1

Stochastic Modeling of Power Transformers HealthCondition Evolution Process to Support theDevelopment of Distribution Grid Renewal

Investment StrategiesSamuel Ramos de Pina, MSc Student, Técnico Lisboa, Pedro M. S. Carvalho, Supervisor, Técnico Lisboa,

Abstract—A country’s electrical power system is a fundamentalcomponent of its sovereignty being therefore essential to ensureits availability and reliability. The transmission and distributionoperators are responsible for the management of the system andthe maintenance of its assets. Among the multitude of distributionassets, the HV/MV and MV/MV power transformers (PTs) areone of the most important. PTs suffer with age a degradationof their condition, reason why it was considered important todevelop an aging model for this kind of asset that would mapage onto health condition in a probabilistic way.

In this thesis, a stochastic aging model is proposed anddeveloped in order to be used to simulate the health conditionevolution of distribution PTs. The model is parameterized withthe data available on health indexes of the installed PTs. Healthindexes are mapped onto expected values of PTs failure ratesin order to allow simulating the reliability degradation togetherwith the health condition evolution with age.

The developed aging model is used as a decision-makingsupport tool. More specifically, it is used to simulate different gridrenewal investment strategies. By understanding consequencesof the failure of each PT, including the consequent energynot supplied (ENS) and simulating the reliability degradationof the stock of PTs, it was possible to evaluate the cost andcorresponding risks of different stock renewal strategies. In thisthesis, simple renewal strategies were evaluated and their outputscompared in terms of ENS and expected total costs

Index Terms—Power Transformers, Aging Models, Power Sys-tem Reliability, Condition Assessment, Monte Carlo Simulation,ENS Analysis.

I. WORK OBJECTIVES

THIS master’s thesis has three defined objectives. The firstone, which was defined by EDPD, is the computation ofan estimate for the current total amount of energy not supplied(ENS), associated with failures in the PTs of the nationaldistribution grid, based on the data collected in recent years.The second objective is to develop an aging model whichestimates the evolution of the PTs health degradation withtime. Based on this, the final goal is to develop a simulatorthat allows to estimate, using the Monte Carlo method, theaverage evolution of the installed PTs condition and extractmultiple output parameters, such as the average ENS for theupcoming years. This simulator can then be used as a tool fortesting different renewal strategies for the grid equipment andcompare the outputs between them.

The aging model should model the evolution of the healthindex of the transformers with time. This aging model should

then be incorporated into an application named the globalstock simulator, which should be able to estimate, using theMonte Carlo Simulation method, the total energy not suppliedfor the upcoming years, simulating the health state of thetransformers global stock. This simulator should incorporatethe option of testing different renewal strategies for the PTsstock. It should be possible to replace, at any time duringthe simulation, some of the transformers installed in the gridfor new ones. This way it is possible to experiment differentstrategies for the substitution and renewal of the PTs installedstock, and estimate the impact of those substitutions in the totalnon delivered energy in the future. This provides an interestingprediction of how the quality of the energy delivery servicewill be affected by the aging of the power transformers.This type of analysis can be used to simulate different strate-gies for investment in the grid renewal.

In the end, the final contribution of this thesis work shouldbe a tool that makes it possible to study and find the besttrade-off between investment in the grid and losses due to thedegradation of the park and consequent non delivered energy.

II. INTRODUCTION

The energy distribution grid, managed and operated by EDPDistribution (EDPD), is a fundamental part of the nationalenergy grid. It is composed by a variety of elements, oneof them being the power transformers (PT), which are re-sponsible for connecting grids with different voltage levels.There are hundreds of this type of transformers installed inthe Portuguese distribution grid, which combine into a totalof 17689 MVA of installed power, according to EDPD [1].These PTs can be divided into two groups. Ones responsiblefor the transformation between high voltage (HV) and mediumvoltage (MV) (E.G. 60/15kV, 60/10/kV). The majority of thePTs installed are in this first group and are defined as HV/MV.The other ones make the transformation between differentlevel of medium voltage (E.G. 30/10 kV) and are calledMV/MV. These represent a minority of the installed stock ofPTs.

The aforementioned PTs can have rated powers up to 40MVA, therefore each time one of these transformers fails, therecan be a considerable amount of energy not supplied to theconsumers. Even though some substations can have more thanone transformer working in parallel, in order to increase the

2

Fig. 1: Evolution of the maintenance importance with time.Adapted from: [2]

redundancy, a PT failure can still have a major impact on thesupplied energy.

The Portuguese distribution grid has a total of 739 installedPTs, with a current average age of 30.5 years. The expectedlifetime defined by the manufacturers for the PTs is between25 and 40 years, depending on the manufacturer. Each ofthem can cost over 500 000 Euros. This means that it iscrucial to develop, a strategic plan for the grid renewal, inorder to guarantee that its implementation is technically soundand economically feasible. The main motivation for this thesisis the development of this grid renewal strategy. The maincontributions of this thesis will be the development of technicaltools that may work as a support for the EDPD decisionmaking process, regarding the strategic investment plan.

A. Equipment Reliability and Preventive Maintenance

With the second industrial revolution, new technologicaladvancements initiated the emergence of large factories andorganizational models of production as envisioned by Taylorand Ford. Since machinery and technical equipment are themain protagonists of production processes, a technical failurecan have massive consequences. This is the reason for the largeinvestment in studies and research associated with equipmentreliability. This lead to the development of a new field calledmaintenance engineering, usually related with the mechanicaland electrical engineering. The first breakthroughs in this fieldoccurred during the second world war and the investment andresearch continued to grow until today 1. The work developedin this field can be applied not only to industrial processes,but on any kind of process that includes equipment aging anddegradation. In this case, it will be applied to the reliabilitystudy of the PTs of the distribution grid.

From the beginning, the main goal of these studies hasbeen to find mathematical procedures that can improve theassessment or the estimation of equipment reliability, in orderto reduce the time of inoperability, as explained by [3]. Manydifferent methods were developed as a support for this field ofresearch. One of the major examples is the use of the Weibulland log-normal distributions applied to this scope [4, 5].

