Numerical Assessment of Thermal Effects in Bundled Overhead Conductors for Dynamic Line Rating

Featured Application

The proposed numerical methodology allows transmission system operators to assess the thermal and current-carrying capacity of overhead lines, accounting for single and bundled conductor configurations. Bundle-specific effects, which are neglected in traditional standards (IEEE and CIGRE) calculations, are incorporated to provide more accurate dynamic line rating evaluations. This approach can support real-time ampacity management, operational planning, contingency analysis, and transmission line optimization, offering a scalable framework for diverse networks.

Abstract

Dynamic Line Rating (DLR) is increasingly important for maximizing capacity of existing overhead transmission lines. Conventional thermal rating methods, such as IEEE 738 and model conductors as single, isothermal cylinders and offer limited guidance for multi-conductor bundles, not fully capturing the complex aerodynamic and thermal interactions present in high-voltage networks. This study addresses these limitations by presenting a high-fidelity, two-dimensional coupled thermal-fluid model developed in COMSOL Multiphysics 4.3b. Single and bundled configurations (two-conductor, three-conductor and four-conductor) are analyzed under steady-state conditions using the Shear Stress Transport (SST) turbulence model, accounting for sub-conductor spacing, wind speed, and interactions between temperature distribution and airflow. Simulation results are compared with ampacity calculations from relevant standards to evaluate limitations of simplified models. Results show that leeward conductors reach temperatures up to ~4 °C higher than windward conductors, forming the thermal bottleneck, with peak temperatures of ~103.3 °C versus ~99 °C for single conductors. For bundled conductors, the current required to keep the maximum temperature at 100 °C was calculated, and this value was found to be approximately 3% lower than the current predicted by IEEE 738. The study emphasizes the importance of multiphysics, position-aware simulations to prevent overloading and optimize transmission line utilization.

1. Introduction

Overhead transmission lines are critical components of modern power systems, ensuring the efficient and reliable transfer of electricity over long distances. However, expanding transmission capacity through new line construction is increasingly constrained by social, environmental, regulatory, and economic challenges. Large-scale projects often encounter lengthy approval processes, high capital costs, and public opposition due to their environmental footprint. In parallel, power system operators are confronted with rapidly changing operating conditions: increasing electricity demand, growing peak loads, deregulated energy markets, and, most critically, the large-scale integration of variable Renewable Energy Sources (RES) such as wind and solar [1,2]. These developments frequently push transmission networks to operate close to their thermal and stability limits, raising concerns about congestion, reduced reliability, and higher operational risks.

To address these challenges without the need for costly new infrastructure, operators are increasingly adopting advanced strategies to enhance the utilization of existing transmission assets, with reconductoring to increase line capacity and DLR to adjust operational limits based on real-time conditions emerging as particularly promising approaches [3]. Reconductoring involves replacing existing conductors with high-capacity alternatives, but its implementation is capital-intensive and disruptive. DLR, by contrast, represents a more flexible, adaptive, and cost-effective approach. Unlike static line rating (SLR) methods that rely on conservative worst-case assumptions, DLR continuously adjusts the current-carrying capacity of lines according to real-time environmental conditions such as ambient temperature, wind speed, and solar irradiance [4,5,6]. This dynamic adjustment increases available capacity, facilitates the integration of variable renewable generation, reduces congestion and curtailment, and improves overall grid flexibility [5,7,8]. Recent advances in sensor technologies, weather forecasting, and data analytics have accelerated the deployment of DLR systems, with studies reporting 10–30% additional available capacity and significant improvements in operational reliability [5,9,10].

Recent studies have further emphasized the practical applications and benefits of DLR. It has been shown that DLR-based monitoring enables real-time assessment of operational limits, enhancing safety and reducing the risk of conductor overheating [11]. The technology has been demonstrated to optimize transmission capacity and increase grid flexibility without requiring additional infrastructure investments [12]. Moreover, Dynamic Thermal Rating has been successfully applied to increase allowable current on existing lines while maintaining safety and reliability, contributing to congestion reduction and more efficient integration of renewable energy sources [5]. These findings highlight the importance of investigating DLR methods, particularly for high-voltage lines with bundled conductor configurations.

Despite these advantages, the practical application of DLR still faces significant technical challenges. The most widely used industry standards, IEEE 738 [13] and CIGRE Technical Brochure 601 [14], provide thermal rating methods that model conductors as single, isothermal cylinders. While effective for individual conductors, these simplified models fail to account for the aerodynamic and thermal interactions present in multi-conductor bundles, which are commonly deployed in high-voltage transmission systems. In practice, bundle ampacity is often estimated by multiplying the ampacity of a single conductor by the number of conductors in the bundle. However, experimental and numerical studies demonstrate that leeward conductors in a bundle may experience elevated temperatures—sometimes several degrees higher than windward conductors—making such simplifications non-conservative [15,16]. This thermal imbalance can reduce the effective ampacity of the entire bundle and lead to underestimated risks of overheating, mechanical degradation, and accelerated aging.

Recent research has attempted to refine the modeling of overhead conductors by considering non-isothermal behavior, turbulence effects, and the influence of environmental conditions. While progress has been made for single-conductor systems [17,18,19,20], relatively few studies have focused on bundled configurations despite their widespread use in high-voltage networks. Moreover, existing analyses often rely on simplified turbulence models or partial thermal considerations, leaving a gap in high-fidelity multiphysics approaches that couple fluid dynamics with thermal conduction and radiation.

While numerous studies have employed k-ε, k-ω, and SST turbulence models for heat transfer and fluid flow simulations in various engineering applications, nearly all of these investigations focus on domains other than overhead conductors or bundled configurations. However, SST-based approaches specifically targeting the heat transfer in conductors or bundled conductors with consideration of DLR appear to be largely unexplored. Although some relevant studies may exist, the literature still shows a noticeable gap in high-fidelity multiphysics analyses that integrate SST turbulence modeling with conductor thermal behavior and DLR considerations.

Furthermore, existing studies in this area have largely concentrated on the thermal behavior of individual conductors or bundled systems, while paying little attention to how these temperature differences influence DLR [16,21]. In many cases, the non-uniform heating among sub-conductors is not considered, and the real-time current-carrying capacity of the line is assumed uniform. Ignoring these factors can result in non-conservative ratings and may reduce the reliability and efficiency of transmission system operation.

To provide a more accurate assessment of conductor ampacity under DLR conditions, this study presents several key contributions, as summarized below:

• A comprehensive 2D coupled thermal–fluid model for overhead conductors was developed using COMSOL Multiphysics 4.3b with the SST turbulence model, enabling more accurate representation of convective heat transfer compared to conventional approaches.
• The study extends the analysis to bundled conductors (two-, three-, and four-conductor configurations), commonly used in high-voltage transmission systems, which have received limited attention in terms of ampacity and thermal behavior under DLR conditions.
• Unlike simplified industry practices that assume isothermal behavior or calculate bundle ampacity by multiplying single-conductor ratings, the proposed model explicitly considers thermal and aerodynamic interactions among subconductors, highlighting the temperature imbalance between windward and leeward conductors.
• By comparing the numerical results with IEEE 738 and CIGRE TB601 standards, the study reveals important differences between simplified analytical methods and high-fidelity multiphysics modeling, particularly under DLR conditions.
• The findings demonstrate that ignoring non-uniform heating in bundled conductors can lead to non-conservative ratings, which may underestimate the risks of overheating, mechanical degradation, and reduced transmission reliability.

