How to Calculate the Steady-state Temperature Field and Ampacity of Underground Cable System

1. Introduction

The calculation of underground cable ampacity was first studied by A.E.Kennelly in 1893 followed by Neher and McGrath. The IEC-60287 is based on their research. Based on Kennelly’s assumption, the thermal field of direct-buried cables meets the following requirements: ① the ground is isothermal surface; ② the surface of cable is isothermal surface; ③ the superposition principle is applicable. However, the actual temperature distribution does not conform to the assumption and calculation results of IEC-60287 tend to be conservative. As the cable is widely used in the field of power transmission and distribution, the precise measurement of ampacity is significant for improving the economy of cable lines. It is necessary to determine the temperature of cable conductor and environment if adjusting the cable load flexibly and dynamically according to the seasonal variation. The numerical calculation is to analyze the whole temperature field under given cable laying, permutation and load conditions. The temperatures of ground surface and cable surface are unknown variables. Hence, calculation results via numerical method are closer to the actual situation and the ampacity of cable is more accurate than the one by IEC-60287.

The numerical methods of analyzing temperature field include the finite-difference method, finite element method, boundary element method and finite volume method. The finite element method can deal with all boundaries and complex shapes. The finite element method combined with thermal circuit and harmonic mean methods are used to analyze and calculate the temperature field of underground cable in the paper; the validity of the method is verified by means of heating pipe test model. Finally, the ampacity of underground cable system is calculated by the secant method. 

2. The Basic Principle of Finite Element Method

The steady-state temperature field of underground cable system belongs to the issue of two-dimensional steady-state heat conduction. For the temperature control equation of areas with thermal sources (such as cable conductor, metal shield layer and armor layer), see below:

Where:

T -- temperature of a point (x,y) ℃

qv -- Volumetric heat generation rate W/m3

The temperature control equation of areas without thermal sources (such as other layers of cable or soil) is listed below:

The boundary conditions of any heat conduction issue can be summarized into three categories. The first category is known boundary temperature; the second category is known boundary heat flux; the third category is known coefficient of convective heat transfer and fluid temperature. The control equations of three kinds of boundaries are shown in (3)-(5).

Where:

-- Coefficient of heat conductivity, W/(m·℃)

q2 -- Heat flux, W/m2

-- Coefficient of convective heat transfer， W/(m2·℃)

Tf --Fluid temperature

-- Boundary integral, ℃

The commonly used unit of finite element method is triangle. The formulas (3)-(5) are treated by the weighted residual approach and Galerkin method. The corresponding line integral equations are listed below:

l=i, j, m. Combine the whole area and formula (6)-(8):

The integral of triangle area is used to calculate the Kij and Pi. i=1-n, j=1-n (n--number of nodes).

Solve the formula (9) according to the iterative method or Gaussian method.

3. Cable System Temperature Field Model

Take single circuit direct-buried cable as an example to build the model of cable system temperature field, which is shown in Fig.1. 

Fig.1 Temperature field model of single circuit direct-buried cables

The cable system is directly buried 700mm under the ground and surrounded by sandy soil at 100mm. Then the soil is filled again.

3.1 Boundary determination

The whole area is one and a half infinite temperature field. It is necessary to convert one open domain field into close field. The upper boundary, lower boundary, left boundary and right boundary can be determined by the following methods.

The surface shown in Fig.1. is the third category of boundary condition. The coefficient of convective heat transfer and temperature should be known. The formula is given below:

-- Grashof number

-- Prandtl number

-- Coefficient of volume expansion

g-- Gravitational acceleration

--Temperature difference, ℃

l --Linearity size, m

v -- Kinematic viscosity

Nu-- Nusselt number

c,n -- Coefficient

The temperature changes significantly only around the cables. When the soil is far away from the cables, the soil temperature is the same as environmental temperature. The soil, which is 2000mm away from the cable, is free from the effect of cable. Therefore, the lower, left and right boundaries can be straight lines 3000mm from the nearest cables, which is shown in Fig.1. The left and right soil boundaries are the second category while the deep soil boundary is the first category.

3.2 Handling of metal shield layer loss and armor layer loss

The losses of metal shield layer and armor layer are important components of cable loss. Given that these layers are thin, the subdivision has a big effect on calculation results.

The cable is axial symmetric structure, so thermal resistances of all directions within the cable area are the same. How to handle the loss of metal shield layer via thermal circuit model is given below. For the imputation theory, see Fig.2. 

Fig.2 The imputation theory of losses

Q1--conudctor loss

Q2--loss of metal shield layer

RT--conductor shield, insulation layer and insulation shield equivalent thermal resistance

R’T --thermal resistance after loss imputation

The theory of imputation ensures that the total output energy, conductor temperature and temperature of metal shield layer stay the same. R’T can be calculated by means of the formula (13).

The loss of armor layer can be attributed to the conductor according to the above method.

3.3 Handling of thin layer of cables

Take the XLPE cable as an example. The cable is composed of conductor, conductor shield, insulation, insulation shield, metal shield, inner sheath, armor and outer sheath. The conductor shield, insulation shield and metal shield are thinner than other layers. During the grid generation, the change of width of adjacent grids should be kept within a reasonable range, usually between 1.5 and 2.0. Hence, due to the existence of thin layers, the grid generation must be dense. As a result, it takes longer time to calculate. 

In order to ensure accurate calculation and short calculation time, the harmonic mean method is used to deal with all layers except cable conductors. For the formula, see below: 

--Coefficient of equivalent heat conduction

i--Cable structure layer, i=1conductor shield layer; i=n outer sheath

--Coefficient of heat conduction which corresponds to i layer

--the radius which corresponds to i layer

Each layer is equivalent to the same media and equal division is performed based on the radius during the grid generation. By doing so, the effect of thin layers can be eliminated.

