A Novel Dynamic Ampacity Assessment Method for Direct Burial Cables Based on an Electro-Thermal-Fluid Multiphysics Coupling Model

Abstract

Traditional ampacity evaluation methods for direct burial cables, like the correction factor method and the IEC 60287 analytical method, suffer from large calculation errors when dealing with complex installation environments. This paper investigated the influence of multiple environmental factors and proximity effects on the ampacity of 35 kV YJLV22-26/35 3 × 400 mm² direct burial cables using an electro-thermal-fluid coupling FEM model. The results indicate that when accounting for surface temperature and burial depth, the correction factor method may overestimate ampacity by up to 7%, while the analytical method may underestimate it by up to 24%. When soil thermal resistance variations are considered, the correction factor method could overestimate ampacity by 14%, whereas the analytical method may underestimate it by 10%. Due to neglecting solar radiation and air convection effects, these two methods can introduce calculation errors of 23% and 34%, respectively. The ampacity of multi-circuit parallel configurations increases with greater circuit spacing. Based on FEM simulation results, a new dynamic ampacity evaluation method has been proposed that comprehensively considers multiple environmental variables including ambient temperature, burial depth, soil thermal resistivity, solar radiation intensity, wind speed, the number of parallel circuits, and circuit spacing. This method can be directly applied to guide engineering design.

1. Introduction

Direct burial cable installation is characterized by low construction and maintenance costs as well as high safety [1], leading to its widespread adoption in photovoltaic power stations [2], wind farms [3], substations, and urban distribution networks [4]. Consequently, accurately evaluating the cable ampacity is of significant importance. Firstly, an overestimated cable ampacity may result in the selection of an undersized cable cross-section, causing the cable to operate under prolonged overload conditions, which compromises cable safety and shortens its service life [5]. Secondly, an underestimated cable ampacity may cause selection of an oversized cable cross-section, resulting in unnecessary investment and material waste [6]. For power operation departments, there is an increasing need for precise management of the ampacity of underground cables [7]. Traditional methods for evaluating cable ampacity in engineering practice include the correction factor method [8] and the IEC 60287 analytical method [9,10]. However, no literature currently reports on the accuracy and applicability of these traditional methods under multiple environmental factors such as the thermal conductivity of surrounding media, cable burial depth, surface temperature, solar radiation intensity, wind speed, and proximity to adjacent conductors. Guided by this motivation, this paper systematically compares the calculation discrepancies between traditional methods and multiphysics finite element method (FEM) [11,12,13] for the ampacity of direct burial cables under multiple environmental factors. Furthermore, from an engineering application perspective, the complex modeling process of FEM is difficult for general engineering technicians to master. Therefore, proposing an improved correction factor method based on FEM simulation results that simultaneously considers multiple environmental factors, which both ensures calculation accuracy and enables easy and rapid application, would greatly benefit engineering design.

The correction factor method calculates the permissible cable ampacity under actual conditions by multiplying the cable’s rated ampacity under standard conditions by a correction factor. This method is simple and practical, making it widely adopted by power design engineers. However, it suffers from inadequate consideration of environmental factors, leading to significant calculation inaccuracies [14]. Research by Möbius et al. [15] has confirmed that meteorological parameters significantly influence the accuracy of ampacity calculations, underscoring the importance of accurate meteorological data input.

The IEC 60287 analytical method analogizes heat transfer problems to electrical circuit problems by equating the hindering effect of the media surrounding the cable on heat flow to the resistance exerted by an electrical resistor on current. This analogy enables approximate calculation of temperature field distribution using Ohm’s law and Kirchhoff’s law. This method offers clear physical interpretation and accounts for the influence of thermal resistances from cable insulation, sheathing, and surrounding media on the thermal field distribution. Consequently, it facilitates quantitative analysis of the impact of cable structure and thermal characteristics of the surrounding media on the ampacity of cables [11,16]. However, the IEC 60287 analytical method faces challenges in addressing computational problems involving three-phase unbalanced loads [17] or asymmetric cable structures [18,19]. Shabani and Vahidi [20] contend that traditional IEC 60287 analytical method, which evaluate ampacity based on worst-case scenarios, can lead to overdesign. They therefore proposed a probabilistic approach for ampacity assessment.

FEM has become the most prevalent approach for simulating cable temperature field distribution and evaluating ampacity. Klimenta et al. [21] employed an FEM model to investigate how solar radiation influences the current-carrying capacity of a 0.4 kV cable with different types of ground pavement layers. However, this study only examined the effects of surface solar absorption and heat emission rates on the ampacity of underground cables, and did not explain the basis for selecting the convective coefficient. Nie et al. [22] employed the FEM model to calculate the thermal field and current of cables under different placement mode, and the results demonstrate the cable placement mode greatly affects the ampacity. However, the study neglected the influence of solar radiation on the current of direct-buried cables. Aras et al. [23] used the FEM model to calculate the ampacity of a 154 kV XLPE direct-buried power cable under steady-state conditions and compared the results with those obtained by traditional analytical methods, and the findings demonstrate that the FEM offers superior computational accuracy when addressing complex installation environments. However, this model assumes a constant surface temperature of 20 °C and ignores the impact of air convection on heat dissipation, which inevitably compromises the accuracy of the calculations. Bustamante et al. [8] utilized a FEM to investigate the effect of soil thermal resistivity and burial depth on the ampacity of medium-voltage direct-buried cables and compared it with the calculated values from the IEC 60287 analytical method. It was found that in very dry soil conditions, the IEC 60287 analytical method yielded values 7.53–10.49% lower than the simulated values. However, this model neglects the effects of solar radiation, wind speed, and adjacent circuits on the ampacity of the cable.

