconfigured, taking into account the passive filter, the nonlinear load, and the frequency effect at all the buses, as follows:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${Y}_{l,i}^{h}= \\frac{{P}_{l,i}}{{\\left|{{V}_{i}}^{1}\\right|}^{2}}-j\\frac{{\\text{Q}}_{l,i}}{{h\\left|{{V}_{i}}^{1}\\right|}^{2}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$${Y}_{c,i}^{h}=h{Y}_{c,i}^{1}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$${Y}_{f,i}^{h}= \\frac{1}{{{{Z}_{f,i}}^{h}}^{ }}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$${Y}_{i,i+1}^{h}=\\frac{1}{{R}_{i,i+1}+jh{X}_{i,i+1}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe impedance for single tuned filter is\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$${Z}_{f}^{h}=R+j\\omega L+ \\frac{1}{j\\omega C}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eEfficient simulation is essential for large systems, prompting the recommendation to employ decoupled harmonics power flow for system parameter calculations. In the case of the i\u003csup\u003eth\u003c/sup\u003e bus, where P\u003csub\u003enl,i\u003c/sub\u003e and Q\u003csub\u003enl,i\u003c/sub\u003e denote the nonlinear active and reactive power at the fundamental frequency, the nonlinear current can be computed using the provided equations.\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$${I}_{nl,i}^{1}= {\\left[\\frac{{P}_{nl,i}+j{\\text{Q}}_{i,nl}}{{V}_{i}^{1}}\\right]}^{\\text{*}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$${I}_{nl,i}^{h}=c\\left(h\\right){I}_{nl,i}^{1}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e1.3 Constraints\u003c/h2\u003e \u003cp\u003eThe fundamental active and reactive power at bus (i) is considered as constraints during this optimization method and it can be calculated by:\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$${P}_{G,i}-{P}_{l,i}=\\left|{V}_{i}^{\\left(1\\right)}\\right|\\sum _{j=1}^{N}\\left|{V}_{i}^{\\left(1\\right)}\\right|\\left|{Y}_{i,j}^{\\left(1\\right)}\\right|\\text{cos}\\left({{\\varnothing}}_{i,j}^{\\left(1\\right)}- {\\delta }_{i}^{\\left(1\\right)}+{\\delta }_{j}^{\\left(1\\right)}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$${Q}_{G,i}-{Q}_{l,i}=\\left|{V}_{i}^{\\left(1\\right)}\\right|\\sum _{j=1}^{N}\\left|{V}_{i}^{\\left(1\\right)}\\right|\\left|{Y}_{i,j}^{\\left(1\\right)}\\right|sin({{\\varnothing}}_{i,j}^{\\left(1\\right)}- {\\delta }_{i}^{\\left(1\\right)}+{ \\delta }_{j}^{\\left(1\\right)})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere: i\u0026thinsp;=\u0026thinsp;1\u0026hellip;N is the bus number,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({P}_{G,i},{P}_{l,i}{,Q}_{G,i}and{Q}_{l,i}\\)\u003c/span\u003e\u003c/span\u003e are the generated and load active and reactive power respectively. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({Y}_{i,j}^{\\left(1\\right)}\\)\u003c/span\u003e\u003c/span\u003eis the fundamental admittance between bus i and j \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ {\\varnothing}}_{i,j}^{\\left(1\\right)}, {\\delta }_{i}^{\\left(1\\right)}and {\\delta }_{j}^{\\left(1\\right)}\\)\u003c/span\u003e\u003c/span\u003e are the admittance phase angle and the voltage angle at bus i and j at the fundamental frequency.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e1.4 Optimization Techniques\u003c/h2\u003e \u003cp\u003eThis study proposes a combination of two global optimization algorithms, GA and PSO, to solve nonlinear optimization problems. The rationale behind this lies in both algorithms working with an initial population of solutions, making the integration of their search capabilities a sensible strategy. The subsequent sections outline the steps and flowchart of the GA, PSO and the hybrid GA-PSO optimization techniques.\u003c/p\u003e \u003cdiv id=\"Sec6\" class=\"Section3\"\u003e \u003ch2\u003e1.4.1 Particle Swarm Optimization\u003c/h2\u003e \u003cp\u003eSwarm behavior can be replicated through a set of straightforward rules, allowing for the modeling of schools of fish and flocks of birds. Even with individual agents following simple behavior rules, the collective behavior of the entire swarm can become intricate. Agents make decisions based on both their own experiences and the experiences of others, contributing to the fundamental elements that underlie PSO.\u003c/p\u003e \u003cp\u003eEach agent is aware of its personal best value (p\u003csub\u003ebest\u003c/sub\u003e) and its corresponding x, y position, akin to individual experiences. Additionally, each agent is informed about the best value within the group (g\u003csub\u003ebest\u003c/sub\u003e) among the p\u003csub\u003ebest\u003c/sub\u003e, resembling knowledge about how neighboring agents have performed. Agents seek to modify their positions based on its current position, velocity, p\u003csub\u003ebest\u003c/sub\u003e and g\u003csub\u003ebest\u003c/sub\u003e. This modification is conceptually represented by the notion of velocity, where the velocity of each agent can be adjusted using a specific equation.