Estimation of the Current Uncertainty in the Dielectric Shoe Test According to the ISO/IEC 17025 Standard in the High Voltage Laboratory LABAV of the Escuela Politécnica Nacional

Abstract

The High Voltage Laboratory (LABAV) at the Escuela Politécnica Nacional conducts dielectric tests on safety shoes in accordance with the ASTM F2412-18 standard. Additionally, as per the NTE INEN ISO 17025 standard, the laboratory must estimate the uncertainty of its measurements. Despite the scarcity of examples in the existing literature, this work provides a real-world example to assist other laboratories in replicating the uncertainty estimation process. In this article, we systematically present the calculation of leakage current uncertainty in shoes using both the traditional “Guide to the Expression of Uncertainty in Measurement” (GUM) method and the Monte Carlo method (MCM) for validation. The results from both approaches yield a similar uncertainty value of u = 0.0733 mA. Finally, we highlight the advantages that the MCM method offers in this context.

1. Introduction

In the field of personnel safety for those working with electricity, dielectric shoes are a vital element in safeguarding workers’ lives. Consequently, standards ASTM F2412-18a and ASTM F2413-18 [1,2] establish the methods and requirements that various types of safety shoes must meet. For dielectric shoes, three randomly selected shoes from a production lot must withstand a voltage of 18,000 volts for 1 min, with the current passing through the shoe not exceeding 1 mA. If the shoe burns, punctures, or exceeds a current of 1 mA, it does not meet the standard’s requirements, and the entire associated production lot is deemed non-compliant with safety requirements.

Measuring the current in the shoe is crucial during testing because it is a relatively low value, meaning that measurement uncertainty can affect the compliance criterion of 1 mA. For instance, if the test had an uncertainty of 0.5 mA, the laboratory could only guarantee compliance for shoes that reach a value of 0.5 mA, as the current measurement would be 0.5 mA ± 0.5 mA. For higher measurement values, the uncertainty would exceed the compliance limit, and a conformity statement could not be issued. Hence, accurately estimating and achieving low uncertainty is paramount.

To ensure the reliability of laboratory test results, the guidelines of the NTE INEN-ISO/IEC 17025:2018 standard “General requirements for the competence of testing and calibration laboratories” [3] are followed. This standard serves as the foundation for implementing and maintaining a quality management system (QMS) in a testing laboratory. Section 7.2.1.1 of this standard states, “The laboratory shall use appropriate methods and procedures for all laboratory activities and, where appropriate, for the evaluation of measurement uncertainty, as well as statistical techniques for data analysis”. This requirement is also harmonized with the Ecuadorian Accreditation Service (SAE) in its document “PL02-Policy for the estimation of measurement uncertainty” [4].

For estimating uncertainty, the “Guide to the Expression of Uncertainty in Measurement” (GUM) or alternative methods accepted by the International Bureau of Weights and Measures (BIPM) are available [5]. The GUM defines measurement uncertainty as the inability to know the exact value of the measurand. Uncertainty is affected by numerous factors or sources and is generally categorized into two types based on the evaluation method: “Type A” and “Type B”. Type A uncertainty evaluation is determined in most cases by a statistical analysis of a series of observations, obtaining the experimental standard deviation of the measurement, usually from an averaging procedure or regression analysis. In contrast, Type B uncertainty uses non-statistical means, applying scientific judgment based on available information and experience regarding the potential variability of the variable under study.

The High Voltage Laboratory (LABAV) of the Escuela Politécnica Nacional conducts dielectric shoe testing for the public. Based on the presented background, this article aims to present a real example of uncertainty estimation in dielectric testing of safety shoes according to GUM guidelines, validated through Monte Carlo simulations. The goal is to provide a bibliographic source for other laboratories requiring similar uncertainty estimations, as no articles or reference sources with procedures or numerical examples to guide LABAV were found during the estimation process.

2. Materials and Methods

The applied methodology is divided into two phases:

Phase 1: Uncertainty estimation using the GUM Guide [5].

Phase 2: Monte Carlo simulation estimation [6].