In summary, the study provides a multiphysics-based methodology that enhances the accuracy of DLR estimation, thereby contributing to safer, more reliable, and more efficient operation of transmission systems.

This study develops a comprehensive 2D coupled thermal-fluid model for both single and bundled conductors under steady-state conditions, implemented in COMSOL Multiphysics using SST turbulence model. The model considers the convective and radiative heat transfer mechanisms, as well as aerodynamic interactions among subconductors in multi-conductor bundles. Environmental parameters such as wind speed, wind direction, ambient temperature, and solar irradiance are included as boundary conditions to closely simulate real operating conditions of overhead transmission lines. For bundled conductors, the model accounts for the thermal imbalance between windward and leeward subconductors, providing a more realistic estimation of temperature distribution along the line. Additionally, an analysis was conducted on the two-conductor bundle configuration with reduced spacing from 40 cm to 30 cm to demonstrate how closer conductor spacing could further increase the difference between numerical results and standard IEEE ampacity predictions.

Simulation results are compared with ampacity predictions from IEEE 738 and CIGRE TB601 standards to highlight the differences between simplified analytical approaches and numerical results. The comparison includes not only maximum conductor temperatures but also detailed spatial temperature distributions and local convective heat flux variations, providing insight into the limitations of standard single-conductor approximations. Moreover, the analysis considers how non-uniform heating in bundles affects DLR and how closer spacing in the two-conductor bundle can increase the difference between numerical results and IEEE predictions. The main objective of this study is to provide a more accurate, multiphysics-based methodology for estimating the current-carrying capacity of overhead transmission lines, with particular emphasis on two-conductor, three-conductor, and four-conductor bundles under DLR conditions. This approach allows for a more reliable prediction of line capacity under realistic operating conditions.

The remainder of the paper is organized as follows. Section 2 provides an overview of IEEE 738 and CIGRE TB601, describes the modeling of stranded conductors and multi-conductor arrangements, and presents the details and assumptions of the computational model. Section 3 presents and discusses the temperature, wind, and pressure distributions around single and bundled conductors, provides a quantitative comparison with IEEE 738 and CIGRE TB601, and a qualitative discussion in relation to other models reported in the literature, highlighting the differences and advantages of the proposed model. Finally, Section 4 summarizes the key findings of the study.

2. Materials and Methods

Accurate estimation of conductor ampacity requires a thorough understanding of conductor geometry and its interaction with environmental conditions. This study examines the effects of conductor structure and bundle configurations on convective cooling and ampacity, aspects that are not fully captured in standard methods.

Numerical simulations were conducted using COMSOL Multiphysics to model coupled thermal and fluid flow phenomena for single, two-conductor, three-conductor, and four-conductor bundles. This section outlines the methodology and introduces the main components of the numerical framework. Subsequent subsections describe overview of IEEE and CIGRE standards, conductor modeling strategy, computational model details and assumptions. A structural diagram of the proposed solution is presented in Figure 1 to illustrate the overall methodology.

2.1. Overview of IEEE 738 and CIGRE TB601

IEEE 738 and CIGRE TB 601 are two widely adopted references that provide comprehensive methodologies for estimating the current-carrying capacity of overhead conductors under varying environmental and operating conditions. These standards serve as the foundation for DLR practices, offering formulations for heat balance equations, environmental inputs, and conductor-specific parameters that govern thermal performance. Both documents encompass a broad range of considerations, including solar heating, radiative and convective cooling, electrical resistance variations with temperature, and the influence of conductor surface characteristics [22]. Due to the extensive scope of these standards, this section focuses on presenting a general overview while highlighting the elements most relevant to the present study. Among the various thermal mechanisms addressed, convective cooling plays a particularly critical role, especially in bundle configurations where airflow distribution is affected by conductor arrangement. Detailed examination of the convective cooling formulations and assumptions in both standards is provided, as they form a critical reference for validating the numerical simulations.

As a foundation, both standards rely on the principle of thermal equilibrium as given in Equation (1), which forms the basis for calculating the maximum current-carrying capacity under steady-state conditions.

$$\underset{Heat gain }{\underbrace{q_j\left(I, R_{AC}\left(T_c\right)\right)+q_s\left(Q_s,\theta , D\right)}}=\underset{Heat loss }{\underbrace{q_c\left(T_c, T_a, V_w,\Phi \right)+q_r\left(T_c, T_a\right)}}$$

(1)

This principle assumes that the conductor reaches a stable operating temperature when the total rate of heat input is equal to the total rate of heat dissipation. In this framework, the primary sources of heat gain include Joule heating due to the electrical current and solar radiation, while heat is dissipated through convective and radiative cooling. The ampacity I_max is determined by balancing these mechanisms and can be expressed as:

$$I_{max}= \sqrt{{\displaystyle \frac{q_r+q_c-q_s}{{R(T}_c)}}}$$

(2)

In these equations, I_max is the maximum allowable current in amps. q_j (joule heating), q_s (solar heat gain), q_c (convective heat loss), and q_r (radiative heat loss) are all expressed in watts per meter. R(T_c) is the AC resistance of the conductor at temperature T_c, measured in Ω/m, and T_c is the conductor surface temperature in °C. The temperature-dependent AC resistance is calculated using linear interpolation between reference resistance values at two known temperatures, as shown in Equation (3).

$$R_{AC}\left(T_c\right)=\underset{Linear Interpolation }{\underbrace{\left[{\displaystyle \frac{R\left(T_{high}\right)-R\left(T_{low}\right)}{T_{high}-T_{low}}}\right] \left(T_c-T_{low}\right)+R(T_{low})}}$$

(3)

The values of R(T_high) and R(T_low) represent the conductor resistance at specified high and low reference temperatures. These parameters are typically provided by the conductor manufacturer or sourced from standard handbooks for Aluminum Conductor Steel Reinforced (ACSR) conductors. Utilizing these reference values, resistance at any intermediate temperature between T_low and T_high can be accurately estimated through linear interpolation, as outlined in IEEE 738.

The convective cooling mechanisms described in IEEE 738 and CIGRE TB601 are quantified through specific empirical correlations that vary according to wind speed and flow conditions.

Table 1. Convective Heat Transfer Equations for Transmission Line Conductors in IEEE and CIGRE Standards.

