*2.2. Distance-Based Criterion*

Lee considered a cable configuration technique for the balance of current distribution in parallel three-phase cables in [18]. The method requires the computation of self- and mutual impedances of the cables, which are geometry-dependent. Figure 4 depicts schematically the couplings between individual strands.

**Figure 4.** The couplings between individual strands (taken from [19]).

The problem with the method is that the resulting impedance matrix is a full matrix containing imaginary numbers (the elements on the main diagonal are complex numbers, since they include the strand resistances). Moreover, the computed values are relatively small in most practical cases, which causes some numerical problems with the precise determination of the inverse matrix needed in the subsequent computation step. This has led us to the formulation of a simplified method to determine current distribution, relying only on mutual distances between the strands. We have assumed that the current in the individual strand may be approximately proportional to the sum of distances of the strand to its neighbors. We have used the Graph and Network MATLAB toolbox [17] in order to introduce an abstract, geometry-independent layer, which facilitates the computations. An exemplary graph corresponding to model 2 or 4 is depicted in Figure 5a. The strand centers are represented as graph nodes and the distances between them may be written as appropriate weights for the edges (Figure 5b).

**Figure 5.** (**a**) An exemplary graph corresponding to model 2 or 4. (**b**) An exemplary list of edge connections and distances (in mm) between graph nodes.

The number of all graph edges in the considered case is 52 = 10 (the appropriate MATLAB command is nchoosek (5, 2)). Let us notice that in this case, the size of adjacency matrix is moderate (i.e., 10 × 10), but in practice for fewer cables per bundle in three-phase systems, it becomes large and the procedure of writing down the connection list becomes error-prone.

Some MATLAB code snapshots are given below: points = [points\_x points\_y]; % node coordinates distances = squareform(pdist(points)); G = graph(distances); G.Nodes.x=points(:,1); G.Nodes.y=points(:,2); A = full(adjacency(G, G.Edges.Dist)); Currents = zeros(length(points\_x),1); for i = 1:length(Currents) Currents(i)=sum(A(i,:)); end Currents = Currents./sum(sum(A));

A comparison of the values of computed and experimentally determined currents for the preset excitation current 200 A is given in Table 3. The discrepancies between the corresponding values do not exceed 21%.


**Table 3.** The percentage error values for the maximal attainable current values.

An interesting case is when the cables are placed on a circle radius (model 5), when the uniform current distribution was obtained. The simplified computational model also indicates that such spatial configuration is close to optimum. The results are the premise for further examination of cases, where the cables are placed in the nodes of equilateral polygons.