As evidenced by the above review, correction factor methods overlook certain environmental factors, resulting in limited computational accuracy and restricted applicability. The IEC 60287 analytical method employs simplifications in its calculations, making it suitable only for simple installation environments and inadequate for ensuring accuracy in complex scenarios, thus failing to meet the requirements for precise cable ampacity management. In contrast, FEM can fully overcome the limitations of traditional methods and accurately simulate the temperature distribution of cables, thereby enabling precise calculation of cable ampacity. However, the FEM models mentioned in the above literature do not comprehensively consider the impact of multiple environmental factors on cable ampacity. The main contributions of this paper are as follows:

• An electric-thermal-fluid coupling FEM model has been developed which can simultaneously account for solar radiation, convective heat dissipation, and surface radiation, thereby enabling precise calculation of the ampacity of direct burial cables;
• A quantitative analysis was conducted on the influence of multiple environmental factors (ambient temperature, cable burial depth, soil thermal resistivity, solar radiation intensity, wind speed) and proximity effects (number of circuits, circuit spacing) on the ampacity of direct burial cables, revealing the underlying patterns of influence;
• A systematic analysis of the calculation discrepancy of traditional methods under multiple environmental factors was performed;
• Based on the simulation results of the FEM model, an improved correction factor method that takes into account both multiple environmental factors and proximity effects has been proposed, which can greatly benefit engineering design.

This paper built an electro-thermal-fluid coupling model using COMSOL Multiphysics 6.2 to simulate the effects of multiple environmental factors and proximity effects on the ampacity of 35 kV YJLV22-26/35 3 × 400 mm² direct burial cables and analyzed computational discrepancies of traditional methods. The paper consists of four chapters. Section 1 reviews current research progress and gaps in this field. Section 2 details the principles of the correction factor method and the IEC 60287 analytical method, followed by the presentation of a modeling approach for the electro-thermal-fluid multiphysics FEM model. Section 3 simulates in detail the effects of multiple environmental factors and proximity effects on the ampacity of direct burial cables, while analyzing the computational deviations of traditional methods. Section 4 summarizes the main findings and contributions of this paper.

2. Cable Ampacity Calculation Methods

2.1. Correction Factor Method

The calculation formula of the correction factor method recommended by the Chinese standard GB 50217-2018 [24] is shown in (1), where I_B is the rated ampacity of the cable under reference conditions (25 °C soil temperature, 1.2 K·m·W⁻¹ soil thermal resistivity), I is the actual operating current of the circuit, K_t is the temperature correction factor, K_r is the thermal resistivity correction factor, and K_N is the correction factor for multiple cables laid in parallel.

$$K_{\mathrm{t}}\cdot K_{\mathrm{r}}\cdot K_{\mathrm{N}}\cdot I_{\mathrm{B}}\ge I$$

(1)

When the soil temperature differs from the reference conditions, the temperature correction factor K_t can be calculated using (2), where T_M is highest permissible temperature of the conductor, T₂ is the actual soil temperature, and T₁ is the reference soil temperature corresponding to the rated ampacity.

$$K_{\mathrm{t}}=\sqrt{{\displaystyle \frac{T_{\mathrm{M}}-T_2}{T_{\mathrm{M}}-T_1}}}$$

(2)

K_r can be calculated by (3), where ρ_s is the coefficient of thermal resistance of soil in units of K·m/W. Parallel circuit correction factor K_N is given in

Table 1. The correction factor for the ampacity of cables when multiple cables are directly buried in parallel in soil from GB 50217-2018.

Circuit Spacing/mm	Number of Three-Core Cable Circuits
	1	2	3	4	5	6
100	1	0.9	0.85	0.80	0.78	0.75
200	1	0.92	0.87	0.84	0.82	0.81
300	1	0.93	0.90	0.87	0.86	0.85

.

$$K_{\mathrm{r}}=-0.231\ln \left({\rho }_{\mathrm{s}}\right)+1.0195$$

(3)

2.2. IEC 60287 Analytical Method

The 35 kV XLPE high-voltage power cable is composed of a conductor, semi-conductive layer, XLPE insulation, metallic shielding, filler, wrapping tape, inner sheath, steel tape armor, and outer sheath, as illustrated in Figure 1.

The heat sources within the cable are generated by conductor losses, dielectric losses, metallic sheath losses, and armor losses. Heat flux propagates from the interior of the cable to the exterior. Heat transfer can be analogized to electric current flow in a circuit. Therefore, IEC 60287-1-1 provides an equivalent thermal circuit model for directly buried cables, as shown in Figure 2, where P₁ represents conductor resistance loss, P₂ denotes dielectric loss, P₃ signifies metallic sheath loss, and P₄ indicates armor loss. R_t1 is the insulation thermal resistance, R_t2 is the thermal resistance of the inner sheath, R_t3 is the thermal resistance of outer sheath, and R_t4 is the soil thermal resistance. The variable n represents the amount of conductor with the same sectional area in a cable, T_M is the conductor operating temperature, and T₀ is the surface temperature.