Replacing or repairing an equipment too often might dras-tically increase the production costs. Any decision-makingpolicy that defines when and how to do maintenance of anequipment is defined as a maintenance strategy. If maintenanceis performed before the equipment starts to malfunction it

is called predictive maintenance. Based on this idea, a dif-ferent concept was developed named Strategic Replacementof Equipment, introduced by P. C. I. Crooks in an articlefrom the Journal of the Operational Research Society [6].The goal of this concept is to try to anticipate, by predictingthe degradation process of the equipment, the replacementand consequent investment in critical equipment before itfails. Thus avoiding the consequences and costs associatedwith the failure. Most recently some new theories have beendeveloped, regarding the subject of preventive maintenance.These new theories incorporate concepts such as Bayesianfailure-rate modeling, combined with gamma processes, tocompute maintenance schedules and preventive maintenanceoptimization [7].

B. Power Transformers Condition Assessment

The predictive maintenance research can be applied to manydifferent fields including the energy sector. Some of the mostimportant components of the energy distribution grids are thepower transformers (PTs), which are responsible for changingthe voltage level of the power being transmitted throughoutthe grid.

Since the PTs have an important role on the energy distribu-tion system, it is crutial to guarantee a good assessment of theequipment condition. The health condition of a transformeris influenced by several physical parameters, which whencombined can be quantified into a health index [8]. Thishealth index is computed through the combination of operatingobservations, field inspections and laboratory tests. It aimsto describe the overall health condition of the asset. Thehealth index uses data from gases, oil and furan analysis. Thedissolved gases analyzed are hydrogen (H2), methane (CH4),ethane (C2H4), ethylene (C2H6), acetylene (C2H2), carbondioxide (CO2) and carbon monoxide (CO). The parameters ofthe oil that are analyzed are the Break Down Voltage (BDV),Interfacial Tension (IFT), acid and water content [9]. Besidesthese tests, there are some other complementary analysis thatcan be performed like the winding resistance, sweep frequencyresponse analysis (SFRA) and vibration analysis [10]. OnFigure 2 are presented the main tests that can be performed inorder to asses the health condition of a transformer, organizedfrom the most important to the less.

According to [10] the main causes for transformers failuresinclude electrical breakdown, lightning, dielectric fault, looseconnection, incorrect maintenance and excessive loading. Theelectrical breakdown is the most common failure cause, pos-sibly being hasten by contamination, thermal aging, repetitivecycles of excessive voltage stress and mechanical deformation.Other common failure cause is the dielectric failure, which canhave a profound effect on the useful life of the transformers.On Figure 3 the distribution of the failures causes per occur-rence rate are presented. These types of failures can belong totwo categories: repairable and non-repairable. Depending onthe category, there will either be a replacement or a repair time.The non-repairable failures are normally associated with agefailures, as explained in [11]. On Figure 4 are presented thetransformers components mainly responsible for the failures.

3

Fig. 2: The importance of different diagnostic methods to es-timate transformer conditions with color distinctions. Source:[10]

Fig. 3: Transformers failure causes. Adapted from: [10]

EDPD uses an index named Partial Health Index (PHI) thatis similar to the HI previously explained. PHI analyzes, in aglobal way, the same physical conditions aforementioned andassigns a value from zero to one hundred percent. Where zerois total degradation and one hundred is perfect health condi-tion. There is another index designated Global Health Index(GHI), which is computed based on the PHI affected by anage factor, defined internally by EDPD. This GHI can then beconverted into a failure index (FI), which is a value computedbased on the GHI and on an environmental factor. This factorsrepresents the probability of external occurrences enhancingthe probability of a transformer outage. External occurrencescan comprehend nearby trees, possibility of floods, fire, among

Fig. 4: Defective components of a transformer (LTC: load tapchanger). Adapted from: [10]

others. All the transformers installed on the grid are catalogedand have an associated PHI, GHI, EF and FI value.

These internal indexes are deeply connected with the healthcondition of the PTs. It is possible that there is a relationshipbetween the age, the failure rate and the aforementionedindexes of the transformers. If these relationships can bedefined, it might be possible to predict failure probabilities forall the transformers and take action, either on the maintenanceor replacement of the more degraded equipment.

III. AGING MODEL FOR POWER TRANSFORMERS

It is logical that all equipment degrades with age. Thereis a large amount of research regarding aging models forequipment, which studies and models the way the health ofthe machinery evolves with the using time [12]. The alreadydeveloped aging models that estimate the evolution of thecondition of equipment can be based on two types of data[13]: the historical data retrieved from the assessment of theevelution of the equipment condition throughout its workinglifetime; real time acquired data monitored with sensors. Thislast type of data is essential to the processes associated withthe Industry 4.0, where live monitoring is a major concern.

The first aging models developed to estimate the agingbehavior of electrical equipment used the so-called AverageAge Method [14]. This method uses a large amount of dataregarding the mean life of the equipment to compute itsaverage lifetime. Usually this was the method used to computethe value presented in the equipment descriptions and datasheets regarding the life expectancy of each equipment. Thisapproach is usually not accurate, since it does not include realenvironmental and operational conditions of the equipment.Besides, this method is not adequate for power systemscomponents, such as generators or transformers. This type ofequipment has a long useful life, for an electrical equipment,and therefore there is a short amount of end-of-life failurerecords. As an example, the current average failure rate forPTs in the Portuguese distribution grid is around 0.005 failuresper PT per year. Besides, the operating conditions of each PTcan be drastically different depending on the environmentalconditions and efforts supported during its lifetime.

The average age method only uses data regarding thecomponents that have died, which is a disadvantage of themethod. With the condition assessment of the equipment itis possible to take a different approach to the developmentof the aging model. An alternative method is the probability-distribution-based methods, which take into consideration, notonly the dead components, but the operating ones as well.

