# Heat Transfer Simulation and Temperature Rapid Prediction for Trench Laying Cable

Authors: Chen-Zhao Fu; Wen-Rong Si; Ke-Ke Fang; Jian Yang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Heat transfer process for trench laying cable is complex. To guarantee safe operation of the cable, it is necessary to predict the temperature and maximum current capacity of trench laying cable rapidly and accurately. Therefore, in this study, an adaptive optimized particle swarm optimization algorithm (LFVPSO) is proposed based on Levy flight algorithm, and it is used to modify the back propagation neural network algorithm (LFVPSO-BPNN). Then, combined with numerical simulations, a network algorithm for temperature prediction of trench laying cable is developed using LFVPSO-BPNN. Finally, the maximum current capacity of four-loop three-phase trench laying cable is calculated using LFVPSO-BPNN together with genetic algorithm (GA&LFVPSO-BPNN). At first, it is found that the LFVPSO-BPNN algorithm proposed in this study is reliable and accurate to predict the cable maximum temperature for different loops (Tmax,i) in the trench. Furthermore, as compared with other similar algorithms, when LFVPSO-BPNN algorithm is used to predict the temperature of trench laying cable, its computation time would be reduced and the prediction accuracy would be improved as well. Second, it is revealed that the effect of ground air temperature (Tsur) on the maximum current capacity of trench laying cable (It,max) is remarkable. As Tsur increases, the It,max for both flat-type and trefoil-type trench laying cable would significantly decrease. In addition, with the same Tsur, the It,max for the flat-type trench laying cable are obviously higher.


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## Body

## 1. Introduction

Due to the urban development and aesthetic demand, the underground cable distributions are prevailing for urban power distribution [1]. Among these distributions, trench laying cables are often used in power plants or transformer substations. As compared with other underground cable laying methods, trench laying is more convenient for cable construction and maintenance, and it can accommodate multiloop cables [2]. The insulation performance of trench laying cable is closely related to its cable core temperature. When the cable core temperature is higher than 363.15 K, the cable insulation aging will be accelerated and cable service life will be greatly shortened [3]. Therefore, to prevent cable overheating and guarantee cable safe operation, it is necessary to predict the temperature rise of trench laying cable rapidly and accurately.Currently, the temperature prediction and ampacity calculation for underground cable mainly include equivalent thermal circuit method and numerical simulation method. For the equivalent thermal circuit method, it is mainly based on the IEC-60287 standard [4], where the heat transfer process between the cable and environment is equivalent to a thermal circuit for the calculations. This method can predict temperature rise for direct buried cable rapidly and accurately [5]. However, for the trench laying cable, the heat transfer process is much more complex. When the equivalent thermal circuit method is adopted for the calculation, the prediction model would be oversimplified, and it may lead to large prediction errors [6]. For the numerical simulation method, the heat transfer process between the cable and environment is numerically simulated, and the temperature distributions and cable ampacity are obtained based on the finite-element method (FEM) or finite-difference method (FDM). For example, Al-Saud et al. [7] numerically studied the temperature variations of direct buried cables using perturbed FEM. They found that the effect of thermal conductivities of soil and backfill material on the cable temperature was remarkable. The effect of cable layout on the temperature distributions and current capacity of trench laying cable was numerically studied by Xiong et al. [8] with FEM. It was found that as compared with regular arrangement, when cables were irregularly arranged in the trench, the cable temperature was higher and its current capacity was lower. While in this work, the radiation heat transfer in the trench was not considered. Liu et al. [9] numerically studied the heat transfer of three flat-type laying cables in a rectangular trench with different Rayleigh numbers. They found that the effect of cable distance on cable heat transfer performance is remarkable. Furthermore, the cable core temperature of shallow trench laying cable was calculated by Anders et al. [10] using thermal circuit method and FEM, respectively. It was found that compared with the thermal circuit method, the computational results obtained using FEM were more accurate, but the computational time of FEM was much longer. The temperature distribution of direct buried cable was numerically studied by Vollaro et al. [11] with FDM, and a calculation correlation of thermal resistance between cable and ground was obtained. It was showed that the prediction results obtained by the FDM were more accurate than those of IEC-60287 standard. Some other thermal modelling studies were also performed by Foteinopoulos et al. [12], Salonitis et al. [13], and Marano et al. [14]. A two-dimensional finite-difference model of the thermal history of parts manufactured in powder bed fusion additive manufacturing processes was presented by Foteinopoulos et al. [12]. In their study, an algorithm for node birth and distance adaptation over time was used, minimizing computational time and memory. Salonitis et al. [13] performed the thermal modelling of the material removal rate and surface roughness for die-sinking electrical discharge machining. In their study, a simple, thermal based model had been developed for the determination of the material removal rate and the average surface roughness achieved as a function of the process parameters, and the model’s predictions could agree well with experimental results. The thermal behavior of trench-isolated bipolar transistors was thoroughly investigated by Marano et al. [14]. In their study, a novel strategy to analytically evaluate the temperature field was proposed, which accounted for the heat propagation through the trench and the nonuniform heat flux distribution over the interface between the silicon box surrounded by trench and the underlying substrate. This novel strategy could be employed for an accurate, yet fast evaluation of the thermal resistance of a trench-isolated device as well as of the temperature gradients within the silicon box. Most of these researches showed that as compared with the thermal circuit method, when the heat transfer of trench laying cable was studied with numerical simulations, the accuracy of prediction results would be greatly improved, but its computational time would also be much longer. To solve this problem, artificial intelligence algorithms combined with numerical simulations are recommended for rapid and accurate prediction of cable temperature rise. Recently, for underground cable, artificial intelligence algorithms are mainly focused on the fields of cable laying cost optimization, cable maximum current capacity calculation, and cable temperature prediction. For example, a modified JAYA algorithm was proposed by Ocłoń et al. [15] to optimize the laying cost of 400 kV direct buried cables. It was found that as compared with the traditional particle swarm optimization algorithm (PSO), lower cable laying cost could be obtained with this modified JAYA algorithm. A formula for laying cost calculation of direct buried cable was proposed by Cichy et al. [16], where the costs of labor, material, and operation loss were considered. Combing this formula with genetic algorithm (GA), the optimum cable size and filling size of backfill material were obtained. The maximum current capacity of pipe laying cables, containing 12 cables and 15 pipes, was calculated by Moutassem and Anders [17] with vector immune system algorithm. They found that as compared with numerical simulation using FEM, the calculation time could be greatly reduced with vector immune system algorithm. Combing shuffled frog-leaping algorithm with PSO algorithm, the maximum current capacity of pipe laying cable was calculated by Zarchi et al. [18]. They found that current harmonics would have great influence on maximum cable current capacity. The temperature rise of pipe cable was predicted by Zarchi and Vahidi [19] with Monte Carlo method and Hong point estimate method (HPEM), respectively, and the maximum current capacity of pipe cable was also calculated. It was found that as compared with Monte Carlo method, the HPEM requires fewer input parameters and its calculation cost was lower. Furthermore, a temperature prediction network for parabolic-type buried cable filled with fluidized thermal backfill material was proposed by Fu et al. [20] with radial basis neural network algorithm. Combining this temperature prediction network with genetic algorithm (GA), the optimum cable laying cost was obtained. These studies showed that for recent years, artificial intelligence algorithms used for underground cable were mainly focused on the researches about laying cost optimization for direct buried cable or maximum current capacity calculation for pipe laying cable, while the related researches for temperature prediction of underground cable were few, almost no studies were performed on rapid prediction of temperature rise for trench laying cable.Therefore, in this study, to predict the temperature rise of trench laying cable rapidly and accurately, an adaptive optimized particle swarm optimization algorithm (LFVPSO) is proposed based on Levy flight algorithm, and it is used to modify the back propagation neural network algorithm (LFVPSO-BPNN). Then combined with numerical simulations, a network algorithm for temperature prediction of trench laying cable is developed using LFVPSO-BPNN. Finally, the maximum current capacity of four-loop three-phase trench laying cable is calculated using LFVPSO-BPNN together with genetic algorithm (GA&LFVPSO-BPNN). According to the authors’ knowledge, almost no such studies were performed yet, and it would be meaningful for temperature rapid prediction and maximum current capacity calculation of trench laying cable.


## 2. Physical Model and Computational Method

### 2.1. Physical Model and Geometric Parameters

The physical model and geometric structure for trench laying cable are presented in Figure1. In this study, since the trench length is much larger than its cross-sectional dimension, the trench laying cable is simplified as a two-dimensional model for the computations. Four-loop three-phase cables are laid in the trench, including flat-type laying model (Figure 1(a)) and trefoil-type laying model (Figure 1(b)). The dimension of the trench is 2 m (width) × 2.2 m (height). The trench wall is made of concrete, and its thickness is 0.2 m. The inside and outside areas of the trench are filled with air and soil, respectively. It is assumed that the soil temperature far away (20 m) from the trench bottom is fixed at consistent (Tsoil = 283.15 K), and the soil far away (20 m) from trench lateral walls are adiabatic. The top surface of trench is exposed to the environment with temperature of Tsur and the equivalent surface heat transfer coefficient is hf (11 W/(m2·K)). The radiation heat transfer is existed between cable surface and inner surface of trench, where the radiant emissivity on the cable surface is 0.9, and it is 0.6 on the inner surface of the trench. For flat-type trench laying cable models (Figure 1(a)), the distance between the cable core and trench side wall is 0.12 m. The cable core distance between the upper and lower layer cables is 0.4 m. The distance between the lower layer cable core and trench bottom is 0.15 m. And the cable core distance between two adjacent cables in the same loop is 0.15 m. For trefoil-type trench laying cable models (Figure 1(b)), the distance between the cable core and trench side wall is 0.21 m. The cable core distance between the upper and lower layer cables is 0.5 m. And the distance between the lower layer cable core and trench bottom is 0.15 m. The arrangement for a single-loop three-phase trefoil cables is presented in Figure 1(c), where the cables are in a regular triangle arrangement. To improve the quality of computational mesh near the contact points between cables, the cables are assumed to be stacked with small gaps between each other, where the gap width is 1% re [21]. The structure for a single cable (8.7/15 kV YJV 1 × 400) is presented in Figure 1(d). It shows that the cable is mainly composed of cable core, insulation layer, sheath layer, and external layer, and their radii are rc, rins, rs, and re, respectively. Typical geometrical and physical parameters for the trench laying cable are listed in Tables 1 and 2.Figure 1
Physical model and geometrical structure for trench laying cable [22]: (a) flat-type trench laying cable; (b) trefoil-type trench laying cable; (c) trefoil cable arrangement; (d) cable structure.
(a)(b)(c)(d)Table 1
Typical geometric parameters for trench laying cable [22].
rc (mm)rins (mm)rs (mm)re (mm)11.917.818.823.4Table 2
Typical physical parameters for trench laying cable [22].
MaterialThermal conductivity (W/(m·K))Electronic conductivity (S/m)Relative dielectric constantCable coreCopper3775.998 × 1071.0Insulation layerXLPE0.2851 × 10−152.5Sheath layerAluminium2383.774 × 1071.0External layerPolyethylene0.1321 × 10−152.5Concrete trenchConcrete1.731 × 10−151.0Soil layerSoil11 × 10−151.0