The conductor loss P₁ is calculated by (4), where R_dc is the direct current resistance in Ω, R is the alternating current resistance in Ω, α_t is the temperature coefficient of resistance at 20 °C (4.03 × 10⁻³/K for aluminum conductors and 3.93 × 10⁻³/K for copper conductors), ρ is the conductor resistivity in Ω·m, A is the conductor sectional area in m², y_s is the coefficient of skin effect, and y_p is the influence coefficient of adjacent conductors.

$$\left\{\begin{array}{l}{P}_1=I^{2}R\\ R=R_{\mathrm{dc}}\left(1+y_{\mathrm{s}}+y_{\mathrm{p}}\right)\\ R_{\mathrm{dc}}={\displaystyle \frac{\rho \left[1+{\alpha }_{\mathrm{t}}\left(T_{\mathrm{M}}-20\right)\right]}{A}}\end{array}\right.$$

(4)

y_s and y_p are calculated by (5) and (6), respectively, where f is the power frequency in Hz, k_s and k_p are the skin effect factor and proximity effect factor, respectively, both having a value of 1 for round stranded conductors, s is the axial spacing between three-phase conductors in meters, and d_c is the conductor outer diameter in meters.

$$\left\{\begin{array}{l}{y}_{\mathrm{s}}={\displaystyle \frac{{x_{\mathrm{s}}}^4}{192+0.8{x_{\mathrm{s}}}^4}}\\ x_{\mathrm{s}}=\sqrt{{\displaystyle \frac{8\pi f}{R_{\mathrm{dc}}}}{k}_{\mathrm{s}}\times {10}^{-7}}\end{array}\right.$$

(5)

$$\left\{\begin{array}{l}{y}_{\mathrm{p}}={\displaystyle \frac{{x_{\mathrm{p}}}^4}{192+0.8{x_{\mathrm{p}}}^4}}{\left({\displaystyle \frac{d_{\mathrm{c}}}{s}}\right)}^2\times \left[0.312{\left({\displaystyle \frac{d_{\mathrm{c}}}{s}}\right)}^2+1.18/\left({\displaystyle \frac{{x_{\mathrm{p}}}^4}{192+0.8{x_{\mathrm{p}}}^4}}+0.27\right)\right]\\ x_{\mathrm{p}}=\sqrt{{\displaystyle \frac{8\pi f}{R_{\mathrm{dc}}}}{k}_{\mathrm{p}}\times {10}^{-7}}\end{array}\right.$$

(6)

The dielectric loss P₂ is calculated by (7), where ω is the power supply angular frequency in rad/s, tanδ is the insulation’s dielectric loss factor, taken as 0.005 [10], C is the unit length capacitance of cable in F·m⁻¹, ε is the relative permittivity of the insulation, and D_i is the outer diameter of the insulation in meters.

$$\left\{\begin{array}{l}{P}_2=\omega C{U}_0^2\tan \delta \\ C=5.6\times {10}^{-11}{\displaystyle \frac{\epsilon }{\ln \left(D_{\mathrm{i}}/d_{\mathrm{c}}\right)}}\end{array}\right.$$

(7)

The metallic sheath loss P₃ and the steel armor loss P₄ are expressed as multiples of the conductor resistance loss, as shown in (8) and (9):

$$P_3={\lambda }_1{P}_1$$

(8)

$$P_4={\lambda }_2{P}_1$$

(9)

Values of λ₁ and λ₂ can be referred to in the standard IEC 60287-1-1. Consequently, the maximum allowable ampacity of the cable can be calculated as shown in (10).

$$I=\sqrt{{\displaystyle \frac{\Delta T-P_2\left[0.5R_{\mathrm{t}1}+n\left(R_{\mathrm{t}2}+R_{\mathrm{t}3}+R_{\mathrm{t}4}\right)\right]}{R\left[R_{\mathrm{t}1}+n\left(1+{\lambda }_1\right)R_{\mathrm{t}2}+n\left(1+{\lambda }_1+{\lambda }_2\right)\left(R_{\mathrm{t}3}+R_{\mathrm{t}4}\right)\right]}}}$$

(10)

The insulation thermal resistance R_t1 is calculated by (11), where ρ_t1 is the XLPE insulation layer thermal resistivity in K·m·W⁻¹.

$$R_{t1}={\displaystyle \frac{{\rho }_{t1}}{2\pi }}\ln {\displaystyle \frac{D_{\mathrm{i}}}{d_{\mathrm{c}}}}$$

(11)

R_t2 is calculated using (12), where ρ_t2 is the thermal resistivity of the inner sheath in K·m·W⁻¹, and X is the ratio of the inner sheath thickness to its outer diameter.

