General
Duval triangle is one of the tools used to assess the condition of a power transformer given the concentrations of three key gases dissolved in the transformer oil.
Apparently one of the factors affecting the accuracy of the results provided by the Duval triangle is the accuracy three concentrations CH4, C2H2 and C2H4 are measured with. But how rapidly the triangle accuracy decreases with the increase in concentrations uncertainty? This is what we're going to find out in this article.
To do this, this is the methodology we're going to follow:
Determine a set of measurement error we will check through. This is going to be the values running from 0 to 100 in steps of 5.
Define a measurement instrument characteristics. What we need are the measurement ranges and lower detection level of all three key gases.
Generate a set of 1000 true concentrations of three key gases. This is what the actual concentrations in the transformer oil are.
Given the set of true concentration values, apply the measurement uncertainty and get a set of measurement instrument readings. The measurement uncertainty is to be defined from the first error value defined in the step 1.
Determine and compare the Duval triangle results for the true and the measured concentration values.
To make the result statistically significant, repeat steps 3-5 several times.
Repeat steps 3-6 for each remaining measurement error value defined in the step 1.
Set up a plot with the horizontal axis representing the evalue and the vertical axis representing the ratio of the number of tests where the Duval triangle results match for the true and the measured concentrations.
The final plot will show how the Duval triangle results depend on the measurement error of the device used to measure the three key gases concentrations.
All the numeric calculations and visualizations related to the proposed methodology are done in Python with numpy and matplotlib. The source code of all algorithms is available on GitHub by the link in the ‘Sources’ section.
We'll then continue to execution.
Step 1
Here we will simply set up a all the measurement errors we want to examine. They are going to start from 0% (ideal measurement instrument) and proceed to 100% (very poor accuracy) in steps of 2%.
Step 2
Define a measurement instrument characteristics. What we need are the measurement range and lower detection level.
To define the measurement ranges let us examine the gas generation chart. It is clearly visible from the image below, taken from [1], that as a local spot temperature inside a power transformer increases, methane first starts to show off at the temperatures of about 150 deg.C. Both methane and ethylene appear when the temperature rises above 350 deg.C and the presence of both methane, ethylene and acetylene is an indication of local spots temperature exceeding 500 deg.C.
Recommended alarm levels as proposed by IEC 60599 are: 100 ppm for methane, 70 ppm for ethylene and 10 ppm for acetylene. Since each power transformer may have its own standard gassing activity not related to a real fault, we’ll double these proposed levels and will examine the results with methane concentrations sampled from the range 0-200 ppm, ethylene from 0-140 ppm and acetylene from 0-20 ppm.
We will then consider that if the true value of a concentration is below an instrument lower detection level, the measurement error will be equal the lower detection level. Otherwise, if the true value is above the lower detection level, the measurement error is defined directly by the instrument’s manufacturer.
We will also assume all measured values are distributed normally with the mean value being the true concentration value. We will also assume that for our normal distribution three sigma rule is applicable, that is, the measurement uncertainty equals triple standard deviation value and 99.7% of all measured values will not be more distant as the measurement uncertainty from the mean. For the rest 0.3% of measurements, we’ll take them as measurement instrument faults which caused incorrect readings.
One more thing to note is that having this approach, it’s theoretically possible that some of measured values will be negative for the concentrations below the lower detection level. We’ll treat such cases as zero values since negative concentrations is something that never happens.
Having that said, we can define our measured concentration value (mv) as:
where:
tv is the true value;
ac is the measurement measurement error;
N(0; 1) is the standard normal distribution with mean 0 and deviation 1;
ldl is the lower detection level.
To check which values of the error and the lower detection levels we need to pick let us take a look what is offered from the industrial manufacturers. Four large dissolved gases monitors providers with their products are:
Qualitrol, Serveron TM8;
GE, Kelman Transfix;
Lumasense, SmartDGA;
Vaisala, OPT100.
From its specifications it’s clear that all four monitors are able to provide measurements of three gases mentioned above within the ranges far exceeding the ranges we consider. Errors (termed as accuracies in manufacturers' specifications and detection levels are of bigger interest. The table below summarizes these values for all four monitors.
So, for the lower detection levels of methane, ethylene and acetylene we’ll use the values of 2 ppm, 2 ppm and 0.5 ppm. For all the standard accuracies we’ll use 5%.
Step 3
Generate the true concentration values. To do so, we'll take 1000 values for each concentration within the ranges 0-200 ppm for methane, 0-140 ppm for ethylene and 0-20 ppm for acetylene.
To visualize the result, 3D scatter plot with three axes values representing three gases concentrations, is shown on the image below. As you can see, all points are uniformly distributed inside the box bounded by concentration ranges.
Step 4
Apply the measurement uncertainty. As explained earlier, for each of the true values we’ll generate 100 measured values following the formula:
To visualize the result, a set of CH4 concentrations, both true and measured, is shown on the image below. The values on the plot are arrange in ascending order and the measurement error value of 30% was chosen. The vertical axis is the conentration values, the horizontal axis is a simple measurement index.
Steps 5 & 6
For the generated true values of concentrations and the measured values check which region of the Duval triangle each of the points falls into. Record the results, namely which number of points had the Duval triangle regions the same for the true and the measured values and which not.
Then repeat the steps 3-5 with each of the remaining measurement error values defined in the step 1. Ultimately get the final results showing the number of points with coinciding Duval triangle regions and not.
Two final plots for the errors spanning from 0% to 100% in steps of 2% with 1000 true gas concentration values and 100 measured concentrations for each error step are shown on the image below. The first plot show each point as a ratio between the samples where the Duval triangle region matched both for the true and the measured concentration values against the total number of measured concentrations. For our numbers, the total number of measured concentrations 1000 which is a number of points standing for each of the measurement error, and each of these points is the ratio of matched Duval triangle region results from a set of 100 measured concentrations generarted for each of 1000 true concentration values.
The second plot is basically the same, but each ratio for a given measurement error is an arithmetical mean of all of 1oo ratios from the first plot, so representing an average Duval triangle error given the measurement error.
It’s clearly seen the ratio of correct results is about 100% with 0% measurement error and then drops down to about 70% for 100% measurement error.
The link [2] to GitHub repository also contains data.csv file with the statistics showing which number of samples had matched and mismatched Duval regions. In some cases even for the ideal measurement instrument with 0% measurement error the regions were mismatched which happened for low concentrations, when the actual measurement error equaled 5%.
Sources
FIST 3-30. Transformer maintenance. Facilities instructions, standards and techniques. https://www.usbr.gov/power/data/fist/fist3_30/fist3_30.pdf
Algorithms source codes. https://github.com/leonidpospeev/duval-triangle-accuracy