Knowing the estimated mean life of an equipment, togetherwith its current age, can work as an indicator of the risk of anend of life failure. The development of a more complete agingmodel is necessary to quantify the probability of a possiblefailure. There is a common accepted graph that generallyrepresents the evolution of the probability of failure with theage of the equipment. It is the so called Basin Curve, whichis represented in Figure 5. This representation states that thefailure rates in electronic equipment are usually higher whenthe equipment is installed, due to construction or installation

4

Fig. 5: Basin curve for failure rates of eltronic equipment.Source: [15]

failures. Then, the failure rates decreases during the normaloperating lifetime. As the end of life approaches the failurerates of the equipment increases again.

A. Markov Model based aging model

Taking into account the available data, it will be proposedthe development of a model based on Markov Models Theory(MMT). These type of models can be used to determine theevolution of the health condition based only on the currentstate data. The Markov Models are a probability based decisionprocess, where the next state depends only upon the currentstate. Therefore this type of model was selected, since theonly data available regards the current health condition of thePTs. MMT has been extensively applied in other areas such ascivil engineering, where it is used to model the degradation ofbridges, pavement, steel hydraulic structures or water pipingcomponents [16–19].

On this thesis work it is proposed the use of this typeof processes to estimate the evolution of the degradationtrajectory for the health condition of the PTs, based on thecurrent health indexes. On [20], an algorithm is suggested forthe implementation of a Markov predictive model. Althoughthe proposed approach is not a complete fit for the problemat study, it is useful to understand how this kind of problemscan be modeled. In Figure 6 it is presented a diagram thatillustrates the different phases of builiding a Markov model.First the historical data is collected, then some statisticalmethods are applied, e.g. main components analysis and timeseries analysis are applied to extract important data items.After that, the HI is computed using the processed acquireddata. Then the Markovian deterioration model is identifiedand parameterized based on historical observations of theequipment health data. This process allows to establish thelong-run performance of the equipment, possibly predicting it[20].

A Markov decision process is characterized as a memorylessprocess, which in this case predicts the future condition ofequipment as a probabilistic estimate [19, 21, 22]. The MarkovChain process depends on the transition probabilities givenas Pi j [22]. Where Pi j is the probability of the equipmentevolving from state condition i to j, in a specific time interval.A set of transition probabilities can be represented by atransition matrix, P(t).

1) State Discretization: In order to implement a MarkovChain model it is necessary to define discrete intervals, which

Fig. 6: Offline Markov Model diagram. Source: [20]

in this case are the degradation levels. Fifteen degradationstates were defined from more healthy to more degraded level.The PTs depending on the current condition are inserted in oneof the defined intervals.

2) Age discretization: It was also necessary, in order toparameterize the model, to define twelve age intervals. Then,based on the degradation condition of the PTs inserted in eachage interval, different probabilities distribution functions forthe health of the corresponding transformers were defined.This probability distributions will guide the random generationassociated with the evolution of the degradation state with theaging of the PTs.

3) PDI Probabilities Distribution: For the age intervalscorresponding to younger PTs, the degradation probabilitydistribution corresponds to log-normal distributions. The meanvalue of the distribution increases with the age interval itmodels. For the older age intervals, the associated distributionis the Maximum Value Distribution. The set of the distributionsfor all the age intervals generate the stationary PDI probabilitydistribution matrix. This Matrix defines the probability for aPT to have a specific value of PDI depending on its age. Itwas considered that all the PTs above 65 years of active lifehave a probability of total degradation equal to 1.

4) Transition Matrices: The Markov Chain model is basedon the transition between states in time. This transitions area probabilistic process, which is defined by the transitionmatrices. There is one matrix per each defined age interval.All the matrices are N square matrices, where N is the numberof defined PDI states. For the computation of this matrices itwas necessary to define an optimization problem and resort toan optimization software named GAMS in order to solve it.

5) Trajectories Simulation: The objective of the first partof the thesis is to develop an aging model that can representthe evolution of the health/degradation of each PT with time.This evolution is the so-called degradation trajectory. With thedata of the PDI levels distribution according to each age, itwould be possible to generate random numbers that followthat distribution and use them as the degradation level of eachPT along each time interval of the simulation. This way eachPDI level evolution simulated for a PT would be independentof the level it had in the time period before the transition. Thismethod is not valid for the needs of this problem because itcould lead to a PT having a better PDI level from one year to

5

Fig. 7: 1st Markov transition matrix graphical example

the next, meaning the degradation would not be monotonous,which is not acceptable. Even though, in real life there is thepossibility of making interventions on the PTs, which couldlead to an improve in the health status of the transformer, thefrequency of this type of procedure is so low that it was notconsidered during this study.

In order to find an alternative that meets the need for amonotonous degradation of the PT, it was used the MarkovChains theory, which defines that the transition to a state leveldepends on its current state. In the end it was obtained atotal of 12 matrices, 15 by 15, that represent the transitionprobabilities between the 15 PDI levels for the 12 differentfive-year age intervals, up to a maximum of 60 years.

With the obtained transition matrices it is possible to sim-ulate the health condition evolution trajectories for each PT.Take as an example a brand new PT, which has an age equalto zero and consequentially a PDI level equal 1. This will beused to simulate what will the PDI level on age range 1 be.After a five-year period, it is necessary to generate a randomnumber between 0 and 1, which will define according to thevalues in the first transition matrix, what will be transition thatshould be applied and consequently the next PDI level of thePT. For the next age ranges PDI estimate the process is similar.For each new prediction it is generated a new random numberusing the Matlabs rand function. This way it is possible topredict, based on all of the collected and processed data, whatthe evolution of the PDI levels will be like. In Figure 7 it isshown a small example of the application of the 1st Markovtransition matrix to the PTs which are at PDI level zero. Theexample shows how the next level depends on the current one,which is the 1st PDI level since the PTs are brand new.

