Thermogram Based Indirect Thermographic Temperature Measurement of Reactive Power Compensation Capacitors

Abstract

An increase in reactive power consumption results in an increase in electricity costs. This negative phenomenon can be prevented by using reactive power compensation methods. One of them is the installation of capacitors. These capacitors are exposed to external conditions, such as temperature and humidity. As a consequence, the aging process occurs. Another negative phenomenon is the corrosion that occurs inside the capacitor as a result of moisture absorption. As a result of this phenomenon, the capacitor can be damaged. One of the symptoms of the ongoing corrosion of the inside of the capacitor is an increase in temperature. Capacitors designed for reactive power compensation operate at mains voltage. They are often placed in a switchgear. For this reason, the use of contact methods of temperature measurement is difficult and dangerous. An alternative is thermographic measurement. Determining the internal temperature of the capacitor by thermographic measurement of the temperature of the case is possible with the use of numerical methods. One of them is FEM (Finite Element Method). The temperature results on the capacitor housing obtained from the simulation work were verified by comparing them with the result of thermographic temperature measurement. Both values differed by 0.2 °C. On the basis of the defined model, the differences between the temperature inside the capacitor housing and the temperature on the capacitor housing were determined by simulation. A simplification was proposed by replacing the cylinder made of layers with a homogeneous cylinder.

1. Introduction

A polypropylene capacitor is used for reactive power compensation [1]. Reactive power is one of the power components that are supplied to the considered system [2]. The other components are active power and power distortion [3]. Reactive power consumption is caused by the magnetization of the coils in electric motors and transformers [4]. The consumption of reactive power causes the appearance of a reactive current component. As a consequence, the resultant value of the current increases [5], and so does the cost of energy. The reactive power consumed by industrial plants (with a large number of electric motors and other electric devices with coils) is inductive. The use of a capacitor increases the capacitive reactive power. This causes the appearance of the current component of the capacity. As a consequence, the resultant value of the current decreases [6,7].

The capacitors designed for reactive power compensation are placed inside a cabinet [8]. Despite this, they are exposed to environmental conditions, such as humidity and temperature [9,10]. Moisture absorption into the capacitor causes corrosion to progress in the inner layers of the capacitor. As a consequence, an increasing length of metallization is disconnected from the direct connection to the zinc-sprayed end caps (shooting). The corrosion process is not uniform; there remain more corrosion-resistant pieces of metal in various spots, which can bridge the gap between the shooting and metallization. Consequently, it can cause an excessive current density in the vestigial links and localized heating that exceeds the tolerance of the dielectric, resulting in catastrophic failure [11].

An increase in the temperature of the capacitor accelerates the aging of the capacitor dielectric. It is caused by external factors (the temperature in the cabinet) and internal factors [12]. The main internal factor is the increase in power delivered to the capacitor. The temperature of the capacitor also depends on the waveform, frequency, and harmonic content of the working voltage [13]. Taking these facts into account, it is possible to conclude that an increase in the temperature of the capacitor may be an indication of impending failure. Regular monitoring of its temperature also helps to extend the service life of the capacitor.

There are two groups of capacitor temperature measurement methods: contact methods [14] and non-contact methods [15]. The contact methods are based on the application of a temperature sensor to the surface of the sensor’s capacitor. The examples of the sensors used are thermoresistor (e.g., PT1000) and thermocouple. The advantage of contact sensors is the accuracy of ±0.75% in the case of a thermocouple [16] and 0.30 + 0.005$·$|T| for a thermoresistor [17]. Reading the temperature measured by the sensor requires the use of a properly designed measuring system. In some multimeters, a circuit for measuring temperature with a thermocouple is included. This makes it possible to avoid some of the problems associated with the use of thermocouples (e.g., cold junction compensation). To read the temperature measured with a thermoresistor, a dedicated system containing a current source with a sufficiently low amperage (e.g., 1 mA) and a voltmeter is used. The amperage value is selected so as not to cause an increase in the temperature of the thermoresistor. As a consequence, it is possible to quickly measure the temperature of the capacitor case [18].

