# Numerical Study of Heat Transfer in Trefoil Buried Cable with Fluidized Thermal Backfill and Laying Parameter Optimization

Authors: Chen-Zhao Fu; Wen-Rong Si; Lei Quan; Jian Yang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Trefoil buried cable is one of the important cable arrangements for the underground transmission line, and its heat transfer performance is relatively poor. By filling with fluidized thermal backfill material (FTB) around trefoil buried cables, the heat transfer would be efficiently enhanced, while the filling cost should also be considered. In the present study, the heat transfer process in the FTB trefoil buried cables is numerically studied, where the cable core loss and eddy current loss in the cable were coupled for the simulation. The heat transfer performances and ampacities for trefoil buried cables with different back fill materials were analysed and compared with each other. Then, the laying parameters for the parabolic-type FTB trefoil buried cables were optimized with the radial basis function neural network (RBNN) and genetic algorithm (GA). Firstly, it is found that, with FTB material, the maximum temperature in the cable core is obviously reduced, and the cable ampacity is greatly improved as compared with the cables buried around natural soil (NS). Secondly, when compared with flat-type FTB model, the heat transfer rate in the cable with parabolic-type FTB laying method would be slightly reduced, while the FTB amount used for the buried cables is greatly reduced. Finally, as for parabolic-type FTB trefoil buried cables, with proper design of geometric parameters (s1 = 0.290 m, s2 = 0.302 m, and l = 0.3 m with I = 1300 A) for the FTB laying cross section, the overall performance for the cable was optimized.


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## Body

## 1. Introduction

As global economics increases, the demand for electricity supply increases rapidly [1]. The electrical energy is mainly transported through power cables, including overhead power cable and underground power cable. The overhead lines are often used for long distance electric power transmission, while the underground line usually plays an important role in the area near the city or some other special areas. For buried cables, the costs for laying and maintenance are usually much higher than those of the overhead cables [2, 3]. Therefore, for buried cables, the safety and cost issues should be considered simultaneously. During actual operation process, the surrounding soil near the buried cable will be heated and the soil will become dryer. This will drastically reduce the soil conductivity and directly affect the heat transfer in the cable system [4, 5]. In order to avoid soil desiccation and overheating of cables, the fluidized thermal backfill material (FTB) would be filled around the cables, which would efficiently improve heat transfer rate for the buried cable system [6]. FTB material is an engineered slurry backfill mixture, which comprises fine and coarse natural mineral aggregates constituting the bulk volume of the mixture, cement providing the interparticle bond and strength, and fluidizer to impart a homogeneous fluid consistency for ease of placement [4, 6]. However, it should be noted that the cost of FTB laying method is much higher than that of natural soil (NS) laying method. Therefore, when the FTB laying method is used, not only heat transfer process in the cable system should be investigated, but the laying parameters should also be optimized to reduce the FTB laying cost.Recently, the heat transfer performance in the buried cables has been widely studied by many researchers. Ocłoń et al. [6] have numerically studied the heat transfer performance in the flat-type FTB trefoil buried cables. It was found that the heat transfer rate was efficiently improved when the FTB material was used for the buried cables. The effects of the backfill material amount and thermal conductivity on the cable ampacity were numerically studied by León and Anders [7] with the finite element method (FEM). They found that the cable ampacity would be improved as the backfill material amount and thermal conductivity increased. The soil desiccation phenomenon around buried cables has been investigated by Gouda and Dein [8]. The thermal properties of eleven different backfill materials were analysed and the most suitable backfill material was obtained. Rerak and Ocłoń [9] have numerically investigated the heat transfer performance in the trefoil buried cables with FEM method, where the thermal conductivities of soil and backfill material were related to the temperature variations. In their research, the effects of the backfill material thermal conductivity and cable core current on the maximum temperature of cables were carefully analysed. Furthermore, the moisture and temperature distributions of the backfill material surround the buried cables were numerically studied by Anders and Radhakrishna [10], and their numerical results could agree well with experimental data. In addition, researches on the optimization of laying parameters for buried cables are also popular in recent years. Ocłoń et al. [11] have optimized the laying parameters for flat-type buried cables by using improved Jaya algorithm combined with FEM method. It was found that the improved Jaya algorithm would be more accurate and efficient for the optimizations. Cichy et al. [12, 13] have optimized the laying parameters for both the flat-type and the trefoil-type buried cables by using genetic algorithm (GA). With this method, the optimum laying parameters were obtained for the minimum laying cost. Saud [14] has optimized the laying parameters for the flat-type buried cables by using particle swarm optimization method (PSO). With this method, the optimum laying parameters were obtained for different object functions, including the maximum temperature, laying cost, and ampacity of buried cables. Furthermore, Ocłoń et al. [4] have also optimized the laying parameters for the flat-type buried cables by using particle swarm optimization method (PSO). In their study, the thermal conductivities of soil and backfill material were related to the temperature variations, and the optimum laying parameters were finally obtained combined with experimental results [15].Based on above literature survey, it shows that the researches on the heat transfer performance of buried cables and the corresponding optimizations of laying parameters were popular in the recent years. However, all these researches were mainly focused on the flat-type buried cables, and the researches on the heat transfer and optimization for the trefoil-type buried cables were relatively few. Trefoil buried cable is one of the important cable arrangements for the underground transmission line. It is often used in the situation with limited construction space and its heat transfer performance is relatively poor. By filling with fluidized thermal backfill material (FTB) around trefoil buried cables, the heat transfer would be efficiently enhanced [16], while the filling cost of FTB should also be considered. In the present paper, the heat transfer process in the flat-type FTB trefoil buried cables was numerically studied first, and the results were compared with those of flat-type NS trefoil buried cables. Then, the heat transfer process in the parabolic-type FTB trefoil buried cables was numerically studied, and the results were compared with those of flat-type FTB trefoil buried cables. Finally, the laying parameters for the cross section of parabolic-type FTB trefoil buried cables were optimized with the radial basis function neural network (RBNN) and genetic algorithm (GA), and the optimum laying parameters and minimum total cost function were obtained. According to the authors’ knowledge, almost no study was performed on the heat transfer process in the parabolic-type FTB trefoil buried cables before, and the results would be meaningful for the optimal design for the FTB trefoil buried cables.