\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$${V}_{i}^{k+1}={wv}_{i}^{k}+{c}_{1}{rand}_{1}*\\left({{g}_{best}}_{i}- {S}_{i}^{k}\\right)+{c}_{2}{rand}_{2}*({{g}_{best}}_{ }- {S}_{i}^{k})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({V}_{i}^{K}\\)\u003c/span\u003e\u003c/span\u003e is velocity of agent i at iteration k, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(w\\)\u003c/span\u003e\u003c/span\u003e is weighting function, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({c}_{j}\\)\u003c/span\u003e\u003c/span\u003e is weighting coefficients, rand is a random number between 0 and 1, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({S}_{i}^{K}\\)\u003c/span\u003e\u003c/span\u003e is current position of agent i at iteration k, p\u003csub\u003ebesti\u003c/sub\u003e is p\u003csub\u003ebest\u003c/sub\u003e of agent i, and g\u003csub\u003ebest\u003c/sub\u003e is g\u003csub\u003ebest\u003c/sub\u003e of the group. Namely, velocity of an agent can be changed using three vectors. The velocity is usually limited to a certain maximum value [\u003cspan additionalcitationids=\"CR17 CR18 CR19 CR20 CR21 CR22\" citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe following weighting function is usually utilized:\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e\n$$w= {w}_{max}- \\frac{{w}_{max}- {w}_{min}}{{iter}_{max}}\\text{*}iter$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere, w\u003csub\u003emax\u003c/sub\u003e is initial weight, w\u003csub\u003emin\u003c/sub\u003e is the final weight, iter\u003csub\u003emax\u003c/sub\u003e is maximum iteration number, and iter is the current iteration number.\u003c/p\u003e \u003cp\u003eThe three velocity vectors convey the following implications: The initial term represents the agent's previous velocity. The second and third terms are employed to adjust the agent's velocity. Without these latter terms, the agent would persist in moving in the same direction until reaching a boundary. In essence, this implies an exploration of new areas, making the first term synonymous with diversification in the search process.\u003c/p\u003e \u003cp\u003eThe current position (searching point in the solution space) can be modified by the following equation:\u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e\n$${S}_{i}^{K+1}= {{S}_{i}^{k}}_{ }+ {V}_{i}^{k+1}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe general process flow of PSO can be outlined as follows:\u003c/p\u003e \u003cp\u003e \u003cstrong\u003eStep 1\u003c/strong\u003e \u003cp\u003eInitialization of each agent's initial conditions. Initial search points (s\u003csub\u003ei\u003c/sub\u003e\u003csup\u003e0\u003c/sup\u003e) and velocities (v\u003csub\u003ei\u003c/sub\u003e\u003csup\u003e0\u003c/sup\u003e) for each agent are typically randomly generated within the allowable range. The current search point is set as pbest for each agent. The best-evaluated value of pbest becomes gbest, and the corresponding agent number is stored.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eStep 2\u003c/strong\u003e \u003cp\u003eEvaluation of each agent's search point. The objective function value is computed for each agent. If the value surpasses the current pbest of the agent, the pbest value is updated with the current value. If the best value of pbest is superior to the current gbest, gbest is updated with the best value, and the corresponding agent number is stored.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eStep 3\u003c/strong\u003e \u003cp\u003eModification of each search point.\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003cstrong\u003eStep 4\u003c/strong\u003e \u003cp\u003eChecking the exit condition. If the current iteration number reaches the predetermined maximum iteration number, the process exits; otherwise, it returns to Step 2. Figure\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e provides a visual representation of the overall flowchart of PSO [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e, \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e].\u003c/p\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eBased on, [\u003cspan additionalcitationids=\"CR22 CR23 CR24\" citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e], the following parameters were deemed suitable, and their values are found to be independent of specific problem characteristics:\u003c/p\u003e \u003cp\u003eci\u0026thinsp;=\u0026thinsp;2:0 w\u003csub\u003emax\u003c/sub\u003e =0,9 w\u003csub\u003emin\u003c/sub\u003e =0,4\u003c/p\u003e \u003cp\u003eThe values are also proved appropriate for power system problems.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section3\"\u003e \u003ch2\u003e1.4.2 Genetic Algorithm\u003c/h2\u003e \u003cp\u003eUnlike traditional optimization techniques that start with a single candidate and iteratively seek the optimal solution using static heuristics, the Genetic Algorithm (GA) approach employs a population of candidates to simultaneously and adaptively explore various areas within a solution space, [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eWhile some heuristic search methods rely on local search, such as hill climbing, others opt for a nonconvex optimization approach that accepts cost-deteriorating neighbors. GA stands out as one of the most popular methods extending beyond simple local search. Genetic algorithms operate on a population of individuals, where each individual represents a potential solution. Following the generation of an initial population, either randomly or heuristically, the algorithm evolves the population through sequential and iterative application of three operators, culminating in the formation of a new generation at the end of each iteration, [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eSelection.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eCrossover.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eMutation.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"2. Material and Method","content":"\u003cp\u003eIn the realm of heuristic optimization, a robust method for tackling complex nonlinear equations involves the combination of particle swarm optimization (PSO) and genetic algorithms (GA). This hybrid approach, known as GA-PSO, synergizes the strengths of PSO and GA by incorporating PSO mechanisms for generating individuals in a new generation, alongside the traditional crossover and mutation procedures observed in GA, [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eIn the following section the proposed hybrid GA-PSO optimization technique will be defined in steps as well as in a flowchart shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e2.1 GA-PSO Optimization Technique Steps:\u003c/h2\u003e \u003cp\u003e \u003cb\u003eStep 1\u003c/b\u003e:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eInput line data at the fundamental harmonic (i.e: branches impedance).