### 2.2. Governing Equations and Computational Method

In this study, the electromagnetic effect, heat transfer, and air flow are coupled with each other. The computational domain is divided into solid domain and fluid domain for the coupling computations. For solid domain, the governing equations for the heat transfer are as follows:(1)0=∇λ⋅∇T+Qe,Cable core and sheath layer,0=∇λ⋅∇T,The rest solid regions,where T is temperature and λ is the solid thermal conductivity. The thermal conductivity of soil (λsoil) changes with temperature, which is defined as follows [23]:(2)λsoilT=λdry+λwet−λdry⋅exp−a1T−Trefa2⋅Tlim2,a1=TlimTref,a2=1−1a1,where λdry and λwet are the thermal conductivities under dry and wet conditions for soil, which are 0.3 W/(m·K) and 1 W/(m·K), respectively; Tlim is the cable limit temperature, 363 K; Tref is the reference temperature of surrounding soil, 293 K; and Qe is the electromagnetic loss density, which is defined as follows [24]:(3)Qe=Qrh+Qml=12Re⋅J⇀⋅E⇀∗+12Re⋅jωB⇀⋅H⇀∗,∇×A⇀=B⇀,E⇀=−jωA⇀,B⇀=μH⇀,J⇀=σT⋅E⇀.where Qrh and Qml are the electrical loss density and magnetic loss density, respectively; Re is the real part of imaginary number; J⇀ is current density; j is the imaginary unit; ω is the phase angle, 2πf; f is frequency of electrical excitation, 50 Hz; B⇀ is the magnetic induction intensity; A⇀ is the magnetic vector potential; E⇀ is the electric field intensity; H⇀ is magnetic field intensity; E⇀∗ and H⇀∗ are the conjugate complex numbers of E⇀ and H⇀; and σT is the electrical conductivity, which is defined as follows:(4)σT=1ρref1+αT−Tref,where ρref is the reference electrical resistivity and α is the temperature coefficient.The air flow in trench is a two-dimensional natural convection turbulent heat transfer, and governing equations are formulated as follows:(5)∇⋅v⇀=0,ρv⇀⋅∇v⇀=−∇p+∇⋅μf+μt⋅∇v⇀+∇v⇀T+ρ0g1−βT−T0,ρv⇀⋅∇T=∇⋅λfcp+μtσt⋅∇T,where v⇀ is the velocity vector; T0 is the air reference temperature; ρ and ρ0 are the air density at T  and T0, respectively; p, g, and β are the pressure, gravitational acceleration, and volumetric expansion coefficient, respectively; μf and μt are the air dynamic viscosity and turbulence viscosity, respectively. λf is the thermal conductivity of air; cp is the specific heat at the constant pressure of air; and σt is the turbulence Prandtl number in energy equation.As for air turbulent flow, the standardk-ε model is adopted, which are defined as follows:(6)ρv⇀⋅∇k=∇μf+μtσk⋅∇k+Pk−ρε,ρv⇀⋅∇ε=∇μf+μtσε+Cε1εkPk−Cε2ρε2k,where k is the turbulent kinetic energy; ε is the turbulent dissipation rate; Pk is the shear production of turbulence; σk and σε are the turbulence Prandtl numbers in k and ε equations; and Cε1 and Cε2 are the turbulent model parameters in k and ε equations.The boundary conditions are set as follows:(7)bottom soil Boundary:T=Tsoil=283.15K,Left and right soil boundary:∂T∂n=0,Ground surface:λ∂T∂n=hfT−Tsur,Air contact surface inside trench:λi∂T∂n=hiT−Tf+σT4−Ji1−εi/εi;u=v=0,where Tsoil is the deep soil temperature; hf is the equivalent heat transfer coefficient on the ground surface, which is set as 11 W/(m2·K); Tsur is the air temperature on the ground; λi (i = 1, 2) represents the thermal conductivity of cable external layer or concrete layer, respectively; hi (i = 1, 2) represents the convective heat transfer coefficient on the cable surface or trench inner surface; Tf is the air temperature in the trench; εi (εc and εΤ) represents the radiant emissivity on the cable surface or trench inner surface, respectively; Ji (Jc and JΤ) represents the effective radiation on the cable surface or trench inner surface, respectively; and σ is the blackbody radiation constant.In this study, the governing equations are solved with the Pardiso solver embedded in commercial code COMSOL Multiphysics 5.2. The current frequency is set as 50 Hz. The conservative interface flux conditions for mass, momentum, and heat transfer are adopted at the solid-solid and solid-fluid interfaces. For convergence criteria, all residuals of the calculations are less than 10−3.


## 2.1. Physical Model and Geometric Parameters

The physical model and geometric structure for trench laying cable are presented in Figure1. In this study, since the trench length is much larger than its cross-sectional dimension, the trench laying cable is simplified as a two-dimensional model for the computations. Four-loop three-phase cables are laid in the trench, including flat-type laying model (Figure 1(a)) and trefoil-type laying model (Figure 1(b)). The dimension of the trench is 2 m (width) × 2.2 m (height). The trench wall is made of concrete, and its thickness is 0.2 m. The inside and outside areas of the trench are filled with air and soil, respectively. It is assumed that the soil temperature far away (20 m) from the trench bottom is fixed at consistent (Tsoil = 283.15 K), and the soil far away (20 m) from trench lateral walls are adiabatic. The top surface of trench is exposed to the environment with temperature of Tsur and the equivalent surface heat transfer coefficient is hf (11 W/(m2·K)). The radiation heat transfer is existed between cable surface and inner surface of trench, where the radiant emissivity on the cable surface is 0.9, and it is 0.6 on the inner surface of the trench. For flat-type trench laying cable models (Figure 1(a)), the distance between the cable core and trench side wall is 0.12 m. The cable core distance between the upper and lower layer cables is 0.4 m. The distance between the lower layer cable core and trench bottom is 0.15 m. And the cable core distance between two adjacent cables in the same loop is 0.15 m. For trefoil-type trench laying cable models (Figure 1(b)), the distance between the cable core and trench side wall is 0.21 m. The cable core distance between the upper and lower layer cables is 0.5 m. And the distance between the lower layer cable core and trench bottom is 0.15 m. The arrangement for a single-loop three-phase trefoil cables is presented in Figure 1(c), where the cables are in a regular triangle arrangement. To improve the quality of computational mesh near the contact points between cables, the cables are assumed to be stacked with small gaps between each other, where the gap width is 1% re [21]. The structure for a single cable (8.7/15 kV YJV 1 × 400) is presented in Figure 1(d). It shows that the cable is mainly composed of cable core, insulation layer, sheath layer, and external layer, and their radii are rc, rins, rs, and re, respectively. Typical geometrical and physical parameters for the trench laying cable are listed in Tables 1 and 2.Figure 1
Physical model and geometrical structure for trench laying cable [22]: (a) flat-type trench laying cable; (b) trefoil-type trench laying cable; (c) trefoil cable arrangement; (d) cable structure.
(a)(b)(c)(d)Table 1
Typical geometric parameters for trench laying cable [22].
rc (mm)rins (mm)rs (mm)re (mm)11.917.818.823.4Table 2
Typical physical parameters for trench laying cable [22].
MaterialThermal conductivity (W/(m·K))Electronic conductivity (S/m)Relative dielectric constantCable coreCopper3775.998 × 1071.0Insulation layerXLPE0.2851 × 10−152.5Sheath layerAluminium2383.774 × 1071.0External layerPolyethylene0.1321 × 10−152.5Concrete trenchConcrete1.731 × 10−151.0Soil layerSoil11 × 10−151.0


