Volume 10, Issue 4 p. 335-343
Research Articles
Free Access

Particle tracing modelling on moisture dynamics of oil-impregnated transformer

Yi Cui

Yi Cui

School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, 4072 Australia

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Hui Ma

Hui Ma

School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, 4072 Australia

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Tapan Saha

Corresponding Author

Tapan Saha

School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, 4072 Australia

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Chandima Ekanayake

Chandima Ekanayake

School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, 4072 Australia

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Daniel Martin

Daniel Martin

School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, 4072 Australia

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First published: 01 July 2016
Citations: 6


An accurate estimation of moisture in oil-impregnated cellulose of a transformer is difficult due to the complex moisture dynamics inside the transformer, which is highly influenced by the temperature and dimension of the transformer. In this study, a novel particle tracing method is proposed for estimating the moisture in transformers. Different from the conventional approaches using Fick's diffusion law, the proposed method formulates moisture diffusion from a microscopic view of water particles’ motion. Especially, the transmission probability of water particles is employed to correlate the microscopic particles’ motion with the macroscopic moisture distribution. The proposed particle tracing method is applied to model moisture dynamics in both pressboard specimens and a prototype transformer. Extensive moisture diffusion and ageing experiments are carried out to verify the proposed method.

1 Introduction

Life expectancy of a power transformer is highly dependent on the condition of its cellulose materials [1]. However, cellulose can eventually degrade due to electrical, thermal and mechanical stresses. Moisture is one of the most harmful by-products of cellulose degradation. It can reduce the electrical and mechanical strength of the cellulose and also can further accelerate the cellulose degradation. Therefore, it is necessary to accurately estimate the moisture content in transformers and help utilities make informed decisions on their transformers’ operation and maintenance schedules.

During transformer operations, load condition and ambient temperature are always changing. This leads to moisture dynamics in the transformer, including (i) moisture exchanges at the interface between the cellulose and oil due to the vapour pressure difference and temperature variation; and (ii) moisture moves inside the cellulose caused by moisture gradient. Moisture dynamics is a complex heat transfer and mass diffusion process. However, because the time constant of heat transfer is considerably smaller than that of diffusion, moisture dynamics is usually modelled as a diffusion process.

Many researchers studied moisture diffusion using Fick's diffusion law and determined diffusion coefficients by conducting experiments on pressboard samples [2-9]. Instead of adopting Fick's diffusion law, this paper proposes a novel particle tracing method to estimate moisture of oil-impregnated cellulose in transformers. In particle tracing method, moisture diffusion is modelled from a microscopic perspective of water particles’ movements. Water particle is a conceptual entity for describing collective and dynamic behaviour of bunch of water molecules. It contains a huge number of water molecules; however, its physical size is small enough that can be treated at microscopic level.

In particle tracing method, particles’ motion is governed by certain motion principle and the trajectory of each particle is estimated and directly visualised. This method is exemplified in estimating moisture of both pressboard specimens and oil–cellulose system in a prototype transformer. The estimation results are validated by extensive moisture diffusion and accelerated ageing experiments.

This paper is organised as follows. Theories of moisture diffusion are briefly reviewed in Section 2. Section 3 details the formulation of the particle tracing method. Section 4 applies the proposed method to estimate moisture distribution in both pressboard specimens and prototype transformer. This paper is concluded in Section 5.

2 Moisture diffusion in oil-impregnated cellulose of transformers

Moisture diffusion is a water molecule movement process governed by the random motion of individual molecule. In cellulose material, the bindings of water molecules and cellulose cell walls are different and they are affected by local moisture concentration. Therefore, in cellulose water molecules are not equally free to migrate.

In the literature, moisture dynamics is commonly modelled by Fick's law as [2]
where W(x, t) is the moisture content of the cellulose at position x and the tth time step. D denotes the diffusion coefficient [2] which can be expressed as
where D0 = 1.34 × 10−13 m2/s and k = 0.5, Ea is the activation energy which equals 8074 kJ/mol [2], T denotes the measured temperature in Kelvin and T0 equals 298 K.

