Electric cables Calculations for current ratings Finite element method
|Publication Date:||1 June 2003|
|ICS Code (Cables):||29.060.20|
The most important tasks in cable current rating calculations are the determination of the conductor temperature for a given current loading or, conversely, the determination of the tolerable load current for a given conductor temperature. In order to perform these tasks the heat generated within the cable and the rate of its dissipation away from the conductor, for a given conductor material and given load, must be calculated. The ability of the surrounding medium to dissipate heat plays a very important role in these determinations and varies widely because of factors such as soil composition, moisture content, ambient temperature and wind conditions. The heat is transferred through the cable and its surroundings in several ways. For underground installations the heat is transferred by conduction from the conductor, insulation, screens and other metallic parts. It is possible to quantify the heat transfer processes in terms of the appropriate heat transfer equation as shown in Annex A (equation A.1).
Current rating calculations for power cables require a solution of the heat transfer equations which define a functional relationship between the conductor current and the temperature within the cable and its surroundings. The challenge in solving these equations analytically often stems from the difficulty of computing the temperature distribution in the soil surrounding the cable. An analytical solution can be obtained when a cable is represented as a line source placed in an infinite homogenous surrounding medium. Since this is not a practical assumption for cable installations, another assumption is often used; namely, that the earth surface is an isotherm. In practical cases, the depth of burial of the cables is in the order of ten times their external diameter, and for the usual temperature range reached by such cables, the assumption of an isothermal earth surface is a reasonable one. In cases where this hypothesis does not hold; namely, for large cable diameters and cables located close to the ground surface, a correction to the solution has to be used or numerical methods should be applied.
With the isothermal surface boundary, the steady-state heat conduction equations can be solved assuming that the cable is located in a uniform semi-infinite medium.
Methods of solving the heat conduction equations are described in IEC 60287 (steady-state conditions) 1 and IEC 60853 (cyclic conditions), for most practical applications. When these methods cannot be applied, the heat conduction equations can be solved using numerical approaches. One such approach, particularly suitable for the analysis of underground cables, is the finite element method presented in this document. The cases when the use of the finite element method is recommended are discussed next.
Field of application
In classical cable rating calculations, the heat conduction equation is solved under several simplifying assumptions  2. This limits the field of the applicability of the analytical methods. The limitations of the classical methods will be apparent from a few examples. In the analytical methods described in IEC 60287 , IEC 60853-1  and IEC 60853-2 , the case of a group of cables is dealt with on the basis of the restricted application of superposition. To apply this principle, it must be assumed that the presence of another cable, even if it is not loaded, does not disturb the heat flux path from the first cable, nor the generation of heat within it. This allows separate computations to be performed for each cable with the final temperature rise being an algebraic sum of the temperature rises due to cable itself and the rise caused by the other cables. Such a procedure is reasonably correct when the cables are separated from each other. When this is not the case, for example for cables in touching formation, the temperature rise caused by simultaneous operation of all cables should be considered. A direct solution of the heat conduction equation employing the finite element method offers such a possibility.
Numerical methods also permit more accurate modelling of the region's boundaries for example, a convective boundary at the earth surface, constant heat flux circular boundaries for heat or water pipes in the vicinity of the cables, or an isothermal boundary at the water level at the bottom of a trench. Thus, when an isothermal boundary cannot be assumed, for example, for cables installed in shallow troughs or directly buried not far from the ground surface, the finite element method provides a suitable tool for the thermal analysis.
Perhaps the most obvious case when the analytical approximations fail is when the medium surrounding the cable is composed of several materials having different thermal resistivities. Figure 2 shows an example of such situation. This is an actual cable installation where not only were several soil characteristics present, but also, a vertical convective boundary had to be dealt with. The non-uniform soil conditions and non-isothermal boundaries are handled easily by the finite element method. The computational efficiency of this approach is also quite satisfying. With presently available personal computers, calculations involving networks with several thousand nodes can be performed in a matter of minutes.