The transition matrices were computed using an optimiza-tion program named GAMS (General Algebraic ModelingSystem), which solves non linear problems with multiple con-straints in an intuitive and user friendly way. Each computedtransition matrix needs to guarantee that, when multiplied bythe PDI probabilities vector of the starting age interval, theresult is the desired PDI probabilities vector of the next ageinterval. This condition is the main constraint implementedin the optimization problem developed in order to computethese matrices. Besides this, there were others applied inorder to model the PDI evolution in correctly. Since the

optimization problem was not always feasible, there was theneed to loose some of the constraints in for some of thematrices. After computing all the matrices, it was applied atest to check if the PDI probabilities from each age rangemultiplied by the corresponding transition matrix originatedthe desired probabilities for the next age range. The resultswere then compared with the desired probability matrix inorder to find the error associated with the transition matrices,as is demonstrated by:

∀n, T Merror = [PPDI ](n−1) × [T M]n − [PPDI ](n), (1)where T Merror is the error between the initially desired valueand the obtained through the transition matrix, [PPDI ](n−1)is the PDI probability distribution vector for a certain ageinterval n, T M is the transition matrix associated with aparticular age interval n and [PPDI ](n) is the value for theprobability distribution vector obtained at Subsection ??. Theaverage absolute error for the obtained transition matrices isaround 0.02%. The sum of all the absolute errors for eachelement of the probability matrix is 3%, which is an acceptableerror, when taking into account all of the constraints and nonlinearity of the optimization problem.

B. Failure probability estimation based on the PDI

As explained before it is possible to obtain the failure indexof each PT from the PDI level. All the intermediate indexesbetween the FI and the PDI were explained in the introductionchapter, together with the way they are defined. Once thefailure indexes of the transformers are computed based onthe PDI levels generated through the aging model defined inSubsection II-B, it is necessary to obtain a failure probabilityto each PT in order to simulate the failures in the transformersstock and the correspondent ENS.

The proposed method is based in normalizing the averagevalue of the fail probability to average value of the failureindex. The idea is to assume that there is a linear relationshipbetween the average value of the FI and the historic averagefailure rate, as expressed by:

P f(per 5 year) = FI ×FR( f ail\PT\year) × 5

FI, (2)

where P f is the failure probability of each PT in a 5year period, FI is the failure index of each PT, FR is theyearly average failure rate of the installed PTs and FI isthe average failure index of the installed PTs. Notice thatin the developed model the analysis of the PDI, PHI, GHIand FI evolution is made for five-year intervals, meaning thatthe failure probability should also be computed for five-yearintervals.

For every PT installed in the grid, it is generated, in eachtime interval simulated, a random number using the Randfunction from Matlab. This function outputs a random numberbetween zero and one. If this number is smaller than the failureprobability associated with a PT in a specific time interval, itis considered that the PT has failed in that time period. Thisfailure is registered and then, depending on if is a critical

6

Fig. 8: Computation order of the simulation outputs

failure or not, a new PT can be installed as a replacementfor the old one. When a new transformer is installed, it isconsidered to have, in the beginning of the next time intervalof the simulation, zero years of age and a PDI level equal toone.

1) Outputs of the Simulator: The estimation of the PTshealth degradation and consequence failure prediction is in-corporated in the simulator in order to be possible to extracta large number of different outputs in each time period. Someof the more relevant outputs are: Average age of the installedPTs, Average PDI value, Total amount of failures, EnergyNot Supplied, Energy Not Supplied in PTs with high losspercentage, Total Strategy Cost, Cost of the ENS.

In Figure 8 it is represented in a simplified graphical formthe algorithm for the computation of the output values of thesimulator.

IV. STOCK HEALTH CONDITION SIMULATION ANDRELIABILITY EVALUATION

After developing a model to characterize the evolution ofthe PTs health condition along its lifetime and to estimatethe PTs associated probability of failure, in this section itis presented an algorithm to estimate, based on a simulationmethod, the evolution of the state of the global installed stockof PTs. This proposed simulator allows to extract informationabout several different parameters, such as the average age ofthe PTs stock or the average total ENS value for the system.In order to do this simulation, it is applied the Monte CarloMethod, which relies on repeating random sampling in orderto obtain a numeric value for a stochastic problem. This wayit is possible to use randomness to solve problems that mightbe deterministic in principle.

1) Monte Carlo Simulation Method: In order to computethe desired simulation, it is applied the Markov Chain MonteCarlo Method, which relies on repeating random samplingin order to obtain a numeric value for a stochastic problem.This way it is possible to use randomness to solve problemsthat might be deterministic in principle. The Markov Chainmodels when combined with the Monte Carlo Simulationtheory originates the so-called Markov Chain Monte Carlomethod (MCMC). This is a powerful method to generaterandom samples from a probability distribution, which can beused in the computing of statistical estimates. This method isparticularly useful in applications where one is forming anestimate based on multivariable probability distributions ordensity functions, that would not be possible to obtain ana-lytically [23, 24]. The Monte Carlo simulations are effective

when modeling situations that have a high level of uncertaintyin the inputs, for example calculating risks in business, whichis exactly the type of problem that is being addressed. In asimplistic way, this method allows to compute a numericalestimation for processes or functions that depend on randomvariables. These variables could arise organically as part ofthe modeling of a real-life system, such as a complex roadnetwork, the transport of neutrons, or the evolution of the stockmarket. In many cases, however, the random objects in MonteCarlo techniques are introduced artificiality, in order to solvepurely deterministic problems. In this case, the MCMC simplyinvolves random sampling from certain known probabilitydistributions. In either the natural or artificial setting of MonteCarlo techniques the idea is to repeat the experiment a largeamount of times (or use a sufficiently long simulation run)to obtain large quantities of interest using the Law of LargeNumbers [25]. At the end, the average values of the samplebased outputs are computed.

The common problem where MCMC is applied consistson the existence of a stochastic process that generates arandom variable X, and the goal is to compute the expectedvalue of another process, f (X), that depends on X, E[ f (X)].Usually the value for the random variable cannot be modeledby an analytic function, making it impossible to obtain adeterministic value for the E[ f (X)].

