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 ( = 0.290 m, = 0.302 m, and = 0.3 m with = 1300 A) for the FTB laying cross section, the overall performance for the cable was optimized.
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 Figure 1. It includes the flat-type FTB trefoil buried cables (Figure 1(a)) and parabolic-type FTB trefoil buried cables (Figure 1(b)). The length () 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 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 , the distance between the FTB bottom surface and the lower cable core is s2 and the FTB with is . Besides FTB region, the other computational domain is filled with the natural soil (NS).
(a)
(b)
The arrangement of trefoil power cables and cable structure are presented in Figure 2. It shows that the trefoil power cables are stacked in a triangular arrangement (Figure 2(a)), where the cable diameter is . 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 ( = 2%) 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 , , , and , respectively. Typical geometric and physical parameters of trefoil cables are listed in Table 1.
(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:where is the thermal conductivity. is the heat loss of power cable. The thermal conductivities of NS and FTB material ( and ) are dependent on temperature variations, which are defined as follows [6, 17]: where and are the thermal conductivities under dry and wet conditions for NS and FTB material (Table 2). is the limited temperature of power cable (363 K). is the reference temperature of surrounding soil (293 K). The variations of the thermal conductivity of NS and FTB are presented in Figure 3.
The heat loss of power cable () is defined as follows:where is the total current density in the cable and is the electronic conductivity, which are defined as follows:where is the inductive current density. is the source current density. , j, , and are the magnetic vector potential, unit of complex number, angular frequency, and electric scalar potential, respectively. and are the electrical resistivity and temperature coefficient at a reference temperature of = 293 K.
The boundary conditions for the computational domain are set as follows:In 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 ( = 3.5, = 8), 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 ( = 2%) 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 () and heat flux on the cable surface () 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.
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 is h1. 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.
When the cable core current is fixed at 1145 A, the variations of temperature along the symmetry edge are presented in Figure 6(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 () with cable buried depth () are presented in Figure 6(b). It shows that the maximum deviation of between our present computations and those of Ocłoń et al. [6] is 2.3 K.
(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 Figure 7. 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 ( = 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, 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, of the cable core for the NS trefoil buried cables is obviously higher than that for the FTB trefoil buried cables and of the cable core for the FTB trefoil buried cables would be further reduced as the FTB thickness () 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 ( = 0.40 m), 1558.2 A ( = 0.45 m), 1587.2 A ( = 0.50 m), and 1616.9 A ( = 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.
(a)
(b)
(c)
(d)
The flat-type FTB trefoil buried cables as mentioned above (Figure 1(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 () is much lower than that for the flat-type FTB model.
When 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 Figure 9. It shows that the temperature distributions in the flat-type FTB and parabolic-type FTB trefoil buried cables are similar, where the corresponding 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 ) used for the trefoil buried cables can be greatly reduced.
(a) Flat-type FTB buried cables ( = 0.560 m2)
(b) Parabolic-type FTB buried cables ( = 0.351 m2)
The variations of the maximum cable core temperature () 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, 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 ( = 0.351 m2) and 1513.5 A ( = 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 ) 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.
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 Figure 11. It shows that the FTB laying area () or FTB amount is closely related to the FTB laying parameters, mainly including , , and . 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 () as reported by Ocłoń et al. [11] is adopted to access the overall performance for trefoil buried cables, which is defined as follows:where is the FTB laying area. is the maximum temperature of cable core. is the optimum working temperature for the cable ( = 338 K or 65°C). PF is the penalty function. When the value of is higher than that of (338 K), the penalty function (PF) should be set to guarantee the rationality of total cost function (), where the PF is defined as follows:Therefore, it should be noted that when the value of is lower than that of , the total cost function () is depended on . Otherwise, should be both related to and .
In the present study, three geometric parameters of FTB laying cross section, including , , and , were selected as the design variables, and the total cost function () 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 (, , and ), 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 variables, the range of each variable is divided into equal probable intervals. For each variable, design points are randomly selected from each interval. Based on the 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 and are from 0.1 m to 0.4 m, while the range of 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 () of the cable core and the total cost function () for the corresponding samples were calculated based on the simulation results, which are also presented in Table 6.