$$\left\{\begin{array}{l}{R}_{\mathrm{t}2}={\displaystyle \frac{{\rho }_{\mathrm{t}2}}{6\pi }}G\\ G=2\pi \left(0.00022619+2.11429X-20.4762X^2\right)\end{array}\right.$$

(12)

R_t3 is obtained using (13), where ρ_t3 is the external sheath thermal resistivity in K·m·W⁻¹, D_e is the outer diameter of the outer sheath in meters, and D_a is the diameter of the steel armor in meters.

$$R_{\mathrm{t}3}=1.6\times {\displaystyle \frac{{\rho }_{\mathrm{t}3}}{2\pi }}\ln {\displaystyle \frac{D_{\mathrm{e}}}{D_{\mathrm{a}}}}$$

(13)

The thermal resistance of the soil surrounding the cable, R_t4, is calculated using (14), where ρ_t4 is the thermal resistivity of the soil in K·m·W⁻¹, and L is the distance from the cable axis to the ground surface in meters.

$$\left\{\begin{array}{l}{R}_{\mathrm{t}4}=1.5\times {\displaystyle \frac{{\rho }_{\mathrm{t}4}}{\pi }}\left[\ln \left(2u\right)-0.63\right]\\ u=2L/D_{\mathrm{e}}\end{array}\right.$$

(14)

2.3. Finite Element Method (FEM)

The FEM can accurately simulate the thermal field distribution both inside and outside the cable. By constraining the permissible maximum temperature of the conductor, the maximum ampacity of the cable can be inversely determined. The thermal field distribution inside and around the cable satisfies the heat balance equation shown in (15), where m is the medium density in kg/m³, c is the specific heat capacity of the medium in J·kg⁻¹·K⁻¹, T is the spatial and temporal distribution function of the temperature field, t is time in seconds, k is the thermal conductivity in W·K⁻¹·m⁻¹, and Q is the heat generation per unit volume in W/m³.

$$mc{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }t}}=k{\nabla }^{2}T+Q$$

(15)

When a cable reaches equilibrium after carrying a steady-state current for an extended period, the thermal field distribution both inside and outside the cable can be considered as a steady-state field. The internal thermal field distribution of the cable follows the Poisson distribution, as shown in (16).

$${\displaystyle \frac{\mathsf{\partial }{T}^2}{\mathsf{\partial }{x}^2}}+{\displaystyle \frac{\mathsf{\partial }{T}^2}{\mathsf{\partial }{y}^2}}+{\displaystyle \frac{\mathsf{\partial }{T}^2}{\mathsf{\partial }{z}^2}}=-{\displaystyle \frac{Q}{k}}$$

(16)

The external thermal field of the cable follows the Laplace distribution, as shown in (17).

$${\displaystyle \frac{\mathsf{\partial }{T}^2}{\mathsf{\partial }{x}^2}}+{\displaystyle \frac{\mathsf{\partial }{T}^2}{\mathsf{\partial }{y}^2}}+{\displaystyle \frac{\mathsf{\partial }{T}^2}{\mathsf{\partial }{z}^2}}=0$$

(17)

The boundary conditions are set as shown in Figure 3. The deep soil temperature is generally constant [1,5,22] and is set as a Dirichlet boundary condition. When there is no heat source around the cable, the heat flux through the boundary is zero, thus, horizontal boundaries on both sides of the computational domain can be set as Neumann boundary conditions. When the heat generated by a cable reaches the ground surface, it is dissipated into the air through convection, so the surface boundary condition must account for convective heat loss. At the same time, the surface absorbs solar energy, causing its temperature to rise. The hotter surface then loses heat by radiating upward. Thus, the final surface temperature is determined by the balance of three factors: convective heat loss, solar radiation, and surface radiation. This surface temperature ultimately affects the ampacity of the direct burial cable. Therefore, unlike previous models that only considered convective heat loss at the ground surface while neglecting both solar radiation and surface radiation, the present model treats the surface as a radiation–convection boundary condition. The governing equations for the boundary conditions are shown in Equations (18)–(20), where T_g is the deep soil temperature in K, T_a is the ambient air temperature in K, T is the surface temperature in K, q_n is the heat flux density through the boundary in W·m⁻², q_s is the solar power density in W·m⁻², h_c is the convection coefficient in W·m⁻²·K⁻¹, ε is the radiative coefficient, taken as 0.9. σ is Stefan–Boltzmann constant with a value of 5.67 × 10⁻⁸ W·m⁻²·K⁻⁴.

$$\mathrm{Dirichlet} \mathrm{boundary} \mathrm{condition}:T=T_{\mathrm{g}}$$

(18)

$$\mathrm{Neumann} \mathrm{boundary} \mathrm{condition}:-k{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }n}}=q_{\mathrm{n}}$$

(19)

$$\mathrm{Radiation}-\mathrm{convection} \mathrm{boundary} \mathrm{condition}:-k{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }n}}=h_{\mathrm{c}}\left(T-T_{\mathrm{a}}\right)+\epsilon \sigma \left(T^4-T_{\mathrm{a}}^4\right)-q_{\mathrm{s}}$$

(20)

This study employs COMSOL Multiphysics 6.2 to establish an electro-thermal-fluid multiphysics coupling FEM model. The computational domain is shown in Figure 4, with a cable burial depth of L. Length from the cable center to the left, right, and bottom boundaries of the computational domain is set to 5L. To eliminate the influence of boundary effects on the temperature field distribution, the left, right, and bottom boundaries of the computational domain are set as infinite element domains to simulate an infinitely extended space. The mesh was autonomously generated using a physics-controlled mesh approach, employing refined mesh elements. The number of elements is 79,337, and the average element quality is 0.8325.

