Impacts of Harmonic Voltage Distortions on the Dynamic Behavior and the PRPD Patterns of Partial Discharges in an Air Cavity Inside a Solid Dielectric Material

Abstract

The monitoring of partial discharges (PDs) is one of the main methods used worldwide for evaluation and diagnosis of the insulation conditions in equipment powered by medium and high voltages. The occurrence of PDs is usually an indication of the appearance of insulation defects, which over time can compromise the dielectric withstand of the material used, increasing the probability of complete breakdown. In general, laboratory tests for detecting and registering PDs are carried out using purely sinusoidal voltages. However, it is very common for an electrical asset to be subjected at some point in its operating life to voltages distorted by harmonic components. Some studies reported in the literature reveal that harmonic distortions can affect the PDs’ characteristics, nevertheless, the effects of individual harmonic components on PDs still need to be analyzed. In this context, this paper proposes to evaluate the impacts of harmonic voltage distortions on the dynamic behavior and the phase-resolved partial discharge (PRPD) patterns of PDs in an air cavity within a solid dielectric material. For this, a simulation model was implemented, which was used to analyze the effects of applying distorted voltages composed of different harmonic orders (third, fifth, and seventh) and distinct levels of distortion (1%, 3%, 5%, 10%, 15%, and 20%). In addition, the influence of the third harmonic phase angle on PDs is also analyzed. The results extracted from the simulations revealed that the harmonic distortions caused changes in the numbers of PDs per cycle, in the mean apparent charges of the PDs per cycle, and in the PRPD patterns’ characteristics. These changes were very significant for higher distortion levels, which in practice may impair the interpretation of PD measurement records for the diagnosis of the condition of the insulation system.

1. Introduction

Failures in insulation systems are one of the main causes of damage and stoppages of medium and high voltage electrical equipment, such as rotating electrical machines (motors and generators), power transformers, power cables, switchgear, and industrial drives, among others [1]. In this way, monitoring the insulation conditions throughout the equipment’s operating cycle is essential to avoid failures and enable the scheduling of maintenance stops, reducing the generation of losses in the production process of which the electrical asset is part.

A common sign of defects and degradation of the dielectric material that makes up the insulation is the occurrence of partial discharges (PDs). The IEC 60270 standard [2] defines PDs as “electrical discharges that only partially bridge the insulation between conductors and which can or cannot occur adjacent to a conductor”. PDs can appear in different regions of the insulation, due to the existence of imperfections such as cavities and delamination. Over time, the local degradation caused by PDs intensifies and can result in the complete breakdown of the insulation, putting the functioning of electrical equipment at risk [3].

Another factor that contributes to insulation deterioration is the presence of harmonic components that distort the equipment supply voltage waveform [4,5,6,7,8]. In general, harmonic distortions cause increases in electrical and thermal stresses that act on the dielectric material, which can cause changes in its physical properties. Once the insulating capacity of the dielectric material is affected, there are significant reductions in the reliability and service life of the insulation system as a whole [7].

In addition to being an inherently aggravating factor for the degradation and aging of the dielectric material, some studies have reported that harmonic voltage distortions also influence the generation of partial discharges [9,10,11,12,13]. The study carried out by [9] is a pioneer in the reporting of deformations in phase-resolved partial discharge (PRPD) patterns caused by distorted voltages. In this study, variations were also reported in the number of discharges and the PDs charge magnitudes.

The PDs resulting from some distorted voltage waveforms cause acceleration of the aging process of the dielectric material, as presented in [10]. According to [11], the PD measurements in a power transformer fed by two different distorted voltages revealed different PRPD patterns and charge magnitudes for each condition studied. The same study concluded that the changes in PDs characteristics caused by the harmonic components can affect the interpretation of the recorded data, leading to inaccurate diagnoses of the insulation condition. Similar conclusions were also pointed out later in [12], when distorted voltages were applied to an oil-impregnated sample containing an embedded gaseous inclusion.

Furthermore, the impact of voltage harmonics produced by variable frequency drives (VFD) when used to feed electric motors at variable speeds [13] was investigated. In this case, the distorted voltages applied to the stator windings led to large changes in the PDs’ intensity levels, increasing the risk of failure of the motor insulation systems.

