*2.3. Barycenter-Based Criterion*

Let us notice that in this case, the size of adjacency matrix is moderate (i.e., 10 × 10), but in practice for several cables per bundle in three-phase systems, it becomes large and the procedure of writing down the connection list becomes error-prone. When analyzing current distributions in cables with multiple strands, an important factor to be accounted for seems to be the symmetry. Intuitively, the arrangements with certain symmetry lead to even current distribution between the strands. To a certain extent, the symmetry can be taken into account by analyzing the barycenters (gravity centers) of strand groups. Therefore, we propose another simplified criterion for preliminary selection of the most promising arrangements of strands. It is based on minimizing distances between barycenters of strands belonging to individual phases. If (xk, yk) are coordinates of the center of strands consisting of phase A (k = 1, 2, 3, ... n), then the barycenter of phase A equals *xA* = (nk=<sup>1</sup> xk)/n and *yA* = (nk=<sup>1</sup> yk) /n. Analogous formulas are used for phases B and C. Then we build a triangle of vertices (*xA*, *yA*), (*xB*, *yB*), (*xC*, *xC*) and calculate its area as follows:

$$S = \frac{1}{2} \det \begin{vmatrix} \mathbf{x}\_A & \mathbf{y}\_A & 1\\ \mathbf{x}\_B & \mathbf{y}\_B & 1\\ \mathbf{x}\_C & \mathbf{y}\_C & 1 \end{vmatrix} \tag{2}$$

By analyzing several examples offered by literature in the subsequent section, we are convinced that the arrangements with as small *S* as possible are good candidates to keep a high degree of symmetry for current distribution. This criterion has to be modified when the phase barycenters are collinear, because then the triangle area is zero. Without loss of generality, we can assume then that

*YA* = *YB* = *YC*. In such a case, it is convenient to introduce discrepancy between phase barycenters as follows:

$$D = \max(X\_{A\prime}, X\_{B\prime}, X\_{\mathbb{C}}) - \min(X\_{A\prime}, X\_{B\prime}, X\_{\mathbb{C}}) \tag{3}$$

The analysis of several examples in the subsequent section indicates that "good" arrangements have *D* as low as possible.

#### **3. Relevant Case Studies from Literature**

In the paper [3], Dawson and Jain have analyzed chosen methods to balance current distribution in DC circuits (hot wire + earth). The authors have suggested the following methods to obtain more uniform current distributions:


**Figure 6.** Strand transposition. Own work, based on [3].

**Figure 7.** Separation: (**a**) of cables with complementary polarizations, (**b**) of groups of cable pairs. Own work, based on [3].

The authors have also pointed out some examples of optimal spatial configurations, providing uniform current sharing. Some examples are listed below in Figure 8:

**Figure 8.** Spatial configurations considered by Dawson and Jain as optimal ones. Own work, based on [3].

Let us notice that all above-given configurations exhibit a common feature, namely the barycenters of supply and return wires approximately overlap. It is evident that there exists a certain symmetry of cable placement in each considered case. At the same time, highly asymmetric layouts (referred to as asymmetry of the first (Figure 9) and the second (Figure 10) kind) are not recommended.

**Figure 9.** Asymmetric configuration of the first kind. *I*1 = *I*3 > *I*2. Own work, based on [3].

**Figure 10.** Asymmetric configuration of the second kind. *I*2 > *I*1. Own work, based on [3].

Lee has analyzed optimal configurations for three-phase three- and four-wire systems [18]. An exemplary optimal configuration for the flat layout is listed below in Figure 11. Colors brown, black, and red denote successive phases.

**Figure 11.** Optimal flat configuration suggested in [18].

If we assume that the center of the leftmost wire has the coordinate *x* = 0, the distance between the centers of successive wires is equal to unity, and the phase currents have the same amplitudes, then the gravity center for the phase L1 (A) is found at the point 13 (0 + 5 + 6) - 3.67, for the phase L2 (B) at the point 13 (1 + 4 + 7) = 4, and for the phase L3 (C) at the point 13 (2 + 3 + 8) - 4.33. The discrepancy between the barycenter coordinates does not exceed 17%.