Extensive studies have been carried out to determine the diffusion coefficients of cellulose (paper and pressboard). Ast [3] adopted permeation method for determining the diffusion coefficients of kraft paper at different moisture levels and temperatures. Guidi and Fullerton formulated a relationship between the diffusion coefficients and the local water content and temperature [4, 5]. Howe [5] investigated the diffusion coefficients for different types of cellulose materials. Asem found the diffusion coefficients for oil-impregnated pressboard [6]. Foss verified the parameters of diffusion coefficients for both impregnated and non-impregnated kraft paper by using other researchers’ data [4]. By using dielectrometric sensors, Du [7] studied the diffusion coefficients for non-impregnated pressboard. Recently, a series of diffusion coefficients considering pressboard thickness was reported by García [8, 9].

However, considerable difficulties still remain in applying the above results to transformers. This is because transformers may be constructed by different cellulose materials and may have a heterogeneous temperature distribution during their operation. Moreover, Fick's law in one dimension (1D) may not be able to fully describe the diffusion process in the cellulose of a transformer. In addition, for 1D and 2D finite-element modelling (FEM) of moisture distribution, the performance is normally satisfied and without the problem of numerical instabilities. However, the 3D FEM may encounter some difficulties. The first difficulty is related to the Peclet number (refer to Section 3.1). The second difficulty is that 3D FEM models require high level of details about transformer design.

3 Particle tracing method for modelling moisture diffusion

3.1 Formulation of particle tracing method

Particle tracing method employs numerous small quantities (particles) and their collective dynamic behaviour to model the mass transfer and diffusion process [10]. Compared with FEM method, it has several advantages. First, it is not sensitive to the Peclet number in modelling mass transfer process. Peclet number is the ratio between the energy transferred by the fluid convection and that by the fluid conduction. If Peclet number is large (generally above 1000) the FEM may not be applicable to model mass transfer. The Peclet number may reach 107 in 3D modelling of moisture dynamics in transformers [11]. Moreover, particle tracing method does not require a pre-defined mesh as FEM does. Instead, it utilises discrete-element method (DEM) [12] to calculate the position and velocity of each particle at each time instance. With the DEM approach, particle tracing method can avoid numerical instabilities associated with the continuum approach (e.g. FEM). Furthermore, by using particle tracing method, the trajectory of each particle can be visualised, which provides a complementary tool to interpret the modelled results and help to understand moisture diffusion in power transformers.

In particle tracing method, the transferable quantities (e.g. moisture) are discretised as certain amount of particles from the microscopic perspective. The position and velocity of each particle are computed iteratively by using the corresponding motion equation. The macroscopic distribution of the transferrable quantities can be approximated by the state-space equation of the particles. To determine the particle's motion, Newtonian equation is adopted as
where mp denotes particle's mass, urn:x-wiley:17518822:media:smt2bf00282:smt2bf00282-math-0004 is particle's velocity vector and urn:x-wiley:17518822:media:smt2bf00282:smt2bf00282-math-0005 is the collective effect of different driving forces imposed on the particle.

These driving forces include the gravity force of the particles, the drag force from the fluid flow of oil, the diffusion force from the moisture gradient, Brownian force and interaction forces [13]. In this paper, both particle–particle interactions and particle–fluid flow (transformer oil) interactions are considered. Table 1 summarises the driving forces imposed on the water particles.