In practical engineering applications, cable sizes must be selected based on the power transmission requirements of different circuits to optimize investment. Consequently, even within the same substation or power plant, multiple cable sizes are often employed. During the cable design phase, engineers require rapid and precise selection of cable sizes for various circuits, necessitating both accurate and efficient methods for evaluating ampacity. Building upon the concept of the correction factor method and based on FEM simulation results, this paper proposes a more precise correction factor calculation formula, as shown in (21). This new evaluation method comprehensively considers the impact of ambient temperature, cable burial depth, soil thermal resistance, solar radiation, wind speed, and the proximity effect of parallel cables on cable ampacity. In the formula, I_B represents the cable ampacity under reference conditions (soil temperature: 25 °C, burial depth: 0.8 m, soil thermal resistivity: 1 K·m·W⁻¹, wind speed: 0 m/s, solar radiation intensity: 0), the correction factors K_t, K_d, K_r, K_c, and K_N respectively reflect the influences of ambient temperature, cable burial depth, soil thermal resistance, solar radiation and wind speed, and the proximity effect of parallel cables on the cable’s ampacity. Compared to the traditional correction factor method, the new approach proposed in this paper incorporates additional correction factors for cable burial depth and solar radiation, thereby more accurately reflecting the actual environmental impact on cable ampacity. Furthermore, leveraging the advantage of FEM in precisely calculating temperature fields, this paper also revises the correction factors for soil thermal resistance and the proximity effect of parallel cables used in the conventional method.

$$I=K_{\mathrm{t}}\cdot K_{\mathrm{d}}\cdot K_{\mathrm{r}}\cdot K_{\mathrm{c}}\cdot K_{\mathrm{N}}\cdot I_{\mathrm{B}}$$

(21)

3. Results and Analysis

This study takes the aluminum-core directly buried cable YJLV22-26/35 3 × 400 mm² as an example. The ampacity is calculated using the three methods mentioned above, with differences compared and causes analyzed. The cable structural dimensions and material properties are listed in

Table 2. Cable structural dimensions and electrical parameters.

Material	Thickness/mm	Diameter/mm	Thermal Conductivity /W·m−1·K−1
Aluminum conductor	-	22.6	217.7
Inner semiconductor	0.8	24.2	0.29
XLPE insulation	10.5	45.2	0.29
Outer semiconductor	1	47.2	0.29
Metallic shielding	0.1	47.4	400
Wrapping tape	1.5	105.1	0.2
Inner sheath	1.4	107.9	0.2
Armor	0.8	109.5	49.3
Outer sheath	4.9	119.3	0.2

. The soil thermal resistivity is set at 1 K·m·W⁻¹.

3.1. Influent of Ambient Temperature

When the influence of air convection and solar radiation are not considered, the ground surface temperature can be assumed to be equal to the ambient temperature. Assuming a single cable circuit, the cable ampacity under different ambient temperatures calculated by the three methods is shown in Figure 5. When the surrounding ambient temperature warms from 15 °C to 50 °C, the ampacity calculated by both the correction factor method and the IEC 60287 analytical method decreases by 27%, whereas the FEM result shows a 22% drop, indicating that changes in ambient temperature have a noticeable influence on cable ampacity. The rated ampacity of the cable under reference conditions for the correction factor method can be found in the cable selection manual [25]. The results indicate that the FEM calculated values are consistent with those obtained by the factor correction method, verifying the accuracy of the FEM calculations. The calculated values from the IEC 60287 analytical method are approximately 28% lower than the FEM results, suggesting that the analytical method may underestimate the cable ampacity.

The reason why the ampacity calculated by the analytical model is significantly lower than that obtained by FEM can be explained by examining the temperature field distribution, as shown in Figure 6a. It is evident that the temperature is primarily concentrated in the soil, indicating that soil thermal resistance plays a decisive role in cable ampacity. The soil temperature obtained by the analytical method is 38% higher than that obtained by the FEM. This discrepancy occurs because the analytical method treats soil thermal resistance as a lumped parameter, thereby overestimating its effect, whereas the FEM considers it as a distributed parameter, yielding more accurate results. Consequently, the temperature differences in various parts of the cable calculated by the two methods become more pronounced. For instance, the temperature of the cable outer surface computed by the analytical method reaches 79.2 °C, while the FEM result shows only 64.4 °C—a difference of 14.8 °C, as illustrated in Figure 6b. As a result, the cable ampacity calculated by the analytical method is substantially lower than that obtained by the FEM model.