Standard	Cooling Type	Wind Speed, Vw	Equation
IEEE 738	Natural	Zero	$q_c=3.645 · {\gamma }^{0.5}·D^{0.75}· {\left(T_s-T_a\right)}^{1.25}$	(4)
	Forced	Low	${q_{c1}=K}_{angle }·\left[1.01+1.35·N_{Re}^{0.52}\right]·\lambda f ·(T_s-T_a)$	(5)
	Forced	High	${q_{c2}=K}_{angle }·0.754 ·N_{Re}^{0.6}·\lambda f ·(T_s-T_a)$	(6)
CIGRE TB601	Natural	Zero	$q_{c1}=\pi ·{\lambda }_f·\left(T_s-T_a\right)·N{u}_{\beta }$	(7)
	Forced	High	$q_{c2}=\pi ·{\lambda }_f·\left(T_s-T_a\right)·N{u}_{\delta }$	(8)

summarizes these formulations, distinguishing between natural and forced convection modes as defined by each standard. The table highlights the key parameters, applicable wind speed ranges, and corresponding equations used to estimate convective heat loss from overhead conductors. This comparison provides a clear overview of the differences in modeling approaches, serving as a foundation for the subsequent analysis presented in this study.

Both IEEE 738 and CIGRE TB601 address convective cooling by considering wind speed and conductor-air interactions; however, their approaches differ notably in formulation and flow regime classification. IEEE 738 distinguishes between natural convection (V_w = 0 m/s) and forced convection, with separate empirical equations for low (typically below 3 m/s) and high wind speeds (above 3 m/s). These formulations explicitly incorporate the Reynolds number to characterize airflow turbulence and include correction factors for wind angle incidence, reflecting the directional dependency of convective heat transfer. Conversely, CIGRE TB601 employs a unified approach based on the Nusselt number, which implicitly encompasses both natural and forced convection regimes through dimensionless fluid dynamics correlations. Instead of using wind speed thresholds, CIGRE TB601 calculates convective heat loss by applying Nusselt correlations that adapt continuously to varying flow conditions. This method offers greater flexibility, especially under transitional flow regimes, but may abstract away explicit wind speed categories present in IEEE’s model [13,14].

Moreover, while both standards utilize similar environmental parameters such as ambient temperature, conductor surface temperature, and air thermal conductivity, their empirical coefficients and correlation functions differ. For instance, the convection coefficients in IEEE’s forced convection equations tend to increase with Reynolds number raised to powers between 0.52 and 0.6, whereas CIGRE’s Nusselt-based correlations follow established dimensionless numbers calibrated for conductor geometry and flow regime. IEEE’s segmented approach may lead to overestimation or underestimation of cooling under certain wind speeds due to sharp regime transitions, while CIGRE’s continuous model may better capture subtle variations but depend heavily on correlation accuracy. Consequently, the conductor temperature predictions and the resulting ampacity estimates based on IEEE 738 and CIGRE TB601 thermal models are generally comparable, with only slight differences arising from variations in the calculation of convective heat losses and solar radiation gains [5,21].

2.2. Modeling of Stranded Conductor and Multi-Conductor Arrangements

The 795 MCM Drake ACSR conductor was selected as the reference model in this study because it is employed in the example ampacity calculations of both IEEE 738 and CIGRE TB601 standards. This facilitates practical and consistent comparison between the numerical simulation results and those derived from the standards. The ACSR design of Drake conductor features a stranded configuration composed of concentric layers of aluminum wires wound around a central galvanized steel core. While the electrical properties are detailed in Section 2,

Table 2. Properties of the 795 MCM Drake ACSR Conductor [13,14].

Property	Description and Units (SI)
Conductor Cross-Section	795.0 MCM (400.98 mm2)
Number of Strands	26/7 ACSR (Aluminum/Steel)
Aluminum Strand Diameter	4.44 mm
Steel Strand Diameter	3.45 mm
Overall Conductor Diameter, D	28.14 mm
Nominal Current Rating, In	907 A

summarizes the geometric characteristics of the Drake ACSR conductor.

Figure 2 illustrate the stranded conductor geometries considered in this study: single, two-conductor, three-conductor, and four-conductor bundle configurations. The outer-layer strands of the stranded conductors are numbered in the figures to enable easier reference throughout Section 3.

To enhance clarity in the interpretation of temperature distribution results, specific identifiers were assigned to the conductor strands and bundle elements in the geometric representations. In the stranded single conductor, the outer-layer aluminum strands were numbered individually to facilitate direct referencing during 2D thermal distribution analyses. This allows for precise identification of which strand is being evaluated, particularly when discussing localized temperature gradients. In the bundled configurations, each sub-conductor was labeled as C1, C2, C3, or C4 depending on its position within the arrangement. This labeling system is consistently used in subsequent result figures. The spacing between sub-conductors in the bundle, denoted as d, is illustrated in Figure 2. Typical values for sub-conductor spacing range from 30 cm to 65 cm, depending on the design considerations. In addition, several studies in the literature have also adopted a spacing of 40 cm for similar transmission configurations. Based on these references, a spacing of 40 cm was used in this study [23].

2.3. Computational Model Details and Assumptions

A coupled multiphysics model was developed in COMSOL Multiphysics to simulate thermal behavior of overhead bundled conductors, including the influence of bundle configuration as well as convective, radiative, and solar heat transfer mechanisms. The simulation platform and selected physics interfaces are described in the following subsection. Material properties of the conductor components are then defined, followed by the boundary and environmental conditions representing typical operating scenarios.

2.3.1. Physics Interfaces, Material Properties and Boundary Conditions

As shown in Figure 3, the DLR model is implemented in COMSOL Multiphysics using coupled electromagnetic, heat transfer, and fluid flow interfaces. The electromagnetic interface evaluates Joule heating arising from current flow, while the heat transfer interface accounts for conduction within the conductor, convection to ambient air, and surface-to-surface radiation.

The fluid flow interface incorporates non-isothermal and turbulent wind conditions to represent convective cooling. These coupled interfaces are solved simultaneously, and the resulting conductor surface temperature (T_c) serves as the governing parameter for real-time ampacity estimation. In addition, the material properties of all conductor components, including electrical, thermal, and radiative parameters, are defined and summarized in

Table 3. Key material properties defined for conductors and surrounding medium in the model.

Parameter	Unit	Steel	Aluminum	Air
Relative Permeability	-	1	1	1
Relative Permittivity	-	1	1	1
Density	kg/m3	7780	2703	${\displaystyle \frac{1.293-1.525 \ast {10}^{-4} \ast H_e+6.379 \ast {10}^{-9} \ast {H_e}^2}{1+0.00367 \ast T_{film}}}$
Heat Capacity	J/(kg⋅K)	481	955	Cp(T)
Thermal Conductivity	W/(m⋅K)	44.5	240	$\mathrm{k}\left(\mathrm{T}\right)=2.424 \ast {10}^{-2}+7.477 \ast {10}^{-5} \ast T_{film}-4.407 \ast {10}^{-9} \ast T_{film}^2$
Electrical Conductivity	S/m	3.48 × 106	3.54 × 107	0
Dynamic Viscosity	Pa⋅s	-	-	$\mathsf{\eta }\left(\mathrm{T}\right)={\displaystyle \frac{1.458 \ast {10}^{-6} \ast {(T_{film}+273)}^{1.5}}{T_{film}+383.4}}$
Reference Resistivity	Ω⋅m	2.8736 × 10−7	2.8264 × 10−8	-
Temperature Coefficient	1/K	0.0045	0.00403	-
Reference Temperature	K	293.15	293.15	-

, providing the essential inputs for the multiphysics simulation.