Recently, a new method for assessing the behavior of PDs was applied in a needle-hemisphere electrode configuration immersed in mineral oil [14]. This method is based on the identification of the harmonic components of the leakage current measured on the grounded electrode of the experimental system. For this, the experimentally measured data were analyzed using the fast Fourier transform (FFT), which made it possible to obtain the distortion level of each harmonic order that composes the leakage current signal. The results obtained in [14] showed that the harmonic components can be used as a possible source of information about the PDs’ activity, and that there may be a relation between the rise of harmonic components in the leakage current and the increase in the PDs’ intensity, with the fifth harmonic being the most dominant.

The studies mentioned above have shown that distorted voltages affect the production and behavior of PDs. However, few works in the literature analyze the individual contribution of each voltage harmonic order (h) in the production and behavior of PDs. Based on these aspects, the objective of this study is to investigate the following dataset that characterizes PDs: the number of PDs generated per cycle, mean PD apparent charges per cycle, and PRPD patterns. In order to compare the effects on the production and characteristics of partial discharges, distorted voltages with individual harmonic orders (third, fifth, and seventh) and different magnitudes of total harmonic distortion (THD) (1%, 3%, 5%, 10%, 15%, and 20%) were applied in a solid dielectric material with an internal air cavity. For the third harmonic order, the effects of the harmonic phase angle (${\phi }_h$) on the PDs characteristics were also analyzed. The simulations were performed through a dynamic computational simulation model that applies finite element analysis (FEA). The FEA results were used to determine the time-dependent physical quantities that are necessary for modeling the PD phenomenon.

2. FEA Model Geometry and Fundamental Equations

The model used in this work was originally proposed by [15] to simulate PDs that occur in solid dielectric materials due to the presence of internal air cavities. This model has been experimentally validated and extensively tested in different case studies, in which the PDs’ behavior was evaluated by varying the size and location of the cavity [16], the amplitude and frequency of the applied voltage [17], the material temperature [18], and among other circumstances [19,20,21]. To the best of our knowledge, there are no studies on the application of this model to analyze the effects of distorted voltages on the behavior of PDs. Thus, the present work proposes to fill this knowledge gap.

The geometry of the simulation model consists of a cylinder with a height (h_dm) of 2 mm and a radius (r_dm) of 5 mm, filled with a homogeneous dielectric material (epoxy resin) with relative permittivity (ε_dm) of 4.4, into which was inserted a spherical air cavity with a radius (r_c) of 0.775 mm and a surface thickness (h_cs) of 0.05 mm. The cavity surface was implemented in the geometry to model the surface charge decay through conduction along the cavity wall, defining its conductivity as greater than that of the dielectric material [15], as will be explained later in Section 3.3.

Figure 1 shows the geometry details of the model implemented in the finite element simulation software COMSOL Multiphysics^® (Stockholm, Sweden). As shown in Figure 1, axial symmetry was used to transform the three-dimensional (3D) model into a two-dimensional (2D) axisymmetric model. Using axial symmetry, the cylinder and sphere are represented by a rectangle and a semicircle, respectively. This transformation made it possible to reduce the finite element mesh size and, consequently, the computational simulation time, without losing precision in the solution [22].

In the simulations presented in this work, alternating voltages were applied to the high-voltage (HV) electrode located on the upper face of the geometry, while the lower face of the geometry was grounded. To model PDs, it is necessary to know the electric potential (V) and current density ($\overrightarrow{J}$) distributions in the geometry, which are found by solving Equation (1) of Gauss divergence law and Equation (2) of continuity:

$$\nabla ·\overrightarrow{D}={\rho }_v$$

(1)

$$\nabla ·\overrightarrow{J}+\frac{\partial {\rho }_v}{\partial t}=0$$

(2)

where $\overrightarrow{D}$ is the electric displacement field and ρ_v is the volume charge density. Assuming that the dielectric is linear, homogeneous and isotropic, then $\overrightarrow{D}=\epsilon \overrightarrow{E}$, where ε is the permittivity of the medium and $\overrightarrow{E}$ is the electric field [15]. Furthermore, since $\overrightarrow{E}=-\nabla V$ and $\overrightarrow{J}=\sigma \overrightarrow{E}=-\sigma \nabla V$, given that σ is the electrical conductivity, Equation (1) can be re-written in the form of Equation (3):

$$-\nabla ·\left(\sigma \nabla V\right)-\nabla ·\frac{\partial }{\partial t}\left(\epsilon \nabla V\right)=0$$

(3)

Equation (3) is solved for the electric potential by applying the finite element method in the COMSOL Multiphysics^® (Stockholm, Sweden) software. The other electrical quantities such as electric field and current density are calculated from the values obtained for the electric potential in all parts of the model geometry.

