Resonance-Aware Power Factor Correction in Transmission Networks Using Weighted Indices and Tuned Passive Filters for Harmonic Mitigation

Abstract

Power factor correction in transmission networks with nonlinear loads cannot be addressed solely from the viewpoint of reactive compensation because harmonic distortion and resonance may compromise the expected technical benefits. In this context, this study proposes a resonance-aware and decision-oriented methodology that integrates nonlinear-load screening, weighted bus prioritization based on power factor degradation and harmonic severity, and tuned passive-filter design validated through impedance-frequency analysis and IEEE 519 compliance criteria. The methodology was implemented in DIgSILENT PowerFactory using the IEEE 14-bus test system, where nonlinear loads were allocated at buses 9 and 14 to emulate converter-dominated operating conditions. Under this scenario, the power factor decreased to 0.78271 and 0.85875, while total harmonic distortion increased to 22.01% and 20.07%, respectively. After the implementation of tuned passive filters, the power factor improved to 0.83023 at bus 9 and 0.90414 at bus 14, whereas total harmonic distortion was reduced to 4.61% and 5.22%, respectively, thus restoring compliance with IEEE 519. In addition, load currents decreased by approximately 16–19%. These results demonstrate that the proposed framework provides a technically consistent procedure for identifying critical buses, mitigating dominant harmonics, improving power factor, and avoiding adverse resonance conditions within a unified compensation workflow.

1. Introduction

The technological transformation of modern power systems has substantially altered the structural and operational paradigms of conventional electrical networks. This transition has been driven mainly by the increasing penetration of renewable energy sources, the widespread integration of power-electronic-based loads, and the deployment of energy storage systems. Although these developments improve flexibility and controllability, they also intensify several power-quality challenges, among which harmonic proliferation and power factor deterioration are particularly significant [1].

Power factor (PF) is one of the most relevant indicators for assessing the effective utilization of electrical energy and, consequently, the energy efficiency of a power system. A low PF reflects an increased reactive power demand, which leads to higher electrical losses, transformer overloading, and a reduction in the effective transmission capacity of the network [2]. In systems supplying inductive loads and equipment with a high penetration of power electronic interfaces, such as variable frequency drives and switched-mode power supplies, harmonic currents are introduced into the network. These distortions not only degrade the PF, but also create operating conditions that may promote resonance phenomena, thereby amplifying waveform distortion and further compromising power quality [3].

In this context, IEEE Std 519-2022 [4] establishes recommended limits for harmonic distortion in both current and voltage. Compliance with these limits is essential to prevent equipment damage, preserve electromagnetic compatibility, and ensure the proper operation of interconnected electrical devices without causing harmful interference to the rest of the system [4]. Nevertheless, recent studies have shown that, in transmission networks with a considerable concentration of nonlinear loads, total harmonic distortion (THD) commonly exceeds 10%, while the PF may fall below 0.9, thus violating internationally accepted operating criteria and power-quality requirements [5,6].

According to the International Energy Agency (IEA, 2023), approximately one-fifth of the energy losses in low-voltage networks is associated with poor PF conditions and the presence of harmonic distortion in the electrical system [7]. This issue is particularly critical in industrial environments, where more than 60% of the installed infrastructure relies on electronic devices based on power converters or switched-mode supplies, which typically introduce harmonic distortion levels in the range of 10% to 15% [8]. Therefore, the combined effect of reactive power demand and waveform distortion has become a central concern in the analysis and operation of contemporary power systems.

The problem becomes even more complex in networks containing multiple nonlinear loads. In such conditions, the interaction between network inductances and the capacitances introduced by compensation devices or filters may produce resonant circuits capable of amplifying specific harmonic components [9]. Consequently, reactive power compensation through capacitor banks cannot be addressed solely from a steady-state reactive power perspective. If the system impedance is not characterized beforehand, an improper tuning decision may intensify harmonic distortion and compromise the transient and dynamic performance of the network [10].

This technical contradiction has been evidenced in previous investigations based on the IEEE 14-bus test system, where the installation of non-tuned passive filters, despite improving the global PF, may significantly increase THD at particular buses of the network [11]. Such findings demonstrate that PF correction cannot be treated independently from harmonic behavior and resonance conditions. Instead, a coordinated methodology is required to jointly address reactive power compensation, harmonic mitigation, and network resonance constraints under a unified technical framework.

From an economic standpoint, the consequences are also substantial. Industrial users may face a twofold financial penalty: first, due to utility charges associated with low PF operation; and second, due to additional technical losses, which may account for between 3% and 7% of the monthly electricity bill [12]. From a technical perspective, low-order harmonics, especially the 5th and 7th components, are particularly harmful because they overload transformers, reduce their service life by more than 20%, increase conductor heating losses, and may trigger unnecessary protective relay operations, thereby affecting service continuity and system reliability [13].

Accordingly, the central challenge is to improve the PF without introducing resonance phenomena while simultaneously reducing the harmonic distortion caused by nonlinear loads. Addressing this dual requirement demands an integrated and technically consistent decision-making framework. In this regard, the use of weighted indices based on PF and THD provides an effective mechanism for assessing the electrical criticality of each bus and prioritizing corrective actions according to measurable power-quality indicators [14].

Based on this premise, the present study adopts the IEEE 14-bus system as a benchmark test network for the analysis of compensation strategies and harmonic mitigation. The main objective is to develop and validate a methodology capable of improving the PF, reducing harmonic distortion, and ensuring compliance with IEEE 519 [4], while explicitly considering the risk of harmonic resonance under nonlinear loading conditions. In this way, the study aims to provide a decision-oriented framework that integrates regulatory compliance, technical effectiveness, and operational feasibility [15].

Despite the considerable progress reported in the literature, existing studies usually address power factor correction, harmonic mitigation, and resonance control in a partial manner rather than within a unified decision framework. Recent contributions on resonance suppression and impedance-stability improvement mainly emphasize converter control and active damping strategies, which are technically valuable but are not directly formulated as bus-prioritization procedures for network-wide compensation planning [16,17,18]. In parallel, hybrid and active filtering approaches have demonstrated strong harmonic-elimination capability, particularly under variable operating conditions, but these solutions generally involve higher implementation complexity and are not always presented together with a systematic criterion for selecting the most critical buses requiring intervention [19,20,21]. In this context, the harmonic-control limits, benchmark-system data, and six-pulse converter spectrum adopted in this study are supported by standard and benchmark references [22,23,24].

At the same time, recent reviews on harmonic mitigation in converter-dominated systems confirm that passive, active, and hybrid solutions must be selected according to the dominant harmonic spectrum, network characteristics, and implementation constraints [25,26,27]. For systems dominated by well-defined low-order harmonics, such as the 5th and 7th components typically associated with six-pulse converters, tuned shunt passive filters remain an attractive option because of their selective attenuation capability, simple structure, and practical feasibility [28,29]. In addition, IEEE guidance for harmonic filters, shunt capacitors, and practical capacitor-bank selection provides a technical basis for designing such solutions under realistic implementation constraints rather than purely theoretical compensation targets [30,31]. More broadly, recent power-quality enhancement strategies also confirm the need to select compensation solutions according to the specific distortion scenario and implementation constraints [32].

Therefore, the research gap addressed in this work is not merely the need for another compensation device, but the lack of an integrated methodology capable of answering, in a technically consistent manner, three practical questions: which buses should be prioritized first, which measurable indicators should govern that decision, and how the selected compensation strategy can improve power factor and harmonic performance without aggravating resonance conditions. In this context, the novelty of the present study lies in combining nonlinear-load screening, weighted bus prioritization based on PF deterioration and harmonic severity, impedance-frequency resonance assessment, and tuned passive-filter design within a single reproducible framework validated against IEEE 519 compliance criteria.

To provide a more explicit comparison with representative previous studies,

Table 1. Comparative summary of representative previous studies related to power factor correction, harmonic mitigation, and resonance-aware compensation.

Reference	Main Emphasis/Reported Contribution	Solution Family	Main Limitation Relative to the Present Work
[16,17,18]	Resonance suppression, impedance-frequency behavior, and resonance identification in distorted electrical environments.	Resonance analysis and control-oriented strategies	These works clarify resonance mechanisms and mitigation needs, but they are not formulated as a complete bus-prioritization and compensation-design framework for transmission networks.
[19,20,21]	Comparative discussion of passive, active, and hybrid compensation alternatives, including dynamic filtering capability and resonance-aware design considerations.	Passive, active, and hybrid filtering	They provide valuable compensation insights, but they do not integrate candidate-bus screening, weighted prioritization, and post-compensation validation within a unified procedure.
[25]	Recent review of harmonic elimination and mitigation techniques in converter-based power systems.	State-of-the-art review	It offers a broad technical overview, but it does not propose a reproducible methodology for selecting where compensation should be applied first in a networked transmission setting.
[28]	Demonstrates the effectiveness of passive harmonic filters for mitigating dominant low-order harmonics in an industrial power-system application.	Tuned passive filtering	It supports the technical feasibility of passive filtering, but it is application-specific and does not combine resonance assessment with weighted electrical prioritization of buses.
[29,30,31]	IEEE guidance for harmonic filters and shunt capacitors, including practical design and specification criteria.	Engineering standards and application guides	These standards support component selection and implementation practice, but they do not define a network-level decision methodology for ranking critical buses under nonlinear loading.
[32]	Review of modern strategies for power quality enhancement, from conventional devices to advanced and AI-based approaches.	Broad review of power-quality solutions	It confirms the diversity of available solutions, but it does not establish an integrated resonance-aware workflow linking diagnosis, prioritization, filter design, and validation.

summarizes the main emphasis, solution family, and scope limitations of selected references that are directly relevant to harmonic mitigation, resonance control, and compensation design under nonlinear loading conditions.

As shown in Table 1, previous studies have contributed important knowledge on harmonic mitigation devices, resonance behavior, and power-quality improvement. However, the literature still lacks a compact and reproducible workflow that jointly determines the most critical buses, prioritizes them through measurable electrical indicators, and designs tuned passive filters under explicit resonance and IEEE 519 compliance constraints. This is the specific methodological space addressed by the present work.

The main contributions of this research are summarized as follows:

• A systematic methodology is proposed for power factor correction in transmission networks with nonlinear loads, integrating harmonic assessment, weighted prioritization indices, and resonance-aware passive filter design.
• A bus-ranking strategy is developed by combining PF- and THD-based criteria, allowing the identification of the most critical locations for compensation.
• The proposed methodology is validated in the IEEE 14-bus test system implemented in DIgSILENT PowerFactory, considering nonlinear disturbances associated with six-pulse converter behavior.
• Tuned passive filters are designed and assessed not only from the viewpoint of reactive compensation, but also in terms of harmonic attenuation and resonance prevention.
• The obtained results demonstrate that the proposed framework improves the PF, reduces THD below the IEEE 519 admissible limit, and decreases load currents, thereby enhancing both network efficiency and power quality.

The novelty of the present work lies not in the isolated use of tuned passive filters, which are well established in the literature, but in the formulation of an integrated and reproducible methodology that combines candidate-bus screening, weighted prioritization, resonance-aware design, and IEEE 519-based validation within a single decision framework for transmission networks under nonlinear loading.

The remainder of this paper is organized as follows. Section 2 presents the Theoretical Foundation, including the concepts of power factor, nonlinear loads, harmonic sources, resonance phenomena, compensation techniques, and harmonic control standards. Section 3 describes the Methodology, including the test-system configuration, harmonic assessment procedure, multi-criteria prioritization approach, and passive filter design strategy. Section 4 provides the Results and Discussion, where the performance of the proposed compensation scheme is evaluated under the considered operating scenarios. Finally, Section 5 summarizes the main findings and presents the Conclusions.

2. Theoretical Foundation

This section establishes the theoretical basis required to analyze power factor correction in transmission networks affected by nonlinear loading. Its purpose is not to provide a general or exhaustive review of common concepts, but to introduce only the technical elements that are strictly necessary to support the proposed methodology. In particular, the section provides the conceptual basis for: (i) interpreting the deterioration of power factor under nonsinusoidal conditions; (ii) justifying the six-pulse converter spectrum adopted for nonlinear-load representation; (iii) understanding the impedance-frequency mechanisms that govern resonance risk; and (iv) supporting the selection of tuned passive filters as the compensation strategy evaluated in this work.

Accordingly, the discussion is organized around five technical pillars: the relation between power factor and power quality, the behavior of nonlinear loads, the main harmonic sources and their effects, the resonance phenomenon in networks containing reactive compensation devices, and the principal compensation techniques used for harmonic mitigation and power factor improvement. This foundation is necessary to justify the proposed methodology, particularly the combined use of power-factor- and THD-based indices together with resonance-aware passive filter design.

2.1. Power Factor and Power Quality in Electrical Networks

Power factor is one of the most relevant indicators for evaluating the effective use of electrical energy and the overall operating efficiency of a power system. In modern networks, however, the increasing penetration of power electronic devices degrades the power factor through two simultaneous mechanisms: the phase displacement between voltage and current at the fundamental frequency, and waveform distortion caused by harmonic components introduced by nonlinear converters [33]. As a result, the total power factor may be significantly lower than unity, even when the displacement power factor remains within acceptable limits.

Under nonlinear operating conditions, the power factor can be rigorously interpreted according to IEEE 1459-2022 [34] as the product of a displacement component and a distortion-related component. This decomposition is important because it separates the contribution of reactive power exchange at the fundamental frequency from the contribution associated with harmonic distortion [34]. In practical terms, a nonsinusoidal current increases the apparent power without a proportional increase in useful active power, which directly deteriorates the overall power factor. This physical interpretation is conceptually illustrated in Figure 1, where the distortion of the current waveform increases the apparent power demand of the load. Previous studies have reported that the simultaneous operation of multiple converters may reduce the power factor by up to 12% relative to nominal conditions, particularly in unbalanced operating scenarios [35].

