lifetime modeling and management of transformers653948/fulltext02.pdfinvestigation of transformer...

Lifetime Modeling and Management ofTransformers

JOHANNA ROSENLIND

Doctoral ThesisKTH Royal Institute of Technology

School of Electrical EngineeringDivision of Electromagnetic Engineering

Stockholm, Sweden 2013

TRITA-EE 2013:037 ISBN 978-91-7501-883-6

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Abstract

This work have studied and developed lifetime estimation methods for the powertransformer and how these could be used for asset management purposes. It is re-search performed in the intersection of the fields of reliability theory, statistical anal-ysis, and stochastic process theory applied to lifetime estimations and management oftransformers.

The chosen approaches are the following. The thesis assess the effect of thermalstresses on the lifetime of the transformer. The effect of hotspot temperature is as-sessed with a loss of life measure. Within this study, improvements to the thermalmodel have been made.

The thesis moves on to an alternative method for lifetime estimation in whichdiagnostic measurements are forecasted using a stochastic process and iterative real-izations of this stochastic process is used to estimate a probability distribution for thetransformer.

The thesis moves on to study the loss of life measure from a system perspectiveby calculating the loss of life estimate from the load profiles of cold load pickup,increased electric vehicle penetration, and normal operation. These are applied insuch a way that they can be used for asset management purposes.

Then, the thesis uses estimates of failure times with the aim to reach a proba-bilistic, dynamic capacity rating selection method which use a failure rate which isconditioned on the time-dependent load level.

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Sammanfattning

Det här arbetet har studerat och utvecklat metoder för estimering av transforma-torlivslängden och hur dessa skulle kunna användas för hantering av tillgångar1. Dettaär forskning framförd inom skärningspunkten mellan fälten: tillförlitlighetsteori, sta-tistisk analys och statistisk process teori. Detta har sedan blivit applicerat påhanteringav transformatorer som tillgång.

En beskrivning av de valda ansatserna följer. Den här avandlingen skattar effektenav termisk påfrestning pålivslängden av en transformator. Effekten av hotspot tempe-raturen estimeras med ett mått som skattar livslängdsförlusten för given drift. Inomden här studien såhar förbättringar påden termiska modellen gjorts.

Avhandlingen går vidare till att utveckla en alternativ metod för livslängsestime-ring som nyttjar diagnostiska mätningar genom att prognosticera dessa med hjälp aven stokastisk process och genom iterativa realisationer av dessa, estimera sannolik-hetsfördelningen för transformatorns livslängd.

Avhandlingen går vidare till att positionera livslängsförlustsmåttet i ett system ge-nom att beräkna livslängdsförlusten under de olika driftförhållandena kallastpåslag2,ökad penetration av elektriska fordon, och normal drift. Dessa implementeras såatt dekan användas med syftet att hantera tillgångar.

Slutligen sågår avhandlingen vidare till att estimera feltider med syftet att välja enprobabilistisk, dynamisk kapacitet som använder en felintensitet som är betingad pÃ¥en tidsberoende belastningsnivå.

1Författarens översättning av asset management.2Författarens översättning av cold load pickup.

Acknowledgements

This thesis was written as part of the Ph.D. project "Lifetime modeling and managementof transformers" at the Department of Electromagnetic Engineering, School of ElectricalEngineering, KTH Royal Institute of Technology. The three first years of the project werefunded by the Swedish Center of Excellence in Electric Power Engineering (EKC2), the re-maining year by Controllable and Intelligent Power Components (CIPOWER) and SwedishCentre for Smart Grids and Energy Storage (SweGrids). The financial support is gratefullyacknowledged.

I thank my two supervisors, Assistant Professor Patrik Hilber and Professor RajeevThottappillil, for their invaluable support, encouragement, and guidance throughout theproject. I thank my current and former colleagues and collaborative partners for providinga stimulating and fun work environment. I thank my family for always pushing me to bebetter. Finally, this thesis is dedicated to my friends for all the love and support; you areand will always be my safe foundation.

Johanna

Stockholm, October 2013

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List of Publications

The appended publications to this doctoral thesis are the following:

Publication I, P. Hilber, C.J. Wallnerström, J. Rosenlind, S. Babu, and P. Westerlund.Benefits of reliability centred asset management. In Proceedings of the 22nd In-ternational Conference on Electricity Distribution (CIRED), Stockholm, Jun., 2013.

Publication II, F. Josue, I. Arifianto, R. Saers, J. Rosenlind, P. Hilber, and Suwarno.Transformer hot-spot temperature estimation for short-time dynamic loading. In pro-ceedings of the IEEE International Conference on Condition Monitoring and Diag-nosis (CMDM), Bali, Aug., 2012.

Publication III, I. Arifianto, F. Josue, R. Saers, J. Rosenlind, P. Hilber, and Suwarno.Investigation of transformer top-oil temperature considering external factors. In Pro-ceedings of the IEEE International Conference on Condition Monitoring and Diag-nosis (CMDM), Bali, Aug., 2012.

Publication IV, J. Rosenlind, and P. Hilber. On improvements of power transformer con-dition monitoring by considering thermal models. In Proceedings of the 12th Inter-national Conference on Probabilistic Methods Applied to Power Systems (PMAPS),Istanbul, Turkey, Jun., 2012.

Publication V, P. Grahn, J. Rosenlind, P. Hilber, K. Alvehag, and L. Söder. A method forevaluating the impact of electric vehicle charging on transformer hotspot tempera-ture. In Proceedings of the Second European Conference and Exhibition on Innova-tive Smart Grid Technologies (ISGT-EUROPE 2011), Manchester, United Kingdom,Dec., 2011.

Publication VI, J. Rosenlind, H. Tavakoli, and P. Hilber. Frequency response analysis(FRA) in the service of reliability theory. In Proceeding of the International Con-ference on Condition Monitoring, Diagnosis and Maintenance 2011 (CMDM 2011),Bucharest, Romania, Sep., 2011.

Publication VII, J. Rosenlind, and P. Hilber. Prerequisites for transformer lifetime mod-eling towards a better understanding. In Proceedings of the 11th International Con-ference on Probabilistic Methods Applied to Power Systems (PMAPS), Singapore,Jun., 2010.

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Publication VIII, J. Rosenlind, F. Edström, P. Hilber, and L. Söder. On selecting dy-namic thermal rating for transformers by correlating failure rate with mean load, loadvolatility and ambient temperature. Submitted to International Journal of ElectricalPower & Energy Systems (Elsevier).

Publication IX, F. Edström, J. Rosenlind, K. Alvehag, P. Hilber, and L. Söder. Influenceof ambient temperature on transformer overloading during cold load pickup. IEEETransactions on Power Delivery, vol. 28(1), 153-161, Jan., 2013.

Publication X, F. Edström, J. Rosenlind, P. Hilber, and L. Söder. Modeling impact of coldload pickup on transformer aging using Ornstein-Uhlenbeck Process. Published inIEEE Transactions on Power Delivery, vol. 27(2), 590-595, Apr., 2012.

Publication XI, J. Rosenlind, J. Setréus, and P. Hilber. Reliability screening for identify-ing critical power transformers in the GB transmission system. Submitted to Energies(ISSN 1996-1073; CODEN: ENERGA).

The division of the work between the authors can be found in Sections 1.3.1.

List of variables

The main notation used throughout the thesis is stated below for quick reference. Othersymbols have been defined as required.

Chapter 2L loss of lifev aging rateθh hotspot temperatureµ oil viscosityA, B material constantsveq equivalent oil viscosityvx oil viscosity at height xρ oil densityθi topoil temperature (measured inside cooling ducts)θb bottom oil temperatureR ratio of load losses to no load lossesK load factor∆θo,R topoil to ambient temperature gradient at rated loadk11, k22, k21 thermal constantsτo oil time constantτw winding time constantθo topoil temperatureθa ambient temperatureCoil thermal capacity of oilq f e no load lossesqcu load lossesRth,oil convection thermal resistance of oilRth,air convection thermal resistance of airθtank tank temperaturex oil exponenth convection heat transfer coefficientA surface area through which convection heat transfer takes place∆θH hotspot temperature rise∆θHS,R hotspot temperature rise at rated load

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x

y winding exponenti time incrementθµ oil viscosity temperaturePW,pu winding lossesPDC,pu DC lossesPE,pu eddy lossesθHS,R hotspot temperature rise at rated loadθK temperature factor for loss correctionA,B model parametersν Constant for viscosity exponent for all cooling modes in the modified modelH Hotspot temperature factor

Chapter 3

µ mean of the stochastic processβ uncertainty of the stochastic processβ estimation of the uncertaintyσ volatility of the stochastic processσ estimation of the volatilityBt Brownian incrementX0 initial value of the stochastic processN number of measurementsDt diagnostic measurement∆ the degree of deformationW Wiener incrementh time stepw0 initial value of the stochastic processµFRA linear leaning of the stochastic processσFRA variance of the stochastic process

Chapter 4

ηM Montsinger aging estimateηA Dakin-Arrhenius aging estimateθh transformer hotspot temperatureA, B material constants in the aging modelPv

t,i load from electric vehicle i at time step tT ci the time when the vehicle returns homeCp charging powerEsoc

i the energy level in the batteryPv,tot

t total electric vehicle loadPn,h

t normalized load from householdsH total number of households

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C average consumption of each householdPh,tot

t total household loadSD diversified value of the loadSU undiversified value of the loadγ the decay rate of the exponentialu Heaviside step functionσ volatility of Ornstein-Uhlenbeck processβ drift parameter of Ornstein-Uhlenbeck processα mean value of Ornstein-Uhlenbeck processλ j expected outage rate of component jr j expected outage duration for component jISM

j,k risk index for component j and load scenario kQCT S

i,z,k outage events is consequence on CTS zUi expected unavailability for outage event iIEENS

j,k risk index for causing interruption of supply at system load scenario kQPNS

i,k power not supplied for outage event iIDG

j,k risk index for causing disconnected generationQDG

j,k disconnected generation due to outage event i for load scenario k

Chapter 5Θbubble the temperature of bubble formationΘA the ambient temperatureWWP weight of moisture in paperPPres total pressureVg gas content of oilm the mean load level covaritem0 the threshold value for the mean load level where the

covariate have a significant impact on the failure rateλt failure rate for thermally-related failuresλa failure rate for ageing-related failuresc coefficient of the polynomialε the minimum value of the failure rate for thermally-related failuresη the shape parameter of the Weibull distributionγ the scale parameter of the Weibull ditributionµ the shift parameter of the Weibull distributiontage the age of the componentψ the acceleration parameter for the Weibull distribution

Contents

Abstract (eng.) iii

Abstract (swe.) iv

Acknowledgement v

List of Publications vii

List of Variables ix

Contents xii

1 Introduction 11.1 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Ph.D. project objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Author’s contribution in appended publications . . . . . . . . . . . 51.3.2 Additional publications not appended . . . . . . . . . . . . . . . . 7

1.4 Thesis Outline and Guidelines for Reading . . . . . . . . . . . . . . . . . . 7

2 Transformer Thermal Modeling 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Thermal Model Research Challenges . . . . . . . . . . . . . . . . . . . . . 112.3 Thermal Model Improvements . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Incorporating weather effects in the top-oil temperature model . . . 122.3.2 Hot-spot temperature estimation for short-time dynamic loading . . 16

3 Lifetime Estimation 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Deterioration modeling with diagnostic measurement values . . . . . . . . 233.3 Demonstration of Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.1 Frequency Response Analysis . . . . . . . . . . . . . . . . . . . . 253.3.2 Degradation model of FRA measurements with stochastic process . 26

3.4 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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CONTENTS xiii

4 Transformer Asset Management Methods 294.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.1 The increasing penetration of electric vehicles . . . . . . . . . . . . 314.2.2 The cold load pick-up situation . . . . . . . . . . . . . . . . . . . . 324.2.3 Normal operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Results and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Probabilistic Dynamic Capacity Rating 395.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2.1 Time-dependent covariates . . . . . . . . . . . . . . . . . . . . . . 405.2.2 Loading-related failures . . . . . . . . . . . . . . . . . . . . . . . 405.2.3 Aging-related failures . . . . . . . . . . . . . . . . . . . . . . . . 425.2.4 The survivor function and expected life time . . . . . . . . . . . . 435.2.5 Capacity rating selection . . . . . . . . . . . . . . . . . . . . . . . 43

5.3 Results and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3.1 Capacity rating . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.4 Some thoughts on the perspective in the estimation procedure . . . . . . . . 47

6 Closure 496.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.2.1 Thermal modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 506.2.2 Lifetime forecasting using diagnostic measurements . . . . . . . . 516.2.3 Asset management in three-loading scenarios . . . . . . . . . . . . 516.2.4 Probabilistic capacity selection method which takes into account

the volatility of the load . . . . . . . . . . . . . . . . . . . . . . . 51

Bibliography 53

Chapter 1

Introduction

This chapter provides a general background to the topic and the motivation for the research.The objective of this Ph.D. project is presented along with the main contributions of thisthesis.

