This paper formulates the exact analytical probability density function (PDF) for the ratio of two independent dissolved gas analysis (DGA) measurements that include individual gas measurement errors. It is demonstrated that for small DGA gas measurement errors, the correct two-gas ratio PDF approaches a conventional Gaussian distribution. As the measurement accuracy decreases, the ratio PDF becomes non-Gaussian with the maximum likelihood value of the PDF deviating from the true underlying value. For larger errors, the maximum likelihood estimate of the gas ratio deviates significantly from presumed Gaussian statistics. A method for de-biasing measured gas ratio values is presented and a simple application is used to demonstrate the proposed approach

Aizpurua, J. I.

Stewart, B. G.

University of Strathclyde Institutional Repository

Uncertainty Analysis of Two Gas Measurement DGA Ratios for Improved Diagnostics Applications B. G. Stewart1 and J. I. Aizpurua2,3 1 Institute of Energy and Environment, University of Strathclyde, 204 Cowcaddens Road, Glasgow, Scotland, UK 2Mondragon University, Electronics & Computer Science Department - Signal Theory & Communications, Arrasate, Spain, 3Ikerbasque, Basque Foundation for Science, Bilbao, Spain brian.stewart.100@strath.ac.uk Abstract: This paper formulates the exact analytical probability density function (PDF) for the ratio of two independent dissolved gas analysis (DGA) measurements that include individual gas measurement errors. It is demonstrated that for small DGA gas measurement errors, the correct two-gas ratio PDF approaches a conventional Gaussian distribution. As the measurement accuracy decreases, the ratio PDF becomes non-Gaussian with the maximum likelihood value of the PDF deviating from the true underlying value. For larger errors, the maximum likelihood estimate of the gas ratio deviates significantly from presumed Gaussian statistics. A method for de-biasing measured gas ratio values is presented and a simple application is used to demonstrate the proposed approach. I. INTRODUCTIONTransformers are key assets for the correct operation of the power grid [1], and accordingly, transformer health monitoring activities are crucial to ensure reliable grid operation [2]. Dissolved gas analysis (DGA) is an important industry-accepted method for assessing the health of the electrical insulation of oil-filled power transformers [3]-[6]. DGA involves extracting oil samples and making measurements of key combustible gases including hydrogen (H2), methane (CH4), acetylene (C2H2), ethylene (C2H4), ethane (C2H6), carbon monoxide (CO) and carbon dioxide (CO2), measured in μlitres/litre or parts-per-million (ppm). Diagnostic health assessment is often performed through determining the levels of these gases and then evaluating a set of key gas ratios, labelled from R1 to R5, as defined in Table I. TABLE I KEY GAS RATIO DEFINITIONS Ratio R1 R2 R3 R4 R5 Gases CH4/H2 C2H2/C2H4 C2H2/CH4 C2H6/C2H2 C2H4/C2H6 The IEEE C57.104 and IEC 60599 standards outline methods for the measurement and analysis of key gas ratios such as Rogers ratios, Doernenburg ratios and the Duval triangle [7], [8]. These methods define boundary gas levels and gas-ratio threshold levels that correspond to insulation fault types. When measuring gas levels, hardware equipment and laboratory measurement procedures induce gas measurement errors. For an analytical detection limit S, the IEC 60599 states that above 10 x S the uncertainty on gas measurements is 15% and for measurements below 10 x S the uncertainty can increase up to 30% [8]. Duval and Dukarm highlight that gas measurement errors can range from 5% to 30%, and if measurement accuracies are unknown, they recommend the adoption of a default 15% measurement error [9]. Though many modern gas measurement instruments are capable of attaining 5% measurement errors or better, many other gas measurements still have large measurement uncertainties. Indeed, historical gas measurements collected by legacy measurement equipment will likely have larger or unknown measurement errors. It is therefore crucial to know how best to evaluate the accuracy of gas ratio measurements for these situations, and accordingly, interpret the outcome of any DGA diagnostic methods. In this context, a general understanding of the correct theoretical and analytical framework for determining a two-gas ratio