Table 1. Domains of different driving forces imposed on the water particles
Domain Mineral oil Interface between cellulose and oil Cellulose bulk
gravity force
drag force
Brownian force
diffusion force
particle–fluid flow interactions
particle–particle interactions
The gravity force urn:x-wiley:17518822:media:smt2bf00282:smt2bf00282-math-0006 is expressed as (4). For submicron or even smaller size particles, the drag force from fluid flow and other external forces dominate while gravity has little effect on particle's trajectories
where mp = 3.75 × 10−18 kg denotes the mass of particles (moisture), ρp = 103 kg/m3 denotes the density of particles and ρ = 861 kg/m3 denotes the density of the surrounding fluid (transformer oil), urn:x-wiley:17518822:media:smt2bf00282:smt2bf00282-math-0008 denotes the gravity vector.
The drag force urn:x-wiley:17518822:media:smt2bf00282:smt2bf00282-math-0009 due to the fluid flow is expressed as
where τp is the velocity response time constant, i.e. the time for a particle to respond to a change in the flow velocity of the carrier fluid [14]. urn:x-wiley:17518822:media:smt2bf00282:smt2bf00282-math-0011 is the fluid flow velocity vector.
The velocity response time for spherical particles in a laminar flow is defined as
where μF = 8.2 × 10−6 m2/s denotes the oil viscosity at 55°C. dp = 2 × 10−7 m denotes the diameter of spherical moisture particles in both oil and cellulose at submicron scale and it is determined by considering the model accuracy and the computational cost.
In particle tracing method, Brownian force as (7) is applied to the particles to take account for diffusion of suspended water particles in a fluid. In Brownian forces, each particle is imposed with a force at each time step to describe particles’ random movements
where Δt is the time step, rp = 10−7 m is the particle's radius, TF is the fluid temperature, kB is the Boltzmann's constant and ζ is a random number, which follows normal distribution (zero mean, unit standard variation). In the calculation, ζ is chosen in all directions in space. At each time step, a unique value of ζ is created for each particle.
In particle tracing modelling, diffusion force is treated as a nominal force that depends on the gradient of moisture concentrations as
where P(x, t) denotes the particle distribution density at position x and time t, Dt = kBT/(6πηrp), representing the interaction between spherical water particles and the cellulose medium. η denotes a scale factor which is determined by P(x, t). urn:x-wiley:17518822:media:smt2bf00282:smt2bf00282-math-0015 denotes the velocity vector of fluid flow of mineral oil.
The interactions between fluid flow and particles are considered in such a way that the particles are imposed by drag force from the fluid and in turn particles exert a momentum force on the fluid. It allows performing an accurate coupling between the water particles’ movements and the velocity field of mineral oil. The mineral oil is treated as continuous medium rather than particles. Oil characteristics are temperature dependent [15]. They are used as the input in fluid flow modelling (Table 2). The total volume force exerted by all particles on the fluid at position urn:x-wiley:17518822:media:smt2bf00282:smt2bf00282-math-0016 can be expressed as
where δ is the delta function, urn:x-wiley:17518822:media:smt2bf00282:smt2bf00282-math-0018 is the ith particle position vector, and the sum is taken over all particles.
Table 2. Temperature-dependent oil characteristics for coupling fluid flow with particle tracing modelling
Temperature, K Viscosity, m2/s Density, kg/m3 Thermal conductivity, W/m °C Specific heat, J/kg °C
258 4.5 × 10−4 9.02 × 102 0.134 1.7 × 103
268 1.8 × 10−4 8.96 × 102 0.133 1.74 × 103
278 8.5 × 10−5 8.9 × 102 0.132 1.785 × 103
288 4.5 × 10−5 8.84 × 102 0.131 1.825 × 103
298 2.7 × 10−5 8.79 × 102 0.131 1.87 × 103
308 1.7 × 10−5 8.73 × 102 0.13 1.91 × 103
318 1.15 × 10−5 8.67 × 102 0.129 1.95 × 103
328 8.2 × 10−6 8.61 × 102 0.128 1.995 × 103
338 6.1 × 10−6 8.55 × 102 0.128 2.04 × 103
348 4.7 × 10−6 8.49 × 102 0.127 2.08 × 103
358 3.8 × 10−6 8.44 × 102 0.126 2.12 × 103
373 3 × 10−6 8.35 × 102 0.125 2.18 × 103
For particle–particle interactions, Lennard-Jones force and universal gravitation between two particles are taken into consideration [10]. The Lennard-Jones interaction is expressed as (10), representing the intermolecular potential function to determine the transport property of a particle
where d denotes the distance between the particles, ɛ denotes the interaction strength and rc denotes the collision diameter.
On the basis of (10), the Lennard-Jones force of ith particle urn:x-wiley:17518822:media:smt2bf00282:smt2bf00282-math-0020 can be computed as

3.2 Boundary conditions

In particle tracing method, the solid material of cellulose insulation is modelled as continuous medium at the macroscopic level. Two boundary conditions are considered to simulate the effect of cellulose properties (e.g. relative magnitude of the paper fibres and the voids between the fibres) on the moisture diffusion as follows:
  1. Stick boundary: It is assumed that all the water particles contacting the surfaces of the cellulose (boundaries) will be forced to stay steadily upon the boundaries without any movement. Consequently, the velocities of these particles are set to zero and their positions are kept unchanged (Fig. 1a).
  2. Diffuse-reflection boundary: When large amount of water particles move towards the boundaries, some proportion of water particles will be bounced back. If there is no energy loss in the reflection, such bounce back of water particles is described by specular reflection. If energy loss is considered, there will be a reduction in velocities of particles and Knudsen's cosine law is employed [10] (Fig. 1b).
During moisture diffusion both two boundary conditions can occur with equal possibility. Thus, a probability fraction of 50% is assigned to the water particles reaching the boundaries to determine which boundary condition governs the further motion of these particles. When modelling particle movement in different materials and conditions, this fraction can be altered. Moreover, for diffuse-reflection boundary, the probability that particles can bounce back from walls of cellulose can be also changed. The reflection can be done either by specular reflection or according to Knudsen's cosine law. Different from FEM method, the permittivities and the diffusion coefficients of the oil–cellulose interfaces need to be continuous (smoothed using linear interpolation) in particle tracing method.
Details are in the caption following the image