For the "two shelves" layout, the optimal spatial configuration is suggested as in Figure 12.

**Figure 12.** Optimal configuration "two shelves with offset" suggested in [18].

If, similar to the previous case, we assume that the distance between the cable centers is equal to unity and we set the coordinate of the center of the leftmost cable on the lower shelf as (0,0), then it is trivial to compute (e.g., the coordinates of the center of the leftmost cable on the upper shelf as ( √3/2, 1)). The barycenter of the phase L1 is located approximately in the point (1.956, 0.333), for the phase L2 in the point (2.244, 0.667), for the phase L3 in the point (2.577, 0.333). The whole area of interest is the area of the rectangle with dimensions 5 × 2. The area of the triangle determined by the barycenters of individual phases may be computed using the well-known dependence given by Equation (1). In the considered case, S = 0.0503, which may be referred to as the area of the rectangle (5·2 = <sup>10</sup>), means the triangle area is only 0.5%. Let us notice the vertical symmetry axis passing through the center of the black wire on the lower shelf. The horizontal symmetry could not be fulfilled because of the odd number of considered wires.

Another recommended layout for the nine-cable setup is shown below in Figure 13.

**Figure 13.** Optimal configuration "two shelves" suggested in [18].

The coordinates of the barycenter for the phase L1 (A) are xA = 13 (0 + 2 + 3) - 1.667, yA = 13 (0 + 1 + 0) - 0.333, for the phase L2 (B) xB = 13 (0 + 2 + 3) - 1.667, yB = 13 (1 + 0 + 1) - 0.667, and for the phase L3 (C) xC = 13 (1 + 1 + 4) = 2, yC = 13 (0 + 1 + 0) - 0.333. The area determined by the barycenters of individual phases is slightly higher than in the previous case, S = 0.0556, however, it remains at an acceptable level. It should be remarked that this configuration is slightly worse than the previous one, which can be confirmed with the results obtained by Lee himself (cf. the values of indicators δT and δM listed in Tables 2 and 3 in the paper [12]).

In the case of three-phase systems with neutral wire, the recommended layouts are depicted below in Figures 14–18:

**Figure 14.** Optimal flat configuration (with neutral wires) suggested in [18].

**Figure 15.** Optimal configuration "two shelves with offset" (with neutral wires) suggested in [18].

**Figure 16.** Optimal configuration "two shelves without offset" (with neutral wires) suggested in [18].

**Figure 17.** Fully optimal configuration "two shelves without offset" suggested in [18].

**Figure 18.** Fully optimal configuration "two shelves without offset" (with neutral wires) suggested in [18].


> (a) The flat system

$$\mathbf{x\_A} = \frac{1}{3}(0+7+8) = 5, \; \mathbf{x\_B} = \frac{1}{3}(1+6+9) \cong 5.333, \; \mathbf{x\_C} = \frac{1}{3}(2+5+10) \cong 5.667.$$

$$\mathbf{x\_N} = \frac{1}{3}(3+4+11) = 6.$$

The maximal discrepancy between the gravity centers for individual phases is equal to 0.667, which, referring to the shelf width (11), is only 6.06%. The gravity center for the neutral wire may be a little distant from the other ones, because under operating conditions (under standard assumptions concerning symmetries of supply and load), the current flowing through the neutral strands takes negligible values with respect to the phase currents.

(b) the layout "two shelves with offset"

$$\mathbf{x}\_{\mathcal{A}} = \frac{1}{3} (0 + 4 + 3 + \sqrt{3}/2) \cong 2.622, \ \mathbf{y}\_{\mathcal{A}} \cong 0.333, \ \mathbf{x}\_{\mathcal{B}} = \frac{1}{3} (\sqrt{3}/2 + 3 + 4 + \sqrt{3}/2) \cong 2.911,$$

$$\mathbf{y}\_{\mathcal{A}} \cong 0, \mathbf{667}, \mathbf{x}\_{\mathcal{C}} = \frac{1}{3} (1 + 5 + \sqrt{3}/2 + 2) \cong 2.955, \ \mathbf{y}\_{\mathcal{C}} \cong 0.333,$$

$$\mathbf{x}\_{\mathcal{N}} = \frac{1}{3} (\sqrt{3}/2 + 1 + 2 + \sqrt{3}/2 + 5) \cong 3.244, \ \mathbf{y}\_{\mathcal{N}} \cong 0.667.$$

The area determined by the barycenters for individual phases is equal to 0.0073, which, referring to the area of the region with dimensions 5 + √3/2 × 1, is only 0.12%.