error, based on individual gas measurement errors is required. Normally practitioners presume gas ratio values are Gaussian distributed around the true value (e.g. [10]). However, this is not the case as will be presented below. The statistical distributions of the ratio of two independent Gaussian distributed measurement variables has previously been studied [11-14]. However, these papers do not provide an analysis of best estimate methods for establishing the maximum likelihood (ML) ratio probability for applications such as DGA ratio-based diagnostics. To this end, the main contributions of this paper are twofold. Firstly, the formulation of the correct probability density function (PDF) for two-gas ratio dissolved gas evaluations with measurement errors. Secondly, the conception of a methodology to elicit the best estimate of the true underlying gas ratio through ML estimation methods. In the evaluations, it is shown that the correct PDF for a two-gas ratio determination is non-Gaussian. Both the correct gas ratio PDF and a presumed standard Gaussian formulated PDF are compared. A simple application to DGA data demonstrates the underlying principles in relation to best-estimate gas-ratio values. The paper structure is as follows. Section 2 derives the PDF for two measurement gas ratios with individual gas measurement uncertainties and compares the correct PDF with a Gaussian PDF formulation as a function of measurement errors. Section 3 outlines a method for determining the best estimate of the true underlying ratio based on the correct PDF and measured ratio value. Section 4 provides some examples to demonstrate the principles outlined in the paper. Finally, Section 5 summarizes the main conclusions. This is a peer reviewed, accepted author manuscript of the following research paper: Stewart, B. G., & Aizpurua, J. I. (2022). Uncertainty analysis of two gas measurement DGA ratios for improved diagnostics applications. 1-4. Paper presented at International Conference on High Voltage Engineering, Chongqing, China.II. UNCERTAINTY OF DGA DIAGNOSTICS: DERIVATION OF GAS RATIO PROBABILITY DENSITY FUNCTIONS Let any measured gas ratio measurement be defined as R = x/y, where x and y are the numerator and denominator gas measurements respectively. Also, let R0 = x0/y0 represent the true underlying gas ratio, where x0 and y0 are the true gas levels with no measurement errors. If x and y are independent Gaussian (Normal) distributed variables, then the resulting individual PDFs are expressed as: (1) where z = {x, y}, is the gas measurement, z0 is the underlying true gas level and σz is the standard measurement error on z. A. Gas Ratio PDF with Measurement ErrorsThe PDF for the ratio R, PR(R, R0), is determined from thefollowing formulation: (2) which results in (3) Substituting the relevant terms from (1) and (2) into (3), then (4) Expanding terms and performing the integration results in: (5) Substituting R0 = x0/y0 and defining a = σx/x0 and b = σy/y0 as the fractional errors on x0 and y0, then (5) can be expressed as: . (6) It can be seen that PR(R, R0) is a function of R0, a and b. Equation (6) can be normalized in relation to R0 by defining a scaling factor α = R/R0. The corresponding PDF distribution of α, denoted Pα(α), is defined as follows: . (7) Formulating (7) leads to the resulting PDF for , i.e.  . (8) Equation (8) depends solely on a and b, and permits the evaluation of gas ratio probabilities for all values of R0 and all fractional errors on gas measurements through the scaling process. That is, the relationship R = αR0 permits scaling of all α values to R and R0 through Equation (8). Figure 2 shows some examples of the PDF Pα(α) for different measurement error values of a and b, and demonstrates that Pα(α) is generally non-Gaussian in nature. Figure 1. Example P() values as a function of for a = 15% and b = {5%; 10%; 15%; 20%; 25%; 30%}. B. Standard Gaussian Gas Ratio PDFFor simplicity, the usual practice when analyzing a gas ratiomeasurement, R, is to adopt a Gaussian PDF [10]. Forcomparison purposes, the PDF for a presumed Gaussiandistribution, PG(R,R0), is described by:(9) where σG is determined through:  (10) which results in: (11) Substituting α = R/R0 and noting that dR/dα = R0, then (9) can be transformed into the following Gaussian form for variable α: (12) where . It can be seen in (12) that PαG(α) is independent of R0 and