3.2. The Influence of Cable Burial Depth

When the soil surface temperature is 25 °C, the impact of burial depth on cable ampacity is shown in Figure 7a. It can be observed that the ampacity obtained by both FEM and the analytical method reduces with increasing burial depth, while the correction factor method neglects the impact of burial depth on ampacity. As the burial depth increases from 0.5 m to 1.2 m, the cable ampacity obtained by FEM decreases by 7%, whereas that obtained by the IEC 60287 analytical method decreases by 12%. The variation in burial depth has a more significant impact on the results calculated by the analytical method, which can be attributed to the overestimation of soil thermal resistance in this approach.

Taking the FEM calculation results as the benchmark, it can be observed that the computational error of both the correction factor method and IEC 60287 analytical method increase as the cable burial depth increases, as shown in Figure 7b. With the increase in burial depth from 0.5 m to 1.2 m, the calculation error of the correction factor method rises from 0.1% to 7%, overestimating the ampacity, while the calculation error of the IEC 60287 analytical method increases from 24% to 28%, significantly underestimating the ampacity.

Taking a soil temperature of 25 °C and a burial depth of 0.8 m as the baseline conditions, the curves of ampacity correction factors versus burial depth for the FEM and IEC analytical methods are shown in Figure 8. Through curve fitting, the expression for the FEM burial depth correction factor can be obtained as shown in (22), where d represents the cable burial depth in meters.

$$K_{\mathrm{d}}=0.9821d^{-0.086}$$

(22)

3.3. The Influence of Soil Thermal Resistance

Soil thermal resistivity is influenced by factors including soil density, soil constituents, moisture content, porosity, and temperature and can vary within the range of 0.4–4 K·m·W⁻¹ [12,26,27]. Using a ground surface temperature of 25 °C and a cable burial depth of 0.8 m as the baseline conditions, the correction factors K_r obtained by the three calculation methods, as the soil thermal resistivity increases from 0.5 K·m/W to 3 K·m·W⁻¹, are shown in Figure 9. It can be observed that the K_r decreases as the soil thermal resistivity increases. At soil thermal resistivity below 1 K·m·W⁻¹, the IEC 60287 analytical method yields the highest K_r value, while the K_r values from the FEM and the correction factor method are close. At soil thermal resistivity exceeds 1 K·m·W⁻¹, the correction factor method yields the highest K_r value, the IEC analytical method yields the lowest, and the FEM yields an intermediate value. Therefore, for cases where the thermal resistance is greater than 1 K·m·W⁻¹, the correction factor method tends to overestimate the cable ampacity, while the IEC 60287 analytical method tends to underestimate it. Specifically, when the soil thermal resistivity increases from 1 K·m·W⁻¹ to 3 K·m·W⁻¹, the correction factor method overestimates the ampacity by 14%, whereas the IEC 60287 analytical method underestimates it by 10%. Among the three methods, the IEC 60287 analytical method is most sensitive to soil thermal resistance. This is because it represents soil thermal resistance using lumped parameters, which overestimate the soil’s resistance to heat flow. Consequently, the higher the soil thermal resistivity, the lower the ampacity assessed by the IEC 60287 analytical method, indicating a greater margin of error.

Through curve fitting, the expression for the thermal resistance correction factor of the FEM can be obtained, as shown in (23), where ρ_t represents the soil thermal resistivity coefficient in K·m·W⁻¹:

$$K_{\mathrm{r}}=-0.299\ln \left({\rho }_{\mathrm{t}}\right)+0.9939$$

(23)

3.4. The Influence of Solar Radiation and Air Convection

From the ampacity calculation Formula (10), it is clear that the ground surface temperature is a critical factor limiting the ampacity of directly buried cables—the higher the surface temperature, the lower the ampacity. Therefore, for outdoor direct burial cable, the ampacity of cables is constrained by the ground surface temperature. The ground surface heats up after absorbing solar radiation, as illustrated in Figure 10. At 25 °C ambient temperature, if the effect of solar radiation is neglected, the ground temperature will match the ambient temperature. However, when solar radiation is considered, the ground surface temperature increases with rising solar radiation intensity. Under natural convection conditions, the surface temperature can reach up to 54 °C when the maximum solar radiation intensity reaches 500 W/m², with a growth rate exceeding 116%. This demonstrates the significant impact of solar radiation on earth surface temperature.

The sharp increase in surface temperature, in turn, greatly reduces the cable’s ampacity. Consequently, previous models often neglected the influence of solar radiation, leading to substantial computational errors. When the surface temperature exceeds the air temperature, convection occurs, causing the surface temperature to decrease. In the case of solar radiation power at 100 W/m², when the convection coefficient increases from 10 W·m⁻²·K⁻¹ to 30 W·m⁻²·K⁻¹, the surface temperature decreases by 9.4%. In the case of solar radiation intensity at 500 W/m², the same increase in the convective coefficient produces a 28.7% drop of surface temperature. Evidently, the stronger the solar radiation, the more pronounced the cooling effect of convection becomes.