3. PD Modeling

3.1. PD Inception Criteria

The occurrence of PDs in an air cavity depends on two conditions being satisfied [23]. First, the electric field inside the cavity (E_c(t)) must be greater than the inception electric field (E_inc). According to [23], E_inc can be estimated through Equation (4). The terms p_c and r_c are, respectively, the cavity air pressure and cavity radius. (E/p_c)_cr, B and n are parameters used to characterize the gas ionization processes.

$$E_{inc}={\left(\frac{E}{p_c}\right)}_{cr}{p}_c\left[1+\frac{B}{{\left(2r_c{p}_c\right)}^n}\right]$$

(4)

Second, there must be at least one free electron in the cavity to initiate the ionization process and trigger the electron avalanche that will propagate through the cavity, constituting the PD [23]. The main sources of free electrons for the discharge initiation are surface emission and volume ionization [15]. Thus, the total electron generation rate (N_e,tot(t)) can be determined mathematically through Equation (5):

$$N_{e,tot}\left(t\right)=N_{e,cs}\left(t\right)+N_{e,cv}$$

(5)

In Equation (5), N_e,cs(t) represents the electron generation rate due to surface emission, and N_e,cv the electron generation rate due to ionization of the cavity volume. In this simulation model, N_e,cv was adopted as a constant, while N_e,cs(t) was considered dependent on the detachment of electrons from charges that were trapped on the cavity surface in previous PDs [15]. In this way, electrons will only be emitted from the cavity surface after the first discharge has occurred.

The number of electrons generated per second due to a previous PD ($N_{e,pre{v}_{pd}}\left(t\right)$) is defined by Equation (6):

$$N_{e,pre{v}_{dp}}\left(t\right)=N_{e,cs0}\left(t\right)\left|\frac{E_c\left(t_{pre{v}_{pd}}\right)}{E_{inc}}\right|$$

(6)

where $E_c\left(t_{pre{v}_{pd}}\right)$ is the electric field in the cavity at the instant of occurrence of the last PD, and $N_{e,cs0}\left(t\right)$ is the rate of electrons generated per second for the initial inception field. Furthermore, the term $N_{e,cs0}\left(t\right)$ is dependent on the polarity of the electric field in the cavity, being subdivided into two other terms: $N_{e,cs0H}$ and $N_{e,cs0L}$, which represent the electron generation rates due to high and low surface emissions, respectively. This subdivision is necessary because when the polarity of the electric field changes between two consecutive PDs, it becomes more difficult to remove electrons from the cavity surface [15]. Thus, $N_{e,cs0}\left(t\right)$ can be expressed by Equation (7):

$$N_{e,cs0}\left(t\right)=\{\begin{array}{c}{N}_{e,cs0H}, \mathrm{if} \frac{E_c\left(t\right)}{E_c\left(t_{pre{v}_{pd}}\right)}>0\\ N_{e,cs0L}, \mathrm{if} \frac{E_c\left(t\right)}{E_c\left(t_{pre{v}_{pd}}\right)}<0\end{array}$$

(7)

Considering that the number of electrons that can be detrapped from the cavity surface decreases over time, then Equation (8) is used to determine $N_{e,cs}\left(t\right)$:

$$N_{e,cs}\left(t\right)=N_{e,pre{v}_{pd}}\left(t\right) exp\left(-\left(\frac{t-t_{pre{v}_{pd}}}{{\tau }_{dec}}\right)\right)exp\left(\frac{E_c\left(t\right)}{E_{inc}}\frac{T_{dm}}{T_{amb}}\right)$$

(8)

where ${\tau }_{dec}$ is the effective charge decay time constant, $T_{dm}$ is the dielectric material temperature, and $T_{amb}$ is the ambient temperature.

