Modeling and Simulation of Rotating Machine Windings Fed by High-Power Frequency Converters for Insulation Design Modeling and Simulation of Rotating Machine Windings Fed by High-Power Frequency Converters for Insulation Design

Modern power systems include a considerable amount of power electronic converters related to the introduction of renewable energy sources, high-voltage direct current (HVDC) systems, adjustable speed drives, and so on. These components introduce repet- itive pulses generated by the commutation of semiconductor switches, resulting in overvoltages with very steep fronts and high dielectric stresses. This phenomenon is one of the main causes of accelerated insulation aging of motors in power electronic-based systems. This chapter presents state-of-the-art computational tools for the analysis of motor wind- ings excited by fast-front pulses related to the use of frequency converters based on pulse-width modulation (PWM). These tools can be applied for the accurate prediction of over- voltages and dielectric stresses required to propose insulation design improvements. In the case of the stress-grading system used in medium-voltage (MV) motors, transient finite-element method (FEM) is used to study the effect of fast pulses. It is shown how, by controlling the material properties and the design of the stress-grading systems, solutions to reduce the adverse effects of fast pulses from PWM-type inverters can be proposed.


Introduction
The detrimental effect of steep voltage surges propagating along the windings of power components such as transformers, reactors, motors, generators, and so on has been a topic of great interest for almost a century (see for instance [1][2][3][4][5][6]). In the case of motors, these surges have been traditionally associated with energization, re-energization, or incoming lightning pulses [7]. Over the last decades, this has become a larger issue with the increasing use of frequency converters for speed and torque control, which introduce repetitive steep-front voltage impulses and associated partial discharges into the machine windings (see Figure 1). The insulation of this type of devices is subjected to high and sustained electric and thermal stresses [8,9]. This effect is further amplified by the use of long connection cables [10].
Industrial surveys and other studies show that 20-40% of rotating machine failures are due to stator winding problems, and 70% of these are due to insulation failures [11][12][13][14][15][16]. Dielectric stress along motor's windings associated with the propagation of fast-front voltage surges is a major source of premature deterioration or failure of the insulation system of these devices [9,17].
Early motor representations for fast-transient studies consisted of approximating the surge impedance of the motor as a simple lumped termination of the feeding cable [18]. This allowed understanding and predicting the transient overvoltages produced at the cable/motor connection. However, overvoltages can appear at several points inside the machine coils during the transient period. Models that are more detailed were later introduced to consider the transient potential distribution along windings. These models are known as white box models and can be classified in lumped and distributed parameter representations [7]. Although the former are easier to implement and less computationally expensive, the latter are more accurate since the wave propagation along the winding coils is introduced in a more precise manner.
Rudenberg first introduced the application of traveling wave theory to study the fast-transient behavior of windings [19]. A more rigorous approach based on a multi-conductor line model and applied to transformer windings was described by Rabins in 1960 [20]. Oraee [4] and Guardado [5] independently developed this approach for electrical machines in the 1980s. The main idea of their modeling approach is that a coil can be approximated by a multi-conductor transmission line (MTL): each conductor represents a turn (or group of turns) of the coil, and the continuity between the end of one turn and the beginning of the next one is preserved by means of a topological connection between the corresponding nodes. This is called a zig-zag connection and is shown in Figure 2.
The distributed parameter model of an MTL is completely characterized by its electrical parameter matrices: series impedance matrix Z and shunt admittance matrix Y. For the case of machine windings, these matrices are obtained from the geometrical configuration of the machine. Considering the typical stator coil diagram depicted in Figure 3, the incident surge   will propagate along two different regions: slot and overhang. Due to the variation in the electromagnetic field distribution in these two regions, the corresponding parameter matrices will also be different [5].
Another challenging aspect of machine-winding modeling is the computation of electrical parameters. Common approaches followed for this task are (1) measuring the parameters, (2) applying analytical expressions, and (3) applying numerical methods. Analytical and numerical methods are the only possible choices for new designs. However, since machine windings have complex geometrical features, the available analytical expressions rely on numerous approximations and assumptions and are therefore restricted to particular geometries. Numerical methods are applicable in a more general sense as long as detailed geometrical and electrical data are available, which is the case at the design stage [22].
One of the first insulation problems stemming from the use of power frequency converters with medium-voltage (MV) motors was associated with the electric stresses on the surface of the coils. Partial discharge (PD) and heat can erode the coatings of the stress-grading (SG) system [23], aggravating the problem and, perhaps, eventually destroying the ground-wall insulation. This process can take a long time to produce a critical failure, but the ozone produced as a byproduct of surface PD [24] may accelerate the degradation and hence the need to rewind the motor [23].
The SG system in MV-rotating machines is composed of two coatings: the conductive armor coating and the semi-conductive stress-grading coating. The conductive armor coating is a relatively conductive layer that is usually applied in the form of a tape to the surface of the ground-wall insulation in form-wound coils that lies inside the stator slot of a rotating machine. The function of this coating is to limit the surface potential of the coil to a value equal or close to ground potential, thereby avoiding the possibility of discharges between the coil insulation and the slot wall. The length of the conductive armor coating extends beyond the end of the slot [25], as illustrated by part A in Figure 4. The semi-conductive-grading coating starts at the end of the conductive armor coating, part B, also in the form of a tape, and they are often overlapped by a couple of centimeters, shown as region E. The purpose of the semiconductive-grading coating is to produce a smooth transition from the potential in the conductive armor coating to the high voltage (HV) on the surface of the coil insulation outside of the slot, shown as part C in Figure 4. The graded voltage drop along the surface of the coil avoids harmful high electric field concentrations.
The conductive armor coating is commonly a composite of either varnish or polyester resin filled with graphite or carbon black. It is considered as a constant conductivity layer with values between 10 and 0.01 S/m. The semi-conductive-grading coating of coils is a composite of silicon carbide (SiC) and varnish or polyester resin, with a field-dependent conductivity. A profile of the voltage from the slot exit to the end of the semi-conductive-grading tape (SGT) measured using an alternating current (AC) electrostatic voltmeter is shown in Figure 5. The coil was energized with 8 kVrms at 60 Hz. The conductive armor tape (CAT) is at ground potential (first 10 cm in the plot of Figure 5) while along the SGT the voltage increases smoothly (after 10 cm), something that is accomplished by the generation of resistive heat as seen in the inserted infrared image of Figure 5, where the section of the SG presents a hot spot. The slope of the voltage defines the tangential electric field on the surface of the stress-grading region. Overvoltages and pulses having steep fronts, like those from PWM-type inverters, considerable modify the stress relief in the SGT and in the portion of the CAT that is outside of the slot. The design of the stress grading and the conductive armor coatings has become difficult under this condition, and modeling had resulted in a useful tool in understanding the influence of the various design parameters.  The remaining of this chapter is divided as follows: Section 2 describes the modeling and parameter determination of motor windings excited by the fast pulses produced by an inverter. Section 3 describes a finite-element method (FEM)-based approach to analyze the performance of stress-grading systems. Finally, Section 4 presents the conclusions and final remarks of this chapter.