4. Loss Calculation

In the whole temperature field domain, only the cable contains thermal sources, which includes conductor loss, dielectric loss, losses of metal shield layer and armor layer. The AC resistance of multi-circuit cable conductors and losses of metal shield layer and armor layer can be calculated by mans of Bessel functions. We do not present that in the paper. 

5. Ampacity Calculation

When loads and sections of cables are equivalent, the load current of each cable is the same. That is, the ampacity to be confirmed is one value. The secant method can be used to calculate the ampacity. The formula is listed below:  

Procedures of the secant method are as follows:

(1) Select the current value xk-1 at random and calculate f(xk-1). If the value meets the requirements, xk-1 is the target ampacity. Otherwise, do the procedure (2). 

(2) Select the current value xk at random and calculate f(xk). If the value meets the requirements, xk is the target ampacity. Otherwise, do the procedure (3). 

(3) Calculate the current value xk+1 and f(xk+1) according to the formula (15). If the value meets the requirements, xk+1 is the target ampacity. Otherwise, do the procedure (4).

(4) xk-1 =xk, f(xk-1)=f(xk), xk =xk+1, f(xk)=f(xk+1 ), return to the procedure (3).

6. Method Verification

According to the similarity principle, three direct-buried heating sticks are used to verify the correctness of finite element method used to calculate the temperature field of direct-buried cable system. The test model is shown in Fig.3.

Fig.3 Model pf heating stick

The diameter of heating stick is 12mm; the resistance is 120Ω; the length is 2.5m; the applied voltage at both ends is 120V; the duration is 10 days. The air temperature is 22℃andsoil temperature is 18℃.Test results are as follows: 1#pipe 63℃; 2#pipe 67℃; 3#pipe 62℃. The soil within the diameter of 10cm around the stick is dry and thermal resistance is 1.66. The thermal resistance of external soil does not change and the resistance is 0.78. The finite element model is built and calculation results are as follows: 1#pipe 63.2℃; 2#pipe 67.3℃; 3#pipe 63.2℃.   

Test results are consistent with the calculation results of finite element method, which proves the effectiveness of finite element in the calculation of temperature field of direct-buried long heating stick. According to the similarity principle, it can be applied to calculate the temperature field of direct-buried cable with effectiveness. 

7. Calculation Examples

Take the 800mm2 YJLW02 XLPE cable as an example to calculate the ampacity under many laying conditions. Parameters of the cable are shown in Tab.1. The environment conditions are listed in Tab.2

Tab.1 Parameters of cable

Tab.2 Environment conditions of buried cables

7.1 Imputation of metal shield layer

Assumptions are as follows: two terminals of cable are grounded and the 500A current goes through the conductors; the thermal resistance of insulation is 3.5K·m/W; losses of three phase conductors are 6.2297W/m, 6.2352W/m and 6.2288W/m; losses of metal shield layers are 13.893W/m, 10.3145W/m and 9.8375W/m.

According to the formula (13), after the loss of metal shield layer is attributed to the conductor, thermal resistances of insulation layers turn into 1.0835K·m/W, 1.3186K·m/W and 1.3569K·m/W. 

The highest temperature of conductor before and after imputation is 65.893℃ and 65.895℃. The result proves that the method is feasible.

7.2 Handling of thin layer of cable

According to the formula (14), all layers except cable conductors can be reduced into one layer.

The thermal resistance of metal shield layer is 1.236K·m/W while that of outer sheath is 6K·m/W. After imputation, each cable has only two layers: conductor and outer sheath. In addition, the external diameter keeps the same. The equivalent thermal resistances of outer sheath layers of three cables are 1.4235K·m/W, 1.6257K·m/W and 1.6586K·m/W. The method can also accelerate the speed of calculation.

7.3 Ampacity calculation

The relation between the ampacity and distance is shown in Fig.4. When the cable distance is 200mm, the relation between the ampacity of two terminals grounding and cable circuit is shown in Fig.5.

Fig.4 Ampacity with different distances between cables

Fig.5 Ampacity with different cable loops

7.4 Analysis of results

Based on Fig.5, it s found that the cable ampacity declines as the number of circuits increase. With an increasing number of circuits, the electromagnetic effect and heat effect enhance, causing the ampacity to decline.

From the Fig.4, when one terminal of cable is grounded, the ampacity of single-circuit cable increases as the distance extends. However, when two terminals of cable are grounded, the ampacity goes down if the distance is short and vice versa. The reason is the law of loss variation caused by the electromagnetic induction. The relation between cable loss and distance is shown in Fig.6. 

Fig.6. Losses with different distances of cables

When one terminal is grounded, the losses of conductors basically keep the same; the metal loss is mainly eddy current loss and the value is in inverse proportion to the distance. Therefore, the ampacity increases as the distance extends.

When two terminals are grounded, the losses of conductors basically keep the same; the metal loss is mainly circulation current loss and the value is proportional to the distance. Hence, the curve must have a middle point. Before the point, the loss effect is more than the heat effect and the ampacity is in inverse proportion to the distance; after the point, the loss effect is less than the heat effect and the ampacity is proportional to the distance.

8. Conclusions

The finite element method (FEM) is used to calculate the temperature field of underground cable system. The accuracy of temperature field analysis and ampacity calculation is improved. Against the problems of complex cable structure, uneven thickness and slow calculation, the cable is equivalent to two layers by means of thermal circuit and harmonic mean methods. Besides keeping the calculation accuracy, the rate of calculation is also enhanced.

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