The disadvantage of contact methods is the unknown value of thermal resistance between the sensor and the case. This problem can be solved by gluing the thermoresistor to the capacitor case using an adhesive with a known value of thermal conductivity. Welding thermocouples to the surface on which the temperature is measured is also used. This results in a lack of mobility of the sensor used. Another disadvantage of using contact methods is the possibility of an electric shock.

The use of non-contact methods allows us to avoid the problems linked to the use of temperature sensors (including the unknown thermal resistance of the temperature sensor—the capacitor case). These methods include measurement with a pyrometer and measurement with a thermographic camera. The use of a thermographic camera enables temperature mapping on the capacitor surface [19]. This is an advantage over the use of a pyrometer, which does not provide such a possibility. The advantages of the contact method include the mobility and lower prices of thermographic cameras. An additional advantage is the ability to perform a thermographic temperature measurement of the capacitor case without having to open the cabinet where it is located. Performing such a measurement is possible only after prior installation of a special viewfinder on the cabinet door made of infrared transmittance material (λ = 1 µm $÷$ λ = 14 µ). Non-contact methods also have disadvantages. Their precision (compared to contact methods) is lower and amounts to ±2 °C or ±2% [20].

The surface temperature of the capacitor measured by non-contact methods depends on the conditions during the measurement. The most important of them include the value of the emissivity coefficient ε [21], reflected temperature T_r [22], distance between the lens and the observed object d [23], ambient temperature T_a [24], transmittance of the atmosphere τ_a, temperature of the external optical system T_l [25], transmittance of the external optical system τ_l [26], and relative humidity ω [27]. Furthermore, the thermographically measured die temperature depends on the viewing angle β [28] and the sharpness of the recorded thermogram T_us [29]. The measured value of the capacitor case temperature also depends on the place on the case where the measurement was made. This is related to the shape of the capacitor case [30].

Other problems related to the thermographic temperature measurement are the unknown value of the emissivity coefficient, the value of which changes during the operation of the capacitor. Another problem is the unknown value of the reflected radiation. Both problems can be solved by using black tape or paint with a known emissivity value in place of the material. When measuring through the viewfinder, the effect of the material from which it is made must be compensated for. Ways to compensate for the influence of the viewfinder have been described in the literature.

The contact and non-contact methods of temperature measurement make it possible to measure the temperature on the surface of the capacitor case. It is impossible to obtain information about the temperature distribution inside the case. In order to obtain this information, an indirect temperature measurement can be performed by detecting the temperature of the capacitor case (using a contact or non-contact method) and performing the simulation work, which results in the distribution of temperatures inside the case.

There are several ways to determine the difference between the temperature of the case and the junction temperature of a semiconductor diode. The difference between the case temperature and the junction temperature can be determined using a one-dimensional (1-D) heat transfer model. In the case of one layer of the element, it is possible to compare the analyzed path of the heat flux flow to the electrical circuit. In this method, the electric potential can be compared to the temperature, the electrical resistance to thermal resistance, and the electric current flowing in the branch to the thermal flux in the analyzed path. In the case of more layers, the thermal resistance of a particular layer can be compared to the series-connected electrical resistances [31]. For the transient analysis, the heat capacity is also taken into account. In this case, the chain of series-connected resistances is replaced by a chain of RC crosses. Foster [32] and Cauer [33] methods are used to analyze such quadrilaterals. It is also possible to determine the difference between the diode junction temperature and the diode case temperature using the Fourier equation. One of the methods that make it possible to simulate the temperature distribution in this case is the Finite Element Method (FEM) [34].

The FEM method consists of dividing the solid of an analyzed element into a finite number of smaller elements. The shape of the elements depends on the software that enables this method to be used. In some of them (e.g., Solidworks), the solid of the component under analysis is divided into a finite number of tetrahedral elements. The temperature values sought are located at the nodes. Knowing the temperature values of the nodes that have been set on the selected surface and the properties of the solid under analysis (e.g., thermal conductivity k), the temperature distribution in the solid under analysis can be determined. The temperature value at an individual node is determined from the temperatures of the neighboring nodes [35,36].

