Probabilistic Hosting Capacity Enhancement in Non-Sinusoidal Power Distribution Systems Using a Hybrid PSOGSA Optimization Algorithm

The high penetration of distributed generation (DG) units with their power-electronic interfaces may lead to various power quality problems, such as excessive harmonic distortions and increased non-sinusoidal power losses. In this paper, the probabilistic hosting capacity (PHC) due to the high penetration of photovoltaic units in a non-sinusoidal power distribution system is investigated. A C-type harmonic filter is proposed, to maximize the harmonic-constrained PHC. An optimization problem is formulated by using a Monte Carlo simulation, taking into account various uncertain parameters, such as the intermittent output power of the DGs, background voltage harmonics, load alteration, and the filter parameters’ variations. In addition, different operational constraints have been considered, such as the bus voltage, line thermal capacity, power factor, and individual and total harmonic distortion limits. A swarm-based, meta-heuristic optimization algorithm known as the hybrid particle swarm optimization and gravitational search algorithm (PSOGSA) has been examined for the optimal design of the proposed filter. Besides, other optimization algorithms were examined for validation of the solution. The PHC results obtained are compared with the conventional deterministic HC (DHC) results, and it is found that the PHC levels are higher than those obtained by conservative HC procedures, practical rules of thumb, and the DHC approaches.


Introduction
Renewable energy resources play a vital role in current energy systems, driven by their sustainability, eco-friendliness, and techno-economic advantages [1,2]. Nowadays, the energy generation mix has been upgraded in many countries such as Germany, Australia, and the United Kingdom, to include more renewable distributed generation (DG) units instead of the conventional fossil-fuel-based resources, to solve various challenges such as future energy needs, remarkable oil price fluctuations, increasing risks of fossil-fuel pollution, and energy conservation strategies, to minimize transmission and to distribution network losses [3][4][5]. Consequently, the liberalization of electric energy markets has led to an augmented integration of DG units, such as photovoltaics (PV) and wind turbines (WT) in today's power systems [6]. However, unplanned and excessive harmonic mitigation techniques have been proposed, to enhance harmonic-constrained HCs to comply with international PQ limits, such as single-tuned passive filters in [8], C-type passive filters in [7], and active harmonic filters in [26]. In addition, inter-and supra-harmonics and their impacts on HC assessment were examined in [27]. However, a dynamic framework that employs numerous uncertain parameters such as variable DG-produced power caused by climate fluctuations, the uncertainty of DG integration location and unit ratings, daily load profile variations, and uncertainties in network modeling in the case of the absence of confirmed real-time measurements is required to express HC better. In this regard, it was found that that deterministic HC (DHC) assessment methodologies only show a conservative (worst-case) figure for a network's capability to host more DG units [28]. For that reason, recent studies have started to use the probabilistic hosting capacity (PHC), unlike DHC studies, which ignore the uncertainty of the electrical parameters [29,30]. The Monte Carlo simulation (MCS) is commonly used to handle the uncertainty of electrical parameters [31].
In this paper, the PHC of a harmonic-distorted power distribution system is explored in the presence of high PV penetration. A proposed C-type passive harmonic filter is proposed to maximize the harmonic-constrained PHC. The C-type harmonic filter is a high-pass passive filter, used in both utility and industrial networks, since it ensures harmonic mitigation for a wide range of harmonics, and it reduces resonance risks. In addition, the C-type filter operates as a standalone shunt capacitor bank at the fundamental frequency, and therefore its fundamental power loss is significantly lower than that of other passive filter types. An optimization problem is formulated by using the MCS framework, considering various uncertainty indices such as the intermittent output power of the DGs, background voltage harmonics, load alterations, and the filter parameters' uncertainties. The bus voltage limits, line thermal capacity, power factor, and individual and total harmonic distortion limits have been considered as the problem constraints. A meta-heuristic swarm-based optimization algorithm known as hybrid particle swarm optimization and gravitational search optimization (PSOGSA) has been used for the optimal design of the proposed filter. The proposed filter design using PSOGSA was compared with other design approaches, using the well-known particle swarm optimization (PSO) and the crow search algorithm (CSA) and it was found that the proposed design outperforms conventional design approaches. Further, the impact of multi-pulse VFDs on the system's PHC is investigated. The main contributions of this paper are as follows: a.
PHC assessment is performed under the presence of numerous uncertain parameters using MCS. b.
Different meta-heuristic optimization techniques are utilized for the optimal design of the proposed filter, taking into account numerous uncertainties of the parameters and the operational power quality indices. c.
A comparative analysis of the DHC and PHC results is presented. d.
The impact of multi-pulse VFDs on the system's PHC is investigated. To the best of the authors' knowledge, no work in the literature has considered this impact. e.
The proposed filter design using PSOGSA was compared to other design approaches, using the PSO and the CSA, and it was found to achieve higher PHC levels than the conventional HC results.
Unauthorized DG units may cause reliability and power quality (PQ) issues and implications. In this regard, the proposed PHC enhancement via harmonic mitigation is believed to provide a way forward for the development of non-sinusoidal power distribution systems with additional nonlinear loads and DG alliance room, while complying with the system operation and reliability requirements.
The rest of the paper is organized as follows: Section 2 introduces the mathematical formulation of the optimization problem. Section 3 presents the problem formulation. In Section 4, the results obtained are presented and discussed for the system under study. The impact of using multi-pulse VFDs is explored, and a comprehensive analysis of the DHC and PHC results is provided. Finally, Section 5 presents the conclusions and future works.  Figure 1 represents the configuration of the distribution system considered in this work. This system is a balanced, symmetric industrial distribution system consisting of a utility substation (slack bus), distribution feeder, and hybrid loads comprised of linear loads (induction motors), and nonlinear loads represented by six-pulse VFDs, and a DG unit (represented by a PV system) that is interfaced with the system through power-electronic interface inverters, and a shunt C-type harmonic filter. In this work, three possible harmonic sources are taken into consideration, namely the background voltage harmonics that are present in the utility, current harmonics injected from the PV system, and the harmonic currents due to the load nonlinearity.