2.1. Phase 1: Identification of Sources of Uncertainty

In the testing of dielectric shoes, the main sources of uncertainty affecting the current measurement are as follows:

• Resolution: The resolution of the equipment is a type B uncertainty and is obtained from the equipment’s catalog. In this case, the shoe tester is a GESTER GT-KB42 model whose ammeter resolution is 0.01 mA.
• Current meter error: The uncertainty of the current meter is obtained from the calibration certificate issued by an accredited laboratory. This uncertainty is considered Type B, as it is based on expert judgment rather than direct measurement. The calibration certificate was obtained from a metrological laboratory accredited under ISO 17025, with an uncertainty of 0.007 mA.
• Repeatability and reproducibility: The repeatability of the dielectric shoe test is primarily ensured by the use of an automated testing system, which minimizes operator influence. In metrology, repeatability refers to the variability in results when the same procedure is performed under identical conditions. In this case, the shoe tester (GESTER GT-KB42) is fully automated, ensuring high repeatability as the operator’s role is limited to preparing the test setup. The operator follows simple and standardized steps: they fill the shoe with a conductive material according to the [2] method, place the shoe inside the tester, attach the voltage electrodes, and close the testing chamber. Once the chamber is sealed, the operator initiates the test through a digital interface, which is pre-programmed to follow the standardized testing procedure. The system then conducts the test autonomously, applying the voltage and measuring the resulting current without further human intervention.
  This automation eliminates variability due to operator handling, which is a critical factor in ensuring consistent and repeatable results. To quantify the repeatability of the test, ten repetitions were performed, yielding a standard deviation of 0.107 mA. This result highlights the precision and stability of the automated system when conducting the dielectric test under controlled conditions.
  In addition to ensuring repeatability through the use of an automated testing system, reproducibility is also a key consideration in dielectric shoe testing. Reproducibility refers to the consistency of test results when conducted under varying conditions, such as by different operators, using different equipment, or in different laboratories. In this case, reproducibility is supported by adherence to standardized test methods, specifically [2]. Since the procedure is well-defined and the equipment is programmed to follow this standard without manual adjustments, variations due to human intervention are minimized.
  While reproducibility testing often involves comparing results across different locations or setups, the use of an internationally recognized standard and an automated system in this test significantly reduces variability between tests conducted in different environments. Moreover, the calibration of the equipment in accordance with ISO 17025 ensures that the system operates consistently across different setups. This calibration, alongside the use of the same ASTM test method, enhances the reproducibility of the test, even when conducted by different laboratories. The system’s capacity for automated, standardized procedures ensures that tests performed in different settings can yield comparable results, providing confidence in the reproducibility of the dielectric shoe test.
• Temperature: Ambient temperature can affect the characteristics of both the measuring equipment and the shoe under test. This uncertainty is considered Type A and is estimated from temperature measurements taken during the test. Ten repetitions were performed, resulting in a standard deviation of 0.000979 mA/°C. Temperature fluctuations can influence the test because voltage is applied to a dielectric material; in this case, the shoe and insulation materials may change their conductivity with temperature. However, it is important to note that according to [2], the shoe must rest for 24 h in the testing area, which must maintain a temperature of $\mathrm{22}\pm 2$ °C. During the test, the temperature should remain between $22\pm 2$ °C, and the test itself lasts only one minute, meaning that temperature fluctuations are minimal when following the standardized method.
• Voltage variability: The electric tester can exhibit voltage fluctuations that affect the measurement. This uncertainty is considered Type A and is estimated from voltage and current measurements in the laboratory during the test. Ten repetitions were performed, resulting in a standard deviation of 0.000979 mA/°C.

These uncertainty parameters are presented in Equation (1) and in

Table 1. Estimated values of the uncertainty components of the proposed model.

Source of Uncertainty	Estimated Value	Units	Type	Distribution	Divisor
Resolution ${∆}_{Res}$	0.01	mA	B	Rectangular	$2\sqrt{3}$
Error $\epsilon$	0.007	mA	B	Normal	$k;k=2$
Repeatability ${∆}_{Rep}$	0.107	mA	A	Normal	$\sqrt{n};n=10$
$∆I_t$ Temperature Correction	0.000979	mA/°C	A	Normal	$\sqrt{n};n=10$
$∆I_v$ Voltage Correction	0.0424	mA/V	A	Normal	$\sqrt{n};n=10$

.

2.2. Phase 1: Selection of the Measurement Model

The measurement model that relates the measured current I to the influencing variables can be represented by Equation (1):

$$I_{fc}=I_m+{\Delta }_{Res}+{\Delta }_{Rep}-(\Delta I_t+\Delta I_v)-\epsilon ,$$

(1)

where

• $I_m$ is the leakage current measured;
• $I_{fc}$ is the leakage current corrected;
• ${\Delta }_{Res}$ is the deviation of the current due to the resolution of the instrument;
• ${\Delta }_{Rep}$ is the deviation of the current due to measurement repeatability;
• $\Delta I_t$ is the deviation of the current due to temperature variation;
• $\Delta I_v$ is the deviation of the current due to voltage variation;
• $\epsilon$ is the error of the instrument due to calibration.