Magnetic Fields Interface

The Magnetic Fields (mf) interface in COMSOL Multiphysics is used to model magnetic fields in conducting and non-conducting domains under both steady-state and transient conditions. It solves Maxwell’s equations for magnetostatics, allowing the computation of magnetic flux density, induced currents, and associated electromagnetic quantities. In the study, the Amper’s Law in Solids node was applied to the conductor domains to account for current flow within solid materials.

Within the Amper’s Law in Solids node, under the Constitutive Relation (J_c–E) section, the Conduction Model was set to Linearized Resistivity. This approach was selected to improve the accuracy of Joule heating calculations in temperature-dependent conductors. By updating the electrical resistivity as the conductor temperature rises, the model ensures that the generated resistive heating reflects the evolving thermal state, allowing the system to reach a physically consistent energy balance. This provides a more accurate representation of the coupled electromagnetic–thermal behavior of the conductors under operating conditions. The governing relations are:

$$J_c=\sigma ·E$$

(9)

$$\sigma ={\displaystyle \frac{1}{{\rho }_0·\left[1+\alpha ·\left(T-T_{ref}\right)\right]}}$$

(10)

where J_c is the current density, E is the electric field, σ is the temperature-dependent conductivity, ρ₀ is the reference resistivity at temperature T_ref, and α is the temperature coefficient of resistivity.

Additionally, two External Current Density nodes were defined: one for aluminum conductors and the other for steel strands. This distinction is necessary because the electrical resistivities of aluminum and steel differ significantly, resulting in the majority of the current flowing through the aluminum strands under the same applied voltage. Drake ACSR conductors were used in the study, and the line’s nominal current was specified as 1025 A following IEEE 738 standards. Current was distributed according to the conductor materials: 97% through aluminum strands and 3% through steel strands. The strand diameters were 4.44 mm for aluminum and 3.45 mm for steel, resulting in current densities of 2.47 × 10⁶ A/m² and 4.70 × 10⁵ A/m², respectively.

These values were directly assigned to the corresponding conductor domains. To model the surrounding air, a rectangular domain measuring 2 m in length and 1 m in width was included, as illustrated in Figure 4, providing a sufficiently large region to avoid boundary effects on the magnetic field solution.

Heat Transfer in Solids and Fluids Interface

The Heat Transfer in Solids and Fluids (ht) interface in COMSOL Multiphysics is employed to simulate conductive and convective heat transfer within solid and fluid domains. It enables the coupling of thermal behavior in conductors with airflow in the surrounding medium, ensuring an accurate representation of temperature distribution. In the model, the conductors were modeled under the Solids node, while the surrounding air was represented using the Fluids node. The reference temperature was set to 20 °C. As illustrated in Figure 4, the right boundary of the rectangular air domain was defined as inflow, and the left boundary was defined as outflow, thereby enabling natural air circulation around the conductors.

Turbulent Flow, SST Interface

In computational fluid dynamics (CFD), turbulence modeling is essential for predicting flow behavior in engineering applications such as aerodynamics, heat transfer, and power systems. The SST model, developed by Menter [24], is widely adopted for its accuracy and robustness, particularly in boundary-layer and separation flows. Within the Reynolds-Averaged Navier–Stokes (RANS) framework, it combines the near-wall precision of the k–ω model with the free-stream stability of the k–ε model, while an additional shear-stress correction enhances its ability to capture flow separation and reattachment [25,26].

The SST model has been shown to perform reliably under thermal gradients and adverse pressure conditions [27,28], and it consistently outperforms the traditional k–ε model in cases involving heat transfer, separation, and transitional turbulence [29]. A comparative summary of the k–ε, k–ω, and SST models is provided in

Table 4. Comparison of RANS turbulence models: strengths, limitations, and applications [30,31,32].

Model	Primary Strengths	Primary Limitations	Typical Suitable Problems
k–ε	Robust, low cost; well validated for free-shear and many industrial flows.	Poor near-wall resolution; degraded accuracy under APG and in boundary-layer separation.	Free-shear flows, jets, mixing layers.
k–ω	Accurate near walls; does not require wall functions for low-Re grids.	Sensitive to free-stream ω specification; may be less stable in free-stream regions.	Wall-bounded flows where detailed boundary-layer resolution is required.
SST (k–ω→k–ε)	Good near-wall accuracy and free-stream robustness; improved separation/reattachment prediction; moderate cost.	Higher cost than standard k–ε; may be unnecessary for very simple free-shear flows.	Flows with boundary-layer separation, APG, heat transfer near walls, rotating/complex flows.

, highlighting their relative accuracy, robustness, and applicability.

For DLR simulations, SST offers distinct advantages: it resolves thin boundary layers around conductors with high fidelity, remains stable in the free stream, and accurately predicts separation and reattachment under adverse pressure gradients. This balance between accuracy and computational efficiency makes SST the most suitable choice for modeling conductor–air interactions in DLR studies [33,34].

Boundary conditions, as illustrated in Figure 4, were defined with the right boundary of the air domain set as the inlet, the left boundary as the outlet, and the upper boundaries as walls, ensuring proper boundary-layer development and convective heat transfer around the conductors.

Surface to Surface Radiation Interface

The Surface-to-Surface Radiation (rad) interface was employed to account for radiative heat transfer between conductor surfaces and the surrounding environment. In high-temperature conditions, radiative losses constitute a significant portion of the total heat exchange, making their inclusion essential for accurate thermal analysis of conductors. Within this interface, the Diffuse Surface node was assigned to all aluminum and steel conductor surfaces to model inter-surface radiation exchange and ensure realistic thermal coupling between different conductor layers. To account for solar effects, an External Radiation Source was implemented with a prescribed heat flux of Q_s = 1027 W/m² and an incident angle (θ) of approximately 74.8°, both specified in accordance with IEEE 738. This parameter represents peak solar loading under clear-sky conditions, ensuring that worst-case scenarios for conductor heating are reflected in the simulation.

The diffuse-gray assumption adopted in the radiation model simplifies surface emissivity treatment while maintaining physical accuracy for metallic conductors. Combined with convective and conductive mechanisms, this interface provides a comprehensive description of the overall heat balance governing conductor temperature in DLR applications.

2.3.2. Model Assumptions and Limitations

The numerical model developed in this study relies on several well-defined assumptions to ensure both computational efficiency and practical relevance. The analysis was conducted in 2D, which simplifies the representation of the conductor while inherently neglecting some 3D effects. In particular, wind-induced turbulence along the helical strands and minor variations in the z-axis geometry are not captured. However, due to the relatively small exposed surface area in the z-direction, these effects are considered negligible and are not expected to significantly influence the model’s accuracy [13].

The 2D modeling approach employed in this study not only simplifies the geometric representation of conductors but also provides significant computational efficiency. Approximate simulation times for single, two-conductor, three-conductor, and four-conductor bundled configurations are summarized in

Table 5. Computational Times for 2D Conductor Bundle Models.

Bundle Type	Simulation Time (mm:ss) *
Single conductor	[02:37]
Two-Conductor	[04:44]
Three-Conductor	[08:46]
Four-Conductor	[11:12]

* Simulation times depend on the computational resources and may vary slightly across different hardware configurations.