Mathematical Model Formulation
When a probabilistic analysis is undertaken, the calculation of the problem's uncertain parameters is done through MCS. The MCS is utilized to develop a large number of probabilities of the studied parameter. For each probability, the 95th percentile of the considered parameter is calculated. In this work, each normal distribution was obtained from MCS, using 1000 samples, and considering a relevant appropriate uncertainty tolerance level for each parameter.  Figure 1 represents the configuration of the distribution system considered in this work. This system is a balanced, symmetric industrial distribution system consisting of a utility substation (slack bus), distribution feeder, and hybrid loads comprised of linear loads (induction motors), and nonlinear loads represented by six-pulse VFDs, and a DG unit (represented by a PV system) that is interfaced with the system through power-electronic interface inverters, and a shunt C-type harmonic filter. In this work, three possible harmonic sources are taken into consideration, namely the background voltage harmonics that are present in the utility, current harmonics injected from the PV system, and the harmonic currents due to the load nonlinearity.

Mathematical Model Formulation
When a probabilistic analysis is undertaken, the calculation of the problem's uncertain parameters is done through MCS. The MCS is utilized to develop a large number of probabilities of the studied parameter. For each probability, the 95th percentile of the considered parameter is calculated. In this work, each normal distribution was obtained from MCS, using 1000 samples, and considering a relevant appropriate uncertainty tolerance level for each parameter.

System Modeling
In this section, the mathematical model of the examined distribution system and its probabilistic parameters are presented. In a probabilistic mathematical environment, various types of uncertainties are present. To handle them, probabilistic distributions, such as the normal probability density function (PDF) are usually utilized to represent the variations of these uncertain parameters. In this work, the normal PDF T(x) is defined by the expected mean value (μ) and the standard deviation (σ) as follows [32]: If xmin and xmax are the minimum and maximum limits of the input random variable x so that (xmin ≤ x ≤ xmax), then the expected mean value for a 95% confidence level (CL) that the random variable x is present within its limits (μ 95 ) is obtained as follows [32]:

System Modeling
In this section, the mathematical model of the examined distribution system and its probabilistic parameters are presented. In a probabilistic mathematical environment, various types of uncertainties are present. To handle them, probabilistic distributions, such as the normal probability density function (PDF) are usually utilized to represent the variations of these uncertain parameters. In this work, the normal PDF T(x) is defined by the expected mean value (µ) and the standard deviation (σ) as follows [32]: If x min and x max are the minimum and maximum limits of the input random variable x so that (x min ≤ x ≤ x max ), then the expected mean value for a 95% confidence level (CL) that the random variable x is present within its limits (µ 95 ) is obtained as follows [32]: To simulate a practical condition, the utility grid is represented as a distorted voltage source that feeds the system with a sinusoidal fundamental supply, in addition to the superimposed background voltage harmonics.

Line Model
For the harmonic load flow calculations, the electrical feeders or lines are represented by their hth harmonic admittances. The hth harmonic admittance of the line (Y h L ) is given as follows: where Z h L is the hth impedance of the line, R h L is the line's resistance, and X h L is the hth inductive reactance of the line.

Load Model
Under the deterministic considerations, the hybrid loads are generally composed of linear and nonlinear loads. Typical linear loads are lighting, small-power, and heating loads. On the other hand, nonlinear industrial loads are typically VFDs. From a mathematical modeling perspective, the typical linear load is modeled by using a parallel arrangement of a resistance and an inductive reactance, which are obtained from the fundamental load flow analysis. Thus, the equivalent admittance of the linear load is established as follows: where P L 95 and Q L 95 are the 95th percentiles of the active and reactive powers of the load, respectively.
α l represents the linear load portion with respect to the total load composition. Consequently, α nl represents the nonlinear load portion, which varies from 0 (pure linear load) to 1 (100% nonlinear load). Accordingly, As recommended by the IEEE standard 519 [33], the nonlinear load is usually modeled by a current source injecting the corresponding harmonic currents. The magnitude of the nonlinear load's fundamental current (I 1 nl ) can be obtained by fundamental load flow analysis, as given below: Thus, for higher-order harmonic currents, the hth nonlinear load's harmonic current (I h nl ) is calculated by: where β h denotes the ratio of the hth harmonic current to the fundamental current.