## 2. Methodology

### 2.1. Model and Geometric Parameters

The model of trefoil buried cables with fluidized thermal backfill material (FTB) is presented in Figure1. It includes the flat-type FTB trefoil buried cables (Figure 1(a)) and parabolic-type FTB trefoil buried cables (Figure 1(b)). The length (L) and height (H) of the computational domain are 20 m and 10 m, respectively. Three-phase cables are installed in a rectangular trench with width of 1.4 m. The distance between the trench top surface and the lower cable core is 1 m. As for the flat-type FTB trefoil buried cables (Figure 1(a)), the FTB thickness is h and the distance between the trench bottom surface and the lower cable core is 0.2 m. As for the parabolic-type FTB trefoil buried cables (Figure 1(b)), the distance between the FTB top point and the upper cable core is s1, the distance between the FTB bottom surface and the lower cable core iss2 and the FTB with is 2l. Besides FTB region, the other computational domain is filled with the natural soil (NS).Figure 1
Model for FTB trefoil buried cables. (a) Flat-type FTB trefoil buried cables. (b) Parabolic-type FTB trefoil buried cables.
(a) (b)The arrangement of trefoil power cables and cable structure are presented in Figure2. It shows that the trefoil power cables are stacked in a triangular arrangement (Figure 2(a)), where the cable diameter is dj. In order to improve the computational mesh quality near the contact points between cable surfaces, the cables were assumed to be stacked with very small gaps (l1 = 2%dj) instead of contact points between each other. Furthermore, as shown in Figure 2(b), the power cable is composed of copper conductor, insulation layer, sheath layer, and Jacket layer, where the corresponding radiuses are rc, ri, rs, and rj, respectively. Typical geometric and physical parameters of trefoil cables are listed in Table 1.Table 1
Typical geometric and physical parameters of trefoil cables (cable quadrature = 1600 mm2).
Cable Radius [mm] Material Thermal conductivity [W/(m·K)] Electronic conductivity [S/m] Cable core 24.8 copper 400 5.998 × 107 Insulation layer 55.3 XLPE 0.38 1.0 × 10−15 Sheath layer 61.7 copper 400 5.998 × 107 Jacket layer 66.8 HDPE 0.38 1.0 × 10−15Figure 2
Power cables in trefoil arrangement and cable structure. (a) Power cables in trefoil arrangement. (b) Power cable structure.
(a) (b)


### 2.2. Governing Equations and Computational Method

In the present study, the heat transfer in the buried cables can be regarded as two-dimensional steady heat conduction process. The power cable is composed of copper conductor, insulation layer, sheath layer, and Jacket layer, and the heat loss is produced in the copper conductor and sheath layer, named as conductor loss and eddy loss, respectively. The heat transfer equation for the computational domain is as follows:(1)k·∇2T=-qv,Cable  core  and  sheath  layer0,Insulation  layer,  jacket  layer,  NS  and  FTB  regionswhere k is the thermal conductivity. qv is the heat loss of power cable. The thermal conductivities of NS and FTB material (kNS and kFTB) are dependent on temperature variations, which are defined as follows [6, 17]: (2)kT=kdry+kwet-kdry·exp⁡-a1T-Trefa2·Tlim2a1=TlimTrefa2=1-1a1where kdry and kwet are the thermal conductivities under dry and wet conditions for NS and FTB material (Table 2). Tlim is the limited temperature of power cable (363 K). Tref is the reference temperature of surrounding soil (293 K). The variations of the thermal conductivity of NS and FTB are presented in Figure 3.Table 2
k d r y and kwet for NS and FTB material [6, 17].
Material k d r y [W/(m·K)] k w e t [W/(m·K)] NS 0.30 1.00 FTB 1.54 4.35Figure 3
Variations ofkNS and kFTB with temperature.The heat loss of power cable (qv) is defined as follows:(3)qv=J⃑2σwhere J→ is the total current density in the cable and σ is the electronic conductivity, which are defined as follows:(4)J⃑=J⃑e+J⃑sJ⃑s>0Cable  coreJ⃑s=0Sheath  layer(5)J⃑s=-σ∇φ,J⃑e=-jωσA⃑,σ=1ρ01+αT-Trefwhere J⃑e is the inductive current density. J⃑s is the source current density. A⃑,j, ω, and φ are the magnetic vector potential, unit of complex number, angular frequency, and electric scalar potential, respectively. ρ0 and α are the electrical resistivity and temperature coefficient at a reference temperature of Tref = 293 K.The boundary conditions for the computational domain are set as follows:(6)∂T∂xx=0orx=L=0,∂T∂yy=0=0,Ty=H=293KIn the present study, the above governing equations are solved with commercial code COMSOL MULTIPHYSICS, and the Pardiso solver is employed for the computations. The current frequency is set to 50 Hz. The conservative interface flux condition for heat transfer is adopted at the cable-FTB and FTB-NS interfaces, as well as the internal interfaces between different layers inside the cable, which means that the heat flux on one-side of the interface was considered to be equal to the heat flux on the other-side of the interface between different computational regions. For convergence criteria, all residuals of the calculations are less than 10−4.


## 2.1. Model and Geometric Parameters

The model of trefoil buried cables with fluidized thermal backfill material (FTB) is presented in Figure1. It includes the flat-type FTB trefoil buried cables (Figure 1(a)) and parabolic-type FTB trefoil buried cables (Figure 1(b)). The length (L) and height (H) of the computational domain are 20 m and 10 m, respectively. Three-phase cables are installed in a rectangular trench with width of 1.4 m. The distance between the trench top surface and the lower cable core is 1 m. As for the flat-type FTB trefoil buried cables (Figure 1(a)), the FTB thickness is h and the distance between the trench bottom surface and the lower cable core is 0.2 m. As for the parabolic-type FTB trefoil buried cables (Figure 1(b)), the distance between the FTB top point and the upper cable core is s1, the distance between the FTB bottom surface and the lower cable core iss2 and the FTB with is 2l. Besides FTB region, the other computational domain is filled with the natural soil (NS).Figure 1
Model for FTB trefoil buried cables. (a) Flat-type FTB trefoil buried cables. (b) Parabolic-type FTB trefoil buried cables.
(a) (b)The arrangement of trefoil power cables and cable structure are presented in Figure2. It shows that the trefoil power cables are stacked in a triangular arrangement (Figure 2(a)), where the cable diameter is dj. In order to improve the computational mesh quality near the contact points between cable surfaces, the cables were assumed to be stacked with very small gaps (l1 = 2%dj) instead of contact points between each other. Furthermore, as shown in Figure 2(b), the power cable is composed of copper conductor, insulation layer, sheath layer, and Jacket layer, where the corresponding radiuses are rc, ri, rs, and rj, respectively. Typical geometric and physical parameters of trefoil cables are listed in Table 1.Table 1
Typical geometric and physical parameters of trefoil cables (cable quadrature = 1600 mm2).
Cable Radius [mm] Material Thermal conductivity [W/(m·K)] Electronic conductivity [S/m] Cable core 24.8 copper 400 5.998 × 107 Insulation layer 55.3 XLPE 0.38 1.0 × 10−15 Sheath layer 61.7 copper 400 5.998 × 107 Jacket layer 66.8 HDPE 0.38 1.0 × 10−15Figure 2
Power cables in trefoil arrangement and cable structure. (a) Power cables in trefoil arrangement. (b) Power cable structure.
(a) (b)