Fast-transient modeling of rotating machine windings for inverter excitation
The computer model described in this section for the prediction of the fast-transient response of the machine winding is a non-uniform, multi-conductor, distributed-parameter model defined in the frequency domain. This model is selected over other alternatives given its good balance between high accuracy and practicality, as explained below: 1. Why non-uniform? The distribution of electric and magnetic fields is different in the slot and overhang regions, resulting in different electrical parameters for each region that should be considered to accurately reproduce the pulse propagation. In this case, the nonuniform model consists of five segments considering the typical geometry of a woundform motor, as shown in Figures 3 and 6.
2. Why multi-conductor? This modeling approach of the winding allows natural inclusion of the inductive and capacitive coupling between turns [21].
3. Why based on distributed parameters? A distributed parameter model based on transmission line theory can represent the propagation of a fast pulse along windings in a more rigorous manner than a lumped parameter model based on circuit theory [27].

4.
Why defined in the frequency domain? The partial differential equations describing wave propagation in time domain become ordinary differential equations in the frequency domain, making the solution more straightforward. In addition, including frequency dependence of the system parameters is substantially easier in the frequency domain than in the time domain [28].
The winding electrical parameters (capacitance, inductance, and losses) in the overhang and slot regions of the coil are calculated using the finite-element method (FEM)-based software COMSOL Multiphysics [29], as described in Section 2.2.

