# Uncertainty of Standardized Track Insulation Measurement Methods for Stray Current Assessment

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## Abstract

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## 1. Introduction

- from seconds to minutes, if applying test signals in off-service conditions: time intervals of seconds are necessary for the polarization of electrolytes in the test circuit to take place, after which, test quantities can be measured with care to reject external noise with sufficiently long observation times;
- from hours to weeks, if using track electric quantities during train service:
- from days to years, considering the normal evolution of track insulation degradation, with aging of insulating materials, pollution of surfaces, stagnation of water, etc.

- method A.2, track insulation with civil structure: the important point that makes this less invasive compared to method A.3 is that the running rails are continuous and do not need to be sectioned; however, the test setup is more complex, with more measured quantities involved and an overall worse accuracy. The evaluation of uncertainty and optimal test conditions is discussed in Section 2;
- method A.3, track insulation without civil structure: the presence or absence of the civil structure is not the relevant point here, as the running rails must be sectioned to the desired length, either by cutting them or exploiting the presence of insulating rail joints; this is a more accurate method and should be preferred whenever possible, especially for high track insulation values;
- method A.4, lateral voltage gradient method in open area sites: this method measures the voltage gradient in the soil caused by running trains, with the field laterally extending from the tracks at two points at different distances; it is suitable for large open areas, but necessitates access to soil as homogeneous as possible and may, thus, be disturbed by buried structures and installations, such as in an urban context.

## 2. Method A.2: Continuous Track, Line Closed to Traffic

#### 2.1. Method Description and Setup

#### 2.2. Practical Factors

- The longitudinal impedance of a running rail is comprised of an internal inductance term and an AC resistance term, both accounting for AC effects, such as the skin effect and hysteresis, and related losses [33,34]; values for low current (< 50 A) amount to about $0.8$ $\mathsf{\mu}$H/m and 100 $\mathsf{\mu}$$\Omega $/m, with the former contributing less than 25 $\mathsf{\mu}$$\Omega $/m of the inductive reactance at 50 Hz.
- The stray capacitance of a running rail amounts to less than 10 nF/m (obtained by multiplying by 2 the values shown in [35]), providing more than 300 $\mathrm{k}$$\Omega $ of capacitive reactance, easily shunted by the transversal track insulation term considered here.

#### 2.3. Variability and Uncertainty Analysis

- a long track section is tested: a 10-times longer track ( 1 km) will provide a 10-times larger ${I}_{l}^{*}$ and the resulting errors will, this time, be about 20% (large, but acceptable);
- a track with poor insulation is tested: similarly, an insulation level of only 10 $\Omega $km will provide a similar distribution of the errors, so a 10-times larger ${I}_{l}^{*}$ will again reach an uncertainty of the order of 20%.

## 3. Method A.3: Sectioned Track, Line Closed to Traffic

#### 3.1. Method Description and Setup

#### 3.2. Practical Factors

- an earth electrode may be used driven into the soil at a convenient distance from the tested track (EN 50122-2 requires 30 m minimum); the reason for such distance is avoiding distortion of the electric field in the soil; the earthing resistance is quite limited anyway, for which, even in good soil, values lower than about 50 $\Omega $ are difficult to achieve, so that this earthing system is suitable for the voltmetric terminals, but not for the test supply;
- using the remaining part of the system before the injection point, earthing the test supply with a resistance ${R}_{0}$ usually of some $\Omega $; with systems of limited length or still under construction, instead, ${R}_{0}$ reaches too high values; the influence of this parameter was evaluated in [39] and is considered later in Section 3.3;
- earthed parts, such as cable trays, sharing the earthing resistance of the power distribution system, usually of the order of 1 $\Omega $ or less, can be used for both purposes (earthing the power supply and providing a reference for voltmetric measurements);
- the concrete structure supporting the track, if provided with reinforcement, cannot be used, being too close to the track under test.

#### 3.3. Variability and Uncertainty Analysis

## 4. Method A.4: Lateral Potential Gradient in Normal Service

#### 4.1. Method Description and Setup

#### 4.2. Practical Factors

^{2}for mechanical robustness).