Accordingly, the total power factor can be expressed as

$$P{F}_{\mathrm{T}}=P{F}_{\mathrm{disp}}P{F}_{\mathrm{dist}},$$

(1)

where $P{F}_{\mathrm{disp}}$ is the displacement power factor and $P{F}_{\mathrm{dist}}$ is the distortion power factor.

The displacement component is defined as

$$P{F}_{\mathrm{disp}}={\displaystyle \frac{P}{\sqrt{P^2+Q^2}}},$$

(2)

where P is the active power and Q is the reactive power at the fundamental frequency.

The distortion component is written as

$$P{F}_{\mathrm{dist}}={\displaystyle \frac{1}{\sqrt{1+{\left(\frac{TH{D}_i}{100}\right)}^2}}},$$

(3)

where $TH{D}_i$ denotes the total harmonic distortion of current, expressed as a percentage. Equations (1)–(3) show that the deterioration of the total power factor in nonlinear systems cannot be attributed exclusively to reactive power, but must also be interpreted in terms of harmonic distortion. In this article, the notation $THD$ is used in a generic sense when referring to total harmonic distortion as a general power-quality concept, whereas $TH{D}_i$ is used when the distortion is explicitly associated with bus i. This convention is adopted consistently in the mathematical formulation, the prioritization procedure, and the discussion of bus-specific results.

2.2. Nonlinear Loads and Their Implications

The presence of total harmonic distortion in power systems is strongly associated with the integration of nonlinear loads, especially in networks with renewable generation, electronically controlled demand, and converter-interfaced equipment. Unlike linear loads, nonlinear devices such as variable frequency drives, inverters, and switched-mode power supplies do not draw a sinusoidal current even when supplied with a sinusoidal voltage [36]. Consequently, the current waveform becomes distorted, the root-mean-square current increases, and the electrical interaction between voltage, current, and apparent power becomes more complex.

As suggested by the conceptual behavior presented in Figure 1, the integration of nonlinear loads does not merely increase harmonic distortion; it also modifies reactive power exchange and enlarges the effective current demanded from the network. Therefore, power factor deterioration in such systems is no longer a purely reactive phenomenon, but rather the result of the combined effect of displacement and distortion.

The severity of the distortion depends on the load composition and on the way nonlinear devices are distributed throughout the network. In systems containing a large number of unfiltered DC/AC converters, the total harmonic distortion may exceed 15% at heavily stressed buses. Under single-phase nonlinear operation, an uneven distribution of nonlinear loads may even raise THD beyond 25%, while also introducing zero-sequence and negative-sequence harmonic components that are not typically present in balanced three-phase nonlinear operation [37].

From a physical viewpoint, nonlinear loads exhibit an effective impedance that varies with the instantaneous value of voltage and with the internal switching state of semiconductor devices. This nonlinear behavior breaks the proportional relation between voltage and current over a cycle, so that a distorted current waveform is produced even when the applied voltage remains nearly sinusoidal. Such behavior is characteristic of rectifier-based structures used in both single-phase and three-phase electrical systems, which are widely employed in complex electronic equipment and have a direct impact on power-quality indices at the point of common coupling [38].

Typical nonlinear loads include industrial electronic equipment, frequency converters, rectifiers, and switched-mode power supplies. These circuits commonly incorporate diodes, thyristors, transistors, and other switching devices. As a result, the most relevant distortions are usually concentrated in the lower-order harmonics, especially the 5th and 7th components, which are known to increase transformer and line heating, distort the supply voltage seen by sensitive loads, and ultimately reduce the reliability and quality of modern power systems [39]. Figure 2 illustrates the relationship between the fundamental waveform and its harmonic distortion.

2.3. Harmonic Sources and Their Effects

Power-electronic-based control equipment is one of the main sources of waveform distortion in modern electrical networks. Advances in power semiconductor technology have enabled high-performance energy conversion and control, improving efficiency and controllability in industrial and utility applications. However, the widespread use of these devices has also intensified the injection of harmonic components into voltage and current waveforms [40].

From a frequency-domain perspective, harmonic-related disturbances can be classified into two broad categories. The first comprises components below the fundamental frequency, commonly referred to as subharmonics, which are associated with low-frequency oscillatory phenomena and flicker. The second comprises components above the fundamental frequency, which are typically introduced by rectifiers, inverters, frequency converters, and static compensation devices. In practical harmonic studies, the second category is of primary interest because it directly affects THD, current loading, losses, and resonance conditions in the network.

In addition, the use of reactive compensation devices near harmonic-producing equipment may intensify the resonance risk, since the interaction between system inductances and local capacitances can amplify the existing harmonic content. For this reason, harmonic source identification must be linked to the electrical environment in which compensation is later implemented.

For the present study, the selected nonlinear source is a static power converter based on solid-state switching devices, namely a controlled three-phase rectifier. This choice is technically justified because such converters are representative of the harmonic behavior commonly observed in industrial networks and in converter-dominated operating conditions.

A controlled three-phase rectifier converts three-phase alternating current into an adjustable direct-current output by controlling the firing angle of thyristors. In contrast to an uncontrolled diode rectifier, this configuration allows output regulation and, under certain operating conditions, reversible power flow. Its applications are particularly relevant in high-power motor drives and industrial processes requiring variable-speed operation and high torque, such as rolling mills, grinders, and cranes [41].

For the case study considered in this work, a six-pulse controlled rectifier is adopted. In this topology, conduction is transferred among switching devices every 60° electrical, and the full-bridge structure uses the three supply phases to generate the DC output. This rectifier is widely used in medium- and high-power applications, including DC motor drives, frequency converters, and uninterruptible power supply systems because it provides a robust and efficient conversion stage with reduced output ripple [42,43]. Nevertheless, its main drawback is the injection of characteristic low-order harmonics, particularly the 5th and 7th components, which may cause transformer overheating, communication interference, and malfunction of sensitive equipment unless suitable filtering is provided in accordance with IEEE 519 requirements [4].

To support this discussion, Figure 3 presents the full-bridge six-pulse rectifier topology adopted in this study. The corresponding non-sinusoidal current waveform, shown in Figure 4, explains the origin of the dominant harmonics generated by this converter and justifies its selection as a representative distortion source in the proposed analysis.

Another relevant source of harmonics is the variable frequency drive (VFD), which controls the speed of AC motors by coordinated adjustment of supply frequency and voltage. Its structure typically consists of three stages: an AC/DC rectifier, an intermediate DC link for energy storage and filtering, and a DC/AC inverter based on PWM switching with IGBTs [44]. This architecture converts a fixed-frequency input into a variable-frequency output and is therefore widely used in pumping systems, ventilation, compressors, and conveyor applications.

From an operational viewpoint, VFDs offer substantial energy-saving benefits. For example, a 20% reduction in motor speed may yield energy savings on the order of 50%, while soft-start capability also reduces current peaks, mechanical stress, and maintenance requirements [43]. However, these benefits are accompanied by power-quality challenges, since VFDs typically inject harmonic components such as the 5th, 7th, and 11th orders. Their mitigation requires the use of passive or active filters, line reactors, or higher-pulse configurations such as 12-pulse or 18-pulse arrangements.

Because the harmonic behavior of a VFD is largely determined by its internal conversion stages, the representative AC–DC–DC link–DC–AC structure is shown in Figure 5. This architecture explains the origin of the most relevant low-order harmonics and their subsequent effect on THD and power factor.

As previously discussed, harmonics increase the apparent power demanded by the load and, consequently, the current required from the network. This condition may force the use of larger conductors, higher-rated protection devices, and transformers with greater power capacity. Since conductor losses are proportional to the square of current, an increase in harmonic current directly raises Joule losses in the system [45].

In addition, higher-frequency harmonic components intensify the skin effect, concentrating current near the outer surface of the conductor and reducing the effective current-carrying cross section. This phenomenon increases the effective resistance of the conductor and may lead to thermal overloading or excessive conductor oversizing [46]. Likewise, certain zero-sequence harmonics may accumulate in the neutral conductor because they are phase-aligned across phases, which justifies neutral conductor oversizing and more careful protection coordination in distorted operating environments.

2.4. Resonance Phenomenon

Harmonic resonance occurs when the frequency of an injected harmonic component approaches or coincides with the natural frequency of an equivalent inductive-capacitive network. In practical power systems, this resonance condition is not fixed, since it depends on system topology, operating point, load composition, and the location of capacitive compensation devices. In modern networks containing converter-based loads, this interaction becomes even more critical because harmonic sources continuously interact with the frequency-dependent system impedance [16].

The installation of capacitor banks in highly distorted systems may cause parallel resonance, a condition in which the equivalent system impedance reaches a maximum at a particular frequency. Under such circumstances, even relatively small harmonic injections may produce large voltage distortions, overcurrents, or overvoltages, thereby damaging sensitive equipment and worsening power quality [17]. This is one of the main reasons why reactive compensation cannot be designed solely from a fundamental-frequency standpoint.

A useful tool for detecting and preventing these conditions is the impedance-versus-frequency analysis, also known as frequency sweep analysis. By scanning the network response over a specified frequency range, this method identifies peaks and valleys in the impedance profile and therefore reveals the possible occurrence of parallel and series resonance. This approach is fundamental when selecting capacitor sizes and reactor values, since the tuning must ensure that the resonant frequencies are displaced away from the dominant harmonic orders [18].

In addition to parallel resonance, series resonance may also arise when the equivalent impedance becomes minimum at a specific harmonic frequency. In such a case, large currents may circulate between capacitive and inductive elements, potentially overloading and damaging the associated components. Therefore, resonance may be understood as a frequency-selective amplification mechanism that alters both current magnitude and system impedance whenever the excitation frequency approaches the natural frequency of the equivalent L-C structure. Figure 6 conceptually illustrates this phenomenon and highlights why no compensation strategy should be formulated without a prior impedance sweep assessment.

In both series and parallel resonance, the key design variable is the relationship between the resonant frequency $f_r$ and the characteristic harmonic frequencies injected by nonlinear loads. For this reason, effective harmonic mitigation requires compensation devices capable not only of supplying reactive power, but also of controlling the frequency response of the network in a predictable and stable way.

2.5. Compensation Techniques and Power Factor Correction

Power factor correction may be implemented through capacitor banks, passive filters, active filters, or hybrid compensation schemes. Conventional compensation systems often use fixed or step-switched capacitor banks to supply reactive power according to load demand. However, in the presence of nonlinear loads, the technical assessment of a compensation device cannot be restricted to its capacitive contribution because its interaction with harmonic sources and network impedance must also be considered.

To provide a comparative view of the main compensation alternatives, Figure 7 summarizes the most common schemes used in practice. This comparison is relevant because it highlights that the best compensation option depends not only on the reactive demand of the load, but also on the distortion profile and the resonance susceptibility of the network.

Passive filters, which are composed of capacitors and reactors tuned to specific harmonic frequencies, are a cost-effective solution for supplying reactive power while attenuating dominant harmonics [19]. By contrast, active filters use digitally controlled power-electronic converters to inject compensating currents that counteract both reactive and harmonic current components. Their main advantage lies in their dynamic response, which allows them to adapt to time-varying operating conditions, fluctuating load levels, and changing harmonic spectra [20].

Despite their advantages, compensation devices must be designed with prior resonance analysis. Recent studies have shown that an improperly tuned passive filter may increase THD by approximately 30% even when the global power factor improves. This adverse behavior occurs when the new compensation elements shift the system resonance toward one of the characteristic harmonic frequencies already present in the network. As a consequence, parallel resonance may amplify the harmonic distortion beyond admissible power-quality limits. To prevent such conditions, recent design methodologies rely on impedance-frequency analysis and predictive modeling to determine appropriate tuning and sizing parameters under different system conditions. In this regard, continuous monitoring further improves the reliability of compensation schemes by enabling early detection of resonance-prone conditions and adaptive reassessment of the design assumptions [21].

In summary, the literature indicates that power factor correction in converter-dominated networks should not be approached as a purely reactive compensation problem. Instead, it must be formulated as a coupled power-quality problem involving harmonic distortion, network impedance behavior, and resonance risk. This theoretical perspective directly supports the methodology proposed in this work, where bus prioritization and passive filter design are jointly formulated under harmonic and resonance constraints.

2.6. Standards for Harmonic Control

IEEE Std 519-2022 establishes the principal framework for harmonic control in electrical power systems [4]. Its purpose is to preserve power quality by limiting harmonic distortion at the point of common coupling (PCC), thereby protecting both utility infrastructure and end-user equipment. The standard adopts a shared-responsibility approach: users are expected to limit the harmonic currents injected into the network, whereas utilities should maintain sufficiently low system impedance to avoid excessive harmonic voltage amplification. In this context, compliance assessment must consider both voltage distortion and current distortion, since they represent complementary manifestations of the same harmonic interaction phenomenon.

With respect to voltage quality, IEEE 519-2022 specifies admissible limits for individual harmonic components and for total harmonic distortion of voltage, $TH{D}_v$, as a function of the nominal voltage level at the PCC. These limits are summarized in

Table 2. Voltage distortion limits at the PCC [22].

Voltage at PCC	Individual Harmonic (%)	Total Harmonic Distortion, ${\mathit{THD}}_{\mathit{v}}$ (%)
$V\le 1.0$ kV	5.0	8.0
1 kV $<V\le 69$ kV	3.0	5.0
69 kV $<V\le 161$ kV	1.5	2.5
161 kV $<V$	1.0	1.5

. Their technical significance lies in the fact that more restrictive thresholds are imposed as the system voltage level increases, reflecting the higher sensitivity of transmission-level operation to harmonic distortion and its broader impact on system-wide performance.

