Application of the hp-finite element method to modeling thermal fields of high voltage underground cables buried in multi-layer soil

In this paper, we investigate the application of the adaptive higher-order Finite Element Method (hp-FEM) to heat transfer problems in electrical engineering. The proposed method is developed based on the combination of the Delaunay mesh and higher-order interpolation functions. In which the Delaunay algorithm based on the distance function is used for creating the adaptive mesh in the whole solution domain and the higher-order polynomials (up to 9 order) are applied for increasing the accuracy of solution. To evaluate the applicability and effectiveness of this new approach, we applied the proposed method to solve a benchmark heat problem and to calculate the temperature distribution of some typical models of buried doubleand single -circuit power cables in the homogenous and multi-layer soils, respectively.


INTRODUCTION
The underground cables system, which is one of main transmission and distribution systems of power systems, is used by the power companies and industry in densely populated cities instead of overhead lines even its installation and maintenance are more expensive and complicated.Moreover, the stability and safety operations of buried power cables are the expectation of power utilities.
The important characteristics in operation and design of the underground power cables are the current-carrying capacity and usable working life.These values very much depend on the maximum operating temperature and the ability to transfer the cables-generated heat to the surrounding soil domain.Therefore, the thermal field computation of buried power cables is a very important task of many power engineers, researchers and manufactures all over the world.
In general, this problem is solved by using the analytical and/or numerical methods.In particular, the results of developments and applications of the numerical methods to engineering problems have gained much attention in recent years.
The numerical methods, such as Finite Difference Method (FDM), Boundary Element Method (BEM) and Finite Element Method (FEM) and Meshfree methods, with their advantage is to provide more accurate simulating than the analytical method in complex geometries have been appling to calculate several practical heat transfer problems in electrical engineering such as the thermal field distribution of underground cables [1]-[4], heat simulation for MEMS design [5] and thermal field of transformer [6].In recent years, the hp-FEM has been strongly developed [7]- [8] and successfully applied to many problems of civil, mechanical and electrical engineering [9] due to this method has given the very high accurate solutions.Besides, the adaptive Delaunay mesh that is still used in the application to FE -[10] creates the flexible FE mesh in the whole solving region.Which means that the small-size elements are much more efficient and distributed in domain where the solution has important features.Thus this algorithm can decrease CPU times but still ensure the high accuracy of numerical solution.
Unfortunately, the application of the hp-FEM to heat transfer problems is very rare.For this reason, we have proposed an approach of the hp-FEM that is the combination of the adaptive Delaunay mesh and higher-order interpolation functions.The advantages of this approach are that it strongly increases the accuracy of solution and can decrease the CPU times compared with the uniform mesh and/or lower-order FEM.In order to demonstrate the advantage and applicability of this method, we have used it to test on the benchmark heat problem and to calculate the steady-state thermal distribution of some typical power cable systems buried in the homogenous and multi-layer soils.

THE hp-FEM FOR THERMAL TRANSFER PROBLEMS
In general, the Poisson equation describes the steady state heat transfer in homogeneous medium can be written as where k( o Cm/W) is the thermal resistivity of medium.Q(W/m) is the heat generation rate in the heat source and T( o C) is the unknown temperature.
In the two-dimensional medium, the unknown temperature function In which [7] we have L L L are the area coordinates of e th element.
In the hp-FE procedure, the whole solving domain is subdivided by the triangular elements, where the total number of nodes in per element depends on the order of interpolation polynomial.

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The hp-FE solution will be obtained by minimizing the numerical integrations of piecewise polynomials.Finally, we have the matrix equation as follows where T is a column vector of temperatures at finite element nodes, A is the heat conductivity matrix with , 1,2, and b is a vector is given by 1, 2,..., NUMERICAL RESULTS

Benchmark heat problem
In order to demonstrate the advantage of the the hp-FEM, we now apply it to the benchmark heat problem which has the following equation where one assumes that k=1 and without internal heat source - [5], and the function values on boundaries are as , 0 The analytical solution of this benchmark problem is given by In this section, we have used the higher-order (up to 6 th -order) FEM to solve this benchmark problem.The contour plot of isothermal lines and the error comparison between the higher-order FE solutions are illustrated in Fig. 1. and TABLE.I., respectively.It has shown that the solutions of the higher-order FEM are much more accurate than the one of the 1 st -order FEM.

Temperature distribution of three-phase cable system buried in homogenous soil
In this section, we present the application of the hp-FEM to calculate the temperature distribution of high voltage underground cables buried in rectangular homogenous soil as in Fig. 2. Besides, the following assumptions are also given as The effect of radiation and convection at the ground surface are neglected, Thermal resistivity of soil is constant, The length of cable is more larger than its buried depth.Fig. 2. Model of the buried three-phase cable system in rectangular soil.
As above introduced, the first step of the adaptive hp-FE procedure is to subdivide the solving domain by mean of adaptive triangular elements corresponding to nodes.In this work, we use the adaptive Delaunay algorithm [10] to create the adaptive elements in the whole solving domain, and then we apply the higher-order interpolation polynomial to each element.
Finally, the hp-FE solutions are obtained by solving (6).In this approach, each conductor of cable system is assumed to be a heat source and data of this problem given by

Temperature distribution of three-phase cable system buried in two-layer soil
In this case, we study the thermal behavior of cables in two-layer medium.We use the cables in the case of double circuit and cables are directly buried at the depth of 1.9m in native soil.The boundary beetwen two layer at the depth of 3.0m.
The values of thermal conductivity of upper and under layer are 1.00 (W/ o Cm), 1.30 (W/ o Cm), respectively.The ground surface is represented by convective boundary (Cauchy condition) with convection loss coefficient of 5 (W/ o Cm 2 ).The phase spacing of each cable circuit is 0.25m and cables still loaded 662A and generate heat rate of 32.029 (W/m).