Secondly, based on the sample data and calculation results as presented in Table 6, the geometric parameters for the parabolic-type FTB trefoil buried cables (, , and ) 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 to the radial basis function is the vector distance between the input vector and the weight vector , multiplied with a threshold value .Gaussian function is employed as the radial basis function:When approaches zero, which means the input vector and weight vector 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 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.
The maximum temperature () of the cable core was predicted with RBNN and validated with commercial code COMSOL MULTIPHYSICS. The predicted values of with RBNN and COMSOL MULTIPHYSICS for 10 sets of random samples are presented in Table 7. It is found that the maximum deviation of with RBNN and COMSOL MULTIPHYSICS is 2 K. Therefore, the prediction results with RBNN for the present study are reliable.
Finally, combined with RBNN, the genetic algorithm (GA) [22, 23] is used to optimize the geometric parameters for the parabolic-type FTB trefoil buried cables (, , and ), where the total cost function () 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 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 (, , and ) 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 , , and are 0.290 m, 0.302 m, and 0.3 m, respectively, and the maximum temperature () of the cable core is 337.998 K. The deviation of predicted with GA and COMSOL MULTIPHYSICS is 0.078 K, and is also quite close to the optimum cable working temperature ( = 338K or 65°C). With the optimum geometric parameters ( = 0.290 m, = 0.302 m, and = 0.3 m), the minimum total cost function () is 0.2420 for parabolic-type FTB trefoil buried cables.
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 (, , and ) 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 , , and are 0.290 m, 0.302 m, and 0.3 m, respectively, the maximum temperature () of the cable core is 337.998 K, and the minimum total cost function () is 0.2420.
Nomenclature
| , : | Model coefficients in (2) |
| : | Threshold value in (9) |
| : | FTB area, m2 |
| : | Magnetic vector potential, Wb/m |
| : | The diameter of cable, m |
| : | Total cost function |
| : | The thickness of the flat-type FTB laying model, m |
| : | The distance between the cables and ground in [6], m |
| : | The height of the computational model, m |
| : | Cable core current, A |
| : | Unit of complex number |
| : | Current density, A/m2 |
| : | Inductive current density, A/m2 |
| : | Source current density, A/m2 |
| : | Dry thermal conductivity of backfill material, W/(m·K) |
| : | Thermal conductivity of FTB, W/(m·K) |
| : | Thermal conductivity of NS, W/(m·K) |
| : | Wet thermal conductivity of backfill material, W/(m·K) |
| : | The distance between the edge of FTB and the axis of symmetry in the parabolic-type FTB laying model, m |
| : | Distance of the gap between cables, m |
| : | Distance between two neighbor cable cores, m |
| : | The width of the computational model, m |
| : | Input to the radial basis function in (9) |
| : | Heat loss, W/m3 |
| : | The radius of conductor, m |
| : | The radius of insulation layer, m |
| : | The radius of jacket layer, m |
| : | The radius of sheath layer, m |
| : | The distance between upper cable core and the top of FTB in the parabolic-type FTB laying model, m |
| : | The distance between the lower cable core and bottom of FTB in the parabolic-type FTB laying model, m |
| : | Temperature, K |
| : | The limited temperature, K |
| : | The maximum temperature, K |
| : | The optimum temperature, K |
| : | The reference temperature, K |
| Weight vector in (9) | |
| Input vector in (9) |
| : | Reference temperature coefficient in (5) |
| : | Reference electrical conductivity, S/m2 |
| : | Electrical conductivity, S/m2 |
| : | Heat flux, W |
| : | Electric scalar potential, V |
| : | Angular frequency, rad/s |
| : | Conductor |
| : | Insulation layer |
| : | Jacket layer |
| max: | The maximum value |
| opt: | The optimum value |
| ref: | The reference value |
| s: | Sheath layer |
| FEM: | Finite element method |
| FTB: | Fluidized thermal backfill |
| GA: | Genetic algorithm |
| LHS: | Latin hypercube sampling |
| NS: | Natural soil |
| PF: | Penalty function |
| PSO: | Particle swarm optimization |
| RBF: | Radial basis function |
| RBNN: | Radial basis neural network |
| SGA: | Standard genetic algorithm |
| XLPE: | Cross-linked polyethylene. |
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was financially supported by the project “Research and demonstration application of temperature rise algorithm for buried (direct buried and pipe laying) cable group” from State Grid Corporation of China (SGCC) under Grant 52094018001K.