PV System Model
From the harmonic load flow perspective, the DG source, represented by a PV system in this work, is usually modeled as a harmonic current source [8]; thus: where I 1 PV and I h PV are the fundamental and the hth harmonic currents of the PV system, respectively. In addition, β h PV denotes the ratio of the hth harmonic current to the fundamental current of the PV system, and S 95 PV is the 95th percentile of the injected apparent power of the PV system.

Harmonic Filter Model
The proposed C-type harmonic filter is a high-pass passive filter that guarantees proper harmonic mitigation for a wide range of harmonics, eliminates resonance risks, and has a very low power loss compared to other passive filter types as it acts as a shunt capacitor bank at the fundamental frequency [7]. Figure 2 represents the equivalent circuit of the proposed harmonic filter.

Harmonic Filter Model
The proposed C-type harmonic filter is a high-pass passive filter that guarantees proper harmonic mitigation for a wide range of harmonics, eliminates resonance risks, and has a very low power loss compared to other passive filter types as it acts as a shunt capacitor bank at the fundamental frequency [7]. Figure 2 represents the equivalent circuit of the proposed harmonic filter. As shown in Figure 2, the proposed C-type shunt passive filter consists of a main capacitor CF1 designed to provide the required reactive power support at the fundamental frequency. This main capacitor is connected in series with a double-arm combination of an inductor (LF), an auxiliary capacitor (CF2), and a resistance (RF). The inductive reactance (XLF) and the auxiliary capacitive reactance (XCF2) should be tuned at the fundamental frequency; that is, XLF = XCF2 = XF, where XF is the filter equivalent reactance at the fundamental frequency, to bypass the resistor and to operate at a remarkably low power loss at the fundamental frequency. Accordingly, the hth harmonic impedance of the filter ( h F Z ) can be calculated as the equivalent impedance of both the impedance of the main capacitor (ZCF1) and the impedance of the double-arm branch (Zshunt), as follows:

Load Flow of the System
In this work, the well-known Newton-Raphson method is used to solve the load flow problem of the test system at the fundamental frequency, in order to obtain the fundamental voltages and currents of the system. Then, harmonic power flow (HPF) analysis is performed to find the nonfundamental harmonic voltages and currents of the system. The electric utility and the studied hybrid load are modeled as the slack and PQ bus, respectively. The PV system is assumed to operate at the unity power factor (PF). As shown in Figure 2, the proposed C-type shunt passive filter consists of a main capacitor C F1 designed to provide the required reactive power support at the fundamental frequency. This main capacitor is connected in series with a double-arm combination of an inductor (L F ), an auxiliary capacitor (C F2 ), and a resistance (R F ). The inductive reactance (X LF ) and the auxiliary capacitive reactance (X CF2 ) should be tuned at the fundamental frequency; that is, X LF = X CF2 = X F , where X F is the filter equivalent reactance at the fundamental frequency, to bypass the resistor and to operate at a remarkably low power loss at the fundamental frequency. Accordingly, the hth harmonic impedance of the filter (Z h F ) can be calculated as the equivalent impedance of both the impedance of the main capacitor (Z CF1 ) and the impedance of the double-arm branch (Z shunt ), as follows:

Load Flow of the System
In this work, the well-known Newton-Raphson method is used to solve the load flow problem of the test system at the fundamental frequency, in order to obtain the fundamental voltages and currents of the system. Then, harmonic power flow (HPF) analysis is performed to find the non-fundamental harmonic voltages and currents of the system. The electric utility and the studied hybrid load are modeled as the slack and PQ bus, respectively. The PV system is assumed to operate at the unity power factor (PF).
From a conventional deterministic perspective, the hth harmonic line current vector ([I h ]) can be obtained as follows:

Problem Formulation
In this work, the main objective of the proposed optimization problem is to maximize the system's PHC, considering various possible uncertainties of parameters under non-sinusoidal operations. Therefore, the proposed C-type is optimally designed to achieve the optimization objectives mentioned earlier.
The concept of uncertainty handling in the PHC analysis is illustrated in a simplified way in Figure 3, while considering the bus voltage as an illustrative index for the PHC assessment.
From a conventional deterministic perspective, the hth harmonic line current vector ([I ]) can be obtained as follows: where [V h ] and [Y h ] are the harmonic voltage and the network's admittance matrices for the hth harmonic, respectively.