## 2.2. Governing Equations and Computational Method

In this study, the electromagnetic effect, heat transfer, and air flow are coupled with each other. The computational domain is divided into solid domain and fluid domain for the coupling computations. For solid domain, the governing equations for the heat transfer are as follows:(1)0=∇λ⋅∇T+Qe,Cable core and sheath layer,0=∇λ⋅∇T,The rest solid regions,where T is temperature and λ is the solid thermal conductivity. The thermal conductivity of soil (λsoil) changes with temperature, which is defined as follows [23]:(2)λsoilT=λdry+λwet−λdry⋅exp−a1T−Trefa2⋅Tlim2,a1=TlimTref,a2=1−1a1,where λdry and λwet are the thermal conductivities under dry and wet conditions for soil, which are 0.3 W/(m·K) and 1 W/(m·K), respectively; Tlim is the cable limit temperature, 363 K; Tref is the reference temperature of surrounding soil, 293 K; and Qe is the electromagnetic loss density, which is defined as follows [24]:(3)Qe=Qrh+Qml=12Re⋅J⇀⋅E⇀∗+12Re⋅jωB⇀⋅H⇀∗,∇×A⇀=B⇀,E⇀=−jωA⇀,B⇀=μH⇀,J⇀=σT⋅E⇀.where Qrh and Qml are the electrical loss density and magnetic loss density, respectively; Re is the real part of imaginary number; J⇀ is current density; j is the imaginary unit; ω is the phase angle, 2πf; f is frequency of electrical excitation, 50 Hz; B⇀ is the magnetic induction intensity; A⇀ is the magnetic vector potential; E⇀ is the electric field intensity; H⇀ is magnetic field intensity; E⇀∗ and H⇀∗ are the conjugate complex numbers of E⇀ and H⇀; and σT is the electrical conductivity, which is defined as follows:(4)σT=1ρref1+αT−Tref,where ρref is the reference electrical resistivity and α is the temperature coefficient.The air flow in trench is a two-dimensional natural convection turbulent heat transfer, and governing equations are formulated as follows:(5)∇⋅v⇀=0,ρv⇀⋅∇v⇀=−∇p+∇⋅μf+μt⋅∇v⇀+∇v⇀T+ρ0g1−βT−T0,ρv⇀⋅∇T=∇⋅λfcp+μtσt⋅∇T,where v⇀ is the velocity vector; T0 is the air reference temperature; ρ and ρ0 are the air density at T  and T0, respectively; p, g, and β are the pressure, gravitational acceleration, and volumetric expansion coefficient, respectively; μf and μt are the air dynamic viscosity and turbulence viscosity, respectively. λf is the thermal conductivity of air; cp is the specific heat at the constant pressure of air; and σt is the turbulence Prandtl number in energy equation.As for air turbulent flow, the standardk-ε model is adopted, which are defined as follows:(6)ρv⇀⋅∇k=∇μf+μtσk⋅∇k+Pk−ρε,ρv⇀⋅∇ε=∇μf+μtσε+Cε1εkPk−Cε2ρε2k,where k is the turbulent kinetic energy; ε is the turbulent dissipation rate; Pk is the shear production of turbulence; σk and σε are the turbulence Prandtl numbers in k and ε equations; and Cε1 and Cε2 are the turbulent model parameters in k and ε equations.The boundary conditions are set as follows:(7)bottom soil Boundary:T=Tsoil=283.15K,Left and right soil boundary:∂T∂n=0,Ground surface:λ∂T∂n=hfT−Tsur,Air contact surface inside trench:λi∂T∂n=hiT−Tf+σT4−Ji1−εi/εi;u=v=0,where Tsoil is the deep soil temperature; hf is the equivalent heat transfer coefficient on the ground surface, which is set as 11 W/(m2·K); Tsur is the air temperature on the ground; λi (i = 1, 2) represents the thermal conductivity of cable external layer or concrete layer, respectively; hi (i = 1, 2) represents the convective heat transfer coefficient on the cable surface or trench inner surface; Tf is the air temperature in the trench; εi (εc and εΤ) represents the radiant emissivity on the cable surface or trench inner surface, respectively; Ji (Jc and JΤ) represents the effective radiation on the cable surface or trench inner surface, respectively; and σ is the blackbody radiation constant.In this study, the governing equations are solved with the Pardiso solver embedded in commercial code COMSOL Multiphysics 5.2. The current frequency is set as 50 Hz. The conservative interface flux conditions for mass, momentum, and heat transfer are adopted at the solid-solid and solid-fluid interfaces. For convergence criteria, all residuals of the calculations are less than 10−3.


## 3. Grid Independence Test and Model Validations

### 3.1. Grid Independence Test

First, the grid independence tests were performed for both flat-type trench laying cable model (Figure1(a)) and trefoil-type trench laying cable model (Figure 1(b)). For the test cases, the air temperature on the ground (Tsur) is set as 293.15 K, and the cable current for different loops are kept the same (I = 800 A, f = 50 Hz). The computational grids for different trench laying cable models are presented in Figure 2. The unstructured mesh is constructed for the computations, and the mesh is intensified near the surfaces of cable and trench wall. The cable maximum temperature (Tmax) and electromagnetic loss (Φloss) with different computational grids are presented in Table 3. It shows that for the flat-type trench laying cable model, when the total element number changes from 27,610 (Grid 3) to 50,188 (Grid 4), the difference of cable maximum temperature (Tmax) is 0.05%, and the difference of electromagnetic loss (Φloss) is 0.16%. Therefore, for the flat-type trench laying cable model, the Grid 3 with total element number of 27,610 should be good enough for the test. For this grid, the minimum size of the grid element in the cable zone (Zone 1), air zone (Zone 2), trench wall zone (Zone 3), and soil zone (Zone 4) are 1.5 mm, 4.2 mm, 11.7 mm, and 111.7 mm, respectively. As for the trefoil-type trench laying cable model, when the total element number changes from 27,798 (Grid 3) to 53,290 (Grid 4), the difference of cable maximum temperature (Tmax) is 0.11%, and the difference of electromagnetic loss (Φloss) is 0.01%. Therefore, for the trefoil-type trench laying cable model, the Grid 3 with total element number of 27,798 should be good enough for the test. For this grid, the minimum size of the grid element in the cable zone (Zone 1), air zone (Zone 2), trench wall zone (Zone 3), and soil zone (Zone 4) are 1.1 mm, 1.2 mm, 12.7 mm, and 103.1 mm, respectively. Based on the aforementioned analysis, similar mesh settings to the Grid 3 (27,610) for flat-type trench laying cable model and Grid 3 (27,798) for trefoil-type trench laying cable model were finally adopted for the following simulations.Figure 2
Computational grids for different trench laying cable models: (a) flat-type trench laying cable and (b) trefoil-type trench laying cable.
(a)(b)Table 3
Cable maximum temperature and electromagnetic loss with different computational grids (Tsur = 293.15 K, I = 800 A, and f = 50 Hz).
Cable modelGridRegionMaximum grid size (m)Minimum grid size (m)Total element numberTmax (K)Φloss (W)Flat-type trench laying cableGrid 1Zone 10.00970.002410,881336.520190.65Zone 20.29710.0049Zone 30.31920.0199Zone 43.10440.2489Grid 2Zone 10.00570.001718,070333.742183.38Zone 20.11370.0049Zone 30.18150.0181Zone 42.18440.1411Grid 3Zone 10.00430.001527,610333.708182.55Zone 20.09380.0042Zone 30.15150.0117Zone 41.83060.1117Grid 4Zone 10.00240.001350,188333.541182.25Zone 20.06820.0020Zone 30.12560.0117Zone 40.99780.0951Trefoil-type trench laying cableGrid 1Zone 10.00600.001514,408342.287190.67Zone 20.11400.0013Zone 30.18150.0260Zone 42.51120.1411Grid 2Zone 10.00590.001522,618343.014189.69Zone 20.09110.0012Zone 30.14570.0227Zone 41.78100.1088Grid 3Zone 10.00470.001127,798341.779188.69Zone 20.08850.0012Zone 30.11730.0127Zone 40.98680.1031Grid 4Zone 10.00330.001153,290341.411188.71Zone 20.08770.0011Zone 30.10930.0103Zone 40.48720.0748


### 3.2. Model Validations

Subsequently, the computational model and methods are validated. Since numerical study on multiphysics coupling simulation for the trench laying cable is few, the numerical simulations for three-phase pipe laying cable [3] are restudied. The physical model for pipe laying cable [3] is presented in Figure 3. Three-phase cables are laid in a polyvinyl chloride (PVC) pipe. Both natural convection and radiation heat transfer are considered on the cable surfaces and pipe inner surface. The cable diameter is 150 mm, the outer diameter of PVC pipe is 620 mm and its thickness is 10 mm. The radiant emissivity on the cable surface is 0.8, and it is 0.5 on the inner surface of PVC pipe. The equivalent heat transfer coefficient on the outer surface of PVC pipe (hr) is 5 W/(m2·K), and the environment temperature on the outside surface of PVC pipe (Tr) is 293.15 K. Typical geometrical and physical parameters for the pipe laying cable [3] are listed in Table 4. The computational results for the cable maximum temperature (Tmax), radiation heat flux (Φrad), convection heat flux (Φconv), and total heat flux (Φtot) on the cable surface are presented in Figure 4. It shows that the present computational results are in good agreement with those as reported by Fu et al. [3], where the maximum deviations for Tmax, Φrad, Φconv, and Φtot are 0.056%, 4.70%, 4.41%, and 4.47%, respectively.Figure 3
Physical model for pipe laying cable and cable structure [3]: (a) physical model for pipe laying cable and (b) cable structure.
(a)(b)Table 4
Typical geometrical and physical parameters for the pipe laying cable [3].
Diameter (mm)Thermal conductivity (W/(m∙K))Electronic conductivity (S/m)Relative dielectric constantCable core60.44005.998 × 1071.0Insulation layer1200.2861 × 10−152.5Sheath layer1402383.774 × 1071.0External layer1500.2801 × 10−152.5PVC pipe6200.1671 × 10−161.0Figure 4
Computational results for pipe laying cable: (a) cable maximum temperature and (b) heat flux on cable surface.
(a)(b)In addition, since the air flow inside PVC pipe is laminar flow in Fu et al.’s study [3], the turbulence natural convection heat transfer in a two-dimensional square cavity [25] is restudied. The physical model for the two-dimensional square cavity [25] is presented in Figure 5. It shows that the top and bottom walls of the cavity are adiabatic. Two vertical walls are high temperature and low temperature walls, where the wall temperature is kept constant and the temperature difference is 40 K. The computational results for the average Nusselt number (Nuave) are presented in Table 5. It shows that the average Nusselt numbers (Nuave) obtained from the present simulations can agree well with those in Kuznetsov and Sheremet’s study [25], and the maximum deviation is 6.26%.Figure 5
Physical model for two-dimensional square cavity [25].Table 5
Computational results for Nuave of present simulation and the study by Kuznetsov and Sheremet [25].
RaNuave (present simulation)Kuznetsov and Sheremet’s study [25]Deviation (%)10718.6017.724.9710835.5033.416.26


