The procedure presented in the previous sections is able to obtain the solution to (2) using a parametric formulation of the PGD, in a 4D domain (spatial dimensions *x*, *y*, frequency *f*, and distance *d* between conductors). The results generated by the PGD allows to obtain the AC resistance and AC reactance of parallel rectangular conductors, for a range of frequencies and distances, which can be used for the harmonic analysis of the AC lines. Nevertheless, from an industrial point of view, it is also necessary to compute the force between conductors, which can damage them in case of very high short-circuit currents. In this case, it is necessary to replace the frequency with the time in the 4D formulation of the PGD, that is,

$$\begin{array}{}{\displaystyle A(x,y,t,d)=\sum _{i=1}^{n}({X}_{i}(x)\cdot {Y}_{i}(y)\cdot {T}_{i}(t)\cdot {D}_{i}(d)).}\end{array}$$(5)

The problem to solve in the 4D domain (*x*,*y*,*t*,*d*) is now
$$\begin{array}{}{\displaystyle \begin{array}{rl}& \frac{{\mathrm{\partial}}^{2}A(x,y,t,d)}{\mathrm{\partial}{x}^{2}}+\frac{{\mathrm{\partial}}^{2}A(x,y,t,d)}{\mathrm{\partial}{y}^{2}}=\\ & \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}=-{\mu}_{0}{J}_{0}(x,y,t,d)+{\mu}_{0}\sigma (x,y,t,d)\cdot \frac{\mathrm{\partial}A(x,y,t,d)}{\mathrm{\partial}t},\end{array}}\end{array}$$(6)

with boundary conditions
$$\begin{array}{}{\displaystyle A(x,y,t,d){\mid}_{x=\mathrm{\infty}}=A(x,y,t,d){\mid}_{y=\mathrm{\infty}}=0,}\end{array}$$(7)

and assuming that the conductivity *σ* does not vary with time.

The configuration analyzed in this work is shown in Figure 11. It represents a single phase line, with two conductors carrying the current in one direction (I-A, I-B), and two conductors carrying the same current in the opposite direction (V-A, V-B).

Figure 11 Single phase line, with two conductors carrying the current in one direction (I-A, I-B), and two conductors carrying the same current in the opposite direction (V-A, V-B)

One of the strengths of the PGD is that post processing the solution is very fast, because the solution is obtained directly in separated form. In this section the force per-unit length that appears on a single conductor is computed as
$$\begin{array}{}{\displaystyle F=\underset{s}{\iint}\overrightarrow{J}\times \overrightarrow{B}\phantom{\rule{thickmathspace}{0ex}}\mathrm{d}S=\underset{s}{\iint}\overrightarrow{J}\times (\mathrm{\nabla}\times A)\phantom{\rule{thickmathspace}{0ex}}\mathrm{d}S,}\end{array}$$(8)

where *S* is the cross-section of the conductor. Taking advantage that the MVP is obtained in separated form by the PGD after solving (6), the components of →*B* can be obtained without computing the MVP in the full 4D domain, which is impractical. Instead, these components can be obtained directly from the separated representation of the MVP, as

$$\begin{array}{}{\displaystyle {B}_{x}=\frac{\mathrm{\partial}A}{\mathrm{\partial}y}=\sum _{i=1}^{n}({X}_{i}(x)\cdot \frac{\mathrm{d}{Y}_{i}(y)}{\mathrm{d}y}\cdot {T}_{i}(t)\cdot {D}_{i}(d)),}\end{array}$$(9)

$$\begin{array}{}{\displaystyle {B}_{y}=-\frac{\mathrm{\partial}A}{\mathrm{\partial}x}=-\sum _{i=1}^{n}(\frac{\mathrm{d}{X}_{i}(x)}{\mathrm{d}x}\cdot {Y}_{i}(x)\cdot {T}_{i}(t)\cdot {D}_{i}(d)).}\end{array}$$(10)

The computation of the force and the corresponding short-circuit current, for conductors I-A and I-B of Figure 11, are represented respectively in Figures 12 and 13, for the particular case of *d* = 50 mm.

Figure 12 Force (top) and short-circuit current (bottom) of conductor I-A of Figure 11, for *d* = 50 mm

Figure 13 Force (top) and short-circuit current (bottom) of conductor I-B of Figure 11, for *d* = 50 mm

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