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A New Methodology for Determining the Moisture

Diffusion Coefficient of Transformer Solid Insulation

D. F. GarcaSchool of Electrical and Electronic Engineering

Universidad del Valle

Cali, Colombia

diego.garcia@correounivalle.edu.co

B. Garca, J. C. Burgos and R. VillarroelElectrical Engineering Department

Carlos III University

Legans (Madrid), Spain

bgarciad@ing.uc3m.es,jcburgos@ing.uc3m,

rvillar@ing.uc3m

AbstractIn this paper, a new methodology for determining

the moisture diffusion coefficient of transformer solid insulation

is presented. Unlike classical methodologies, the proposed one

does not require to measure the evolution of the moisture

distribution inside the material under test, but the overall

moisture evolution is measured, which is easier to do from theexperimental standpoint. In addition, the methodology includes

an optimization process based on genetic algorithms, with an

objective function that incorporates a moisture diffusion model

solved by the finite element method.

KeywordsCellulose insulations; moisture; diffusion

coefficient; genetic algorithms

I. INTRODUCTION

Moisture inside a transformer may mainly appear as aconsequence of external contamination and because of acellulosic insulation degradation process and also, to a lesser

extent, by oil oxidation [1]. Due to the hydrophilic nature ofcellulosic insulation, and hydrophobic nature of oil, moisturemainly remains in the solid insulation. However, thedistribution of moisture between oil and the cellulosicinsulation is not static, but depends on the transformeroperating conditions, and mainly on the operation temperature.

Moisture negatively affects the life expectancy of powertransformers and can lead to hazardous conditions. Tominimize the amount of water in the insulation of newtransformers, they are subjected to a drying process in thefactory, previous to the impregnation of the insulation with oil.Drying processes must be sometimes repeated during atransformers life to remove the moisture generated by the

effect of aging and by contamination. These drying processesare usually performed in the field.

Understanding and properly estimating the moisturedynamics in power transformer insulation is essential forimproving the manufacturing process, operation andmaintenance of these equipments. Moisture dynamics insidethe cellulose insulation can be estimated using a mathematicalmodel of diffusion based on Fick's second law, which for anunidirectional diffusion can take the form of (1), where D isthe effective diffusion coefficient (m

2s

-1), c is the local

moisture concentration (% of the dry material weight) and x

is the coordinate in the direction of the water movement.

c cD

t x x

=

(1)The main parameter of these mathematical models based

on Ficks second law is the moisture diffusion coefficient. Theaccuracy of the simulation models estimations depends on thevalue of the moisture diffusion coefficient.

The experimental determination of the moisture diffusioncoefficient in cellulose insulation is a difficult task, mainly

because of its dependence with moisture concentration. Theexperimental classical methodologies, used in solid materials,for determining the moisture diffusion coefficient, withdependence on moisture concentration, require measure theevolution of the moisture distribution inside the material under

study by means of complicated experiments or by usingsophisticated equipments like nuclear magnetic resonance(NMR) or neutrons images (NI).

In this work, is presented a methodology for determiningthe moisture diffusion coefficient of transformer solidinsulation that is experimentally easier to implement than theclassical ones, because only require to measure the overallmoisture evolution of the material under test, during a drying

process. The methodology includes, in addition to the dryingexperiments, an optimization process based on geneticalgorithms.

The proposed methodology was used in the determinationof the moisture diffusion coefficient of Kraft-paper and

pressboard insulations impregnated and non-impregnated byoil [2], [3]. The diffusion coefficients obtained with the

proposed methodology allowed a better estimation of themoisture dynamics in cellulosic insulations.

II. TECHNIQUES USED FOR DETERMINING DIFFUSIONCOEFFICIENTS OF SOLID MATERIALS

All experimental methodologies used to estimate thediffusion coefficient of solid materials include three steps:

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Perform an experiment that involves moisturedynamic inside the material under study.

Simulate the experiment by means a mathematicalmodel in which the diffusion coefficient is the mainmodel parameter.

Finding the diffusion coefficient value by successivefitting of the simulation results with respect to the

experimental ones.

These three steps can be carried out by using differenttechniques. There are three main criteria by mean of whichthese techniques can be classified:

The type of experiment performed.

The basics of the mathematical model used.

The method used to solving the mathematical model.

In Fig. 1 is shown a tree where the different techniquesused for determining the moisture diffusion coefficients insolid materials are classified.

Experimental methods for D

Gravimetrics

Permeation

Steady state

Time lag

Type of experiment

Adsorption and

desorption kinetics

Drying

Drying tunnel

TGA

Moisture profiles

Sectioning of the specimen

Radiographic techniques

Gamma rays

Neutron images

Nuclear magnetic resonance

Chemical analysis

Karl Fischer

Dielectrometry sensors

Others: colors contrast, radioactive tracers, etc.

Mathematical

basis

Moisture flux

(Ficks first law)

Moisture diffusion

(Ficks second law)

Approximation

Simplified

Constant diffusivity

Variable diffusivity

Method of solution to the

mathematical model

Regular regime

Finite differences

Direct

Numerical methods

Finite elements

Fig. 1.Techniques used for determining the moisture diffusion coefficient insolid materials.