Problem Formulation
In this work, the main objective of the proposed optimization problem is to maximize the system's PHC, considering various possible uncertainties of parameters under non-sinusoidal operations. Therefore, the proposed C-type is optimally designed to achieve the optimization objectives mentioned earlier.
The concept of uncertainty handling in the PHC analysis is illustrated in a simplified way in Figure 3, while considering the bus voltage as an illustrative index for the PHC assessment. As shown in Figure 3, in the PHC analysis, different results can be achieved, such as the pessimistic result (PHCU), which represents the PHC value by using the upper uncertainty level of the bus voltage, the realistic result (PHC95), which represents the PHC value by using a high percentile, such as the 95th percentile of the bus voltage, and the optimistic result (PHCL), which represents the PHC value by using the lower uncertainty level. , , , where PHC 95 represents the 95th percentile of the PHC, and where  As shown in Figure 3, in the PHC analysis, different results can be achieved, such as the pessimistic result (PHC U ), which represents the PHC value by using the upper uncertainty level of the bus voltage, the realistic result (PHC 95 ), which represents the PHC value by using a high percentile, such as the 95th percentile of the bus voltage, and the optimistic result (PHC L ), which represents the PHC value by using the lower uncertainty level.

Objective Function
In this work, the objective function (OF) of the proposed optimization problem is to maximize the system's PHC, as formulated below: where PHC 95 represents the 95th percentile of the PHC, and where X 95 CF1 , X 95 F , and R 95 F are the 95th percentiles of the filter's main capacitive reactance, the filter's equivalent reactance at the fundamental frequency, and the filter's damping resistance, respectively. P 95 PV is the 95th percentile of the injected active power of the PV system.

Constraints
Due to the probabilistic nature of the studied network parameters, the relevant problem constraints should be selected, to account for these parameter uncertainties. Therefore, an appraisal of the problem constraints is performed by using the percentile concept to ensure that the achieved results are bounded within the preset constraints within the considered CL. In this study, five constraints are considered as follows: are the minimum and maximum bus voltages, respectively. In this work, the considered minimum and maximum bus voltage limits are 0.95 and 1.05 pu, respectively.

Line Capacity Constraint
The 95th percentile of the line's current (I 95 L,Rms ) is constrained by its maximum thermal capacity limit (I rms L,max ), as expressed in (13):

DG Capacity Constraint
The 95th percentile of the total active power produced by the DG unit is bounded by the total connected load capacity, to avoid excessive reverse power flows. In this study, 100% penetration is considered as the upper boundary for the total DG penetration [19].

Displacement and True Power Factors Constraints
The 95th percentiles of both the displacement power factor (DPF 95 ) and the power factor (PF 95 ) at the point of common coupling (PCC) have to be sustained in their satisfactory ranges, as follows: where DPF min and DPF max are the minimum and maximum DPF limits, respectively. PF min and PF max are the minimum and maximum PF limits, respectively. In this work, DPF min and PF min are considered as 0.92 lagging, while DPF max and PF max are considered as unity, respectively [34].

Harmonic Distortion Constraints
In the presence of multiple current and voltage harmonic distortion sources in the network, specific constraints should be considered, to avoid operational hazards due to excessive harmonic distortions. In our study, the IEEE Standard 519 recommended limits were followed. Accordingly, TDD 95 , THDV 95 , and the hth harmonic individual current and voltage distortions I HCD 95 h and I HVD 95 h , respectively, should be calculated and constrained as follows: where TDD max , THDV max , I HCD h max , and I HVD h max are the maximum allowable limits recommended by the IEEE 519 for TDD 95 , THDV 95 , I HCD 95 h , and I HVD 95 h , respectively. I fl is the full load current (maximum demand) under normal operating conditions. I 1 L and V 1 L represent the fundamental line current and bus voltage, respectively. V h L is the hth harmonic voltage at the PCC. The 95% percentiles of the harmonic distortion limits have been considered, in accordance with the IEEE standard 519 recommendations [33].

Search Algorithm
Recently, numerous meta-heuristic optimization algorithms have been established, such as PSO, Differential Evolution (DE), Genetic Algorithm (GA), Ant Colony (AC), Gravitational Search Algorithm (GSA), and CSA. The main aim of these algorithms is to achieve the best result (global optimum) from all achieved outcomes, in a time-effective manner. To achieve this, two key features should be included in any optimization algorithm, to help find the desired global optimum, namely exploration and exploitation [35]. In the literature, many trials have been performed, to combine together several heuristic optimization algorithms in a hybrid technique that aims to utilize the advantages of each independent technique, which helps achieve results that are superior to what can be achieved by each algorithm individually [36]. Figure 4 represents a comprehensive flowchart of the proposed problem formulation.