. All simulations were performed on a workstation equipped with an Intel Core i7-12700H, 2.70 GHz processor, 32 GB of RAM, and an NVIDIA RTX 3070 Ti GPU, running Windows 11 Home.

This confirms that the 2D approach is both computationally efficient and accurate for the intended thermal analysis. Environmental conditions used in the simulations are summarized in

Table 6. Reference operating conditions for model validation according to IEEE 738.

Parameter	Description	Value
Ta	Ambient Temperature	40 [°C]
Ts	Max. Surface Temperature	100 [°C]
Vw	Wind Speed	0.61 m/s
Φ	Wind Angle	90°
ε	Surface Emissivity	0.8
Qs	Solar Heat Flux	1027 [W/m2]
θ	Solar Incidence Angle	76.1 [deg]
Imax	DLR Current	1025 A

. These parameters follow the example operating conditions recommended in IEEE 738, providing a standardized reference for validation and comparison. The model is validated against analytical solutions and reference data, enabling accurate calibration of the geometric and thermal representations. This approach enhances confidence in the predictive capability of the model while allowing detailed parametric studies beyond what purely analytical methods can provide [13]. All simulations were conducted under steady-state conditions, consistent with IEEE 738. This assumption simplifies the analysis and aligns with common practice in conductor thermal rating studies, providing reliable estimates of surface temperature and thermal limits without significant loss of accuracy.

3. Results and Discussion

To examine the impact of key environmental and operational parameters on conductor temperature, an example following IEEE methodology was considered, with a load current of 1025 A, ambient temperature of 40 °C, solar radiation of 1027 W/m², surface emissivity (ε) set to 0.8, and wind speed assumed at 0.61 m/s. Building on this scenario, both single and bundled conductor configurations are analyzed using COMSOL Multiphysics. A comparative assessment is then performed to evaluate their thermal performance and calculate the corresponding DLR under identical environmental and electrical loading conditions.

3.1. Single Conductor Configuration

Understanding the thermal behavior of overhead conductors is fundamental to the reliable and efficient operation of power transmission systems. While internal Joule heating provides a relatively uniform baseline of thermal input, it is the external environmental influence, namely wind direction, convective cooling efficiency, and solar radiation, that sculpt the asymmetric temperature distribution along the conductor’s surface. In this study, wind is assumed to blow from right to left, defining the right side as the windward face and the left as the leeward. This orientation elegantly correlates the root causes of thermal variation to aerodynamic flow patterns and solar loading.

The simulations conducted for the single conductor case also serve as a model validation study. In this validation, the results obtained from the coupled multiphysics model are compared with the reference calculations provided in IEEE 738 and CIGRE TB601. During the simulation, the conductor is represented by 16 outer strands surrounding the core, each experiencing slightly different thermal conditions due to local convection and radiation effects. To obtain a representative conductor surface temperature, the temperatures of all 16 outer strands were numbered from 1 to 16 and average surface temperature (T_c) was calculated as:

$$T_c={\displaystyle \frac{1}{N}}\sum _{i=1}^N{T}_i$$

(11)

where T_c is the average surface temperature of the outer strands, T_i is the surface temperature of strand i, and N = 16 is the total number of outer strands. The relative error with respect to the reference standards is defined as:

$$Relative Error \left(\%\right)={\displaystyle \frac{\left|T_c-T_{ref}\right|}{T_{ref}}}·100$$

(12)

Under the IEEE 738 reference operating conditions listed in Table 6, the conductor surface temperature is reported as 100 °C, while the CIGRE TB 601 reference provides a value of 102 °C. The average strand temperature obtained from the simulation is 99.04 °C, corresponding to relative errors of 0.96 % and 2.90 % with respect to IEEE 738 and CIGRE TB 601, respectively.

This close agreement confirms that the numerical model reliably captures the thermal behavior of a single conductor under the defined operating conditions. As shown in Figure 5, the temperature distribution along the conductor surface is highly non-uniform, reflecting the combined influence of aerodynamic and radiative effects.

Strand 2, located in the upper-leeward quadrant, reaches the highest temperature of approximately 102.2 °C, whereas Strand 11, positioned on the lower-windward side, remains at 95.8 °C. The elevated temperatures observed in the leeward strands result from reduced convective heat removal due to low local airflow and simultaneous exposure to direct solar radiation.

In contrast, windward strands experience stronger airflow and lower solar gain, leading to more effective convective cooling. The spatial temperature variation demonstrates the critical role of strand location and local flow conditions in determining thermal response. The strand-level temperature profile is presented in Figure 6, and further illustrates the formation of two distinct thermal zones.

The hot zone, comprising Strands 1–9 on the upper-leeward side, exhibits temperatures ranging from 100 °C to 102 °C. These strands are exposed to low airflow velocities in the wake region and receive significant solar radiation, limiting convective heat transfer. The cool zone, covering Strands 10–15, experiences lower temperatures from 95.8 °C to 98 °C, where airflow accelerates around the conductor surface and convective heat removal is enhanced. A pronounced thermal gradient between Strands 9 and 10 coincides with the flow separation point, marking a clear transition between inefficient and efficient cooling regions. Strand 16, located at the leeward lower sector, shows a high temperature (~101 °C), reinforcing the spatial symmetry of thermal patterns dictated by aerodynamic effects and convective interactions. Airflow mechanisms driving this temperature distribution are visualized in Figure 7, where the wake region is evident around Strands 2–9.

In this region, air velocity drops sharply, often approaching zero, leading to stagnation and reduced convective cooling. Strands 10–12 and 14–16 are exposed to accelerated airflow along the conductor curvature, which increases convective heat transfer and lowers the surface temperature. Strand 13, although experiencing near-zero tangential airflow, is located at the frontal stagnation point. Here, direct wind impingement compresses the thermal boundary layer, enhancing heat transfer despite the absence of significant lateral flow. The figure highlights the interplay between local airflow patterns and strand-specific thermal response.

The strand-level wind velocity distribution is shown in Figure 8, which quantitatively confirms the heterogeneous nature of airflow along the conductor.

Wake-region strands exhibit velocities below 0.1 m/s, leading to limited convective dissipation and higher temperatures. In contrast, strands located in accelerated flow zones experience velocities exceeding 1.2 m/s, significantly enhancing convective heat removal. The velocity profile closely aligns with the observed temperature gradients, demonstrating the critical impact of local aerodynamic conditions on strand-level thermal behavior.

The strand-level pressure profile is illustrated in Figure 9, where a local pressure maximum of +0.25 Pa occurs at Strand 13 due to frontal wind impact, while a trough of −0.25 Pa at Strand 10 reflects flow acceleration, consistent with Bernoulli’s principle.

Persistent low pressure and recirculating airflow in the leeward region (Strands 1–9) further reduce convective cooling. These aerodynamic features, combined with differential radiation exposure, provide a comprehensive explanation for the strand-level temperature variations along the conductor.

The results indicate that local airflow, pressure distribution, and solar radiation jointly control the thermal behavior of individual conductor strands. They emphasize the limitations of conventional average-temperature-based rating methods and demonstrate the necessity of high-resolution, strand-level modeling for accurate DLR assessment.

