The development of a thermal circuit model is essential for accurately calculating the temperature of cables. During the operational phase of the cable, certain critical parameters, such as the core temperature, cannot be directly measured using probes or sensing devices due to environmental constraints. Consequently, it is imperative to derive these parameters through the thermal circuit model (Tao et al., 2023). This paper presents a general framework for constructing the thermal circuit model of power cables (Figure 1).

The transient thermal circuit model utilized for simulating the transverse heat dissipation process of a single-core cable body, as outlined in the IEC 60853 standard, is illustrated in Figure 2 (IEC, 2008). This model incorporates the effects of the heat capacity of each layer on heat dissipation, in contrast to the steady-state thermal circuit model.

Where θ _C is the temperature of the cable core; θ _S denotes the temperature of the outer skin of the cable; θ ₀ is the external ambient temperature; and W _C represents both conductor loss and electromagnetic losses of the cable. The equations for the thermal resistance and heat capacity of each layer are presented as follows: T A = T 1 (1) T B = q s T 3 (2) Q A = Q C + p Q i (3) Q B = 1 − p Q i + Q S + p 1 Q W q s (4) Q B 1 = 1 − p 1 Q W q s (5) p = 1 2 ⁡ ln D i D c − 1 D i D c 2 − 1 (6) p 1 = 1 2 ⁡ ln D e D s − 1 D e D s 2 − 1 (7) q s = 1 + W s W c (8) where W _s is the loss of metal sheath and shield; D _c is the outer diameter of the conductor; D _i is the outer diameter of the insulation layer; D _s is the outer diameter of the metal sheath; D _e is the outer diameter of the cable; Q _c , Q _i , Q _s , and Q _w are the heat capacitance of the cable core, the insulation layer, the metal sheath layer, and the outer jacket layer, respectively; while T ₁, T ₂, T ₃, and T ₄ are the thermal resistance of the cable’s insulating layer, the inner lining layer, the outer sheath layer, and the external environment, respectively. T ₁, T ₂, and T ₃ are can be determined as follows (YU, 2022): T 1 = ρ 1 2 π ln D 1 o D 1 i n (9) T 2 = ρ 2 2 π ln D 2 o D 2 i n (10) T 3 = ρ 3 2 π ln D 3 o D 3 i n (11) where ρ ₁ is the thermal resistance coefficient of the insulation layer; ρ ₂ is the thermal resistance coefficient of the inner lining layer; ρ ₃ is the thermal resistance coefficient of the outer sheath layer (K·m/W); while D _o and D _in are the outer and inner diameters of the corresponding layers, mm, respectively.

The external thermal resistance varies depending on the laying conditions. For a single-core power cable that is directly buried in the ground, T ₄ is obtained as follows: T 4 = ρ T 2 π ln u + u 2 − 1 (12) where ρ _T is the thermal resistance of the soil (K·m/W) and u is calculated as follows: u = 2 L D e (13) where L is the distance from the axis of the power cable to the earth surface (mm) and D _e is the outer diameter of the power cable (mm).

Electromagnetic losses consist of eddy current losses and circulating current losses. According to IEC standards, the eddy current loss coefficient is denoted by λ₁′, while the circulating current loss coefficient is represented by λ₁′′. The total loss coefficient (λ₁) is the sum of the two, i.e.: λ 1 = λ 1 ′ + λ 1 ″ (14)

The magnitude of electromagnetic losses is given by: W s = λ 1 W (15) where W is the total cable losses, λ1′ is the metallic sheath/armor eddy current loss factor and λ1′′ is the metallic sheath/armor circulating current loss factor.

The hysteresis loss in the cable is negligible due to the use of copper as the conductor material, a non-ferromagnetic metal with relative permeability (μ_r) approximately equal to 1 and a hysteresis loop area approaching zero. For cables containing ferromagnetic materials (e.g., steel armor), hysteresis loss must be modeled separately. Since the 110 kV cable studied in this work features a non-magnetic sheath, this component is excluded. Additionally, the dielectric loss tangent (tanδ) of the cable insulation is ≤0.001. The dielectric loss power density, calculated as W_d = ωCU²tanδ, contributes less than 0.2% to the total losses and is therefore also neglected. Consequently, this study focuses exclusively on conductor losses and electromagnetic losses. By applying Equations 14, 15, both conductor losses and electromagnetic losses are introduced as heat sources into the thermal circuit model, thereby achieving electro-thermal multiphysics coupling.