For harmonic currents, the standard defines the admissible distortion limits as a function of the short-circuit ratio $I_{sc}/I_L$, where $I_{sc}$ is the maximum short-circuit current at the PCC and $I_L$ is the maximum demand load current at the fundamental frequency. This ratio is a key indicator of system stiffness: a stronger system can tolerate a larger harmonic current injection without producing excessive voltage distortion, whereas a weaker system is more vulnerable to the same harmonic source. Accordingly, IEEE 519-2022 provides different current distortion limits depending on both the voltage level and the value of $I_{sc}/I_L$.

For systems with nominal voltages between 120 V and 69 kV, the corresponding current distortion limits are presented in

Table 3. Current distortion limits for systems with nominal voltages between 120 V and 69 kV [22].

	Maximum Harmonic Current Distortion as a Percentage of ${\mathit{I}}_{\mathit{L}}$
${\mathit{I}}_{\mathit{sc}}/{\mathit{I}}_{\mathit{L}}$	$\mathbf{3}\le \mathit{h}<\mathbf{11}$	$\mathbf{11}\le \mathit{h}<\mathbf{17}$	$\mathbf{17}\le \mathit{h}<\mathbf{23}$	$\mathbf{23}\le \mathit{h}<\mathbf{35}$	$\mathbf{35}\le \mathit{h}<\mathbf{50}$	TDD
<20	4.0	2.0	1.5	0.6	0.3	5.0
$20<50$	7.0	3.5	2.5	1.0	0.5	8.0
$50<100$	10.0	4.5	4.0	1.5	0.7	12.0
$100<1000$	12.0	5.5	5.0	2.0	1.0	15.0
>1000	15.0	7.0	6.0	2.5	1.4	20.0

. In this voltage range, the standard defines admissible individual harmonic distortion limits for different harmonic-order intervals, together with the corresponding total demand distortion (TDD). This classification is especially relevant for practical studies because lower-order harmonics, particularly the 5th and 7th, are usually the dominant components in networks with six-pulse converter loads.

For systems with nominal voltages between 69 kV and 161 kV, the admissible current distortion limits become more restrictive, as shown in

Table 4. Current distortion limits for systems with nominal voltages between 69 kV and 161 kV [22].

	Maximum Harmonic Current Distortion as a Percentage of ${\mathit{I}}_{\mathit{L}}$
${\mathit{I}}_{\mathit{sc}}/{\mathit{I}}_{\mathit{L}}$	$\mathbf{3}\le \mathit{h}<\mathbf{11}$	$\mathbf{11}\le \mathit{h}<\mathbf{17}$	$\mathbf{17}\le \mathit{h}<\mathbf{23}$	$\mathbf{23}\le \mathit{h}<\mathbf{35}$	$\mathbf{35}\le \mathit{h}<\mathbf{50}$	TDD
<20	2.0	1.0	0.75	0.30	0.15	2.5
$20<50$	3.5	1.75	1.25	0.50	0.25	4.0
$50<100$	5.0	2.25	2.00	0.75	0.35	6.0
$100<1000$	6.0	2.75	2.50	1.00	0.50	7.5
>1000	7.5	3.50	3.00	1.25	0.70	10.0

. This reduction is consistent with the greater operational sensitivity of higher-voltage networks, where harmonic propagation may affect a wider portion of the system and compromise transmission performance.

For systems with nominal voltages above 161 kV, the permissible distortion thresholds are even lower, as summarized in

Table 5. Current distortion limits for systems with nominal voltages above 161 kV [22].

	Maximum Harmonic Current Distortion as a Percentage of ${\mathit{I}}_{\mathit{L}}$
${\mathit{I}}_{\mathit{sc}}/{\mathit{I}}_{\mathit{L}}$	$\mathbf{3}\le \mathit{h}<\mathbf{11}$	$\mathbf{11}\le \mathit{h}<\mathbf{17}$	$\mathbf{17}\le \mathit{h}<\mathbf{23}$	$\mathbf{23}\le \mathit{h}<\mathbf{35}$	$\mathbf{35}\le \mathit{h}<\mathbf{50}$	TDD
<20	1.0	0.5	0.38	0.15	0.10	1.50
$20<50$	2.0	1.0	0.75	0.30	0.15	2.50
≥50	3.0	1.5	1.15	0.45	0.22	3.75

. These limits reflect the stringent power-quality requirements of extra-high-voltage networks, where harmonic contamination must be minimized to preserve stability margins, equipment lifetime, and acceptable operating conditions over large interconnected areas.

For nonlinear loads, the application of IEEE 519-2022 requires harmonic flow studies that explicitly account for the frequency-dependent impedance of the electrical network [22]. These studies should consider the operating scenarios associated with the highest expected distortion levels, since harmonic performance depends not only on the injected spectrum, but also on the system operating point and network configuration. When significant harmonic content is present, the standard and related technical literature recommend the use of tuned passive filters or active power filters to attenuate the dominant harmonic components and maintain compliance with the admissible distortion limits.

From the viewpoint of the present study, these regulatory criteria are particularly relevant because they provide the quantitative basis for evaluating whether the buses affected by nonlinear loads remain within admissible harmonic limits. Therefore, IEEE 519-2022 is not only used here as a reference standard for result validation, but also as a technical benchmark for the prioritization of compensation measures and the subsequent design of harmonic mitigation filters.

3. Methodology

This section presents the methodology proposed in this study to identify the most critical buses under nonlinear loading conditions, prioritize compensation actions, and design tuned passive filters capable of improving the power factor while ensuring harmonic mitigation and resonance avoidance. Rather than being limited to the construction of a simulation model, the methodological contribution lies in defining a sequential and reproducible decision procedure that links network diagnosis, nonlinear-load screening, harmonic assessment, multi-criteria prioritization, impedance-frequency characterization, filter sizing, and post-compensation validation within a single workflow. The IEEE 14-bus test system implemented in DIgSILENT PowerFactory is used as the benchmark platform for applying and validating each stage of this methodology.

From a methodological viewpoint, the proposed framework is based on three principles. First, the location of nonlinear loads and compensation devices must be justified from measurable electrical indicators rather than arbitrary selection. Second, power factor correction in the presence of harmonics must be treated as a coupled problem involving both reactive power compensation and distortion control. Third, the design of passive filters must explicitly account for the frequency response of the network in order to avoid resonance amplification. These principles are operationalized through an ordered sequence of analytical and simulation-based stages, so that the section functions not merely as a description of the test system, but as a formal methodology for diagnosis, prioritization, design, and validation.

3.1. General Methodological Workflow

Let the electrical network be represented by the set of buses

$$\mathcal{B}=\{1,2,\dots ,N_b\},$$

(4)

where $N_b$ is the total number of buses in the system. The subset of candidate buses for nonlinear load allocation is denoted by

$${\mathcal{B}}_{PQ}=\left\{i\in \mathcal{B}|\mathrm{bus} i \mathrm{is} \mathrm{of} \mathrm{type} \mathrm{PQ}\right\},$$

(5)

since buses of type Slack and PV are excluded from the nonlinear-load placement process due to their voltage regulation and system-balancing functions.

For each candidate bus $i\in {\mathcal{B}}_{PQ}$, the methodology evaluates electrical variables obtained from load-flow and harmonic-flow simulations, including nodal voltage $V_i$, load current $I_i$, total harmonic distortion $TH{D}_i$, and power factor $P{F}_i$. These variables are subsequently used to identify the buses with the greatest susceptibility to harmonic deterioration and to define the priority order for passive filter allocation.

The overall methodology comprises the following stages:

• Configuration of the IEEE 14-bus test system in DIgSILENT PowerFactory.
• Establishment of the base operating condition with linear loads only.
• Sequential allocation of nonlinear loads at candidate PQ buses and harmonic response assessment.
• Selection of the buses exhibiting the highest harmonic severity.
• Harmonic-flow analysis of the system under the definitive nonlinear-load scenario.
• Multi-criteria prioritization of candidate buses for passive compensation.
• Frequency sweep analysis and tuned passive filter design.
• Post-compensation validation in terms of voltage profile, current demand, power factor, and harmonic distortion.

This methodological structure is summarized in the flowchart shown in Figure 8. The diagram is included to visually clarify the logical sequence of the proposed workflow before presenting its mathematical and implementation details.

3.2. Algorithmic Formulation of the Proposed Procedure

To ensure methodological clarity and reproducibility, the complete procedure is formalized through Algorithm 1. In contrast to a simplified pseudocode description, this formulation explicitly separates system initialization, candidate-bus screening, harmonic evaluation, bus prioritization, resonance characterization, filter design, and iterative compliance verification.

3.3. Systematic Procedure Adopted in This Study

The systematic implementation of the proposed methodology is described below.

3.3.1. Configuration of the Test System

The electrical network under study corresponds to the IEEE 14-bus test system, which is a widely accepted benchmark for power-system analysis, compensation studies, and harmonic assessment. In its original formulation, the system consists of 14 interconnected buses, 5 generators, 11 load buses, 20 transmission links represented by their series impedance and shunt susceptance, and transformer branches operating at nominal voltage levels of 132 kV, 33 kV, and 11 kV [23]. The single-line representation adopted in this work is shown in Figure 9.

The choice of this test system is justified by two reasons. First, its size is sufficiently compact to allow a detailed interpretation of local harmonic effects at specific buses. Second, it preserves enough topological diversity to assess the propagation of harmonic distortion and the interaction between nonlinear loads, network impedance, and compensation devices.

From a methodological standpoint, the use of the IEEE 14-bus system should be interpreted as a benchmark-scale validation platform rather than as a limitation of the proposed approach. The workflow developed in this study is not tied to a fixed network size, because its main stages—candidate-bus screening, harmonic-flow evaluation, weighted prioritization, frequency-sweep analysis, and bus-specific filter design—are formulated as modular procedures applicable to any transmission or subtransmission network in which the required electrical data are available. In larger systems, the number of candidate buses and simulation runs increases, but the decision logic of the method remains unchanged.

Algorithm 1 Resonance-aware methodology for power factor correction and harmonic mitigation

Require:  IEEE 14-bus test system data; set of buses $\mathcal{B}$; candidate set ${\mathcal{B}}_{PQ}$; harmonic standard limits from IEEE 519; nonlinear load model based on a six-pulse rectifier Ensure:  Set of prioritized buses for compensation; passive filter parameters; validated post-compensation operating condition   1: Configure the IEEE 14-bus system in DIgSILENT PowerFactory   2: Run conventional load flow under linear-load conditions   3: Store base-case variables $\{V_i^{\left(0\right)},I_i^{\left(0\right)},P{F}_i^{\left(0\right)}\}$ for all load buses i   4: for all $i\in {\mathcal{B}}_{PQ}$ do   5:       Temporarily replace the linear load at bus i with the nonlinear six-pulse load model   6:       Run harmonic power flow   7:       Extract $TH{D}_i$, individual harmonic magnitudes, and local electrical response   8:       Restore the original linear load at bus i   9: end for 10: Identify the buses with the highest harmonic severity and select them as the definitive nonlinear-load locations 11: Apply the selected nonlinear loads simultaneously in the system 12: Run harmonic power flow for the definitive nonlinear-load condition 13: Compute $\{V_i,I_i,P{F}_i,TH{D}_i\}$ for all relevant buses 14: for all candidate compensation bus i do 15:       Compute the power-factor deterioration and harmonic-severity indices 16:       Normalize the indices and calculate the global prioritization score $I{P}_i$ 17: end for 18: Rank buses in descending order of $I{P}_i$ 19: Select the top-ranked buses for passive filter design 20: for all selected bus k do 21:       Perform frequency sweep analysis in the range $60\mathrm{Hz}\le f\le 1800\mathrm{Hz}$ 22:       Determine the dominant resonance frequencies and identify whether the critical resonance is series or parallel 23:       for all dominant harmonic order $h\in {\mathcal{H}}_d$ do 24:           Compute the reactive compensation requirement at the fundamental frequency 25:           Estimate the filter parameters $(C_k,L_k,R_k)$ for tuning near $h{f}_1$ 26:           Insert the tuned passive filter model at bus k 27:           Run harmonic power flow with the filter in service 28:           if all relevant IEEE 519 constraints are satisfied then 29:                Store the filter design and validation results 30:                break 31:           else 32:                Adjust the filter parameters and repeat the evaluation 33:           end if 34:       end for 35: end for 36: Compare the pre- and post-compensation conditions in terms of voltage profile, current demand, power factor, and harmonic distortion 37: Generate the final technical assessment

Therefore, the contribution of the present work lies in the formulation and validation of a scalable decision procedure rather than in the exclusive analysis of a 14-bus topology. The IEEE 14-bus system was selected because it allows the full sequence of diagnosis, prioritization, resonance assessment, and compensation validation to be demonstrated in a transparent and technically interpretable manner before extending the same methodology to networks of greater dimensionality.

The generator operating limits and voltage setpoints used in the model are summarized in

Table 6. Generator data and voltage setpoints of the IEEE 14-bus system.

Bus	${\mathit{Q}}_{\mathit{g},min}$ [MVAr]	${\mathit{Q}}_{\mathit{g},max}$ [MVAr]	${\mathit{P}}_{\mathit{g},min}$ [MW]	${\mathit{P}}_{\mathit{g},max}$ [MW]	V [p.u.]
1 (Slack)	−160	300	0	320	1.06
2	−40	50	0	80	1.045
3	0	40	−1	1	1.01
6	−6	24	−1	1	1.07
8	−6	24	−1	1	1.09

. These data define the active and reactive power operating bounds of the generating units and establish the regulated voltage levels at the generation buses.