3.2. Two-Conductor Bundle Configuration

The airflow in the bundled conductor configuration moves from right to left, illustrating the complex thermal and aerodynamic interaction between the two conductors. As shown in Figure 10, the windward conductor (C1) on the right is directly exposed to the ambient wind, allowing effective convective cooling on its windward face. Its leeward face, however, experiences higher temperatures due to the formation of a localized wind shadow. Heat generated by the conductor is transferred to the surrounding air, producing a thermal wake that trails downstream. This thermal wake modifies the local airflow characteristics, increasing turbulence and air temperature, which significantly impacts the cooling performance of the leeward conductor (C2). The leeward conductor receives this pre-heated and disturbed airflow, reducing the temperature gradient between the conductor surface and the air, thus lowering convective heat transfer efficiency. As a result, the downstream conductor accumulates more heat and reaches the highest temperature in the bundle, approximately 107.2 °C. This illustrates that, in bundled configurations, the thermal performance of each conductor is interdependent, and the downstream conductor is subject to less favorable cooling conditions than the windward conductor. The strand-level temperature profiles are shown in Figure 11.

Both conductors exhibit similar profile shapes, indicating that wind direction is the primary factor controlling strand-level temperature distribution. In the windward conductor (C1), the leeward strands (Strands 1–9) reside in a “wind shadow” region and exhibit higher temperatures between 100 °C and 102.6 °C, while the windward strands (Strands 10–14) experience lower temperatures down to 95.6 °C, reflecting enhanced convective cooling. The leeward conductor (C2) shows an overall upward shift in the temperature profile by roughly 4 °C due to thermal shielding, with the leeward strands reaching up to 106.5 °C and windward strands remaining around 99.9 °C. The temperature differentials—7.0 °C for C1 and 6.6 °C for C2—highlight the effect of airflow pre-heating on the downstream conductor. The hot spot consistently occurs at Strand 2 in both conductors, indicating that the leeward region receives the least convective cooling. The temperature distribution demonstrates that both aerodynamic shielding and wake formation must be accounted for in DLR calculations for accurate ampacity assessment. The strand-level wind velocity is presented in Figure 12.

The windward conductor (C1) experiences undisturbed ambient airflow, generating a characteristic velocity distribution with accelerated flow along the conductor sides (peaking at approximately 0.21 m/s at Strand 15) and stagnation at the frontal point (Strand 13) where the flow splits. The downstream conductor, located within the turbulent thermal wake of the upstream conductor, encounters reduced airflow across all strands, with a peak velocity of only about 0.14 m/s. This diminished airflow reduces convective heat transfer efficiency, explaining the higher temperature profile observed on the leeward conductor. Both conductors exhibit zero velocity at the frontal stagnation point, emphasizing that local flow separation and wake formation dictate strand-level cooling efficiency. The figure highlights the interplay between aerodynamic wake effects and strand-specific temperature responses. The strand-level surface pressure distribution is shown in Figure 13.

Both conductors present a high-pressure stagnation point at Strand 13 and a low-pressure trough at Strand 10, consistent with cylindrical body flow characteristics. However, the magnitude of pressure variation differs: the windward conductor exhibits a peak pressure of +0.26 Pa and a minimum of −0.25 Pa, reflecting direct exposure to undisturbed wind. The leeward conductor (C2) experiences a dampened pressure profile, with a peak of +0.16 Pa and a minimum of −0.18 Pa, due to shielding by the windward conductor (C1).

The reduced pressure gradients on the downstream conductor lead to lower local wind velocities, further diminishing convective heat transfer and contributing to elevated temperatures. This clearly demonstrates how aerodynamic shielding modifies both pressure and velocity fields, directly influencing strand-level thermal behavior in bundled conductors.

These results demonstrate that the thermal behavior of bundled conductors is strongly controlled by aerodynamic interactions between the windward and leeward elements. The upstream conductor generates a thermal wake that both increases local air temperature and introduces turbulence, reducing convective cooling efficiency on the downstream conductor. As a result, the maximum temperature—and thus the ampacity limit—of the bundle is determined by the leeward conductor (C2) rather than the windward conductor. Accurate consideration of these wake-induced thermal and aerodynamic effects is therefore essential for reliable DLR and for optimizing the operational safety and efficiency of bundled transmission lines under varying environmental and loading conditions.

3.3. Three-Conductor Bundle Configuration

Figure 14 depicts the temperature distribution in a triple-bundle arrangement under crosswind.

The airflow, moving from right to left, first interacts with the windward conductors (C1 and C3). These conductors are directly exposed to fresh ambient air, which promotes effective convective heat transfer and keeps their surface temperatures relatively low. In contrast, the leeward conductor (C2) is located entirely within the turbulent wakes generated by C1 and C3. Instead of receiving undisturbed air, C2 is surrounded by preheated, low-velocity, and chaotic flow. Consequently, C2 reaches a surface temperature of about 107 °C, which is 4–5 °C higher than the windward conductors despite identical current, geometry, and boundary conditions. This clearly demonstrates that aerodynamic shielding is the dominant factor responsible for the thermal disadvantage of the leeward conductor and establishes it as the limiting element for ampacity in the system. Figure 15 presents the strand-level surface temperature profiles for three sub-conductors.

All conductors exhibit a similar circumferential temperature distribution, with a maximum at the stagnation point (Strand 2) and a minimum near Strand 13, consistent with the effect of wind incidence. Compared with C1 and C3, the temperature profile of C2 is uniformly higher across all strands. This vertical shift indicates that the leeward conductor remains hotter along its entire circumference, not only at localized regions. The persistence of this offset demonstrates the continuous influence of wake effects and confirms that conductor position within the bundle is a key determinant of thermal performance. Figure 16 presents the local wind velocity distribution on the conductor surfaces.

The windward conductors (C1 and C3) reach maximum surface velocities of approximately 0.21 m/s, enabling robust convective cooling. In contrast, the leeward conductor (C2) achieves only ~0.14 m/s at its maximum, a substantial reduction caused by wake-induced turbulence and flow deceleration. This reduction in airflow velocity leads to a lower convective heat transfer coefficient, thereby allowing more heat from Joule losses and solar radiation to accumulate at the surface. The stagnation point at Strand 13 is common to all conductors, indicating uniform wind incidence. Beyond this point, velocity fields diverge due to aerodynamic interaction, leading to asymmetric thermal behavior. Figure 17 depicts the pressure distributions around the conductor surfaces.

All three conductors display a cylindrical pressure pattern, with a high-pressure stagnation region at Strand 13 and a low-pressure area near Strand 10. C3 exhibits the steepest pressure gradient, reflecting its position at the outer edge of the bundle and direct exposure to undisturbed wind. In contrast, C1 and C2 experience more moderate pressure variations due to partial aerodynamic shielding. Despite this dampened pressure profile, C2 remains the thermal hotspot, highlighting that pressure distribution alone cannot reliably predict conductor heating. Instead, the combined effects of reduced velocity, turbulence, and thermal wake interactions govern the downstream conductor’s elevated temperature.