## 3.1. Grid Independence Test

First, the grid independence tests were performed for both flat-type trench laying cable model (Figure1(a)) and trefoil-type trench laying cable model (Figure 1(b)). For the test cases, the air temperature on the ground (Tsur) is set as 293.15 K, and the cable current for different loops are kept the same (I = 800 A, f = 50 Hz). The computational grids for different trench laying cable models are presented in Figure 2. The unstructured mesh is constructed for the computations, and the mesh is intensified near the surfaces of cable and trench wall. The cable maximum temperature (Tmax) and electromagnetic loss (Φloss) with different computational grids are presented in Table 3. It shows that for the flat-type trench laying cable model, when the total element number changes from 27,610 (Grid 3) to 50,188 (Grid 4), the difference of cable maximum temperature (Tmax) is 0.05%, and the difference of electromagnetic loss (Φloss) is 0.16%. Therefore, for the flat-type trench laying cable model, the Grid 3 with total element number of 27,610 should be good enough for the test. For this grid, the minimum size of the grid element in the cable zone (Zone 1), air zone (Zone 2), trench wall zone (Zone 3), and soil zone (Zone 4) are 1.5 mm, 4.2 mm, 11.7 mm, and 111.7 mm, respectively. As for the trefoil-type trench laying cable model, when the total element number changes from 27,798 (Grid 3) to 53,290 (Grid 4), the difference of cable maximum temperature (Tmax) is 0.11%, and the difference of electromagnetic loss (Φloss) is 0.01%. Therefore, for the trefoil-type trench laying cable model, the Grid 3 with total element number of 27,798 should be good enough for the test. For this grid, the minimum size of the grid element in the cable zone (Zone 1), air zone (Zone 2), trench wall zone (Zone 3), and soil zone (Zone 4) are 1.1 mm, 1.2 mm, 12.7 mm, and 103.1 mm, respectively. Based on the aforementioned analysis, similar mesh settings to the Grid 3 (27,610) for flat-type trench laying cable model and Grid 3 (27,798) for trefoil-type trench laying cable model were finally adopted for the following simulations.Figure 2
Computational grids for different trench laying cable models: (a) flat-type trench laying cable and (b) trefoil-type trench laying cable.
(a)(b)Table 3
Cable maximum temperature and electromagnetic loss with different computational grids (Tsur = 293.15 K, I = 800 A, and f = 50 Hz).
Cable modelGridRegionMaximum grid size (m)Minimum grid size (m)Total element numberTmax (K)Φloss (W)Flat-type trench laying cableGrid 1Zone 10.00970.002410,881336.520190.65Zone 20.29710.0049Zone 30.31920.0199Zone 43.10440.2489Grid 2Zone 10.00570.001718,070333.742183.38Zone 20.11370.0049Zone 30.18150.0181Zone 42.18440.1411Grid 3Zone 10.00430.001527,610333.708182.55Zone 20.09380.0042Zone 30.15150.0117Zone 41.83060.1117Grid 4Zone 10.00240.001350,188333.541182.25Zone 20.06820.0020Zone 30.12560.0117Zone 40.99780.0951Trefoil-type trench laying cableGrid 1Zone 10.00600.001514,408342.287190.67Zone 20.11400.0013Zone 30.18150.0260Zone 42.51120.1411Grid 2Zone 10.00590.001522,618343.014189.69Zone 20.09110.0012Zone 30.14570.0227Zone 41.78100.1088Grid 3Zone 10.00470.001127,798341.779188.69Zone 20.08850.0012Zone 30.11730.0127Zone 40.98680.1031Grid 4Zone 10.00330.001153,290341.411188.71Zone 20.08770.0011Zone 30.10930.0103Zone 40.48720.0748


## 3.2. Model Validations

Subsequently, the computational model and methods are validated. Since numerical study on multiphysics coupling simulation for the trench laying cable is few, the numerical simulations for three-phase pipe laying cable [3] are restudied. The physical model for pipe laying cable [3] is presented in Figure 3. Three-phase cables are laid in a polyvinyl chloride (PVC) pipe. Both natural convection and radiation heat transfer are considered on the cable surfaces and pipe inner surface. The cable diameter is 150 mm, the outer diameter of PVC pipe is 620 mm and its thickness is 10 mm. The radiant emissivity on the cable surface is 0.8, and it is 0.5 on the inner surface of PVC pipe. The equivalent heat transfer coefficient on the outer surface of PVC pipe (hr) is 5 W/(m2·K), and the environment temperature on the outside surface of PVC pipe (Tr) is 293.15 K. Typical geometrical and physical parameters for the pipe laying cable [3] are listed in Table 4. The computational results for the cable maximum temperature (Tmax), radiation heat flux (Φrad), convection heat flux (Φconv), and total heat flux (Φtot) on the cable surface are presented in Figure 4. It shows that the present computational results are in good agreement with those as reported by Fu et al. [3], where the maximum deviations for Tmax, Φrad, Φconv, and Φtot are 0.056%, 4.70%, 4.41%, and 4.47%, respectively.Figure 3
Physical model for pipe laying cable and cable structure [3]: (a) physical model for pipe laying cable and (b) cable structure.
(a)(b)Table 4
Typical geometrical and physical parameters for the pipe laying cable [3].
Diameter (mm)Thermal conductivity (W/(m∙K))Electronic conductivity (S/m)Relative dielectric constantCable core60.44005.998 × 1071.0Insulation layer1200.2861 × 10−152.5Sheath layer1402383.774 × 1071.0External layer1500.2801 × 10−152.5PVC pipe6200.1671 × 10−161.0Figure 4
Computational results for pipe laying cable: (a) cable maximum temperature and (b) heat flux on cable surface.
(a)(b)In addition, since the air flow inside PVC pipe is laminar flow in Fu et al.’s study [3], the turbulence natural convection heat transfer in a two-dimensional square cavity [25] is restudied. The physical model for the two-dimensional square cavity [25] is presented in Figure 5. It shows that the top and bottom walls of the cavity are adiabatic. Two vertical walls are high temperature and low temperature walls, where the wall temperature is kept constant and the temperature difference is 40 K. The computational results for the average Nusselt number (Nuave) are presented in Table 5. It shows that the average Nusselt numbers (Nuave) obtained from the present simulations can agree well with those in Kuznetsov and Sheremet’s study [25], and the maximum deviation is 6.26%.Figure 5
Physical model for two-dimensional square cavity [25].Table 5
Computational results for Nuave of present simulation and the study by Kuznetsov and Sheremet [25].
RaNuave (present simulation)Kuznetsov and Sheremet’s study [25]Deviation (%)10718.6017.724.9710835.5033.416.26


## 4. Prediction Algorithm

In this study, an adaptive optimized particle swarm optimization algorithm (LFVPSO) is proposed based on Levy flight algorithm, and it is used to modify the back propagation neural network algorithm (LFVPSO-BPNN). Then combined with numerical simulations, a network algorithm for temperature prediction of trench laying cable is developed using LFVPSO-BPNN. Finally, the maximum current capacity of four-loop three-phase trench laying cable is calculated using LFVPSO-BPNN together with genetic algorithm (GA&LFVPSO-BPNN).