PSOGSA Algorithm
In this work, the PSOGSA optimization algorithm is used for the optimal design of the proposed harmonic filter. The main idea of PSOGSA is to combine PSO's social thinking (exploitation) capability with the GSA's local search (exploration) ability [37][38][39][40]. Gravitational search algorithm (GSA) is one of the recent meta-heuristic optimization algorithms that was developed to mimic the Newtonian laws of gravity and motion [39]. It has shown remarkable search abilities in solving various optimization problems. However, it still has some disadvantages, such as slow convergence, and the tendency to become stuck in the local minima. In order to merge the PSO and GSA optimization algorithms, the position-updating equation of the search agent's velocity (ν j t+1 ) is given as follows [35,37]: where X j t , ν j t , and a j t are the position, velocity, and acceleration of the jth agent at the tth iteration, respectively. r 1 , r 2 , and r 3 are uniformly distributed random numbers in the range of [0, 1]. C X and C Y are positive coefficients. X best is the best position achieved so far. N max is the maximum number of iterations.
In addition, the position of each search agent in the swarm is updated as follows: The parameters of PSOGSA considered in this work are summarized in Table 1. To validate the achieved solution through the PSOGSA algorithm, other optimization algorithms have been examined for solution validation, namely, the PSO and CSA optimization algorithms. The PSOGSA is proposed in this work, due to its superior performance in both the exploitation and exploration processes, and its faster convergence capability when compared to the widely known meta-heuristic algorithms such as PSO and CSA. Due to these benefits, the PSOGSA has been used to resolve various engineering problems in the literature [41,42].

PSO Algorithm
PSO is an evolutionary, heuristic-based optimization algorithm inspired by the social behavior of bird swarms. It is one of the most widely used optimization techniques in the literature. The main idea of the PSO algorithm is that it mathematically mimics the motion of a number of search agents (particles) which hover around in the search space to reach the target food location (best solution) [35,40]. The parameters of PSO considered in this work are summarized in Table 1.

CSA Algorithm
The CSA is a nature-inspired optimization algorithm that simulates the social performance of crows. Crows are considered as one of the smartest birds because they can remember faces, use tools, interconnect in complicated ways, and remember their food hiding places a long time later. In addition, crows are acquisitive birds and each crow attempts to steal other crows' food. The mathematical model of CSA is explained in [43][44][45]. The controlling parameters of the CSA implemented in this paper are presented in Table 1.

System Under Study
The configuration of the power distribution system considered in this work is presented in Figure 2. This system is a balanced, symmetric industrial distribution system consisting of a utility substation (slack bus), distribution feeder, and hybrid loads comprised of linear loads (induction motors) and nonlinear loads, represented by six-pulse variable frequency drives (VFDs), and DG units (represented by a PV system) interfaced with the system through power-electronic interface inverters, and a shunt C-type harmonic filter. The base voltage and apparent power of the examined system are 13.8 kV and 7.5 MVA, respectively. The slack bus voltage is 1 pu. The system harmonic limits are observed at the load bus, and thus it is considered as the PCC. This system has a total load power factor of 0.92 lagging. Thus, the rated load active and reactive power demands are considered as 0.92 and 0.39 pu, respectively. The line and load data are obtained from [7]. The ratio of nonlinear load power to total load power is considered to be 25%. The harmonic spectrum of the considered background voltage harmonics, PV nonlinear current, and nonlinear load current are given in Appendix A. It can be easily noted from [7,34] that the uncompensated distribution system suffers from excess harmonic distortions that exceed the IEEE 519 standard limits, as presented in [34] and summarized below:

•
The THDV level of the uncompensated test system is 5.4563%, which exceeds the acceptable standard limit (5%).

•
The 25th individual current harmonic (I H C D 25 ) of the uncompensated system is 1.0548%, which surpasses the maximum allowable limit (1%).

•
The 5th individual voltage harmonic (I H V D 25 ) of the uncompensated system is 3.22%, which exceeds the maximum acceptable limit (3%).
Due to the above limit violations, the uncompensated system cannot host any DG unit, because of the harmonic distortion in the system. Also, DG penetration may deteriorate the system operational indices further. To solve this problem, a proposed C-type passive harmonic filter is proposed, to mitigate the harmonic distortion problems of the uncompensated system, and consequently strengthen its capability to enhance the system's HC. Different meta-heuristic optimization techniques are investigated for the optimal design of the proposed filter, namely PSOGSA, PSO, and CSA. The following sections present the detailed results of both the DHC and the PHC assessments. Afterwards, a comparative analysis between the achieved DHC and PHC results is presented. Finally, the impact of multi-pulse VFDs on the system's PHC is presented and discussed.