Obtaining the correct FEM results requires knowledge of the material properties (thermal conductiveness) and the construction of the capacitor used. The construction of the capacitor has been described in the literature. The material properties of the materials used for reactive power compensation are also described.

The authors of the work do not know the accuracy with which the temperature of the selected surface inside the capacitor intended for reactive power compensation can be determined from thermographic measurement of the temperature case. The time required to prepare this measurement is unknown. Uknown are also the results of experimental work, which would make it possible to obtain the information sought. For this reason, it was decided to undertake research work that would result in an experimental verification of the obtained temperature distribution inside the housing. It was also decided to check how much the thermally measured temperature of the capacitor casing differs from the temperature of a selected surface (inside the capacitor) obtained by simulation work. For this reason, it was decided to undertake research work, the result of which will be an experimental verification of the temperature distribution obtained inside the case. It was also decided to check how much the temperature of the capacitor case measured by the thermography differs from the temperature of the selected surface (located inside the capacitor) obtained by means of simulation work This article can provide knowledge for practitioners who work with capacitors.

This paper consists of the following sections. Section 2.1 presents the structure of the capacitor. Section 2.2 presents information on the modelling of the temperature distribution inside the capacitor. Section 2.3 describes the verification of the model obtained. Section 3 presents the results of the simulation and experimential work. Section 4 discusses the results obtained. Section 5 contains the conclusions.

2. Materials and Methods

2.1. Dimensions and Internal Structure of the Capacitor

The KJF—0.83/400 polypropylene capacitor (SR Passives, Portarlington, Ireland) [37] was used in the conducted works. In order to know the exact structure and internal dimensions of the capacitor, it was opened, and its internal dimensions were measured. Pictures showing the internal structure of the capacitor are shown in Figure 1.

The thickness of the strips that form a cylinder within the capacitor was measured using a Motic Images Plus 3.0 microscope (Motic, Xiamen, China) [38]. The external and internal dimensions of the capacitor used are shown in Figure 2.

2.2. Modeling the Temperature Distribution Inside the Capacitor

Knowledge of the construction of the capacitor is necessary for the correct calculation of the difference between the internal temperature T_rs and the capacitor’s case temperature T_c.

The simplest method to determine the value of T_rs from the measured value of T_C is based on Fourier law. Heat flow in the cylindrical wall is described in Equation (1) [39].

$$T_{rs}=\frac{P_{in}·ln\frac{d_2}{d_1}}{2·\mathsf{\pi }·\lambda ·l}+T_C$$

(1)

where $P_{in}$—the power dissipated inside the capacitor, d₁—the external capacitor dimension, d₁—the internal capacitor dimension, and l = capacitor high.

Knowledge of the internal dimensions of the capacitor is necessary for the correct implementation of the model, which will enable a calculation of the temperature distribution inside the case, too. To determine the temperature distribution inside the case, simulation work was performed, during which the Finite Element Analysis (FEA) method was used. By definition, FEA is a numerical method to solve problems in engineering and mathematical physics [40]. The software applied in the work performed was Solidworks 2020 SP05 (Dassault Systèmes, Vélizy-Villacoublay, France), which uses FEA, and the simulation was completed with the use of this software.

In order to correctly determine the temperature distribution in a capacitor, it is necessary to consider conductivity, radiation, and convection. Conductivity (also inside a capacitor) is obtained from thermal conductivity. Radiation is linked to the radiation coefficient h_r. The h_r coefficient defines the amount of thermal energy transferred to the environment by radiation per unit time, per unit area, and per unit temperature difference between the body radiating energy and the environment. The h_r value can be determined using Equation (2) [41].

$$h_r=\epsilon ·{\sigma }_c\left(T_S+T_a\right)\left(T_S^2+T_a^2\right)$$

(2)

where ε—emissivity coefficient (−), σ_c is Stefan–Boltzmann constant equal to 5.67 × 10⁻⁸ (W·m⁻²·K⁻⁴), T_S is the surface temperature (K), and T_a is the temperature of the air (ambient) (K).