## 2.2. Governing Equations and Computational Method

In the present study, the heat transfer in the buried cables can be regarded as two-dimensional steady heat conduction process. The power cable is composed of copper conductor, insulation layer, sheath layer, and Jacket layer, and the heat loss is produced in the copper conductor and sheath layer, named as conductor loss and eddy loss, respectively. The heat transfer equation for the computational domain is as follows:(1)k·∇2T=-qv,Cable  core  and  sheath  layer0,Insulation  layer,  jacket  layer,  NS  and  FTB  regionswhere k is the thermal conductivity. qv is the heat loss of power cable. The thermal conductivities of NS and FTB material (kNS and kFTB) are dependent on temperature variations, which are defined as follows [6, 17]: (2)kT=kdry+kwet-kdry·exp⁡-a1T-Trefa2·Tlim2a1=TlimTrefa2=1-1a1where kdry and kwet are the thermal conductivities under dry and wet conditions for NS and FTB material (Table 2). Tlim is the limited temperature of power cable (363 K). Tref is the reference temperature of surrounding soil (293 K). The variations of the thermal conductivity of NS and FTB are presented in Figure 3.Table 2
k d r y and kwet for NS and FTB material [6, 17].
Material k d r y [W/(m·K)] k w e t [W/(m·K)] NS 0.30 1.00 FTB 1.54 4.35Figure 3
Variations ofkNS and kFTB with temperature.The heat loss of power cable (qv) is defined as follows:(3)qv=J⃑2σwhere J→ is the total current density in the cable and σ is the electronic conductivity, which are defined as follows:(4)J⃑=J⃑e+J⃑sJ⃑s>0Cable  coreJ⃑s=0Sheath  layer(5)J⃑s=-σ∇φ,J⃑e=-jωσA⃑,σ=1ρ01+αT-Trefwhere J⃑e is the inductive current density. J⃑s is the source current density. A⃑,j, ω, and φ are the magnetic vector potential, unit of complex number, angular frequency, and electric scalar potential, respectively. ρ0 and α are the electrical resistivity and temperature coefficient at a reference temperature of Tref = 293 K.The boundary conditions for the computational domain are set as follows:(6)∂T∂xx=0orx=L=0,∂T∂yy=0=0,Ty=H=293KIn the present study, the above governing equations are solved with commercial code COMSOL MULTIPHYSICS, and the Pardiso solver is employed for the computations. The current frequency is set to 50 Hz. The conservative interface flux condition for heat transfer is adopted at the cable-FTB and FTB-NS interfaces, as well as the internal interfaces between different layers inside the cable, which means that the heat flux on one-side of the interface was considered to be equal to the heat flux on the other-side of the interface between different computational regions. For convergence criteria, all residuals of the calculations are less than 10−4.


## 3. Grid Independence Test and Model Validations

Firstly, the grid independence test was performed. In the present study, the parabolic-type FTB trefoil cables were adopted for the test (s1 = 3.5rj, l = 8rj), where the cable core current is 1145 A and the current frequency is 50 Hz. As presented in Figure 4, the self-adaptive tetrahedral mesh was used for the computations, and the grids are intensified around cable regions. In order to improve the mesh quality near the contact points between cable surfaces, according to the report of Bu et al. [18], the cables were stacked with very small gaps (l1 = 2%dj) instead of contact points between each other (Figure 2(a)). Four sets of grids were used for the test, and the computational results are presented in Table 3. It shows that the Grid-3 with total element number of 19346 is good enough for the test based on the comparison of the maximum cable temperature (Tmax) and heat flux on the cable surface (ϕ) with different grids. For Grid-3, the minimum length of the grid element in both the cable zone (Zone 1) and FTB zone (Zone 2) is 0.3 mm and it is 0.4 mm in the NS zone (Zone 3). Therefore, similar grid settings to the test grid of Grid-3 were employed for the following simulations.Table 3
Computational results with different meshes.
Grid Region Minimum length of grid element [mm] Maximum length of grid element [mm] Total element number T m a x [K] ϕ [W] Grid 1 Zone 1 1.5 0.4 10228 326.56 68.899 Zone 2 1.5 0.4 Zone 3 1.5 0.5  Grid 2 Zone 1 0.4 0.2 16337 329.557 69.892 Zone 2 1.5 0.4 Zone 3 1.5 0.4  Grid 3 Zone 1 0.3 0.1 19346 329.563 68.903 Zone 2 0.3 0.1 Zone 3 0.4 0.2  Grid 4 Zone 1 0.3 0.1 28394 329.563 68.903 Zone 2 0.3 0.1 Zone 3 0.35 0.15Figure 4
Typical computational mesh for parabolic-type FTB trefoil buried cables.Subsequently, the computational model and methods were validated. The heat transfer process in three-phase buried cables [6] was restudied and the model is presented in Figure 5. It shows that the power cables are parallel arranged and the cable core distance is 0.4 m between each other. The distance between the ground and cable core ish1. In the near-cable region, the NS is used as the backfill material, while the other regions are filled with multilayer soil. The symmetry boundary condition is adopted on the left edge of the computational domain, and the right edge and bottom surface of the computational domain are considered to be adiabatic. Furthermore, the ground temperature is fixed at 293 K. The thermal conductivities of different backfill materials are dependent on temperature variations, which can be calculated with (2) and parameters listed in Table 4.Table 4
Thermal conductivity of different buried material in Ocłoń’s study [6].
Material k d r y [W/(m·K)] k w e t [W/(m·K)] NS 0.30 1.00 MS 0.21 2.28 Gr 0.26 2.67 SCLG 0.26 2.01 SCL 0.17 1.45Figure 5
Model for validations [6].When the cable core current is fixed at 1145 A, the variations of temperature along the symmetry edge are presented in Figure6(a). It shows that the maximum temperature deviation between our present computations and those of Ocłoń et al. [6] is 1.4 K. Meanwhile, the variations of the maximum cable core temperature (Tmax) with cable buried depth (h1) are presented in Figure 6(b). It shows that the maximum deviation of Tmax between our present computations and those of Ocłoń et al. [6] is 2.3 K.Figure 6
Model validations for temperatures with Ocłoń et al. [6]. (a) Temperatures along the symmetry edge. (b) The maximum temperature of cable core.
(a) (b)