(c) the layout "two shelves without o ffset"

xA - 2.333, y A- 0.333, xB - 2.333, yB - 0.667, xC - 2.667, y A- 0.333,

$$\text{x}\_{\text{N}} \cong \text{2.667}, \text{ y}\_{\text{N}} \cong 0.667$$

The area determined by the gravity centers for individual phases is equal to 0.0558, which means 1.12% of the whole area for admissible solutions.


The layout symmetry is evident. For the odd number of strands, it was impossible to achieve an optimal placement for two shelves, but in this case, it is feasible. The presented layouts provide a full load symmetry for the strand wires.


Wu [20] has compiled a summary of configurations recommended by the U.S. and the Canadian standards for 2 ... 6 strands per phase. We have chosen for the analysis those cases, which are most often used in the practice. Analyzing the cases depicted in Figures 19–24, we can draw the following conclusions:


**Figure 19.** Layouts recommended by the U.S. standard. Own work, based on [20].

**Figure 20.** Layouts recommended by the Canadian standard. Own work, based on [20].

**Figure 21.** Layouts recommended by the U.S. standard. Own work, based on [20].

**Figure 22.** Layouts recommended by the Canadian standard. Own work, based on [20].

**Figure 23.** Layouts recommended by the U.S. standard. Own work, based on [20].

**Figure 24.** Layouts recommended by the Canadian standard. Own work, based on [20].

For the first aforementioned configuration (depicted in the leftmost part of Figure 21), we have determined the barycenter location for each phase. If we assume that the origin of the coordinate system (point (0,0)) is placed in the middle of the wire A1, and the distance between the shelves (i.e., 18") is approximately 10 times the wire diameter, then the coordinates of the other wires are: B1(2,0), C1(4,0), A2(10, 0), B2(8,10), C2(6,0), A3(0,10), B3(2,10), C3(6,10), A4(10,10), B4(8,10), C4(6,10), A5(0,20), B5(2,20), C5(4,20), A6(10,20), B6(8,20), C6(6,20).

It is straightforward to compute that the barycenter coordinates for each phase are then the same and they are equal to (5, 10). It means that the proposed criterion for choosing the optimal spatial configuration is fulfilled in this case.

It should be noticed that not all spatial configurations from those depicted in Figures 19–24 result in a fully balanced load of individual strands. A practical conclusion resulting from the analysis of cases listed in [20] is that it is necessary to carry out detailed computations for each scenario, and the entries in the standards should be treated as recommended under certain circumstances.


Two cable groups consisting of 18 flat-laid, single-conductor cables were laid on a metallic tray in free space, each with 300 mm<sup>2</sup> cross-section, cf. Figure 25. We assume a unit distance between the cable centers, a similar distance assumed between the cables on the internal edges and the compartment bar.

**Figure 25.** Preliminary cable configuration] (**a**) and the initial current distribution (**b**) [21].

The "x" coordinates of the barycenters for the case depicted in Figure 25 were as follows:

phase A 15 (3 + 4 + 5 + 6 + 9) = 5.4 phase B 15 (10 + 11 + 12 + 13 + 14) = 12 phase C 15 (15 + 16 + 17 + 18 + 19) = 17

> For the optimized case (see Figure 26), the corresponding values were, respectively,

**Figure 26.** Current and temperature after system layout optimization. Own work, based on [21].

phase A 15 (3 + 10 + 11 + 16 + 17) = 11.4 phase B 15 (5 + 6 + 13 + 14 + 19) = 11.4 phase C 15(4 + 9 + 12 + 15 + 18) = 11.6

The dispersion between the extreme current values was minimized from 11.6 (which was about 61% of total length considered) down to 0.2 (about 0.61%).