The load data used to model the demand buses are listed in

Table 7. Load data used for the IEEE 14-bus system model.

Bus	P [MW]	Q [MVAr]
2	21.7	12.7
3	94.2	19.0
4	47.8	−3.9
5	7.6	1.6
6	11.2	7.5
9	29.5	16.6
10	9.0	5.8
11	3.5	1.8
12	6.1	1.6
13	13.5	5.8
14	14.9	5.0

. These values correspond to the base operating condition and are later modified at selected buses when nonlinear loads are introduced into the system.

The electrical parameters of the interconnecting lines are reported in

Table 8. Transmission-line data of the IEEE 14-bus system.

Link	R [p.u.]	X [p.u.]	${\mathit{B}}_{\mathit{sh}}$ [p.u.]
1–2	0.01938	0.05917	0.0528
1–5	0.05403	0.22304	0.0492
2–3	0.04699	0.19797	0.0438
2–4	0.05811	0.17632	0.0374
2–5	0.05695	0.17388	0.0340
3–4	0.06701	0.17103	0.0346
4–5	0.01335	0.04211	0.0128
4–7	0.00000	0.20912	0.0000
4–9	0.00000	0.55618	0.0000
5–6	0.00000	0.25202	0.0000
6–11	0.09498	0.19890	0.0000
6–12	0.12291	0.25581	0.0000
6–13	0.06615	0.13027	0.0000
7–8	0.00000	0.17615	0.0000
7–9	0.00000	0.11001	0.0000
9–10	0.03181	0.08450	0.0000
9–14	0.12711	0.27038	0.0000
10–11	0.08205	0.19207	0.0000
1–13	0.22092	0.19988	0.0000
13–14	0.17093	0.34802	0.0000

. These data are essential not only for steady-state power-flow computations but also for the frequency-domain behavior of the network, since the impedance distribution strongly influences harmonic propagation and resonance characteristics.

3.3.2. Base Operating Condition

The base-case operating condition, hereafter referred to as Case 1, was obtained by executing a conventional load-flow simulation in DIgSILENT PowerFactory considering only linear and balanced loads. The Newton–Raphson method was adopted for this stage because of its numerical robustness, quadratic convergence in well-conditioned systems, and broad acceptance in medium- and large-scale power-system studies.

This stage provides the electrical reference state against which the impact of nonlinear loads and the effectiveness of the compensation strategy can later be assessed. In particular, the base case yields the nodal voltage magnitudes, the branch and load currents, and the power factor at each load bus. These variables constitute the initial benchmark for the comparative analysis developed in the subsequent stages.

The base-case voltage profile is shown in Figure 10. This figure is introduced at this point because voltage magnitudes must be established before any nonlinear disturbance is applied, thereby allowing the later identification of deviations caused by harmonic-producing loads.

The corresponding load currents and power-factor values are listed in

Table 9. Load current and power factor at the load buses under the base operating condition (Case 1).

Bus	$\mathit{PF}$ (Case 1) [p.u.]	${\mathit{I}}_{\mathbf{load}}$ (Case 1) [kA]
2	0.86306	0.1052
3	0.98026	0.4162
4	0.99669	0.2059
5	0.97855	0.0333
6	0.83091	0.2204
9	0.87150	0.5606
10	0.84057	0.1782
11	0.88929	0.0651
12	0.96728	0.1046
13	0.91879	0.2447
14	0.94805	0.2655

. These results characterize the initial electrical condition of each load bus and provide the quantitative reference required to evaluate the subsequent deterioration caused by nonlinear-load integration.

3.4. Methodological Implementation Under the Adopted Case-Study Conditions

The practical implementation of the methodology follows the same sequence described above, but under the actual modeling assumptions adopted in this study. In particular, nonlinear loads are modeled using the harmonic signature of a six-pulse converter, candidate buses are restricted to PQ nodes, and compensation is ultimately applied at the buses exhibiting the highest combined harmonic severity and power-factor deterioration.

At this stage, two remarks are especially relevant. First, the choice of candidate buses is not arbitrary but constrained by the operating role of each node in the network. Second, the passive filter design is not performed directly after observing a poor power factor, but only after the combined electrical severity has been quantified and the resonance behavior of the network has been examined.

3.4.1. Implementation of Nonlinear Loads

The next stage of the methodology consists of representing the harmonic-producing loads through a physically consistent nonlinear-load model. In this study, nonlinear loads are modeled by means of an ideal and balanced three-phase six-pulse rectifier, which is representative of the harmonic behavior commonly associated with variable frequency drives and switched-mode power supply units. This choice is justified because six-pulse converters constitute one of the most frequent sources of characteristic low-order harmonics in industrial and utility applications.

The harmonic spectrum adopted in this work, summarized in

Table 10. Harmonic spectrum adopted for the three-phase six-pulse rectifier model.

Harmonic Order h	${\mathit{I}}_{\mathit{h}}/{\mathit{I}}_1$ [%]	Phase [°]
5	20.000000	180
7	14.285710	0
11	9.090909	180
13	7.692308	0
17	5.882353	180
19	5.263158	0
23	4.347826	180
25	4.000000	0
29	3.448276	180
31	3.225806	0
35	2.857143	180
37	2.702703	0
41	2.439024	180
43	2.325581	0
47	2.127660	180
49	2.040816	0

, is consistent with the theoretical behavior described for six-pulse converters in IEEE Std 519-2022 [24]. According to the Fourier-series representation of a p-pulse converter, the characteristic harmonic orders are given by

$$h=pn\pm 1,n\in {\mathbb{Z}}^{+},$$

(6)

where p is the number of pulses of the converter. For the case considered here, $p=6$, and therefore the characteristic harmonics appear at orders $h=5,7,11,13,17,19,\dots$. Under ideal balanced operation, non-characteristic harmonics are negligible or tend to be cancelled out.

The relative magnitude of the harmonic current components decreases approximately in inverse proportion to the harmonic order, which may be expressed as

$${\displaystyle \frac{I_h}{I_1}}\approx {\displaystyle \frac{100}{h}}\%,$$

(7)

where $I_1$ is the fundamental current and $I_h$ is the current magnitude of the h-th harmonic component. Equation (7) explains why the 5th- and 7th-order harmonics are typically dominant in six-pulse rectifier applications. In particular, the 5th harmonic exhibits the largest amplitude, followed by the 7th harmonic, while higher-order components progressively decrease in magnitude. This spectral model is suitable for reproducing the expected harmonic behavior in both theoretical analyses and harmonic power-flow simulations.

The adopted harmonic spectrum is listed in Table 10, including the relative magnitude and phase angle of each characteristic harmonic component.

Once the nonlinear-load model was defined, the operational role of each bus in the IEEE 14-bus network was analyzed in order to identify admissible candidate locations. Slack and PV buses were excluded from this process because of their voltage-regulation and power-balancing functions. Consequently, the candidate set was restricted to buses 4, 5, 9, 10, 11, 12, 13, and 14.

To determine the most suitable locations for nonlinear-load placement, the original linear load at each candidate bus was temporarily replaced by the six-pulse nonlinear-load model, and a harmonic power-flow simulation was executed in each case. The objective of this screening procedure was to quantify the local and system-wide harmonic response produced by each candidate location and, in this way, identify the buses with the highest harmonic severity.

The corresponding results are reported in

Table 11. Harmonic indices obtained for the candidate buses during the nonlinear-load screening stage.

Bus	THD [%]	5th Harmonic [%]	7th Harmonic [%]
4	17.37	4.66	0.6390
5	1.87	0.725	0.1836
9	17.08	13.29	10.4000
10	4.76	4.77	2.3900
11	2.06	1.506	0.2590
12	4.57	1.486	0.8460
13	7.56	2.83	1.0570
14	11.08	5.71	1.4744

, where the total harmonic distortion and the dominant low-order components are listed for all candidate buses. These quantities were chosen as screening indicators because the 5th and 7th harmonics are the most representative components of six-pulse converter operation and therefore provide direct information on the severity of the distortion mechanism introduced into the system.

Table 11 shows that buses 9 and 14 exhibit the most severe harmonic response and were therefore selected as the definitive locations for nonlinear-load allocation in the subsequent stage of the analysis. In addition, the 5th harmonic was identified as the most influential component in the system response, which directly supports its use as the main tuning target in the passive-filter design stage.

3.4.2. Harmonic Power-Flow Analysis Under Nonlinear Loading

After selecting buses 9 and 14 as the most critical locations for nonlinear-load allocation, the harmonic power-flow analysis was performed for the definitive nonlinear-load scenario, hereafter denoted as Case 2. In this case, the original linear loads connected at buses 9 and 14 were replaced by nonlinear loads modeled as harmonic current sources following the six-pulse rectifier spectrum defined previously. The purpose of this stage was to quantify the effect of nonlinear loading on waveform distortion, harmonic propagation, power factor deterioration, and the overall electrical response of the network.

Before introducing the quantitative harmonic indices, it is instructive to examine the current and voltage waveforms at the buses directly affected by the nonlinear loads. Figure 11 and Figure 12 show the resulting current and voltage waveforms at buses 9 and 14, respectively. These figures are presented here because they provide the first direct evidence of the nonsinusoidal current injection associated with the selected converter-based load model.

As expected, the current waveform becomes strongly distorted due to the injection of harmonic components produced by the six-pulse converter model. By contrast, the voltage waveform remains closer to sinusoidal behavior, although it is still affected by superimposed distortions caused by the network response to harmonic current injection. This distinction is consistent with the fact that nonlinear loads act primarily as distorted current sources, while the corresponding voltage distortion depends on the system impedance seen at the point of connection.

The influence of the nonlinear loads is not restricted to their connection buses. Because harmonic currents propagate through the network according to the system topology and impedance distribution, adjacent buses may also experience measurable waveform distortion. This propagation effect is illustrated in Figure 13, where the current and voltage waveforms at bus 10 are shown. The response at this neighboring bus confirms that the distortion generated at buses 9 and 14 spreads throughout the electrical network through bus coupling and line impedance interactions.

To evaluate compliance with IEEE 519, buses 9 and 14 were considered as the main power-quality assessment points. The admissible current distortion limits depend on the ratio between the available short-circuit current and the maximum load current at the point of common coupling. For this reason, the short-circuit current and the corresponding fundamental load current were first determined for both buses, as shown in Figure 14 and Figure 15.

$${\displaystyle \frac{I_{sc,9}}{I_{L,9}}}=23.32$$

(8)

$${\displaystyle \frac{I_{sc,14}}{I_{L,14}}}=25.736$$

(9)

Based on (8) and (9), both buses fall within the interval $20<I_{sc}/I_L<50$. Since the nominal voltage at the corresponding point of common coupling is 33 kV, the applicable current-distortion limits are those established by IEEE 519 for systems with nominal voltages between 120 V and 69 kV. The subset of admissible limits used in this study is summarized in

Table 12. IEEE 519 current distortion limits applicable to buses 9 and 14 in Case 2.

${\mathit{I}}_{\mathbf{sc}}/{\mathit{I}}_{\mathit{L}}$	$3\le \mathit{h}<11$	$11\le \mathit{h}<17$	$17\le \mathit{h}<23$	$23\le \mathit{h}<35$	$35\le \mathit{h}<50$	THD
$20<I_{sc}/I_L<50$	7.0	3.5	2.5	1.0	0.5	8.0

.

The detailed harmonic spectrum obtained from the harmonic power-flow simulation is presented in

Table 13. Calculated harmonic distortion at bus 9 in Case 2.

Harmonic Order h	Harmonic Distortion [%]
5	18.515
7	11.646
11	2.027
13	1.211
17	0.446
19	0.326
23	0.262
25	0.249
29	0.224
31	0.211
35	0.180
37	0.162
41	0.125
43	0.107
47	0.075
49	0.061
THD	22.0145

and

Table 14. Calculated harmonic distortion at bus 14 in Case 2.

Harmonic Order h	Harmonic Distortion [%]
5	16.039
7	6.897
11	2.585
13	2.933
17	2.912
19	2.860
23	2.716
25	2.644
29	2.549
31	2.513
35	2.467
37	2.482
41	2.525
43	2.550
47	2.595
49	2.613
THD	20.0658

. These tables are required to compare the simulated distortion profile against the admissible harmonic limits and to identify the dominant harmonic orders that should be targeted by the mitigation strategy.

The results show that both buses are subject to severe current distortion and that the 5th harmonic is the dominant component in the system response. At bus 9, both the 5th and 7th components exceed the admissible IEEE 519 limits by a wide margin, whereas at bus 14 the 5th harmonic remains dominant and higher-order harmonics also become non-negligible. Therefore, the harmonic response observed in Case 2 provides a clear technical basis for the application of a tuned passive filtering strategy focused primarily on the attenuation of the dominant low-order harmonic components.

This comparison is more clearly illustrated in Figure 16 and Figure 17, where the harmonic distortion obtained from simulation is contrasted against the corresponding IEEE 519 admissible limits. In these bar charts, the solid blue bars represent the harmonic distortion values obtained from simulation, whereas the red hatched rectangular markers indicate the corresponding IEEE 519 admissible limits for each harmonic order. This graphical convention is used consistently in all subsequent harmonic-comparison figures in order to distinguish clearly between the measured response of the system and the regulatory threshold adopted as the reference criterion.

To complement the harmonic assessment, the principal electrical variables obtained under Case 2 are summarized in

Table 15. Load current under the nonlinear-load operating condition (Case 2).