In the model, the choice of convective heat-transfer coefficient is critical, as it determines the accuracy of the simulation results. In the COMSOL-based simulation of three-phase cable temperature distribution, Hu et al. used a convection coefficient of 6.5 W·m⁻²·K⁻¹ [17]. Nie et al. adopted a value of 12.5 W·m⁻²·K⁻¹ [22]. Both Ocłoń et al. [28] and Bustamante [8] applied a coefficient of 10 W·m⁻²·K⁻¹ in their FEM models. Klimenta et al. used 12.5 W·m⁻²·K⁻¹ for dry grassland and 8 W·m⁻²·K⁻¹ for paved surfaces [21]. Wind speed significantly influences the convective heat transfer coefficient. Hens [29], Laloui and Rotta Loria [30], Kim et al. [1], Lee et al. [31] have proposed different relationships between the convection coefficient and wind speed, as shown in Figure 11. This study adopts the relationship reported in [30,31], as shown in (24).

$$h_{\mathrm{c}}=10+3.1v$$

(24)

Through numerical simulation, the variation in cable ampacity under the combined impact as a consequence of solar radiation and wind speed can be obtained, as shown in Figure 12.

It can be observed that the cable’s ampacity is lowest under conditions of maximum solar radiation and no wind. As the wind speed increases, the cable ampacity follows a logarithmic pattern of increase. When the wind speed exceeds 6.5 m/s, the influence of wind diminishes, and the cable ampacity tends to stabilize.

The above computational results indicate that solar radiation and wind speed are critical factors influencing the ampacity of directly buried cables. However, both the correction factor method and the IEC 60287 analytical method neglect the impact of solar radiation and wind speed on ampacity, leading to significant computational errors. When the FEM results are taken as the benchmark, the computational errors of the correction factor method and the IEC 60287 analytical method are shown in Figure 13. From Figure 13a, it can be found that the error of the correction factor method progressively increases with rising solar radiation intensity. This occurs because solar radiation reduces the actual cable ampacity, but the correction factor method fails to account for solar radiation effects, leading to an overestimation of the cable ampacity. Meanwhile, the ampacity error shows a decreasing trend as wind speed increases. The maximum ampacity error reaches 23% under conditions of v = 0 and P_s = 500 W/m². The above analysis demonstrates that the correction factor method, by neglecting solar radiation effects, significantly overestimates the actual ampacity of cables. This may lead to the dangerous practice of selecting undersized cable cross-sections, resulting in cables operating under overload conditions. For instance, in mountainous photovoltaic power plants, collector lines typically employ direct burial cables to transmit electricity from the PV array to the step-up substation. During midday in summer, when solar radiation is most intense and the power plant’s generation capacity peaks, the cable lines operate under high loads. If the cable cross-section is undersized, it can cause the operating temperature to exceed permissible limits. Over time, this could pose a threat to the safe operation of the cable lines.

From Figure 13b, it can be observed that the error of the IEC 60287 analytical method gradually decreases with increasing solar radiation intensity. This occurs because the IEC 60287 analytical method inherently tends to underestimate cable ampacity. As solar radiation intensifies, the actual cable ampacity decreases, and the discrepancy between the analytical results and the actual values narrows. Simultaneously, the ampacity error shows an increasing trend with higher wind speeds. The maximum ampacity error reaches 34% under conditions of v = 6.5 m·s⁻¹ and P_s = 100 W·m⁻².

Taking zero solar radiation intensity as the baseline, the expression for the ampacity correction factor under different solar radiation intensities and wind speeds can be derived, as shown in Equation (25):

$$K_{\mathrm{c}}\left(v, P_{\mathrm{s}}\right)=\left\{\begin{array}{l}-0.139v^2+2.323v+498.86,\hspace{1em}{P}_{\mathrm{s}}{=100 \mathrm{W}/\mathrm{m}}^2\\ -0.470v^2+6.325v+477.32,\hspace{1em}{P}_{\mathrm{s}}{=200 \mathrm{W}/\mathrm{m}}^2\\ -0.664v^2+9.256v+456.89,\hspace{1em}{P}_{\mathrm{s}}{=300 \mathrm{W}/\mathrm{m}}^2\\ -0.858v^2+12.56v+435.26,\hspace{1em}{P}_{\mathrm{s}}{=400 \mathrm{W}/\mathrm{m}}^2\\ -1.244v^2+17.04v+411.42,\hspace{1em}{P}_{\mathrm{s}}{=500 \mathrm{W}/\mathrm{m}}^2\end{array}\right.$$

(25)

3.5. The Influence of Parallel Circuits

In the scenario of multiple directly buried cable circuits, each circuit is set to carry the same current load. When three circuits are buried in parallel, the central circuit exhibits the highest cable temperature, as shown in Figure 14. Therefore, this study uses the temperature of the central circuit as the controlling condition for ampacity calculation. As soon as the central circuit temperature reaches 90 °C, the corresponding current is defined as the maximum permissible ampacity.

In the case of multiple cable circuits laid in parallel, the temperature fields between circuits interact synergistically, leading to a reduction in the allowable cable ampacity, which is known as the proximity effect. The proximity effect is influenced by the spacing between circuits. As the circuit spacing increases, the proximity effect diminishes, resulting in an increase in the allowable cable ampacity, as shown in Figure 15, where S stands for the circuit spacing. Figure 15 further indicates that cable ampacity decreases as the number of parallel circuits increases. The reduction in ampacity becomes more pronounced when the circuits are spaced closer together.