1.1 Thesis Overview

The transformer has two functions in the electrical grid: to transfer energy from one circuitto another by utilizing a common magnetic field and to provide isolation between an ACsupply and its load, with the exception of the autotransformer that fulfills only the first [1].Transformers are often manufactured when ordered. In combination with the long manu-facturing time, it could take months before a transformer is replaced if an outage is suddenand unexpected. In addition, the function of the transformers put them on strategically po-sitions in the intersection of different voltage levels, which makes them one of the mostimportant components in the electrical grid. Hence, it is important to predict the trans-former lifetime, so that the number of sudden and unexpected failures can be balanced tothe number of condition improvement activities.

A typical number for the failure rate of the transformer is 0.02 failures/year, i.e., cor-responding to one failure in 50 years time. Hence, a transformer failure is a rare event.Moreover, power transformers are custom-made to fit their particular position in the grid.Hence, the available data are scattered over several transformer types. This causes a lack ofhigh-quality failure data, which makes purely statistical analysis of the transformer lifetimedifficult. Hence, other approaches that are based not only on failure data in themselves butalso use either expert opinion, or diagnostic measurements, are more preferable.

The work carried out within this Ph.D. project is part of the Reliability centred assetmanagement (RCAM) programme at KTH. The research concerning reliability-centeredasset management aims to develop methods for electrical systems that relate maintenanceeffort to system availability, and to reach an optimal maintenance management plan byminimizing the total cost [2]. Lindquist et. al. [3] mentioned that lifetime models intendedto be used within the framework of RCAM should include aging and the effect of mainte-nance. Here Lindquist et. al. refer to a statistical lifetime model that could be used to make

1

2 CHAPTER 1. INTRODUCTION

general conclusions for the lifetime of a transformer population. However, transformer fail-ures are considered to be dependent on random external events and internal weaknesses [4].Instead, a component-specific lifetime model that incorporates more information than thetraditional statistical lifetime model is once again seen as more preferable.

Essential to the lifetime estimation is thermal modeling, because the insulation of thetransformer is considered to provide an upper boundary of its lifetime and the insulationundergoes thermal degradation. The work of improving the thermal modeling for the trans-former is one of the focus areas of this thesis.

A degradation model is a component-specific lifetime model. It is commonly used torelate the failure time of a component with the deterioration during its usage period. Thedegradation of the transformer component is assessed from diagnostic testing during thelifetime of a transformers. It should be noted that difference is made between the degrada-tion of the condition and the absolute condition of the actual component where the formeris the rate of change of the later. The reason for studying the rate of change of the condi-tion, instead of the absolute condition, is that condition assessment techniques struggle toidentify this all-embracing measure that reflect the true condition of the component. Hence,this thesis focuses on forecasts of the rate of condition change with diagnostic measurementvalues as a degradation model. This is one of the focus areas of this research.

Importance indices are used to assign rank to components in a system. Particularlyimportant indices used in this project use information from the system to assign rank to acomponent. By combining importance indices with lifetime models, more precise decisionson maintenance planning or component investments can be made. This is one of the focusareas of this research.

Methods for maintenance planning and component investments, as discussed in theprevious paragraph, are used for an electrical grid that is not under stress, as opposed tomethods for power system restoration that are used when the system is brought back tonormal operation after a set-back. In particular, methods used for the restoration situationthat which allow for component overloading or emergency loading without jeopardizingthe restoration procedure constitute one of the focus areas of this research.

When lifetime models in the context of power system restoration were studied, it wasfound that system aspects were not brought into the dimensioning of component capacity,especially not the rapidly-varying system aspects, such as load volatility. This thesis bringsthe aspects of load volatility into focus and load level into lifetime modeling. This made itpossible to develop a probabilistic dynamic rating method for transformers.

For a background in each previously-discussed research area, the reader is referred tothe introduction to each chapter. The reason for including the background review in theintroduction to each chapter and not here is that the research areas are so intertwined thatthe review had to be made using a perspective were each research area is centered.

1.2 Ph.D. project objective

This thesis is part of the Ph.D. project "Lifetime modeling and management of transform-ers" within the maintenance management research program at the Swedish Center of Excel-

1.3. MAIN CONTRIBUTIONS 3

lence in Electric Power Engineering (EKC2) and, during the later stages, the Controllableand Intelligent power components (CIPOWER) program within the Swedish Centre forSmart Grids and Energy Storage (SweGRIDS) initiative at KTH.

The main objective of this project is to connect the areas of reliability theory and con-dition estimation. To achieve this objective, lifetime models that use diagnostic measure-ments as input are developed. Moreover, thermal modeling that is essential for lifetimemodeling, are developed. As a secondary objective, system aspects are incorporated intotransformer asset management during both power system restoration and regular compo-nent replacement. Moreover, in the next step, the aim is to introduce system aspects duringcomponent rating determination.

1.3 Main ContributionsThe work presented in this thesis focuses on the development of lifetime models and as-set management methods for transformers based on reliability theory. The main scientificcontributions of this thesis are the following:

Method 1. Hotspot temperature determination.Two thermal models are developed, as follows:

1. The first includes the additional cooling brought on by wind.2. The second calculates the hotspot temperature gradient based on a winding oil

temperature.

Introduced in Chapter 2 and presented in Papers II and III.

Method 2. Forecasting diagnostic measurement values to determine the mean time to fail-ure.Diagnostic measurements over a time period are assumed to express the rate of dete-rioration of the transformer. This deterioration will lead to a failure. The diagnosticmeasurements could be forecasted with the aim to determine the time to failure.

Introduced in Chapter 3 and presented and demonstrated in Paper VI.

Method 3. Asset management by combining load flow modeling, importance indices andlifetime model.A three-step procedure that consists of the following steps:

1. The load flow model determines the loading of that particular component.

2. Importance indices express the component’s importance for system reliability.

3. The loss of life model measures the stress that the component has been sub-jected to.

This produces a component ranking scheme.


Introduced in Chapter 4 and presented in Publication XI.

Method 4. Identifying a power restoration strategy that incorporates transformer overload-ing. The load volatility in a cold load pickup situation is modeled with the Ornstein-Uhlenbeck process. The effect on the transformer is modeled with two aging models.This leads to an optimal restoration strategy that ensures a suitable load level on thetransformer.

Introduced in Chapter 4 and presented in Publication X.

Method 5. Probabilistic capacity rating that makes allowance for load volatility.The modeling approach is for:

• thermally-related failures that condition the transformer failure rate on the co-variate load volatility, and

• aging-related failures that are modeled with a three-parameter Weibull distribu-tion.

By inserting a predefined risk level, a probabilistic capacity rating can be determined.

Introduced in Chapter 5 and presented in Publication VIII.

1.3. MAIN CONTRIBUTIONS 5

1.3.1 Author’s contribution in appended publications

All appended publications are the result of collaborative work by the author, supervisors,and co-authors. The following list describes the specific contributions made by the authorin the appended publications.

Publication I [5], All authors collaboratively worked with this publication. This paperprovides an introduction to the current research within the RCAM research group.More specifically, the author contributed with the Section "The value of accuratethermal models of transformers".

Publication II [6], This publication is the result of a Master of Science project presentedin [7] by the student Mr. Fretz Josue. The author initiated the project idea andsupervised the project together with Dr. Robert Saers, ABB Corporate Research.The author and Mr. Fretz Josue wrote the major parts of the paper.

Publication III [8], This publication is the result of a Master Of Science project presentedin [9] by the student Mr. Indera Arifianto. The author initiated the project idea andsupervised the project together with Dr. Robert Saers, ABB Corporate Research.The author did minor contributions to the writing of the paper.

Publication IV [10], The author outlined the study and wrote the paper.

Publication V [11], The author and Ms. Pia Grahn jointly initiated the research idea to-gether, jointly outlined the study, and jointly wrote the paper. Ms. Pia Grahn is a PhDstudent at the department of power systems, her project is entitled "the potential offlexible electronic vehicles on the electricity market". All contributions to the projectwhich can be assigned the topic of electric vehicles alone, have been provided byMs. Pia Grahn. The same goes for the contributions from the topic of transformers,which have been provided by the author.

Publication VI [12], The author initiated the research idea, outlined the study and wrotethe paper while getting major contributions from Dr. Hanif Tavakoli on Section IIand III. Dr. Hanif Tavakolis thesis is entitled "An FRA transformer model with ap-plication on time domain reflectometry". All contributions to the publication whichcan be assigned the topic of frequency response analysis alone have been providedby Dr. Hanif Tavakoli.

Publication VII [13], The author outlined the study and wrote the paper.

Publication VIII [14], The author and Mr. Fredrik EdstrÃ¶m jointly outlined both thestudy and the paper. Mr. Fredrik Edström licentiate thesis is entitled "On risks inpower system restoration". All contributions to the publication which can be as-signed the topic of power system restoration alone have been provided by Mr. FredrikEdström.


Publication IX [15], The author, Mr. Fredrik Edström and Dr. Karin Alvehag jointlyoutlined both the study and wrote the paper. In particular, Dr. Karin Alvehag con-tributed with the Section on customer damage functions. All contributions that canbe assigned power system restoration alone have been provided by Mr. Fredrik Ed-ström.

Publication X [16], The author and Mr. Fredrik Edström jointly outlined both the studyand the paper. All contributions that can be assigned power system restoration alonehave been provided by Mr. Fredrik Edström.

Publication XI [17], The author and Dr. Johan Setréus jointly outlined both the study. Dr.Johan Setréus thesis is entitled "On reliability methods quantifying risks to transfercapability in electric power transmission systems". Dr. Johan Setréus provided withthe knowledge on importance indices, the ranking of the transformers based on theseimportance indices and he also provided with the simulation implementation of theAC power flow model of the power system.

1.4. THESIS OUTLINE AND GUIDELINES FOR READING 7

1.3.2 Additional publications not appendedIn addition to the appended publications, the following publications were written withinthe Ph.D. project.

Publication i , P. Hilber, and J. Rosenlind. Reliability cost of preventive maintenance andother interruptive activities. ELFORSK.

Publication ii , P. Hilber, C.J. Wallnerström, J. Rosenlind, J. Setréus and N. Schönborg.Risk analysis for power systems - overview and potential benefits. In Proceedingsof the International Conference on Electricity Distribution (CIRED) 2010 workshop,Lyon, France, Jun., 2010.

In addition to these publications, [7, 9, 18] were published within the Ph.D. project.

1.4 Thesis Outline and Guidelines for ReadingThis thesis serves as an introduction and extended summary of the appended publications.The content of the chapters in the thesis is outlined as follows.

Chapter 2 focuses on the topic of thermal modeling of transformers and contain an in-troduction to this research area, following is a motivation for the chosen researchapproach, along with two thermal model improvements.

Chapter 3 focuses on lifetime estimation combined with deterioration modeling that usesinformation obtained from diagnostic tools and forecasts those to estimate the oper-ational lifetime of the transformer. The method is demonstrated in a case study.

Chapter 4 focuses on asset management methods applied to the particular loading situ-ations of 1) increasing penetration of electric vehicles, 2) cold load pick-up and 3)normal operation.

Chapter 5 focuses on a method to estimate the probabilistic dynamic capacity rating whenthe volatility of the loading is taken into consideration.

Chapter 6 concludes the thesis and presents areas of future work.

Reading guidelines

It is recommended that each of Chapters 2 to 5 be read, along with the appended paper ofeach chapter. The reader could then continue with Chapter 6, which concludes the thesis,and proposes future work.

Chapter 2

Transformer Thermal Modeling

This chapter introduces Publications II, III, and IV. Section 2.1 explains the importance ofthermal models. Section 2.2 reviews the research challenges for thermal models. Section2.3 describes the developed thermal models from this Ph.D. project. Section 2.3.2 providesconclusions from the model development and presents future work.