### 4.1. LFVPSO-BPNN Algorithm

Back propagation neural network algorithm (BPNN) is a multilayer feed-forward network algorithm trained by back propagation of computational errors [26], and it is often used to predict and analyze nonlinear correlations. As presented in Figure 6, the BPNN topology structure is composed of input layer, hidden layers, and output layer [27].Wherex is the input variable of BPNN; Wh represents the transitive relation between input layer and hidden layer; θh is the threshold of the hidden layer neuron; Wo represents the transitive relation between hidden layer and output layer; θo is the threshold of output layer neuron; and y is the output variable of BPNN.Figure 6
BPNN topology structure [27].Before BPNN training, the weights and thresholds between neurons should be preset. Improper setting of weights and thresholds will lead to unreasonable prediction results. To improve the prediction accuracy of BPNN algorithm, the weights and thresholds of the network can be optimized in advance. Particle swarm optimization algorithm (PSO) has a good advantage in multiobjective optimizations. When large number of weights and thresholds of BPNN need to be optimized, the PSO algorithm can be used for the optimizations [28, 29]. However, it should be noted that the PSO algorithm may lead to local optimum results. Try to avoid this defect of PSO algorithm, the Levy flight algorithm is recommended to improve the performance of PSO [30, 31]. Levy flight trajectory is composed of frequent short-distance random walks and occasional long-distance random jumps. The Levy flight extremum can produce random jumps and extreme speed changes in direction, which is helpful to expand searching space. Therefore, combining PSO algorithm with Levy flight algorithm, the local optimum risk of PSO algorithm might be greatly reduced. Huseyin and Uǧuz [30] combined PSO algorithm with Levy flight algorithm to update the position of particle swarm, and then selected the optimum particle position to calculate the optimum value of the objective function. This improved PSO algorithm (LFPSO) can effectively avoid the local optimum risk of PSO algorithm, but its computation time is relatively long because sometimes the position of particle swarm is needed to be calculated twice for a single iteration during the iterative process. Jensi and Jiji [31] proposed an enhanced PSO algorithm together with Levy flight algorithm by setting a random number (rand ∈ [0, 1]) to combine the PSO algorithm with Levy flight algorithm (PSOLF). When rand ∈ [0.5, 1], the PSO algorithm and Levy flight algorithm were combined with linearly decreasing weight method to update the position of particle swarm and calculate the optimum value of the objective function. When rand ∈ [0, 0.5), only PSO algorithm was used to update the position of particle swarm and calculate the optimum value of the objective function. As compared with LFPSO algorithm [30], the computation time of this algorithm (PSOLF) can be greatly reduced. However, when the weight of the Levy flight algorithm becomes smaller, the local optimum risk of PSOLF algorithm would increase. Therefore, in this paper, based on the study of Jensi and Jiji [31], an adaptive optimized PSO algorithm (LFVPSO) is further proposed together with Levy flight algorithm, and it is used to find the optimum weights and thresholds of BPNN. For the LFVPSO algorithm, when rand ∈ [0.5, 1], the PSO algorithm and Levy flight algorithm are directly combined, and the adaptive mutation idea of genetic algorithm (GA) is also adopted to update the position of particle swarm and calculate the optimum value of the objective function. As compared with PSOLF algorithm [31], the local optimum risk of present algorithm (LFVPSO) would be reduced, and its prediction accuracy would be improved. According to the authors’ knowledge, almost no such studies were performed yet.In LFVPSO algorithm, a random number (rand ∈ [0, 1]) is set first. When rand ∈ [0, 0.5), the PSO algorithm is used to update the position of particle swarm. When rand ∈ [0.5, 1], the PSO algorithm is combined with Levy flight algorithm to update the position of particle swarm. Then, find out the position of the particle with the worst fitness in the current particle swarm and update particle swarm position to far away from the worst particle position at a certain rate. The next particle swarm position is calculated as in equation (8). Meanwhile, the adaptive mutation idea of genetic algorithm (GA) is also adopted, and the adaptive mutation operation is introduced into the PSO algorithm. For this operation, a random number (rand′ ∈ [0, 1]) is set, and the position of some particle in the swarm is randomly updated again with a certain probability as rand′ > 0.95. The flowchart for BPNN optimization with LFVPSO (LFVPSO-BPNN) is presented in Figure 7. In LFVPSO algorithm, the prediction error of BPNN is set as fitness function. The particle swarm position is randomly initialized using Levy flight algorithm and updated with equation (8). The optimum weights and thresholds of BPNN are calculated with iteration method to minimize the fitness function. Based on these operations, the optimum BPNN prediction model is constructed, which is the LFVPSO-BPNN model. The particle swarm position is updated and calculated as follows:(8)Xik+1=Xik+a1Vik+1−a2r1Pwk−Xik,rand∈0,0.5,Xik+1=Xik+a1Vik+1+ϑ0⋅Xik−Pgk⊗Levyβ−a2r3⋅Pwk−Xik,rand∈0.5,1,where Xik+1 is the position of particle i at iteration k + 1; Xik is the position of particle i at iteration k; Vik+1 is the velocity of particle i at iteration k + 1; Pwk is the position of the particle with the worst fitness in the current particle swarm; Pgk is the position of the particle with the best fitness in the previous particle swarm; a1 is the scaling factor of velocity weight, 0.2; a2 and ϑ0 are the step scaling factors, 0.01; r1 is a random number between 0 and 1; and β is the Levy distribution index.Figure 7
The flowchart for BPNN optimization with LFVPSO (LFVPSO-BPNN).


### 4.2. Temperature Prediction Network of Trench Laying Cable

When the trench structure and cable layout are determined, the cable maximum temperature of each loop in the trench (Tmax,i) is mainly affected by the air temperature on the ground (Tsur) and cable current of each loop (Ii), and the maximum current capacity of trench laying cable (It,max) is mainly affected by Tsur. Therefore, in this study, Tsur and Ii are selected as the input variables, Tmax,i is selected as the output variable, and a temperature prediction network of trench laying cable is developed with LFVPSO-BPNN. Then, It,max was calculated using LFVPSO-BPNN together with genetic algorithm (GA), where Tsur is set as the input variable, and the maximum value of Tmax,i should not exceed the safe operating temperature of cable (363.15 K). The flowchart of prediction algorithm for temperature rise of trench laying cable is presented in Figure 8. First, for different trench laying cable models, including flat-type laying model (Figure 1(a)) and trefoil-type laying model (Figure 1(b)), the input variables of training samples (Tsur and Ii) for temperature prediction network of trench laying cable (LFVPSO-BPNN) are selected using random sampling function. With this method, 80 training samples are selected for each cable model. The input variables of typical training samples for LFVPSO-BPNN are listed in Table 6, where Tsur changes from 283.15 K to 313.15 K, and Ii changes from 200 A to 1200 A. Based on the input variables of training samples in Table 6, the heat transfer processes for trench laying cables are numerically simulated with COMSOL Multiphysics, and the computational results of Tmax,i are set as the output variables of LFVPSO-BPNN. Then, the temperature prediction network for trench laying cable (LFVPSO-BPNN) is trained and constructed, and the predicted results of Tmax,i with LFVPSO-BPNN are further validated using COMSOL Multiphysics. Finally, the maximum current capacity of trench laying cable (It,max) under different Tsur is calculated using LFVPSO-BPNN together with genetic algorithm (GA&LFVPSO-BPNN). During the iteration process, the Tsur should be set in advance, and the maximum value of Tmax,i should not exceed the safe operating temperature of cable (363.15 K). After that, the predicted results with GA&LFVPSO-BPNN should also be validated using COMSOL Multiphysics.Figure 8
The flowchart of prediction algorithm for temperature rise of trench laying cable.Table 6
Input variables of typical training samples for LFVPSO-BPNN.
SamplesTsur (K)I1 (A)I2 (A)I3 (A)I4 (A)Flat-type trench laying cable1308.11554.02514.73861.26741.502295.38207.81625.83423.51956.983302.36788.37633.43444.17628.96..................78311.101014.92340.501079.87295.3879312.90356.64993.89841.81539.3880287.40637.42946.18667.911060.83Trefoil-type trench laying cable1310.921099.94724.11320.20377.792292.43929.32711.42404.30270.933293.66285.55267.381088.39433.17..................78311.67764.76716.10641.85374.4979288.16869.65976.86877.1604.8080303.72848.03633.37339.76951.93In this study, there are 5 input variables and 4 output variables in the temperature prediction network for trench laying cable (LFVPSO-BPNN). The input variables areTsur and Ii (i = 1, 2, 3, and 4), and the output variables are Tmax,i (i = 1, 2, 3, and 4). One hidden layer is set in the present temperature prediction network, and the hidden layer is composed of 10 neuron nodes. Therefore, there are 104 weights and thresholds need to be optimized for the prediction network. The weight and threshold for the prediction network are set to be ranged in [−5, 5], and the number of particle swarm for LFVPSO is set to 60. The fitness function of present prediction network is defined as follows:(9)fitness=∑iMy⌢−y⌢′i2,where y⌢ is the normalized simulation value of training samples; y⌢′ is the normalized prediction value of training samples; and M is the number of training samples, M = 80.The maximum current capacity of trench laying cable (It,max) is defined as follows:(10)It,max=Max∑143Ii2,Tmax,imax≤363.15K,where Ii is the single-cable current for loop i and the currents of three cables in the same loop are kept the same and Tmax,i is the cable maximum temperature of loop i in the trench.