## 4. Results and Discussion

### 4.1. Performance Comparison between NS and FTB Trefoil Buried Cables

When the cable core current is fixed at 1500 A, the temperature distributions in the trefoil buried cables with NS and flat-type FTB are presented in Figure7. It shows that, for the NS trefoil buried cables (Figure 7(a)), since the thermal conductivity of NS material is relatively low (Figure 3), the heat transfer rate is relatively low and the maximum temperature of the cable core is high (Tmax = 448 K or 175°C), while for the FTB trefoil buried cables (Figures 7(b)–7(d)), since the thermal conductivity of FTB material is relatively high (Figure 3), the heat transfer rate would be significantly improved and the maximum temperature of the cable core is obviously reduced. When the thickness of FTB is fixed at 0.4 m, 0.45 m, and 0.50 m, Tmax of the cable core are 362 K (89°C), 357 K (84°C), and 353 K (80°C), respectively. The variations of the maximum cable core temperature in the trefoil buried cables with NS and flat-type FTB are presented in Figure 8. It shows that, with the same cable core current, Tmax of the cable core for the NS trefoil buried cables is obviously higher than that for the FTB trefoil buried cables and Tmax of the cable core for the FTB trefoil buried cables would be further reduced as the FTB thickness (h) increases. In the present study, it is shown that the temperature distributions inside three-phase cables are quite similar for the same working condition. Therefore, the ampacities in the three-phase cables would also be similar. Here, the cable ampacity is defined as the corresponding current in the cable core when the maximum cable core temperature reaches 90°C (363 K). It shows that the cable ampacity for the NS trefoil buried cables is 1262.3 A, while the cable ampacities for the FTB trefoil buried cables are 1513.5 A (h = 0.40 m), 1558.2 A (h = 0.45 m), 1587.2 A (h = 0.50 m), and 1616.9 A (h = 0.55 m), respectively. Therefore, it is indicated that, for trefoil buried cables, the heat transfer rate would be efficiently improved by using FTB material, and the cable ampacity would be significantly increased.Figure 7
Temperature distributions in the trefoil buried cables with NS and flat-type FTB. (I = 1500 A). (a) NS buried cables. (b) Flat-type FTB buried cables (h = 0.40 m). (c) Flat-type FTB buried cables (h = 0.45 m). (d) Flat-type FTB buried cables (h = 0.50 m).
(a) (b) (c) (d)Figure 8
Variations of the maximum cable core temperature in the trefoil buried cables with NS and flat-type FTB.The flat-type FTB trefoil buried cables as mentioned above (Figure1(a)) are quite easy to be constructed. However, this FTB laying method needs large amount of FTB laying materials, which would lead to a certain waste. Therefore, in order to reduce the FTB amount used for the trefoil buried cables, the parabolic-type FTB laying method (Figure 1(b)) is adopted, and the heat transfer performance and ampacity of the parabolic-type FTB trefoil buried cables are analysed and compared with those of the flat-type FTB trefoil buried cables. Typical geometric parameters for flat-type FTB and parabolic-type FTB trefoil buried cables are presented in Table 5. It shows that, in the present study, the cross section area for the parabolic-type FTB model (AFTB) is much lower than that for the flat-type FTB model.Table 5
Geometric parameters for flat-type FTB and parabolic-type FTB trefoil buried cables.
FTB laying model s 1 [m] s 2 [m] l [m] h [m] A F T B [m2] FTB (Parabolic) 3.5rj 0.2 8rj / 0.351 FTB (Flat) / / / 0.4 0.560When the cable core current is fixed at 1500 A, the temperature distributions in the flat-type FTB and parabolic-type FTB trefoil buried cables are presented in Figure9. It shows that the temperature distributions in the flat-type FTB and parabolic-type FTB trefoil buried cables are similar, where the corresponding Tmax of the cable core are 362 K (89°C) and 368K (95°C), respectively. This indicates, with parabolic-type FTB laying method, the heat transfer rate in the cable would almost keep the same, while the FTB amount (represented by AFTB) used for the trefoil buried cables can be greatly reduced.Figure 9
Temperature distributions in flat-type FTB and parabolic-type FTB trefoil buried cables. (I = 1500 A). (a) Flat-type FTB buried cables (AFTB = 0.560 m2). (b) Parabolic-type FTB buried cables. (AFTB = 0.351 m2).
(a)
Flat-type FTB buried cables (AFTB = 0.560 m2) (b)
Parabolic-type FTB buried cables (AFTB = 0.351 m2)The variations of the maximum cable core temperature (Tmax) in the flat-type FTB and parabolic-type FTB trefoil buried cables are presented in Figure 10. It shows that, with the same cable core current, Tmax of cable core for the parabolic-type FTB model is little higher than that for the flat-type FTB model, and the corresponding ampacities are 1477.5 A (AFTB = 0.351 m2) and 1513.5 A (AFTB = 0.560 m2), respectively. This indicates that compared with flat-type FTB model, the ampacity in the cable is slightly reduced with parabolic-type FTB laying method (reduced by 2.4% in the present study), while the FTB amount used for the buried cables (represented by AFTB) is greatly reduced (reduced by 37.3% in the present study, Table 5). Therefore, the parabolic-type FTB laying method is recommended for the trefoil buried cable applications.Figure 10
Variations of the maximum cable core temperature in flat-type FTB and parabolic-type FTB trefoil buried cables.