2.1 Introduction

Initially, this introduction describes the importance of transformer thermal modeling andthen moves on to describe how thermal models connect to lifetime modeling of transform-ers.

The role of the transformer for the network’s thermal overload capability is describedby the following quotation [19]:

Power transformers are the "bottlenecks" in a network’s thermal overload ca-pability in the same way as power cables: better use of these produces a directimprovement in the networks thermal overloading capability. The benefits areparticularly noticeable since transformers are widespread in both transmissionand distribution networks.

Moreover, the thermal capacity limit for the transformer is set by combining maximumallowed stresses of the materials and calculations using a worst-case scenario on ambienttemperatures and loading. Table 2.1 summarizes, as an example, common thermal capacitylimits for transformers [20]. It should be noted that there are other thermal limits specifiedfor other loading situations as well, and this particular example was chosen because theselimits have been implemented in the studies included in this thesis.

More accurate thermal models allow utilities to operate the transformer more closelyto its thermal limits, and something that today is considered to be unused capacity in secu-rity margins would be readily available for utility usage. However, to reduce the securitymargin, the credibility and the accuracy of the thermal modeling technique need to be im-proved.

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10 CHAPTER 2. TRANSFORMER THERMAL MODELING

Table 2.1: Suggested thermal limits of temperature and loading for the particular case ofloading above nameplate rating for a power transformer with 65°C rise. [20]

Top-oil temperature 110°CHotspot conductor temperature 180°CMaximum loading 200%

The most severe consequence of increased thermal stress is accelerated aging of thesolid insulation. The process of aging is a chemical reaction, where heat works as a catalyst.Hence, an elevated temperature accelerates the chemical reaction, i.e., the aging of thetransformer. Following this reasoning, it becomes intuitively clear that the hottest partof the solid insulation undergoes the most rapid aging. That part of the solid insulationis referred to as the hotspot. Its corresponding temperature is referred to as the hotspottemperature. The thermal models studied within this project aim to improve estimations ofthe hotspot temperature value.

Thermal models connect to lifetime modeling in the following way. Within one philos-ophy of transformer lifetime modeling it is assumed that the consumed lifetime could bedetermined from the hotspot temperature on the reasoning that the thermal stress continu-ously consumes the lifetime. According to [21], the loss of life L over a certain period oftime is equal to

L =

∫ t2

t1V dt, (2.1)

where V is the aging rate given by either

V = 2(θh−98)/6, (2.2)

for non-thermally upgraded paper, or

V = e15000

110+273−15000

θh+273 (2.3)

for thermally upgraded paper, where θh is the hotspot temperature. This loss of life valuecould be used as an estimate for the condition of the transformer, and in turn, also as ameasure of the consumed lifetime.

When working with thermal models, a commonly-used analysis tool is the thermal-electrical analogy. In the thermal-electrical analogy the similarities between electrical andthermal properties are used to solve for temperatures using tools that normally are usedto solve for electrical properties. Using the thermal-electrical analogy, heat flow is seenas current, and the force that generates heat transfer, i.e., temperature difference, is seenas voltage difference. Thermal resistance, which is the inherent ability of a material towithstand heating, has the analogy of electric resistance. Thermal capacitance, which isthe inherent ability to store heat, has the analogy of electrical capacitance. Table 2.2 showsthe particular circuit components used to represent thermal characteristics. This analogy isused to explain the upcoming thermal models.

2.2. THERMAL MODEL RESEARCH CHALLENGES 11

Table 2.2: The thermal components along with their electrical counterparts as seen in theelectrical-thermal analogy.

Thermal component Electrical component

Heat source Voltage sourcePower source Current sourceHeat transfer CurrentTemperature Voltage differenceThermal resistance Electrical resistanceThermal capacitance Electrical capacitance

2.2 Thermal Model Research ChallengesPublication IV summarize the current research challenges associated with thermal model-ing. The challenges are repeated here for convenience.

• The thermal model in itself should

– respond to environmental factors;– incorporate the dependence of winding loss on temperature;– incorporate the dependence of oil viscosity changes on temperature.

• Validation of the thermal model requires hotspot temperature measurements.

The first three challenges connect to the actual model itself. The last one connects totechnology advancements of temperature measurement devices. Within the scope of thePhD project, the first three are addressed. What follows here is a discussion of each ofthese research opportunities. The final research challenge is not studied within the project.Hence, it will not be discussed in either this text or the future work (in chapter 6) becauseit requires a study with another technical character.

The environmental impact on the transformer could be derived from the meteorologicalconditions, i.e., the climate of the region where the transformer is located. More specif-ically, these conditions are characterized by the presence of rain, wind, sun radiation,and snow. The complexity concerning the thermal modeling aspects of the meteorologi-cal conditions arises from there being a time constant between the temperature effects ofthe surrounding and the hotspot temperature, which is nonzero. For certain large powertransformers, this time constant equals several hours. This makes the effect on the hotspottemperature difficult to quantify.

The benefit from an accurate thermal response to an environmental condition is that theoperation of the transformer could be adjusted to suit the particular climate region. Accord-ing to [22], the world can be classified into five zones, namely (1) Equatorial climates, (2)Arid climates, (3) Warm temperate climates, (4) Snow climates, and (5) Polar climates.

Each zone is characterized by the temperature, the amount of rain or snow, and theair pressure that could give an indication on what type of weather the transformer couldbe subjected to. This weather information could be used as input for the operation. For


instance, in regions with a colder temperature where the transformer would be offeredadditional cooling, a larger capacity could be used without risking unnecessary wear on thecomponent.

The second research challenge is the dependence of the winding losses on the actualtemperature of the windings. Using the thermal-electrical analogy representation, the lossesfrom the transformer winding is represented by a current source in the thermal model. Thestrength of this current source is highly dependent on the actual temperature at the positionwhere it is evaluated. Early thermal models treated the winding loss as a linear dependenceon the height of the hotspot winding disk. Later models have a nonlinear dependence forthis. Current research strives to find a suitable representation.

The third research challenge is recognized as the dependence of oil viscosity on tem-perature. This is the property of the transformer oil that has been shown to have the mostprominent variance with temperature. In [23] this dependence is described as

µ = A× eB/(T+273), (2.4)

where the material constants A = 0.0013573 and B = 2797.3 and T is the average betweenthe hotspot temperature and the winding temperature. Moreover, in [24] more values ofA and B are found. In [25] Aubin et.al. use the equivalent viscosity to represent thisdependence according to

veq =∫ 1

0vx dx, (2.5)

where vx is specific for the transformer oil. For the transformer oil Voltesso, the expressionof the integrand vx reads as

vx =1000

ρ

[10

308.44.2617Tx

−2.17], (2.6)

where Tx is the absolute temperature given by the expression

Tx = (θi −θb)x+θb +273.15, (2.7)

and θi is the top oil temperature (measured inside the cooling ducts), θb is the bottom oiltemperature, and x is the relative height at which the temperature is evaluated. Current re-search strives to incorporate the effect of transformer oil viscosity variation into the thermalmodel.

2.3 Thermal Model Improvements

2.3.1 Incorporating weather effects in the top-oil temperature model

Publication III is a study on the effect of wind on the hotspot temperature. What follows isa summary of the developed method and the results from that study.

2.3. THERMAL MODEL IMPROVEMENTS 13

(a)

(b)

Figure 2.1: Equivalent thermal circuits for a) Eq. (2.8) and b) Eq. (2.10).

Method

The differential equation for the top-oil temperature is [21][1+RK2

1+R

]x

∆θo,R = k11τodθo

dt+[θo −θa], (2.8)

where R is the ratio of load loss to no-load loss, τo is the oil time constant, ∆θo,R is thetop-oil to ambient temperature gradient at rated load, K is the load factor, k11 is a thermalconstant, and x is the oil exponent.

Within this study, Eq. (2.8) was alternated to incorporate the effect of the surroundingair and the effect of wind. The modeling procedure for this will be explained next. Thecircuit in Fig. 2.1.a. represents the original thermal model in Eq. (2.8). Figure 2.1.b.represents the circuit of the altered model. The factors introduced in circuit b) are theeffects of:

• the surrounding air and• aerodynamics.

These are seen in Fig. 2.1 as the two parallel resistances, placed in series with the thermalresistance of the oil Rth,oil .

Kirchoff’s current law applied to node θo in Fig. 2.1.b, gives

q f e +qcu =Coildθo

dt+

(θo −θa)

Rth,T. (2.9)


Rearrangement and solving of Eq. (2.9) gives[1+RK2

1+R

]∆θo,R

Rth,T

Rth,T,R=

Rth,T

Rth,T,Rτo

dθo

dt+(θo −θa). (2.10)

To demonstrate the improvement of the thermal model that incorporates the wind speed,the alternations that were made to the thermal model in Eq. (2.8), have been divided intoModel 1 and Model 2. Model 1 incorporates only the effects of the surrounding air. Model2 incorporates the effects of the surrounding air and the effect of aerodynamics. The circuitin Fig. 2.1.b has a switch that alternates between the situations of either no-wind or wind.In principle, this means that the expression for the total thermal convection resistance Rth,Talternates between the following expressions

Rth,T =

1

hoil,NA + 1hair,NA , no-wind

1hoil,NA + 1

hair,FA , wind,(2.11)

where the subscripts stand for either natural convection (N) or forced convection (F). ForModel 1 the switch is only in the upper position. The switch in Model 2 alternates betweenthe upper and the lower positions. Publication III provides the procedure for deriving thecoefficients hair and hoil and other details.

Results

The model was applied to a data set ranging from July 6th to October 19th 2011, with mea-surements of top-oil temperature, load, and ambient temperature for a 63 MVA, oil-natural-air-forced (ONAF), 55/140 kV transformer unit. During normal operation the transformeris in ONAN cooling mode.

Figure 2.2 shows the topoil temperature calculations using Model 1 and Model 2 for awindy period. Figure 2.3 shows the corresponding calculations for a non-windy period.

Table 2.3 lists the percentage of time during which the model underestimates the topoil temperature with units if 2°C, 3°C, and 5°C, i.e., the underestimation measure ∆Tu hasbeen calculated according to

∆Tu = θo,meas −θo. (2.12)

Consequently, Table 2.4 lists the overestimation measure ∆To which has been calculatedaccording to

∆To =−∆Tu. (2.13)

In Tables 2.3 and 2.4 Model 2 shows better results than Model 1. Moreover, the fractionof time when Model 1 provides an overestimate of the top oil temperature value is alwayshigher than the corresponding underestimation. Hence, this model has a slight tendencytowards a conservative value in the same manner as the reference model.

Conclusions from the case study

The thermal models presented by IEC and IEEE do not consider operation during reducedcooling. The thermal model in this study does this by using a correction factor.


Figure 2.2: The topoil temperature calculation using Model 1 and Model 2 on the studiedwindy period compared to the IEC differential method and temperature measurements.

Figure 2.3: The topoil temperature calculation of Model 1 and Model 2 on the studied non-windy period compared to the IEC differential method and temperature measurements.


Table 2.3: The underestimation of calculated top-oil temperature.

∆Tu IEC diff. [%] Model 1 [%] Model 2 [%]

> 2°C 12 11 9> 3°C 3.8 4.6 2.8> 5°C 0.01 0 0

Table 2.4: The overestimation of calculated top-oil temperature.

∆To IEC diff. [%] Model 1 [%] Model 2 [%]

> 4°C 14 14 12> 7°C 4.4 3.9 3.3> 10°C 1.3 0.7 0.4

The foremost conclusion is that the results of the top-oil temperature calculations con-sidering external factors by looking into the weather aspects, i.e., wind, yield a slight im-provement compared with the IEC differential method calculation results. The improve-ments are more significant on a windy period.

2.3.2 Hot-spot temperature estimation for short-time dynamic loading

Publication II studies the dependence of winding losses on temperature and oil-viscosityon temperature. What follow is a summary of the developed method and the results of thatstudy.

Method

The IEC Std. 60076-7 [21] provides the following two approaches for hotspot temperaturecalculations:

• Approach 1 - based on an exponential function to express temperature variations;• Approach 2 - based on a differential function.

Approach 2 is particularly suitable for online monitoring due to its quick response to loadchanges. Within this study improvements have been made to Approach 2 (Publication II).Hence, this approach is repeated here for convenience.