Winding model
Wave propagation along the non-uniform multi-conductor transmission system representing the winding is defined by the Telegrapher equations in the Laplace domain as follows [30,31]: In Eq. (1), U(x,s) and I(x,s) are the voltage and current vectors at point x along the line, Z(x,s) and Y(x,s) are the series impedance and shunt-admittance matrices of the line per unit length, respectively. According to Eq. (1), the winding electrical parameters are a function of both space and time. Voltages and currents at the terminals of the segment can be related using the chain matrix definition, assuming constant parameters over a segment Δx and applying boundary conditions at x and x + Δx [31]: where the chain matrix of the segment is defined as In Eq. (3), ψ is the propagation constant matrix of the line, given by ψ ¼ M ffiffiffi λ p M À1 , where M and λ are the eigenvalue and eigenvector matrices of the Z(x,s)Y(x,s) product, respectively. In addition, Y 0 is the characteristic admittance matrix of the line segment, given by The two-port representation defined by Eq. (2) is the basis of the non-uniform model of the machine-winding coil, according to the following procedure [21]: 1. The coil is divided into five segments, as shown in Figures 2 and 6.
2. Each of the five chain matrices is obtained from Eq. (3) as a function of parameters Z and Y, which are different in the geometrical regions identified as overhang and slot.
3. The chain matrices are multiplied according to the cascaded connection shown in Figure 6 in order to obtain a single chain matrix representing the complete coil.
Voltages and currents at the sending (S) and receiving (R) nodes of the complete coil are related as follows: In Eq. (4), Φ 11 , Φ 12 , Φ 21, and Φ 22 are the elements of the complete chain matrix representation of the coil. This representation is transformed into an equivalent admittance matrix form to include the zig-zag connection in order to preserve continuity between turns as a pulse propagates along the coil (see Figure 2) [32,33]. This yields The admittance matrix and chain matrix elements are related according to the following expressions: In Eq. (5), admittance submatrices Y con11 , Y con12 , Y con21 , and Y con22 are included to connect the end of each turn to the beginning of the next one (zig-zag connection), as well as the source and load admittances representing winding terminal connections.
The Laplace-domain voltages at each turn of the coil are obtained from Eq. (5). These voltages are finally transformed into transient voltage responses in the time domain by means of the application of the inverse numerical Laplace transform [28].

Parameter computation
FEM-based simulation program COMSOL Multiphysics is used to compute the capacitance and inductance matrices in the overhang and slot regions of the machine winding [29]. For the calculation of inductance in the slot region, it is assumed that the slot walls behave as magnetic insulation due to eddy currents at the high frequencies related to the pulses produced by frequency converters, meaning that the flux is normal to the boundary and does not penetrate the core. In the overhang region, these walls are replaced by an open-boundary condition.

Capacitance
Capacitance matrix C is calculated using the electrostatics module of COMSOL using the forced voltage method as follows [22]: The charge in all n elements due to conductor i is obtained by exciting conductor i with a fixed voltage while defining all other conductors at zero potential, thus obtaining the i-th column of the capacitive matrix. This process is repeated for each conductor to obtain the complete matrix.

Inductance
Inductance matrix L is computed from the magnetic energy method [22]. The current is nonzero in one or two terminals at a time and the energy density is integrated over the whole geometry. Self-inductance L ii is obtained from the magnetic energy W m, i due to the injection of a current I i to turn i: Mutual inductance L ij is obtained from the magnetic energy W m, ij due to the simultaneous injection of current to turns i and j: When using this method, all self inductances must be computed first and then applied for the computation of mutual inductances.

Losses
The series losses matrix (R) of the winding is considered frequency-dependent and obtained from the concept of complex penetration depth [22]. The dielectric losses matrix (G) is computed using the "electric currents" module in COMSOL [29]. Finally, the series impedance and shunt admittance matrices required by the winding model are computed according to Z = R + sL and Y = G + sC.

Case study
A schematic cross-section of the coil considered in this study is shown in Figure 7. The main parameters of the stator coil are summarized in Table 1. Figure 8 shows a schematic representation and a picture of the experimental setup. Besides the MV form-wound coil under test, it includes a waveform generator (Keysight 33500B), an oscilloscope (Agilent DSO-X 2014A), and a 100-Ω load connected at the end of the coil. Steel plates were included to emulate the EM field distribution in the slot region [34]. The experimental setup was placed in a laboratory facility free of EM interference. More details of this test case can be found in [35].
The capacitance and inductance matrices are computed from FEM simulations using COMSOL Multiphysics, as explained in Section 2.2. Sample simulations are shown in Figures 9  and 10 for the first turn. Figure 9 shows the distribution of electric potential when turn 1 (top turn) is excited with 1 V, while all other turns and the slot walls are grounded. This allows the calculation of the first column of the capacitance matrix of the coil using Eq. (7). Figure 10 shows the distribution of magnetic energy density when turn 1 is excited with 1 A, while all other turns are left unexcited. Integrating this magnetic energy density results in the magnetic energy required in Eq. (8) to compute the self-inductance of turn 1.
The winding model is validated considering a PWM-type excitation with different rise times between 100 and 500 ns connected to the first turn of the coil. The rest of winding is represented       The effect of the excitation rise time is analyzed in a more general manner in Figure 13, which shows the potential difference between turns for different rise times. According to this figure, the potential difference is inversely proportional to the rise time of the excitation.