- accessibility of the area to place the test electrodes in a line, as prescribed by the Wenner method (four electrodes in a line, with external ones for the test current ${I}_{t}$ and the inner ones for the voltage reading ${V}_{t}$, spaced by s); the resulting apparent soil resistivity value can be calculated from the resistance reading $R={V}_{t}/{I}_{t}$ as $\rho =2\pi sR$; the resistivity value refers to the depth s, so that, to double the probed depth, the electrodes span is doubled as well;
- often, the Schlumberger method is used instead, because it requires the movement of two electrodes only, keeping the inner ones for voltage more compact; keeping their separation s and calling p the separation between each external one and the nearest voltage electrode (with $p>2s$), the resistivity may be estimated again from the calculated resistance value as $\rho =\pi p(p+s)/sR$ and the depth is $p+s/2$, deeper than the previous one; in other words, for a given target depth, the Schlumberger method is more compact and faster;
- specifically focusing on the track geometry and roads nearby, keeping s of the order of 1 m to 2 m, the separation p may increase to what is allowed by the areas nearby (e.g., 5 m to 20 m); the depth values to focus on are in this range and they should be supported by a careful analysis of the resistivity values behavior to determine abnormal distributions and lack of homogeneity;
- it is, in fact, observed that interference by other metallic/conductive buried structures is almost certain in an urban/suburban context and larger volumes of soil (going deeper) help to average the contributions.

#### 4.3. Variability and Uncertainty Analysis

- first, a practical example of an extensive test campaign carried out along a tramway line is considered in order to focus on data dispersion, determination of the linear regression slope ${m}_{sr}$, etc.; the results are reported in the next Section 4.4 for consistency with previous sections;
- then, formulations are analyzed for sensitivity to the parameters and to robustness to extreme situations caused by practical issues, such as issues in placing electrodes;
- last, propagation of uncertainty is calculated using the given formulations, having already evaluated the behavior for uncommon values of parameters.

- For ${m}_{sr}$, it is a matter of propagating the uncertainty of ${V}_{R2,2}$ and ${V}_{1,2}$ through the least mean square (LMS) regression, as performed in [43] for the determination of stray capacitance (as the intercept and not the slope, as in the present case).
- For $\rho $, it is not a matter of uncertainty alone: the measurement itself is carried out by automatic volt-amperometric measurements at undisturbed frequencies, and the calibration with reference resistors indicates an instrumental uncertainty of the order of 1% to 2%, depending on the resistance values. The variability in the soil resistivity instead should be accounted for depending on the location, depth and environmental/seasonal conditions. The latter may be ruled out if the soil resistivity is measured immediately before (or after) the track measurements. The former can be accounted for by repeated measurements and then taking a weighted average as the $\rho $ value and their dispersion as a Type A estimate of their uncertainty.

#### 4.4. Application to a Tramway System

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Sketch of the method A.2 setup for a track insulation measurement: the current measuring circuits in blue, the voltage measuring circuits in red.

**Figure 2.**Modification in method A.2 to the rail voltage drop measuring terminals and to the track-to-earth voltage measuring terminal in order to obtain a unique potential reference node (circled in green) and avoid ground loops.

**Figure 3.**Sketch of the method A.3 setup for a rail insulation measurement: the current measuring circuits in blue, the voltage measuring circuits in red.

**Figure 4.**Track insulation ${G}_{re}$ for different rail resistance values ${R}_{r}=20\mathrm{m}\Omega /\mathrm{km}$, 40 m$\Omega $/km, 60 m$\Omega $/km (shown from dark to light color) vs. voltmetric terminal position P (varying between 50 m and ${L}_{t}/2$ from the injection point). Track section of variable length ${L}_{t}=500\mathrm{m}$ (blue), ${L}_{t}=1000\mathrm{m}$ (green) and ${L}_{t}=1500\mathrm{m}$ (red). Earthing resistance at the injection point ${R}_{0}=5\Omega $. Reference ideal value of rail insulation ${G}_{re}^{*}=10\mathrm{mS}/\mathrm{km}$.

**Figure 5.**Sketch of the method A.4 setup for a track insulation measurement, showing the double track and the two electrodes (E1 and E2) and related geometrical quantities.