This modeling approach is found in IEC Std. 60076-7 [21]. The increment increase ofthe top-oil temperature is expressed as

Dθo,i =Dt

k11τo

[[1+R×K2

1+R

]x

×∆θo,R − [θo −θa]

]. (2.14)

The top-oil temperature is expressed as

θo,i = θo,i−1 +Dθo,i. (2.15)

The hotspot temperature rise is expressed as

∆θH,i = ∆θH1,i −∆θH2,i, (2.16)

whereD∆θH1 =

Dtk22τw

× [k21 ×∆θHS,RKy −∆θH1] , (2.17)

andD∆θH2 =

Dt1

k22τo

× [(k21 −1)×∆θHS,RKy −∆θH2] . (2.18)

Using Eqs. (2.15) and (2.16), the final expression for the hotspot temperature reads as

θH,i = θo,i +∆θH,i. (2.19)

The transformer oil characteristics govern the heat dissipation process. More specifi-cally, it is the transformer oil temperature that affects the efficiency in this heat dissipationprocess. A low transformer oil temperature means a high oil viscosity. Under this con-dition the heat dissipation is less efficient than for the case of the higher transformer oiltemperature, when heat is dissipated more efficiently [26]. During transient loading of thetransformer, the thermal dynamics will not be in a steady-state condition. Heated oil movesmore easily. Thus, certain oil paths will arise inside the oil ducts where oil travels faster. Inhotspot temperature measurements, this will manifest itself as the thermal overshoot phe-nomenon, which is the slight overshoot in temperature following a transient load beforethe temperature drops to a steady-state value. This temperature drop results from the oilpaths being not as prominent as before. To implement this into the thermal model, withinthis study, the following dependence of the transformer oil viscosity on temperature will beused:

µ =(1.36×10−6)× e

2797.3θµ+273 . (2.20)

In Eq. (2.20), θµ is the oil viscosity temperature. Within the study presented here, thistemperature was investigated further so that the most appropriate one is selected for thethermal modeling. The temperature considered to provide the best result was the hotspottemperature, i.e.,

θµ = θHS. (2.21)

A remark here could be made on the physical interpretation of having the oil viscosityevaluated at the hotspot temperature. A direct physical interpretation of this model implies


that, at high hotspot temperatures, the transformer oil would be flammable. This is not thecase, even for the overload situation. However, one should keep in mind that the systemis not in a steady-state condition. Within this study this temperature also expresses theuncertainties in oil viscosity associated with this condition.

The winding losses dependence on the hotspot temperature reads as

PW,pu = PDC,pu ×(

θH +θK

θHS,R +θK

)+PE,pu

(θHS,R +θK

θH +θK

), (2.22)

where PDC,pu represents the DC losses, PE,pu the eddy losses, θHS,R the hotspot temperaturerise at rated load, and θK the temperature factor for the loss correction.

In Eq. (2.17) and Eq. (2.18) a modification is made to the thermal constants k21 and k22using parameters that represent the dependence of the oil viscosity and the winding losseson temperature. The altered expressions for Eq. (2.17) and Eq. (2.18) are

D∆θH1 =

(Dtτw

)×[(

K2 × (A−θHS,i)×PPW,pu ×θ vµ ×θHS,i

)−∆θH1

](2.23)

and

D∆θH2 =

(Dt

τo ×θ vµ

)×[(

K1.3 ×B×θ vµ)−∆θH2.

](2.24)

The hotspot temperature rise is calculated with Eq. (2.16) and the final hotspot temperaturewith Eq. (2.19). Table 2.5 gives the model parameters A and B for each of the studied sce-narios of this model. The first modification proceeds from the assumption that the hotspottemperature sensor is located at the hotspot location. The second modification proceedsfrom the assumption that the hotspot temperature sensor is not located at the hotspot loca-tion.

Table 2.5: The chosen model parameters for the first and the second modification.

ModificationParameter First Second

A θHS,R θw,av

B θo,R θo,av

Results

The transformer is a 40 MVA, oil-forced-air-forced (OFAF) cooled, 21/115 kV component.The measured top-oil temperature is an input for the final hotspot calculations.

Figures 2.4 and 2.5 show the hotspot temperature calculations using the modified ther-mal models. The calculated hotspot temperature using the first modified thermal modeland the second modified thermal model captures the overshoot phenomenon during normal


0 5 10 15 20−10

0

10

20

30

40

50

60

70

80

90

100

Tem

pera

ture

[°C

]

Time [h]

0 5 10 15 200

50

100

Load

[%]

Ambient temperatureCalculated hot−spot temperature with H=1.3Measured top oil temperatureMeasured hot−spot temperatureCalculated hot−spot temperature with H=1Load

Figure 2.4: Hot-spot temperature calculation by the modified difference equation methodduring normal loading.

load and overloading. On the other hand, the calculated hotspot temperature with the sec-ond modified thermal model has only a slightly different value compared with the measuredhotspot temperature during normal load. It shows a lower overestimation to the measuredhotspot temperature during overloading than the first modified difference equation model.

Table 2.6: The overestimation during normal loading.

∆To Diff method Diff method First mod Second modH = 1 [%] H = 1.3 [%] [%] [%]

> 3°C 19 69 89 8> 5°C 7 10 14 0.53> 10°C 0.0 2 0.0 0.0

Conclusions from the case study

This study proposes a thermal model for hotspot temperature determination that incorpo-rates the dependence of oil-viscosity and winding losses on temperature. The proposedthermal model generates better results than the reference model (the thermal model of IEC


0 5 10 15 20−10

0

10

20

30

40

50

60

70

80

90

100

Tem

pera

ture

[°C

]

Time [h]

0 5 10 15 200

20

40

60

80

100

120

Load

[%]

Ambient temperatureCalculated hot−spot temperature with H=1.3Calculated hot−spot temperature with H=1Measured hot−spot temperatureMeasured top oil temperatureLoad

Figure 2.5: Hot-spot temperature calculation by the modified difference equation methodduring overloading.

Table 2.7: The overestimation during overloading.

∆To Diff method Diff method First mod Second modH = 1 [%] H = 1.3 [%] [%] [%]

> 3°C 39 73 66 35> 5°C 19 46 49 11> 10°C 5 11 11 0.52

Std. 60076-7). However, the proposed thermal model should be validated for more trans-formers. As a consequence, further investigation is required for a more general conclusion.

From the calculation it can be concluded that the sensor for measuring the hotspottemperature is not located at the actual hottest spot. Later this was verified in a discussionwith Hasse Nordman [27] who has extensively studied the transformer at hand.

Chapter 3

Lifetime Estimation

This chapter introduces Publication VI. Section 3.1 discusses several lifetime definitionsand defines the operational lifetime. Section 3.2 describes an estimation procedure for theoperational lifetime. Section 3.3 demonstrates the estimation procedure in a numericalexample.

3.1 IntroductionCentral to this chapter is the operational lifetime of the transformer. This introductionserves to define and motivate the use of this operational lifetime by presenting lifetimedefinitions found in the literature and to present how this lifetime can be used by relating itto condition-based maintenance plans.

In the literature, several lifetime definitions are found. These definitions are been sum-marized in [28] and are hereby repeated for convenience.

Economical lifetime is a theoretic concept that refers to the time up to the replacementtime that gives the optimal profit from an alternative investment [29].

Technical lifetime for a component is defined as the time until the component is unable toperform the desired function and it is impossible to repair the component [30].

Strategic lifetime is the time until the component is replaced because of structural changesmade in the power system. The old transformer is not able to transform betweenwanted voltage levels; either it is not designed for these voltages and currents or thepower supplier wants a different transformer in that place for other reasons [28].

Equivalent lifetime is an overall sum of the components age, the stresses from lifetimeconsuming factors, and the stresses from lifetime intensifying factors [30].

It should be mentioned that the three first definitions are discussed in [31] as end of lifecriterions.

The lifetime definitions could be classified as either relative measures or absolute mea-sures. The former measure serve to produce a ranking of the studied components. The

21

22 CHAPTER 3. LIFETIME ESTIMATION

latter measure constitute an all-embracing absolute measure of the actual lifetime. The dif-ficulties associated with an all-embracing measure lead to the development of the relativemeasures. It is the study at hand that decides which lifetime measure should be used.

Statistical methods for lifetime determination of transformers have been presented infor instance [32, 33, 34]. One step to incorporate more information in the modeling proce-dure is achieved with the Bayesian approach which incorporate subjective information suchas expert opinion as well [35, 36, 37]. However, both of these approaches require an exten-sive failure data set in order to generate reliable results, as is demonstrated in [38], whichis difficult in the case of the power transformer population. Moreover, with these methods,only general conclusions about the transformer population as a whole could be made, andnot conclusions about individual components. The later is critical in the decision-processregarding precise maintenance activities. One step toward this is taken in [39] which usesregression analysis to express the dependence of the lifetime on insulation and cooling.However, a fully component-based lifetime model should take into account the conditionof the component.

The approach suggested in this text is a condition-based approach for lifetime deter-mination. Condition assessment is defined as [40] a comprehensive assessment of thecondition of an equipment taking into account all relevant information, e.g. design in-formation, service history, operational problems, results of condition monitoring and otherchemical and electronic tests. Condition assessment techniques for transformers are for in-stance [41, 42, 43, 44, 45, 46, 47, 48, 49], and an extensive review can be found in [50]. Thedefinition of the term condition [40] is an expression of the state of health of an equipmentwhich takes into account its aged state as well as any inherent faults. A note is also added tothis definition that says that this condition normally is the perceived condition determinedfrom the result of measurements, which may not be a complete and accurate representa-tion of the actual condition since there might be a hidden deterioration process inside theequipment; hidden in the sense that it is not possible to follow its progression by measur-ing parameters. Based on the hidden deterioration process, hidden Markov models havepreviously been used to model the lifetime of the transformer. The more general methodis presented in [51], and applied to the special case of transformers in [52, 53]. Commoncritique to these methods is that the state transition probability depends only on the stateof origin and state destination which means that the transition rate of the condition is nottaken into account. However, the trend of the deterioration is an important consideration inlifetime determination of transformers.

For the method suggested in this text, the definition of a technical lifetime needs to befurther refined. This will be motivated in terms of soft and hard failures that are the eventsthat govern the termination of the suggested lifetime. This thesis sees a hard failure as oneevent that terminates the technical lifetime. At the same time, a soft failure is one eventthat necessarily does not result in the termination of the technical lifetime of the componentbut terminates the time of suitability for operation. The latter concept, which is terminatedby a soft failure, will here be referred to as the operational lifetime of the component.In practice, this means that after the operational lifetime has terminated, a maintenanceactivity could put the component back into operation again. This implies that this methodcould be used as input to a maintenance policy.

3.2. DETERIORATION MODELING WITH DIAGNOSTIC MEASUREMENT VALUES 23

The approach suggested in this thesis is a condition-based approach to operational life-time estimation which incorporates the trend of the deterioration in the modeling procedure.This is done by forecasting diagnostic measurement values with a stochastic process up un-til a point when these measurements reach a threshold value which is considered to be thetermination of the operational lifetime. The fact that the diagnostic measurements doesnot represent the complete and accurate condition of the component, as suggested in theabove-mentioned definitions, is not crucial because this method aims to forecast the op-erational lifetime as input to a maintenance policy, and it does not attempt to forecast thefunctional lifetime of the component which would require another approach due to the factthat the consequences of a component failure is more inherent in the latter case. However,it should not be denied that this method does not attempt to quantify the time to sudden andunexpected failures that happen due to random reasons which cannot be captured by thediagnostic tools at hand.

The remaining of this Chapter is organized as follows. Section 3.2 describes the sug-gested modeling approach by defining the stochastic process and account for the estimationprocedure for the parameters in the stochastic process. Finally Section 3.3 demonstrates themethod by accounting for an initial case study made within the project (Publication VI). Itshould be noted that this method is implemented by using the diagnostic tool frequency re-sponse analysis (FRA) which might not be the most suitable for this application. However,this demonstration only serves the purpose of outlining the methodology.

3.2 Deterioration modeling with diagnostic measurementvalues

This section introduces the theory for the suggested modeling procedure.In this text, diagnostic measurements are forecasted using the Ornstein-Uhlenbeck

(OU) process. These forecasted values are used to estimate the operational lifetime byestimating the point in time when these forecasts reach a predefined threshold value, i.e.,the threshold value is the stopping criterion for the mathematical calculation.