### 4.2. Laying Parameter Optimization for Parabolic-Type FTB Trefoil Buried Cables

The laying parameters for the parabolic-type FTB trefoil buried cables are presented in Figure11. It shows that the FTB laying area (AFTB) or FTB amount is closely related to the FTB laying parameters, mainly including s1, s2, and l. In order to enhance the heat transfer in trefoil buried cables and reduce the use of FTB material, the laying parameters should be optimized for the parabolic-type FTB trefoil buried cables. In the present study, the total cost function (Fc) as reported by Ocłoń et al. [11] is adopted to access the overall performance for trefoil buried cables, which is defined as follows:(7)Fcl,s1,s2=AFTB+PFTmax-Topt>0Topt=338KAFTBTmax-Topt≤0Topt=338Kwhere AFTB is the FTB laying area. Tmax is the maximum temperature of cable core. Topt is the optimum working temperature for the cable (Topt = 338 K or 65°C). PF is the penalty function. When the value of Tmax is higher than that of Topt (338 K), the penalty function (PF) should be set to guarantee the rationality of total cost function (Fc), where the PF is defined as follows:(8)PF=10×Tmax-ToptTherefore, it should be noted that when the value of Tmax is lower than that of Topt, the total cost function (Fc) is depended on AFTB. Otherwise, Fc should be both related to AFTB and Tmax.Figure 11
The geometric parameters of parabolic-type FTB trefoil buried cables.In the present study, three geometric parameters of FTB laying cross section, includings1, s2, and l, were selected as the design variables, and the total cost function (Fc) is adopted as the objective function for the optimization. The flowchart for the optimization procedure is presented in Figure 12. Firstly, based on the design variables (s1, s2, and l), the geometric parameter combination samples were designed with the Latin hypercube sampling method (LHS) [19, 20]. LHS is an effective method to design sample points within the design space, by which the design points are equally spaced of each variable in design space [19]. When sampling a function of n variables, the range of each variable is divided into m equal probable intervals. For each variable, design points are randomly selected from each interval. Based on the m points for each variable, the sample points are obtained by the random combination. Hence, the design points are equally spaced of each variable in design space [19]. With this method, 50 sets of random samples were selected as the design samples for the optimizations, which are presented in Table 6. As shown in Table 6, both the ranges of s1 and s2 are from 0.1 m to 0.4 m, while the range of l is from 0.3 m to 0.7 m. Based on these 50 sets of samples, the heat transfer processes in the parabolic-type FTB trefoil buried cables were simulated with commercial code COMSOL MULTIPHYSICS. Then, the maximum temperature (Tmax) of the cable core and the total cost function (Fc) for the corresponding samples were calculated based on the simulation results, which are also presented in Table 6.Table 6
Design samples with LHS method and the computational results ofTmax and Fc (I = 1300 A).
Samples s 1 [m] s 2 [m] l [m] A F T B [m2] T m a x [K] F c 1 0.3843 0.3769 0.3117 0.3233 332.3360 0.3233 2 0.3181 0.3218 0.3565 0.3183 334.6860 0.3183 3 0.1562 0.2716 0.6180 0.4077 355.5410 175.8177 4 0.3818 0.3014 0.5370 0.5316 330.0740 0.5316 5 0.2238 0.2232 0.4746 0.3155 338.1350 1.6655 6 0.1128 0.2896 0.3806 0.2221 342.2830 43.0521 7 0.1604 0.3373 0.5287 0.3921 334.6920 0.3921 8 0.3733 0.1883 0.3684 0.2918 338.1560 1.8518 9 0.3564 0.1961 0.4594 0.3687 335.8680 0.3687 10 0.3014 0.2172 0.4194 0.3140 337.4600 0.3140 11 0.2902 0.3613 0.6789 0.6546 328.3190 0.6546 12 0.1001 0.1387 0.6094 0.2479 343.8940 59.1879 13 0.1322 0.1752 0.4716 0.2255 343.8250 58.4755 14 0.1508 0.2070 0.5538 0.3093 339.4900 15.2093 15 0.3458 0.2273 0.4407 0.3641 335.3500 0.3641 16 0.2356 0.3250 0.4092 0.3281 335.4600 0.3281 17 0.1934 0.3718 0.4042 0.3261 335.1620 0.3261 18 0.1763 0.1190 0.5951 0.2859 341.9510 39.7959 19 0.1854 0.1328 0.5425 0.2735 342.0180 40.4535 20 0.2569 0.1070 0.3248 0.1667 347.9650 99.8167 21 0.1383 0.1705 0.3164 0.1380 349.9190 119.3280 22 0.2917 0.1161 0.3972 0.2364 343.2150 52.3864 23 0.3084 0.2500 0.5218 0.4286 333.7690 0.4286 24 0.3242 0.1794 0.4314 0.3155 338.0940 1.2555 25 0.2048 0.3864 0.6011 0.5264 330.6140 0.5264 26 0.3900 0.3173 0.6881 0.7151 327.4050 0.7151 27 0.1692 0.2671 0.6513 0.4393 334.7680 0.4393 28 0.3359 0.1279 0.5774 0.4059 336.8790 0.4059 29 0.2382 0.2923 0.6391 0.5106 332.1900 0.5106 30 0.3548 0.3105 0.6294 0.6152 329.0470 0.6152 31 0.2127 0.3023 0.5580 0.4290 333.8130 0.4290 32 0.3296 0.2413 0.6464 0.5518 335.1400 0.5518 33 0.2035 0.1481 0.5005 0.2713 341.5770 36.0413 34 0.3635 0.3457 0.5641 0.5801 328.6610 0.5801 35 0.3443 0.1867 0.3929 0.2980 338.2780 3.0780 36 0.2591 0.2779 0.4456 0.3472 335.6430 0.3472 37 0.2461 0.3439 0.3002 0.2413 337.7180 0.2413 38 0.2195 0.2851 0.4870 0.3622 335.5940 0.3622 39 0.2712 0.2026 0.3484 0.2329 341.1970 32.2029 40 0.2303 0.3541 0.5037 0.4297 332.5610 0.4297 41 0.1175 0.1257 0.3597 0.1312 351.9180 139.3112 42 0.2642 0.3307 0.6724 0.5971 329.9080 0.5971 43 0.3056 0.3823 0.6227 0.6271 328.0290 0.6271 44 0.1912 0.2420 0.3305 0.2009 342.9110 49.3109 45 0.2527 0.2565 0.6677 0.5163 332.6100 0.5163 46 0.1246 0.1592 0.3740 0.1583 348.6010 106.1683 47 0.2746 0.3970 0.3427 0.3188 333.4970 0.3188 48 0.1778 0.3630 0.5872 0.4737 332.2980 0.4737 49 0.1092 0.2612 0.6948 0.4105 336.4770 0.4105 50 0.3695 0.2309 0.6618 0.5919 330.8430 0.5919Figure 12
Flow chart for the optimization procedure.Secondly, based on the sample data and calculation results as presented in Table6, the geometric parameters for the parabolic-type FTB trefoil buried cables (s1, s2, and l) were trained with the radial basis function neural network (RBNN) [19–21]. RBNN is a two-layer network with strong ability to approximate objective function and rapid convergence rate, including a hidden layer of radial basis function (RBF) and a linear output layer [19]. Figure 13 shows the model of radial basis function neuron. The input n to the radial basis function is the vector distance between the input vector x and the weight vector w, multiplied with a threshold value b.(9)n=dist·b=w-x·bGaussian function is employed as the radial basis function:(10)y=radbasn=e-n2When n approaches zero, which means the input vector x and weight vector w are approximative, the output of the radial basis function will be a relatively large value close to 1; thus the network has a strong ability of local perception. Figure 14 shows the whole structure of the radial basis neural network. The output of the network is a weighted sum of the outputs of hidden layers. The RBNN is established by using mathematical software MATLAB. The function of RBNN in MATLAB is newrb. Besides the sample data of inputs and outputs, a spread constant and the maximum number of neurons need to be determined. Spread constant is related to the threshold value b that controls the sensitivity of radial basis function neuron. Its value should be large enough to ensure the response within all the design space. However it should not be so large that all the neurons have significant response.Figure 13
The model of radial basis function neuron.Figure 14
The structure of radial basis neural network.The maximum temperature (Tmax) of the cable core was predicted with RBNN and validated with commercial code COMSOL MULTIPHYSICS. The predicted values of Tmax with RBNN and COMSOL MULTIPHYSICS for 10 sets of random samples are presented in Table 7. It is found that the maximum deviation of Tmax with RBNN and COMSOL MULTIPHYSICS is 2 K. Therefore, the prediction results with RBNN for the present study are reliable.Table 7
The comparison ofTmax with RBNN and COMSOL (I = 1300 A).
Samples Design variables T m a x [K] s 1 [m] s 2 [m] l [m] RBNN COMSOL 1 0.3152 0.2035 0.3393 340.7534 340.095 2 0.1936 0.1294 0.5348 342.1368 342.075 3 0.3079 0.3779 0.4763 328.3667 330.314 4 0.3557 0.2250 0.5911 332.6455 332.348 5 0.2578 0.2849 0.5785 333.4514 332.856 6 0.2280 0.2790 0.4530 336.7406 336.332 7 0.3755 0.3126 0.4107 332.0261 332.229 8 0.1740 0.3672 0.6314 333.6398 331.612 9 0.1432 0.1645 0.6692 340.7285 339.486 10 0.1196 0.1443 0.3564 349.3775 350.652Finally, combined with RBNN, the genetic algorithm (GA) [22, 23] is used to optimize the geometric parameters for the parabolic-type FTB trefoil buried cables (s1, s2, and l), where the total cost function (Fc) is adopted as the objective function for the optimizations. Genetic algorithm (GA) can be easily used to get the maximum value or minimum value of a specific function. In the optimization, when GA is performed, the RBNN acts as the objective function to be minimized. GA uses the theory of biological evolution, and it obtains the optimized point by a number of generations evolving from the initial population. Individuals of each generation are decided by the value of fitness function of each individual of the last generation and the randomness of selection, crossover, and mutation. The standard genetic algorithm (SGA) is conducted using the genetic algorithm toolbox of MATLAB. The predicted results of Tmax with GA were also validated with commercial code COMSOL MULTIPHYSICS. When the cable core current is fixed at 1300 A, the optimum results with GA for the parabolic-type FTB trefoil buried cables are presented in Table 8. It shows that the values of the optimum geometric parameters (s1, s2, and l) of FTB cross section with GA are almost the same at populations of 100 and 200. When the population number is 200, the optimum values of s1, s2, and l are 0.290 m, 0.302 m, and 0.3 m, respectively, and the maximum temperature (Tmax) of the cable core is 337.998 K. The deviation of Tmax predicted with GA and COMSOL MULTIPHYSICS is 0.078 K, and Tmax is also quite close to the optimum cable working temperature (Topt = 338K or 65°C). With the optimum geometric parameters (s1 = 0.290 m, s2 = 0.302 m, and l = 0.3 m), the minimum total cost function (Fc) is 0.2420 for parabolic-type FTB trefoil buried cables.Table 8
The optimum results with GA for parabolic-type FTB trefoil buried cables (I = 1300 A).
Population Optimal design variables (GA) T m a x [K] Fc s 1 [m] s 2 [m] l [m] GA COMSOL GA 100 0.298 0.303 0.3 337.799 337.675 0.2456 200 0.290 0.302 0.3 337.998 338.076 0.2420


