1 Introduction

Polymers are commonly preferred in electrical insulation systems with superb physical and chemical properties as well as easy-to-manufacture structure, lightweight, high tensile stress, and low-costs. Testing the adopted insulation system for complying with the working conditions is important in terms of a healthy electrical system operation. There are certain tests for safety and reliability of electrical systems. One of these tests is comparative tracking index test determined by International Electrotechnical Commission (IEC). This test aims to determine the surface tracking performance of the polymer under humid and polluted weather conditions.

Carbon is the main component of polymer materials. Thus, the phenomenon called tracking on the surface of the polymer dielectric material is unavoidable [1,2,3,4]. Tracking causes a conductive path on the surface of the polymer material [4]. As a result, there is a failure in the electrical insulation system and becomes non-functional. Thus, testing the tracking resistance of a polymer insulating material used in any insulation system has a vital importance. Surface tracking failure is formed under moist and polluted conditions as follows [5]:

  • Polluted and wet are formed in surface

  • Dry-band is formed by joule heat

  • Partial discharge is formed across dry-band

  • Carbonization is begun in surface due to the discharge.

  • Propagation of the carbonization

  • Formation of tracking breakdown [5]

Historically, underground transmission systems were used in underwater crossings which are now adopted in urban life since the higher buildings are constructed and aesthetic concerns within the city are present. Production of commercial cables in a certain length requires high cost and is a compelling job. For example, we can assume an island 100 km away from coastline. Producing, transporting, and storing an underwater energy system cable that will be delivered to the island within 100 km distance in a single piece is a difficult job. Therefore, underground transmission and distribution lines require joints in certain points. The study of Bolat and Kalenderli investigated electric field distribution in the cable joints using finite elements method [6]. The result of this study showed that since there are disruptions and discontinuities in these joints, the electrical field is non-uniformly distributed. They stated that there are high electric fields in the joints and these high electric fields are leading to leakage currents, discharges, and breakdown on the cable. Underground transmission systems are exposed to moisture and polluted conditions both under the ground and water. Since the joints are subjected to higher electric field and environmental conditions, these points should be investigated in terms of surface tracking of polymer materials used in these points.

In this study, tracking process on the joints of underground energy transmission and distribution systems are investigated using 3D finite elements method. Test method adopted in this investigation as surface tracking test model is IEC 60112 comparative tracking index (CTI). As suitable accordance with the test standard, a 3D model is generated using the finite elements method. Electric field and electric potential distributions between the electrodes are investigated depending on the different positions of the solution droplet on the sample surface. The results are calculated with finite elements method and compared to the tracking on sample surface which is aged under laboratory conditions.

2 Materials and method

2.1 Finite elements method

Finite elements method is a numerical method commonly adopted in modeling the engineering applications [7,8,9,10,11]. At the present time, finite elements method is used in different calculations in various fields including electric engineering [12,13,14,15]. Nowadays, there are even package software that enable to solve multiphysics problems [16]. Using package software for solving the problem provides an ease of solution as well as more accurate results. The operations that will be applied when a problem is solved with a finite element package software can be as follows:

  • The problem is geometrically drawn.

  • Materials and properties of the region are defined for the geometry.

  • Boundary conditions for the problem are identified.

  • The geometry is divided into smaller subregions and equations for these subregions are defined according to the boundary conditions by the software.

  • The mathematical model is solved by the package software by considering all the stated conditions [17].

Figure 1 shows a triangle finite element solution region in 2D. For a triangle element in the finite triangle element solution region, a local node is determined and numbered. For each triangle element, a coefficient matrix is formed (Eq. 1). The next step involves generating the global coefficient matrix (Eq. 2) by merging the coefficient matrix of each element. By using this coefficient matrix and known node point values, the values of the unknown nodes are calculated. When this calculation is made, minimum energy principle is used as a basis (Eq. 3). The analysis with finite elements method uses Maxwell equations in differential form. These equations can be stated as Maxwell–Ampere Law (Eq. 4), Faraday Law (Eq. 5), Electric Gauss Law (Eq. 6), and Magnetic Gauss Law (Eq. 7).