The OU process is a stochastic process that satisfies the following stochastic differentialequation (SDE):

dXt =−β (µ(t)−Xt)dt +σdBt , t ≥ 0, (3.1)

where Bt , t ≥ 0 denotes the Brownian motion, µ(t) is the time-dependent mean of theprocess, and β and σ are two constants that describe the uncertainty and volatility. More-over, the OU process has been used to describe, for instance, temperature evolution [54, 55],load [56] and fluctuations in the stock market. The solution to Eq. (3.1), which also definesthe OU process, is given by

Xs = Xte−β (s−t)+β∫ s

tµ(u)e−(β (s−u))du+σ

∫ s

te−β (s−u)dBu, (0 ≤ t < s) (3.2)

where the initial value of the process X0 is the initial deviation of the diagnostic measure-ment from its mean value and where the integral is taken in the Ito sense. Since there is no


information about the diagnostic measurements before t = 0, X0 ∼ N(0, σ2

2β ). If the Brown-ian increment is ignored in Eq. (3.1), it can be seen that Xt has a drift towards its mean valueµ with rate β , with a magnitude proportional to the distance between the current value andzero.

Using this setup, the diagnostic measurements Dt can be modeled using Eq. 3.2

Dt = Xt ∀t. (3.3)

The parameters of the stochastic process µ(t), σ and β needs to be estimated for the di-agnostic measurements values. In [57], an estimation method for these is presented. Thesame method is applied here with diagnostic measurements as input data. This estimationprocedure will be summarized next. An estimate for σ from the quadratic variation of themeasurement values is expressed as [57]

σ 2 =1N

N−1

∑i=0

(Di+1 −Di)2, (3.4)

where D0, D1, ..., D j, ..., DN , are discrete measurements taken at a time difference ∆t.Furthermore, an estimate for β is expressed as [57]

β =− log(

∑Ni=1Yi−1(Di −µt)

∑Ni=1Yi−1(Di−1 −µt)

), (3.5)

whereYi−1 =

µt −Di−1

σ 2 . (3.6)

By inserting the estimate of σ from Eq. (3.4) into Eq. (3.5), an estimate of β isobtained.

It should be noted that a realization of the stochastic process should be made using thetime step ∆t, as was used for the recording of the diagnostic measurements.

The estimation procedure for the operational lifetime is described next. Iteratively, thestochastic process in Eq. (3.1) is used to generate realizations of the time evolution of thediagnostic measurements up until the threshold value. The point in time when the diagnos-tic measurement forecast reaches the threshold is takes as the estimate for the operationallifetime. The different estimations is obtained from a Monte Carlo simulation. These es-timations are treated as observations of the operational lifetime and are used to obtain anempirical distribution for the operational lifetime, defined as follows [58]. Let Fn be thedistribution that distributes the mass 1/n in each of the observations x1,x2, ...,xn. This isthe empirical distribution for the observations. Moreover, the estimate of the first moment,which is an estimate of the mean value of the operational lifetime, expressed in terms ofthe empirical distribution, reads as

θ(x1,x2, ...,xn) =∫ ∞

−∞xdFn(x) =

1n

n

∑i=1

xi. (3.7)

This estimate will be used as input to the developed maintenance policy.

3.3. DEMONSTRATION OF METHOD 25

3.3 Demonstration of MethodThis section demonstrates the modeling procedure by discussing the results of the numeri-cal demonstration in Publication VI. The reason for including the results from the study inPublication VI in this chapter is to provide a general demonstration of the modeling tech-nique. It should be emphasized, however, that the stochastic process used in PublicationVI, is different from the process initially discussed in this chapter (i.e., Eq. (3.2)). Thereason for this is mainly that the study was an initial study on this topic and it was seen asthe most suitable modeling technique at the time when the study took place.

3.3.1 Frequency Response Analysis

Frequency response analysis (FRA) is a diagnostic method that characterizes a system byanalyzing its phase and magnitude (i.e., frequency) response when the system is subjectedto sinusoidal inputs. The frequency response is uniquely defined by the geometrical pa-rameters of the system. This means that FRA can be used either to design a system witha specified geometry or to analyze an existing system to identify geometrical deviations.FRA can be used to evaluate the mechanical condition of the windings and the clampingstructure applied as a diagnostic tool for power transformers [59, 60, 61].

Generally, the frequency range of FRA is within 10 Hz - 10 MHz. For a transformer,the frequency response within this interval is governed by the capacitance and inductancethat, in turn, is determined by the geometrical construction and inherent material of thetransformer. Mechanical deformations change the capacitive and inductive parameters, andyield deviations in the FRA spectrum. This means that FRA is a comparative method, inwhich a fingerprint measurement taken at an earlier stage is compared with a measurementtaken at a later stage, often after a relocation or short circuit.

FRA has the capability of identifying faults of the following types [61, 62, 63]:

• core movement;• winding deformation;• winding movement;• damages of loose winding or clamping structure;• partial collapse of the winding;• short-circuited turns;• open circuit windings.

Unfortunately, the interpretation of FRA results as used for detecting mechanical changesinside a transformer has neither been standardized nor fully agreed upon among researchers.As a consequence, several transfer functions are in use.

Essentially, FRA is a comparative method. Therefore, a fingerprint of the responsemeasurement either from the transformer or a sister unit of the studied transformer needsto be available for a comparison of the present measurement. The changes in frequencyresponse that can be assigned to mechanical faults, are the following [64]:

• abnormal shifts in the existing resonances;


Figure 3.1: An example FRA diagram.

• emergence of new resonances;• evaporation of previous resonances; and• considerable changes in the overall shape of the frequency response.

Figure 3.1 shows an example of a typical FRA diagram. It should be noted that thisdiagram display one measurement only, and it is not a demonstration of two or more mea-surements

3.3.2 Degradation model of FRA measurements with stochasticprocess

The deviation of the reference measurement and the new measurement of the response canbe mathematically modeled in the following way: If the deviation of the current measure-ment from the reference measurement is denoted by ∆, make the assumption that the timeevolution of this parameter can describe the mechanical degradation of the windings. Then,by modeling ∆(t) it is possible to determine the degradation path, expressed as

∆(t) : t0 > t > 0,∆(t0) = z0, (3.8)

where t0 is the operational lifetime as defined previously, and z0 is the threshold value forthe parameter ∆(t0).

For the study in Publication VI, the stochastic process used in the modeling procedureis a specific Wiener process, defined as follows:

• If the starting value W (0) = 0,• an increment W(t+h)-W(t) follows a normal distribution N(µFRAh,σ2

FRAh),

3.4. DISCUSSION AND CONCLUSION 27

• then the process W (t) = µFRAt + σFRAW0(t) is called a Wiener process of linearleaning µFRA and variance σ2

FRA, where W0 is a standard Wiener process.

This Wiener process is used to define the stochastic process used in the modeling procedurein this study. The stochastic process

zi j : t j > 0, (3.9)

consists of the termszi j = x0 +µFRAti j +σFRAW0(ti j), (3.10)

where x0 is the initial value of the process and W0 is, as before, a standard Wiener process.In the study presented in Publication VI, the parameters µ and σ are obtained by max-

imum likelihood estimation using the FRA measurements of the degree of deformation.Figure 3.2 shows a realization of the degradation path. The parameters of the stochasticprocess were chosen by the authors to provide reasonable estimations of the lifetime.

The threshold value that governs the failure criterion is provided in [62], which says thatwhen the degree of deformation is equal to or larger than 5 dB, the windings are consideredto be in such a severe condition that the transformer should be taken out of operation. Thus,the time t0 given by

ti0 : zit0 = z0 (3.11)

is seen as the failure time of the transformer with respect to winding faults.In the last step, the empirical probability distribution is determined with a Monte Carlo

simulation, which is described next. In an iterative manner, a new realization of the degra-dation path is used to estimate the failure time. The inherent stochasticity of the failuretimes originates from either measurement errors, or uncertainties in the forecasted value.

3.4 Discussion and ConclusionThis chapter suggests a deterioration modeling approach which utilizes forecasted diagnos-tic measurement values to predict the operational lifetime of the transformer. The methodis demonstrated in a case study (Publication IV), in which the methodology to model thedeviation between continuous FRA measurements with a monotonous Wiener process isoutlined. Three things needs to be discussed here:

• the choice of stochastic process,• the choice of diagnostic tool, and• the choice of threshold value for the simulation.

A comment here could be done on the appearance of the time evolution of the FRA mea-surement. First, the FRA measurements are not the type of measurements that is taken ata constant time frequency throughout the lifetime of the component. At most, those mea-surements are performed when the component is installed, when it has been relocated or ifany serious fault has been discovered. Second, the FRA provides a measure of the degreeof deformation. Deformations inside a transformer occur in bursts; hence this degree of


0 5 10 15 20−4

−2

0

2

4

6

8

t (years)

∆ (d

B)

A realization of the degradation path

z0 = 5 dB

Figure 3.2: One realization of a degradation path (blue), and the Wiener process (black)along with the threshold value for failure z0 = 5 dB (dashed). The failure time for thisparticular degradation is seen at the intersection of the threshold value and the realizationof the degradation path. In this particular study, the failure time is around 14 years.

deformation will not have smooth and continuous appearance. Third, there is no consen-sus in the threshold value for this parameter. Hence the FRA tool is not the most suitabletool for this modeling approach. However, there are others that would serve this purposebetter as, for instance, dissolved gas analysis (DGA). DGA is performed on a regular basisthroughout the operational lifetime of the transformer suggesting that more measurementvalues would be available for the simulations. In addition, the increase and the decrease ofthe concentration of the dissolved gases in the oil have a more smooth time evolution thanthat of the FRA measurement. DGA is a more established diagnostic tool, hence there isa better consensus on critical oil concentration levels. All three reasons suggest that DGAwould be a better choice than the FRA for this modeling procedure.

Chapter 4

Transformer Asset ManagementMethods

This chapter introduces Publications V, X, and XI. ï»¿Section 4.1 introduces the asset man-agement concept. Section 4.2 describes the developed methods in each Publication. Section4.3 demonstrates the methods in a numerical example.

4.1 Introduction

Central to this chapter is the concept of asset management. This text defines asset manage-ment as [65]

“the systematic and coordinated activities and practices through which an or-ganization optimally manages its assets and their associated performance, risksand expenditures over their lifecycle for the purpose of achieving its organiza-tional strategic plan.”

Applying this definition to the power system transformer, asset management involves thefollowing activities: (1) replacing/refurbishing, (2) maintaining, (3) operating, and (4) dis-posing the transformer. An overview of research made within each of these areas willfollow.

During the post war economy, countries that had suffered war damages had a need tobuild themselves up again and were also given a chance to do so. Hence, large investmentswere made in the power system infrastructure. This, in combination with the considerablelifetime that the transformer can have, the consequence today from these investments isthat many transformers are reaching a considerable age and the risk associated with aging-related failures are becoming more and more significant. A significant part of the researchaimed at transformer asset management have been focusing on finding replacement meth-ods in order to address the challenge of an aging transformer population. Transformer assetmanagement methods with a focus on replacement have been presented in [34, 37, 48, 66].Two partitions can be seen here, one part of the research focuses on statistical analysis

29

30 CHAPTER 4. TRANSFORMER ASSET MANAGEMENT METHODS

of the remaining lifetime [34, 37] and the other focus is on health index and conditionassessment [48, 49, 66, 67].

Today, the combination of the development of sophisticated sensors and the abilityto store and manipulate large amounts of data allows for a more significant conditionmonitoring and more sophisticated diagnostics. This manifests itself in activities in linewith smart grid development. In terms of transformer asset management this involves theactivities of maintaining and operating the transformer population more efficiently. Re-search aimed at transformer asset management with a focus on maintenance can be foundin [68, 69, 70, 71, 72, 73].

Proceeding from these activities, this chapter mainly focuses on the processes of replac-ing and/or refurbishing, and maintaining the transformers in a population. For the work inthis text, deterioration and/or aging have been the key consideration. Models of this havebeen used to set limitations on allowable thermal stress for the transformer. In this text, thefollowing deterioration (e.g. aging) models are used to assess the stress subjected to thetransformer:

• the loss of life measure (introduced in Chapter 2); and• the relative loss of life measure estimated with

– the Montsinger model, and– the Dakin-Arrhenius model

(both of which are introduced later in the current chapter).