Modeling of stress-grading systems from rotating machines
Modeling of stress-grading systems of MV rotating machines is an essential tool for their design and optimization. An accurate model can serve several purposes, such as [36]: 1. describing how the system is likely to behave under certain conditions;   2. modifying the material characteristics to match specific requirements; 3. predicting the effects of aging and determining the material's life time in order to establish a maintenance policy.
FEM has become one of the most commonly used techniques for modeling stress-grading systems [37][38][39]. With FEM, the geometry can be considered including most of the details that influence the electric field distribution. To determine how the electric field stress is distributed during PWM voltage excitation, the problem must be solved in the time domain with considerably small time steps. This, together with the high nonlinearity of the semi-conductive stress-grading coating, makes modeling of SG systems a complicated task [40,41].

Modeling of stress-grading systems
In the general case, problems with stress-grading systems can be represented with subdomains of conductors, subdomains of perfect dielectrics (air or another surrounding medium and main insulation), and subdomains of stress-grading materials [42]. Conductors can be considered perfect conducting regions of known potential. Neglecting the dielectric loss component of the materials, the equation to be solved is of the form [43] where σ is the electrical conductivity, ε r is the relative permittivity, and ε 0 is the vacuum permittivity. The electric field dependency (σ E ð Þ) of the semi-conductive-grading coating helps to limit the maximum electric field on the surface of the coil giving adaptability to different designs and voltage levels [44]. For proper FEM simulations, it is required to know the nonlinearity of the electrical conductivity at high electric stress for the semi-conductive-grading material. One of the most common expressions used to define the electric field dependency in S/m of this coating is given by [45,46] where σ 0 = 1.85 x 10 À9 S/m, α= 0.00112, and n = 0.67. Eq. (10) can be solved analytically or numerically. FEM or an Equivalent Electric Circuit Model [36] can be used for this purpose. FEM is widely used to model the SG coatings due to some advantages over, for example, equivalent circuit models. With circuit models, some geometric aspects of the SG systems, like overlapping of coatings or the sharp end of the slot exit, are difficult to be considered. In addition, with FEM, the thermal field problem can be solved using the same geometry, taking as a source of heat the ohmic losses from the electric solution.
Transient FEM with efficient algorithms needs to be used in order to solve the highly nonlinear problem associated with SG coatings. Nowadays, FEM software provides new capabilities that allow performing highly nonlinear simulations. The "electric currents" module of the FEMbased software COMSOL Multiphysics [29] is used in the next sections to compute the electric field distribution and resistive heat produced in the CAT and SGT of form-wound coils under the application of fast rise time pulses.

FEM modeling of stress-grading coatings under PWM voltage waveforms
With the aid of an infrared camera, it is possible to observe the heat generated because of the electric stress grading under fast pulses. In order to show this condition, a 4.2-kV form wound coil was energized with 4-kV squared pulses with a repetition rate of 2000 pulses per second, and the temperature was registered using an infrared camera model Flir-SC500. As it can be seen in Figure 14, the heat distribution is different in comparison to that presented in Figure 5 for power frequency (60 Hz); under this condition, the high electric stress moves from the SGT to the CAT at slot exit. Something important to notice in the infrared image is the presence of hot spots that follow the patterns of the CAT tapping; this will be discussed in Section 3.4.
Usually, modeling of the conductive armor coating and the semi-conductive-grading coatings is done separately; however, under fast pulses, it is important to understand the combined performance of both coatings. Therefore, the analysis of coil ends working with PWM voltages requires the simultaneous simulation of both coatings.