\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eInput the fundamental active ve reactive bus power bus voltages amplitude and phase bus code ( slack, load or generating bus).\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eInsert Sbase, maximum number of iteration\u0026thinsp;=\u0026thinsp;1000 and the accuracy\u0026thinsp;=\u0026thinsp;0.001.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eGenerate Ybus, Jacobian Matrix.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eUse Newton Raphson method to calculate the power solution at the fundamental frequency.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eCalculate active ve reactive power at the slack bus for the fundamental frequency.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eCalculate the reactive power required by the filter that is the total fundamental reactive power.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 2\u003c/b\u003e \u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eInput harmonic current spectrum and its frequency order.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eEnter active and reactive power for the non linear load and its position.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eCalculate I1 for the non linear load.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eCalculate Ih using C(h) and I1 for the non linear load.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eCalculate Ih and Vh at all buses.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 3\u003c/b\u003e \u003c/p\u003e \u003c/div\u003e\n\u003ch3\u003e1. Give a first assumption for the filter size and tuned frequency.\u003c/h3\u003e\n\u003cp\u003e \u003cb\u003eStep 4\u003c/b\u003e \u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eInput PSO parameters (c\u003csub\u003ei\u003c/sub\u003e =2:0 w\u003csub\u003emax\u003c/sub\u003e =0,9 w\u003csub\u003emin\u003c/sub\u003e =0,4).\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eDefine objective functions (minimum P\u003csub\u003eloss\u003c/sub\u003e and THD).\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eDefine constrains (fundamental power equation).\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eGenerate initial filter position and tuned harmonic order for each agents.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eEvaluate the searching point for each agents.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eModification for each searching agents.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eMaximum iteration number reached.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003e \u003cb\u003eStep 5\u003c/b\u003e \u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eInput GA parameters.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eUse the PSO output and generate a random value for the filter size and set the frequency order for each chromosome in the population.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eCalculate the power equations in at the fundamental frequency then recalculate V\u003csub\u003eh\u003c/sub\u003e\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eEvaluate the objective functions of minimum active power losses with the constraints of not exceeding THD limits for each chromosome in the population.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eMaximum Number of Iterations.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eIf not, selection, crossover and mutation.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eNew Population.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003ePrint filter size, position and set frequency.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003e \u003cb\u003e2.2 The Hybrid GA‑PSO Test on the IEEE 14 Bus System and on a Real Electrical Network\u003c/b\u003e \u003c/p\u003e \u003cp\u003eTo explore the optimal placement and dimensions of a passive filter within a power system, aiming to reduce total harmonic distortion to the IEEE standard-defined legal limits, minimize active power losses in branches, and improve the system power factor using a hybrid optimization technique integrating PSO and GA; two tests had been done. In the following sections the details of the tested systems were identified.\u003c/p\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e2.2.1 The IEEE 14 Bus System\u003c/h2\u003e \u003cp\u003eThe 14-bus IEEE standard system was selected as the basis for the optimization study. The optimization task involved addressing the passive filter design and allocation problem. The GA-PSO hybrid optimization technique previously defined was applied to assess solutions for the IEEE 14-bus system. This undertaking aimed to evaluate the effectiveness, robustness, and quality of the proposed technique.\u003c/p\u003e \u003cp\u003eThe IEEE 14-bus system comprises four generation buses, one slack bus, and nine load buses, all operating at the same voltage level (1kV). The characteristics of each bus, including its type, rated voltage amplitude and angle, active and reactive power for loads and generators, as well as reactive power limits, are detailed in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. The single line diagram for the IEEE 14 bus is represented using ETAP program as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eIn the assumption that the system is initially free from harmonics, a nonlinear load was introduced to serve as the harmonic source in the system. Specifically, a nonlinear load in the form of an IEEE typical 6-pulse rectifier was added to bus number 10. This load contributes 18.75 MW and 16.53 Mvar of active and reactive power, respectively. The harmonic spectrum associated with this load is depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eGeneration Bus Information for IEEE 14 Bus System.