There are also advantages in using the finite element method in the transient analysis. The analytical approach for transient calculations is described in IEC 60853-1 and IEC 60853-2. In this document, separate computations are performed for the internal and the external parts of the cable. Coupling between internal and external circuits was achieved by assuming that the heat flow into the soil is proportional to the attainment factor of the transient between the conductor and the outer surface of the cable. The validity of the methods did not rest on an analytical proof, but on an empirical agreement of the responses given by the recommended circuits and the temperature transients calculated by more lengthy but more accurate computer-based methods. Here, again, the finite element method offers a solution with minimal simplifying assumptions.
It should be noted that the value selected for the thermal resistivity of the soil, and its temperature, will have a significant influence on any calculated current rating or cable temperature. In many cases there is little to be gained by using a 'more accurate' method of calculation if soil conditions are not known with a degree of certainty.
Information obtained from the finite element method
The usual cable rating problem is to compute the permissible conductor current so that the maximum conductor temperature does not exceed a specified value. Numerical methods, on the other hand, are used to compute the temperature distribution within the cable and its surroundings given heat generated within the cable (this is particularly useful when we need to determine the temperature field and specific isotherms around the cable). However, when numerical methods are used to determine cable rating, an iterative approach has to be used for the purpose. This is accomplished by specifying a certain conductor current and calculating the corresponding conductor temperature. Then, the current is adjusted and the calculation repeated until the specified temperature is found convergent within a specified tolerance.
An explanation of the finite element method is given in Clause 2 followed by the discussion of input requirements in Clause 3. In Clause 4, several examples where the application of the finite element approach is advisable are presented.
Although this report concentrates on the use of finite element methods for the calculation of heat transfer through the materials surrounding buried cables, other numerical methods are available. These include finite difference methods, boundary element methods, the superposition method described in Electra 87  and the approaches combining conformal transformation and the finite difference method.
Finite difference methods (FDM) are frequently used in the study of electric stress distribution in cable joints and terminations. It has been shown that FDM is more suitable than FEM for three dimensional cable problems. This is because difficulties can arise when using FEM to model long thin objects, such as cables, in three dimensions. However, FDM is intended for use with rectangular elements and hence is not well suited for modelling curved surfaces.
Boundary element methods need less effort in defining the input data and use less computer time than FEM. However, transient analysis cannot be performed using boundary element methods.
The superposition method described in Electra 87 for the calculation of the response of single core cables to a step function thermal transient has a number of advantages over FEM. These include the following:
a) it requires relatively little modelling data, typically less than 100 nodes compared with 1000 nodes for FEM. The method is therefore more suitable for real time rating systems. The one-dimensional temperature field can be derived using numerically stable methods. Hence, relatively large time steps can be used without introducing significant errors;
b) approximate methods can be developed to use this approach when two different cable backfills exist;
c) the method can be used as a basis for calculating transient temperatures for three dimensional problems such as occur in cable joint bays and systems with separate water cooling;
d) it can be used to calculate mutual heating between crossing cables;
e) it is suitable for studying the effect of temperature dependant material properties such as conductor resistance, dielectric losses and soil thermal resistivity.
Although this superposition method is suitable for many cable rating problems, it is not well suited to problems involving a large number of cables and complex geometry.
The approach applying conformal transformation was proposed by CIGRE WG 21.02 and is described in Electra 98 . Germay and Mushamalirwa  compared the finite element method with four approaches based on a conformal transformation of the z-plane perpendicular to the cable axes into a w-plane, in order to transform the circular boundaries of the cables into horizontal straight segments to facilitate the solution. However, the conformal transformation method has several drawbacks. The major one is that the equations describing the transformed network are equivalent to finite difference equations obtained by discretising the heat equation in the w-plane and, hence, the complexity of a numerical solution of the heat conduction problem is not avoided. Another drawback is that both the earth and cable surfaces are assumed to be isothermal. In addition, transformation of the boundaries between regions with different resistivities point by point is very laborious and the resulting computer software cannot efficiently handle more than four cables in one installation.
1 IEC 60287 has been withdrawn and replaced by a series of publications (see item 2 of the Bibliography).
2 Figures between brackets refer to the bibliography.