**Figure 6.**Comparison of method A4 formulas of track-to-earth conductance given in EN 50122-2 versions 2010 (blue) and 2022 (light brown): (

**a**) single-track case, (

**b**) double-track case. The reference parameters are: $\rho =50\Omega \mathrm{m}$, ${m}_{sr}=0.001$, ${s}_{g}=1.5\mathrm{m}$ and ${s}_{tt}={s}_{g}+2\mathrm{m}$. The difference between curves is of the order of 10% to 18% for the various a values.

**Figure 7.**Results of method A.4 measurements for three positions along a tramway route in an urban context: (

**a**,

**c**,

**e**) voltages of the track and electrode E1 with respect to electrode E2, in blue and red respectively; (

**b**,

**d**,

**f**) estimated angular coefficient (stray current ratio ${m}_{sr}$) by linear regression (black circles are original samples, the orange line is the resulting linear regression).

Brand/Model | Uncert. Expression | $\mathit{u}\left\{{\mathit{V}}_{\mathbf{rX}}\right\}$ @ 1 mV | $\mathit{u}\left\{{\mathit{V}}_{\mathbf{rX}}\right\}$ @ 3 mV |
---|---|---|---|

Weilekes Elektronik MiniLog2 | 0.5% + 10 $\mathsf{\mu}$V | 1.5% | 0.83% |

National Instruments USB 6210 | 0.05% FS + 12 $\mathsf{\mu}$V | 8.9% | 3.0% |

Gossen Metrawatt H29S | 0.02% + 0.01% FS + 5 cts. | 0.02% + 0.01% 300 mV + (2 × 300 mV/30,000)/1 mV = 0.02% + 3% + 2% = 5.02% | 0.02% + 0.01% 300 mV + (2 × 300 mV/30,000)/3 mV = 0.02% + 3% + 0.66% = 3.68% |

Fluke 117 | 0.5% + 2 cts. | 0.5% + (2 × 600 mV/6000)/1 mV = 20.5% | 0.5% + (2 × 600 mV/6000)/3 mV = 7.17% |

**Table 2.**Worked out method A4 on three locations of the same tramway system: geometry parameters and main results.

Location | $\mathit{\rho}$ ($\mathsf{\Omega}\phantom{\rule{0.166667em}{0ex}}\mathbf{m}$) | a ($\mathbf{m}$) | b ($\mathbf{m}$) | ${\mathit{s}}_{\mathit{g}}$ ($\mathbf{m}$) | ${\mathit{s}}_{\mathit{tt}}$ ($\mathbf{m}$) | ${\mathit{m}}_{\mathit{sr}}$ | ${\mathit{G}}_{\mathbf{TE}}$ ($\mathbf{S}\phantom{\rule{0.166667em}{0ex}}\mathbf{km}$) |
---|---|---|---|---|---|---|---|

1 | 17.1 | 11.1 | 37.8 | 1.5 | 6.7 | 0.000605 | 0.0436 |

2 | 19.8 | 14.2 | 46.2 | 1.5 | 3.7 | 0.0026 | 0.1525 |

3 | 38.6 | 8.6 | 45.7 | 1.5 | 3.9 | 0.0038 | 0.0949 |

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**MDPI and ACS Style**

Bhagat, S.; Bongiorno, J.; Mariscotti, A.
Uncertainty of Standardized Track Insulation Measurement Methods for Stray Current Assessment. *Sensors* **2023**, *23*, 5900.
https://doi.org/10.3390/s23135900

**AMA Style**

Bhagat S, Bongiorno J, Mariscotti A.
Uncertainty of Standardized Track Insulation Measurement Methods for Stray Current Assessment. *Sensors*. 2023; 23(13):5900.
https://doi.org/10.3390/s23135900

**Chicago/Turabian Style**

Bhagat, Sahil, Jacopo Bongiorno, and Andrea Mariscotti.
2023. "Uncertainty of Standardized Track Insulation Measurement Methods for Stray Current Assessment" *Sensors* 23, no. 13: 5900.
https://doi.org/10.3390/s23135900