Let us assume, for example, that X is a continuous randomvector with an associated probability density function. Theproblem can then be described as

E[ f (X)] =∫

f (x)p(x)dx, (3)

where the integral is computed over the domain of X. Thesame example can be applied to discrete random variables,as the health indexes that will be studied during this work.The equivalent to the f function for the thesis will be all theoutputs computed based on the health index, such as: averageENS, failure probability per PT, average number of failures,ENS costs, among others.

Since there is a large amount of estimates obtained, it isimportant to compute the expected value and the variance ofthe outputs. Let us consider the probability density of y asf(y), then the expected value for y is

µy = E(y) =∫ +∞−∞

y f (y)dy. (4)

Considering the N trials ran, an estimator for µy can be asimple mean, which can be demonstrated to have an expectederror of zero, making it an unbiased estimator [26],

y =1N

∑yi . (5)

Since the Monte Carlo simulation involves pseudo-randomdraws of the inputs, a different results will be obtained eachtime the probabilistic analysis is performed. That is, each timethe Monte Carlo simulation ran, slightly different results for ywill be obtained. In the following analysis, it is shown that thevariability in the results of y (i.e., how much the mean estimate

7

varies between Monte Carlo simulation runs) depends on N,the number of trials in each Monte Carlo simulation.

The variance of the obtained estimates is obtained by

V(y − µy) =σ2y

N. (6)

By having an estimator for the output value and for itsvariance, it is possible to estimate the probability distributionof the error. For large values of N, the central limit theorem canbe applied to approximate the distribution of y. This theoremstates that for a sufficiently large N, the distribution of y − µywill approach a normal distribution with mean value of zeroand variance obtained by

σ2yN .

This way, it is possible to define confidence intervals, wherethe size of the interval depends on the number of N trials. Forexample, for a 95% confidence on the values of µy , and sincethe normal distribution has approximately 95% of its valueswithin ± two standard deviations of the mean, it is possibleto state

P{−1.96σy√

N≤ y − µy ≤ 1.96

σy√N} ∼ 0.95. (7)

It is not possible to calculate the real value for the aboveerror estimate, since it is depend on σy , which is unknown.Although it is possible to use an estimate for its value throughan unbiased estimator for σy2:

sy2 ≡ ωy2 =1

N − 1

N∑i1

(yi − y) (8)

The MCMC method will be vastly applied throughout thedeveloped work, in order to obtain all of the output valuesof the simulator, which depend directly or indirectly on theevolution of the health index. This index will be modeledby the Markov Model defined and parametrized based on theavailable data regarding the condition of the PTs installed inthe Portuguese grid.

The global PTs stock condition simulator presented in thissection does not analyze each PT individually, it analyses theglobal stock of PTs as a whole. This means that, if only onetrial was preformed the obtained results could be drasticallydifferent each time the model was run. Even though theamount of failures, for example, could be the same betweenexecutions, the ENS could be drastically different because thePTs that failed would probably not be the same. Consequentlythe ENS would not be the same, which would generate amassive empiric error in the obtained outputs. The variancebetween results would be rather significant if the amount ofrepetitions was small.

The Monte Carlo simulation method allows to minimizethe empiric error by making a large amount of samples. Aspreviously stated, with the increase of number of trials foreach execution of the program and for the same confidenceinterval, the error will have a tendency to decrease. It is alsoexpected that the errors associated with the simulated variablesgenerate a normal distribution around theirs mean values.

A. ENS Estimation due to Power Transformers Failures

It is important to establish a method to compute the totalamount of ENS due to failures in the PTs (HV/MV andMV/MV). The ENS value is one of the most important valuesthat can be extracted as an output from the defined model.The ENS is a computed based on the failure power, thestoppage time and the rate of failure associated with eachtransformer.

Whenever there is a failure in a PT, it takes time toreconfigure the grid schema in order to redistribute as muchas possible the power load that was associated with thedamaged transformer. After this first task is completed,thetransformer replacement process begins. Usually, the timeassociated with the redistribution of the load from the damagedPT to others, that can handle the increase in their loads, iscalled reconfiguration time and in average corresponds to 15minutes. . The replacement of a failed transformer by a newone, either it is a backup PT or a mobile station, takes around24 hours. The two time values aforementioned are the onesconsidered for simulation purposes.

ENSn = ((RT × N L) + (RP × SDL)), (9)

where ENSn is the energy not supplied associated with eachPT that fails, RT is the reconfiguration time, N L is the naturalload, RP is the replacement period and SDL is the shut download.

To obtain the an estimation for the average total ENSvalue its necessary to apply the Equation 9 to all the powertransformers and sum the absolute value from all of them. Itis important to sum the absolute value in order to correct thenegative sign that some PTs have for the ENS. This processis represented by:

ENStotal =739∑n=1|ENSn |, (10)

where |ENSn | is the absolute value of energy not suppliedfrom each PT and n is the PT index. The estimation obtainedfor the current energy not supplied due to power transformersfailure is 242.8 MVAh per year.

V. RESULTS

A. Global Stock Renewal Strategies

The developed global stock simulator allows to estimate theaverage PDI evolution for the installed stock of PTs. This toolmakes it is possible to simulate different renewal strategies forthe global stock and analyze its output parameters in order tocompare the strategies. This can be a useful tool when planningthe investment strategy for the distribution grid PTs.

On the thesis different renewal strategies were tested andare presented in detail. There are many different replacementcriteria such as the age, the PDI level, the risk associated witha failure, the remaining useful life or even hybrid criteria.

As an example, the results obtained from the simulation oftwo hybrid strategies are presented. This type of strategies canbe based in any of the above mentioned criteria or in any others

8

that can be defined in the future. The presented hybrid strategyis based on the "Over X" criteria mixed with the "Top possibleENS" criteria. The "Over X" criteria consist on replacing allthe PTs with an age above an X value. The "Top possibleENS" criteria consist in replacing the PTs with the highestprobability of failure together with the corresponding ENSconsequences. The examples presented show the combinationof the "over 55 years" together with the "top 20 possible ENS"which is called hybrid1 strategy, together with a differentstrategy named hybrid2, which corresponds to the "over 60years" strategy together with "the top 40 possible ENS".