In the case of the Solidworks software, the h_r value is calculated on the basis of the equation implemented in the software and the ε value of the surface entered by the user. The value of ε depends on the condition of the surface. This value is also affected by aging processes (for example, metal oxidation). Since a brand new capacitor was used, the average ε value of the case from the table, ε = 0.15, was adopted as the value for aluminum (for the purposes of simulation work). For the black fragments of the capacitor case (in the top and bottom of the case), the value of ε for plastic was assumed to be 0.94. The same value was taken for the plastic caps inside the capacitor housing.

The value of material emissivity is possible to find with the comparative method. In the first step, the value of the marker (which was put on the surface by Velvet Coating 811-21 paint) emissivity has been set. In the second step, the thermographic measurement of the surface temperature (with marker’s emissivity, the measurement point was situated on the marker) has been done. In the third step, the measurement point was situated on the surface with the unknown emissivity, as close to a marker as possible. In the last, the fourth step, the temperature measurement has been done at the new measurement point. The value of emissivity has been changed in the thermographic camera software as long as the result of the thermographic temperature measurement on the marker and the result of the thermographic temperature on the surface have not been equal. The emissivity value sought is the value at which both thermographic measurement results are equal.

The second coefficient to be calculated is the convection coefficient h_c. The h_c coefficient defines the amount of thermal energy transferred to the environment by convection per unit time, per unit area, and per unit temperature difference between the body (releasing energy) and the environment. The h_c factor can be determined by carrying out the experimental work. An alternative way to determine the h_c coefficients is using the theory of similarity to physical phenomena. Relationships with physical quantities that characterize a given phenomenon were described using the criteria of Nusselt, Grashof, and Prandtl. Note that this is a method that allows you to determine the approximate values of the h_c coefficient. Based on the data presented in the literature, it can be concluded that the results obtained in the simulation work, in which the h_c coefficients were used, are aligned with the measurement results (difference lower than 5 °C). The value of the convection coefficient for a cylindrical surface can be obtained from Equation (3) [42].

$$h_{cr}=0.48{(\frac{g·Pr·\alpha ·d^3}{{\nu }^2}·\left(T_S-T_a\right))}^{\frac{1}{4}}\frac{k}{d_r}$$

(3)

where h_cr is the convection coefficient of the cylindrical surface (W·m⁻²·K⁻¹), g is the gravitational acceleration of 9.8 (m·s⁻²), d_r is equal to the diameter of the work roll (a characteristic size in this case) (m), Pr is the Prandtl number (−), α is an expansion coefficient equal to 0.0034 (K⁻¹), and ν is kinematic viscosity equal to 1.9 × 10⁻⁵ (m²·s⁻¹).

The Prandtl number can be obtained from Equation (4) [43].

$$P_r=\frac{c·\eta }{k}$$

(4)

where c is the specific heat of air equal to 1005 (J·kg⁻¹·K⁻¹) in 293.15 (K), η is the dynamic air viscosity equal to 1.75 × 10⁻⁵ (kg·m⁻¹·s⁻¹) in 273.15 (K), and k is thermal conductivity (W/m k).

To determine the convection coefficient for a flat surface, h_cf, Equation (5) [44] should be used.

$$h_{cf}=\frac{Nu·k}{L}$$

(5)

where h_cf is the convection coefficient of flat surfaces, Nu is the Nusselt number (−), and L is the characteristic length in meters (for a vertical wall, it is its height).

The Nusselt number is determined from Equation (6) [45].

$$Nu=a{\left(Pr·Gr\right)}^b$$

(6)

where a and b are dimensionless coefficients, whose values depend on the shape and orientation of the analyzed surface and the product Pr·Gr, and Gr is the Grashof number.