3.4. Four-Conductor Bundle Configuration

Figure 18 shows the temperature distribution within a four-conductor bundle configuration under crosswind conditions.

The airflow approaches from right to left and first encounters the windward conductors (C1 and C4). These conductors are directly exposed to undisturbed ambient wind, which enables strong convective cooling and prevents excessive temperature rise. In contrast, the leeward conductors (C2 and C3) are located fully within the thermal wakes generated by C1 and C4. The airflow reaching them is slower, turbulent, and preheated, reducing its cooling capacity.

As a consequence, C2 reaches a peak surface temperature of 107.1 °C, while C1 and C4 remain near 102.5 °C. This temperature difference of about 5 °C occurs even though all conductors carry the same current and share identical geometry and boundary conditions.

This finding clearly shows that a conductor’s position within the bundle strongly influences its thermal behavior. In particular, C2 consistently reaches the highest temperature, making it the limiting factor for the system’s overall ampacity. Figure 19 presents the circumferential temperature variation in each conductor as a function of strand position.

All conductors show a similar temperature trend, with a maximum at Strand 2 on the leeward side and a minimum near Strand 13. This pattern reflects the effect of crosswind on the conductors. The leeward conductors, C2 and C3, have temperature curves that are consistently higher than those of C1 and C4. This upward shift occurs along the entire circumference, showing that aerodynamic shielding affects the full perimeter of the leeward conductors. Among them, C2 reaches the highest temperature at all strand positions, confirming it as the thermal hotspot. Overall, these trends indicate that wake-induced heating is a continuous and circumferentially uniform phenomenon that elevates the temperature of leeward conductors. Figure 20 shows the local wind velocity at the conductor surfaces.

The windward conductors (C1 and C4) achieve maximum values of approximately 0.21 m/s, providing effective convective cooling. In contrast, the leeward conductors (C2 and C3) reach only ~0.15 m/s, reflecting the reduced airflow momentum within the wake region. Although all velocity profiles intersect at the stagnation point (Strand 13), which confirms uniform wind incidence, they diverge significantly around the conductor circumference. The velocity decay on the leeward conductors leads to a lower convective heat transfer coefficient, which, when combined with identical Joule and solar heating, results in higher operating temperatures. This analysis establishes a direct causal link: wake-induced velocity reduction is the key aerodynamic mechanism driving the thermal disadvantage of leeward conductors. Figure 21 presents the surface pressure distribution of the four conductors.

Each exhibits the classical cylindrical flow pattern, with a high-pressure stagnation point at Strand 13 and a low-pressure trough near Strand 10. The windward conductors (C1 and C4) exhibit the steepest pressure gradients, with peak values around +0.27 Pa and minima near −0.26 Pa, reflecting their direct exposure to the incoming airflow. In contrast, the leeward conductors (C2 and C3) show gentler pressure gradients, with maximum values near +0.17 Pa and minimum values around −0.18 Pa. This attenuation reflects the shielding effect: the leeward conductors are aerodynamically protected, but at the cost of reduced airflow momentum and weaker convective cooling. Importantly, despite the smaller pressure differentials, C2 and C3 still emerge as the hottest conductors, demonstrating that pressure alone cannot capture thermal risk. The combined evaluation of pressure, velocity, and temperature confirms that leeward conductors suffer from aerodynamic shielding, making them the critical elements that determine bundle ampacity.

3.5. Discussion

This study developed and validated a detailed 2D coupled thermal–fluid model to examine aerodynamic and thermal interactions in single and bundled conductors under DLR conditions. By implementing SST turbulence model in COMSOL Multiphysics, the work systematically evaluated the limitations of conventional standards such as IEEE 738 and CIGRE TB 601, which neglect position-dependent effects within bundled conductors. Validation against these standards for a single conductor demonstrated the reliability of the proposed numerical framework.

The simulation results, summarized in

Table 7. Comparison of maximum conductor temperatures predicted by IEEE 738, CIGRE TB 601, and the proposed COMSOL model.

Bundle Configuration	DLR Current [A]	IEEE 738 Maximum Temp. [°C]	CIGRE TB601 Maximum Temp. [°C]	COMSOL Maximum Temp. [°C]	Calculated DLR Current [A]
Single Conductor	1025	100	102	TC1 = 99.040	-
Two-Conductor				TC1 = 99.410 TC2 = 103.379	994.04
Three-Conductor				TC1 = 99.340 TC2 = 103.250 TC3 = 98.750	993.87
Four-Conductor				TC1 = 99.280 TC2 = 103.310 TC3 = 102.670 TC4 = 99.130	993.27

, demonstrate that bundled conductors exhibit significant temperature non-uniformities driven by aerodynamic shielding, wake effects, and solar radiation.

As expected, the single conductor exhibited temperatures consistent with IEEE 738 and CIGRE TB601 values. In the two-conductor bundle, the windward conductor (C1) showed nearly identical behavior to the isolated conductor, while the leeward conductor (C2) reached ~103 °C due to wake effects—namely the reduced wind speed and pre-heated airflow caused by the upstream conductor. In the three-conductor arrangement, C1 and C2 behaved similarly to the two-conductor case, whereas C3, positioned beneath them, exhibited a slightly lower temperature (~98.7 °C). Although a similar temperature to C1 might be expected, the oblique solar incidence (76.1° relative to the x-axis) caused C1 to partially shade C3, reducing its absorbed solar radiation and thus limiting its heating. In the four-conductor configuration, C1 and C4 (windward) again matched the single-conductor values, while C2 consistently emerged as the hottest conductor (~103.3 °C), acting as the thermal hotspot. Conductor C3 exhibited an intermediate value, slightly cooler than C2, again due to partial shading. These findings confirm that downstream conductors systematically govern the thermal limit of the bundle, with temperature elevations of up to 3–6 °C compared to windward or isolated conductors.

Considering the maximum temperature observed in the bundled configurations (~103.3 °C at the DLR current of 1025 A according to IEEE 738), an additional analysis was performed to determine the current at which the hottest sub-conductor reaches the critical limit of 100 °C. The simulation results indicate that this occurs at approximately 993.27 A, corresponding to a reduction of about 3.1% compared to the standard IEEE 738 DLR estimate. The calculated DLR currents are very close for all bundled configurations, since the maximum temperatures of the sub-conductors are nearly identical (around 103 °C). Therefore, only minor differences are observed between the bundle cases, which are provided for clarity. These observations highlight that, despite the general reliability of IEEE 738, the allowable current it predicts for bundled conductors may be slightly higher than what is observed under realistic thermal conditions. It should be emphasized that this adjustment is specific to the conditions considered in this study; for other operational scenarios—including variations in sub-conductor spacing, conductor type, wind speed, and solar radiation—separate analyses are necessary, and further investigation is recommended.

To illustrate that the DLR difference can vary when operational parameters change, an additional simulation was performed in which all conditions were kept constant except for the sub-conductor spacing, which was reduced from 40 cm to 30 cm. The results showed a decrease of approximately 4.1% compared to the IEEE 738 estimate, providing a concrete example that even small variations in operational parameters can lead to noticeable changes in the calculated ampacity.