Bus	${\mathit{I}}_{\mathbf{load}}$ (Case 2) [p.u.]
2	0.95704
3	0.99049
4	0.98633
5	0.98131
6	0.93488
9	0.98838
10	0.95308
11	0.94727
12	0.94900
13	0.95286
14	1.00799

,

Table 16. Nodal voltage magnitudes under the nonlinear-load operating condition (Case 2).

Bus	V (Case 2) [p.u.]
1	1.0604
2	1.0461
3	1.0110
4	1.0225
5	1.0232
6	1.0735
7	1.0728
8	1.0930
9	1.0816
10	1.0708
11	1.0669
12	1.0597
13	1.0561
14	1.0564

and

Table 17. Power factor under the nonlinear-load operating condition (Case 2).

Bus	$\mathit{PF}$ (Case 2)
2	0.86137
3	0.97881
4	0.99203
5	0.97412
6	0.82579
9	0.78271
10	0.81185
11	0.87421
12	0.96055
13	0.90986
14	0.85875

. These quantities allow the effect of nonlinear loading to be evaluated not only in terms of harmonic distortion, but also in terms of its impact on current demand, nodal voltage profile, and power factor.

Under nonsinusoidal conditions, the total power factor must be computed by simultaneously considering the displacement power factor and the distortion factor. Using the formulation introduced in Section 2, and linking the variables obtained from DIgSILENT PowerFactory, the total power factor at bus i is evaluated as

$$P{F}_i^{\left(2\right)}={\displaystyle \frac{P_i}{S_i}}·{\displaystyle \frac{1}{\sqrt{1+{\left(\frac{TH{D}_i}{100}\right)}^2}}},$$

(10)

where $P_i$ and $S_i$ are the active and apparent powers at bus i, respectively, and $TH{D}_i$ denotes the total harmonic distortion evaluated at bus i, expressed as a percentage. The resulting power-factor values for Case 2 are listed in Table 17.

The results confirm that nonlinear loading produces a substantial deterioration in both harmonic quality and power factor, particularly at buses 9 and 14. These findings justify the need for a systematic prioritization strategy in order to determine where passive compensation should be installed first and how its design should be guided by the combined effect of power-factor degradation and harmonic severity.

3.4.3. Multi-Criteria Prioritization of Passive-Filter Locations Using AHP

To identify, in an objective and technically consistent manner, the buses with the greatest need for harmonic compensation, a multi-criteria prioritization methodology was formulated on the basis of the electrical variables obtained from the base-case power flow and the nonlinear-load harmonic-flow analysis. The prioritization stage is built upon three measurable indicators: the power factor under linear-load conditions, the power factor under nonlinear-load conditions, and the total harmonic distortion at each bus.

Based on these quantities, a global prioritization index is constructed from normalized technical criteria. In order to avoid an arbitrary assignment of criterion weights, the Analytic Hierarchy Process (AHP) is adopted to determine the relative importance of each criterion in a structured manner. In the present study, the prioritization of candidate buses for passive compensation is based on the following three criteria:

• normalized harmonic severity;
• relative power-factor deterioration;
• absolute power-factor deficiency.

The mathematical formulation of these criteria and the derivation of the AHP-based prioritization index are presented in the next subsection.

3.4.4. Definition of Evaluation Criteria

In order to identify, in an objective and technically consistent manner, the buses with the highest need for passive compensation, three electrical criteria were defined from the results of the base-case load-flow analysis and the nonlinear-load harmonic study. Let i denote a candidate bus for compensation. For each candidate bus, the following input variables are considered:

• The power factor under linear-load conditions, denoted by $P{F}_i^{CL}$;
• The power factor under nonlinear-load conditions, denoted by $P{F}_i^{CNL}$;
• The total harmonic distortion, denoted by $TH{D}_i$.

These variables provide a direct quantitative characterization of the electrical severity at each bus. Based on them, three prioritization criteria are defined: relative power-factor deterioration, normalized harmonic severity, and absolute power-factor deficiency.

The first criterion quantifies the loss of power factor caused by nonlinear loading. It is defined as

$$\Delta P{F}_i=max\left(0,P{F}_i^{CL}-P{F}_i^{CNL}\right),$$

(11)

where the $max(·)$ operator ensures that only effective deteriorations are retained. To make this quantity comparable across candidate buses, it is normalized with respect to the maximum observed deterioration:

$$I_{PF,i}={\displaystyle \frac{\Delta P{F}_i}{{max}_j(\Delta P{F}_j)}}.$$

(12)

By construction, $I_{PF,i}\in [0,1]$, and values close to unity indicate buses with the highest relative loss of power factor.

The second criterion measures the harmonic severity of each bus relative to the admissible distortion limit. This criterion is designed to reflect not only the magnitude of the distortion, but also its degree of non-compliance with the applicable standard. The harmonic exceedance is therefore defined as

$$E_{THD,i}=max\left(0,{\displaystyle \frac{TH{D}_i}{TH{D}_{\lim }}}-1\right),$$

(13)

where $TH{D}_{\lim }$ is the applicable admissible limit according to IEEE 519. The corresponding normalized harmonic-severity index is given by

$$I_{THD,i}={\displaystyle \frac{E_{THD,i}}{{max}_j\left(E_{THD,j}\right)}}.$$

(14)

If a given bus does not exceed the admissible distortion threshold, then $E_{THD,i}=0$, and its harmonic contribution to the prioritization process is null.

The third criterion evaluates the absolute deficiency of the power factor under nonlinear-load conditions. This metric is introduced to distinguish buses that, regardless of their relative deterioration, still operate far from unity power factor. It is defined as

$$D_{PF,i}=1-P{F}_i^{CNL},$$

(15)

and normalized as

$$I_{D,i}={\displaystyle \frac{D_{PF,i}}{{max}_j\left(D_{PF,j}\right)}}.$$

(16)

This criterion complements $I_{PF,i}$, since it captures the absolute operating condition of each bus rather than only the deterioration relative to the base case.

3.4.5. AHP-Based Hierarchical Decision Structure

Once the technical criteria were defined, the Analytic Hierarchy Process (AHP) was used to determine their relative importance in a structured and reproducible manner. The hierarchy adopted in this study comprises three levels:

• Level 1: global objective, namely, to identify the most suitable bus for passive filter installation;
• Level 2: technical criteria, defined as

  $$C_1=I_{THD},C_2=I_{PF},C_3=I_D;$$
• Level 3: alternatives, corresponding to the candidate buses of the IEEE 14-bus test system.

The pairwise comparison of criteria is represented by the reciprocal square matrix

$$\mathbf{A}=\left[a_{ij}\right]\in {\mathbb{R}}^{n\times n},n=3,$$

(17)

where each entry $a_{ij}$ expresses the relative importance of criterion i with respect to criterion j, according to Saaty’s fundamental scale. The comparison matrix satisfies

$$a_{ij}>0,a_{ii}=1,a_{ji}={\displaystyle \frac{1}{a_{ij}}}.$$

(18)

For the present problem, the comparison matrix is written as

$$\mathbf{A}=\left[\begin{array}{ccc}1& a_{12}& a_{13}\\ {\displaystyle \frac{1}{a_{12}}}& 1& a_{23}\\ {\displaystyle \frac{1}{a_{13}}}& {\displaystyle \frac{1}{a_{23}}}& 1\end{array}\right],$$

(19)

where $a_{12}$, $a_{13}$, and $a_{23}$ represent the relative importance of harmonic severity over power-factor deterioration, harmonic severity over absolute deficiency, and power-factor deterioration over absolute deficiency, respectively.

The criterion-weight vector is defined as

$$\mathbf{w}=\left[\begin{array}{c}{w}_1\\ w_2\\ w_3\end{array}\right],w_i>0,\sum _{i=1}^3{w}_i=1,$$

(20)

and is obtained from the principal eigenvector of $\mathbf{A}$, namely,

$$\mathbf{A}\mathbf{w}={\lambda }_{max}\mathbf{w},$$

(21)

where ${\lambda }_{max}$ is the maximum eigenvalue of the comparison matrix.

For practical implementation, the weights were estimated through column normalization. First, the sum of each column is computed as

$$s_j=\sum _{i=1}^n{a}_{ij},$$

(22)

then each element is normalized according to

$${\overline{a}}_{ij}={\displaystyle \frac{a_{ij}}{s_j}},$$

(23)

and finally the weight of each criterion is obtained as the average of the corresponding normalized row:

$$w_i={\displaystyle \frac{1}{n}}\sum _{j=1}^n{\overline{a}}_{ij}.$$

(24)

3.4.6. Consistency Verification

To verify that the pairwise judgments are logically coherent, the consistency of the AHP matrix was evaluated. The maximum eigenvalue is estimated as

$${\lambda }_{max}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\displaystyle \frac{{\left(\mathbf{A}\mathbf{w}\right)}_i}{w_i}},$$

(25)

from which the consistency index is computed as

$$CI={\displaystyle \frac{{\lambda }_{max}-n}{n-1}}.$$

(26)

The consistency ratio is then obtained through

$$CR={\displaystyle \frac{CI}{RI}},$$

(27)

where $RI$ is Saaty’s random index. For $n=3$, the adopted value is

$$RI=0.58.$$

(28)

The comparison matrix is considered acceptable when

$$CR<0.10.$$

(29)

3.4.7. Global Prioritization Index

Once the criterion weights are obtained, the global prioritization index for each candidate bus is defined as

$$I{P}_i=w_1{I}_{THD,i}+w_2{I}_{PF,i}+w_3{I}_{D,i}.$$

(30)

The buses are then ranked in descending order of $I{P}_i$. The buses with the highest values are interpreted as the most critical locations and, therefore, as the most appropriate locations for the installation of tuned passive filters.

3.4.8. Numerical Values Obtained for the IEEE 14-Bus System

The candidate-bus data used in the prioritization analysis are summarized in

Table 18. Electrical data of candidate buses used in the prioritization stage.

Bus	$\mathit{PF}$ (Case 1)	$\mathit{PF}$ (Case 2)	THD [%]
4	0.99669	0.99203	8.69
5	0.97855	0.97412	7.58
9	0.87150	0.78271	22.01
10	0.84057	0.81185	19.35
11	0.88929	0.87421	13.65
12	0.96728	0.96055	9.18
13	0.91879	0.90986	10.37
14	0.94805	0.85875	20.07

. These values include the power factor under the base operating condition, the power factor under nonlinear-load conditions, and the total harmonic distortion at each candidate bus.

From these values, the individual indices were calculated. For clarity and reproducibility, the normalized results are reported in

Table 19. Normalized indices used for the prioritization process.

Bus	${\mathit{PLF}}_{\mathit{i}}$	${\mathit{PLF}}_{\mathbf{norm},\mathit{i}}$	${\mathit{I}}_{\mathit{THD},\mathbf{norm},\mathit{i}}$	${\mathit{PFA}}_{\mathit{i}}$	${\mathit{PFA}}_{\mathbf{norm},\mathit{i}}$
4	0.005	0.052	0.395	0.008	0.037
5	0.004	0.050	0.345	0.026	0.119
9	0.089	0.994	1.000	0.217	1.000
10	0.029	0.322	0.879	0.188	0.866
11	0.015	0.169	0.620	0.126	0.579
12	0.007	0.075	0.417	0.039	0.182
13	0.009	0.100	0.471	0.090	0.415
14	0.089	1.000	0.911	0.141	0.650

, while the underlying quantities $\Delta P{F}_i$, $E_{THD,i}$, and $D_{PF,i}$ are listed in

Table 20. Criterion values and normalized indicators used in the AHP-based ranking.

Bus	$\Delta {\mathit{PF}}_{\mathit{i}}$	${\mathit{I}}_{\mathit{PF},\mathit{i}}$	${\mathit{E}}_{\mathit{THD},\mathit{i}}$	${\mathit{I}}_{\mathit{THD},\mathit{i}}$	${\mathit{D}}_{\mathit{PF},\mathit{i}}$	${\mathit{I}}_{\mathit{D},\mathit{i}}$
4	0.00466	0.05218	0.00000	0.00000	0.00797	0.03668
5	0.00443	0.04961	0.00000	0.00000	0.02588	0.11910
9	0.08879	0.99429	1.75181	1.00000	0.21729	1.00000
10	0.02872	0.32161	1.41946	0.81028	0.18815	0.86589
11	0.01508	0.16887	0.70619	0.40312	0.12579	0.57890
12	0.00673	0.07536	0.14721	0.08403	0.03945	0.18155
13	0.00893	0.10000	0.29595	0.16894	0.09014	0.41484
14	0.08930	1.00000	1.50822	0.86095	0.14125	0.65005

.

The three evaluation criteria used in the pairwise comparison matrix were therefore defined as

$$C_1=I_{THD},C_2=I_{PF},C_3=I_D.$$

(31)

Using Saaty’s fundamental scale, the pairwise comparisons were established as follows:

$$C_1\mathrm{versus}{C}_2=3,C_1\mathrm{versus}{C}_3=5,C_2\mathrm{versus}{C}_3=3.$$

Thus, the comparison matrix becomes

$$\mathbf{A}=\left[\begin{array}{ccc}1& 3& 5\\ {\displaystyle \frac{1}{3}}& 1& 3\\ {\displaystyle \frac{1}{5}}& {\displaystyle \frac{1}{3}}& 1\end{array}\right].$$

(32)

The criterion-weight vector obtained from AHP is

$$\mathbf{w}=\left[\begin{array}{c}0.6333\\ 0.2605\\ 0.1062\end{array}\right],$$

(33)

or, equivalently,

$$w_{THD}=0.6333,w_{PF}=0.2605,w_D=0.1062.$$

(34)

The estimated maximum eigenvalue is

$${\lambda }_{max}=3.04,$$

(35)

the consistency index is

$$CI={\displaystyle \frac{3.04-3}{3-1}}=0.0194,$$

(36)

and the consistency ratio is

$$CR={\displaystyle \frac{0.0193}{0.58}}=0.0334.$$

(37)

Since

$$CR=0.0334<0.10,$$

(38)

the pairwise comparison matrix is considered acceptably consistent.