The multilayer soil model is capable of accurately simulating the diffusion process of heat generated by cables within the soil. Yu (2022) illustrates the applicability of the multilayer soil model through the research and analysis of a five-layer soil thermal circuit model, which facilitates precise calculations of the temperature field associated with a single-core cable. In this layered soil model, each layer is represented by a T-type thermal circuit model, where the thermal resistance of each layer is R and the heat capacity is C. When the soil is stratified into five layers, the corresponding equivalent thermal circuit model is depicted in Figure 3.

The thermal resistance and heat capacity of each layer are calculated as follows: R m = ρ T 2 π log 1 + t h r int (16) C m = π r e x t 2 − r int 2 · C p (17) R s 1 = 1 2 R 1 (18) R s i = 1 2 R i − 1 + R i (19) R s 6 = 1 2 R 5 (20) where m is the number of soil layers (m = 1, 2, 3, 4, and 5); th is the thickness of each soil layer; r _ext and r _int are the outer and inner diameters of each soil layer, respectively; C _p is the volumetric specific heat capacity of the soil (J/mm³·°C); and i = 2, 3, 4, and 5.

Soil modeling plays a critical role in assessing the temperature rise effects associated with underground cables. After the dimensions and configuration of the cable have been established, the soil emerges as the primary factor affecting the distribution of the cable’s temperature field. However, the model is predicated on the assumption of a uniform soil environment, which presents certain limitations when confronted with the complexities and variations of the actual operational environment of the cable. Therefore, it is unable to accurately replicate the soil conditions at different locations surrounding the cable.

The integration of the soil-layered thermal circuit model with the thermal circuit model of the IEC standard cable body offers a more comprehensive understanding of the thermal characteristics of the cable. This approach more accurately reflects the complex thermal environment encountered in practical applications, thereby enhancing the reliability and safety of the system while optimizing the design and installation of the cable. This methodology is widely acknowledged due to the continuity in the structure and temperature distribution across each layer of the cable (Qin et al., 2024). The construction of the integrated thermal circuit model is illustrated in Figure 4, where the parameters of the cable body model are developed in accordance with the IEC standard, and the parameters of the soil model are derived from the soil layering model. The integrated thermal circuit model is capable of simulating the heat exchange process between the cable and the surrounding soil environment more comprehensively, thereby providing a more robust theoretical foundation for engineering practice.

In practical applications, cables are typically installed in clusters within the soil, and the mutual thermal impacts among these cables affect the temperature distribution in the surrounding environment. When the effects of neighboring cables are incorporated into the thermal circuit model, the temperature field of all underground cables can be efficiently determined using analytical methods, which holds substantial engineering significance (Niu et al., 2023). This study simulates the mutual thermal impact between cables by incorporating the losses from adjacent cables at designated locations within a multilayer soil thermal circuit model. Furthermore, the collective thermal interactions of all neighboring cables are integrated into the RC trapezoidal thermal circuit model of the cables under analysis. Consequently, this approach allows for a more precise modeling of the transient temperature of the cables. The transient thermal circuit model that accounts for the mutual thermal impacts is depicted in Figure 5.

The losses W _k generated by neighboring cables should be precisely incorporated into their respective locations within the soil model. If an external cable k is situated within the soil thermal resistance (e.g., R _S4), this thermal resistance must be subdivided into two sub-resistances (i.e., R _S4a and R _S4b ). The values of these sub-resistances will be allocated proportionally based on the position of the external cable, and a heat source W ₁ will be introduced at the junction of the two sub-resistances to ensure the model’s accuracy in calculating the temperature increase of the cable. If the external cable is positioned at a node between two layers of the soil model, it can be effectively simulated by incorporating an additional heat source W ₂ into the thermal circuit diagram. Given that the soil thermal circuit model operates on the principle of superposition, it is adequate to add the losses from each cable to the corresponding location in the thermal circuit model that represents the soil. Furthermore, the thermal impacts of other heat sources, such as steam pipes and heating pipes, on the cables can be addressed in a similar manner.

In real-world operating environments, cables are typically affected by a multitude of heat sources. These sources encompass not only the internal heating of the cable and variations in soil temperature but also include surrounding equipment, climatic conditions, and other external factors. This results in a highly complex heat dissipation environment. In contrast, traditional thermal circuit models often operate under the assumption of a simple and uniformly distributed environment, which renders them inadequate for effectively addressing the complexities associated with actual operating conditions.