The total electron generation rate presented above is associated with the stochastic nature of the partial discharge phenomenon. To consider this statistical aspect in the model, a probability function ($P\left(t\right)$) is defined in order to confirm the occurrence of a PD. In [15], it was verified that $P\left(t\right)$ is proportional to $N_{e,tot}\left(t\right)$ and to the integration time step ($\Delta t$), as expressed by Equation (9):

$$P\left(t\right)=N_{e,tot}\left(t\right)·\Delta t$$

(9)

The value of $P\left(t\right)$ calculated in Equation (9) for each time instant is compared with a random value (between 0 and 1) generated by a random function ($R\left(t\right)$). If $P\left(t\right)$ is greater than $R\left(t\right)$, and $E_c\left(t\right)$ is greater than $E_{inc}$, the PD is confirmed, otherwise there is no PD occurrence.

3.2. Cavity Conductivity

After confirmation of a PD occurrence, the air cavity becomes conductive throughout the discharge period. Thus, in the implemented model, the cavity conductivity (${\sigma }_c\left(t\right)$) is changed from its initial value (${\sigma }_{c0}$) (air conductivity value), to a maximum value (${\sigma }_{cmax}$), while a PD is occurring. Whilst the value of ${\sigma }_c\left(t\right)$ is increasing, $E_c\left(t\right)$ is reduced until reaching the value of the extinction electric field ($E_{ext}$), the instant that the PD phenomenon ends. After the PD extinction, ${\sigma }_c\left(t\right)$ is set back to its initial value. According to [15,24], the value of ${\sigma }_{cmax}$ can be estimated using the electron conductivity in the plasma through Equation (10):

$${\sigma }_{cmax}=\frac{{\alpha }_e{e}^2{N}_e{\lambda }_e}{m_e{c}_e}$$

(10)

In Equation (10), ${\alpha }_e$ is a coefficient related to the electron energy distribution and mean free path, $e$ is the elementary charge, $m_e$ is the electron mass, $c_e$ is the electron thermal velocity, ${\lambda }_e$ is the electron mean free path, and $N_e$ is the electron density defined by Equation (11):

$$N_e=\frac{q/e}{4/3·\pi r_c^2}$$

(11)

where $q/e$ corresponds to the number of electrons in the discharge channel [15].

3.3. Surface Charge Decay

Following the PD extinction, the charges that remain free on the cavity surface will decay through recombination that occurs along the cavity wall, before the next PD occurs [15]. In the implemented model, the movement of charges along the cavity wall was considered to be dependent on the directions of $E_c\left(t\right)$ and of the electric field due to surface charges ($E_{cs}\left(t\right)$), which is calculated using Equation (12):

$$E_{cs}\left(t\right)=E_c\left(t\right)-E_{c0}\left(t\right)$$

(12)

The term $E_{c0}\left(t\right)$ in Equation (12) represents the electric field inside the air cavity when no PD occurs. If an alternating voltage is applied, the direction of $E_c\left(t\right)$ changes over time. When $E_c\left(t\right)$ is opposite to $E_{cs}\left(t\right)$, the charges accumulated on the cavity surface due to the previous PD tend to move towards the centers of the upper and lower surfaces of the cavity, as shown in Figure 2a. In this way, there is no recombination of positive and negative charges and hence no charge decay. On the other hand, when $E_c\left(t\right)$ has the same direction as $E_{cs}\left(t\right)$, the charges accumulated on the cavity surface tend to move in the opposite direction to the regions where they were accumulated, as shown in Figure 2b. In this situation, due to the movement of positive and negative charges in opposite directions along the cavity surface, there is recombination of these charges, which results in charge decay.

In order to consider the effects of the surface charge decay, the simulation model uses a cavity surface conductivity (${\sigma }_{cs}\left(t\right)$) that depends on the directions of $E_c\left(t\right)$ and $E_{cs}\left(t\right)$, and on the intensity of $E_c\left(t\right)$, as presented in Equation (13).

$${\sigma }_{cs}\left(t\right)=\{\begin{array}{c}{\sigma }_{cs0}exp\left(\alpha \left|E_c\left(t\right)\right|\right), \mathrm{if} \frac{E_c\left(t\right)}{E_{cs}\left(t\right)}>0\\ {\sigma }_{cs0}, \mathrm{if} \frac{E_c\left(t\right)}{E_{cs}\left(t\right)}<0\hfill \end{array}$$

(13)

In Equation (13), ${\sigma }_{cs0}$ is the initial cavity surface conductivity value and $\alpha$ is the stress coefficient. It can be seen from Equation (13) that as $E_c\left(t\right)$ increases, ${\sigma }_{cs}\left(t\right)$ can assume very high values. To avoid convergence problems in the simulation, ${\sigma }_{cs}\left(t\right)$ is limited by a maximum value (${\sigma }_{csmax}$) that is estimated from experimental data [15].