is an explicit function of a and Uncertainty analysis of two gas measurement DGA ratios for improved diagnostics applications1b. Scaling to R0 is again performed through R = αR0 noting thatPG(R,R0) = PαG(α|R)/R0.C. Comparison of Pα(α) and PαG(α)As examples, Figure 2 shows the differences in the PDFsand the ML values between Pα(α) and PαG(α) for: (a) a=5%, b=15% and (b) a=10%, b=30%. Pα(α) is seen to be asymmetric compared to PαG(α,) and also possesses a wider PDF skirt. It is also noted that the ML α value, αML, of Pα(α) is always < 1, while for PG(α), which is symmetric, αML = 1. It is also noted that when b is small, or when both a and b are small, then Pα(α) tends towards the Gaussian form PαG(α). For diagnostics purposes the ML estimate of a PDF is often adopted as the most likely measured value. As a consequence, the measured value, R, requires to be “de-biased” to the best estimate of Ro based on the ML value from Pα(α) (see below). III. INFERENCE OF TRUE GAS RATIO R0As the ML value of Pα(α) < 1, a bias correction process is required to estimate the true underlying value of R0, i.e. R0. The measured value R is always an underestimate of the best estimate of the true value R0, i.e. the underlying value of R0 must be larger to ensure R is the peak (ML value) of the correct underlying PDF PR(R, R0). The correction process involves calculating αML using Pα(α) and then adjusting the measured value R by dividing it by αML to obtain the corrected or unbiased estimate R0. The method for inferring R0 is as follows. Using the values of a and b, determine analytically the ML value of α from Pα(α) through: (13) Differentiating (13) and solving for αML, results, after simplification, in the following quadratic equation:  (14) where A4 = 2b6, A3 = b4(1+3a2), A2 = b2(a2b2+2a2-b2), A1 = a2(a2 + 3a2b2 - 2b2), and A0 = - a4(1+b2). Equation (14) may be solved through numerical iteration methods, e.g. Newton-Raphson iteration. However, there are some special cases. If b << a or b = 0, then (14) reduces to a4αML - a4 = 0 resulting in αML = 1. If a << b or a = 0, then (14) reduces to 2b2α2ML + αML - 1 = 0. Solving directly andtaking the positive root gives .Once αML has been evaluated and with the measured value R = x/y presumed to be the ML value of PR(R, R0), then the true underlying R0 (which is always greater than R) is determined though the relationship: (18) Example calculations of αML as a function of gas measurement errors a and b are shown in Figure 3. As can be seen, when errors increase, αML reduces. In addition, αML is more susceptible to variations in a when b is constant.  (a) a = 5%, b=15%(b) a = 1 0%, b = 30%Figure 2. Example plots of Pα(α) and PG(α) IV. NUMERICAL EXAMPLEDifferent gas ratio determinations exist for DGA-based power transformer diagnostics [7, 8]. To demonstrate the principles of the proposed ML correction method, this section focuses on the application of Rogers ratios that classify faults according to the ratio values shown in Table II. As an example, consider the following key gas measurements taken from a power transformer: CH4 = 156 ppm, H2 = 164 ppm, C2H2 = 15 ppm, C2H4 = 167 ppm and C2H6 = 58ppm. With the selected gas values, the corresponding ratio values are R1 = 0.9512, R2 = 0.0898 and R5 = 2.8793. Assuming gas measurements without errors, the transformer health is classified as a low temperature thermal fault according to Table II. Assuming a Gaussian PDF, the ML estimates of the true underlying values R10, R20 and R50 are the measured values, i.e. G0 = R1, G0 = R2 and G0 = R5. For the correct PDF, PR(R, R0), the ML estimates, i.e. R0, R0 and R0, are obtained through solving (14), then applying (18). Corrected R0 value estimates as a function of a = b from 0 to 30% are shown in Figure 4. Depending on the gas errors, the diagnostic estimates cross over Rogers ratio boundaries, potentially changing the fault diagnosis outcome. Uncertainty analysis of two gas measurement DGA ratios for improved diagnostics applications2Figure 3. Example ML plots as a function of gas errors a and b TABLE II ROGERS RATIO FAULT DIAGNOSTICS [7], [8] Case R2 R1 R5 Fault Diagnosis 0 < 0.1 0.1-1 < 1 Unit normal 1 < 0.1 < 0.1 < 1 Low energy arcing - PD 2 0.1-3 0.1-1 > 3 Arcing – high energy discharge 3 < 0.1 0.1-1 1-3 Low temperature thermal 4 < 0.1 > 1 1-3 Thermal < 700 0C 5 < 0.1 > 1 > 3 Thermal > 700 0C Figure 4. Unbiased estimates of the