## 4.1. LFVPSO-BPNN Algorithm

Back propagation neural network algorithm (BPNN) is a multilayer feed-forward network algorithm trained by back propagation of computational errors [26], and it is often used to predict and analyze nonlinear correlations. As presented in Figure 6, the BPNN topology structure is composed of input layer, hidden layers, and output layer [27].Wherex is the input variable of BPNN; Wh represents the transitive relation between input layer and hidden layer; θh is the threshold of the hidden layer neuron; Wo represents the transitive relation between hidden layer and output layer; θo is the threshold of output layer neuron; and y is the output variable of BPNN.Figure 6
BPNN topology structure [27].Before BPNN training, the weights and thresholds between neurons should be preset. Improper setting of weights and thresholds will lead to unreasonable prediction results. To improve the prediction accuracy of BPNN algorithm, the weights and thresholds of the network can be optimized in advance. Particle swarm optimization algorithm (PSO) has a good advantage in multiobjective optimizations. When large number of weights and thresholds of BPNN need to be optimized, the PSO algorithm can be used for the optimizations [28, 29]. However, it should be noted that the PSO algorithm may lead to local optimum results. Try to avoid this defect of PSO algorithm, the Levy flight algorithm is recommended to improve the performance of PSO [30, 31]. Levy flight trajectory is composed of frequent short-distance random walks and occasional long-distance random jumps. The Levy flight extremum can produce random jumps and extreme speed changes in direction, which is helpful to expand searching space. Therefore, combining PSO algorithm with Levy flight algorithm, the local optimum risk of PSO algorithm might be greatly reduced. Huseyin and Uǧuz [30] combined PSO algorithm with Levy flight algorithm to update the position of particle swarm, and then selected the optimum particle position to calculate the optimum value of the objective function. This improved PSO algorithm (LFPSO) can effectively avoid the local optimum risk of PSO algorithm, but its computation time is relatively long because sometimes the position of particle swarm is needed to be calculated twice for a single iteration during the iterative process. Jensi and Jiji [31] proposed an enhanced PSO algorithm together with Levy flight algorithm by setting a random number (rand ∈ [0, 1]) to combine the PSO algorithm with Levy flight algorithm (PSOLF). When rand ∈ [0.5, 1], the PSO algorithm and Levy flight algorithm were combined with linearly decreasing weight method to update the position of particle swarm and calculate the optimum value of the objective function. When rand ∈ [0, 0.5), only PSO algorithm was used to update the position of particle swarm and calculate the optimum value of the objective function. As compared with LFPSO algorithm [30], the computation time of this algorithm (PSOLF) can be greatly reduced. However, when the weight of the Levy flight algorithm becomes smaller, the local optimum risk of PSOLF algorithm would increase. Therefore, in this paper, based on the study of Jensi and Jiji [31], an adaptive optimized PSO algorithm (LFVPSO) is further proposed together with Levy flight algorithm, and it is used to find the optimum weights and thresholds of BPNN. For the LFVPSO algorithm, when rand ∈ [0.5, 1], the PSO algorithm and Levy flight algorithm are directly combined, and the adaptive mutation idea of genetic algorithm (GA) is also adopted to update the position of particle swarm and calculate the optimum value of the objective function. As compared with PSOLF algorithm [31], the local optimum risk of present algorithm (LFVPSO) would be reduced, and its prediction accuracy would be improved. According to the authors’ knowledge, almost no such studies were performed yet.In LFVPSO algorithm, a random number (rand ∈ [0, 1]) is set first. When rand ∈ [0, 0.5), the PSO algorithm is used to update the position of particle swarm. When rand ∈ [0.5, 1], the PSO algorithm is combined with Levy flight algorithm to update the position of particle swarm. Then, find out the position of the particle with the worst fitness in the current particle swarm and update particle swarm position to far away from the worst particle position at a certain rate. The next particle swarm position is calculated as in equation (8). Meanwhile, the adaptive mutation idea of genetic algorithm (GA) is also adopted, and the adaptive mutation operation is introduced into the PSO algorithm. For this operation, a random number (rand′ ∈ [0, 1]) is set, and the position of some particle in the swarm is randomly updated again with a certain probability as rand′ > 0.95. The flowchart for BPNN optimization with LFVPSO (LFVPSO-BPNN) is presented in Figure 7. In LFVPSO algorithm, the prediction error of BPNN is set as fitness function. The particle swarm position is randomly initialized using Levy flight algorithm and updated with equation (8). The optimum weights and thresholds of BPNN are calculated with iteration method to minimize the fitness function. Based on these operations, the optimum BPNN prediction model is constructed, which is the LFVPSO-BPNN model. The particle swarm position is updated and calculated as follows:(8)Xik+1=Xik+a1Vik+1−a2r1Pwk−Xik,rand∈0,0.5,Xik+1=Xik+a1Vik+1+ϑ0⋅Xik−Pgk⊗Levyβ−a2r3⋅Pwk−Xik,rand∈0.5,1,where Xik+1 is the position of particle i at iteration k + 1; Xik is the position of particle i at iteration k; Vik+1 is the velocity of particle i at iteration k + 1; Pwk is the position of the particle with the worst fitness in the current particle swarm; Pgk is the position of the particle with the best fitness in the previous particle swarm; a1 is the scaling factor of velocity weight, 0.2; a2 and ϑ0 are the step scaling factors, 0.01; r1 is a random number between 0 and 1; and β is the Levy distribution index.Figure 7
The flowchart for BPNN optimization with LFVPSO (LFVPSO-BPNN).


## 4.2. Temperature Prediction Network of Trench Laying Cable

When the trench structure and cable layout are determined, the cable maximum temperature of each loop in the trench (Tmax,i) is mainly affected by the air temperature on the ground (Tsur) and cable current of each loop (Ii), and the maximum current capacity of trench laying cable (It,max) is mainly affected by Tsur. Therefore, in this study, Tsur and Ii are selected as the input variables, Tmax,i is selected as the output variable, and a temperature prediction network of trench laying cable is developed with LFVPSO-BPNN. Then, It,max was calculated using LFVPSO-BPNN together with genetic algorithm (GA), where Tsur is set as the input variable, and the maximum value of Tmax,i should not exceed the safe operating temperature of cable (363.15 K). The flowchart of prediction algorithm for temperature rise of trench laying cable is presented in Figure 8. First, for different trench laying cable models, including flat-type laying model (Figure 1(a)) and trefoil-type laying model (Figure 1(b)), the input variables of training samples (Tsur and Ii) for temperature prediction network of trench laying cable (LFVPSO-BPNN) are selected using random sampling function. With this method, 80 training samples are selected for each cable model. The input variables of typical training samples for LFVPSO-BPNN are listed in Table 6, where Tsur changes from 283.15 K to 313.15 K, and Ii changes from 200 A to 1200 A. Based on the input variables of training samples in Table 6, the heat transfer processes for trench laying cables are numerically simulated with COMSOL Multiphysics, and the computational results of Tmax,i are set as the output variables of LFVPSO-BPNN. Then, the temperature prediction network for trench laying cable (LFVPSO-BPNN) is trained and constructed, and the predicted results of Tmax,i with LFVPSO-BPNN are further validated using COMSOL Multiphysics. Finally, the maximum current capacity of trench laying cable (It,max) under different Tsur is calculated using LFVPSO-BPNN together with genetic algorithm (GA&LFVPSO-BPNN). During the iteration process, the Tsur should be set in advance, and the maximum value of Tmax,i should not exceed the safe operating temperature of cable (363.15 K). After that, the predicted results with GA&LFVPSO-BPNN should also be validated using COMSOL Multiphysics.Figure 8
The flowchart of prediction algorithm for temperature rise of trench laying cable.Table 6
Input variables of typical training samples for LFVPSO-BPNN.
SamplesTsur (K)I1 (A)I2 (A)I3 (A)I4 (A)Flat-type trench laying cable1308.11554.02514.73861.26741.502295.38207.81625.83423.51956.983302.36788.37633.43444.17628.96..................78311.101014.92340.501079.87295.3879312.90356.64993.89841.81539.3880287.40637.42946.18667.911060.83Trefoil-type trench laying cable1310.921099.94724.11320.20377.792292.43929.32711.42404.30270.933293.66285.55267.381088.39433.17..................78311.67764.76716.10641.85374.4979288.16869.65976.86877.1604.8080303.72848.03633.37339.76951.93In this study, there are 5 input variables and 4 output variables in the temperature prediction network for trench laying cable (LFVPSO-BPNN). The input variables areTsur and Ii (i = 1, 2, 3, and 4), and the output variables are Tmax,i (i = 1, 2, 3, and 4). One hidden layer is set in the present temperature prediction network, and the hidden layer is composed of 10 neuron nodes. Therefore, there are 104 weights and thresholds need to be optimized for the prediction network. The weight and threshold for the prediction network are set to be ranged in [−5, 5], and the number of particle swarm for LFVPSO is set to 60. The fitness function of present prediction network is defined as follows:(9)fitness=∑iMy⌢−y⌢′i2,where y⌢ is the normalized simulation value of training samples; y⌢′ is the normalized prediction value of training samples; and M is the number of training samples, M = 80.The maximum current capacity of trench laying cable (It,max) is defined as follows:(10)It,max=Max∑143Ii2,Tmax,imax≤363.15K,where Ii is the single-cable current for loop i and the currents of three cables in the same loop are kept the same and Tmax,i is the cable maximum temperature of loop i in the trench.