Temperature distribution of double-circuit three-phase cable system buried in multi-layer soil
In this case, we test the double-circuit threephase underground cables system buried at the depth of 1.9m in cable bedding in multi-layer soil.The upper layer is trench backfill and surrounding medium is native soil.The values of thermal conductivity of cable bedding, trench backfill and native soil are 1.00 (W/ o Cm), 1.25 (W/ o Cm) and 0.80 (W/ o Cm), respectively.The width and height of cable bedding are 1.80m and 0.6m, respectively.The ground surface represented by convective boundary (Cauchy condition) with convection loss coefficient of 5 (W/ o Cm 2 ).The phase spacing of each cable circuit is 0.25(m) and the nearest distance between two circuits is 0.35(m).Cables still loaded 662A and generated heat rate of 32.029 (W/m).The results are illustrated in Figs.14.-.16.and TABLE IV.

REMARK
In three tested cases of A, B, C and D of Section .III., we have firstly created the adaptive triangular mesh in the whole solving domain, we have then applied the higher-order interpolation and shape functions (up to 9 th order) [7]- [8] to each triangular element.Thus the total numbers of triangles were 722 in case A, 465 of case B, 866 of case C and 1190 of case D corresponding to the total numbers of nodes have increased from 400 to 13225 in case A, from 259 to 19087 in case B, from 474 to 35458 in case C, and from 654 to 48766 in case D. The calculated results have been presented in many Figures and TABLES.They can give some remarks as follows TABLE.I. of the benchmark problem has been shown that the hp-FE solutions are in good agreement with the analytical one.Thus the proposed method is very strong and efficient for solving the engineering problems defined by the Poisson equation, including the heat transfer and electromagnetic problems, etc.
Due to the mutual effect of adjacent conductors, the temperature of the mid conductor of any cable system is always highest.Besides, due to the characteristic of adaptive mesh, the total number of nodes on per circle modeled conductor is equal but their distributed location compared between circles is different, we can see the maximum temperatures of conductor 2 and 3 of case B are different while those of FDM and BEM are equal (see TABLE. .III.).This also appears in case D (see TABLE .IV.).
The hp-FEM is successfully applied to the buried cables in multi-layer soil with the convective boundary.It shows the high applicability and effectiveness of the proposed method in complex engineering problems.
The temperature results obtained by the adaptive hp-FEM are more accurate than those of BEM, FDM and COMSOL.Thus they are the very good datum for design and operation of underground power cables.

CONCLUSION
This paper has applied for the first time the hp-FEM to calculating the benchmark heat problem and the temperature distribution of underground cable systems in homogeneous and multi-layer soils.The results of the proposed method are compared to those of other methods, it has been seen that the hp-FE solutions are much more accurate and the hp-FEM is efficiently applied to complex geometrical problems.

Fig. 3 .
Fig. 3. Zoom solving domain surrounding cables is discretised by using the adaptive 1 st -order elements.

Fig. 4 .
Fig. 4. Contour plot of isothermal lines surrounding cables is solved by the adaptive 1 st -oder FEM.

Fig. 5 .
Fig. 5. Zoom solving domain surrounding cables is discretised by using the adaptive 9 th -order elements.

Fig. 6 .
Fig. 6.Contour plot of isothermal lines surrounding cables obtained by using the adaptive 9 th -oder FEM.

Fig. 6 .
Fig.6.Temperature of underground cable system is obtained by using the adaptive 9 th -oder FEM.

Fig. 7 .
Fig. 7. Comparison of temperature solutions of underground cable system are solved by the adaptive hp-FEM.

Fig. 8 .
Fig. 8. Domain surrounding cables buried in two-layer soil is discretised by using the adaptive 1 st -order elements.

Fig. 9 .
Fig. 9. Domain surrounding cables buried in two-layer soil is discretised by using the adaptive 9 th -order elements.

Fig. 10 .
Fig.10.Temperature of underground cable system buried in two-layer soil is obtained by using the adaptive 1 stoder FEM.

Fig. 11 .
Fig. 11.Temperature of underground cable system buried in two-layer soil is obtained by using the adaptive 9 thoder FEM.

Fig. 12 .
Fig. 12.Comparison of temperature solutions of underground cable system buried in two-layer soil are solved by the adaptive hp-FEM.

Fig. 16 .
Fig. 16.Comparison of temperature solutions of double-circuit cable system buried in multi-layer soil solved by the hp-FEM.

Table I .
Error of HP-FE Solutions Of Benchmark Problem

Table II .


Table III .
TABLE.II.The results are illustrated in Figs.3.-.7.The comparison between the hp-FE solutions and those of FDM, BEM and COMSOL software is presented in TABLE.III.Comparison of Numerical Solutions of Cable Temperatures

Table IV .
Comparison between HP-FE Solutions of Buried Power cables in Multi-Layer Soil