The temperature of a cable is affected not only by radial heat transfer, which refers to the heat conduction between the conductor and the insulation layer, but also by axial heat flow along the length of the cable. This axial heat flow can create temperature gradients at various locations along the cable. Traditional thermal circuit models frequently overlook the significance of axial heat dissipation when analyzing heat conduction between the conductor and the insulation layer. This oversight can result in inaccuracies in the calculated results and may restrict the model’s precision under certain complex operational conditions.

The schematic diagram of the model, which illustrates the installation of the underground power cable along the center of the street, is presented in Figure 6.

In Figure 6, the area designated for the installation of underground cables is longitudinally divided into two sections. The central portion of the street (Area 1) is characterized as an unfavorable heat dissipation zone with a specified width of b₀. Conversely, the remaining section is Area 2, which represents a standard installation area. Due to the differing soil thermal resistance coefficients in these two areas, a temperature gradient is established along the axial direction of the cable, resulting in the generation of axial heat flow. To analyze the three-dimensional thermal field, the cable is discretized along its length (Z-axis) into N sections, with each segment represented as a small section of the conductor ∆z. The heat flow is illustrated in Figure 7.

Where W _C is the conductor loss generated by the current passed through the cable; W _r is the heat dissipated along the conductor in the radial direction; and W _L is the heat dissipated along the conductor in the axial direction. Due to the different external environmental parameters, there is heat exchange along the cable’s axial direction. Compared with other existing schematic diagrams of heat flow in cable conductor segments, Figure 7 simultaneously considers the influence of radial heat dissipation (W_r) and axial heat dissipation (W_L), achieving a coupled analysis of the cable’s lateral and longitudinal heat conduction. This enables a more accurate simulation of heat exchange in cables under different heat dissipation environments.

According to the law of conservation of energy, the total heat entering a node, combined with the heat generated by the conductor, is equal to the total heat exiting the node. Therefore, the thermal equilibrium equation for the conductor section can be expressed as follows: W C z Δ z + W L z − Δ z = W r z Δ z + W L z (21)

In comparison to the thermal equilibrium equation of the conventional thermal circuit model, the equation presented in this paper accounts for axial heat dissipation within the cable conductor. The enhanced thermal circuit model is developed using a single-core cable situated in various soil environments as a case study (Figure 8). The neighboring cable segments are connected by axial thermal resistance (T_L) in Figure 8, and each node of each cable segment branch from bottom to top represents the structure of each layer of the cable from the conductor to the soil surface, respectively. This model simultaneously incorporates both radial interlayer conduction and axial intersegment conduction, providing a complete characterization of three-dimensional heat diffusion processes. Given that the heat capacity of the cable conductor is relatively small in comparison to other materials (e.g., insulation and sheath materials), it can be disregarded during the establishment of the thermal circuit model. Furthermore, this model is also applicable to three-core cables, where the transient thermal circuit model is constructed by partially modifying the cable body model to accommodate the three-core configuration (Liu et al., 2016).

This thermal circuit model delineates the entire area into two regions: unfavorable and normal heat dissipation zones. The coordinate z = 0 is positioned at the center of the unfavorable heat dissipation zone. The axial thermal resistance of the conductor is calculated as follows: Δ T L = ρ A (22) where ρ is the thermal resistance of the conductor (K·m/W) and A is the cross-sectional area of the conductor (mm³).

W is related to the temperature of the conductor, which can be determined as follows: W = W 20 1 + α 20 θ − 20 = W c o n s t + δ W · θ (23) W c o n s t = W 20 1 − 20 α 20 (24) δ W = α 20 W 20 (25) where α ₂₀ is the conductor’s resistivity temperature coefficient at 20°C, typically valued at 3.93 × 10⁻³ K⁻¹ for copper; and W ₂₀ is the conductor loss at the reference temperature: W 20 = I 2 R 20 (26) where R ₂₀ is the AC resistance of the conductor at 20°C, incorporating skin and proximity effects: R 20 = R 20 ′ 1 + y s + y p (27) where R’₂₀ is the DC resistance of the conductor at 20°C. For copper conductors, R^’ ₂₀ = 1.72 × 10⁻⁸ Ω⋅m; y _s is the skin effect factor, and y _p is the proximity effect factor.

The operation of cables involves various metal materials (e.g., conductors, metal sheaths, and armor) along with insulating materials. These components generate losses and emit heat, thereby establishing a heat flow field. The heat flow field exhibits a correspondence with the physical quantities present in both the thermal circuit and the current field. Consequently, it is feasible to employ circuit theory and the principles of the current field to analyze and calculate the thermal circuit model of the cable.