Three loading scenarios are studied:

• the increased loading due to penetration of electric vehicles;• the cold load pickup during the restoration after a power outage; and• the normal operation of a transformer fleet obtained from a power flow analysis.

Transformer operation will be the focus of the next chapter in this thesis (Chapter 5).For a literature review of this research area the reader is referred to the introduction ofChapter 5.

Section 4.2 describes the calculation procedure. A note should be made on the differ-ent output measures. The output measures from these calculations differ slightly in eachstudy, but they all serve the asset management activities of operating and refurbish/replacetransformers in a population. Commonly, the information needed to support the asset man-agement decision process could be subdivided into three general categories: [74] (1) tech-nical, (2) economic, and (3) societal. Publications V, X, and XI contribute to Category 1. Aremark could be done on Publication IX (covered in the following chapter) that contributesto Categories 2 and 3. The reason for not including the result from Publication IX in thischapter is that the method is intended for operational purposes instead of replace and/orrefurbish and maintain the transformer fleet, which is the intended focus of the currentchapter.

4.2. METHOD 31

4.2 MethodIn Chapter 2, the loss of life measure was introduced in Eq. (2.1). In this work, thismeasure is used for aging estimation purposes. Other aging models, such as the Montsingermodel, provide a measure of the relative loss of life during a time period t according to thefollowing expression [75]

ηM =1t

∫ t

02(ΘHS(t)−98)/6dt. (4.1)

This model was derived empirically from observations of the correlation between the tem-perature and the tensile strength of cellulosic paper aged in oil and air. Moreover, theDakin-Arrhenius model is based on thermal laws for degradation and is expressed as [76]

ηA =1

100

∫ t

0

dt10A/ΘHS(t)+B

, (4.2)

where the parameters A and B are specific for the type of material used as solid insulation.Table 4.1 summarizes the studies that form a basis for this chapter. It lists the particular

loading scenarios and the corresponding aging measures applied in each of those studies.What follows in this section is a brief description of the three loading scenarios and thechosen modeling approach for each load.

Table 4.1: A summary of the loading scenarios and corresponding aging estimate that forma basis for this chapter.

Aging estimate Loading scenario PublicationEq. (2.1) Increased EV penetration VEq. (4.1)/(4.2) Cold load pick-up XEq. (2.1) Normal loading XI

4.2.1 The increasing penetration of electric vehiclesPublication V studies the effect of increased electric vehicle penetration on existing com-ponents in the power grid. More specifically, it motivates the use of controlled charging toreduce the wear of the transformer.

Electric vehicles with external charging possibility are classified into plug-in electricvehicles (PEVs), and plug-in hybrid electric vehicles (PHEVs).

The charging of an electric vehicle is modeled as a load profile in discrete time andis based on stochastic parameters. For particular assumptions, the reader is referred toPublication V. Each electric vehicle i corresponds a load Pv

t,i for time step t, and v is anindex implying a vehicle load, expressed as

Pvt,i =

Cp if T ci ≤ t < T ci +

Esoci

Cp

0 otherwise, (4.3)


where Cp is the charging power, T ci is the time when the vehicle returns home and is parkedand connected, and Esoc

i is the energy level in the battery. The expected value E[Pvt ] of the

electric vehicle load Pvt is estimated with Monte Carlo simulations for n samples according

to

E[Pvt ] =

1n

n

∑i=1

Pvt,i. (4.4)

The total electric vehicle load at time t for N vehicles is thus

Pv,tott = N ×E[Pv

t ]. (4.5)

This load is added to the household load Pht which is estimated as the normalized load curve

Pn,ht multiplied with a total number of households H, and the assumed average consumption

C kWh per day and place of resident

Ph,tott = H ×Pn,h

t ×C. (4.6)

Finally, the total load profile Pt at time t that was used as input to the transformer thermalmodel, is calculated as:

Ptott = Pv,tot

t +Ph,tott . (4.7)

The following charging scenarios are studied:

1. uncontrolled charging in a residential area;2. uncontrolled charging in a parking lot;3. uncontrolled charging in a residential area with a parking lot;4. controlled charging with price-regulated tariff; and5. controlled charging with external unit distributed for valley filling.

A more thorough description of Scenarios 1 to 5 is found in Publication V. These loadingscenarios are used as input for Eq. (2.1).

4.2.2 The cold load pick-up situationPublication X studies the effect of load modeling on transformer aging using the Montsingerand Dakin-Arrhenius aging models. Moreover, the use of failure criteria makes it possibleto determine a probability for failure, given a certain loading situation.

The cold load pickup arises when the aggregated load for residents during restorationbecomes several times higher than the level before the power outage. It can be even higherthan the peak loading conditions experienced during heavy loading. This occurs becausethe thermostatic load loses diversity. The peak load and its duration after a power outagedepend on the length of the power outage and the ambient temperature.

In Publication X, the loading profile for cold load pick-up (CLPU) is assumed to havethe characteristic shape of the decaying exponential, as seen in Fig. 4.1.a, which has themathematical expression [77]

S(t) =[SD(SU −SD)e−γ(t−t ′)

]u(t − t ′)+SU

[1−u(t − t ′)

]u(t −T ′), (4.8)

4.2. METHOD 33

0 5 10 15 200

1000

2000

3000

4000

5000

6000

7000

Time [h]

Load

[kV

A]

(a)

0 5 10 15 200

1000

2000

3000

4000

5000

6000

7000

Time [h]

Load

[kV

A]

(b)

Figure 4.1: a) Delayed exponential model for the CLPU load profile, with the parametersSD = 3000 and SU = 6000. b) Realization of the stochastic load profile described by theOrnstein-Uhlenbeck process for the modeling parameters σ = 0.08, β = 0.0002, and thevalue of α as seen in Fig. 4.1.a.

where SD is the diversified value of the load, SU is the undiversified value of the load, γ isthe decay rate of the exponential, t ′ denotes a point in time that specifies the end of the coldload peak, and T ′ is the point in time of the load pickup.

To model the probabilistic nature of the aggregated load, a stochastic component isintroduced in the model. The stochastic component has its origin in the manually controlledload governed by switches that are turned on and off in a random fashion.

Mathematically, the load profile are modeled by the OU process defined by Eq. (3.2).The theory of this process was covered in Chapter 3, and thus will not be repeated here.A realization of the mathematically modeled load profile of Fig 4.1.a is seen in Fig. 4.1.busing the following values for the modeling parameters σ = 0.08, β = 0.0002, and thevalue of α as seen in Fig. 4.1.a.

The load profile in Fig. 4.1.b is used as input to the thermal models that, in turn are theinput for the aging models in Eqs. (4.1) and (4.2).

4.2.3 Normal operationThe method in Publication XI combines the loss of life measure in Eq. (2.1) with im-portance indices and outputs a transformer ranking with respect to both importance andaging. The concept of importance for transformers have been used before in [68] but theapproach in this thesis use importance indices to use this. This study uses an electric, ACpower flow model for the load. For power transmission systems, interruptions of supplyor blackouts seldom occur due to pure topology reasons. However, they occur due to thedynamics of the power system. In the power system model, the substation configurationsare modeled in detail. This enables a more accurate model for cascading outage events.


The transmission system model is defined at component level. The expected outage rate λ j

and expected outage duration r j are estimated for all components j in the system based onhistorical events. In the case study, the model has been applied to the Great Britain (GB)transmission system. Data on this system are found in the GB Seven Year Statement (GBSYS).

To make the description of the method more comprehensive the importance indices willbe described next.

The component risk index ISMj,k is defined as

ISMj,k = ∑

i∈Ω j

∑z

QCT Si,z,k Ui, (4.9)

which is the sum of risks for all outage events i at load scenario k where component jcaused the event, QCT S

i,z,k is the consequence of the outage event i on critical transfer section(CTS) z after the event and at load scenario k, and Ui is the expected unavailability foroutage event i.

The index expressing the risk of causing interruption of supply at system load scenariok is defined as

IEENSj,k = 8760 ∑

i∈Ω j

QPNSi,k Ui. (4.10)

This index follows the same notation as the index ISMj,k defined in Eq. (4.9), except that the

consequence QPNSi,k is the power not supplied (PNS) for outage event i.

The index expressing the risk of component j causing disconnected generation is de-fined as

IDGj,k = 8760 ∑

i∈Ω j

QDGj,k Ui. (4.11)

This index follows the same notation as the index ISMj,k defined in Eq. (4.9), except that the

consequence QDGi,k is the disconnected generation due to outage event i and for load scenario

k.

4.3 Results and ConclusionsIn each of the Publications V, X, and XI the described methods are demonstrated in casestudies. This section serves to provide the main results and conclusions from those casestudies.

Publication V presents a method to evaluate the impact of electric vehicle chargingbehavior on transformer hotspot temperature and hence the loss of life measure in Eq.(4.2). The results indicate that the scenarios of uncontrolled charging and tariff chargingcould lead to increased peak powers, negatively affecting the transformers’ loss of life whenexponential aging behavior occurs. The results also imply that charging control could helpin utilizing the capacity of transformers more efficiently.

Publication X presents an aging study for the CLPU loading situation. The Montsingerand Dakin-Arrhenius aging models are used in the study. Figure 4.4 shows the estimated

4.3. RESULTS AND CONCLUSIONS 35

0 200 400 600 800 1000 1200 14000

20

40

60

80

100

Hot

spot

tem

pera

ture

[C d

egre

e]

Time [min]0 200 400 600 800 1000 1200 1400

020406080100

Loss

of l

ife [m

in]

0 200 400 600 800 1000 1200 14000

20

40

60

80

100

120

140

Hot

spot

tem

pera

ture

[C d

egre

e]

Time [min]0 200 400 600 800 1000 1200 1400

0

200

400

600

800

1000

1200

1400

Loss

of l

ife [m

in]

0 200 400 600 800 1000 1200 14000

100

200

Hot

spot

tem

pera

ture

[C d

egre

e]

Time [min]0 200 400 600 800 1000 1200 1400

0

200

400

Loss

of l

ife [m

in]

0 200 400 600 800 1000 1200 14000

100

200

Hot

spot

tem

pera

ture

[C d

egre

e]

Time [min]0 200 400 600 800 1000 1200 1400

0

1000

2000

Loss

of l

ife [m

in]

0 200 400 600 800 1000 1200 14000

20

40

60

80

100

120

140

160

Hot

spot

tem

pera

ture

[C d

egre

e]

Time [min]0 200 400 600 800 1000 1200 1400

0

1000

2000

3000

4000

5000

6000

7000

8000

Loss

of l

ife [m

in]

0 200 400 600 800 1000 1200 14000

50

100

Hot

spot

tem

pera

ture

[C d

egre

e]

Time [min]0 200 400 600 800 1000 1200 1400

0

200

400

Loss

of l

ife [m

in]

Figure 4.2: The calculated hotspot temperature and the calculated loss of life for Scenarios1 to 6.


0 0.1 0.2 0.3 0.4 0.50

50

100

150

200

250

300

350

400

450

500

Aging [%]

Rel

ativ

e fr

eque

ncy

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

100

200

300

400

500

600

700

800

Aging [%]

Rel

ativ

e fr

eque

ncy

(b)

Figure 4.3: Estimated distribution of the transformer aging according to a) the Dakin-Arrhenius model and b) the Montsinger model, as a result of Monte Carlo simulation.

distribution of the transformer aging according to a) the Dakin-Arrhenius model and b) theMontsinger model, as a result of Monte Carlo simulation. The conservative nature of theMontsinger estimate and its smaller variance, in relation to the Dakin-Arrhenius model, isseen.

Figure 4.4 shows the aging estimate as a function of the drifting parameter β of theOU process. Since β strongly affects the drift of the process, it is reasonable to investigatethe influence of β on aging. The estimated mean value decays exponentially toward thedeterministic aging value.

The Monte Carlo simulation yields an estimate for the distribution of the maximumhottest-spot temperature, as seen in Fig. 4.5. The recommended hottest-spot temperaturelimitations are 140°C and 180°C. An operation above these limitations increases the riskof accelerated insulation aging and possible evolution of free gas from the insulation. Ac-cording to a survey performed by CIGRÉ, there is no single safe limit for the hottest-spottemperature. In this paper, however, the limit of 180°C was chosen, because operationabove this limit causes significant damage to the solid insulation. From the distributionof the hottest-spot temperature in Fig. 4.5, it is possible to determine the probability forexceeding 180°C, i.e.,

P(supΘhs> 180°C) . (4.12)

The results for this particular numerical demonstration show that this probability is esti-mated at 3.6%.