Two boundary conditions. In the figure, dots denote particles and solid lines denote particles’ trajectories

a Stick boundary

b Diffuse-reflection boundary

3.3 Initialisations

After determining the driving forces and boundary conditions, the system is initialised by releasing certain amount of water particles from the cellulose surface, which is in contact with oil. The initial velocities of these particles are set to zero.

The number of the released water particles can be a random integer and it is independent from the flow pattern and mass of the fluid. If this number is too small, the diversity of particles movement cannot be guaranteed. Then it is possible that eventually all the particles stick on the boundaries without moving further. Consequently, this leads to an incorrect moisture distribution. On the other hand, if the number of the released water particles is too large, more computation resources are required. It is found that the accuracy of modelled moisture almost reach saturation level if the number of particles exceeds certain threshold value. After careful comparisons of the calculated moisture distribution using different number of initial water particles, 3000 water particles are chosen to be released with uniform distribution on the cellulose surface (Fig. 2). It should be notable that the optimal number of particles is not unique, which can vary depending on the geometry, boundary conditions and physics to be modelled.

Details are in the caption following the image

Initialisation of particles’ density and positions (3000 particles are depicted in grey dots). In the figure, x and y axes denote the geometry of the cellulose in mm. The grey circle denotes the outer edge of the pressboard (boundary), while the grey triangles denote the mesh for guaranteeing a uniform releasing of the particles at time instance t = 0 h

3.4 Post-processing and evaluations

After being released from the cellulose surface, all water particles are excited by the combined driving forces and move along the depth of the cellulose to its drier portion. The movement of water particles from wet area to dry area is ruled by both driving forces and boundary conditions. The transmission probability (can be regarded as an indicator of moisture distribution) of water particles inside the cellulose can be computed by counting the number of particles in the cellulose and dividing it by the total particles (distributed in both cellulose and oil). Finally, moisture distribution of the cellulose can be computed. The detailed procedures for deriving the moisture distribution from the transmission probability are discussed in the next section.

4 Results analysis and discussions

4.1 Application to pressboard specimens

Oil-impregnated pressboard specimens with 1 mm thickness are chosen in this study. The density of the pressboard is 1.04 g/mm3.The pressboard is brand new without being subjected to ageing before conducting this study. Necessary pre-treatment procedures were performed on the pressboard specimens including vacuum drying at 100°C inside an oven for 24 h. After the above treatments, pressboard samples have an initial moisture concentration <0.5%. Then, these specimens were immersed in mineral oil at a constant temperature of 25°C for 72 h to ensure complete impregnation.

The prepared oil-impregnated pressboards were put into a specially designed chamber with proper seals [16]. The surrounding of oil-impregnated pressboard is air with constant humidity maintained by saturated salt solution [17]. Temperature-dependent water solubility in oil may affect the boundary conditions and in this study only one surface (top surface) of the pressboard was exposed to the moisture source with a relative humidity of 3.8%. The other surfaces of the pressboard surface were sealed. The temperature of moisture diffusion was kept constant at 50°C.

Moisture distribution of the pressboard along its thickness is estimated by using both Fick's law and the particle tracing method for comparison. Fig. 3a presents the 3D moisture distribution of the pressboard after 12 h diffusion by solving Fick's law as in (1) and (2) [2]. To provide a better illustration, in Fig. 3a the pressboard is sliced into a number of thin pieces so that the moisture distribution of the whole cross-section area can be visualised. Fig. 3b presents the moisture gradient at the local cross-section area (black arrows). As it can be seen from Fig. 3b, there is a decreasing trend in the moisture gradient when moisture migrates into the depth of the pressboard.