From the plots presented in Figures 9 to 13, it is possibleto see that, in terms of the global installed stock, there is amajor improvement in the parameters related with the averagehealth of the installed PTs, namely the average PDI, age andnumber of failures per time interval, when compared with the"over 60 years" strategy. On the other hand, there is a biggeroverhead related with the replacement of transformers, whichhas a similar profile to the one from the "top 60 possible ENS".

Fig. 9: Average PDI evolution with time

Fig. 10: Average PTs age evolution with time

In the Tables I and II presented in the appendix it issummarized an ENS and cost analysis of the global resultsalong the time frame of the simulation. From there, it ispossible to see that in terms of ENS, the results from the hybridstrategies are better than the ones from the "over 60 years" andthe "Top 60 ENS" strategies. On the cost analysis it is clear thatwith the hybrid strategies the costs are lower than with the "top60 ENS" strategy but higher than for the "over 60 years" one.Notice that the value of failures in terminal branches is slightlylower for the hybrid strategies but the ENS in those branches

Fig. 11: Evolution of the average amount of failures with time

Fig. 12: Average number of new PTs installed per each timeinterval (Failures + strategic replacements)

is significantly lower, which can be a significant criteria whenevaluating the strategies to implement in the grid. Anotherinteresting result is the weight that the ENS appreciation hasin the cost of each strategy. Even though the general cost ofthe "over 60 years" strategy is significantly lower than othersstrategies, the weight of the ENS in the global cost is aroundthe double than for the others analyzed strategies. This is anindicator of how the hybrid strategies, are more efficient inimproving the ENS than the "over X years strategy".

The results obtained for these hybrid strategies prove thatthe most efficient strategies will be probably obtained througha strategy developed based in multiple criteria and parameters.

1) Pareto Optimality Frontier: Any renewal strategy isassociated with a cost. The cost results from the sum ofdifferent parts, such as: the investment in new equipment, thecost of fixing damaged equipment and the valuation of theENS.

The decision about the best strategy to be implemented will,most likely, be defined mainly based on these two values: ENSand total strategy cost. The decision making process is usuallycomplex, which is why it is important to present the data tosupport the decision in a clear and intuitive form. Even thoughthe final goal is to find the best strategy to be implemented,it will not always be possible to achieve the global optimalsolution. The decision makers might not be interested in theoptimal solution, particularly if these solutions are sensitiveto the variable perturbations which cannot be avoided in reallife scenarios. In such cases, decision makers are interested infinding the so-called robust solutions, which are less sensitive

9

Fig. 13: Average ENS value and respective monetary appreci-ation per each time interval

Fig. 14: Pareto Frontier example. Source: [30]

to small changes in variables [27].The problem being analyzed is a multi-objective opti-

mization problem (MOO), where the objectives are the totalstrategy cost and the ENS output values. A particular set ofoptimal solutions is referred to as the Pareto optimal set orPareto frontier. By definition, Pareto solutions are consideredoptimal, since there are no other designs that are superior inall objectives [28, 29]. The Pareto Optimality Frontier graphis a useful tool when comparing different solutions, in orderto make a decision about which one is more efficient. If asolution is part of the frontier, it means that there is no otherdiscrete solution that can provide the same output with thesame value for one of the variables and a lower value for theother.

The results obtained from the different strategies simulated,allow to withdraw some conclusions regarding the efficiency ofthe strategies. To do so in Figure 15 is presented a graph withthe Pareto efficiency frontier obtained through the simulationof the above mentioned strategies.

The Pareto efficiency frontier shows that there are manysimulated strategies that are not efficient, which are the onesto the right and above the blue line that represents the frontier.The strategies on top of the frontier are the ones that aremore efficient in terms of the binomial ENS-total cost. The"over 40 years" and the "zero strategy" have a very highvalue for one of those two parameters, which in this caseare the total cost for the "over 40 years" strategy and theENS for the "zero strategy". On the other hand both hybridstrategies, together with the "over 60 years" strategy are themost balanced strategies, since there is an equilibrium betweenthe ENS and the total cost associated with the strategies along

(a) Average PDI evolution with time

Fig. 15: ENS values as a function of the strategy cost forthe simulated renewal strategies together with the respectivePareto Efficiency Frontier

the total simulation time frame.Another interesting result obtained through all the simulated

strategies is the fact that the amount of PTs that are repairedafter failing is not significant, since the recovered transformershave an average weight permanently below 2% of the totalstrategy cost. This happens since the vast majority of thetransformers fail after 50 years of life, which makes themunavailable for recovery, while, on the other hand, only halfof the ones that fail before the 50 years are recovered. Makingthis result exactly what was expected to be obtained.

VI. CONCLUSION

A. Achievements Summary

The thesis work resulted in the development of an agingmodel to estimate the evolution of the health condition of eachPT with time. The model was developed using the currentcondition data from over seven hundred PTs installed in thePortuguese distribution grid. The health condition is assessedthrough physical tests performed by EDPD and is representedby a health degradation index (PDI).

For the development of the aging model, it was necessaryto define a stochastic process based on Markov Chains. Thesetype of processes are particularly useful when there is a needto define the probability for the evolution of states, dependingonly upon the current state. In this case, the current state is adefined health index (PDI) associated with each PT. The nextstate is a PDI value associated with the same transformer, aftera five year period. The use of a Markov Model was essential,since the only available data regarding the PTs was the staticcurrent health condition. This is, there was no informationrelated with the historical dynamic evolution of the healthcondition of the transformers.