The experiments types can be gravimetric, where themoisture evolution is evaluated in overall form; or moisture

profiles experiments, where is necessary determining theevolution on time of the moisture distribution inside thematerial under test. The first type is suitable in which casewhere D is independent of the moisture concentration whilethe second type is appropriate when it depends on moistureconcentration.

As to the basis of the mathematical model, the moistureflux o Ficks first law should be used only when the diffusioncoefficient has not dependence on the moisture concentration.Otherwise should be use models based on Ficks second law.

If the mathematical models are based in moisture flux,then they are expressed by means of an analytical expressionand therefore can be solved by direct methods. On the otherhand, if the mathematical models are based on moisturediffusion, they may be solved by the so-called approximationmethods or by numerical methods.

When the diffusion coefficient has dependence onmoisture concentration, the diffusion model consists in asecond order and non-linear partial derivate equation. Thedirect resolution method consists in approximating thediffusion model using an analytical expression, what canintroduce a high uncertainty when the moisture diffusioncoefficient is calculated. Therefore in this case it is advisableto solve the mathematical diffusion model using a numericalmethod.

III. TECHNIQUES FOR DETERMINING D OF CELLULOSEINSULATIONS USED BY OTHER AUTHORS

Several authors used some of the techniques shown inFig. 1, to determine the moisture diffusion coefficient of

cellulose materials [6-9]. In Table I, the techniques used bythese authors are summarized.

TABLE I. TECHNIQUES USED FOR DETERMINING D

Author

Experiments type Basis of the

mathematical

model

Solution

methodGravimetricsConcentration

profiles

Permeation

Dissection

Karl-Fischer

tritation

Dielectrometrics

Ficksfirstlaw

Fickssecond

law

Direct

Analytical

approximation

Numerical

approach

Ast X X X

Guidi X X X XHoweandAsem

X X X X

Fossa X X

Du X X X

Li X X X

a.The detail of the experiments used for t his author not is explained in his work.

Ast [6] used a simple experiment of permeation thatallowed to quantify the moisture flux through a sheet of Kraft

paper. For this reason he used a model based on Ficks firstlaw, which was solved by direct form. However, this modeldoes not represent appropriately the moisture dynamics in this

kind of hygroscopic material. Additionally in his model, heassumed a linear distribution of moisture along the paperthickness which increases the uncertainty in determining themoisture diffusion coefficient.

Because of the fact that the moisture diffusion coefficientof cellulose materials depends on moisture concentration, themathematical models based on Ficks second law are moreaccurate, and therefore the others authors used this techniquein their works. However this approach requires using someexperiment aimed to determine the moisture profile of thetested sample material. This can make the experimental part,

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the more complex step in the determination of the moisturediffusion coefficient.

As was shown in Table 1, the type of experiment morecommon for obtaining the moisture profiles consists indetermining the local moisture content in some specific pointsinside the material under test, during drying or wetting

processes. This may be done by dissection of the sample, andlater by determination of the moisture concentration using theso-called Karl-Fischer chemical method. However, thistechnique is quite complicated and requires care enough toavoid having inaccurate determinations.

The other type of experiment allowing to obtain themoisture profiles was the one employed by Du [8] that used aninterdielectrometric sensor that related the moisture contentwith an electrical variable, in this case the complexcapacitance. The problem of this type of technique is that thesensor requires a previous calibration to achieve a correctrelation between the moisture content and the electricalvariable.

IV. METHODOLOGY PROPOSED FOR DETERMINING D INCELLULOSE INSULATIONS

As was aforementioned, the classical approaches fordetermining of the moisture diffusion coefficient require touse a diffusion model based on Ficks second law, solved by anumerical method, and otherwise an experiment aimed toobtain the moisture profiles. These conditions can make moredifficult to determine the moisture diffusion coefficient ofcellulose insulations.

The three steps of the proposed methodology to obtain themoisture diffusion coefficient are:

Measuring the global moisture evolution, or average

moisture concentration ( mC ) of the test materialsample during a drying process, i.e. to obtain thedrying curve.

Simulating the experiment by means of a diffusionmodel solved by the finite element method (FEM) inwhich the diffusion coefficient is the main parameter.

Finding the diffusion coefficient value by using anoptimization process based in genetic algorithms(GA).

A. Determination of drying curvesThe drying curves, as shows Fig. 2, can be determined in

different ways depending on the insulation type. When theinsulation is non-impregnated by oil, the drying curve can beobtained by means of a gravimetric experiment, for exampleusing a thermogravimetric analizer (TGA), which records thelost of mass of the wet insulation sample during an isothermaldrying process.

In the case of impregnated cellulose insulations, the dryingcurve can be got by measuring the moisture concentration atdifferent times during a drying experiment with the so-calledhot-oil drying method. This method consists in circulating hotand dry oil, over the wet insulations samples.

