Estimating stray current interference from DC traction lines on buried pipelines by means of a Monte Carlo algorithm

Abstract

The paper presents a non-deterministic approach to the study of stray current effects generated by direct current railway or tramway lines on buried pipelines which are located in the nearby area. The potential shift induced (via conductive coupling) on the latter ones by the stray current dispersed in the soil is evaluated by means of a suitable equivalent circuit combined with a Monte Carlo procedure. The algorithm proposed has to be intended as a tool for the estimation of the corrosion risk for a pipeline (or, more in general, for long buried metallic structures) due to the stray current generated by electrified railway/tramway lines. Different from the existing models present in literature, which are based on a deterministic approach, a method based on the random and statistical aspects of stray current, which are captured by Monte Carlo approach, is proposed. This is the novelty of the method.

Appendix

In case of a single track line, it is possible to use a simplified analytical method to calculate current and potential along the track itself.

Let us assume the following hypotheses:

• the track is very long so that it can be considered of infinite length (in practice, the following condition must be fulfilled for the length $L$ of the line: $\gamma _{{t}}{L}\ge 2$)

• the rails are metallically connected between them along the traction line route so that they can be modelled as a single conductor with earth return.

• only one locomotive is present in the traction line section under study

According to [5] and by considering the shunt current energisation of the track-earth circuit (see Fig. 13 relevant to the simplest case of one locomotive fed by one substation; positive sign for current injected into the track and negative sign for current flowing out from the track) we have the following results that can be obtained by means of the superposition principle of the sources:

Fig. 13

One feeding substation and locomotive and its equivalent

1.1 Case A: section fed by one substation

$$\begin{aligned} I\left( S \right)= & {} -\frac{J}{2}\mathrm{sign} \left( {S-S_1 } \right) e^{-\gamma _t \left| {S-S_1 } \right| }\nonumber \\&+\frac{J}{2}\mathrm{sign} \left( {S-S_\mathrm{loc} } \right) e^{-\gamma _t \left| {S-S_\mathrm{loc} } \right| }\end{aligned}$$

(14)

$$\begin{aligned} V\left( S \right)= & {} \sqrt{\frac{r_t }{g_t }}\left( -\frac{J}{2}e^{-\gamma _t \left| {S-S_1 } \right| } +\frac{J}{2}e^{-\gamma _t \left| {S-S_\mathrm{loc} } \right| } \right) \end{aligned}$$

(15)

$$\begin{aligned} I_l \left( S \right)= & {} \gamma _t \left( -\frac{J}{2}e^{-\gamma _t \left| {S-S_1 } \right| } +\frac{J}{2}e^{-\gamma _t \left| {S-S_\mathrm{loc} } \right| } \right) \end{aligned}$$

(16)

where $I(S)$, $V(S)$ and $I_{l}(S)$ are the track current, the track potential and the track leakage current to soil versus abscissa $S$, respectively; moreover, $r_{t}$, $g_{t}$ and $\gamma _{t}$ are the p.u.l. resistance, p.u.l. conductance and propagation constant of the track-earth circuit while $J$ is the current absorbed by the locomotive which is equal and opposite to the one supplied by the substation.

1.2 Case B: section fed by two substations at the same voltage

In this case, it is possible to show that the following relations exist between the currents $J_{1}$ and $J_{2}$ supplied by substations 1 and 2, respectively, and the current $J_\mathrm{loc}$ absorbed by the locomotive:

$$\begin{aligned} \left\{ {{\begin{array}{l} {\frac{J_1 }{J_2 }=\frac{S_2 -S_\mathrm{loc} }{S_\mathrm{loc} -S_1 }} \\ {J_1 +J_2 =J_\mathrm{loc} } \\ \end{array}}} \right. \end{aligned}$$

(17)

By solving system (17), we obtain:

$$\begin{aligned} J_1= & {} -\frac{S_2 -S_\mathrm{loc}}{S_2 -S_1 }J_\mathrm{loc}\end{aligned}$$

(18)

$$\begin{aligned} J_2= & {} -\frac{S_\mathrm{loc} -S_1 }{S_2 -S_1 }J_\mathrm{loc} \end{aligned}$$

(19)

Thus, from (18) and (19), we can deduce the formulas relevant to the track current, track potential and track leakage current:

$$\begin{aligned}&\!\!\!I\left( S \right) =\frac{J_1 }{2}\mathrm{sign} \left( {S-S_1 } \right) e^{-\gamma _t \left| {S-S_1 } \right| }\nonumber \\&+\frac{J_\mathrm{loc} }{2}\mathrm{sign} \left( {S-S_\mathrm{loc} } \right) e^{-\gamma _t \left| {S-S_\mathrm{loc}} \right| }\nonumber \\&+\frac{J_2 }{2}\mathrm{sign} \left( {S-S_2 } \right) e^{-\gamma _t \left| {S-S_2 } \right| }\end{aligned}$$

(20)

$$\begin{aligned}&\!\!\!V\left( S \right) \nonumber \\&=\sqrt{\frac{r_t }{g_t }}\left( {\frac{J_1 }{2}e^{-\gamma _t \left| {S-S_1 } \right| }\!+\!\frac{J_\mathrm{loc} }{2}e^{-\gamma _t \left| {S-S_\mathrm{loc} } \right| }\!+\!\frac{J_2 }{2}e^{-\gamma _t \left| {S-S_2 } \right| }} \right) \nonumber \\ \end{aligned}$$

(21)

$$\begin{aligned}&\!\!\!I_l \left( S \right) \nonumber \\&=\gamma _t \left( {\frac{J_1 }{2}e^{-\gamma _t \left| {S-S_1 } \right| }+\frac{J_\mathrm{loc} }{2}e^{-\gamma _t \left| {S-S_\mathrm{loc} } \right| }+\frac{J_2 }{2}e^{-\gamma _t \left| {S-S_2 } \right| }} \right) \nonumber \\ \end{aligned}$$

(22)