Details are in the caption following the image

Moisture distribution of the pressboard after 12 h diffusion at T = 50°C

a Moisture distribution of the pressboard bulk

b Moisture gradient in the pressboard (as indicated by the black colour arrows). Calculated using Fick's diffusion law

c Moisture distribution of the pressboard along its depth (thickness) at different time instances at T = 50°C. Calculated using Fick's diffusion law

Fig. 3c shows the moisture distribution of the pressboard at different time instances. From Fig. 3c it can be observed that the moisture migration inside the cellulose is a quite slow process. Even at a constant temperature of 50°C, the moisture of another surface of the pressboard only reaches 1.5% after 12 h diffusion.

Moisture diffusion of the above pressboard is also modelled by using particle tracing method. The trajectories of water particles during the diffusion process are calculated as shown in Fig. 4a (only 24 h diffusion is presented). In Fig. 4a, each water particle is defined as a sphere with certain mass and radius. The trajectory of each water particles is represented as solid lines and the moving direction can be recognised from the ‘tail of the comet’.

Details are in the caption following the image

Modelled trajectories of water particles in the pressboard

a Trajectories of water particles in the pressboard during diffusion process (3000 particles) after 24 h diffusion at T = 50°C

b Poincare map of water particles of pressboard's top and bottom surfaces after 24 h diffusion at T = 50°C. In the figure, x and y axes denote the geometry of the pressboard in mm

Particle tracing method describes hypothetical trajectories of water. It is worth mentioning that it is hard to exactly capture the particles’ exact positions since moisture migration exhibits continuous behaviour, which is highly influenced by moisture and temperature gradients. In particle tracing method, the predication on water particles’ positions is largely dependent on the fineness of the time steps in the calculation. In Fig. 4a, the velocity of each particle during diffusion is quantified by a grey bar on the left (in mm/s). The maximum velocity of water particles is 15 mm/s. These particles have enough energy to move and they are more likely to leave the pressboard and enter the oil. By contrast, certain amounts of particles are bonded on the surface of the pressboard and they cannot move further.

To further investigate the moisture distribution in the above pressboard, Poincare map is adopted in the particle tracing method. Poincare map preserves many properties of periodic and quasi-periodic orbits of the particles and has a lower dimensional state space. Thus, it is often used to provide an insight into particles’ movements [10]. To construct a Poincare map, several observation planes (Poincare sections) are predefined and a reference plane with already known moisture concentration needs to be selected. They can be located at any position in the coordinate system. In this paper, the observation planes are paralleled with the pressboard surface and they are located at different depth from the pressboard surface. Therefore, the pressboard is ‘sliced’ into several layers by the observation planes. The reference plane is chosen as the pressboard surface which is in contact with oil.

At a particular time instance, if a water particle passes one of the above observation planes, a dot will be recorded on this plane to track the movement of the particles. By collecting the number of particles penetrating this plane, the transmission probability of the particles can be computed. Subsequently, the moisture concentration at this observation plane (e.g. a particular depth of the pressboard) can be calculated as
where Wi(x, t) denotes the moisture concentration at the ith layer of the pressboard with the distance of x to the reference plane, Pt(α) denotes the transmission probability (aforementioned in Section 3.4) and Wref(t) denotes the moisture concentration of the reference plane. In the above calculation, if the layer number i is large enough (the pressboard is sliced as infinite thin), the overall moisture distribution of the whole bulk volume of the pressboard can be obtained.

The Poincare map of water particles during the diffusion (after 24 h) is shown in Fig. 4b. In Fig. 4b an observation plane and a reference plane are presented. The observation plane is located at the bottom of the pressboard (this side is in touch with the electrode, drier portion). The reference plane is with horizontal axis (denoting vertical Z-direction) of zero. The particle traces penetrating the reference plane are presented as black dots, while the grey dots denote the water particles that penetrate the observation plane. From Fig. 4b it can be observed that all the water particles diffuse within the pre-defined boundaries of the pressboard specimen. After 24 h diffusion, most of the water particles have penetrated the pressboard and reached the drier part of the pressboard.