DHC Results
The DHC of the studied system is investigated under the prescribed operating conditions. The proposed C-type harmonic filter design, using the PSOGSA algorithm was compared with other design methodologies, using the PSO and CSA algorithms. Table 2 presents the optimal filter parameters and the system main operational indices, using the three examined optimization techniques.
It can be concluded from the results presented in Table 2 that the proposed filter design using the PSOGSA algorithm outperformed the other designs, using PSO and CSA, as it led to higher HC levels, an enhanced voltage profile of the system, lower TDD levels, reduced THDV levels, and increased PF and DPF values. In addition, it was noticed that the optimal filter parameters achieved by the PSOGSA led to slightly higher reactive power support from the filter than the other approaches. However, this reactive power support attained a higher bus voltage at 0.991 pu. Furthermore, the filter resistance (R F ) achieved by using the PSOGSA was smaller than those obtained by other approaches, resulting in a lower filter loss, and consequently, the lowest total power loss among the studied approaches. Although a higher HC result was obtained in [7] using the conventional GA compared to the HC level obtained using PSOGSA (1% HC increase), the PF value of the uncompensated system considered in [7] was 77.43%, which was far below the system value in Table 2. Therefore, the results presented in [7] were not included in the above comparative analysis, as they were obtained by using different reference values.
The convergence of the different design algorithms in achieving the maximum DHC is shown in Figure 5. It was shown that the PSOGSA algorithm can converge to their global solution in fewer iterations, compared with the PSO and CSA algorithms. Besides, the statistical appraisal of the results achieved using the different design algorithms indicates that the PSOGSA provides better and more stable solutions compared to the PSO and CSA.
One can notice that the DHC assessment procedure performed relies on fixed values of the system parameters, with no variations in its parameters, which could be unrealistic, as the electric parameters in the real systems are subject to perpetual variations. Accordingly, the PHC is evaluated in this work, to consider the various uncertainties present in the system such as the intermittent output power of the DGs, background voltage harmonics, load alteration, and the filter parameters' variations.
Although a higher HC result was obtained in [7] using the conventional GA compared to the HC level obtained using PSOGSA (1% HC increase), the PF value of the uncompensated system considered in [7] was 77.43%, which was far below the system value in Table 2. Therefore, the results presented in [7] were not included in the above comparative analysis, as they were obtained by using different reference values.
The convergence of the different design algorithms in achieving the maximum DHC is shown in Figure 5. It was shown that the PSOGSA algorithm can converge to their global solution in fewer iterations, compared with the PSO and CSA algorithms. Besides, the statistical appraisal of the results achieved using the different design algorithms indicates that the PSOGSA provides better and more stable solutions compared to the PSO and CSA. One can notice that the DHC assessment procedure performed relies on fixed values of the system parameters, with no variations in its parameters, which could be unrealistic, as the electric parameters in the real systems are subject to perpetual variations. Accordingly, the PHC is evaluated in this work, to consider the various uncertainties present in the system such as the intermittent output power of the DGs, background voltage harmonics, load alteration, and the filter parameters' variations.