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"9\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colspan=\"3\" nameend=\"c3\" namest=\"c1\"\u003e \u003cp\u003eGeneration Bus\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e \u003cp\u003eVoltage\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c7\" namest=\"c6\"\u003e \u003cp\u003eGeneration\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c9\" namest=\"c8\"\u003e \u003cp\u003eMvar Limits\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eID\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ekV\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eType\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e% Mag.\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003eAngle\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003eMW\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003eMvar\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003eMax\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003eMin\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBus1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eVoltage Control\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e109.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e1.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-1.000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBus 1 H_1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSwing\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e106.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBus 2 H_2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eVoltage Control\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e104.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e29.145\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e29.145\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-29.145\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBus 3 H_3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eVoltage Control\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e101.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e23.400\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e0.000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBus 6 L_6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eVoltage Control\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e107.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.000\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e12.200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e-6.000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe harmonics orders are [5 7 13 15 17 19 23 25 29 32 35 37 42 45 47 49]. The harmonic current can be determined using Eq.\u0026nbsp;\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e10\u003c/span\u003e where C(h) represents a fraction of the fundamental current components or it is defined based on real values from its power spectrum. The coefficients C(h) for different harmonic orders (h) are specified as follows:\u003c/p\u003e \u003cp\u003eC(h)=[ 0.2 0.1429 0.0769 0.0667 0.0588 0.0526 0.0435 0.04 0.0345 0.0313 0.0286 0.0270 0.0238 0.0222 0.0213 0.0204]\u003c/p\u003e \u003cp\u003eIn this work, optimization algorithms were employed to determine the optimal locations and sizes for one and two single-tuned filter. The results obtained were then compared with those obtained without any filters. Prior to the initiation of the optimization algorithm, the total reactive power for the filters was calculated using the Newton-Raphson power flow analysis method, resulting in a value of 73.299 Mvar, representing the total fundamental reactive power.\u003c/p\u003e \u003cp\u003eThe THD at each bus of the system before the filter installation is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. It is evident that the THD has surpassed IEEE limits at buses 10 and 9, and that the THD at buses 1 and 14 is very close to the IEEE limit. Bus 9 exhibits the highest THD value at 5.61%.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cdiv id=\"Sec12\" class=\"Section3\"\u003e \u003ch2\u003e2.2.2 The Second Case Study a Real Electrical Network that is Tubas District Electricity Company:\u003c/h2\u003e \u003cp\u003eThe Tubas District Electricity Company (TDECO), a publicly limited company, was established on 2006 with a capital of \u003cspan\u003e$\u003c/span\u003e3,491,320. It operates in Tubas city, a small governorate covering an area of 402 km\u003csup\u003e2\u003c/sup\u003e in the Middle East of Palestine. In 2002, the company commenced its operations by providing electricity to its initial 23 member shareholders, comprising municipal and village councils. Over the years, the company expanded its services to include electricity departments in municipalities and local authorities, serving a total of 15,000 customers by 2011.\u003c/p\u003e \u003cp\u003eWith the goal of supplying electricity to 20,000 customers with a daily consumption rate of 25 mega-volt amperes, the company relies on two connection points receiving electricity from the Israeli company. The primary feeding point is located in Tayasir, delivering more than 20 MW, while the secondary point is in Al-Jalamah.