## 5. Results and Discussion

### 5.1. Temperature Prediction for Trench Laying Cable

The prediction and simulation results of cable maximum temperature for different loops in the trench (80 training samples for the flat-type trench laying cable model) are presented in Figure9. It shows that the correlation coefficient (R2) between the prediction and simulation results of cable maximum temperature for different loops in the trench (flat-type trench laying cable model) is higher than 0.9990. Similarly, the prediction and simulation results of cable maximum temperature for different loops in the trench (80 training samples for the trefoil-type trench laying cable model) is presented in Figure 10. It shows that the correlation coefficient (R2) between the prediction and simulation results of cable maximum temperature for different loops in the trench (trefoil-type trench laying cable model) is higher than 0.9992. This means the temperature prediction network (LFVPSO-BPNN) for both flat-type and trefoil-type trench laying cable models trained in this study is reliable, which can successfully predict the cable maximum temperature of different loops in the trench. Meanwhile, five additional random test samples are also selected to verify the accuracy of the temperature prediction network (LFVPSO-BPNN). The prediction and simulation results of cable maximum temperature for different loops in the trench (5 random test samples for different trench laying cable models) are presented in Tables 7 and 8. They shows that the predicted cable maximum temperatures for different loops (Tmax,i) in the trench can agree well with those obtained by COMSOL Multiphysics, and the maximum deviation is less than 2 K. This confirms that the LFVPSO-BPNN algorithm proposed in this study can accurately predict the cable maximum temperature for four-loop trench laying cables.Figure 9
The prediction and simulation results of cable maximum temperature for different loops in the trench (80 training samples for the flat-type trench laying cable model): (a) Loop 1; (b) Loop 2; (c) Loop 3; (d) Loop 4.
(a)(b)(c)(d)Figure 10
The prediction and simulation results of cable maximum temperature for different loops in the trench (80 training samples for the trefoil-type trench laying cable model): (a) Loop 1; (b) Loop 2; (c) Loop 3; (d) Loop 4.
(a)(b)(c)(d)Table 7
The prediction and simulation results of cable maximum temperature for different loops in the trench (5 random test samples for the flat-type trench laying cable models).
CaseTsur (K)I1 (A)I2 (A)I3 (A)I4 (A)Input1304.33941.11825.35485.391055.202284.10231.00594.87543.40888.973291.46915.68368.19640.79601.034310.67337.14943.24684.511096.525305.59275.99559.80433.88920.06CaseLFVPSO-BPNNCOMSOL MultiphysicsTmax,1 (K)Tmax,2 (K)Tmax,3 (K)Tmax,4 (K)Tmax,1 (K)Tmax,2 (K)Tmax,3 (K)Tmax,4 (K)Output1355.53348.99336.81364.49354.91348.70336.03363.272297.51307.24304.27322.88298.35307.83305.95322.323331.44309.58319.49315.85331.42309.83319.21317.434337.21360.46346.99371.73336.68360.09347.39371.045318.92326.98322.51344.59319.33326.85322.64343.92Table 8
The prediction and simulation results of cable maximum temperature for different loops in the trench (5 random test samples for the trefoil-type trench laying cable models).
CaseTsur (K)I1 (A)I2 (A)I3 (A)I4 (A)Input1293.54596.05765.13394.86828.062305.35866.33643.07633.3375.243306.64552.56793.42785.18867.684305.80885.1521.95454.42686.915309.71456.111086.56646.81015.99CaseLFVPSO-BPNNCOMSOL MultiphysicsTmax,1 (K)Tmax,2 (K)Tmax,3 (K)Tmax,4 (K)Tmax,1 (K)Tmax,2 (K)Tmax,3 (K)Tmax,4 (K)Output1323.46331.00314.61336.96322.27331.47313.84337.532347.95336.14336.14324.74348.10335.48335.70325.093337.69350.01349.72357.11338.18350.24350.07357.034349.50330.27328.46338.26350.35329.16328.80337.645342.75378.23350.32375.5343.07377.40350.45376.17Based on the same training samples (Table6) and same number of particle swarm (60), the convergence curves of fitness function for the modified BPNN with LFPSO [30], PSOLF [31] or LFVPSO algorithm are presented in Figure 11. It shows that, for the flat-type trench laying cable model, when the iteration reaches convergence, the computation times for the LFPSO-BPNN, PSOLF-BPNN, and LFVPSO-BPNN algorithm are 5651 s, 3801 s, and 3977 s, respectively, and their convergence values of fitness function are 0.2307, 0.2223, and 0.1005, respectively. Similarly, for the trefoil-type trench laying cable model, when the iteration reaches convergence, the computation times for the LFPSO-BPNN, PSOLF-BPNN, and LFVPSO-BPNN algorithm are 5918 s, 3977 s, and 4195 s, respectively, and their convergence values of fitness function are 0.1672, 0.2070, and 0.1008, respectively. It is indicated that, under the same restrictions, as compared with LFPSO-BPNN algorithm, the computation time would be great reduced by using PSOLF-BPNN and LFVPSO-BPNN algorithm. Meanwhile, as compared with LFPSO-BPNN and PSOLF-BPNN algorithm, the prediction accuracy would be further improved using LFVPSO-BPNN algorithm. Therefore, as compared with LFPSO-BPNN and PSOLF-BPNN algorithm, when LFVPSO-BPNN algorithm is used to predict the temperature of four-loop three-phase trench laying cables, its computation time would be reduced and the prediction accuracy would be improved as well.Figure 11
The convergence curves of fitness function for the modified BPNN with LFPSO [30], PSOLF [31], or LFVPSO algorithm: (a) flat-type trench laying cable and (b) trefoil-type trench laying cable.
(a)(b)


### 5.2. Prediction of Maximum Current Capacity for Trench Laying Cable

The prediction and simulation results of maximum current capacity for different trench laying cables under differentTsur are presented in Table 9. The air temperature on the ground (Tsur) is set as 283.15 K and 313.15 K, which represents moderate environment temperature and hot summer environment temperature, respectively. To reduce the prediction error of GA&LFVPSO-BPNN algorithm, the cable current of each loop (Ii) and the maximum current capacity of trench laying cable (It,max) were calculated three times and averaged under different Tsur. Meanwhile, the Tsur and Ii predicted using GA&LFVPSO-BPNN algorithm are input into COMSOL Multiphysics, and the maximum value of the cable maximum temperature for different loops in the trench ((Tmax,i)max) is computed and compared with the safe operating temperature of cable (363.15 K). As is shown in Table 9, for different cases, the maximum deviation between the simulation results of (Tmax,i)max and safe operating temperature of cable (363.15 K) is 0.58 K, which means the prediction results obtained using GA&LFVPSO-BPNN algorithm should be reliable.Table 9
The prediction and simulation results of maximum current capacity for different trench laying cables (Tsur = 283.15 K and 313.15 K).
CasesTsur(K)GA&LFVPSO-BPNNCOMSOLI1(A)I2(A)I3(A)I4(A)It,max (A)(Tmax,i)max (K)(Tmax,i)max (K)Flat-type trench laying cable1283.151111.221136.971093.901102.139427.61363.15—2283.151110.861138.371092.231102.139426.28363.15—3283.151111.161137.481093.261102.869428.76363.15—Averaged283.151111.081137.611093.131102.379427.55363.15363.731313.15892.48905.78877.03875.877533.15363.15—2313.15893.95905.64876.60874.777532.72363.15—3313.15894.37903.90876.00875.957531.15363.15—Averaged313.15893.60905.10876.54875.537532.32363.15363.00Trefoil-type trench laying cable1283.151043.161080.67997.771012.598769.94363.15—2283.151043.521080.57998.081011.928769.73363.15—3283.151044.681080.74997.251011.618770.13363.15—Averaged283.151043.781080.66997.701012.048769.92363.15362.951313.15848.16852.47813.57833.917102.41363.15—2313.15847.29852.94812.61833.797099.27363.15—3313.15848.20852.45813.55833.937102.46363.15—Averaged313.15847.89852.62813.25833.887101.42363.15363.32In Table9, it also shows that, under different Tsur, when the current capacity of trench laying cable (It) reaches its maximum value (It, max), the cable current of Loop 2 (I2) is the highest, and the cable currents of upper layer loops (I1, I2) are higher than those of lower layer loops (I3, I4). This may indicate that, for the case under It,max, when both natural convection and radiation heat transfer are considered on the cable surfaces, the overall heat transfer on the cable surfaces of Loop 2 should be the best, and the heat transfer on the cable surfaces of upper layer loops should be better than that of lower layer loops. For the flat-type trench laying cable model, as Tsur increases from 283.15 K to 313.15 K, the maximum current capacity of trench laying cable (It,max) decreases from 9427.55 A to 7532.32 A, which decreases by 1895.23 A (20%). Similarly, for the trefoil-type trench laying cable model, as Tsur increases from 283.15 K to 313.15 K, the maximum current capacity of trench laying cable (It,max) decreases from 8769.92 A to 7101.42 A, which decreases by 1668.5 A (19%). This may indicate that when Tsur increases from 283.15 K to 313.15 K, the overall heat transfer of trench laying cable should be weakened, and its maximum current capacity (It,max) is decreased significantly. Furthermore, when air temperature on the ground (Tsur) is fixed at 283.15 K, as compared with the trefoil-type laying trench cable model, the It,max for the flat-type trench laying cable model can increase by 657.63 A (7.5%). Similarly, when air temperature on the ground (Tsur) is fixed at 313.15 K, as compared with the trefoil-type laying trench cable model, the It,max for the flat-type trench laying cable model can increase by 430.9 A (6.1%). This means, under the same Tsur, the overall heat transfer performance for the flat-type trench laying cable should be better than that for the trefoil-type laying trench cable, and the maximum current capacity (It,max) for the flat-type trench laying cable are obviously higher. Therefore, with the same working conditions, the flat-type laying method is recommended for the cable arrangement in the trench.