The Prandtl number can be obtained using Equation (4), while the Grashof number can be obtained using Equation (7) [45].

$$Gr=\frac{g·\alpha ·\left(T_S-T_a\right)·L^3·{\rho }^2}{{\eta }^2}$$

(7)

where ρ is the air density equal to 1.21 (kg·m⁻³) in 273.15 (K).

The values of the coefficients a and b in Equation (6) depend on the shape and orientation of the heat transfer surface. The values of the a and b coefficients are presented in

Table 1. Natural convection correlation coefficients; a_lam is the value of coefficient a for laminar flow, b_lam is the value of coefficient b for laminar flow, a_turb is the value of coefficient a for turbulent flow, and b_turb is the value of coefficient b for turbulent flow.

Shape	Gr·Pr	alam	blam	aturb	bturb
Vertical flat wall	109	0.59	0.25	0.129	0.33
Upper flat wall	108	0.54	0.25	0.14	0.33
Lower flat wall	105	0.25	0.25	NA	NA

NA: Not applicable.

.

By inserting the result of Equations (4) and (7) into Equation (6), and then the result of Equation (5) into Equation (4), the h_cf value for the natural convection can be obtained.

2.3. Verification of the Received Model

The temperature distribution inside the case, which was obtained on the basis of simulation work, can be verified experimentally. In the conducted work, the temperature on the surface of the capacitor case was measured using the Optris Xi 400 thermographic camera (Optris, Berlin, Germany) with a telephoto lens (Instantaneous Field of View = 0.9 mrad). The camera is equipped with an IR detector array with a spatial resolution of 382 × 288 pixels. The detectors operate in the LWIR band (Long Wave Infrared) λ = 8 μm $÷$ λ = 14 μm and are characterized by NEDT (Noise Equivalent Differential Temperature) 80 mK. They enable the measurement with a maximum frequency of 80 Hz. The accuracy is equal to ±2 °C or ±2% [46].

The camera was placed in a plexiglass chamber with dimensions of 45 cm × 45 cm × 33 cm. The interior of the chamber is lined with polyurethane foam. The foam used is of a porous structure, and its single pore idealizes the black body model. This chamber structure is characterized by a high value of ε = 0.95 and a small value of the reflectance factor r. This enabled the optical separation of the interior. The camera was connected to the computer via USB. The diagram of the measuring system is shown in Figure 3.

Due to the low value of the emissivity coefficient, a marker was applied to the part of the case where the thermographic temperature measurement was performed, which was made with Velvet Coating 811-21 paint with a known emissivity coefficient value ranging from 0.970 to 0.975 for temperatures within the limit from −36 °C to 82 °C. The uncertainty with which the emissivity coefficient value was determined was 0.004 [47].

To verify the temperature inside the capacitor case, a hole with a diameter of φ = 2.5 mm was made. A k-type thermocouple was inserted into the hole. The thermocouple was placed in the capacitor case in such a way that the welded metal alloy was placed in the polyurethane layer as close to the inside as possible. The distance between the capacitor case and the location of the welded metal alloy was 11 mm. After the thermocouple was placed, the hole was sealed. The free ends of the thermocouple were connected to a UT-55 multimeter (UNI-T, Dongguan City, China). The multimeter was equipped with a system that allowed temperature measurement with the use of a thermocouple. The measured temperature value is subject to an accuracy of 16%. The location of the thermocouple in the capacitor case is shown in Figure 4.

The capacitor was connected to the grid under nominal conditions (230 V, 1.6 A). A photo showing the location of the thermocouple and the marker is shown in Figure 5.

3. Results

Based on data from the technical documentation and the available literature, materials with the properties described in

Table 2. Materials assigned to the individual parts of the capacitor, the selected value of thermal conductivity k (W/m k), and the selected values of the convection coefficients (for the external parts of the case).