Publication XI presents a three-step power transformer ranking method that combinessystem-oriented component reliability indices with component-specific aging estimates.The method provides an overall system screening for identifying critical transformers inasset management activities. Figure 4.6 shows the overall ranking of the 27 transformersfrom the presented case study that includes 369 power transformers. This screening en-


0 0.2 0.4 0.6 0.8 1

x 10−3

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

β

E∗

(ζ)

E*(ζ

A)

E*(ζM

)

Figure 4.4: Estimated mean value of aging as a function of the drifting constant β .

110 120 130 140 150 160 170 180 190 2000

50

100

150

200

250

300

350

Maximum hottest spot temperature [C]

Rel

ativ

e fr

eque

ncy

Figure 4.5: Distribution of the maximum transformer hottest-spot temperature as a resultof a Monte Carlo simulation.


System importance of transformer

Tra

nsfo

rmer

loss

of l

ife [h

rs/y

r]

TR_THOM

20−T

HOM40

−1

TR_THOM

20−T

HOM40

−2

TR_NEEP20

−NEEP4A

TR_BRAW

20−B

RAW4A

TR_ROCH20

−ROCH4A

TR_BRIN

22−B

RIN40

TR_DRAK21

−DRAK41

TR_WALP

12−W

ALP40

TR_CAPE21

−CAPE4B

TR_STAL2

1−STAL4

A

TR_DRAX11

−DRAX41

TR_ROCH1M

−ROCH20

TR_INCE12

−FROD2B

TR_SM

AN10−S

MAN20

−1

TR_SM

AN10−S

MAN20

−2

TR_PENW

12−P

EWO21

TR_WASF1A

−WASF2A

TR_WASF1B

−WASF2B

TR_IRON11

−IRON40

TR_SHRE10

−SHRE4A

TR_DRAX12

−DRAX41

TR_FID

F13−F

IDF21

TR_FID

F14−F

IDF22

TR_CASK10

−ROCK40

TR_PENE11

−PENE40

TR_WALP

13−W

ALP40

TR_WALP

11−W

ALP40

108 hrs 117 hrs 169 hrs

23792 hrs

0

10

20

30

40

50

Figure 4.6: The overall ranking of the 27 transformers, i.e. the combination of loss of lifeand their relative importance to system reliability.

ables a more comprehensive analysis of a few components in the population, where theloss of life gives an indication of the transformers’ condition. The result from the GB casestudy can be used by the transmission system operator as one input to allocate specific assetmanagement activities such as grid reinforcement and maintenance activities.

Chapter 5

Probabilistic Dynamic Capacity Rating

This chapter introduces Publications VIII and IX. Section 5.1 introduces included factorsin dynamic capacity rating and the modeling approach for them. Section 5.2 describesa method for probabilistic capacity rating. Section 5.3 shows the results of a numericaldemonstration of the method. Section 5.4 provides a socioeconomic perspective on dynamicrating.

5.1 Introduction

Dynamic rating is a broad term that refers to the capacity of a component that is dependenton factors external to that component. It is dynamic in the sense that the external factorseither increase or decrease the available capacity continuously either in accordance with areal-time model or in accordance with ready-made calculations. This allows for a more effi-cient capacity usage. Another term for dynamic rating of transformers are dynamic loading,which has previously been suggested in [78]. However, this model does not incorporate thevolatility of the load, which is a stochastic property that cannot be incorporated into a de-terministic modeling approach. Hence, in this text a probabilistic dynamic capacity ratingmethod is developed.

Previously, probabilistic dynamic capacity rating has been developed for over headlines in [79, 80, 81, 82] but it has not yet been suggested for transformers. As mentionedin [79], the probabilistic dynamic capacity rating method has the advantage of not requiringreal-time input data since this approach already accounts for the dynamic characteristics ofthe external properties, by allowing uncertainty.

For the particular case of dynamic rating of transformers, the external factors of partic-ular interest are as follows: ambient conditions, historic loading, and real-time loading.

With regard to ambient conditions, large power transformers have a time parameter ofseveral hours, i.e., the effect of ambient conditions on the hotspot temperature will only benoticeable after several hours. Ambient conditions are discussed in Chapter 2. Thermalmodels were improved for the ambient weather conditions of the transformer, but the effecton the overall capacity rating of the transformer was not discussed in this particular case.

In the same way as ambient conditions affect the inert system of the transformer, his-

39

40 CHAPTER 5. PROBABILISTIC DYNAMIC CAPACITY RATING

torical loading affects the transformer because of the inherent inertia of the system. Thiseffect is not studied within this project.

First, the effect of real-time loading on transformer capacity rating is studied within thisproject. A probabilistic capacity rating method has been developed. This method makesuse of a failure rate conditioned on loading. Common approaches within the field of relia-bility theory involve approximating a time-varying phenomenon with a piecewise-constantfunction. For instance, in [83] the transformer failure rate is conditioned on total dissolvedcombustible gases (TDCG), which is the summation of concentration of hydrogen, ethy-lene, acetylene, methane, ethane, and carbon-monoxide. In the study presented in [83]the failure rate is approximated by a piecewise constant function for particular intervalsof TDCG separated into four classes expressing the condition of the oil. This conditionclassification is described in [84]. This is a rational approach if the covariate of the study issufficiently smooth. However, real-time loading is a relatively rapid, time-varying, exter-nal factor that requires an alternative modeling approach. Hence, the modeling approachsuggested here uses a stochastic simulation with time-dependent covariates.

5.2 Method

5.2.1 Time-dependent covariates

Time-dependent covariates fall into the two broad classes [85, 86]: internal, and external.An external covariate influences the rate of failures over time, and its future path up to anypoint in time is not affected by the occurrence of a failure at an earlier point in time. For theexternal covariate, it is possible to define a survivor function conditional on the covariatepath,

R[t;X(t)] = P[T ≤ t|X(t)]. (5.1)

It should be emphasized that the usual relationship between the conditional survivor func-tion and the conditional hazard function hold in the particular case of the external covariate.For the other case, if the failure rate is assessed on internal covariates, Eq. (5.1) is not ful-filled. Hence, failure rates conditioned on these covariates are not of interest to reliabilitystudies.

5.2.2 Loading-related failures

To estimate the failure rate with respect to thermally-related failures, a failure criterionfor these failures is expressed next. The greatest risk associated with loading beyond thenameplate rating is the evolution of free gas from the insulation of winding and lead con-ductors [20]. In a survey [87], utilities were questioned on their view on critical operatingtemperatures. At a hotspot temperature equal to or above 180°C, ongoing operation isnot common practice because of the inherent risk of gassing and bubbling. However, thistemperature limit is highly dependent on the moisture content of the solid insulation. Forexample, a moisture content of 0.5 % requires a hotspot temperature above 200°C for bub-

5.2. METHOD 41

ble formation. Analytically this threshold is expressed as [20]

Θbubble =

[6996.7

22.454+1.4495lnWWP − lnPPres

]−

[(e0.473WWP)

(V 1.585

g

30

)]−273. (5.2)

The failure criterion is that the transformer fails when the hotspot temperature reaches thetemperature threshold in Eq. (5.2). For the simulation, the definition of the failure rateis the number of failures per unit of time (defined more thoroughly below). Figure 5.1visualizes this failure criterion. This realization of the hotspot temperature is based on thedata σ = 0.08, β = 0.00008, ΘA = 30°C, and m = 6000.

3 4 5 6 7 8 9120

130

140

150

160

170

180

190

Time [h]

Hot

test

spo

t tem

pera

ture

[C]

Figure 5.1: A demonstration of the failure criterion. At 180°C the transformer assumedlyfails.

For a certain period of time, the ratio of the number of times the stochastic processreaches the temperature threshold in Eq. (5.2) and the length of the particular time period,approaches the failure rate as time goes to infinity. This is expressed as

n∆t

→ λt as ∆t → ∞, (5.3)

where n is the number of failures due to transformer overloading, and t is the length ofthe time period. These values are used as failure rate estimates in the ongoing estimationprocess.

This study is aimed towards conditioning the failure rate on overloading for the par-ticular transformer, the ambient temperature ΘA, the mean load m, and the load volatilityspecified by the parameters σ and β . Therefore, for a particular transformer, the failurerate is assumed to be given by the nonparametric regression model, expressed as

λt = f (t,m,ΘA,β ,σ), (5.4)


where the right hand of the equation is an arbitrary nonnegative deterministic hazard func-tion. The approach in this text assumes that the ambient temperature ΘA, β , and σ areconstants. This way, the failure rate could be parameterized as

λt(t|m,c)≈

ε m < m0

∑ni=0 ci(m−m0)

i + ε m ≥ m0(5.5)

5.2.3 Aging-related failuresThe concept of aging of a transformer is described as [31]

Irreversible deleterious changes to the serviceability of the transformer.

Moreover,

such changes are characterized by a failure rate which increases with time.

Based on this, the aging-related failures are modeled with a three-parameter Weibull distri-bution. In that setting, the failure rate is expressed as [88]

λa =

(ηγ

)(tage −µ

γ

)η−1

, (5.6)

where γ is the scale parameter, η is the shape parameter, µ is the shift parameter, and tage isthe actual age of the transformer. Equation (5.6) models the effect of general aging only. Itis essential, however, for the modeling approach to capture the effect of wear or degradationas well. To capture this in the model, an acceleration parameter ϕ is multiplied with thetime variable in Eq. (5.6), which indicates that a more degraded transformer will enter theaging state more rapidly, and willalso age faster. The expression for the failure rate, withthe acceleration parameter ϕ , is

λa =

(ηγ

)(ϕ tage −µ

γ

)η−1

. (5.7)

It is common practice to equalize the degradation of the transformer with that of thesolid insulation. This is reasonable in the sense that the solid insulation is an upper boundfor the condition of the transformer. The degradation process of the solid insulation is acomplex degradation process described by the following subprocesses:

• hydrolysis - governed by carboxylic acids dissociated in water;• oxidation - governed by hydroxyl radicals;• pyrolysis - no agent needed to initiate the decomposition; and• the interaction between the aforementioned subprocesses.

Each of the agents that initiate the decomposition is either a by-product of aging or orig-inates from the surrounding and is deposited on the solid insulation continuously. It isnot unreasonable to suggest that the acceleration parameter should be dependent on theinsulation contamination [89, 90].

5.2. METHOD 43

The underlying mechanism that governs the deterioration of the solid insulation is de-scribed by the decrease of the degree of polymerization (DP) value. At first, when thetransformer is put into operation, its DP-value equals 1200, and throughout its lifetime itdecreases until it reaches a level of 250, at which point the ability to withstand additionalstresses is critically low. Therefore, it could be suggested that this acceleration parametercould be a proportional ratio between the current DP-value for the transformer and 1200,i.e.,

ϕ(DP) ∝1200DP

. (5.8)

5.2.4 The survivor function and expected life timeThe survivor function conditioned on the covariates is given by [91]

R(t|m,c) = e[−∫ t

0 λdu]. (5.9)

According to the failure criterion used in this study, the survivor function expresses theprobability of not exceeding Θbubble°C at a particular load level.

Moreover, the expected lifetime (or mean time to failure) can be expressed in terms ofthe survival function according to [91]

E(t) =∫ ∞

0R(t|m,c)dt. (5.10)

5.2.5 Capacity rating selectionTo compensate for possible uncertainties in the estimation procedure, an assigned expertcan offer an opinion that could make up for uncertainties. This could be done by assigninga value to a risk level ε . Based on this risk level, the capacity rating could be selectedaccording to the following procedure:

1. Estimate σ and β using historical data (from the last hour or year).2. Calculate the failure rate of failure-related failures.3. Select appropriate risk level ε .4. Estimate m0 - a good choice is the actual nameplate rating.5. Choose an acceptance interval for failure rate estimation.6. Make a realization of the stochastic process and count the number of times the oil

temperature exceeds Θbubble°C.7. Sum the failure rate from aging- and loading-related failures.8. Calculate the survivor function from Eq. (5.9) from the failure rate and compare this

with the value of ε .9. Update m0:

a) If the modeled failure rate is too high, reduce m0 and repeat the procedure fromStep 5.

b) If the modeled failure rate is too low, increase m0 and repeat the procedure fromStep 5.


