**1. Introduction**

Delivery of electrical energy often requires cables with high current-carrying capacity, that is, ampacity. In some situations it is necessary to use multistrand cables connected in parallel. Since 1957, the publication year of the seminal paper by Neher and McGrath [1], much attention was paid by the engineering community to the problems related to computations of current distribution in such systems. Murgatroyd developed a method to compute total proximity loss per unit length in multistrand bunch conductors of arbitrary shape [2]. Dawson et al. carried out a simplified analysis of nonuniform current sharing based on a coupled circuit model [3]. An analysis of their results, pertaining to DC systems, allowed us to formulate a useful criterion for selecting the most promising spatial configurations of cables after a generalization to three-phase systems. Petty presented a matrix algebra method for solving current distribution among phase conductors [4]. Ghandakly et al. provided current sizing information for bundled cables in high-current applications in the form of ampacity tables [5]. Sellers and Black proposed several improvements to the Neher–McGrath model: a new parameter accounting for unequal heating among the cables, some improved relationships for thermal resistance for the fluid layer existing in pipe-type cables and cables installed in ducts, as well as a modified model for thermal resistance of concrete duct banks with nonsquare cross-sections [6]. Du and Burnett developed a general prediction method for current distribution and examined its usefulness on data concerning supply of Hong Kong office building installations [7]. The textbook by Anders [8] is a comprehensive source of information on steady-state and transient ampacity calculations for electric power cables, in particular for the cases not covered in IEC 60287 standard (derived from the Neher–McGrath paper). Special attention was paid to such problems as derating considerations for cables crossing thermally unfavorable regions and for deeply buried cables. The thermal effect on cable ampacity in multistrand cable systems has also been considered in several recent papers published in MDPI Energies [9,10].

Generally speaking, current distribution in multistrand cable systems is uneven due to several factors, such as skin and proximity effects related to strand geometry [3,5], the conditions of heat exchange with the neighborhood [8–10], soil inhomogeneity, moisture, installation depth (if the cables

are buried), harmonic spectrum of currents flowing through the cables (cf. e.g., [11]), and the presence of buried metal objects, to mention but a few. Analytical treatment of all these factors is hardly possible [12], therefore numerical methods, in particular the Finite Element and the Finite Volume methods, have found wide use for engineering purposes.

At the same time, it should be stated that the problem is not merely a theoretical one; as pointed out by Ciegis et al. [ ˇ 13], many existing power lines are overdimensioned by up to 60% in terms of transmitted power, which means additional costs and waste of deficit core material (e.g., copper). Desmet et al. [14] estimate that cable losses reach 5% of the total power consumption. On the other hand, the infrastructure grid has to be redesigned constantly in order to adapt to new conditions resulting from installation of new distributed generating capacities like wind farms and so forth. The wide use of power electronics devices both in industry and at households results in the presence of highly distorted current spectra. All these challenges have to be faced by the designers of cable supply systems. Their negligence may lead to serious malfunctions [11,15] or even fires of cable installations [9]. Such a situation has occurred during professional activities of the first author of the present paper. Despite considerable e fforts aimed at a correct choice of cables supplying the main low-voltage switchboard in a car hood factory, after putting the installation into operation, a fire resulting from overloading one phase of the supply system has destroyed completely the cable installation (Figure 1). The considered system was designed in accordance with the guidelines included in the Polish standard PN-IEC 60364-5-523:2001 (derived from translation from IEC 60364-5 part 52 International Standard) and was supposed to withstand a 10% safety margin, however, as pointed out previously, the system experienced a serious fault soon after being put into operation. This fact has inspired the first author to carry out additional studies on current distribution of multibundle cable systems. It was found during a "postmortem" analysis that in the considered system, considerable di fferences between phase current values were present due to the proximity and skin e ffects [16].

**Figure 1.** A destroyed cable supply system [16].

The aim of the present paper is to propose a simple approach to identify the most promising configurations of low-voltage multibundle cable systems, for which the skin and proximity e ffects are minimized. The original concept of the paper is to avail of some routines available in MATLAB Graph and Network Toolbox [17] in order to facilitate the description of interactions between the cables. It should be remarked that in this manuscript, a simplified model for current distribution is considered. It assumes that current distribution is a function of spatial geometry only. Therefore, the use of procedures related to graph theory is somewhat limited, that is, we use nonoriented graphs as a smart, easily scalable method to separate the description of cable geometry in real-life conditions from the abstract layer that specifies interactions between graph nodes (represented by the cables in the bundle). In this context, we believe the use of adjacency function allows one to simplify the notation and to avoid mistakes. Moreover, we think that the use of notation borrowed from the graph theory allows one to simplify the analysis in those cases when the direct application of barycenter criterion (described in detail in Section 2.3 of the present paper) is not possible (e.g., due to constraints introduced by real-life geometry of the ducts, limited space for cable location, etc.). Then the description of mutual interactions between the strands in the form of a graph may be incorporated as a fragment of some relevant optimization algorithm applied (e.g., to a circuit-based model similar to the one considered by Lee [18]). The full potential of graph theory, which incorporates powerful, highly optimized algorithms for computation of maximal flow between graph nodes (e.g., the Ford–Fulkerson algorithm), is still to be explored in a subsequent publication.

The paper is structured as follows. Section 2 describes the developed laboratory stand. The results of measurements carried out using a single-phase excitation for several cable configurations are presented. A simplified method to determine current distribution using the distance criterion is proposed. Section 3 includes a brief literature review concerning methods of computation of current distribution in multibundle cable systems. Subsequently, a generalization of the results is presented, indicating that in the cases considered as optimal ones, some spatial symmetry patterns may be found. A practical conclusion is the formulation of a criterion, which allows one to choose preliminary configurations to be examined in subsequent steps during an in-depth FEM-based analysis. Section 4 provides a verification of the proposed criterion using the freeware FEMM software for some simple cases.

The ultimate goal of the paper is to provide the designers of cable systems a really simple criterion, which can be easily implemented in a spreadsheet or verified "by hand" computations, allowing them to choose the most promising spatial configurations for subsequent analysis using coupled electromagnetic-thermal FEM calculations.