## 4.1. Performance Comparison between NS and FTB Trefoil Buried Cables

When the cable core current is fixed at 1500 A, the temperature distributions in the trefoil buried cables with NS and flat-type FTB are presented in Figure7. It shows that, for the NS trefoil buried cables (Figure 7(a)), since the thermal conductivity of NS material is relatively low (Figure 3), the heat transfer rate is relatively low and the maximum temperature of the cable core is high (Tmax = 448 K or 175°C), while for the FTB trefoil buried cables (Figures 7(b)–7(d)), since the thermal conductivity of FTB material is relatively high (Figure 3), the heat transfer rate would be significantly improved and the maximum temperature of the cable core is obviously reduced. When the thickness of FTB is fixed at 0.4 m, 0.45 m, and 0.50 m, Tmax of the cable core are 362 K (89°C), 357 K (84°C), and 353 K (80°C), respectively. The variations of the maximum cable core temperature in the trefoil buried cables with NS and flat-type FTB are presented in Figure 8. It shows that, with the same cable core current, Tmax of the cable core for the NS trefoil buried cables is obviously higher than that for the FTB trefoil buried cables and Tmax of the cable core for the FTB trefoil buried cables would be further reduced as the FTB thickness (h) increases. In the present study, it is shown that the temperature distributions inside three-phase cables are quite similar for the same working condition. Therefore, the ampacities in the three-phase cables would also be similar. Here, the cable ampacity is defined as the corresponding current in the cable core when the maximum cable core temperature reaches 90°C (363 K). It shows that the cable ampacity for the NS trefoil buried cables is 1262.3 A, while the cable ampacities for the FTB trefoil buried cables are 1513.5 A (h = 0.40 m), 1558.2 A (h = 0.45 m), 1587.2 A (h = 0.50 m), and 1616.9 A (h = 0.55 m), respectively. Therefore, it is indicated that, for trefoil buried cables, the heat transfer rate would be efficiently improved by using FTB material, and the cable ampacity would be significantly increased.Figure 7
Temperature distributions in the trefoil buried cables with NS and flat-type FTB. (I = 1500 A). (a) NS buried cables. (b) Flat-type FTB buried cables (h = 0.40 m). (c) Flat-type FTB buried cables (h = 0.45 m). (d) Flat-type FTB buried cables (h = 0.50 m).
(a) (b) (c) (d)Figure 8
Variations of the maximum cable core temperature in the trefoil buried cables with NS and flat-type FTB.The flat-type FTB trefoil buried cables as mentioned above (Figure1(a)) are quite easy to be constructed. However, this FTB laying method needs large amount of FTB laying materials, which would lead to a certain waste. Therefore, in order to reduce the FTB amount used for the trefoil buried cables, the parabolic-type FTB laying method (Figure 1(b)) is adopted, and the heat transfer performance and ampacity of the parabolic-type FTB trefoil buried cables are analysed and compared with those of the flat-type FTB trefoil buried cables. Typical geometric parameters for flat-type FTB and parabolic-type FTB trefoil buried cables are presented in Table 5. It shows that, in the present study, the cross section area for the parabolic-type FTB model (AFTB) is much lower than that for the flat-type FTB model.Table 5
Geometric parameters for flat-type FTB and parabolic-type FTB trefoil buried cables.
FTB laying model s 1 [m] s 2 [m] l [m] h [m] A F T B [m2] FTB (Parabolic) 3.5rj 0.2 8rj / 0.351 FTB (Flat) / / / 0.4 0.560When the cable core current is fixed at 1500 A, the temperature distributions in the flat-type FTB and parabolic-type FTB trefoil buried cables are presented in Figure9. It shows that the temperature distributions in the flat-type FTB and parabolic-type FTB trefoil buried cables are similar, where the corresponding Tmax of the cable core are 362 K (89°C) and 368K (95°C), respectively. This indicates, with parabolic-type FTB laying method, the heat transfer rate in the cable would almost keep the same, while the FTB amount (represented by AFTB) used for the trefoil buried cables can be greatly reduced.Figure 9
Temperature distributions in flat-type FTB and parabolic-type FTB trefoil buried cables. (I = 1500 A). (a) Flat-type FTB buried cables (AFTB = 0.560 m2). (b) Parabolic-type FTB buried cables. (AFTB = 0.351 m2).
(a)
Flat-type FTB buried cables (AFTB = 0.560 m2) (b)
Parabolic-type FTB buried cables (AFTB = 0.351 m2)The variations of the maximum cable core temperature (Tmax) in the flat-type FTB and parabolic-type FTB trefoil buried cables are presented in Figure 10. It shows that, with the same cable core current, Tmax of cable core for the parabolic-type FTB model is little higher than that for the flat-type FTB model, and the corresponding ampacities are 1477.5 A (AFTB = 0.351 m2) and 1513.5 A (AFTB = 0.560 m2), respectively. This indicates that compared with flat-type FTB model, the ampacity in the cable is slightly reduced with parabolic-type FTB laying method (reduced by 2.4% in the present study), while the FTB amount used for the buried cables (represented by AFTB) is greatly reduced (reduced by 37.3% in the present study, Table 5). Therefore, the parabolic-type FTB laying method is recommended for the trefoil buried cable applications.Figure 10
Variations of the maximum cable core temperature in flat-type FTB and parabolic-type FTB trefoil buried cables.