2.3. Phase 1: Estimation of Uncertainty Components

2.3.1. Type A Uncertainty

Type A uncertainty is obtained from a series of repeated measurements. The experimental standard deviation s is calculated by Equation (2):

$$s=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{(m_i-\overline{m})}^2}{n-1}}},$$

(2)

where

• $m_i$ is the value in the i-th measurement;
• n is the number of measurements;
• $\overline{m}$ is the average value.

Type A uncertainty ($u_A$) is calculated by Equation (3):

$$u_A={\displaystyle \frac{s}{\sqrt{n}}}.$$

(3)

2.3.2. Type B Uncertainty

Type B uncertainty is estimated from the available information about the sources of uncertainty, such as calibration certificates, previous studies, or expert knowledge. An appropriate probability distribution is used to represent the variability of each source of uncertainty.

2.4. Phase 1: Combination of Uncertainty Components

The combined standard uncertainty $u_c$ is calculated by combining the Type A and Type B uncertainty components using Equation (4):

$$u_c=\sqrt{\sum _{i=1}^N{u}_A^2+\sum _{j=1}^z{u}_B^2}=\sqrt{\sum _{i=1}^M{(C_i.u_i)}^2},$$

(4)

where

• N is the number of Type A uncertainty sources;
• ${\mu }_A$ is the Type A uncertainty of the i-th source;
• z is the number of Type B uncertainty sources;
• ${\mu }_B$ is the Type B uncertainty of the j-th source;
• M is the number of total uncertainty sources;
• $u_i$ are the individual standard uncertainties.

To determine $u_c$ in Equation (4), we need to find a sensitivity coefficient $C_i$ by taking the partial derivative of the mathematical model from Equation (1) with respect to the variable for which the contribution is sought according to Equation (5).

$$C_i={\displaystyle \frac{\partial y}{\partial x_i}},$$

(5)

where

• $C_i$ represents the sensitivity coefficient for the input variable $x_i$;
• y is the output of the model;
• $x_i$ is the specific input variable;
• $\partial y/\partial x_i$ represents the partial derivative of output y with respect to input $x_i$.

The contributions $C_i.u_i$ from Equation (4) are calculated individually by multiplying each uncertainty by its sensitivity coefficient obtained using Equation (5).

Expression of Uncertainty

The total uncertainty is expressed with a confidence interval and a confidence level. The confidence interval represents the range within which the true value of the measurement is found with a certain probability. The confidence level indicates the probability that the true value lies within the confidence interval. This value is known as expanded uncertainty in Equation (6):

$$u=u_c·t_{0.95,n-1},$$

(6)

where

• $t_{0.95,n-1}$ is the value of the Student’s t-distribution with a 95% confidence level and $n-1$ degrees of freedom. This value will be known as coverage factor k.

To determine the total degrees of freedom of the uncertainty, we must calculate the contributions of the degrees of freedom for each uncertainty $u_i$. For Type A uncertainties, the degrees of freedom are equal to $n-1$, with n representing the number of measurements taken to estimate that uncertainty. In this study, for Type B uncertainties, which originate from the calibration certificate and the instrument resolution, a very high value is used; in this case, 200.

Once the individual uncertainties $u_i$ are known and the combined standard uncertainty $u_c$ is calculated from Equation (4), the total degrees of freedom are determined using the Welch–Satterthwaite equation, as in Equation (7).

$${\nu }_{\mathrm{eff}}={\displaystyle \frac{u_c^4}{{\sum }_{i=1}^N\left(\frac{{(C_i·u_i)}^4}{{\nu }_i}\right)}},$$

(7)

where

• ${\nu }_{eff}$ is the effective degrees of freedom;
• $u_c$ is the combined standard uncertainty;
• $u_i$ are the individual standard uncertainties;
• ${\nu }_i$ are the degrees of freedom associated with each $u_i$;
• N is the number of uncertainty components.

With the total degrees of freedom, based on the Student’s t-distribution, we determine the coverage factor k for a 95% confidence level.

2.5. Phase 2: Monte Carlo Simulation Estimation

The Monte Carlo method (MCM) is a widely used numerical method for estimating measurement and calibration uncertainties. This method generates random values of the measured variable (in this case, current), considering the probability distributions of influencing variables and their correlations. By repeating the simulation a considerable number of times, a distribution of the measured current is obtained, allowing for uncertainty calculation and comparison with the estimation obtained using the GUM Guide [6].