• The case of Teatro Regio in Turin, Italy [15]

The configuration recommended by the authors for the system with six strands per phase plus three neutral wires is depicted in Figure 27.


**Figure 27.** Configuration recommended by the authors of [15].

Let us assume that the coordinates of B wire in the leftmost downward edge are (0; 0). Let us assume that the distance between the strand centers in the vertical direction is equal to unity, similarly for the horizontal direction, apart from the upmost row, where the distance is equal to two units, that is, the coordinates of phase A wires are (0; 4), (2; 4), and (4; 4), respectively. The barycenter coordinates are: for phase A (2; 2.5), for B and C phases (2; 5.1), for neutral wires (3; 1). The state close to the optimal one was achieved, and the full symmetry was unreachable due to the presence of neutral wires, which occupied some space. For neutral wires, the barycenter does not have to match the barycenter for phase wires, since under normal operating conditions and supply symmetry, the currents flowing through neutral wires are insignificant.

• Canova et al. have considered the optimal layout of cables using the Vector Immune System algorithm [22]. The case considered was six strands per phase, lack of neutral wire. The configuration indicated by the authors as the optimal one, featuring both the most uniform current partition and the lowest value of magnetic induction, was: BCAACBBCA ACBBCAACB

If we assume that the coordinates of the center of the leftmost downward A wire were (0;0) and the distance between successive wire centers was equal to three units both for the vertical and the horizontal directions, it is straightforward to compute that the barycenter location for each phase was (12; 1.5). It means that the configuration found by the optimization algorithm is indeed the optimal one and the symmetry state was achieved.

• Lee has considered a similar case (six wires per phase, two shelves), but with neutral wires [18]. Let us assume that the origin of the coordinate system was placed in the center of the leftmost downward wire (phase A) and that the distance was equal to unity. ABCNNCBANABC ABCNNCBANABC

The barycenter coordinates for phases A and B were equal to (5.67; 0.5), for phase C (6; 0.5), for the neutral wire (5; 0.5). The state close to the optimal one was achieved, and the full symmetry was unreachable due to the presence of neutral wires, which occupied a part of available space. For the neutral wires, the barycenter location does not have to correspond to the barycenter locations for phases since under normal operating conditions and for supply symmetry, the current values flowing though neutral wires are insignificant.

#### **4. Verification of the Proposed Criterion for Optimal Current Distribution Using FEM**

Taking into account the computation results from the last section, we can draw a conclusion that the criterion based on barycenter location might be useful for quick selection of potentially optimal spatial configurations, which might be helpful for the designers of cable systems. Additional verification for some simple scenarios is provided in this section, using the freeware FEMM software [23].

Comparing three cases depicted in Figures 28–30, it can be stated that the current uniformity is obtained for the system ABCCBA, for which the phase barycenter locations overlap. For the other two configurations, the current distribution is nonuniform. Table 4 contains the results of FEM-based computations for individual phases. The subscript "1" denotes the leftmost wire, the subscript "2" the rightmost one.

**Figure 28.** Current densities for the flat ABCCBA configuration.

**Figure 29.** Current densities for the flat ABCABC configuration.

**Figure 30.** Current densities for the flat AABBCC configuration.



Comparing three cases depicted in Figures 31–33, it can be stated that the current uniformity is obtained for the system ABC/CBA, for which the phase barycenter locations overlap. For the other two configurations, the current distribution is nonuniform. Table 5 contains the results of FEM-based load computations for individual phases. The subscript "1" denotes the upper wire, the subscript "2" the lower one.

**Figure 31.** Current densities for the ABC/CBA configuration.

**Figure 32.** Current densities for the ABC/CAB configuration.

**Figure 33.** Current densities for the ABC/ABC configuration.


**Table 5.** The current percentage values in individual phases ("two shelves" configurations).

If the loss distribution shall be taken into account for individual phases, the best configuration of the six ones considered in this section shall be the layout ABC/CBA. Fractions of total loss dissipated in the wires are depicted graphically in Figure 34. Wire numbering goes from left to right for the flat layouts, for "two shelves" it is given as: wire 1–6.

**Figure 34.** Loss distribution in individual wires for the configurations depicted in Figures 28–33.