## 4.2. Laying Parameter Optimization for Parabolic-Type FTB Trefoil Buried Cables

The laying parameters for the parabolic-type FTB trefoil buried cables are presented in Figure11. It shows that the FTB laying area (AFTB) or FTB amount is closely related to the FTB laying parameters, mainly including s1, s2, and l. In order to enhance the heat transfer in trefoil buried cables and reduce the use of FTB material, the laying parameters should be optimized for the parabolic-type FTB trefoil buried cables. In the present study, the total cost function (Fc) as reported by Ocłoń et al. [11] is adopted to access the overall performance for trefoil buried cables, which is defined as follows:(7)Fcl,s1,s2=AFTB+PFTmax-Topt>0Topt=338KAFTBTmax-Topt≤0Topt=338Kwhere AFTB is the FTB laying area. Tmax is the maximum temperature of cable core. Topt is the optimum working temperature for the cable (Topt = 338 K or 65°C). PF is the penalty function. When the value of Tmax is higher than that of Topt (338 K), the penalty function (PF) should be set to guarantee the rationality of total cost function (Fc), where the PF is defined as follows:(8)PF=10×Tmax-ToptTherefore, it should be noted that when the value of Tmax is lower than that of Topt, the total cost function (Fc) is depended on AFTB. Otherwise, Fc should be both related to AFTB and Tmax.Figure 11
The geometric parameters of parabolic-type FTB trefoil buried cables.In the present study, three geometric parameters of FTB laying cross section, includings1, s2, and l, were selected as the design variables, and the total cost function (Fc) is adopted as the objective function for the optimization. The flowchart for the optimization procedure is presented in Figure 12. Firstly, based on the design variables (s1, s2, and l), the geometric parameter combination samples were designed with the Latin hypercube sampling method (LHS) [19, 20]. LHS is an effective method to design sample points within the design space, by which the design points are equally spaced of each variable in design space [19]. When sampling a function of n variables, the range of each variable is divided into m equal probable intervals. For each variable, design points are randomly selected from each interval. Based on the m points for each variable, the sample points are obtained by the random combination. Hence, the design points are equally spaced of each variable in design space [19]. With this method, 50 sets of random samples were selected as the design samples for the optimizations, which are presented in Table 6. As shown in Table 6, both the ranges of s1 and s2 are from 0.1 m to 0.4 m, while the range of l is from 0.3 m to 0.7 m. Based on these 50 sets of samples, the heat transfer processes in the parabolic-type FTB trefoil buried cables were simulated with commercial code COMSOL MULTIPHYSICS. Then, the maximum temperature (Tmax) of the cable core and the total cost function (Fc) for the corresponding samples were calculated based on the simulation results, which are also presented in Table 6.Table 6
Design samples with LHS method and the computational results ofTmax and Fc (I = 1300 A).
Samples s 1 [m] s 2 [m] l [m] A F T B [m2] T m a x [K] F c 1 0.3843 0.3769 0.3117 0.3233 332.3360 0.3233 2 0.3181 0.3218 0.3565 0.3183 334.6860 0.3183 3 0.1562 0.2716 0.6180 0.4077 355.5410 175.8177 4 0.3818 0.3014 0.5370 0.5316 330.0740 0.5316 5 0.2238 0.2232 0.4746 0.3155 338.1350 1.6655 6 0.1128 0.2896 0.3806 0.2221 342.2830 43.0521 7 0.1604 0.3373 0.5287 0.3921 334.6920 0.3921 8 0.3733 0.1883 0.3684 0.2918 338.1560 1.8518 9 0.3564 0.1961 0.4594 0.3687 335.8680 0.3687 10 0.3014 0.2172 0.4194 0.3140 337.4600 0.3140 11 0.2902 0.3613 0.6789 0.6546 328.3190 0.6546 12 0.1001 0.1387 0.6094 0.2479 343.8940 59.1879 13 0.1322 0.1752 0.4716 0.2255 343.8250 58.4755 14 0.1508 0.2070 0.5538 0.3093 339.4900 15.2093 15 0.3458 0.2273 0.4407 0.3641 335.3500 0.3641 16 0.2356 0.3250 0.4092 0.3281 335.4600 0.3281 17 0.1934 0.3718 0.4042 0.3261 335.1620 0.3261 18 0.1763 0.1190 0.5951 0.2859 341.9510 39.7959 19 0.1854 0.1328 0.5425 0.2735 342.0180 40.4535 20 0.2569 0.1070 0.3248 0.1667 347.9650 99.8167 21 0.1383 0.1705 0.3164 0.1380 349.9190 119.3280 22 0.2917 0.1161 0.3972 0.2364 343.2150 52.3864 23 0.3084 0.2500 0.5218 0.4286 333.7690 0.4286 24 0.3242 0.1794 0.4314 0.3155 338.0940 1.2555 25 0.2048 0.3864 0.6011 0.5264 330.6140 0.5264 26 0.3900 0.3173 0.6881 0.7151 327.4050 0.7151 27 0.1692 0.2671 0.6513 0.4393 334.7680 0.4393 28 0.3359 0.1279 0.5774 0.4059 336.8790 0.4059 29 0.2382 0.2923 0.6391 0.5106 332.1900 0.5106 30 0.3548 0.3105 0.6294 0.6152 329.0470 0.6152 31 0.2127 0.3023 0.5580 0.4290 333.8130 0.4290 32 0.3296 0.2413 0.6464 0.5518 335.1400 0.5518 33 0.2035 0.1481 0.5005 0.2713 341.5770 36.0413 34 0.3635 0.3457 0.5641 0.5801 328.6610 0.5801 35 0.3443 0.1867 0.3929 0.2980 338.2780 3.0780 36 0.2591 0.2779 0.4456 0.3472 335.6430 0.3472 37 0.2461 0.3439 0.3002 0.2413 337.7180 0.2413 38 0.2195 0.2851 0.4870 0.3622 335.5940 0.3622 39 0.2712 0.2026 0.3484 0.2329 341.1970 32.2029 40 0.2303 0.3541 0.5037 0.4297 332.5610 0.4297 41 0.1175 0.1257 0.3597 0.1312 351.9180 139.3112 42 0.2642 0.3307 0.6724 0.5971 329.9080 0.5971 43 0.3056 0.3823 0.6227 0.6271 328.0290 0.6271 44 0.1912 0.2420 0.3305 0.2009 342.9110 49.3109 45 0.2527 0.2565 0.6677 0.5163 332.6100 0.5163 46 0.1246 0.1592 0.3740 0.1583 348.6010 106.1683 47 0.2746 0.3970 0.3427 0.3188 333.4970 0.3188 48 0.1778 0.3630 0.5872 0.4737 332.2980 0.4737 49 0.1092 0.2612 0.6948 0.4105 336.4770 0.4105 50 0.3695 0.2309 0.6618 0.5919 330.8430 0.5919Figure 12
Flow chart for the optimization procedure.Secondly, based on the sample data and calculation results as presented in Table6, the geometric parameters for the parabolic-type FTB trefoil buried cables (s1, s2, and l) were trained with the radial basis function neural network (RBNN) [19–21]. RBNN is a two-layer network with strong ability to approximate objective function and rapid convergence rate, including a hidden layer of radial basis function (RBF) and a linear output layer [19]. Figure 13 shows the model of radial basis function neuron. The input n to the radial basis function is the vector distance between the input vector x and the weight vector w, multiplied with a threshold value b.(9)n=dist·b=w-x·bGaussian function is employed as the radial basis function:(10)y=radbasn=e-n2When n approaches zero, which means the input vector x and weight vector w are approximative, the output of the radial basis function will be a relatively large value close to 1; thus the network has a strong ability of local perception. Figure 14 shows the whole structure of the radial basis neural network. The output of the network is a weighted sum of the outputs of hidden layers. The RBNN is established by using mathematical software MATLAB. The function of RBNN in MATLAB is newrb. Besides the sample data of inputs and outputs, a spread constant and the maximum number of neurons need to be determined. Spread constant is related to the threshold value b that controls the sensitivity of radial basis function neuron. Its value should be large enough to ensure the response within all the design space. However it should not be so large that all the neurons have significant response.Figure 13
The model of radial basis function neuron.Figure 14
The structure of radial basis neural network.The maximum temperature (Tmax) of the cable core was predicted with RBNN and validated with commercial code COMSOL MULTIPHYSICS. The predicted values of Tmax with RBNN and COMSOL MULTIPHYSICS for 10 sets of random samples are presented in Table 7. It is found that the maximum deviation of Tmax with RBNN and COMSOL MULTIPHYSICS is 2 K. Therefore, the prediction results with RBNN for the present study are reliable.Table 7
The comparison ofTmax with RBNN and COMSOL (I = 1300 A).
Samples Design variables T m a x [K] s 1 [m] s 2 [m] l [m] RBNN COMSOL 1 0.3152 0.2035 0.3393 340.7534 340.095 2 0.1936 0.1294 0.5348 342.1368 342.075 3 0.3079 0.3779 0.4763 328.3667 330.314 4 0.3557 0.2250 0.5911 332.6455 332.348 5 0.2578 0.2849 0.5785 333.4514 332.856 6 0.2280 0.2790 0.4530 336.7406 336.332 7 0.3755 0.3126 0.4107 332.0261 332.229 8 0.1740 0.3672 0.6314 333.6398 331.612 9 0.1432 0.1645 0.6692 340.7285 339.486 10 0.1196 0.1443 0.3564 349.3775 350.652Finally, combined with RBNN, the genetic algorithm (GA) [22, 23] is used to optimize the geometric parameters for the parabolic-type FTB trefoil buried cables (s1, s2, and l), where the total cost function (Fc) is adopted as the objective function for the optimizations. Genetic algorithm (GA) can be easily used to get the maximum value or minimum value of a specific function. In the optimization, when GA is performed, the RBNN acts as the objective function to be minimized. GA uses the theory of biological evolution, and it obtains the optimized point by a number of generations evolving from the initial population. Individuals of each generation are decided by the value of fitness function of each individual of the last generation and the randomness of selection, crossover, and mutation. The standard genetic algorithm (SGA) is conducted using the genetic algorithm toolbox of MATLAB. The predicted results of Tmax with GA were also validated with commercial code COMSOL MULTIPHYSICS. When the cable core current is fixed at 1300 A, the optimum results with GA for the parabolic-type FTB trefoil buried cables are presented in Table 8. It shows that the values of the optimum geometric parameters (s1, s2, and l) of FTB cross section with GA are almost the same at populations of 100 and 200. When the population number is 200, the optimum values of s1, s2, and l are 0.290 m, 0.302 m, and 0.3 m, respectively, and the maximum temperature (Tmax) of the cable core is 337.998 K. The deviation of Tmax predicted with GA and COMSOL MULTIPHYSICS is 0.078 K, and Tmax is also quite close to the optimum cable working temperature (Topt = 338K or 65°C). With the optimum geometric parameters (s1 = 0.290 m, s2 = 0.302 m, and l = 0.3 m), the minimum total cost function (Fc) is 0.2420 for parabolic-type FTB trefoil buried cables.Table 8
The optimum results with GA for parabolic-type FTB trefoil buried cables (I = 1300 A).
Population Optimal design variables (GA) T m a x [K] Fc s 1 [m] s 2 [m] l [m] GA COMSOL GA 100 0.298 0.303 0.3 337.799 337.675 0.2456 200 0.290 0.302 0.3 337.998 338.076 0.2420