\u003c/p\u003e \u003cp\u003eThe electrical network in Tubas comprises 674 buses, 267 transformers, 408 cables and transmission wires, including 33 underground cables. To visualize and analyze this network, a single-line diagram was created using ETAP software, capturing the network's structure and components. The single-line diagram is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFor the TUBAS network, the total reactive power used to determine the filter sizes was 6 Mvar, calculated through the Newton-Raphson power flow analysis method before initiating the optimization algorithm. This value represents the total fundamental reactive power.\u003c/p\u003e \u003cp\u003eTo assess the impact of passive filters on the network's behavior, a nonlinear load was introduced to one of the buses. Given the absence of specific harmonic distortion data, except for its exceedance of legal limits in multiple buses and the main feeding point, a nonlinear load was added to the bus with the maximum PV system, considering photovoltaic systems as potential sources of harmonics. At Bus 5, where the largest PV system with 5 MWp is installed (the biggest PV system in the network exceeding 19 MWp), a nonlinear load in the form of an IEEE typical 6-pulse rectifier was added. This load contributes 3.3 MW and 3.7 Mvar of active and reactive power, respectively.\u003c/p\u003e \u003cp\u003eThe Total Harmonic Distortion (THD) at each bus of the system before installing the filters is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e. It is evident that the THD has exceeded IEEE limits at buses 459, 550, 565, and 717, and that the THD at buses 163, 164, 168, and 716 is very close to the IEEE limit. Bus 549 exhibits the highest THD value at 6.34%.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"3. Results and Discussion","content":"\u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e3.1 IEEE 14 Bus System Test Results\u003c/h2\u003e \u003cp\u003e \u003cdiv class=\"BlockQuote\"\u003e \u003cp\u003eThe effect of installating a single passive filter with a capacity of 21.6 Mvar at bus number 9, on the 14 bus IEEE system, could be notice over the THD as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e as well as over the power factor improvment and active power loss reduction. The maximum THD has seen a reduction of 27.6%, decreasing from 5.61\u0026ndash;4.06%. Additionally, the total active power losses at branches have experienced a 2.4% reduction, decreasing from 46.3 MW to 45.2 MW. It is noteworthy that these reductions could further enhance if more than filter is installed. The results of installing one and two single tunde filter is summrized in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eIEEE 14 Bus System Original Case and after Installing One and Two Single Tuned Passive Filters.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eObjective Function values with the Design parameters\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eOriginal case (without Filters)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eOne filter solution\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e \u003cp\u003eTwo filter solution\u003c/p\u003e \u003cp\u003eFilter 1 Filter 2\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFilter size Mvar\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e21.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e37\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e32\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eActive power losses MW\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e46.3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e45.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e \u003cp\u003e42.6\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTHD%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e5.61\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e4.06\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e \u003cp\u003e4.4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFilter Location\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e13\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTuned Frequency\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e7.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e7.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e7.14\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFilter Inductance (mH)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3.96\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e4.56\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFilter capacitance (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\mu F)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e65529\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e47509\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e41089\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFilter Resistance (mΩ)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e97.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e134\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e113.5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ Factor\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e66\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e66\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e90\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePower factor at the main connection point\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e96.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e97.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e \u003cp\u003e99.06\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003e3.2 The Second Case Study Tubas District Electricity Company Test Results\u003c/h2\u003e \u003cp\u003eThe Total Harmonic Distortion (THD) after installing a single passive filter with a capacity of 5 Mvar at bus number 165 in the TUBAS network is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e. Notably, the maximum THD has undergone a reduction of 48% decreasing from 6.34\u0026ndash;3.0%. Additionally, the active power losses at branches have experienced a 8% reduction, decreasing from 1.63 MW to 1.4 MW. Moreover, the power factor at the main connection point has increased by 11% rising from 61.7 to 68.5.