Using the weights given above and the global prioritization index defined in (30), the resulting criticality ranking is

$$\begin{array}{cc}\hfill \mathrm{Bus}9& \to I{P}_9=0.9964,\hfill \\ \hfill \mathrm{Bus}14& \to I{P}_{14}=0.9268,\hfill \\ \hfill \mathrm{Bus}10& \to I{P}_{10}=0.5067,\hfill \\ \hfill \mathrm{Bus}11& \to I{P}_{11}=0.2735,\hfill \\ \hfill \mathrm{Bus}13& \to I{P}_{13}=0.1514,\hfill \\ \hfill \mathrm{Bus}12& \to I{P}_{12}=0.0889,\hfill \\ \hfill \mathrm{Bus}5& \to I{P}_5=0.0441,\hfill \\ \hfill \mathrm{Bus}4& \to I{P}_4=0.0369.\hfill \end{array}$$

(39)

The resulting ranking confirms that buses 9 and 14 are the most critical locations and therefore the priority buses for the installation of tuned passive filters.

3.4.9. Design and Implementation of Tuned Passive Filters

Shunt passive filters were selected in this study because their technical characteristics are consistent with the electrical conditions identified in the IEEE 14-bus case under nonlinear loading. The harmonic assessment revealed that the distortion is dominated by characteristic low-order components, especially the 5th harmonic, associated with the adopted six-pulse converter model. Under this condition, a tuned passive branch provides selective attenuation at the dominant harmonic frequency while simultaneously supplying reactive compensation at the fundamental component, which makes it a technically coherent solution for the buses prioritized through the proposed index [19,25].

Although active and hybrid filters offer important advantages under highly variable operating conditions or broadband harmonic spectra, their main strengths are linked to dynamic adaptation, converter-based control, and broader compensation flexibility [20,21,32]. In the present study, however, the compensation problem is defined by fixed network conditions, clearly identified priority buses, and a harmonic spectrum dominated by known low-order components. Therefore, the additional control complexity and implementation requirements of active or hybrid alternatives were not necessary to demonstrate the proposed resonance-aware prioritization and design methodology. Instead, tuned shunt passive filters were preferred because they allow a direct integration of harmonic selectivity, reactive-power support, practical sizing criteria, and impedance-based resonance verification within a single and reproducible design procedure [21,29,30].

Accordingly, the choice of passive filters should not be interpreted as a claim that they are universally superior to active or hybrid solutions, but rather as the most appropriate option for the specific network conditions, distortion profile, and methodological objective considered in this work. Their use is fully aligned with the aim of validating a compensation workflow that links bus screening, weighted prioritization, resonance assessment, and filter implementation under IEEE 519 compliance constraints.

Frequency Sweep Analysis

Prior to filter sizing, the impedance-frequency response of the network was characterized in DIgSILENT PowerFactory over the range from 60 Hz to 1800 Hz. This frequency sweep analysis was carried out at buses 9 and 14 in order to identify the dominant resonance conditions associated with the interaction between the inductive and capacitive elements of the system. The resulting impedance curves are shown in Figure 18 and Figure 19.

In both cases, the resonance behavior is identified as parallel resonance because it appears as a pronounced peak in the impedance profile. At resonance, the inductive and capacitive reactances cancel each other in the admittance domain, producing a high equivalent impedance. This observation is important because it confirms that the passive compensation strategy must be designed carefully to avoid further amplification of the dominant harmonic components. Based on the harmonic spectrum and the frequency sweep results, the filter design was focused on the 5th harmonic, which was previously identified as the dominant component.

Filter Sizing Criteria

The design and parameter selection of the passive filters were based on IEEE Std 1531-2020 [29], which provides guidance for the application and specification of harmonic filters. According to this guide, a harmonic filter bank is primarily composed of capacitors, reactors, and damping resistors, whose values must be selected not only from the standpoint of reactive power compensation, but also considering the harmonic spectrum, voltage stress, thermal effects, and component tolerances.

Tuned Passive Filter at Bus 9

For the filter connected at bus 9, the nominal voltage is 33 kV and the fundamental frequency is 60 Hz. The first step is to calculate the effective reactive power required to improve the power factor from its existing value to the target value. Let $P{F}_{\mathrm{ex}}$ denote the existing power factor and $P{F}_{\mathrm{des}}$ the desired power factor. Then, the required reactive compensation is computed as

$$Q_{\mathrm{eff}}=P\left[\tan \left(\arccos \left(P{F}_{\mathrm{ex}}\right)\right)-\tan \left(\arccos \left(P{F}_{\mathrm{des}}\right)\right)\right],$$

(40)

where P is the active power of the corresponding load.

For bus 9, the values $P{F}_{\mathrm{ex}}=0.78271$ and $P{F}_{\mathrm{des}}=0.95$ were adopted, leading to

$$Q_{\mathrm{eff}}=13.983\mathrm{MVAr}.$$

(41)

Using the correction-factor guidance from IEEE Std 1036-2020 [30], the equivalent correction factor was taken as $f{c}_x=0.474$, which yields

$$Q_{\mathrm{eff}}=P·f{c}_x.$$

(42)

Considering capacitor tolerance and practical design margins, the compensated requirement becomes

$$Q_{\mathrm{eff}}=15.825\mathrm{MVAr}.$$

(43)

However, because commercially available capacitor banks must be selected from standardized ratings, the final installed compensation was chosen as

$$Q_{\mathrm{installed}}=4\times 3600\mathrm{kVAr}=14.4\mathrm{MVAr},$$

(44)

which lies within the practical design interval and provides a suitable compromise between technical adequacy and economic implementation [31].

The equivalent reactive impedance is then obtained from

$$X_{\mathrm{eff}}={\displaystyle \frac{V_{\mathrm{sys}}^2}{Q_{\mathrm{eff}}}},$$

(45)

resulting in

$$X_{\mathrm{eff}}=75.625\mathsf{\Omega }.$$

(46)

The filter was tuned below the nominal 5th harmonic frequency in order to improve attenuation robustness and compensate for parameter drift due to capacitor aging. Following the recommendations reported in [29,30,32], the tuning frequency was selected at approximately $6\%$ below 300 Hz, corresponding to a tuning ratio of $h_r=4.7$.

The capacitive and inductive reactances are therefore calculated as

$$X_{\mathrm{cap}}={\displaystyle \frac{h_r^2}{h_r^2-1}}{X}_{\mathrm{eff}},$$

(47)

$$X_{\mathrm{ind}}={\displaystyle \frac{X_{\mathrm{cap}}}{h_r^2}},$$

(48)

while the filter capacitance, inductance, and damping resistance are given by

$$C={\displaystyle \frac{1}{2\pi f{X}_{\mathrm{cap}}}},$$

(49)

$$L={\displaystyle \frac{X_{\mathrm{ind}}}{2\pi f}},$$

(50)

$$R={\displaystyle \frac{\sqrt{X_{\mathrm{cap}}{X}_{\mathrm{ind}}}}{Q}},$$

(51)

where Q is the quality factor at the tuning frequency. Following IEEE Std 1531-2020, a design quality factor of 20 was adopted.

The resulting RLC parameters for the filter at bus 9 are listed in

Table 21. RLC parameters of the tuned passive filter installed at bus 9 for 5th-harmonic mitigation in Case 2.

Parameter	Value
$X_{\mathrm{cap}}$	$79.2108\mathsf{\Omega }$
C	$100.4629\mathsf{\mu }\mathrm{F}$
$X_{\mathrm{ind}}$	$3.58582\mathsf{\Omega }$
L	$9.51169\mathrm{mH}$
R	$0.84267\mathsf{\Omega }$

.

Tuned Passive Filter at Bus 14

The same design procedure was applied to bus 14. Using the active power at that bus and the value $P{F}_{\mathrm{ex}}=0.85875$, the required reactive compensation was obtained as

$$Q_{\mathrm{eff}}=3.9485\mathrm{MVAr},$$

(52)

which, after including the design tolerance, becomes

$$Q_{\mathrm{eff}}=4.592\mathrm{MVAr}.$$

(53)

A commercially feasible capacitor-bank arrangement was then selected as

$$Q_{\mathrm{installed}}=3600+500\mathrm{kVAr}=4100\mathrm{kVAr}\approx 4.1\mathrm{MVAr}.$$

(54)

The resulting RLC parameters of the tuned passive filter connected at bus 14 are reported in

Table 22. RLC parameters of the tuned passive filter installed at bus 14 for 5th-harmonic mitigation in Case 2.

Parameter	Value
$X_{\mathrm{cap}}$	$278.2039\mathsf{\Omega }$
C	$28.6040\mathsf{\mu }\mathrm{F}$
$X_{\mathrm{ind}}$	$12.5941\mathsf{\Omega }$
L	$33.4069\mathrm{mH}$
R	$2.95961\mathsf{\Omega }$

.

3.4.10. Validation of the Compensated System

To assess the effectiveness of the designed passive filters, the computed RLC values were incorporated into the DIgSILENT PowerFactory model as shunt passive filters connected at buses 9 and 14. The resulting compensated operating condition is denoted as Case 3. Harmonic power-flow simulation was then performed in order to evaluate the resulting harmonic distortion, current demand, corrected power factor, and bus-voltage magnitudes.

Using the admissible harmonic limits previously defined, the distortion obtained at buses 9 and 14 under Case 3 was compared against the IEEE 519 thresholds. The corresponding results are shown in Figure 20 and Figure 21.

To complement the spectral assessment, the current and voltage waveforms at buses 9, 14, and 10 were also examined. Figure 22, Figure 23 and Figure 24 show that the current waveforms remain distorted because the nonlinear loads are still present, but the voltage waveforms exhibit a smoother profile than in Case 2. In addition, the propagation effect observed previously at bus 10 becomes less pronounced, which indicates that the passive filters reduce the impact of harmonic injection on adjacent buses.

The principal electrical variables obtained under the compensated scenario are summarized in

Table 23. Load current under the compensated operating condition (Case 3).

Bus	${\mathit{I}}_{\mathbf{load}}$ (Case 3) [p.u.]
2	0.95695
3	0.99016
4	0.95164
5	0.96047
6	0.93459
9	0.80447
10	0.79941
11	0.86313
12	0.92265
13	0.90552
14	0.81880

,

Table 24. Bus-voltage magnitudes under the compensated operating condition (Case 3).

Bus	V (Case 3) [p.u.]
1	1.0602
2	1.0455
3	1.0103
4	1.0530
5	1.0420
6	1.0702
7	1.1827
8	1.0902
9	1.2992
10	1.2520
11	1.1591
12	1.0842
13	1.1048
14	1.2769

and

Table 25. Power factor under the compensated operating condition (Case 3).

Bus	$\mathit{PF}$ (Case 3)
2	0.86224
3	0.97966
4	0.99540
5	0.97730
6	0.83055
9	0.83023
10	0.83918
11	0.88852
12	0.96678
13	0.91816
14	0.90414

. These results allow a direct comparison with Case 2 and provide the quantitative basis for evaluating the effect of the tuned passive filters.

The total power factor for the compensated scenario was evaluated using the same formulation previously introduced for nonlinear operating conditions, namely Equation (10). The resulting values are listed in Table 25.

The Case 3 results confirm that the tuned passive filters improve the harmonic response and increase the power factor at the buses where compensation is applied. In particular, buses 9 and 14 remain the most relevant locations from the viewpoint of harmonic mitigation, and the post-compensation results provide the quantitative foundation for the comparative analysis developed in the following section.

4. Results and Discussion

4.1. Case-by-Case Analysis of the System Response

4.1.1. Case 1: Linear-Load Operating Condition

Case 1 corresponds to the base operating condition in which only linear loads are considered. Under this scenario, the IEEE 14-bus system exhibits a stable and electrically well-behaved response in terms of power flow, power factor, and nodal voltage profile. This case therefore serves as the reference condition for assessing the impact of nonlinear-load integration and, subsequently, the effectiveness of the proposed passive compensation strategy.

At the buses of interest, the power-factor values remain relatively close to unity, which indicates a balanced relationship between active and reactive power demand. In particular, bus 9 presents a power factor of 0.87150, whereas bus 14 reaches 0.94805. These values are representative of an undisturbed operating condition and establish the baseline against which the deterioration caused by harmonic-producing loads can be quantified.

The current waveforms in the lines connected to buses 9 and 14 preserve an essentially sinusoidal shape, confirming the absence of significant harmonic components under the base operating condition. This behavior is illustrated in Figure 25 and Figure 26, which are introduced here to document the fundamental waveform behavior before nonlinear disturbances are imposed on the network.

With regard to the nodal voltage profile, the system remains within normal operating margins, with voltage magnitudes approximately between 1.03 p.u. and 1.08 p.u. This behavior is consistent with adequate voltage regulation and a balanced network operating state. For example, bus 9 remains close to 1.0563 p.u., whereas bus 14 is approximately 1.0358 p.u. These values are sufficiently stable to support their use as reference quantities for the subsequent comparative analysis.

Overall, Case 1 represents the benchmark operating condition. The corresponding power-factor values, line currents, nodal voltages, and absence of relevant harmonic distortion provide the technical baseline required to quantify the perturbations caused by nonlinear loads and, later, the mitigation achieved through tuned passive filters.