PHC Results
HC uncertainty may arise, due to many aspects such as unknown DG sites and capacities, the DG's output power intermittency, load alteration, and the absence of confirmed data that is used to conduct HC analysis. Accordingly, the HC assessment should not be a handled as a deterministic problem with no randomness. However, it has to be tackled from a probabilistic perspective, considering relevant parameter uncertainties. This conclusion has been recently verified by numerous studies, as it was concluded that DHC studies, which ignore the uncertainty of electrical parameters, resulting in conservative HC levels that usually lead to a noticeable underestimation of the HC levels [28][29][30].
To perform a PHC analysis, the MCS is utilized to generate an appropriate number of probabilities for the studied uncertain parameters. For each parameter, the 95th percentile is calculated over the entire number of examined MCS iterations. In this work, 1000 MCS iterations were considered for each uncertain parameter.
Afterwards, a scenario-based framework was formulated to execute the fundamental load flow calculations, and the HPF utilizing the obtained high percentiles of the examined parameters. The detailed procedure of the PHC analysis performed is presented in Figure 4. Various tolerance levels have been considered for the relevant uncertain parameters depending on the practical variation probabilities of each parameter. Table 3 shows the considered tolerances that are used for the deviations of the system parameters and filter parameters from their mean values. The probabilistic load active and reactive powers, and their corresponding 95th percentiles, are shown in Figure 6. As shown in Table 3, large uncertainty tolerances equal to ±10% have been considered for the load active and reactive powers, to reflect the practical uncertainty encountered with the daily load alteration. The 95th percentiles of the load active and reactive powers have been calculated by the MCS approach, considering 1000 MC samples, as presented in Figure 6.   Figure 7a,b represent the histograms and relevant PDFs of both the probabilistic TDD and THDV results, with means of 7.44% and 4.429%, and standard deviations of 0.1446 and 0.036, respectively. The probabilistic results of both DPF and PF are constrained by the pre-set problem's constraints. Thus, some samples were discarded, as they violated the problem constraints, which resulted in non-normally distributed results that were skewed (non-symmetric) to the left, as presented in Figure 7c,d, respectively. Based on the best-fit probability distribution procedure for the DPF and PF distributed functions; the smallest extreme value probability distribution was chosen as it best fits their probabilistic distribution in terms of the well-known measures such as AD, and p-value. For the DPF distribution, the location and scale values were 99.32 and 1.041, respectively, while for the PF distribution, the location and scale values are 95.87 and 1.49, respectively.
The probabilistic rms bus voltage (VB rms ) results are shown in Figure 7e, with a mean of 0.9929 pu and a standard deviation of 0.0023 pu. For the sake of clarity, the CDF of the PHC results is presented in Figure 7f. (b) The probabilistic rms bus voltage (VBrms) results are shown in Figure 7e, with a mean of 0.9929 pu and a standard deviation of 0.0023 pu. For the sake of clarity, the CDF of the PHC results is presented in Figure 7f.
As shown in Table 4, under the probabilistic approach, the proposed filter design using the PSOGSA algorithm outperformed the other designs, using PSO and CSA, as it attained higher PHC levels. Besides, the DPF, PF, and voltage profile obtained, using the filter designed using the PSOGSA, were higher than those obtained with other approaches, and thus, an enhanced voltage level was achieved using this design. Furthermore, the filter-optimized resistance (RF) obtained using As shown in Table 4, under the probabilistic approach, the proposed filter design using the PSOGSA algorithm outperformed the other designs, using PSO and CSA, as it attained higher PHC levels. Besides, the DPF, PF, and voltage profile obtained, using the filter designed using the PSOGSA, were higher than those obtained with other approaches, and thus, an enhanced voltage level was achieved using this design. Furthermore, the filter-optimized resistance (R F ) obtained using the PSOGSA was lower than that obtained by other approaches, which resulted in lower filter loss, and therefore the lowest total power loss among the studied approaches. It can also be noticed that the CSA and PSO succeeded in achieving better TDD and THDV levels, respectively. In addition, comparing the DHC results in Table 2 and the PHC results in Table 4, one can clearly notice that the PHC levels achieved were higher than the DHC levels, as described earlier. Figure 8 presents an overall comparison between the achieved DHC and PHC results achieved using the three filter design approaches. As shown, it can be concluded that the PHC level achieved using the PSOGSA was 39.7% higher than the DHC result achieved by using the same design approach.   Furthermore, the system performance was examined under different operational conditions considering the proposed filter design using PSOGSA. Two operational parameters were been considered in this test, namely, the load nonlinearity level (NLL) and the background utility voltage distortion (BVD) level. The effect of varying the NLL and the BVD on the corresponding PHC level is presented in Figure 9a, while the effect of varying the operational parameters TDD and THDV on the corresponding PHC level is presented in Figure 9b.
In Figure 9a, the NLL varies from 20 to 30% with a step of 2.5%, whereas the original design is performed with 25% NLL, as explained earlier. In addition, the BVD varies from 0 to 100%m with a step of 25%, whereas the original design performed at 100% BVD level. It can be noticed that the PHC level decreases, with increases in the NLL. For example, at a BVD level of 0%, the PHC level decreased from 73% at 20% NLL to 29% at 30% NLL. It can also be concluded that the proposed C-type harmonic filter succeeded in operating satisfactorily at heavy harmonic signatures (high BVD and high NLL) simultaneously. Finally, it is clear that varying the NLL has a larger impact on the resulting PHC than variation of the BVD.
The TDD and THDV levels resulting from the above validation are presented in Figure 9b with the corresponding PHC. One can notice that the PHC level was reduced gradually with simultaneous increases of both the TDD and THDV levels. For example, the PHC level was reduced from 76% at Furthermore, the system performance was examined under different operational conditions considering the proposed filter design using PSOGSA. Two operational parameters were been considered in this test, namely, the load nonlinearity level (NLL) and the background utility voltage distortion (BVD) level. The effect of varying the NLL and the BVD on the corresponding PHC level is presented in Figure 9a, while the effect of varying the operational parameters TDD and THDV on the corresponding PHC level is presented in Figure 9b.
In Figure 9a, the NLL varies from 20 to 30% with a step of 2.5%, whereas the original design is performed with 25% NLL, as explained earlier. In addition, the BVD varies from 0 to 100%m with a step of 25%, whereas the original design performed at 100% BVD level. It can be noticed that the PHC level decreases, with increases in the NLL. For example, at a BVD level of 0%, the PHC level decreased from 73% at 20% NLL to 29% at 30% NLL. It can also be concluded that the proposed C-type harmonic filter succeeded in operating satisfactorily at heavy harmonic signatures (high BVD and high NLL) simultaneously. Finally, it is clear that varying the NLL has a larger impact on the resulting PHC than variation of the BVD.
The TDD and THDV levels resulting from the above validation are presented in Figure 9b with the corresponding PHC. One can notice that the PHC level was reduced gradually with simultaneous increases of both the TDD and THDV levels. For example, the PHC level was reduced from 76% at 7.1% TDD and 3.9% THDV to 25% at 7.1% TDD and 3.9% THDV respectively, as shown in Figure 9b.