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe impact of installing two passive filters, with capacities of 1.62 Mvar and 2.8 Mvar, at bus numbers 163 and 717 in the TUBAS network is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e. The results indicate a significant improvement in system performance: The maximum THD has been reduced to 4.4%, representing a reduction of 30.6% compared to the original THD of 6.34%. Active power losses at branches have been reduced by 5.5%, decreasing from 1.63 MW to 1.54 MW.\u003c/p\u003e \u003cp\u003eThe power factor at the main connection point has increased by 14.3%, rising from 61.7 to 72%. These improvements highlight the effectiveness of strategically placing passive filters in the network to mitigate harmonic distortions, decrease active power losses, and enhance power factor. The result of installing one and two single tuned passive filters is summarized in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eTDECO Original Case and after Installing One and Two Single Tuned Passive Filters.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eObjective Function values with the Design parameters\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eOriginal case (without Filters)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eOne filter solution\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e \u003cp\u003eTwo filter solution\u003c/p\u003e \u003cp\u003eFilter 1 Filter 2\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFilter size Mvar\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e2.8\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eActive power losses MW\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e \u003cp\u003e1.54\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTHD%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e6.34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.0\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e \u003cp\u003e4.4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFilter Location\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e165\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e163\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e717\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTuned Frequency\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5.125\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5.14\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5.26\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFilter Inductance (mH)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e7.1 \u0026micro;\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.016\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFilter capacitance (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\mu F)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e65529\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e53978\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e23077\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFilter Resistance (mΩ)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e97.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.124\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.26\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ Factor\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e66\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e99\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e102\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePower factor at the main connection point\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e61.7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e68.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c5\" namest=\"c4\"\u003e \u003cp\u003e72\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eGA and PSO optimization methods were used to select the position of two single tuned passive filter for this system. Using PSO decreasing the THD by 30.6% and reducing the active power losses by 4.3% while enhancing the power factor by 17%, while using GA decreasing the THD by 27.4% and reducing the active power losses by 8.6% while enhancing the power factor by 17.7% as summarized in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. In one hand, the heuristic optimization methods used on this research shows satisfied results, on the other hand the hybrid GA-PSO methods shows a robust effective and high-quality solutions specially for large systems.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eTDECO Original Case and after Installing Two Single Tuned Passive Filters Using PSO and GA.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eObjective Function values with the Design parameters\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eOriginal case (without Filters)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c4\" namest=\"c3\"\u003e \u003cp\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003ePSO\u003c/span\u003e\u003c/p\u003e \u003cp\u003eFilter 1 Filter 2\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eGA\u003c/span\u003e\u003c/p\u003e \u003cp\u003eFilter 1 Filter 2\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFilter size Mvar\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2.4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e1.