Finite-element dimension reduction on the stress-grading coatings
A useful tool in FEM modeling is the ability of reducing the dimension of the elements used to discretize the stress-grading coating subdomain [29]. These coatings are applied in thin layers that require a very fine element discretization; otherwise, if a coarse discretization of the subdomains is used, numerical instability may result [47]. Assuming that the conductivity of the semi-conductive coating depends only on the tangential component of the electric field, the elements used in the SG coatings can be reduced to one dimension (1D) [48]. If the geometry of the problem is considered in two dimensions (2D), as shown in Figure 15(a), the subdomains for the CAT and SGT can be discretized using one-dimensional elements (electric shielding boundary in COMSOL), thereby considerably reducing the total number of elements and, in most cases, making it easier for the solution to converge.
Consider for simulation the 2D geometry of the coil end shown in Figure 15(a) and the transient voltage of a leading edge of one PWM pulse, shown in Figure 15(b). The pulse has a rise time of 1 μs and an overshoot of around of 1.6 times the nominal phase to ground peak voltage for a 6.6-kV coil. The thickness of the ground-wall insulation is 3 mm, the CAT and SGT are 0.5 mm thick, and their lengths are 50 and 100 mm, respectively. The CAT conductivity is 0.01 S/m, and the conductivity of the SGT is given by Eq. (11). The relative permittivities (ε r ) of the CAT and the SGT are, respectively, 50 and 20. A comparison of the results for the electric field on the CAT and SGT using 1D and 2D elements is presented in Figure 16. As it can be seen in this figure, this 1D simplification does not modify the results when compared to the solution obtained considering 2D elements.
In addition, the simplification from a 2D to a 1D modeling can be indeed useful, especially when non-axial-symmetric geometries need to be modeled in three dimensions. For example, form-wound coils can be modeled in 3D, when considering 2D elements in the stressgrading subdomains [49]. One problem that occurs with this simplification is that geometrical details like overlapping in multilayer systems are sometimes difficult to implement.  However, simulations using discretization of the CAT and SGT with one dimension elements reduction can give a sufficiently good approximation when these details are not required in the simulations.
When the response of an SG system during a PWM waveform needs to be modeled, a large computing time is generally required, since the time step is defined by the fast rise time of the pulses (1 μs or less) and the simulation time is in the order of several milliseconds. Using reduced dimension elements in the CAT and SGT domains makes the simulation computationally more efficient. One example of the convenience of this dimension reduction is shown with the transient simulation of electric field and heat in a coil with 3-mm thickness of the ground-wall insulation, and 1 mm CAT and SGT thickness; the length of the CAT is 50 mm from the slot exit. The CAT conductivity is 0.01 S/m, and the relative permittivity is 100. The length of the SGT is 100 mm. The SGT has a relative permittivity of 20 and a nonlinear conductivity given by Eq. (11). Considering a phase-to-ground three-level PWM voltage [50], which waveform is shown in the inset of Figure 17(a), the magnitude of the tangential electric stress versus time is present along both coatings, as it can be observed in Figure 17(a). The magnitude of the resistive heat density along the CAT and SGT as a function of time and electric field is presented in Figure 17(b). According to this figure, the heat is concentrated mainly in the CAT.
The tangential component of the electric field in the CAT is well below the tangential electric field in the SGT (Figure 17(a)), but the heat generated in this coating is considerably higher (Figure 17(b)). Experimental work on form-wound coils stressed by fast pulses has shown that this condition can damage the CAT, allowing surface PD to appear right at the slot exit [51,52]. The next section introduces the use of FEM modeling to investigate possible solutions to this problem.