## 5. Conclusions

In the present paper, the heat transfer process in the parabolic-type FTB trefoil buried cables is numerically studied, where the cable core loss and eddy current loss in the cable were coupled for the simulation, and the thermal conductivity of the backfill material was considered to be related to temperature variations. The heat transfer performances and ampacities for the flat-type NS, flat-type FTB, and parabolic-type FTB trefoil buried cables were carefully analysed and compared with each other. Then, the laying parameters for the cross section of parabolic-type FTB trefoil buried cables were optimized with the radial basis function neural network (RBNN) and genetic algorithm (GA). The main findings are as follows:(1) Since the thermal conductivity of FTB material is relatively high, the heat transfer rate in the FTB trefoil buried cables would be significantly improved and the maximum temperature in the cable core is obviously reduced. Therefore, the ampacity for the FTB trefoil buried cables is much higher than that for the NS trefoil buried cables.(2) When compared with flat-type FTB model, the heat transfer rate in the cable with parabolic-type FTB laying method would be slightly reduced, while the FTB amount used for the buried cables would be greatly reduced. When the cable core current is fixed at 1500 A, the ampacity in the cable is only reduced by 2.4% with parabolic-type FTB laying method, while the FTB amount is reduced by 37.3%. Therefore, the parabolic-type FTB laying method is recommended for the trefoil buried cable applications.(3) For parabolic-type FTB trefoil buried cables, with proper design of geometric parameters (s1, s2, and l) for the FTB laying cross section, the overall performance of the cable would be optimized. When the cable core current is fixed at 1300 A, the optimum values of s1, s2, and l are 0.290 m, 0.302 m, and 0.3 m, respectively, the maximum temperature (Tmax) of the cable core is 337.998 K, and the minimum total cost function (Fc) is 0.2420.



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*Source: 4741871-2019-02-25.xml*