The Part of the Capacitor	Material	k W/(m·K)	The Range of the Used k Values * W/(m K)	hcf, hcr (−)
Case	Aluminum	200	190–230	4.27
Plastic caps (outside)	Plastic	0.54	0.41–0.59	14.55
Plastic caps	Plastic	0.54	0.41–0.59	-
Electrolyte	Polyurethane	0.3	0.24–0.38	-
Zinc-sprayed	Zinc	116	104–119	-
Roller (metal foil)	Aluminum	200	190–230	-
Roller (plastic)	Polypropylene	0.2	0.18–0.33	-

* Depending on the alloy, the chemical composition, etc.

were assigned to the individual surfaces of the capacitor. This table also includes the values used for the convection coefficients h_c, which were determined on the basis of Equations (2)–(6).

The range of h_cf, h_cr coefficients variability is shown in Figure 6.

For the upper flat part of the case, the same value of the convection coefficient was assumed as for the lower plastic part (h_cf = 14.55).

To simplify the model, heat transfer in a cylinder made of layers of aluminum and polypropylene was checked. It was assumed that in the case of a sufficiently small area (unit area) of 1 mm², the model made is approximately a cube or cuboid, one of whose bases is a fragment of the side surface of the cylinder, and the height depends on the number of polypropylene and aluminum layers. Comparative models composed of alternating layers of aluminum and polypropylene consisting of 15, 30, and 60 layers were made. Models were also created (with the same dimensions) composed of a single material, whose parameters are the averaged parameters of the aluminum and polypropylene. The models created and the resulting distributions are shown in Figure 7.

In order to check the results obtained from the analysis of unit area (Figure 6), another set of simulations was prepared. It was checked whether the layered roller could be changed to a homogeneous roller (Figure 8).

Based on the temperature distributions obtained (shown in Figure 7 and Figure 8), a simplified capacitor model can be adopted, in which the cylinder is made of a single material whose properties (e.g., thermal conductivity) are the averaged values of aluminum and polypropylene.

For the created model, a grid with meshes of 14 mm, 7 mm, and 1 mm edges was defined. The duration of the simulation was measured for each mesh. The mesh created and the duration of the simulation are shown in Figure 9.

Based on the data presented in Figure 9, it was found that the most optimal grid with a mesh is that with an edge of 7 mm. For the selected grid and parameters presented in Table 2—ε for plastic equal to 0.94, ε for aluminum equal to 0.15, and ambient temperature equal to 20 °C—it was possible to carry out the simulation work, the results of which were compared with the results of the experimental work. The experimental work was performed with the use of a stand shown in Figure 3. The comparison of the thermographic measurement of the temperature T_c, the surface temperature of the capacitor obtained as a result of the simulation work T_cs, the temperature measured with a thermocouple T_t, and the temperature obtained as a result of the simulation work for the location of the thermocouple T_ts is shown in

Table 3. Comparison of the thermographic temperature measurement T_c, the surface temperature of the capacitor obtained as a result of the simulation work T_cs, the temperature measured with a thermocouple T_t, and the temperature obtained as a result of the simulation work for the location of the thermocouple T_ts.

Tc (°C)	Tcs (°C)	Tc − Tcs (°C)	Tt (°C)	Tts (°C)	Tt − Tts (°C)
22.2	22.1	0.1	22.0	22.2	0.2

.

The differential between the results of the internal temperature T_rs and T_c temperature is equal to 0.72 °C

The recorded thermograms are shown in Figure 10.

By analyzing the data collected in Table 3, it can be concluded that the temperatures gathered on the basis of the obtained model correspond to the real temperatures. Using the real model, the simulation work was carried out, as a result of which the distribution of temperatures inside the case was determined, when the temperature of the capacitor case equaled 22.2, 34.0, and 54.0 °C. The obtained distributions are shown in Figure 11. The source of heat was electrodes.