$$\begin{aligned}&\left[ {K^{\left( e \right) }} \right] =\left[ {{\begin{array}{l@{\quad }l@{\quad }l} {k_{11}^{\left( e \right) } }&{} {k_{12}^{\left( e \right) } }&{} {k_{13}^{\left( e \right) } } \\ {k_{21}^{\left( e \right) } }&{} {k_{22}^{\left( e \right) } }&{} {k_{23}^{\left( e \right) } } \\ {k_{31}^{\left( e \right) } }&{} {k_{32}^{\left( e \right) } }&{} {k_{33}^{\left( e \right) } } \\ \end{array} }} \right] \end{aligned}$$
(1)
$$\begin{aligned}&\left[ K \right] =\left[ {{\begin{array}{lll} {k_{11} }&{} \cdots &{} {k_{15} } \\ \vdots &{} \ddots &{} \vdots \\ {k_{51} }&{} \cdots &{} {k_{55} } \\ \end{array} }} \right] \end{aligned}$$
(2)
$$\begin{aligned}&W=\frac{1}{2}\left[ V \right] ^\mathrm{T}\left[ K \right] \left[ V \right] \end{aligned}$$
(3)
$$\begin{aligned}&\nabla \times H=J+\partial {D} / \partial {t} \end{aligned}$$
(4)
$$\begin{aligned}&\nabla \times E=-\,\partial {B} / \partial {t} \end{aligned}$$
(5)
$$\begin{aligned}&\nabla \times D=\uprho \end{aligned}$$
(6)
$$\begin{aligned}&\nabla \times B = 0 \end{aligned}$$
(7)
Fig. 1
figure 1

Triangle finite element solution region in 2D

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In these equations: \(\nabla \), differential operator (nabla); H, magnetic field intensity (A/m); E, electric field intensity (V/m); D, electric flux density (\(\hbox {C/m}^{2}\)); B, magnetic flux density (\(\hbox {Wb/m}^{2}\)); \(\rho \), electric charge density (\(\hbox {C/m}^{3}\)); J, current density (\(\hbox {A/m}^{2}\))

Structural equations that provide information about the material are used as complimentary equations. \(\varepsilon \), \(\mu \), and \(\sigma \) in these equations are variables that contain information about the structural properties of the material. The equations that provide information about the material are as follows.

$$\begin{aligned}&D=\varepsilon \times E \end{aligned}$$
(8)
$$\begin{aligned}&B=\mu \times H \end{aligned}$$
(9)
$$\begin{aligned}&J=\sigma \times E \end{aligned}$$
(10)

In these equations: \(\varepsilon \), dielectric constant (permittivity) (F/m); \(\mu \), magnetic permeability (H/m); \(\sigma \), electric conductivity (S/m)

2.2 CTI test method analysis using finite elements method

This section specifies the method of test for the determination of the proof and comparative tracking indices of solid insulating materials on pieces taken from parts of equipment and on plane material samples using alternating voltages. The standard provides for the determination of erosion when required [18]. Comparative tracking index test method is first presented by IEC in 1959 [19]. In 1979, the standard has the final form of IEC 60112 standard [20]. In 2003, the fourth version of the test standard is published. This version is revised in 2009 [18]. The aim of CTI test is to determine the surface tracking performance of the polymer insulation material under polluted and moist conditions. 0.1% NH\(_{4}\)Cl (ammonium chloride) solution is dropped at 30 ± 5 s intervals on the sample surface to model the environmental conditions. The solution that will be dripped should have a volume of \(20 + 5\) mm\(^{3}\) [18]. The droplet from the needle of the drop should fall approximately in the middle of the electrodes. According to the test standards, the electrodes should have 2 mm thickness and 5 mm width. The dimensions of the test samples should be 20 mm\(^{2}\) \(\times \) 20 mm\(^{2}\) and the width of the sample should be 3 mm [20]. The copper electrodes used should contact the sample surface horizontally at an angle of 60\(^{\circ }\). The distance between the electrodes should be 4 mm. In this test standard, the tracking performance of the sample is tested by applying voltages between 0 and 600 V to the sample. Electrode arrangement of established CTI test setup in Istanbul University High Voltage Laboratory is shown in Fig. 2.

Fig. 2
figure 2

Electrode arrangement of established CTI test setup in Istanbul University High Voltage Laboratory

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In this study, CTI test configuration is modeled with Comsol Multiphysics v. 5.2 simulation software based on finite element method [21]. The electrode arrangement shown in Fig. 2 is used to draw the model. After this geometry is generated, an ellipsoid droplet is drawn on the surface of the sample. The changes in size and shape of the droplet are analyzed depending on the sample surface and electric potential and electric field are investigated. 3D geometry compliant with IEC 60112 test standards is shown in Fig. 3. After the model geometry is drawn, materials of the CTI experiment set are defined on the shapes. Properties of the materials used in the model are given in Table 1. The dimensions of the droplet are drawn as 2 mm\(^{3}\) \(\times \) 2 mm\(^{3}\) \(\times \) 1.59 mm\(^{3}\) which corresponds the droplet dimensions in when it short-circuits the electrodes.