## 5.1. Temperature Prediction for Trench Laying Cable

The prediction and simulation results of cable maximum temperature for different loops in the trench (80 training samples for the flat-type trench laying cable model) are presented in Figure9. It shows that the correlation coefficient (R2) between the prediction and simulation results of cable maximum temperature for different loops in the trench (flat-type trench laying cable model) is higher than 0.9990. Similarly, the prediction and simulation results of cable maximum temperature for different loops in the trench (80 training samples for the trefoil-type trench laying cable model) is presented in Figure 10. It shows that the correlation coefficient (R2) between the prediction and simulation results of cable maximum temperature for different loops in the trench (trefoil-type trench laying cable model) is higher than 0.9992. This means the temperature prediction network (LFVPSO-BPNN) for both flat-type and trefoil-type trench laying cable models trained in this study is reliable, which can successfully predict the cable maximum temperature of different loops in the trench. Meanwhile, five additional random test samples are also selected to verify the accuracy of the temperature prediction network (LFVPSO-BPNN). The prediction and simulation results of cable maximum temperature for different loops in the trench (5 random test samples for different trench laying cable models) are presented in Tables 7 and 8. They shows that the predicted cable maximum temperatures for different loops (Tmax,i) in the trench can agree well with those obtained by COMSOL Multiphysics, and the maximum deviation is less than 2 K. This confirms that the LFVPSO-BPNN algorithm proposed in this study can accurately predict the cable maximum temperature for four-loop trench laying cables.Figure 9
The prediction and simulation results of cable maximum temperature for different loops in the trench (80 training samples for the flat-type trench laying cable model): (a) Loop 1; (b) Loop 2; (c) Loop 3; (d) Loop 4.
(a)(b)(c)(d)Figure 10
The prediction and simulation results of cable maximum temperature for different loops in the trench (80 training samples for the trefoil-type trench laying cable model): (a) Loop 1; (b) Loop 2; (c) Loop 3; (d) Loop 4.
(a)(b)(c)(d)Table 7
The prediction and simulation results of cable maximum temperature for different loops in the trench (5 random test samples for the flat-type trench laying cable models).
CaseTsur (K)I1 (A)I2 (A)I3 (A)I4 (A)Input1304.33941.11825.35485.391055.202284.10231.00594.87543.40888.973291.46915.68368.19640.79601.034310.67337.14943.24684.511096.525305.59275.99559.80433.88920.06CaseLFVPSO-BPNNCOMSOL MultiphysicsTmax,1 (K)Tmax,2 (K)Tmax,3 (K)Tmax,4 (K)Tmax,1 (K)Tmax,2 (K)Tmax,3 (K)Tmax,4 (K)Output1355.53348.99336.81364.49354.91348.70336.03363.272297.51307.24304.27322.88298.35307.83305.95322.323331.44309.58319.49315.85331.42309.83319.21317.434337.21360.46346.99371.73336.68360.09347.39371.045318.92326.98322.51344.59319.33326.85322.64343.92Table 8
The prediction and simulation results of cable maximum temperature for different loops in the trench (5 random test samples for the trefoil-type trench laying cable models).
CaseTsur (K)I1 (A)I2 (A)I3 (A)I4 (A)Input1293.54596.05765.13394.86828.062305.35866.33643.07633.3375.243306.64552.56793.42785.18867.684305.80885.1521.95454.42686.915309.71456.111086.56646.81015.99CaseLFVPSO-BPNNCOMSOL MultiphysicsTmax,1 (K)Tmax,2 (K)Tmax,3 (K)Tmax,4 (K)Tmax,1 (K)Tmax,2 (K)Tmax,3 (K)Tmax,4 (K)Output1323.46331.00314.61336.96322.27331.47313.84337.532347.95336.14336.14324.74348.10335.48335.70325.093337.69350.01349.72357.11338.18350.24350.07357.034349.50330.27328.46338.26350.35329.16328.80337.645342.75378.23350.32375.5343.07377.40350.45376.17Based on the same training samples (Table6) and same number of particle swarm (60), the convergence curves of fitness function for the modified BPNN with LFPSO [30], PSOLF [31] or LFVPSO algorithm are presented in Figure 11. It shows that, for the flat-type trench laying cable model, when the iteration reaches convergence, the computation times for the LFPSO-BPNN, PSOLF-BPNN, and LFVPSO-BPNN algorithm are 5651 s, 3801 s, and 3977 s, respectively, and their convergence values of fitness function are 0.2307, 0.2223, and 0.1005, respectively. Similarly, for the trefoil-type trench laying cable model, when the iteration reaches convergence, the computation times for the LFPSO-BPNN, PSOLF-BPNN, and LFVPSO-BPNN algorithm are 5918 s, 3977 s, and 4195 s, respectively, and their convergence values of fitness function are 0.1672, 0.2070, and 0.1008, respectively. It is indicated that, under the same restrictions, as compared with LFPSO-BPNN algorithm, the computation time would be great reduced by using PSOLF-BPNN and LFVPSO-BPNN algorithm. Meanwhile, as compared with LFPSO-BPNN and PSOLF-BPNN algorithm, the prediction accuracy would be further improved using LFVPSO-BPNN algorithm. Therefore, as compared with LFPSO-BPNN and PSOLF-BPNN algorithm, when LFVPSO-BPNN algorithm is used to predict the temperature of four-loop three-phase trench laying cables, its computation time would be reduced and the prediction accuracy would be improved as well.Figure 11
The convergence curves of fitness function for the modified BPNN with LFPSO [30], PSOLF [31], or LFVPSO algorithm: (a) flat-type trench laying cable and (b) trefoil-type trench laying cable.
(a)(b)


## 5.2. Prediction of Maximum Current Capacity for Trench Laying Cable

The prediction and simulation results of maximum current capacity for different trench laying cables under differentTsur are presented in Table 9. The air temperature on the ground (Tsur) is set as 283.15 K and 313.15 K, which represents moderate environment temperature and hot summer environment temperature, respectively. To reduce the prediction error of GA&LFVPSO-BPNN algorithm, the cable current of each loop (Ii) and the maximum current capacity of trench laying cable (It,max) were calculated three times and averaged under different Tsur. Meanwhile, the Tsur and Ii predicted using GA&LFVPSO-BPNN algorithm are input into COMSOL Multiphysics, and the maximum value of the cable maximum temperature for different loops in the trench ((Tmax,i)max) is computed and compared with the safe operating temperature of cable (363.15 K). As is shown in Table 9, for different cases, the maximum deviation between the simulation results of (Tmax,i)max and safe operating temperature of cable (363.15 K) is 0.58 K, which means the prediction results obtained using GA&LFVPSO-BPNN algorithm should be reliable.Table 9
The prediction and simulation results of maximum current capacity for different trench laying cables (Tsur = 283.15 K and 313.15 K).
CasesTsur(K)GA&LFVPSO-BPNNCOMSOLI1(A)I2(A)I3(A)I4(A)It,max (A)(Tmax,i)max (K)(Tmax,i)max (K)Flat-type trench laying cable1283.151111.221136.971093.901102.139427.61363.15—2283.151110.861138.371092.231102.139426.28363.15—3283.151111.161137.481093.261102.869428.76363.15—Averaged283.151111.081137.611093.131102.379427.55363.15363.731313.15892.48905.78877.03875.877533.15363.15—2313.15893.95905.64876.60874.777532.72363.15—3313.15894.37903.90876.00875.957531.15363.15—Averaged313.15893.60905.10876.54875.537532.32363.15363.00Trefoil-type trench laying cable1283.151043.161080.67997.771012.598769.94363.15—2283.151043.521080.57998.081011.928769.73363.15—3283.151044.681080.74997.251011.618770.13363.15—Averaged283.151043.781080.66997.701012.048769.92363.15362.951313.15848.16852.47813.57833.917102.41363.15—2313.15847.29852.94812.61833.797099.27363.15—3313.15848.20852.45813.55833.937102.46363.15—Averaged313.15847.89852.62813.25833.887101.42363.15363.32In Table9, it also shows that, under different Tsur, when the current capacity of trench laying cable (It) reaches its maximum value (It, max), the cable current of Loop 2 (I2) is the highest, and the cable currents of upper layer loops (I1, I2) are higher than those of lower layer loops (I3, I4). This may indicate that, for the case under It,max, when both natural convection and radiation heat transfer are considered on the cable surfaces, the overall heat transfer on the cable surfaces of Loop 2 should be the best, and the heat transfer on the cable surfaces of upper layer loops should be better than that of lower layer loops. For the flat-type trench laying cable model, as Tsur increases from 283.15 K to 313.15 K, the maximum current capacity of trench laying cable (It,max) decreases from 9427.55 A to 7532.32 A, which decreases by 1895.23 A (20%). Similarly, for the trefoil-type trench laying cable model, as Tsur increases from 283.15 K to 313.15 K, the maximum current capacity of trench laying cable (It,max) decreases from 8769.92 A to 7101.42 A, which decreases by 1668.5 A (19%). This may indicate that when Tsur increases from 283.15 K to 313.15 K, the overall heat transfer of trench laying cable should be weakened, and its maximum current capacity (It,max) is decreased significantly. Furthermore, when air temperature on the ground (Tsur) is fixed at 283.15 K, as compared with the trefoil-type laying trench cable model, the It,max for the flat-type trench laying cable model can increase by 657.63 A (7.5%). Similarly, when air temperature on the ground (Tsur) is fixed at 313.15 K, as compared with the trefoil-type laying trench cable model, the It,max for the flat-type trench laying cable model can increase by 430.9 A (6.1%). This means, under the same Tsur, the overall heat transfer performance for the flat-type trench laying cable should be better than that for the trefoil-type laying trench cable, and the maximum current capacity (It,max) for the flat-type trench laying cable are obviously higher. Therefore, with the same working conditions, the flat-type laying method is recommended for the cable arrangement in the trench.


## 6. Conclusion

In this study, an adaptive optimized particle swarm optimization algorithm (LFVPSO) is proposed based on Levy flight algorithm, and it is used to modify the backpropagation neural network algorithm (LFVPSO-BPNN). Then combined with numerical simulations, a network algorithm for temperature prediction of trench laying cable is developed using LFVPSO-BPNN. Finally, the maximum current capacity of four-loop three-phase trench laying cable is calculated using LFVPSO-BPNN together with genetic algorithm (GA&LFVPSO-BPNN). The main findings are as follows:(1)
The correlation coefficient between the prediction and simulation results of cable maximum temperature for different loops (Tmax,i) in the trench is higher than 0.9990, and the maximum deviation of Tmax,i is less than 2 K. Therefore, the LFVPSO-BPNN algorithm proposed in this study can accurately predict Tmax,i for four-loop trench laying cables. Furthermore, under the same restrictions, as compared with LFPSO-BPNN and PSOLF-BPNN algorithm, when LFVPSO-BPNN algorithm is used to predict the temperature of four-loop three-phase trench laying cable, its computation time would be reduced, and the prediction accuracy would be improved as well.(2)
The effect of ground air temperature (Tsur) on the maximum current capacity of trench laying cable (It,max) is remarkable. When Tsur increases from 283.15 K to 313.15 K, the overall heat transfer of trench laying cable should be weakened, and its maximum current capacity (It,max) is decreased significantly. For the flat-type trench laying cable model, the It,max decreases from 9427.55 A to 7532.32 A, which decreases by 20%. And for the trefoil-type trench laying cable model, the It,max decreases from 8769.92 A to 7101.42 A, which decreases by 19%.(3)
Under the same ground air temperature (Tsur), the overall heat transfer performance for the flat-type trench laying cable should be better than that for the trefoil-type laying trench cable, and the maximum current capacity (It,max) for the flat-type trench laying cable is obviously higher. When Tsur is fixed at 283.15 K, as compared with the trefoil-type laying trench cable model, the It,max for the flat-type trench laying cable model can increase by 7.5%. Meanwhile, when Tsur is fixed at 313.15 K, as compared with the trefoil-type laying trench cable model, the It,max for the flat-type trench laying cable model can increase by 6.1%.These results would be meaningful for temperature rapid prediction and maximum current capacity calculation of trench laying cable. Furthermore, it should also be noticed that, the service life of a trench laying cable is closely related to the cable temperature. Therefore, as next step, the prediction work for the service life of a trench laying cable based on the predicted cable temperature should also be performed in the near future.



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*Source: 9271283-2021-10-29.xml*