Effect of Utilizing Multi-Pulse VFD Configurations on the System's PHC
A multi-pulse VFD is a drive that utilizes special transformer connections to cancel specific harmonics. On the market, common types of multi-pulse VFDs are available, such as the 6-, 12-, 18-, and 24 VFDs. The main advantage of using a multi-pulse VFD is that the harmonics are eliminated from their source, thus avoiding their propagation into the electrical network. However, the higher the number of pulses of a VFD, the higher the complexity and cost of the drive [46]. The impact of multi-pulse VFDs on the system's PHC is investigated in this work; to the best of the authors' knowledge, this has not been studied before in the literature. Three configurations of multi-pulse VFDs were studied, namely 6-, 12-, and 18-pulse VFDs. Table 5 presents the current harmonic

Effect of Utilizing Multi-Pulse VFD Configurations on the System's PHC
A multi-pulse VFD is a drive that utilizes special transformer connections to cancel specific harmonics. On the market, common types of multi-pulse VFDs are available, such as the 6-, 12-, 18-, and 24 VFDs. The main advantage of using a multi-pulse VFD is that the harmonics are eliminated from their source, thus avoiding their propagation into the electrical network. However, the higher the number of pulses of a VFD, the higher the complexity and cost of the drive [46]. The impact of multi-pulse VFDs on the system's PHC is investigated in this work; to the best of the authors' knowledge, this has not been studied before in the literature. Three configurations of multi-pulse VFDs were studied, namely 6-, 12-, and 18-pulse VFDs. Table 5 presents the current harmonic spectrum of the examined multi-pulse VFDs. It can be concluded from the results presented in Table 6 that the use of multi-pulse VFDs enhances the system's PHC. Using 12-pulse VFD resulted in a PHC enhancement of 3.11%, compared to the conventional 6-pulse VFD. In addition, using the 18-pulse VFD led to a slight PHC enhancement of 3.44%, compared to the 6-pulse VFD. One can notice that the achieved PHC enhancement was bottle-necked by the system's DPF constraint, as almost reached its maximum allowable limit (100%) in the cases of both the 12-pulse VFD and the 18-pulse VFD. This slight enhancement of the PHC level cannot be practically feasible from an economic perspective, as the cost of an 18-pulse VFD is approximately 2.5 times the cost of a 6-pulse VFD. However, further PHC enhancements can be achieved in other case studies, depending on the system data.

Conclusions
In this study, the PHC of a harmonic-distorted power distribution system subjected to high PV penetration is evaluated utilizing the Monte Carlo simulation. A proposed C-type harmonic filter is proposed to maximize the PHC of the examined system. An optimization problem is formulated considering various uncertainty indices, such as the intermittent output power of the DGs, background voltage harmonics, load alterations, and the filter parameters' uncertainties. The bus voltage limits, line thermal capacity, power factor, and harmonic distortion limits have been considered as the problem constraints. The PSOGSA optimization algorithm has been used for the optimal design of the proposed filter. The proposed filter design using PSOGSA was compared with other design approaches using PSO and CSA, and it was found that the proposed design outperforms the compared design approaches. Further, the impact of multi-pulse VFDs on the system's PHC is investigated. The PHC results obtained were compared with the conventional DHC results and it was found that the PHC level achieved using the PSOGSA design approach was 39.7% higher than the DHC level achieved by using the same design approach. Recently, it was concluded that DHC studies, which ignore the uncertainty of electrical parameters, result in optimistic results that cause a noticeable underestimation to the HC levels that are achieved from probabilistic studies.
Our study was limited to fixed, time-independent loading profiles and PV output power, and their effect on the performance of a balanced power distribution system. Furthermore, the consideration of a chronological demand profile with time-dependent DG systems in unbalanced power distribution systems, utilizing multi-objective decision-making techniques was beyond the framework of this study, and this will be included in future studies.

Acknowledgments:
The authors would like to thank the reviewers for their constructive comments and suggestions.

Conflicts of Interest:
The authors declare no conflict of interest.  The ratio of the hth harmonic current to the fundamental current I 1

PV
Fundamental harmonic current of the PV system I h

PV
The hth harmonic current of the PV system β h PV The ratio of the hth harmonic current to the fundamental current of the PV system S 95

PV
The 95th percentile of the injected apparent power of the PV system C F1 Filter's main capacitor L F Filter's inductor C F2 Filter's auxiliary capacitor R F Filter's resistance X LF Filter's inductive reactance X CF2 Auxiliary capacitive reactance of the filter X F Filter's equivalent reactance at the fundamental frequency Z h F The hth harmonic impedance of the filter Z shunt Impedance of the double-arm branch Z CF1 Impedance of the main capacitor PHC 95 The 95th percentile of the PHC X 95

CF1
The 95th percentiles of the filter's main capacitive reactance X 95 The 95th percentile of the displacement power factor PF 95 The 95th percentile of the power factor DPF min Minimum DPF limit DPF max Maximum DPF limit PF min Minimum PF limit PF max Maximum PF limit TDD max Maximum limit for the 95th percentile of total demand distortion TDD 95 THDV max Maximum limit for the 95th percentile of total harmonic voltage distortion THDV 95 Acceleration of the jth agent at the tth iteration r 1 , r 2 , r 3 Uniformly distributed random numbers in the range of [0, 1] C X , C Y Positive coefficients X best Best position achieved so far N max Maximum number of iterations