Fig. 3
figure 3

3D CTI model

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Table 1 Properties of materials used in the model
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After the materials are defined on the geometry, the boundary conditions for the model is set. 600 V potential is applied to one of the electrodes and ground conditions are applied to the other electrode. After the boundary conditions of the model are defined, the model geometry is divided into finite number of triangle elements. This operation divided the model to finite number of small triangle elements and a basis equation for each triangle element are defined. The equations used in electrical model are Eqs. 11, 12, and 13, respectively. \(Q_{j }\)and \(J_{e}\) values in the solution of the problem are set to zero since there no external current source is present. As the finite triangle elements are merged, an equation system for the solution of the problem is obtained. The results are obtained by solving the equation system.

$$\begin{aligned}&\nabla \times J=Q_j \end{aligned}$$
(11)
$$\begin{aligned}&J=\sigma E+J_e \end{aligned}$$
(12)
$$\begin{aligned}&E=-\,\nabla V \end{aligned}$$
(13)

In these equations; \(\nabla \), differential operator (nabla); V, applied voltage (V); E, electric field intensity (V/m); J, current density (\(\hbox {A/m}^{2}\)); \(Q_{j}\), current source (\(\hbox {A/m}^{3}\)); \(J_{e}\), external current density (\(\hbox {A/m}^{2}\)); \(\sigma \), electric conductivity (S/m)

In this study, CTI experiment set is analyzed as stationary study using the finite element method. The effects of the droplet of solution dropped on the surface of the polyurethane sample from the beginning to the evaporation high potential are investigated in terms of electric potential and electric field changes on the polymer surface. Here the droplet’s contact angle for left and right contact points with polyurethane sample is neglected and simulated as ideal. The state of the solution droplet is analyzed under three conditions. The first condition evaluates the changes between the time droplet falls on the surface and disperses between the electrodes. The second condition evaluates the time when the droplets short-circuit the surface between the electrodes. The last condition evaluates the dry-band state between the electrodes when the droplet starts to evaporate. The results of the model were compared with the tracking results of the surface of aged polyurethane resin sample in accordance with CTI test standard under laboratory conditions.

3 Results

3.1 Dropping and dispersing model of the droplet

In this part of the work, a drop is assumed to begin to fall from the dropper into a spherical shape. This droplet is drawn to be fall the middle of the electrodes, and the volume is plotted as 20 \(\hbox {mm}^{3}\). The electric potential and electric field distributions on the sample surface are sliced when the sample touches the sample surface at 1 \(\upmu \)m from the sample surface of the droplet, starts dispersing to the sample surface, and short-circuits the electrodes. To see the electric field changes on the sample surface in a clearer way, a line is drawn to the middle of the sample surface. This line starts at 2 mm behind the high-potential electrode in the y-axis and ends at 2 mm beyond the ground electrode. This line also passes from the middle point of the electrodes. Electric field intensity changes are plotted on the graph throughout the line.

A sphere droplet with a radius of 1.684 mm and 1 \(\upmu \)m from the sample surface are drawn. Meshed form of the model is shown in Fig. 4. Electric potential and electric field distributions for this state of the model are shown in Fig. 5. Electric potential and electric field values in this state are zero. Figure 6 shows the changes in electric field on the line drawn to the middle of the sample surface. In the case where the water droplet does not contact with the sample surface, the electric field intensity on the sample surface at 2 mm and 6 mm point reaches the peak points at which the high-potential and ground electrodes touch. Electric field intensity value between the electrodes has a smaller value.

Fig. 4
figure 4

Mesh form of droplet at 1 \(\upmu \)m on electrode surface

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Fig. 5
figure 5

When the distance of the droplet is 1 \(\upmu \)m to the sample surface a electric potential distribution, b electric field distribution

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Fig. 6
figure 6

Electric field change in the middle of the sample surface when the distance of the droplet is 1 \(\upmu \)m to the sample surface

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Fig. 7
figure 7

When the droplet contacts the sample surface. a Electric potential distribution, b electric field distribution

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A sphere droplet with a radius of 1.684 mm and in contact with the sample surface is drawn. Electric potential and electric field distribution for this state of the model are shown in Fig. 7. Figure 7a shows that very low electrical potential is applied to the surface of the sample by contact with the sample surface of the droplet. With this potential loaded, an electric field change in the middle of the electrodes (Figure 7b) is observed. Figure 8 shows the changes in electric field on the line drawn to the middle of the sample surface. It was observed that the electric field intensity at the point of 4 mm contacted by the droplet was higher than 2 and 6 mm points where the electrodes contacted.