These findings highlight that although IEEE 738 provides generally reliable estimates, its assumptions are limited when applied to bundled configurations. To place these results in a broader context,

Table 8. Comparison of standard, empirical, and numerical approaches for DLR calculation.

Study	Method Type	Conductor Configuration	Environmental Effects	Strengths	Limitations
IEEE 738	Analytical	Single conductor	Ambient wind, ambient temperature, solar (simplified); no wake	Simple, widely used standard	Limited under complex weather; does not account for multi-conductor/bundle effects
CIGRE TB601	Analytical	Single conductor	Similar to IEEE (simplified solar & wind); no wake	Industry reference guide	No detailed flow or bundle effects; single conductor only.
[11]	Empirical	Single conductor	Weather-station data (wind, temp, solar); no wake	Real-world validation; sensor reliability	Limited generalizability; based only on localized measurements; cannot capture conditions in other regions
[16]	Numerical (3D, RANS k-ε)	Multi-conductor bundle	Strong wind, ambient temperature, solar	Considers geometric effects and allows detailed 3D thermal analysis of conductor bundles	High computational demand; complex 3D model. k-ε is simpler but less accurate near walls and in wake regions; may underestimate local hotspots
Present Study	Numerical (2D FEM-based, SST k-ω)	Multi-conductor bundle	Fully coupled wind, solar, wake	Enables accurate prediction of local temperatures and hotspots, with improved accuracy in near-wall and wake regions using SST k-ω.	Higher computational demand compared to analytical and 3D numerical methods

compares the strengths and limitations of analytical, empirical, and numerical approaches, including the present study.

Table 8 provides a comparative overview of analytical, empirical, and numerical approaches for ampacity and DLR evaluation. Standard analytical methods, such as IEEE 738 and CIGRE TB601, are widely adopted due to their simplicity and ease of application. These methods, however, are limited to single conductors and rely on simplified representations of environmental conditions, neglecting complex interactions such as those between bundled conductors or localized thermal variations. Empirical approaches, as in [11] make use of real-world measurements and provide reliable localized predictions under actual weather conditions. Nevertheless, their applicability is confined to the specific regions where measurements are conducted, and they cannot fully capture broader variations along the line. Numerical simulations allow for more detailed thermal assessments. For instance, in [16] the authors employ a 3D RANS k-ε model to analyze multi-conductor bundles, accounting for geometric effects and strong wind conditions. While this approach provides comprehensive thermal insights, it comes at the expense of significant computational effort. In contrast, the present study utilizes a 2D FEM-based approach with the SST k-ω turbulence model to simulate up to four conductors. This methodology enables accurate prediction of local temperature distributions and hotspot locations while maintaining computational efficiency. By combining detailed thermal modeling with a manageable computational cost, the present approach effectively bridges the gap between classical analytical methods and complex 3D numerical simulations, offering a practical and reliable tool for DLR analysis across different conductor configurations.

While this study shows that a 2D simulation provides accurate and computationally efficient results for the thermal analysis of bundled conductors under the considered conditions, some cases may require a 3D simulation. This is particularly relevant for complex conductor geometries, such as spacers, non-uniform strand arrangements, or structural irregularities, and for scenarios with highly non-uniform wind or localized thermal effects. The 2D approach remains advantageous due to its computational efficiency, simpler geometry, and ease of implementation, making it suitable for routine analyses. By clarifying these assumptions and limitations, the study guides when a more detailed 3D simulation may be necessary, allowing a balance between accuracy and computational cost.

The results presented in Table 7 indicate that the temperatures of leeward conductors in bundled configurations are consistently higher than those of windward or isolated conductors. This leads to a slight reduction in allowable current compared to the classical IEEE 738 estimates. For instance, in the two-, three-, and four-conductor bundles, the IEEE 738-based DLR current is 1025 A, whereas the proposed model predicts approximately 993 A for the hottest sub-conductor, corresponding to a reduction of about 3.1%. In an additional analysis where all operational parameters were kept constant but the sub-conductor spacing was reduced from 40 cm to 30 cm, the IEEE 738 DLR remained 1025 A, while the model predicted 983 A, representing a decrease of approximately 4.1%.

Such refinements in temperature assessment have both technological and economic implications. Technically, neglecting the thermal hotspot may result in overloading the conductor beyond its maximum allowable temperature, potentially accelerating aging, compromising mechanical stability, and reducing line reliability. Clearance distances may also be affected due to increased conductor sag, which can further compromise operational safety. From an operational perspective, an inaccurate understanding of the line’s true ampacity may lead to suboptimal energy management decisions, particularly in systems with high penetration of distributed generation, where load flow and demand balancing rely on accurate line ratings. Economically, these factors can translate into increased maintenance costs, unexpected service interruptions, or early conductor replacement, highlighting the importance of precise thermal modeling. The proposed methodology thus provides a framework that can support practical applications, guiding the development of ampacity adjustment strategies and improving operational decision-making in real-world transmission line management.

4. Conclusions

This study investigated the thermal behavior of bundled conductors under realistic environmental conditions by combining finite element modeling with advanced turbulence closure. A systematic comparison was carried out against analytical standards, highlighting the limitations of conventional methods in predicting local temperature variations within conductor bundles.

In summary, the findings establish that the common practice of estimating bundle ampacity by simply scaling single-conductor ratings is fundamentally non-conservative. The true thermal limit of a bundled line is dictated by its hottest leeward conductor, and neglecting this positional dependency can lead to overestimated ampacity, thermal overloading, and accelerated conductor aging. The study demonstrated that temperature non-uniformities within bundles, driven by wake effects, aerodynamic shielding, and partial solar shading, can elevate local conductor temperatures by up to 3–6 °C compared to windward or isolated conductors. This positional dependency introduces systematic deviations that analytical standards cannot capture.

A direct comparison against IEEE 738 revealed that the proposed model consistently predicts slightly lower allowable currents—approximately 3–4% reductions—highlighting the importance of localized thermal assessment in bundle configurations. Such deviations, while numerically small, can translate into meaningful operational impacts, particularly in terms of sag increase, accelerated aging, and overall system reliability. By bridging the gap between simplified analytical methods and more computationally demanding 3D numerical studies, the 2D FEM-based framework adopted here provides both accuracy and efficiency, making it a practical tool for DLR analysis in real-world applications.

To facilitate practical application in field operations, the model results were further processed to derive ampacity correction factors and functional relationships. These allow standard DLR estimates to be adjusted without performing full model simulations for every scenario. Additionally, data from simulations, supplemented by experimental measurements when necessary, can be used to train machine learning algorithms capable of predicting the required ampacity correction factors for diverse operational conditions. This methodology provides a systematic way to translate complex numerical model outcomes into actionable guidelines for real-world transmission line management while maintaining consistency with the underlying physical and thermal behavior of the conductors.

Future studies should explore transient and 3D simulations, supported by experimental validation to reinforce the findings. Additionally, systematically varying wind, solar incidence, loading, and bundle geometry could generate a large dataset for developing an artificial intelligence framework. Such a model could provide data-driven correction factors to supplement existing standards, offering transmission operators a practical tool to account for bundle-specific effects without relying on computationally intensive simulations in routine operation.