Modeling for the design of stress-grading coatings under PWM voltages
As mentioned before, the SG systems in motor coil ends can fail because of the excessive heat generated when they operate under fast rise time pulses like those produced by adjustable speed drives. The excessive heat produced in the CAT can be considered the first problem to be solved in order to improve the performance of these coatings under fast pulses. Figure 18 shows the resistive heat density for the same case presented in Figure 16.
It can be noticed that the heat is produced mainly in the CAT. The heat and electric field in the CAT can be reduced by increasing the conductivity of this coating; however, the high stress will be now moved to the SGT. A possible solution consists of controlling both the material properties and the designs of stress-grading system [26]. For example, a sectionalized stress-grading system consisting of two coatings with different conductive properties can be used in the CAT in addition to the SGT in order to divide the electric field and ohmic losses at the coil end. The first part, starting from the slot end, is a highly conductive coating, and the second part is now referred to as stress-grading coating for high-frequency components (SG HF ). The SG HF coating is designed to filter and relieve the fast rise front of the pulses, while a third coating (SG LF ) is used to grade the lower frequency components of the PWM voltage waveform.
As an example, consider a sectionalized stress-grading system where the thickness of the SG HF and SG LF layers is duplicated (1 mm), and the electric properties of the first layer are conductivity of 0.5 S/m, and ε r = 50, for the second conductive layer (SG HF ): conductivity of 0.05 S/m, and ε r = 50. The third layer SGT is an with the same nonlinear characteristic given by Eq. (11), but with a value of σ 0 = 1.85 Â 10 À8 S/m.
The distribution of resistive heat and electric field is modified, as it can be seen in Figure 19.
With the first conductive layer, the maximum heat is moved to the SG HF and SG LF layers and reduced in around 50%, see Figure 19(a). The electric field is not increased in the SG LF in comparison with the electric field in the original design presented in Figure 16, as shown in Figure 19(b). This is an example of how modeling allows modifying the properties and dimensions of the stress-grading coatings in order to obtain the desired stress grading. Figure 18. Resistive heat distribution along the stress-grading coatings for the same case of the results presented in Figure 14.

Simulation of the tapping of stress-grading systems under fast pulses
As mentioned before, an important characteristic of FEM modeling of stress-grading systems is the possibility of taking into account some details of the geometry that can influence the stress distribution under fast rise time pulses. In tape-based SG systems, the SGT usually overlaps the CAT a couple of centimeters, with one or two full-lapped turns followed by halflapped turns until it reaches the desired length. During FEM modeling, the overlapping interfaces can be reproduced as in the real condition; however, it is not usual to consider the tape interfaces. Optical micrographs of the overlapping tape sections, obtained from formwound coils, have shown that there are transition sections of the coatings with only one tape thickness between segments with two tape thickness [54].
Two-dimensional FEM simulations of the stress-grading system are performed considering the coatings configuration and the voltage waveform from Figure 15, but now the geometrical shape of the half overlapping between tape turns is included. For this case, the conductivity of the CAT was considered as 0.01 S/m, with a relative permittivity of 50, Figure 19.   and the CAT-CAT overlappings occurring every centimeter. The conductivity of the SGT is given by Eq. (11). The shape considered for the tape overlappings is shown in the inset of Figure 20. In the same figure, the simulation results show how the CAT-CAT overlappings present high values of heat, behavior that is experimentally confirmed by the temperature distribution presented in Figure 14. These results demonstrate how the quality of the application of these tape-based coatings must be guaranteed to avoid early problems with the machines fed by PWM drives.

Conclusions
Two crucial topics for the proper insulation design of MV motors excited by PWM drives have been reviewed in this chapter: modeling of the windings for pulse propagation analysis and modeling of the stress-grading system.
A frequency domain non-uniform multi-conductor transmission line approach has been used to study the fast-front transient response of a machine-winding coil when a PWM-type excitation from an inverter is applied. The parameters of the coil were calculated using the finiteelement method. The results when applying a PWM-type excitation to the coil show that the rise time of the source and the length of the cable has an important effect on the transient overvoltage produced at different turns of the coil, as well as in the potential difference between adjacent turns. The comparison between simulation and experimental results, in terms of oscillatory behavior and magnitude, demonstrates that both the winding model and the cable model selected result in a very accurate prediction of the fast-transient response related to the use of inverters in medium-voltage induction machines. Since the winding model is considered as linear for high frequencies, a frequency domain-modeling approach is a very good option to study the fast-transient response of machine windings.
On the other hand, the conductive armor and semi-conductive-grading coatings are important parts of the insulation system in rotating machines. The correct design of these coatings becomes difficult when PWM voltages feed the machine from adjustable speed drives. The stress-grading coatings often cannot effectively relieve the stress during repetitive surge voltages and, in the extreme case, hot spots and/or PDs will gradually destroy the coatings, making the problem worse. FEM modeling is a useful tool in understanding the influence of the various design parameters eliminating the trial and error methods that usually require more time and other resources. By reducing the dimension of the elements used to discretize the stress-grading coatings subdomains, it is possible to reduce the computational burden, but when a detailed geometry of the stress-grading system needs to be included in the simulation, this option cannot be considered. By using FEM modeling, it is possible to compute the maximum electric field, power loss, or voltage at the end of the SGT and use these variables in conjunction with optimization subroutines to obtain improved designs.