Fig. 5a presents the transmission probability of the water particles along the depth of the pressboard. Here, the pressboard is sliced into ten layers and the transmission probability for each layer is computed by counting the number of particles penetrating each layer (observation planes) and dividing it by the number of total released particles. Fig. 5b shows the calculated moisture distribution of the pressboard at different diffusion time by using the particle tracing method. Table 3 compares the calculated average moisture concentration of the pressboard based on Fick's law and particle tracing method. Since moisture diffusion coefficients have a significant influence on the moisture distribution, a comparison of the moisture distribution for oil-impregnated pressboard by using Foss, Guidi [2] and García's diffusion coefficients [8] is provided in Table 3.

Details are in the caption following the image

Transmission probability and moisture distribution of the pressboard calculated using particle tracing method

a Transmission probabilities of water particles in the pressboard during diffusion process at T = 50°C

b Moisture distribution of pressboard along its depth (thickness) direction derived by particle tracing method at T = 50°C

Table 3. Comparison of calculated average moisture (%wt) in pressboard
Time, h Foss Guidi García Particle tracing
4 1.77 1.37 0.83 2.34
8 2.24 1.83 0.94 2.57
12 2.58 2.15 1.03 2.78
16 2.85 2.4 1.1 2.96
20 3.07 2.58 1.16 3.12
24 3.26 2.72 1.22 3.26

From Table 3 it can be seen that García's coefficient produces the lowest moisture content in the pressboard. Guidi's diffusion coefficient produces a relatively lower moisture content compared with Foss's coefficient. The moisture content calculated by Foss's coefficient is the closest to that obtained by particle tracing method. Thus, Foss's coefficient is adopted to solve Fick's equation in the rest of this paper.

From Figs. 3c5b and Table 3 it can be concluded that with the increase in the diffusion time and moisture content of the pressboard, both Fick's law and particle tracing method give quite similar values of moisture concentration (after 12 h diffusion). However, if the pressboard is not wet enough or the moisture is not sufficiently diffused, particle tracing method tends to overestimate moisture of the pressboard (e.g. moisture distribution at 4 h). This may be due to the boundary selection in particle tracing method under low moisture circumstance. In this situation, the possibility fraction between the stick boundary and reflection boundary may not be kept constant at 50% as previously set in the paper (Section 3.2).

4.2 Application to a prototype transformer

Particle tracing method is also applied to estimate the moisture content in a prototype transformer (5 kVA, 240/2000 V). The transformer was subjected to accelerated ageing and moisture diffusion experiments. The prototype transformer was designed and manufactured by ABB and its ratio between the paper and oil in the transformer was maintained at a level similar to a real transformer (represented by a X–Y model [18] with ratios X = 38.32%, Y = 16.45%). A heater was installed on the bottom of the transformer to control the temperature for ageing and moisture diffusion experiments. Table 4 summarises prototype transformer's geometry. The solid cellulose materials consist of kraft paper, mouldable pressboard and spacers.

Table 4. Geometry information of the model transformer
Low voltage (LV) conductors, mm High voltage (HV) conductors, mm LV windings HV windings
1.6 × 7.1 1.4 × 1.4 22 turns/layer, four layers 110 turns/layer, seven layers
Thickness of layer insulation, mm Thickness of core-LV insulation, mm Thickness of HV–LV insulation, mm Thickness of HV-tank insulation, mm
0.25 1.75 9.7 2.75

Due to the complex geometry of the transformer, multi-physics modelling is adopted to model 3D moisture dynamics in the transformer. The model integrates the effects of electromagnetic, thermal, fluid flow and moisture migration physics on the moisture diffusion. Especially it considers the coupling and interactions of these physical phenomena [11]. The details of multi-physics modelling for moisture estimation will be provided in another paper.

After the commission of the prototype transformer, it was subjected to both electrical and thermal loading to attain a certain degree of ageing of its cellulose and oil (equivalent to 34 years of life consumption based on the degree of polymerisation measurement of paper samples collected from this transformer). Electrical loading was provided by using the load bank with maximum power capacity of 6 kW. The prototype transformer was kept at 110°C (using the abovementioned heater) with 30 A load current for a time period equivalent of 35 days (the transformer was kept at 50°C during night and weekend).

After 35 days accelerated ageing on the prototype transformer, experiments were arranged to study the moisture diffusion in the transformer. A sinusoidal temperature profile was imposed on the transformer by using the heater to simulate the operating conditions of a field transformer. A capacitive moisture-in-oil sensor (Vaisala MMT 330) [19] was installed in the transformer. The sensor was inserted into the transformer through a valve on the lid and its tip is close to the cellulose of the winding. The sensor is used to continuously measure the water content in oil (in ppm). Then, water content in cellulose can be derived from the measured moisture concentration in oil based on cellulose isotherms [20]. By using this setup, the moisture content of the transformer could be continuously monitored.