According to the transient thermal circuit model (Figure 8), a thermal balance analysis is conducted for each node. Each node can be represented by a thermal equilibrium equation analogous to Equation 21. In this context, the heat flow between two adjacent nodes is expressed as a function of the temperature difference between the nodes and the thermal resistance. Therefore, Equation 21 can be reformulated as follows: C 1 d θ c n d t + 2 θ c n Δ T L − θ c n − 1 Δ T L − θ c n + 1 Δ T L + θ c n Δ T A − 2 θ i n Δ T A = Δ W c n (28)

By expressing thermal resistance in terms of thermal conductance, the complete thermal circuit model can be represented as a system of first-order differential equations following collation: C d θ d t + Z · θ = Δ W (29) where θ is the M × 1 temperature matrix to be solved; C is the M × 1 heat capacity matrix; Z is the M × M node conductivity matrix; and ∆W is the M × 1 heat source matrix. In summary, the transient temperature variation at each node of the entire cable can be determined by solving this system of differential equations through thermal node analysis. The procedure for utilizing the thermal node analysis method to assess the temperature rise effect in the cable is depicted in Figure 9.

The parameters of the running cable (e.g., the thermal resistance coefficient and ambient temperature data) are initially incorporated into Equations 1–20 and Equation 22 to determine the thermal resistance and heat loss for each layer within the thermal circuit model. Then, Equations 23–27 are used to compute the heat source magnitude, and these values are subsequently integrated into the transient thermal circuit model of the cable (Figure 8). Finally, the thermal equilibrium equations of the thermal circuit model are derived using Equations 28, 29. The fourth-order Runge-Kutta algorithm is employed to programmatically solve this system of equations, thereby generating the transient temperature rise curve for the entire cable at each node. Finally, the fourth-order Runge-Kutta algorithm is again utilized to program and resolve the equations, yielding the transient temperature rise curve for each individual node of the complete cable system.

Reference (Zeng et al., 2022) established an experimental system for temperature measurement of a YJLW03 64/110 1 × 500 power cable, with structural parameters detailed in Table 1. The cable was buried at a depth of 800 mm in soil characterized by a thermal resistivity of 1.2 K·m/W and a surface temperature of 20°C.

The experimental setup comprised the following components: 380 V power supply, voltage regulator (for input voltage adjustment), PLC control cabinet (for current regulation and data acquisition), computer workstation (for real-time data recording and analysis), current booster (to generate high currents), current transformer (for real-time monitoring of cable load current), compensation capacitor bank (to ensure stability of the test circuit) and the test cable. Thermocouples were employed to measure temperatures across the cable layers. For internal temperature profiling, holes were drilled into the cable body to embed thermocouples within specific structural layers. Surface temperatures were measured by affixing thermocouples directly to the cable exterior. Prior to testing, all thermocouples underwent calibration experiments, and drill holes were sealed with epoxy putty to minimize measurement errors.

To validate the accuracy of the constructed thermal circuit model, a transient thermal circuit model of the experimental cable was established based on the configuration shown in Figure 8. During the modeling process, the soil environment was divided into five equal layers to calculate equivalent thermal resistance and capacitance. Numerical analysis was employed to obtain the hot-spot temperature distribution of the conductor, which was then compared with the experimental results from Reference (Zeng et al., 2022). As shown in Figure 10:

Through a comparative analysis utilizing the experimental method, the conductor temperature predicted by this model is marginally higher than the temperature measured experimentally, with a maximum relative error of 7.4%. This discrepancy primarily arises from the thermal circuit model, which is typically constructed based on several simplified assumptions, notably the exclusion of the air gap layer’s effects. This omission results in an overestimation of the thermal resistance value within the model, thereby leading to elevated calculation results. However, the air gap layer does not come into direct contact with the cable conductor, thereby exerting minimal effect on the conductor’s temperature and not compromising the overall accuracy of the results.