To calculate the uncertainty estimation using the GUM Guide, the free application MCM Alchimia was used, which implements the Monte Carlo method to estimate measurement and calibration uncertainties according to the JCGM 101 Guide [6]. This version includes a complete uncertainty budget for the GUM framework and allows for the handling of correlated quantities and regression curves.

Steps of Monte Carlo Simulation

Definition of influencing variables: Identify all variables that can affect the measurement of the current, such as equipment calibration, electric network variability, temperature, and humidity.

Selection of probability distributions: Assign an appropriate probability distribution to each influencing variable, considering available information such as calibration certificates, previous studies, or expert knowledge. For example, a normal distribution can be used for equipment calibration, and uniform distributions can be used for temperature and humidity.

Generation of random values: Use Monte Carlo simulation software to generate random values of influencing variables according to selected probability distributions.

Calculation of current: For each set of random values of influencing variables, calculate the corresponding current value using the defined measurement model.

Repetition of simulation: Repeat steps 3 and 4 a large number of times (e.g., 10,000 or 100,000) to obtain a representative distribution of the measured current.

Analysis of distribution: Analyze the distribution of the measured current, including the mean, standard deviation, and other measures of dispersion.

Estimation of uncertainty: Calculate the total uncertainty $u_c$ by combining Type A and Type B uncertainty components, as described in Section 2.4 of this article.

Comparison with GUM estimation: Compare the estimated total uncertainty from the Monte Carlo simulation with the estimation obtained using the GUM Guide. If the estimations are consistent, the methodology used is validated.

3. Results

3.1. Uncertainty Results Using the GUM Method

The variables from the utilized model contributing to uncertainty are presented in Table 1. The estimated resolution value comes from the instrument’s certificate, while the error originates from the instrument’s calibration certificate. For repeatability and temperature corrections, ten repetitions are performed, and the standard deviation is obtained. The same table specifies the type of uncertainty, the distribution function they follow, and the divisor used to calculate the standard uncertainties $u_i$, which is defined as the division of the estimated value by the divisor.

In

Table 2. Sensitivity coefficients of each uncertainty component of the proposed model.

Source of Uncertainty	Sensitivity Coefficient Calculation	${\mathit{C}}_{\mathit{i}}$	Standard Uncertainties, ${\mathit{u}}_{\mathit{i}}$
Resolution ${\Delta }_{Res}$	${\displaystyle \frac{\partial I_{fc}}{\partial \Delta \mathrm{Res}}}$	1	0.002886751
Error $\epsilon$	${\displaystyle \frac{\partial I_{fc}}{\partial \epsilon }}$	−1	0.003500000
Repeatability ${\Delta }_{Rep}$	${\displaystyle \frac{\partial I_{fc}}{\partial \Delta \mathrm{Rep}}}$	1	0.033836371
$\Delta I_t$ Temperature Correction	${\displaystyle \frac{\partial I_{fc}}{\partial \Delta It}}$	−1	0.000309587
$\Delta I_v$ Voltage Correction	${\displaystyle \frac{\partial I_{fc}}{\partial \Delta Iv}}$	−1	0.013408057

, the sensitivity coefficients $C_i$ calculated from the mathematical model and the standard uncertainties $u_i$ are presented.

Finally, in

Table 3. Contributions and degrees of freedom of each uncertainty component of the proposed model.

Source of Uncertainty	Contribution of Uncertainty ${\mathit{C}}_{\mathit{i}}·{\mathit{u}}_{\mathit{i}}$	${({\mathit{C}}_{\mathit{i}}·{\mathit{u}}_{\mathit{i}})}^2$	${({\mathit{C}}_{\mathit{i}}·{\mathit{u}}_{\mathit{i}})}^4$	Degrees of Freedom ${\mathit{\nu }}_{\mathit{i}}$
Resolution ${\Delta }_{Res}$	0.002886751	$8.33333\times {10}^{-6}$	$6.94444\times {10}^{-11}$	9
Error $\epsilon$	−0.003500000	$1.2225\times {10}^{-5}$	$1.50063\times {10}^{-10}$	200
Repeatability ${\Delta }_{Rep}$	0.033836371	$1.1449\times {10}^{-3}$	$1.3108\times {10}^{-6}$	200
$\Delta I_t$ Temperature Correction	−0.000309587	$9.58771\times {10}^{-8}$	$9.18609\times {10}^{-15}$	9
$\Delta I_v$ Voltage Correction	−0.013408057	$1.79776\times {10}^{-4}$	$3.23194\times {10}^{-8}$	9

, the contributions $C_i·u_i$ from each uncertainty variable are determined, and degrees of freedom are assigned.