The moisture diffusion experiments were conducted in the following steps. First, the transformer was heated up and maintained at 55°C for 7 days to attain equilibrium status of moisture. Then a sinusoidal temperature profile was applied to the transformer for 7 days. One cycle of this temperature profile was 24 h and the temperature variations were between 30 and 80°C. After that the transformer was again kept at 55°C for 7 days to let it reach moisture equilibrium status.

Fig. 6 shows one complete cycle of the moisture measurement under the sinusoidal temperature. In Fig. 6, the temperature (in top figure) and moisture in oil (in middle figure) were directly recorded from the above moisture-in-oil sensor. The water content variation of cellulose interface (in bottom figure) was calculated by Fessler equation [21] as (13). The average water content in cellulose is calculated by averaging the data shown in black curve
where W denotes the water content, Pv denotes the vapour pressure of water and T denotes the temperature.
Details are in the caption following the image

Sinusoidal variation of temperature (in top figure), moisture in oil (in middle figure) and water content of cellulose surface contacting with oil (in bottom figure) of the prototype transformer

By using COMSOL Multiphysics software, 3D moisture distribution of the prototype transformer under the above sinusoidal temperature was obtained as shown in Fig. 7 (t = 172 h). From Fig. 7 it can be seen that moisture is not evenly distributed in the transformer. Most part of the cellulose attains a relatively low moisture level (<1.5%). However, the moisture of cellulose surface in contact with oil may reach up to 4–7%. Along the transformer's axial direction, the moisture also presents a non-uniform distribution. This is caused by the variation in the temperature and oil flow inside the transformer.

Details are in the caption following the image

Moisture distribution of the prototype transformer under sinusoidal temperature (at time instance t = 172 h). The results are obtained by multi-physics modelling method

On the basis of the estimated moisture in Fig. 7, the overall moisture of the whole cellulose bulk can be obtained as
where Vc is the volume of cellulose bulk.

The calculated overall moisture of the cellulose bulk at different diffusion times is shown in Fig. 8a. As can be seen from Fig. 8a, the overall moisture varies as the sinusoidal shape is within the range from 1.45 to 2.3%. The average moisture of the transformer's cellulose is 1.76% by taking the average of data presented in Fig. 8a.

Details are in the caption following the image

Comparison of moisture concentration between multi-physics model and particle tracing method

a Average moisture concentration of the cellulose in the prototype transformer at different times under sinusoidal temperature profile shown in Fig. 6. The results are obtained by multi-physics modelling method

b Average moisture concentration of the cellulose in the prototype transformer at different times under sinusoidal temperature profile shown in Fig. 6. The results are obtained by particle tracing method. The water particles’ transmission probability is also shown (in black colour)

Particle tracing method is also applied to estimate the moisture content in the transformer. The transmission probability of water particles and average moisture concentration of the transformer are calculated as shown in Fig. 8b.

From Fig. 8b it can be seen that there is a high correlation between the moisture distribution inside the cellulose and the particles transmission behaviour. When the moisture diffusion is excited by the sinusoidal temperature, the particles transmission probability exhibits a sinusoidal variation. The moisture in the cellulose also shows a sinusoidal variation but in the opposite direction of particles transmission probability.

The above phenomenon in Fig. 8b can be explained as follows. Particles transmission probability changes with the variations in the temperature. When the temperature reaches the peak value of 80°C (112 h diffusion in Fig. 8b), the maximum transmission probability rises to 0.46 and this implies that the water particles have the highest action energy at this moment. With the highest action energy, the water particles are more prone to leave the host (cellulose) and enter oil or reflect at the boundaries rather than to stay steadily. Consequently, the lowest moisture concentration in cellulose occurs at this time. When the temperature drops, water particles have less action energy and most of them tend to reside inside the cellulose without movement. Therefore, the overall moisture contents of the bulk cellulose material increases. By comparing Figs. 8a and 8b it can be seen that multi-physics modelling and particle tracing method show good agreement in estimating moisture content in the cellulose of the transformer.