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2.4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eActive power losses MW\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.63\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c4\" namest=\"c3\"\u003e \u003cp\u003e1.56\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e1.49\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTHD%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e6.34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c4\" namest=\"c3\"\u003e \u003cp\u003e4.41\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e4.6\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eFilter Location\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e163\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e722\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e163\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e716\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTuned Frequency\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e-\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e5.13\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5.29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e5.39\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e5.26\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eQ Factor\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e99\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e99\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e101\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePower factor at the main connection point\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e61.7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c4\" namest=\"c3\"\u003e \u003cp\u003e72.29\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c6\" namest=\"c5\"\u003e \u003cp\u003e72.6\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4. Conclusion","content":"\u003cp\u003eThe results achieved through the hybrid GA-PSO approach align with the predefined objective functions, focusing on minimizing active power losses, and comply with the IEEE standard for Total Harmonic Distortion limits, serving as specified constraints met within this proposed methodology. The results shows that the THD could be reduced to a proximately 50% of its initial value. The reduction on the active power loss had a lower reduction percentage and that\u0026rsquo;s due to the large number of buses in the time where the algorithm had restricted by allocating two filters as a max.\u003c/p\u003e \u003cp\u003eComparative analysis between the outcomes of the hybrid GA-PSO approach and those of GA or PSO reveals that the hybrid GA-PSO method offers efficient, and high-quality solutions for extensive systems, whereas PSO demonstrates exceptional solutions for smaller systems. The accuracy and efficiency of the hybrid GA-PSO approach are successfully validated through the utilization of ETAP and MATLAB programs, illustrated effectively by both the standard IEEE 14 and the real system of Tubas Electric Company. Irrespective of the objective function's structure, the suggested hybrid GA-PSO approach can be easily and promptly applied to any other distribution system featuring linear and/or nonlinear loads.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eConflict of interest\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that there is no conflict of interest regarding the publication of this paper.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding Statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that there was no funding for this work.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eT.S. Chung, H.C. 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DOI: \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1109/TDC.2002.1177567\u003c/span\u003e\u003cspan address=\"10.1109/TDC.2002.1177567\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Passive filter, harmonic mitigation, Genetic algorithm, Particle swarm optimization, optimal harmonic power flow","lastPublishedDoi":"10.21203/rs.3.rs-4447556/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4447556/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis work presents a hybrid Genetic Algorithm-particle swarm optimization based technique for optimal placement and sizing of a passive power filters to reduce harmonics distortion and mitigate the total active power losses in distribution lines with linear and nonlinear loads. The problem was formulated as a nonlinear multi-objective optimization problem with equality and inequality constraints using hybrid Genetic Algorithm with practical swarm optimization. Recommendations by IEEE standards 519\u0026ndash;1992 and 18-2002 are considered as the main constraints in this study. Based on the minimum total harmonic distortion and active power losses as the main optimization criteria the optimal solutions were selected. The proposed approach is validated on the IEEE 14 bus system and on a real power system. The results confirmed that proposed method provides robust, effective and high-quality solutions for large power systems.\u003c/p\u003e","manuscriptTitle":"Hybrid Genetic Algorithm-Particle Swarm Optimization Based Approach for Optimal Sizing and Placement of a Passive Filter","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-06-12 19:37:11","doi":"10.21203/rs.3.rs-4447556/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"6a3f3652-ca9d-4b25-9539-9bcc69ae8b97","owner":[],"postedDate":"June 12th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-10-01T15:08:05+00:00","versionOfRecord":[],"versionCreatedAt":"2024-06-12 19:37:11","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4447556","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4447556","identity":"rs-4447556","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}