By analyzing the temperature distributions obtained (Figure 11a), the temperature of the capacitor case T_cs and the temperature of the cylinder (made of aluminum and polypropylene) T_rs were compared. The list of temperatures is shown in

Table 4. Comparison of capacitor case temperatures T_cs with temperatures of the cylinder (made of aluminum and polypropylene) T_rs for T_cs = 22.0 °C.

Tcs (°C)	Trs (°C)	Tcs (°C)	Trs (°C)	Tcs (°C)	Trs (°C)
22.0	22.2	34.0	34.5	54.0	55.1

. Other results (Figure 11b,c) were used to obtain the characteristics in Figure 12.

Based on further simulation work, it was noticed that between the temperature T_cs of the case and the point inside the case T_rs, there existed the following relation: $T_{rs}=1.0295·T_{cs}-0.4775$ (Figure 12).

4. Discussion

The convergence between the measured temperatures of the case and the inside of the capacitor and the temperatures determined on the basis of the simulation work depends on the accuracy with which the simulations are performed. In turn, the precision of the simulation depends on the assumed size of the mesh, the accuracy of the input data (e.g., thermal conductivity), and the precision with which the values of convection coefficients are determined. To determine the exact values of the convection coefficient, experimental work should be performed.

Based on the analysis of the results obtained (Table 3), it can be seen that the difference between the temperature of the capacitor case determined on the basis of the simulation work and the temperature of the capacitor measured by thermography is 0.1 °C. It can also be seen that the difference between the temperature inside the capacitor determined by the simulation and the temperature measured inside the capacitor is 0.2 °C. This is indicative of the selection of a grid with an appropriate mesh size and the correct assignment of materials (and their properties) to individual parts of the defined capacitor model. It is also possible to conclude that the values of the convection coefficients determined analytically are sufficiently accurate.

The convergence of the results obtained on the basis of the simulation work and the results obtained from the measurements depends on measurement precision. In the case of thermal imaging measurement, a number of factors affect the camera readout. Particular attention should be paid to the correct selection of the emissivity factor.

5. Conclusions

Corrosion occurring inside the capacitor due to the absorption of moisture causes an increase in temperature. A potential failure can be predicted by regular temperature measurements and regular operational tests. As a consequence, it is possible to avoid additional costs.

When modeling the temperature distribution inside the capacitor with the use of numerical methods, a simplification can be applied by modeling the roller inside the capacitor (composed of aluminum and polypropylene layers) as a uniform cylinder, whose thermal conductivity is the average value of the thermal conductivity of aluminum and polypropylene (Figure 7 and Figure 8). The use of a simplified model saves time related to the creation of the model and mathematical calculations performed during the simulation. The number of meshes and, consequently, the number of operations needed to obtain the temperature distribution inside the capacitor are also reduced.

In addition, the time required to perform these calculations is also reduced, as shown in the description of Figure 8. The simulation time needed to obtain the temperature distribution in a roller made of 10 layers is 1220 s, while the time needed to obtain the temperature distribution in the uniform roller is 11 s. Note that the simulation time required depends on the number and shape of the roller layers analyzed.

Based on the comparison of the results obtained from the measurements and the results obtained during the simulation work (Table 3), it can be concluded that the results are convergent, and the largest difference between them was 0.2 °C. As a consequence, it can be concluded that, on the basis of the obtained model, it is possible to determine the temperature of a point located inside the capacitor case. When comparing the measurement results and the results of the simulation work, the accuracy of the thermographic temperature measurement (2 °C) and the accuracy of the simulation should be taken into account.

The results obtained (Figure 10 and Table 4) allow us to conclude that the largest difference between the temperature of the case and the temperature of the point inside the case was 1.1 °C. Based on simulation work, the relationship between the temperature inside the capacitor case (in the polypropylene near the roller) and the temperature on the capacitor case was determined. This relationship can be described by the following equation: $y=1.0295·x-0.4775$.

The differential between the internal capacitor temperature and the temperature on the capacitor case depends on the methods. In order to simplify the 1-D method (Equation (1)) the difference is equal to 0.72 °C. For FEM, the difference is equal to 0.2 °C.