Otherwise, quit and use the current value of m0 as capacity rating.10. Stop

5.3 Results and conclusions

The capacity selection method described in Section 5.2.5 is demonstrated in PublicationVIII in a numerical example. What is provided here are the main results from this demon-stration.

The contributions from aging-related failures are assumed to be negligible in this par-ticular case. The volatility of the load is assumed to be determined from historical data.

From a realization of the OU process at different load levels, the number of times thehotspot temperature exceeds 150°C is counted for a certain time interval. This temperaturethreshold is determined from Eq. (5.2), assuming a moisture content of 1.5%. Then thefailure rate values at a certain load level are estimated from the number of failures, dividedby the length of the time interval, from Eq. (5.3).

Figure 5.2 shows a realization of the hottest spot temperature for a particular meanof the load level. The length of this simulation is 32 hours, and the hotspot temperatureexceeds 150°C four times. Based on these numbers, the failure rate could be estimatedat 0.089 failures per hour at this particular mean load level. It should be stated that thisparticular case, with this short time period, is given as a demonstrating example.

10 15 20 25 30 35 40 45100

110

120

130

140

150

160

170

180

190

t

ΘH

S

Figure 5.2: A realization of the hot spot temperature, ΘHS, for the given transformer for thecase where ΘA = 30°C, m = 7000, σ = 0.08 and β = 0.0005.

Similar failure rate estimates, but with a longer simulation time, are determined for


discrete mean load levels belonging to the interval equal to

[1.033R;1.300R] MVA. (5.11)

This interval was chosen by the authors, since the load covariate has a substantial effect onthe failure rate when its value belongs to this interval. Other mean load intervals could beof similar importance but are not included in this study.

From realizations of the load profile, for each mean load level of the interval in Eq.(5.11), the failure rate values are estimated.

For this particular case study, there is less than 1 failure per 2000 hours of operationwhen the mean load is 6.2 MVA. This will be used to assign the value of both ε and m0.Furthermore, the functional dependence of the failure rate on the load was determined in aleast square sense and has the following appearance:

λt(t|m,c) = 0.0425399(m−6.2)4 (5.12)

−0.06526(m−6.2)3

+0.078181(m−6.2)2

−0.0197759(m−6.2)+0.00151116.

The fitted polynomial in Eq. (5.12) and the estimated failure rate values are plotted inFig. 5.3.a.

For an ambient temperature equal to 0°C, the failure rate is expressed as

λt(t|m,c) = 0.045196(m−7.0)4 (5.13)

−0.12758(m−7.0)3

+0.163767(m−7.0)2

−0.03789(m−7.0)+0.00050345.

Equation (5.13) is visualized in Fig. 5.3.b along with estimated failure rate values.

5.3.1 Capacity ratingIn this section the capacity rating for the transformer is presented as a function of theambient temperature. Figure 5.4.a shows the variation of the average ambient temperature.Hence, it is assumed that the average temperature varies between 0°C and 15°C duringa 24-hour time period. The particular climate regions where this ambient temperature is areasonable assumption could be determined using the Köppen-Geiger climate classificationscheme found in [22].

Moreover, two load types are studied. The first load type is characterized by the pa-rameter values σ = 0.08 and β = 0.0005. The second load type is characterized by theparameter values σ = 0.10 and β = 0.0005. Hence, the second load type has a largervolatility.

Using the suggested capacity selection procedure, the following capacity rating as afunction of time is given. Here, it is seen how the capacity varies with the ambient temper-ature and the volatility of the load. For the second load type, a higher security margin isneeded to attain the same risk level that is reflected in a lower capacity rating.


6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 80

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Load [MVA]

λ

(a)

7 7.2 7.4 7.6 7.8 8 8.2 8.4 8.60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Load [MVA]λ

(b)

Figure 5.3: a) The polynomial fit (red curve) of Eq. (5.12) to the failure rate values (stars)where σ = 0.08, β = 0.0005, and ΘA = 30°C. b) The polynomial fit (red curve) ofEq. (5.13) to the failure rate values (stars) where σ = 0.08, β = 0.0005 and ΘA = 0°C.

0 5 10 15 20 25−2

0

2

4

6

8

10

12

14

16

Time [h]

Am

bien

t tem

pera

ture

[C]

(a)

0 5 10 15 20 256600

6700

6800

6900

7000

7100

7200

7300

Time [h]

Cap

acity

Rat

ing

[MV

A]

Case 1Case 2

(b)

Figure 5.4: a) The 24-hour variation of the ambient temperature. b) The capacity ratingvariation during a 24-hour period for two load types. The blue curve is obtained for theload type with σ = 0.08 and β = 0.0005. The red curve is obtained for the load type withσ = 0.10 and β = 0.0005.

5.4. SOME THOUGHTS ON THE PERSPECTIVE IN THE ESTIMATION PROCEDURE 47

Figure 5.5: Expected socioeconomic cost as a function of ambient temperatures and timeof second load pick up for a 3 hrs power outage.

It should be noted here that for these two load types, the same level of risk for thermaloverload is taken. Since the second type has a more volatile load, the capacity is reducedby approximately 6%.

5.4 Some thoughts on the perspective in the estimationprocedure

This probabilistic dynamic capacity rating is established with a viewpoint from tempera-ture effects from operational stresses. However, there are other perspectives that could beincorporated as well, such as the socioeconomic perspective. In Publication IX this socioe-conomic perspective is taken into account by the use of customer damage functions. Themain aim of the study was to identify optimal restoration strategies, but the method couldbe used for the purpose of determining the limits for transformer operation. As suggested inPublication IX, the optimal threshold temperature for transformer overloading (the OTTOthreshold) exists for those ambient temperatures that offer a minimum in the expected costfor overloading. Below this threshold it is socioeconomically optimal to overload the trans-former.

As seen in Fig. 5.5, there seems to be a temperature threshold between -5 and 10°C.Below this temperature threshold, it is socioeconomically optimal to pick up the load simul-taneously and overload the transformer. Above the threshold, it is not. The OTTO thresholdis general and not limited to the particular parameters given in this study. Identifying the


OTTO threshold would be a fast and easy measure of weather. It is economically feasible tooverload the transformer during a short period of time in a load restoration situation. Here,in the particular example of a 3-hour power outage, the OTTO threshold is about 7°C.

Chapter 6

Closure

This chapter concludes the thesis and presents areas of future work.

6.1 ConclusionsThis thesis has presented quantitative methods for component reliability estimation and ap-plied these to lifetime modeling, asset management, and capacity rating. Hence, this thesisis written in the intersection of the fields of thermal modeling, lifetime modeling, powersystem modeling and power system reliability, and applies that theory on the developmentof methods for lifetime modeling, asset management, and dynamic rating of power trans-formers. This application is done using tools from mathematical statistics, particularly,reliability theory. The studied loading scenarios are normal operation, increased penetra-tion of electric vehicles, and the restoration situation during CLPU.

This thesis contributes with the following methods:

1. improved thermal model;2. lifetime forecasting using diagnostic measurements;3. asset management in three loading scenarios;4. combination of system reliability and component reliability; and5. probabilistic capacity selection method that takes into account the volatility of the

load.

Method 2 to 5 make up the core of the thesis. What follows here are conclusions from eachof the methods.

Thermal modeling improvements are made using either a factor representing wind ve-locity or factors representing temperature-dependent oil viscosity and temperature-dependentwinding losses. The thermal model that includes the wind velocity factor allows the modelto capture the environment more fully. Hence, the improved model show slight improve-ment when compared with reference models and measurements. The thermal models thatinclude the temperature dependence of oil viscosity and winding losses allow the thermalmodel to respond to rapid load changes. Hence, the improved thermal model shows a slightimprovement when compared with reference models and measurements.

49

50 CHAPTER 6. CLOSURE

It should also be mentioned that these thermal models were developed with the mind-set of attaining a close-as-possible temperature value. Here, a contradicting philosophyaims to calculate conservative values for the temperature and have security margins thatcompensates for the uncertainty in the modeling technique. Having such a security margininherent in the modeling technique is counter-productive. The thermal model should beused for a real-time dynamic rating, because the fact that the estimated values would beoff. Instead, the security margin should be incorporated in the allowed capacity, since thisallows for a more conscious approach for handling the thermal stress.

The degradation model is a component-specific lifetime model that has been used to re-late a component’s failure time with the wear during its period of use. The forecast partiallyor fully models the degradation of the underlying deterioration mechanism that eventuallyleads to failure. Lifetime forecasting has been made using diagnostic FRA measurements.

Specific asset management methods for particular loading scenarios were developed.The particular loading scenarios developed are 1) the increased loading caused by an in-creased electric vehicle penetration, 2) the restoration situation of CLPU, and 3) the normalday-to-day loading.

A probabilistic capacity selection method that takes into account the volatility of theload was developed. This model also incorporates the thermally-related and aging-relatedfailures of the transformer. All of the different parts of the model are connected throughthe central concept of failure rate.

6.2 Future WorkOverall, the major challenge with the continuing work in this thesis consists of verifying thedeveloped methods. The scarcity of failure statistics in reliability studies is a well-knownchallenge. This scarcity becomes even more severe in the case of power transformers,which often are custom-made for their particular position in the grid. Hence, the availablefailure statistics are scattered over more component types. It is always possible to developother methods that model the same phenomenon. This is one explanation for the interest inBayesian statistics where the statistical values are replaced by subjective knowledge of thetransformer at hand. However, we cannot deny the importance of failure statistics for thispurpose.

The challenge associated with reliability studies of the transformer is mainly that thedeterioration of the component has less dependence on the actual technical age of the com-ponent than on another parameter that consumes the lifetime that depends on operation anddamaging events, and some other parameter that prolongs the lifetime that is dependent onmaintenance. The appearance of this dependence needs to be further studied.

What follows are future works specific for each of the listed methods in the previoussection.

6.2.1 Thermal modelingOne challenge with loss of life estimation according to Eq. (2.1) is the uncertainty in thelocation of the hotspot. The temperature profile of the windings is a result of generated

6.2. FUTURE WORK 51

and dissipated heat. The heat originates from current losses in the windings, in particularthe loading level. The heat is taken away by either forced or natural cooling. The coolingmode of the transformer is primarily dependent on the transformer type. Secondarily, itis dependent on loading level. Hence the location of the hotspot is dependent on loadingand transformer type. However, this is not considered in the thermal models studied in thisthesis.

It would be interesting to make the loss of life measure location dependent.

6.2.2 Lifetime forecasting using diagnostic measurementsThe study presented in Chapter 3 applied the method to the FRA diagnostic technique. Alook at the evolution of the modeled parameter ∆ shows that this propagates in terms ofsteps caused by, for instance, short circuits or transportation. At the bottom line, the timeevolution of the parameter ∆ is not gradual. However, it could be perceived as gradual dueto the time steps between the observations of this parameter, so that the overall degradationpath still shows gradual deterioration. Nonetheless, it would be interesting to apply thismethod to another diagnostic tool, as for instance dissolved gas analysis (DGA) whichcertainly has a more prominent gradual-deterioration-behavior than that of the FRA. Thisdiscussion is also included in the end of Chapter 3.

Moreover, the difference between the forecasted and estimated operational lifetimefrom actual failure statistics could be seen as a measure of the suitability of the diagnostictool to evaluate the lifetime of the component. The correlation between the estimated dis-tribution of the operational lifetime with the forecasts, and the estimated distribution of theoperational lifetime using the failure statistics will be the the difference between the plug-inestimates of the first moments for each distribution according to the following expression:

θ1 − θ2 =∫ ∞

−∞x dFn1 −

∫ ∞

−∞x dFn2 (6.1)

This could be used to evaluate the suitability of the diagnostic tool and also aid utilities sothat certain diagnostic tools could be recognized as redundant.

6.2.3 Asset management in three-loading scenariosWithin this project, asset management methods for three loading scenarios have been stud-ied. These modeling approaches differs in the load modeling approach. It would be ofinterest to also modify the loss of life measure to suit the characteristics of each loadingsituation.

6.2.4 Probabilistic capacity selection method which takes into accountthe volatility of the load

Within this project, the capacity selection method which has been developed, uses outputfrom a simulation to estimate the values of the failure rate with respect to overloading.Instead, it would be of interest to use failure rate estimates from a statistical analysis of thefailure behaviour of the transformer.

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