true Rogers ratios R10, R20 and R50 as a function of gas measurement errors a = b For example, in Figure 4, R0 crosses the 1.0 boundary at 16.3% error, R0 crosses the 0.1 boundary at 24.7% error. and R0 crosses the 3.0 boundary at 14.7% error. At larger errors, the fault appears to translate into a Thermal fault < 700 0C. Obviously, the closer the R value to the boundary the higher the likelihood of the underlying true value crossing the boundary level when individual gas measurement errors are large. V. CONCLUSIONSThis paper has presented the correct PDF for DGA two-gas ratios under measurement uncertainty. The analysis has shown that for accurate or low level gas measurement errors, the exact ratio PDF is very close to a Gaussian PDF. However, with less accurate gas measurements, the ratio PDF differs from a Gaussian distribution. The correct ratio PDF is a skewed distribution, where the measured or calculated ratio value is not representative of the best ML estimate of the true gas ratio value, as it would be for a Gaussian PDF. An analytical correction method was presented based on ML PDF criteria, permitting determination of the most-likely estimate of the true gas ratio value R0. It was also shown that individual gas measurement errors may influence potential gas ratio boundary decisions for diagnostic decision-making. Based on the non-Gaussian nature of the correct PDF, future work will present: (i) the construction and determination of precise confidence intervals for two-gas ratio errors, and (ii) the development of suitable analytical formulae for fault classification probabilities based on specified or defined two-gas ratio fault-boundary levels.  REFERENCES [1] M. Liserre, G. Buticchi, M. Andresen, G. Carne, L. Costa, and Z. Zou,“The smart transformer: Impact on the electric grid and technologychallenges,” IEEE Ind. Electron. Mag., vol. 10, no. 2, pp. 46–58, 2016.[2] A. Ashrafian, M. Mirsalim, and M. A. S. Masoum, Application of a recursive phasor estimation method for adaptive fault component baseddifferential protection of power transformers, IEEE Trans. Ind. Informat.,vol. 13, no. 3, pp. 1381–1392, 2017.[3] A. Abu-Siada and S. Islam, A new approach to identify powertransformer criticality and asset management decision based on dissolved gas-in-oil analysis, IEEE Trans. Dielectr. Electr. Insul., vol. 19, no. 3, pp. 1007-1012, 2012. [4] S. W. Kim, S. J. Kim, H. D. Seo, J. R. Jung, H. J. Yang, and M. Duval,New methods of DGA diagnosis using IEC TC 10 and related databases Part 1: Application of gas ratio combinations, IEEE Trans. Dielectr.Electr. Insul., vol. 20, no. 2, pp 685-690, 2013. [5] S. J. Lee, Y. M. Kim, H. D. Seo, J. R. Jung, H. J. Yang, and M. Duval,New methods of DGA diagnosis using IEC TC 10 and related databases Part 2: Application of relative content of fault gases, IEEE Trans.Dielectr. Electr. Insul., vol. 20, no. 2, pp. 691-696, 2012.[6] N. A. Bakar, A. Abu-Siada, and S Islam, A review of dissolved gas analysis measurement and interpretation techniques, IEEE Electr. Insul.Mag., vol. 30, no. 3, pp. 39-49, 2014. [7] Guide for the interpretation of gases generated in mineral oil-immersedtransformers, IEEE Standard C57.104-2019, 2019-06-13[8] Mineral oil-filled electrical equipment in service - Guidance on the interpretation of dissolved and free gases analysis, IEC Standard.60599-2015, 2015-09-16. [9] M. Duval and J. Dukarm, Improving the reliability of transformer gas-in-oil diagnosis, IEEE Electr. Insul. Mag., vol. 21, no. 4, pp. 21–27, 2005 [10] I. B. M. Taha, D. E. A. Mansour, S. S. M. Ghoneim, and N. I. Elkalashy,Conditional probability-based interpretation of dissolved gas analysis fortransformer incipient faults, IET Generation, Transmission andDistribution, vol. 11, no. 4, pp. 943–951, 2017.[11] R. C. Geary, The distribution of ‘student’s’ ratio for non-normal samples,Suppl. J. Royal Statistical Society, vol. 3, no. 2, pp. 178–184, 1936.[12] D. V. Hinkley, On the ratio of two correlated normal random variables,Biometrika, vol. 56, no. 3, pp. 635–639, 1969. [13] G. 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Uncertainty analysis of two gas measurement DGA ratios for improved diagnostics applications

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Uncertainty analysis of two gas measurement DGA ratios for improved diagnostics applications

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