4.1.2. Case 2: Nonlinear-Load Operating Condition

Case 2 corresponds to the scenario in which nonlinear loads are connected at buses 9 and 14. Under this condition, the electrical response of the system changes significantly, especially in terms of power factor, current demand, harmonic distortion, and local voltage behavior. These changes are a direct consequence of the nonsinusoidal current injection associated with the six-pulse converter model adopted in the study.

One of the most immediate effects of nonlinear loading is the deterioration of the power factor at the buses where the harmonic-producing loads are connected. At bus 9, the power factor decreases from 0.87150 in Case 1 to 0.78271 in Case 2, which corresponds to a reduction of approximately 10.19%. Similarly, at bus 14, the power factor decreases from 0.94805 to 0.85875, representing a reduction of approximately 9.42%. This behavior is consistent with the theoretical formulation presented previously, since harmonic distortion increases the apparent power without a proportional increase in useful active power. The complete comparison of power-factor values is presented in

Table 26. Comparison of power factor between Case 1 and Case 2. The symbol ↓ indicates a decrease in the power factor from Case 1 to Case 2.

Bus	$\mathit{PF}$ (Case 1)	$\mathit{PF}$ (Case 2)	Variation
2	0.86306	0.86137	↓ 0.20%
3	0.98026	0.97881	↓ 0.15%
4	0.99669	0.99203	↓ 0.47%
5	0.97855	0.97412	↓ 0.45%
6	0.83091	0.82579	↓ 0.62%
9	0.87150	0.78271	↓ 10.19%
10	0.84057	0.81185	↓ 3.42%
11	0.88929	0.87421	↓ 1.70%
12	0.96728	0.96055	↓ 0.70%
13	0.91879	0.90986	↓ 0.97%
14	0.94805	0.85875	↓ 9.42%

, and the corresponding visual comparison is shown in Figure 27.

The load current also increases under nonlinear operating conditions because harmonic distortion raises the root-mean-square current demanded by the system. For example, at bus 9, the load current increases from 0.94666 p.u. in Case 1 to 0.98838 p.u. in Case 2, which corresponds to an increase of approximately 4.41%. At bus 14, the current increases from 0.96544 p.u. to 1.00799 p.u., also representing an increase of about 4.41%. Although the magnitude of the current variation is smaller than that observed in the power factor, its physical implications are highly relevant because increased current directly intensifies Joule losses, thermal stress in equipment, and conductor loading. The corresponding comparison is summarized in

Table 27. Comparison of load current between Case 1 and Case 2. The symbol ↑ indicates an increase in the load current from Case 1 to Case 2.

Bus	${\mathit{I}}_{\mathbf{load}}$ (Case 1) [p.u.]	${\mathit{I}}_{\mathbf{load}}$ (Case 2) [p.u.]	Variation
2	0.95694	0.95704	↑ 0.0104%
3	0.99010	0.99049	↑ 0.0394%
4	0.98172	0.98633	↑ 0.4696%
5	0.98014	0.98131	↑ 0.1194%
6	0.93458	0.93488	↑ 0.0321%
9	0.94666	0.98838	↑ 4.4071%
10	0.95118	0.95308	↑ 0.1998%
11	0.94600	0.94727	↑ 0.1342%
12	0.94767	0.94900	↑ 0.1403%
13	0.95198	0.95286	↑ 0.0924%
14	0.96544	1.00799	↑ 4.4073%

.

From the harmonic point of view, the distortion levels obtained in Case 2 confirm that the nonlinear loads severely compromise the quality of the current waveform. As previously reported in the harmonic analysis, the 5th harmonic reaches approximately 18.5% at bus 9 and 16.03% at bus 14, whereas the total harmonic distortion rises to 22.01% and 20.06%, respectively. These values clearly exceed the admissible limits established by IEEE 519 and therefore justify the need for a dedicated mitigation strategy.

This conclusion is also supported by the current waveforms of the lines connected to the disturbed buses. Figure 28 and Figure 29 show that the waveforms are no longer sinusoidal and instead exhibit periodic deformation consistent with the presence of characteristic low-order harmonics. The distortion pattern is therefore observable not only in the spectral domain, but also directly in the time-domain current response.

The nodal voltages also exhibit measurable deviations under nonlinear operation. According to

Table 28. Comparison of nodal voltage magnitudes between Case 1 and Case 2. The symbol ↑ indicates an increase in the nodal voltage magnitude from Case 1 to Case 2.

Bus	V (Case 1) [p.u.]	V (Case 2) [p.u.]	Variation
1	1.0600	1.0604	↑ 0.04%
2	1.0450	1.0461	↑ 0.11%
3	1.0100	1.0110	↑ 0.10%
4	1.0186	1.0225	↑ 0.38%
5	1.0203	1.0232	↑ 0.28%
6	1.0700	1.0735	↑ 0.33%
7	1.0620	1.0728	↑ 1.02%
8	1.0900	1.0930	↑ 0.28%
9	1.0563	1.0816	↑ 2.40%
10	1.0513	1.0708	↑ 1.85%
11	1.0571	1.0669	↑ 0.93%
12	1.0552	1.0597	↑ 0.43%
13	1.0504	1.0561	↑ 0.54%
14	1.0358	1.0564	↑ 1.99%

, bus 9 increases from 1.0563 p.u. in Case 1 to 1.0816 p.u. in Case 2, which represents an increase of approximately 2.40%. Similarly, bus 14 rises from 1.0358 p.u. to 1.0564 p.u., equivalent to an increase of approximately 1.99%. Although these voltage deviations remain moderate when compared with the harmonic distortion levels, they still indicate that the harmonic-producing loads modify the reactive-power distribution and the equivalent network response seen at the affected buses.

Figure 30 provides a global comparison of the voltage profile in the two operating conditions and confirms that the introduction of nonlinear loads modifies the electrical state of the system beyond the directly connected buses.

In summary, Case 2 reveals a clear deterioration of power quality. The reduction in power factor, the increase in rms current demand, the large exceedance of harmonic distortion limits, and the measurable changes in the nodal voltage profile demonstrate that the system operates under compromised electrical conditions when nonlinear loads are introduced at buses 9 and 14. These results provide the technical justification for the implementation of harmonic mitigation measures and support the subsequent use of tuned passive filters as the selected compensation strategy.

4.1.3. Case 3: Nonlinear-Load Condition with Tuned Shunt Passive Filters

Case 3 corresponds to the compensated operating condition in which tuned shunt passive filters are installed at buses 9 and 14. From an operational perspective, the improvement observed in this case is not explained solely by a static supply of reactive power, but also by the simultaneous attenuation of the distorted current component. In Case 2, part of the power-factor deterioration arose from the increase in rms current caused by harmonic distortion. Consequently, even under similar active-power demand, the apparent power increased and the power factor decreased. Once the tuned passive filters were incorporated, the waveform distortion contribution to the apparent power was reduced, which explains why the intervened buses recovered part of their power-factor performance while simultaneously exhibiting lower current demand and restored compliance with IEEE 519.

The most immediate improvement in Case 3 is observed in the power factor of the compensated buses. At bus 9, the power factor increases from 0.78271 in Case 2 to 0.83023 in Case 3, which represents a relative improvement of approximately 6.07%. Similarly, at bus 14, the power factor rises from 0.85875 to 0.90414, corresponding to an improvement of approximately 5.27%. Neighboring buses also exhibit partial recovery toward their base-case values. For example, bus 13 reaches 0.91816 in Case 3, compared with 0.90986 in Case 2, thus approaching the value observed under the linear-load condition. These results indicate that the designed filters restore, to a substantial extent, the fundamental current component while reducing the distortion-related contribution to the total power factor.

The complete comparison of power-factor values for the three operating scenarios is presented in

Table 29. Power-factor values for the three operating cases.

Bus	$\mathit{PF}$ (Case 1: Linear Load)	$\mathit{PF}$ (Case 2: Nonlinear Load)	$\mathit{PF}$ (Case 3: Nonlinear Load with Shunt Passive Filter)
2	0.86306	0.86137	0.86224
3	0.98026	0.97881	0.97966
4	0.99669	0.99203	0.99540
5	0.97855	0.97412	0.97730
6	0.83091	0.82579	0.83055
9	0.87150	0.78271	0.83023
10	0.84057	0.81185	0.83918
11	0.88929	0.87421	0.88852
12	0.96728	0.96055	0.96678
13	0.91879	0.90986	0.91816
14	0.94805	0.85875	0.90414

, and the corresponding visual comparison is shown in Figure 31. These results confirm that the proposed filtering strategy improves the power-factor profile not only at the buses where compensation is directly installed, but also at several adjacent buses influenced by harmonic propagation.

A second important effect of the passive filters is the reduction in load current magnitude. According to the values reported in

Table 30. Comparison of load current between Case 2 and Case 3.

Bus	${\mathit{I}}_{\mathbf{load}}$ (Case 2) [p.u.]	${\mathit{I}}_{\mathbf{load}}$ (Case 3) [p.u.]
2	0.95704	0.95695
3	0.99049	0.99016
4	0.98633	0.95164
5	0.98131	0.96047
6	0.93488	0.93459
9	0.98838	0.80447
10	0.95308	0.79941
11	0.94727	0.86313
12	0.94900	0.92265
13	0.95286	0.90552
14	1.00799	0.81880

, the current at bus 9 decreases from 0.98838 p.u. in Case 2 to 0.80447 p.u. in Case 3, which corresponds to an absolute reduction of 0.18391 p.u. and a relative decrease of 18.61%. At bus 14, the current falls from 1.00799 p.u. to 0.81880 p.u., yielding a relative decrease of 18.77%. The adjacent buses also benefit from the compensation strategy: for example, bus 10 exhibits a reduction of approximately 16.12%, whereas bus 13 shows a decrease of approximately 4.97%. These results indicate that the localized action of the filters at buses 9 and 14 mitigates harmonic propagation and reduces the rms current in neighboring portions of the network. From a physical viewpoint, this reduction is highly relevant because it implies lower Joule losses, reduced thermal stress in transformers and lines, and less severe operating conditions for protection devices.

Although the harmonic distortion is effectively reduced in Case 3, the nodal-voltage response reveals that local overvoltage effects may emerge due to the interaction between the tuned filters and the network impedance. In particular,

Table 31. Comparison of nodal voltage magnitudes for the three operating cases.

Bus	V (Case 1) [p.u.]	V (Case 2) [p.u.]	V (Case 3) [p.u.]
1	1.0600	1.0699	1.0602
2	1.0450	1.0476	1.0455
3	1.0100	1.0124	1.0103
4	1.0186	1.0288	1.0530
5	1.0203	1.0280	1.0420
6	1.0700	1.0806	1.0702
7	1.0620	1.0921	1.1827
8	1.0900	1.0984	1.0902
9	1.0563	1.1255	1.2992
10	1.0513	1.1053	1.2520
11	1.0571	1.0851	1.1591
12	1.0552	1.0691	1.0842
13	1.0504	1.0684	1.1048
14	1.0358	1.0744	1.2769

shows that bus 9 reaches 1.2992 p.u. and bus 14 reaches 1.2769 p.u. in Case 3. Significant voltage increases are also observed at neighboring buses, such as bus 10 with 1.2520 p.u. and bus 13 with 1.1048 p.u. These results indicate that, while the proposed filters are effective from the standpoint of harmonic mitigation and power-factor recovery, their interaction with the system impedance may lead to substantial local voltage amplification. Therefore, the compensated condition should not be interpreted as a uniformly improved electrical state in every variable, but rather as a condition in which harmonic and power-factor performance improve at the expense of a more stressed voltage profile in certain network locations.

This observation is particularly important because it highlights a practical trade-off in filter-based compensation: the mitigation of dominant harmonics and the recovery of the power factor do not automatically guarantee acceptable voltage magnitudes. The global voltage comparison for the three cases is presented in Table 31 and illustrated in Figure 32.

The line-current waveforms in Case 3 provide additional qualitative evidence of harmonic mitigation. Figure 33 and Figure 34 show that the current waveforms in the links connected to buses 9 and 14 recover a morphology much closer to sinusoidal behavior when compared with Case 2. In particular, the critical links associated with buses 9 and 14 exhibit visibly lower harmonic deformation, which is consistent with the reduction in THD and rms current reported previously. From an operational perspective, this improvement in waveform quality is beneficial because it reduces dynamic stress on breakers, metering systems, and protection devices.

The most conclusive indicator of filter effectiveness is the reduction in total harmonic distortion. As shown in

Table 32. Comparison of total harmonic distortion between Case 2 and Case 3.

Bus	THD (Case 2) [%]	THD (Case 3) [%]
1	2.8005	1.9476
2	4.5436	3.1050
3	4.3386	2.5914
4	6.7543	3.8519
5	7.5842	3.7563
6	8.0749	2.0836
7	14.3537	3.1022
8	7.4355	1.7888
9	22.0145	4.6114
10	19.3557	4.1129
11	13.6495	2.9901
12	9.1777	2.3501
13	10.3676	2.6735
14	20.0658	5.2156

, the THD at bus 9 decreases from 22.0145% in Case 2 to 4.6114% in Case 3, whereas at bus 14 it decreases from 20.0658% to 5.2156%. Similar reductions are also observed at adjacent buses, including bus 10, where THD decreases from 19.3557% to 4.1129%, and bus 13, where it drops from 10.3676% to 2.6735%. Therefore, from the viewpoint of IEEE 519 compliance, the proposed filter design is effective in restoring admissible harmonic conditions not only at the compensated buses, but also at several neighboring buses influenced by harmonic propagation.