With the calculations from Table 3, we obtain a combined standard uncertainty of $u_c=0.036679084$.

We apply (7) and obtain a total of 12 effective degrees of freedom.

Using the Student’s t-distribution, we determine factor k for 95% confidence, resulting in $k=2.179$. However, for testing laboratories, it is recommended to use factor $k=2$.

Therefore, the expanded uncertainty is $u=k·u_c$, which results in $\mathit{u}=\mathbf{0}.\mathbf{0733581673}\mathbf{mA}$ for this analysis.

3.2. Uncertainty Results Using the MCM Method

For this section, we utilize the MCM Alchimia program. The procedure is straightforward:

• Input the set of equations for the model (the same model as in Equation (1));
• Define each variable of the model along with its units and values;
• Specify the distribution function for each variable (as shown in Table 1);
• Assign infinite degrees of freedom to each variable;
• Finally, run 500,000 simulations for this case.

The software generates a result of $\mathit{u}=\mathbf{0}.\mathbf{07336}\mathbf{mA}$ with a coverage factor of $\mathit{k}=\mathbf{2}$ and a 95.45% confidence interval.

MCM Alchimia proposes in Figure 1 a comparison of the results between the MCM method and the GUM method.

3.3. Expression of the Measurement

The expression of the measurement should include uncertainty. For example, if the measurement in the test yields 0.84 mA, it should be expressed as $\mathit{I}=\mathbf{0}.\mathbf{84}\mathrm{mA}$ with uncertainty $\mathit{u}=\mathbf{0}.\mathbf{733}\mathrm{mA}$ and a coverage factor of $\mathit{k}=\mathbf{2}$ at a confidence level of 95%.

In the classical form, the current would be expressed as I = (0.84 ± 0.733 ) mA.

4. Discussion

4.1. On Uncertainty

When observing the results from Section 3.1 and Section 3.2, it is evident that the uncertainty is practically the same for both methods. This validation allows us to use the result for laboratory test reports in accordance with the NTE INEN ISO 17025 standard.

4.2. Advantages of Monte Carlo Simulation

When calculating uncertainty using the GUM method, the traditional approach involves creating a set of tables in a spreadsheet and following all the steps described in Section 2.1. However, when using the MCM method, there is significant ease in the calculation process. Essentially, the problem reduces to determining the mathematical model of the measurement and its variables in the best possible way. The rest of the simulation is rapid, and the results are practically the same as those indicated by the GUM.

Additionally, for more complex measurements with intricate partial derivatives, the MCM method simplifies the process even further. In general, we recommend using this method due to the following advantages:

• Flexibility: Monte Carlo simulation allows for handling a wide variety of measurement models and probability distributions. This flexibility is beneficial in modeling complex scenarios and diverse sources of uncertainty.
• Consideration of correlations: It can incorporate correlations between influencing variables, providing a more accurate estimation of uncertainty. This feature is crucial as it reflects real-world conditions where variables often influence each other.
• Validation of GUM estimation: Monte Carlo simulation serves as a tool to validate uncertainty estimations obtained using the GUM Guide. By comparing simulated results with theoretical estimations, it helps us to ensure the robustness and reliability of uncertainty assessments.
• Visualization of distribution: It enables the visualization of the distribution of the measured current, offering valuable insights into measurement variability. This visualization aids in understanding the range of potential outcomes and the likelihood of different measurement results under varying conditions. These advantages make Monte Carlo simulation a powerful tool for uncertainty analysis in measurement and calibration, particularly in contexts where complex interactions and diverse sources of uncertainty need to be considered.

5. Conclusions

The uncertainty estimation for current measurement in dielectric shoes has been carried out according to the NTE INEN ISO 17025 standard. The overall estimation result indicates an uncertainty of $\mathit{u}=\mathbf{0}.\mathbf{0733}\mathrm{mA}$ with a coverage factor of $\mathit{k}=\mathbf{2}$ at a confidence level of 95%. It can be observed from Table 3 that the largest contribution to the uncertainty ${\mathit{C}}_{\mathit{i}}·{\mathit{u}}_{\mathit{i}}$ comes from the repeatability variable. In other words, the equipment applying voltage to the shoe experiences some difficulty in reproducing test conditions.

Considering that, according to the ASTM F2413-18 standard, the current limit for a dielectric shoe subjected to 18,000 V for one minute is 1 mA, the most reliable measurement ensuring compliance with the standard would be 0.9267 mA.