Fig. 8
figure 8

Electric field change in the middle of the sample surface when the droplet contacts the sample surface

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Fig. 9
figure 9

When the droplet disperses on the sample surface. a Electric potential distribution, b electric field distribution

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Fig. 10
figure 10

Electric field change in the middle of the sample surface when the droplet disperses on the sample surface

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In Fig. 9, a partially cut sphere contacting the sample surface is drawn to achieve some dispersion at the surface without short-circuiting the droplet and electrodes. As the droplet is not in contact with the electrodes, it is assumed that the volume of the droplet is same. Therefore, using Eq. 14, a height in millimeters was calculated to determine the embarked piece from a 1.75 mm droplet. This calculation shows that 0.7179 mm height piece will be embarked from a 1.75-mm droplet and a partially cut sphere is drawn. Electric potential and electric field distributions for this state of the model are shown in Fig. 10. The figure shows that an electric field is created when the outer surface is in contact with boundary surface. Figure 10 shows the changes in electric field on the line drawn to the middle of the sample surface. An electric field increase is observed on the boundary surface of the droplet. A higher electric field values is calculated at the point where the electrodes contact compared to the boundary surface of the droplet.

$$\begin{aligned} V=\pi \left[ {\frac{4}{3}r^{3}-h^{2}\left( {r-\frac{1}{3}h} \right) } \right] \end{aligned}$$
(14)

In this equation: V, droplet volume (\(\hbox {m}^{3}\)); r, sphere radius (m); h, height of the embarked piece (m).

When the droplet falls on the sample surface, it will continue to disperse. At the end, the droplet will fully disperse between the two electrodes and a current will flow on the droplet. To model this state, a semi-ellipsoid sample surface with 2 \(\hbox {mm}^{3}\) \(\times \) 2 \(\hbox {mm}^{3}\) \(\times \) 1.59 \(\hbox {mm}^{3}\) dimensions is selected. Electric potential and electric field distributions for this state of the model are shown in Fig. 11. At this state, intense electric field has been observed only when the electrodes are in contact. Figure 12 shows the change of electric field across the center of the sample surface in case of short-circuit between the electrodes and the droplet. As the graph shows, peak electric field intensity value is calculated at the contact point with the electrodes, and there is a sudden decrease in intensity as moved away from the electrodes. Maximum electric field values calculated around high-potential and ground electrodes are 3 and 3.6 MV/m, respectively.

Fig. 11
figure 11

When the droplet short-circuits the electrodes. a Electric potential distribution, b electric field distribution

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3.2 Dry-band formation model

Dry-band is formed in sample surface due to Joule heat. Formation of dry-band gives rise to initiation of carbon deposit in sample surface. It is one of the causes in degradation of surface insulation [5]. As the current occurs on the droplet, the temperature of the solution used increases. The increase in temperature triggers the evaporation of the droplet. As a result, a dry-band formation will begin at the center of the droplet. To model this state, a quarter ellipsoid is drawn for two electrodes. The size of the ellipsoids are altered to observe different dry-band sizes on the sample surface. The information about the dimensions of these ellipsoids is given in Table 2.

Fig. 12
figure 12

Electric field change in the middle of the sample surface when the droplet short-circuits the electrodes

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Table 2 Droplet sizes for dry-band region
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Calculations for 1, 20 \(\upmu \)m, 0.5, and 3 mm are made for the dry-band region between the droplets, respectively. The electric potential and electric field intensity values for dry-band distances 1, 20 \(\upmu \)m, 0.5, and 3 mm are plotted in Figs. 13, 14, 15, and 16, respectively. In Figs. 13a, 14a, 15a, and 16a, the droplet particles on both sides of the electrodes appear to approach to the potential of the electrode. Electric field distributions of the states with dry-band spacings of 1, 20 \(\upmu \)m, 0.5, and 3 mm are shown in Figs. 13b, 14b, 15b, and 16b, respectively. As seen in these figures, the surface exposed to electric field intensity is expanding with the increase in the dry-band gap. The electric field changes along the center of the sample surface when the dry-band region is 1, 20 \(\upmu \)m, 0.5, and 3 mm apart are given in Figs. 17, 18, 19, and 20, respectively. In the case of a dry-band distance of 1 \(\upmu \)m, larger electric field intensity was calculated in the dry-band region between the droplet particles compared to the point where the electrodes contacted. The calculated maximum electric field intensity in this region is about four times higher than when no droplet is present. When the dry-band distance is 20 \(\upmu \)m due to the evaporation of the droplets, the maximum electric field intensity with respect to 1 \(\upmu \)m distance is reduced to half. The maximum electric field intensity in this region is almost 30 times higher than the contact point of the electrodes. When the dry-band region is 0.5 mm, the maximum electric field intensity compared to 20 \(\upmu \)m is approximately 0.7 times higher. Yet, the area exposed to the electric field has expanded. In this case, the electric field intensity at the beginning of the dry-band region is equal to about 1.5 times higher the point where the electrode contacts the sample surface. When the dry-band region is 3 mm, the maximum electric field intensity at the point where the electrodes contact the sample surface is approximately 1.5 times higher the electric field intensity at the point where the dry-band begins.