To further verify the particle tracing method, extensive moisture diffusion experiments have been performed on the transformer. The diffusion experiments were divided into five stages and each stage took 21 days, including: (i) the transformer was maintained at 55°C for 7 days to attain moisture equilibrium; (ii) the sinusoidal temperature profile (the same as aforementioned earlier in this section, refer to Fig. 6) was applied to the transformer for 7 days; and (iii) the transformer was kept at 55°C for 1 week to facilitate moisture equilibrium. After the above procedures, dummy paper samples were collected from three locations corresponding to upper, medium and bottom height of transformer winding. This takes the non-uniform moisture distribution into consideration. Karl Fischer titration (KFT) [22] was used to validate the calculation results from both multi-physics modelling and particle tracing method. Table 5 summarises the modelled and KFT measured moisture at different stages.

Table 5. Results comparison of moisture concentration (%wt) in model transformer
Time, days Multi-physics Particle KFT
0 1.55 1.60 1.52
7 1.63 1.68 1.60
14 1.65 1.71 1.37
21 1.66 1.73 1.73
28 1.66 1.73 1.25
35 1.70 1.75 1.55

From Table 5 it can be seen that the moisture in transformer's cellulose estimated by particle tracing method and multi-physics are quite close. These results are also similar to that measured by KFT except some discrepancies at 14 days and 28 days of ageing experiments. This is mainly because the procedures of collecting pressboard specimens for KFT measurement may not be consistent at different collections. The above results demonstrate that particle tracing method can provide an alternative for estimating the moisture in the transformers from the microscopic view.

4.3 Discussion on the complexity of particle tracing method

The challenges of applying particle tracing method for estimating moisture concentration of transformers lay in two aspects:
  1. Comprehensive geometric information of transformer's insulation system: If the geometry of the transformer insulation is known, particle tracing method can be applied to estimate the moisture distribution in the cellulose of the transformer. However, if the detailed geometry information of the transformers is not available, it will cause certain difficulties in its implementation. In this case, some approximation needs to be made. It is assumed that the transformer's insulation construction follows common engineering practice and meets certain criteria (standards) to withstand electromagnetic, thermal and mechanical stresses during the transformer's operation. Thus, by using the nameplate information of the transformer (e.g. rating, core type, temperature rise, oil volume etc.), the prototype insulation geometry of the transformer will be used. Based on the approximated geometry, the particle tracing method can then be applied for moisture estimation.
  2. Complex relationship between oil/cellulose properties and simulation configuration: Moisture migration between oil and cellulose insulation is a complex process. Moisture diffusion can be highly influenced by the properties of oil (density, viscosity, thermal conductivity etc.) and cellulose (density, ageing condition etc.) medium. Changes in oil/cellulose characteristics will result in variation of different driving forces and boundary conditions in particle tracing modelling.
Taking boundary configuration as an example, the stick boundary describes whether the water particles can move further when they enter the cellulose while the diffuse-reflection boundary models the water particles’ reflection when they collide with the cellulose. When different cellulose materials are used in transformers or the cellulose becomes degraded, the possibility fraction between the stick boundary and reflection boundary needs to be altered. When the cellulose is aged, the possibility fraction of stick boundary needs to be increased to more than 50%. This implies that the water particles are more prone to stay with the cellulose instead of being bounced back from cellulose into oil.

Moreover, for diffuse-reflection boundary, the probability that water particles can bounce back from cellulose also needs to be adjusted based on the properties of oil and cellulose. Therefore, further studies are required to investigate the complex relationship between oil/cellulose properties and simulation configuration to provide an accurate moisture estimation of transformers.

5 Conclusion

This paper proposed particle tracing method to estimate the moisture contents of the oil-impregnated cellulose in transformers. Particle tracing method can avoid convergence difficulties in the FEM method. Particle trajectories are computed in a Lagrangian reference frame, removing the restriction on the ranges of the Peclet number. In particle tracing method, water particles’ motion can be visualised and the moisture distribution inside the cellulose can be derived. The experiment results of moisture diffusion on the pressboard samples and a prototype transformer verified the particle tracing method. The particle tracing method could be used as a complement to conventional methods (Fick's law, FEM method) for moisture estimation in transformer oil–cellulose system, especially when conventional methods encounter numeric instabilities.

6 Acknowledgments

Supports from Australian Research Council, and industry partners Powerlink Queensland, Energex, Ergon Energy and TransGrid are gratefully acknowledged.