Finally, to find the dependence of the moisture diffusioncoefficient on temperature and insulation thickness isnecessary to obtain the drying curves at different dryingtemperatures, and also over samples of different thickness.

0 50 100 150 200 250 300 350 4000

1

2

3

4

5

6

Time (min)

Cm(

%)

Fig. 2.Drying curve of non-impregnated Kraft-paper insulation sample of3 mm thick, drying at 60 C.

As can be seen, the proposed experiments do not require tomeasure the evolution of the moisture distribution inside thematerial under test, but the overall moisture evolution ismeasured, which is easier to determine from the experimental

point of view.

B. Simulating the experiment by the FEM drying modelThe second step of the proposed methodology consists in

simulating the drying experiment by means of a FEM dryingmodel.

In the FEM drying model the sample of the material under

study is represented by means of a one-dimensional geometry,that is the thickness in the direction that the water is movinginside the sample during drying.

Boundary conditions in the FEM model depend of theparticular conditions of the experiment, although the mostsuitable can be the equilibrium moisture concentration in theinterface between the insulation sample and the surroundingmedia.

Finally the material is characterized by the moisturediffusion coefficient that can be represented by a general

expression as the one shown in (2). In that expression k is a

dimensionless factor and0

D is the pre-exponential factor in

(m

2

s

-1

).

0

k cD D e

= (2)

If the moisture diffusion coefficient has dependence onvariables as the insulation temperature and thickness, this

dependence should be include in the parameter0

D .

C. Optimization processsThe last step of the proposed methodology is to determine

the value of the moisture diffusion coefficient. This is carried

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out by means of the optimization process based in the GeneticAlgorithms which are illustrated in Fig. 3.

Fig. 3.Flowchart of the optimization process.

The mentioned optimization process works as follows:

1. By means of genetic operations, the GA optimization

function generates different pairs of parameters0

D

and k.

2. Each pair of0

D and k values is introduced into the

FEM drying model obtaining from this, them

C values

of an estimated drying curve.

3. The GA function compares the estimated drying curvewith the experimental curve to determine if the

obtained values of0

D and k are accurate. Otherwise,

another pair is evaluated until the optimization processconverges.

Finally, the values of0

D and k obtained for each

experimental drying curve are correlated with the dryingtemperature and sample thickness, and the mathematicalexpression of the moisture diffusion coefficient is obtained.

The proposed methodology has been successfully appliedto the determination of moisture diffusion coefficients of Kraf-

paper as can be seen in [2, 3].

V. CONCLUSIONS

The methodology proposed in this work, for determiningthe moisture diffusion coefficient of cellulose insulations iseasier to implement than the classical approaches, becauseonly requires to quantify the evolution of the global moisturedesorption in the sample under test, instead of determining themoisture profiles, which is more complicated from theexperimental point of view.

The moisture diffusion coefficients determined by theproposed methodology are accurate enough because a modelbased on Ficks second law was used in their determinationsolved with a numerical technique based on the finite elementsmethod.

ACKNOWLEDGMENT

This work has been supported by the Spanish Governmentby means of the projects DPI2009-07093 and DPI2012-35819.

REFERENCES

[1] Cigr Brochure 349. Moisture equilibrium and moisture migrationwithin transformer insulation systems. Working Group A2. 30; Cigr2008.

[2] D. F. Garca, B. Garca, J. C. Burgos and N. Garca-Hernando.Determination of moisture diffusion coefficient in transformer paperusing thermogravimetric analysis. International Journal of Heat andMass Transfer, 55(4), pp. 1066-1075. 2012.

[3] D. F. Garca, B. Garca and J.C. Burgos. Determination of MoistureDiffusion Coefficient for Oil-Impregnated Kraft-paper Insulation.International Journal of Electrical Power & Energy Systems,unpublished.

[4] P. F. Ast, Movement of moisture through a50p281 kraft paper (dry andoil impregnated. General Electric, Test report HV-ER-66-41, 1966.

[5] W. W Guidi and H, P, Fullerton. Mathematical methods for predictionof moisture take-up and removal in large power t ransformers. Presented

atProceedings of IEEE Winter Power Meeting. 1974.

[6] A. S. Asem and A. F. Howe. Drying of power-transformerinsulation.IEE Proceedings Generation, Transmission andDistribution [See also IEE Proceedings-Generation, Transmission andDistribution] 129(5), pp. 228-232. 1982.

[7] S. D. Foss and L. Savio. Mathematical and experimental analysis of thefield drying of power transformer insulation.IEEE Transactions onPower Delivery. 8(4), pp. 1820-1828. 1993.

[8] Y. Du. Measurements and modeling of moisture diffusion processes intransformer insulation using interdigital dielectrometry sensors. Ph.Ddissertation Dept. of Electrical Engineering and Computer Science,Massachusetts Institute of Technology. 1999.

[9] J. Li, X. Chen, Z. Zhang, R. Liao and L. Yang. Characteristics ofmoisture diffusion in vegetable oil-paper insulation.Gaodianya Jishu/High Voltage Engineering36(6), pp. 1379-1384. 2010.