The developed Markov Model allowed to simulate of theso-called PDI trajectories, which are an estimate of the waythe PDI value of each transformer evolves with time. Theestimated PDI level can then be converted into a failureindex, defined according to the EDPD criteria, and thenconverted into a failure probability. The relationship between

10

the failure index and the failure probability is obtained througha normalization between the current PTs average failure indexand the average historical failure rate.

The Markov Model was used as a base for the developmentof the global stock simulator. This simulator is a programdeveloped to test the evolution of the average condition ofthe installed PTs. This program simulates the evolution ofthe average health condition of the installed PTs, based onthe samples obtained from the Markov Model estimation.Besides the health condition of the PTs, the simulator allowsto estimate other variables, such as: average ENS per year;total cost of the implementation of a renewal strategy; averagenumber of failures per time interval, among others.

The three initially proposed objectives were achieved suc-cessfully. The developed aging model for each individualtransformer models, on a feasible way, the evolution of thehealth state of the PTs. The model can be improved in thefuture, if dynamic historical data regarding the PTs is madeavailable. The global stock evolution simulator turned out tobe versatile, allowing to test and simulate any desired renewalstrategy and retrieving the corresponding estimation for theoutputs values. This can be used by EDPD in the future asa tool to provide guidance when discussing and defining therenewal strategy to be implemented.

B. Practical Implications

The most relevant future implication drawn from this disser-tation are the values obtained from the simulations regardingthe evolution of the grid PTs health status. At the end, fromall of the simulations it is possible to conclude that there willbe an extreme need for investment in the renewal of the stockof PTs in the grid, in order to avoid a possible collapse in thedistribution service quality. 1

There are over seven hundred PTs installed in the grid. It isa simple conclusion that, considering an optimistic lifespan of55 years for each, the replacement rate in order to maintain thegrid operational, without taking into account the early failures,would be of over sixty five transformers per five year interval.

This work can indicate to EDPD the need to develop along term strategic plan for the grid renewal, in order to avoidmajor problems in the service and reduce the correspondinginvestment effort. The main contribution of this work is thepossibility to test the implementation of different renewalstrategies. This way it is possible to develop less expensiveand more efficient strategies, regarding the associated ENSbased on the retrieved results.

The main differences in the results of the implementedrenewal strategies are the time periods when the investmentin the grid will have to be applied. Although there arealso some differences in the way the investment is made,which consequently results in some differences in the financialburden associated with each strategy. At the end, the mainconclusion is that there is no alternative to the investment inthe grid renewal. The real question is how and when to do

1phenomenon happening in Infraestruturas de Portugal, which is the Por-tuguese company responsible for managing the trains infrastructures, whichsaw a collapse in the service quality due to lack of investment.

this investment, which, based on the performed simulations,should be a custom made hybrid strategy.

The simulations results also show that, if there is not animplementation of an adequate renewal strategy, it is likelythat in the future there might occur a large amount of failuresin a same time period. A major concern, besides the ENS andthe cost associated with these possible failures, is the lack ofcapacity from the suppliers to provide the amount of new PTsneeded to replace the ones that will fail in a short time interval.It might be difficult to find companies capable of supplyingover a hundred PTs in a five year period, which should be aconcern for EDPD, when developing a grid renewal plan.

At the end, this work can be used as a support for thedecision-making process regarding the grid renewal strategy,together with the creation and development of the long termstrategic plan of investment in the renewal of the distributiongrid PTs.

C. Future Work

The work developed for this thesis can be used as a basefor future studies regarding the evolution of the distributiongrid health status.

1) Aging Model Improvements: In order to improve the re-liability of the aging model developed for each PT individuallythere are some modification that can be made. If, in the future,it is possible to obtain data regarding the dynamic evolutionof the PHI of each individual transformer it would be possibleto improve the defined Markov Process by better defining therelationship between the age and the health states transitionsfor the transformers. Instead of defining this relationship basedon the static data of the PHI values, it would be possible toanalyze dynamic historical data and better define this process.

The estimation for the current ENS value can increase its re-liability with the obtainment of data regarding the failure ratesof each individual PT. This would be a major improvement tothe current used data, regarding the average failure rates ofthe PTs per county. This new data would also be extremelyimportant to more accurately define the relationship betweenthe failure index of each transformer and its failure probability.This means that this new data would drastically improve thefeasibility and reliability of the developed aging model.

In the developed model it is considered that all the mal-functions result in a transformer failure. If there was more dataregarding the type of failures, and corresponding consequencesfor the transformer health status, the aging model could berefined in order to reflect, in a more realistic way, the behaviorof the transformers.

2) Global Installed Stock Analysis: In the future, the de-veloped model can be used to study custom made renewalstrategies in order to obtain a more efficient approach, than theones presented in this thesis. Based on the developed GlobalStock Simulator, it is possible to define any desired criteriafor the renewal strategies and test the influence on the outputresults. The ideal strategy might be difficult to find, since thereare an infinite number of possible criteria that can be defined.It might be possible, although, to obtain new non dominatedpoints and with that redefine a new Pareto Optimallity Frontier.

11

APPENDIX A

TABLE I: Hybrid type strategy ENS analysis

Strategy ENS(TVAh)

Yearly ENS(MVAh/year)

Number offailures (#)

Non redundantfailures (#)

ENS due to nonredundant failures

(TVAh)

New PTs(#)

Hybrid1 6.27 179.21 119 7.0 1.02 596Hybrid2 5.52 157.75 122 7.0 0.91 661Over 60 8.83 252.37 135 9.0 1.62 462

TOP 60 ENS 8.10 231.46 383 23.0 1.45 762

APPENDIX B

TABLE II: Hybrid type strategy cost analysis

Strategy Global Cost(Me)

ENS Cost(Me)

ENS Cost(%)

New PTsInvestment

(Me)

New PTsInvestment (%)

Repaired PTsCost (Me)

RepairedPTs Cost

(%)Hybrid1 102.1 6.8 6.69% 93.3 91.42% 1.9 1.89%Hybrid2 106.1 6.0 5.66% 98.3 92.65% 1.8 1.69%Over 60 66.8 9.0 13.49% 55.9 83.68% 1.9 2.84%

TOP 60 ENS 128.2 7.7 5.98% 118.8 92.68% 1.7 1.34%

REFERENCES[1] Regulamento de Acesso às Redes e às Interligações do Setor Elétrico,

EDP Distribuição - Energia, S.A., Mar. 2018.[2] D. Shenoy and B. Bhadury, Eds., Maintenance Resource Management:

Adapting Materials Requirements Planning (MRP). Bristol, PA, USA:Taylor & Francis, Inc., 1997.