To assess the efficacy of the developed thermal circuit model, an analysis was conducted using three single-core YJLW02 64/110 1 × 630 cables as a case study. These cables were arranged horizontally, with their metal sheaths interconnected for grounding purposes. The spacing between the cables was established at 250 mm, and the burial depth was set at 700 mm. It is assumed that the cables traverse a street intersection, leading to the division of the underground cable laying area along the axial direction into two regions. The section of cable located within the street intersection is Region 1, which is characterized by unfavorable heat dissipation conditions. Conversely, Region 2 represents a standard laying environment (Figure 6). The soil thermal resistance coefficients for Regions 1 and 2 are 2.5 K·m/W and 1.0 K·m/W, respectively. The temperature rise effect of the cables was calculated utilizing the thermal circuit model developed in this study, as well as a traditional thermal circuit model and the FEM for comparison. The ambient soil temperature was maintained at 20°C. The structural parameters of the cables are outlined in Table 2.

According to national standards, the maximum allowable temperature for the conductor of YJV cable should not exceed 90°C. In the aforementioned calculation example, the cable was initially energized with a 900 A current until reaching steady-state conditions (where the steady-state temperature distribution serves as the initial condition for transient analysis). Subsequently, the current was abruptly increased to 1045 A. A transient thermal circuit model was constructed based on Figure 8, with the cable discretized into 100 segments (0.1 m per segment). Thermal analysis was performed, and the computational results are presented in Figure 11. The model calculates temperatures at all nodes across the 100 segments. For clarity, Figure 11 selectively displays transient temperature rise curves for 10 representative conductor nodes. These curves are arranged in order from top to bottom and correspond to the temperatures of the intermediate conductor nodes that are further and further away from the center of the street (1-m intervals). The horizontal axis represents the elapsed time after the transient current application, while the vertical axis indicates conductor temperature. Transient temperatures for other nodes can be directly extracted from the solution vector θ.

In Figure 11, an extended duration of exposure to the step current correlates with an increase in conductor temperature, which stabilizes after approximately 6 h. The steady-state temperature is recorded at 94.1°C, with the highest temperature observed at the center of the street. Furthermore, the conductor temperature exhibits a decreasing trend as the distance from the street center increases. The transient temperature rise curve for this scenario was calculated using both the traditional thermal circuit model and the FEM. A comparative analysis of the temperature change curve at the conductor’s hottest point over time, derived from the three methodologies, is depicted in Figure 12.

Table 3 summarizes a comparison of the temperature change rule at the hottest point of the conductor over time, as determined by three different calculation methods.

In Table 3, the calculated temperature derived from the thermal circuit model is higher than that obtained through the FEM. This discrepancy arises from the thermal circuit model’s omission of the air gap layer’s effect during its construction. Using the FEM’s calculation results as a reference point, a comparison is made between the temperature results at the hottest point of the conductor, as determined by both the traditional thermal circuit model and the thermal circuit model developed in this study:the maximum temperature of the conductor, as determined by the thermal circuit model developed in this study, is 94.1°C. This value exhibits a deviation of 4.4°C when compared to the results obtained through the FEM, resulting in a relative error of 4.91%. Furthermore, the maximum discrepancy recorded is 7.2°C, with a corresponding maximum relative error of 8.63%. In comparison to traditional models (Table 3), the error associated with the proposed model is relatively minor, thereby indicating that the computational results are more accurate. Furthermore, while conventional thermal circuit models are limited to simulating and calculating the temporal temperature variations at specific nodes, the thermal circuit model proposed in this paper is capable of assessing the temperature rise effects at any node throughout the entire cable. This capability renders it particularly suitable for a diverse range of cables operating in various heat dissipation environments and multi-heat source scenarios. The comparative analysis of the results from these three methodologies substantiates the efficacy of the constructed thermal circuit model in accurately calculating the transient temperature rise effects of cables, particularly in contexts involving varying heat dissipation conditions.

To further demonstrate the advantages of the proposed model, its computational efficiency, environmental adaptability, and dynamic analysis capabilities are compared with several recent models. Reference (Stojanović et al., 2023) employed the finite element method for multi-field coupling under solar radiation, wind speed, and soil moisture conditions, accurately calculating the transient temperature rise of cables under extreme summer and typical winter conditions. However, compared with our model, the finite element method requires longer computation time and necessitates re-modeling for different cable configurations and installation environments, making it diffcult to meet real-time monitoring demands in modern power systems. Reference (Henke and Frei, 2023) proposed a method for determining coupling parameters, which accounts for increasingly complex heat dissipation conditions but is only applicable to parallel-arranged multi-cable scenarios and introduces errors under non-uniform environmental conditions. In contrast, by inputting complex and variable parameters into our thermal circuit model, the proposed model can determine real-time temperature variations at any node along the entire cable, making it suitable for diverse heat dissipation environments and multi-heat-source scenarios.