Fig. 13
figure 13

When the gap of the dry-band region is 1 \(\upmu \)m. a Electric potential distribution, b electric field distribution

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Fig. 14
figure 14

When the gap of the dry-band region is 20 \(\upmu \)m. a Electric potential distribution, b electric field distribution

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Fig. 15
figure 15

When the gap of the dry-band region is 0.5 mm. a Electric potential distribution, b electric field distribution

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Fig. 16
figure 16

When the gap of the dry-band region is 3 mm. a Electric potential distribution, b electric field distribution

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In Fig. 21, a comparative tracking index test set prepared according to IEC 60112 test standard was used to scan the surface of polyurethane sample which is aged under laboratory conditions. This sample is subjected to 600 V voltage stress. Fifty drops of 0.1% NH\(_{4}\)Cl solution were dropped on the sample surface. Namely, the process which is modeled based on the finite element method is repeated 50 times on the sample surface. In the dry-band region, as the gap between the electrodes and the water droplets approach to the same potential, the distance between the electrodes also gets smaller as they are short-circuit. The electric field value (Eq. 15) between the planar electrodes depends on the applied voltage and the inter-electrode gap [22]. Voltage level selected for this study is constant at 600 V. As the gap between the electrodes get narrow in dry-band state, causes the electric field intensity to increase. For this reason, although the exact midpoint of the electrodes is the furthest point, it is expected that the deepest tracking will appear at this point due to the high electric field intensity in the dry-arc state. The effect of the arc formed in the dry-band region on the deterioration of electrical insulation materials has been studied in several studies [23,24,25,26].

$$\begin{aligned} E=-\frac{U}{a}\left[ {V/m} \right] \end{aligned}$$
(15)

In this equation: V, applied voltage (V); a, gap between electrodes (m).

Figure 21 shows that the area exposed to high electric field intensity in the middle of the electrodes is the deepest point and the depth of the tracking is getting closer to the electrodes.

Fig. 17
figure 17

Electric field change in the middle of the sample surface when the distance of the dry-band region is 1 \(\upmu \)m to the sample surface

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Fig. 18
figure 18

Electric field change in the middle of the sample surface when the distance of the dry-band region is 20 \(\upmu \)m to the sample surface

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Fig. 19
figure 19

Electric field change in the middle of the sample surface when the distance of the dry-band region is 0.5 mm to the sample surface

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Fig. 20
figure 20

Electric field change in the middle of the sample surface when the distance of the dry-band region is 3 mm to the sample surface

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Fig. 21
figure 21

A photograph of aged polyurethane resin sample

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4 Conclusion

In this study, the process of tracking on polyurethane sample surface is modeled as a three-dimensional process through a software using the finite element method. The results of this study are as follows:

  • In CTI test standard, tracking process was evaluated according to the state of drops dropped on the sample surface. Dry-band is influence in propagation of carbon path and initiation of carbon deposit. Results of our paper are in harmony with this acceptance.

  • In the case of short-circuit between the electrodes, the electric field intensity at the point of contact of the electrodes is calculated as approximately 3.5 MV/m. Given that the droplet started to evaporate, a dry-band region with a gap of 1 \(\upmu \)m was formed. With the formation of this dry-band region, the calculated electric field intensity around the electrodes is almost zero. On the other hand, the electric field intensity was calculated to approximately 5.6 MV/m in dry-band region with 1 \(\upmu \)m gap.

  • With the expansion of the dry-band region, the intensity of the electric field from the beginning of the dry-band is reduced. However, the area exposed to high electric field intensity is enlarged.

  • The experiment results and the results from finite elements method are compared. As a result, deeper traces are observed as the intensity of the electric field increases.

  • It has been estimated that the value of the electric field decreases as the point of the dry-band region approaches the electrodes. In the aged sample, deeper traces are observed in the middle part of the sample which are at the farthest point to the electrodes.

  • In the paper, electric stress in sample surface was analyzed with FEM called numerical method. In this way, surface tracking process in the underground cable joints was modeled depending on behavior of electrolyte droplet.