[3] A. Jardine and J. Buzacott, “Equipment reliability and maintenance,”European Journal of Operational Research, vol. 19, no. 3, pp. 285 –296, 1985.

[4] P. LOWE and W. LEWIS, “Reliability analysis based on the weibulldistribution: an application to maintenance float factors,” InternationalJournal of Production Research, vol. 21, no. 4, pp. 461–470, 1983.

[5] M. A. K. Malik, “A note on the physical meaning of the weibulldistribution,” IEEE Transactions on Reliability, vol. 24, no. 1, pp. 95–95,1975.

[6] R. C. Woodman, “Replacement strategies,” Journal of the OperationalResearch Society, vol. 18, no. 2, pp. 193–195, 1967.

[7] D. Belyi, E. Popova, D. P. Morton, and P. Damien, “Bayesian failure-ratemodeling and preventive maintenance optimization,” European Journalof Operational Research, vol. 262, no. 3, pp. 1085 – 1093, 2017.

[8] I. G. N. S. Hernanda, A. C. Mulyana, D. A. Asfani, I. M. Y. Negara, andD. Fahmi, “Application of health index method for transformer conditionassessment,” in TENCON 2014 - 2014 IEEE Region 10 Conference, Oct2014, pp. 1–6.

[9] G. C. Stone, “The statistics of aging models and practical reality,” IEEETransactions on Electrical Insulation, vol. 28, no. 5, pp. 716–728, Oct1993.

[10] X. Zhang and E. Gockenbach, “Asset-management of transformers basedon condition monitoring and standard diagnosis [feature article],” IEEEElectrical Insulation Magazine, vol. 24, no. 4, pp. 26–40, July 2008.

[11] “Incorporating aging failures in power system reliability evaluation,”IEEE Transactions on Power Systems, vol. 17, no. 3, pp. 918–923, Aug2002.

[12] J. A. Nachlas and C. R. Cassady, “A general framework for modelingequipment aging,” in 2010 Proceedings - Annual Reliability and Main-tainability Symposium (RAMS), Jan 2010, pp. 1–6.

[13] A. Chen and G. Wu, “Real-time health prognosis and dynamic preventivemaintenance policy for equipment under aging markovian deterioration,”International Journal of Production Research, vol. 45, no. 15, pp. 3351–3379, 2007.

[14] and E. Vaahedi and P. Choudhury, “Power system equipment aging,”IEEE Power and Energy Magazine, vol. 4, no. 3, pp. 52–58, May 2006.

[15] M. Ali, “Reliability of information and communication technology (ict)equipment in power system – study review,” 09 2014.

[16] A. A. Butt, M. Y. Shahin, K. J. Feighan, and S. H. Carpenter, Pavementperformance prediction model using the Markov process, 1987, no. 1123.

[17] S. Ranjith, S. Setunge, R. Gravina, and S. Venkatesan, “Deteriorationprediction of timber bridge elements using the markov chain,” Journalof Performance of Constructed Facilities, vol. 27, no. 3, pp. 319–325,2011.

[18] Q. Li, T. Zhao, L. Zhang, and J. Lou, “Mechanical fault diagnostics ofonload tap changer within power transformers based on hidden markovmodel,” IEEE Transactions on Power Delivery, vol. 27, no. 2, pp. 596–601, 2012.

[19] Y. Jiang, M. Saito, and K. C. Sinha, Bridge performance predictionmodel using the Markov chain, 1988, no. 1180.

[20] A. Chen and G. S. Wu, “Real-time health prognosis and dynamicpreventive maintenance policy for equipment under aging markoviandeterioration,” International Journal of Production Research, vol. 45,no. 15, pp. 3351–3379, 2007.

[21] R. Yates and D. Goodman, Probability and Stochastic Processes, J. W.. S. Inc, Ed., 2005, no. 2.

[22] C. E. Shannon, “A mathematical theory of communication,” Bell systemtechnical journal, vol. 27, no. 3, pp. 379–423, 1948.

[23] W. K. Hastings, “Monte carlo sampling methods using markov chainsand their applications,” 1970.

[24] J. C. Spall, “Estimation via markov chain monte carlo,” IEEE ControlSystems Magazine, vol. 23, no. 2, pp. 34–45, 2003.

[25] D. P. Kroese, T. J. Brereton, T. Taimre, and Z. I. Botev, “Why the montecarlo method is so important today,” 2014.

[26] M. Ruiz Espejo, M. Delgado Pineda, and S. Nadarajah, “Optimalunbiased estimation of some population central moments,” METRON,vol. 71, no. 1, pp. 39–62, Jun 2013.

[27] K. Deb and H. Gupta, “Searching for robust pareto-optimal solutions inmulti-objective optimization,” in Evolutionary Multi-Criterion Optimiza-tion, C. A. Coello Coello, A. Hernández Aguirre, and E. Zitzler, Eds.Berlin, Heidelberg: Springer Berlin Heidelberg, 2005, pp. 150–164.

[28] V. Pareto, Cours d’économie politique. Librairie Droz, 1964, vol. 1.[29] K. Miettinen, “Nonlinear multiobjective optimization,” JOURNAL-

OPERATIONAL RESEARCH SOCIETY, vol. 51, no. 2, pp. 246–246,2000.

[30] C. A. Mattson and A. Messac, “Pareto frontier based concept selectionunder uncertainty, with visualization,” Optimization and Engineering,vol. 6, no. 1, pp. 85–115, 2005.