For 35 kV directly buried high-voltage cables, it is uncommon to have more than three circuits laid in parallel. Therefore, this study considers a maximum of three circuits. Taking the ampacity of a single circuit as the benchmark, the ampacity correction factors K_N for two and three parallel circuits relative to circuit spacing are shown in Figure 16. It can be observed that K_N increases logarithmically with the circuit spacing. This relationship can be expressed by (26), where S represents the circuit spacing in millimeters, and N denotes the number of circuits.

$$K_{\mathrm{N}}\left(S,N\right)=\left\{\begin{array}{l}0.0403\ln \left(S\right)+0.6687\\ 0.0572\ln \left(S\right)+0.4886\end{array}\right.$$

(26)

3.6. Discussion

From the above research, it can be observed that the traditional correction factor method does not account for the influence of cable burial depth on ampacity, limiting its applicability. The IEC 60287 analytical method overestimates soil thermal resistance, thereby exaggerating the impact of burial depth and soil thermal resistance on ampacity, resulting in conservatively low ampacity evaluations. Both methods neglect the effects of solar radiation intensity and wind speed on cable’s allowable ampacity, leading to significant errors when assessing the ampacity of outdoor directly buried cables. FEM, based on electro-thermal-fluid multiphysics coupling models, can adapt to diverse operating conditions and offers high computational accuracy, making it a better choice for practical engineering applications in cable ampacity assessment. Based on FEM simulation results, this paper analyzed the effects of ambient temperature, cable burial depth, soil thermal resistance, solar radiation, wind speed, and the proximity effect of parallel cables on cable ampacity, and formulated the influence factors. Compared to traditional methods, the novel approach proposed in this paper can comprehensively account for the influence of multitude environmental factors, demonstrating better environmental adaptability. Unlike the IEC 60287 analytical method, which employs lumped thermal resistance to calculate thermal field distribution, FEM calculates the thermal field distribution by discretizing thermal resistance into nodes, representing a more precise approach. The assessment of cable ampacity is carried out based on the condition that the conductor temperature does not exceed 90 °C. Therefore, the new method presented in this paper achieves higher computational accuracy. The new method expresses the influence of various environmental factors as correction coefficients, thereby eliminating the need for complex FEM modeling processes and lengthy computation times. Consequently, the new method is as easy to use as the traditional correction factor method. A comparison of performance between the proposed new method and traditional methods is summarized in

Table 3. Comparison of performance among three methods.

Performance	Correction Factor Method	IEC 60287 Analytical Method	The New Method in This Paper
Environmental adaptability	N	N	Y
High computational accuracy	N	N	Y
Ease of use	Y	N	Y

.

Overall, the novel method presented in this paper combines strengths of both the correction factor method and the FEM, ensuring both computational accuracy and ease of use, which is highly beneficial for engineering applications.

Although this paper only focused on analyzing the 35 kV YJLV22-26/35 3 × 400 mm² cable, the analysis in Section 3.1 indicates that the temperature drop on the cable’s internal insulation is minimal compared to that in the soil. In other words, variations in the thickness of the internal insulation layer have a small impact on the cable’s ampacity. Therefore, the new evaluation method for direct burial cables proposed in this paper can be applied to other voltage levels.

4. Conclusions

This paper investigated the influence of multiple environmental factors and proximity effects on the ampacity of 35 kV YJLV22-26/35 3 × 400 mm² direct burial cables using an electro-thermal-fluid coupling FEM model. The study shows that within the surface temperature variation range of 15 °C to 50 °C, the outcomes obtained by the correction factor method align with those of FEM, whereas the IEC 60287 analytical method underestimates the ampacity by approximately 28%. When the burial depth increases from 0.5 m to 1.2 m, the calculation error of the correction factor method rises from 0.1% to 7%, overestimating the ampacity, while the calculation error of the IEC 60287 analytical method increases from 24% to 28%, significantly underestimating the ampacity. As the soil thermal resistivity rises from 1 K·m·W⁻¹ to 3 K·m·W⁻¹, the correction factor method overestimates ampacity by 14%, whereas the analytical method underestimates it by 10%. With increasing solar irradiance, cable ampacity decreases significantly, whereas it shows an increasing trend with higher wind speed. Both the correction factor method and the analytical method neglect the effects of solar radiation and wind speed on ampacity, leading to maximum calculation errors of 23% and 34%, respectively. As the number of parallel circuits increases, the ampacity of the directly buried cable decreases, with the reduction rate being related to circuit spacing.

This study has made the following major contributions:

(1)

The proposed FEM model simultaneously accounts for the effects of solar radiation, convective heat dissipation, and surface radiation on cable ampacity, thereby enhancing computational accuracy.

(2)

It has identified the influence patterns of multiple environmental factors and proximity effects on the ampacity of direct burial cables.

(3)

The computational errors of traditional methods were analyzed, revealing their applicable scope.

(4)

Based on the simulation results of the FEM model, an novel correction factor method that takes into account both multiple environmental factors and proximity effects has been proposed, which can greatly benefit engineering design.

The present study is limited to homogeneous and isotropic soil conditions, without considering backfilling scenarios. A study on the impact of layered soil structures using high-thermal-conductivity backfill soil on the cable ampacity will be conducted in subsequent research.