To illustrate this effect beyond the directly compensated buses, Figure 35 shows the harmonic distortion at buses 10 and 13 under nonlinear-load conditions, whereas Figure 36 presents the corresponding harmonic distortion after passive filtering. The comparison confirms that the tuned filters connected at buses 9 and 14 also improve the harmonic response at adjacent buses, which indicates that their effect is not purely local but propagates through the surrounding network.

An additional effect of the tuned filters is observed in the impedance-frequency response of the network. After the filters are integrated at the buses where the nonlinear loads are located, the equivalent impedance associated with the critical buses is reduced, as shown in Figure 37 and Figure 38. This reduction in impedance at the relevant harmonic range is consistent with the attenuation mechanism of the tuned shunt branches and supports the observed decrease in harmonic distortion.

In summary, the results of Case 3 confirm that the tuned passive filters effectively mitigate the dominant harmonic components, especially the 5th harmonic, while simultaneously improving the power factor and reducing current demand. Moreover, the proposed design restores IEEE 519 compliance at the buses that originally exhibited severe distortion. At the same time, the compensated scenario reveals that harmonic mitigation and power-factor recovery must be interpreted together with the voltage response of the network, since the appearance of local overvoltage conditions indicates that the final design should be evaluated not only from the perspective of THD reduction, but also in terms of voltage acceptability and overall operating security.

4.2. Comparative Analysis

A comparative assessment of the three operating scenarios provides a clear understanding of the impact of harmonic distortion on system performance and the effectiveness of the proposed compensation strategy. In particular, the transition from Case 1 to Case 2 quantifies the electrical degradation caused by nonlinear loads, whereas the transition from Case 2 to Case 3 reveals the extent to which the tuned passive filters restore harmonic compatibility and improve energy-performance indicators.

4.2.1. Comparison of Power Factor

Table 33. Comparative power-factor analysis for the most relevant buses.

Bus	$\mathit{PF}$ (Case 1)	$\mathit{PF}$ (Case 2)	$\Delta$ (%) C1→C2	$\mathit{PF}$ (Case 3)	$\Delta$ (%) C2→C3
9	0.87150	0.78271	−10.19%	0.83023	+6.07%
14	0.94805	0.85875	−9.42%	0.90414	+5.27%
10	0.84057	0.81185	−3.42%	0.83918	+3.37%
13	0.91879	0.90986	−0.97%	0.91816	+0.91%

shows that the power factor deteriorates after nonlinear loads are introduced into the system, with the most severe reductions occurring at the buses where these loads are directly connected, namely buses 9 and 14. After the passive filters are installed, the power factor recovers partially and approaches the values of the base operating condition. This behavior confirms that the filters mitigate the combined effect of harmonic distortion and the additional reactive-power demand associated with converter-based nonlinear loading.

The trend is consistent across the most affected buses. At bus 9, the reduction from 0.87150 in Case 1 to 0.78271 in Case 2 represents a deterioration of 10.19%, while the subsequent increase to 0.83023 in Case 3 corresponds to a recovery of 6.07% with respect to the distorted scenario. Likewise, bus 14 decreases from 0.94805 to 0.85875 and then recovers to 0.90414, yielding a compensation-driven improvement of 5.27%. Neighboring buses such as 10 and 13 also experience partial recovery, which indicates that the benefit of the proposed filters is not confined to the compensated nodes alone. This behavior is fully consistent with the reduction in THD and the decrease in rms current: as the distortion content is attenuated, the apparent power associated with non-useful harmonic components is reduced, and the power factor improves accordingly.

4.2.2. Comparison of Load Currents

The comparison of load currents further confirms the adverse impact of nonlinear loading and the corrective action of the passive filters. As shown in

Table 34. Comparative analysis of load current for the most relevant buses.

Bus	${\mathit{I}}_{\mathbf{load}}$ (Case 2) [p.u.]	${\mathit{I}}_{\mathbf{load}}$ (Case 3) [p.u.]	Reduction C2→C3
9	0.98838	0.80447	−18.61%
14	1.00799	0.81880	−18.77%
10	0.95308	0.79941	−16.12%
13	0.95286	0.90552	−4.97%

, the current magnitude increases in Case 2 due to the higher rms value produced by harmonic distortion. This increase directly affects Joule losses and thermal loading in transformers, conductors, and associated protective devices. After the tuned passive filters are incorporated, the effective current decreases markedly, confirming that the filters divert part of the harmonic current away from the main network.

At bus 9, the current decreases from 0.98838 p.u. in Case 2 to 0.80447 p.u. in Case 3, which corresponds to a reduction of 18.61%. At bus 14, the current decreases from 1.00799 p.u. to 0.81880 p.u., yielding a reduction of 18.77%. Significant improvements are also observed at adjacent buses, such as bus 10, where the current decreases by 16.12%, and bus 13, where the reduction reaches 4.97%. These results indicate that the effect of the compensation is not purely local, but propagates through the network by reducing the harmonic burden and the rms current in nearby electrical paths.

4.2.3. Indicative Energetic and Economic Implications of Current Reduction

Although the present study does not include a utility-specific tariff model or an annualized economic dispatch assessment, the reduction in rms current obtained after compensation allows an indicative evaluation of the energetic benefit associated with the proposed filters. Under the assumption that the equivalent resistance of the affected electrical paths remains unchanged between Case 2 and Case 3, the resistive losses are proportional to the square of the current magnitude, i.e.,

$$P_{\mathrm{loss}}\propto I^{2}R.$$

(55)

Therefore, a normalized loss-reduction index can be defined for each bus i as

$${\eta }_{\mathrm{loss},i}=\left(1-{\left({\displaystyle \frac{I_i^{\left(3\right)}}{I_i^{\left(2\right)}}}\right)}^2\right)\times 100\%,$$

(56)

where $I_i^{\left(2\right)}$ and $I_i^{\left(3\right)}$ are the load-current magnitudes in Case 2 and Case 3, respectively. This index does not represent the total system loss reduction in absolute units, but it provides a technically consistent estimate of the local decrease in Joule losses associated with the compensation-induced current reduction.

Using the values reported in Table 34, the estimated relative loss reduction reaches approximately 33.68% at bus 9 and 34.04% at bus 14. For the adjacent buses, the same indicator yields reductions of approximately 29.75% at bus 10 and 9.69% at bus 13. These values confirm that the decrease in current observed after passive compensation is not only beneficial from the viewpoint of power quality, but also from the standpoint of thermal loading and resistive energy dissipation in the most affected electrical paths.

From an economic perspective, these results indicate that the proposed compensation strategy has the potential to reduce operating costs associated with excessive current demand, technical losses, and poor power factor. However, the exact monetary benefit depends on external parameters that were not modeled in this study, such as the local electricity tariff, the structure of reactive-power penalties, the annual operating hours, and the equivalent resistive parameters of the compensated network. Therefore, the present work reports a rigorous technical-economic indicator based on normalized loss reduction, while a full annualized cost analysis is identified as a relevant extension for future work.

4.2.4. Consolidated Verification of the Main Compensation Benefits

To provide a compact and explicit verification of the benefits achieved by the proposed methodology,

Table 35. Consolidated summary of the main benefits obtained after compensation at the priority buses.

Bus	PF (Case 2)	PF (Case 3)	THD (Case 2) [%]	THD (Case 3) [%]	Current Reduction [%]	Loss-Reduction Index [%]
9	0.78271	0.83023	22.01	4.61	18.61	33.68
14	0.85875	0.90414	20.07	5.22	18.77	34.04

summarizes the main electrical indicators before and after compensation at the two priority buses. This consolidated comparison is included to highlight, in a single view, the simultaneous improvement in power factor, harmonic distortion, current magnitude, and indicative resistive-loss behavior obtained through the proposed resonance-aware workflow.

As shown in Table 35, the proposed methodology does not produce an isolated improvement in a single indicator. Instead, the compensation strategy simultaneously improves the power factor, reduces harmonic distortion below the admissible IEEE 519 threshold at the priority buses, decreases the rms current demand, and yields a substantial indicative reduction in resistive losses. This consolidated evidence strengthens the validation of the proposed method from both the power-quality and energetic viewpoints. These combined results confirm that the effectiveness of the proposed methodology is supported not by a single favorable metric, but by a coherent multi-indicator improvement observed consistently after compensation.

4.2.5. Comparison of THD and Dominant Harmonic Behavior

The most direct evidence of the effectiveness of the proposed filter design is provided by the comparison of total harmonic distortion. According to

Table 36. Comparative analysis of THD values at the most affected buses.

Bus	THD (Case 2) [%]	THD (Case 3) [%]	IEEE 519 Compliance (≤8%)
9	22.01	4.61	Yes, in Case 3
14	20.07	5.22	Yes, in Case 3
10	19.36	4.11	Yes, in Case 3
13	10.37	2.67	Yes, in Case 3

, Case 2 exhibits severe harmonic contamination, with THD values that clearly exceed the IEEE 519 limit of 8% at the critical buses. In contrast, after the passive filters are installed, the THD values fall below the admissible threshold at all buses listed in the comparison. This result indicates that the filters are properly tuned with respect to the dominant harmonic component, namely the 5th harmonic, and that the final design is effective from the standpoint of harmonic compatibility.

At bus 9, THD decreases from 22.01% in Case 2 to 4.61% in Case 3, while at bus 14 it decreases from 20.07% to 5.22%. Similar improvements are also observed at buses 10 and 13, where the distortion is reduced to 4.11% and 2.67%, respectively. Therefore, the reduction in THD exceeds 70% at the critical buses, demonstrating that the tuned passive filters restore compliance with IEEE 519 in the buses that originally exhibited the most severe harmonic distortion.

Taken together, the comparative analysis demonstrates that nonlinear-load integration degrades power quality in a coupled manner, simultaneously affecting power factor, current magnitude, and harmonic distortion. In contrast, the installation of tuned passive filters substantially improves the electrical response of the system by reducing THD, lowering the rms current demand, and partially restoring the power factor. Therefore, from the perspective of harmonic mitigation and regulatory compliance, the proposed compensation strategy proves effective and technically consistent. Nevertheless, as discussed previously, these benefits should be interpreted together with the voltage response of the system, since the compensated scenario also exhibits significant local overvoltage conditions that must be considered in the final engineering assessment.

5. Conclusions

This study proposed and validated a resonance-aware methodology for power factor correction in transmission networks affected by nonlinear loads. The main contribution of the work is not limited to the use of tuned passive filters as compensation devices, but lies in the integrated decision framework that links nonlinear-load screening, weighted bus prioritization, impedance-frequency resonance assessment, and post-compensation validation under IEEE 519 compliance criteria. In this way, the proposed approach provides a structured and reproducible procedure for determining where compensation should be applied, how the critical buses should be ranked, and how the final design should be technically verified.

The results obtained in the IEEE 14-bus system confirmed that the integration of nonlinear loads at buses 9 and 14 produced a significant deterioration in the electrical performance of the network. In the uncompensated nonlinear scenario, the power factor decreased to 0.78271 at bus 9 and 0.85875 at bus 14, while the total harmonic distortion increased to 22.01% and 20.07%, respectively, clearly exceeding the admissible IEEE 519 limit. These results validated both the severity of the selected operating condition and the effectiveness of the proposed screening and prioritization stages for identifying the buses with the greatest need for corrective intervention.

The prioritization framework also proved to be technically consistent. By combining normalized harmonic severity, relative power-factor deterioration, and absolute power-factor deficiency through AHP-based weighting, the methodology avoided arbitrary compensation decisions and correctly identified buses 9 and 14 as the priority locations for intervention. This result is especially relevant because it demonstrates that the proposed method is not based on a single indicator, but on a coordinated assessment of complementary electrical criteria.

After the installation of the tuned shunt passive filters, substantial improvements were observed in the most relevant performance indicators. The power factor increased from 0.78271 to 0.83023 at bus 9 and from 0.85875 to 0.90414 at bus 14. At the same time, the total harmonic distortion was reduced from 22.01% to 4.61% at bus 9 and from 20.07% to 5.22% at bus 14, thereby restoring compliance with IEEE 519 at the buses that originally exhibited the most severe harmonic contamination. In addition, the compensated scenario produced current reductions of approximately 18.61% at bus 9 and 18.77% at bus 14, which supports the conclusion that the proposed design improves not only harmonic performance, but also the energetic use of the network through a lower rms current demand.

A further relevant finding is that the benefits of the compensation were not purely local. Neighboring buses also exhibited reductions in harmonic distortion and partial recovery of power-factor performance, which indicates that the designed filters mitigated harmonic propagation through adjacent portions of the network. This behavior reinforces the practical value of the proposed framework for network-level compensation studies, where the electrical effect of a corrective action extends beyond the installation bus itself.

At the same time, the study also revealed an important engineering limitation that must be considered in the final assessment. Although harmonic distortion and power factor were significantly improved, the compensated scenario showed substantial local overvoltage conditions at some buses. This result confirms that passive harmonic compensation in transmission networks should not be evaluated exclusively through THD and power-factor indicators. Instead, resonance-aware compensation must always be accompanied by a careful verification of post-compensation voltage magnitudes and operating security in order to ensure that the final solution is globally acceptable from a system-operation perspective.

Overall, the proposed methodology proved effective for identifying critical buses, prioritizing compensation actions, designing tuned passive filters, restoring IEEE 519 compliance, and improving power factor under nonlinear-load conditions. Future work should extend the application of the proposed framework to larger transmission and subtransmission networks, compare its performance against active and hybrid filtering alternatives under broader operating variability, and incorporate additional technical and economic criteria so that compensation decisions can be evaluated from a more comprehensive planning perspective.