3.4. Calculation of True and Apparent Charges of the PDs

When partial discharges occur, short-duration current pulses in the order of nanoseconds are generated. However, it is very common for PDs to be quantified by the magnitudes of the electrical charges involved, which are classified into two types: true charge ($q_{true}$) and apparent charge ($q_{app}$). The true charge is the charge effectively accumulated on the cavity surface due to the PD occurrence. The apparent charge corresponds to the charge that is induced on the electrode where the measuring equipment was installed, which is usually the ground electrode [15].

The magnitudes of the true and apparent charges can be obtained by integrating in time domain the currents $I_c\left(t\right)$ and $I_{elec}\left(t\right)$ that flow during the PD through the cavity and electrode surfaces, $S_c$ e $S_{elec}$, respectively. As the FEA software provides the current densities ${\overrightarrow{J}}_c\left(t\right)$ and ${\overrightarrow{J}}_{elec}\left(t\right)$, it is first necessary to integrate them over the areas of the respective surfaces to determine the currents $I_c\left(t\right)$ and $I_{elec}\left(t\right)$. Next, the time integrations are performed, starting at the moment of the PD inception ($t_{inc}$), until the moment when the PD is extinguished ($t_{ext}$), as described in Equations (14) and (15).

$$q_{true}={{\displaystyle \int }}_{t_{inc}}^{t_{ext}}\left(\underset{S_c}{\overset{}{{\displaystyle \int }}}{\overrightarrow{J}}_c\left(t\right)·d\overrightarrow{S}\right)dt$$

(14)

$$q_{app}={{\displaystyle \int }}_{t_{inc}}^{t_{ext}}\left(\underset{S_{elec}}{\overset{}{{\displaystyle \int }}}{\overrightarrow{J}}_{elec}\left(t\right)·d\overrightarrow{S}\right)dt$$

(15)

3.5. PD Model Parameters

All parameters used to implement the PD simulation model are shown in

Table 1. PD model parameters.

Parameter	Definition	Value
$h_{dm}$	Dielectric material thickness	2 mm
$r_{dm}$	Dielectric material radius	5 mm
$r_c$	Air cavity radius	0.775 mm
$h_{cs}$	Cavity surface thickness	0.05 mm
$V_{applmax}$	Applied voltage amplitude	14 kV
$f$	Applied frequency	60 Hz
$T_{dm}$	Dielectric material temperature	293.15 K
$T_{amb}$	Ambient temperature	293.15 K
$E_{inc}$	Cavity inception field	3.02 kV/mm
$E_{ext}$	Cavity extinction field	0.6265 kV/mm
$\Delta t_H$	Time step between PD events	33.3333 µs
$\Delta t_L$	Time step during PD events	1 ns
${\epsilon }_{dm}$	Relative permittivity of the dielectric material	4.4
${\epsilon }_c$	Relative permittivity of the air cavity	1
${\epsilon }_{cs}$	Relative permittivity of the air cavity surface	4.4
${\sigma }_{dm}$	Conductivity of the dielectric material	1 × 10−13 S/m
${\sigma }_{c0}$	Air cavity conductivity between PD events	0 S/m
${\sigma }_{cmax}$	Air cavity conductivity during PD events	0.005 S/m
${\sigma }_{cs0}$	Initial conductivity of the air cavity surface	1 × 10−13 S/m
${\sigma }_{csmax}$	Maximum conductivity of the air cavity surface	6.2891 × 10−10 S/m
${\tau }_{dec}$	Effective charge decay time constant	0.002 s
$N_{e,cv}$	Electron generation rate due to volume ionization	137.71 s−1
$N_{e,cs0H}$	Electron generation rate due to a high surface emission	2100 s−1
$N_{e,cs0L}$	Electron generation rate due to a low surface emission	300 s−1

. The values of $N_{e,cv}$, $N_{e,cs0A}$, $N_{e,cs0B}$ and $E_{ext}$ were obtained and validated experimentally [15].

3.6. PD Modeling Algorithm

Figure 3 presents the flowchart of the dynamic process of the PDs’ occurrence. The algorithm was implemented in MATLAB^® (Natick, MA, USA) integrated with COMSOL Multiphysics^® (Stockholm, Sweden) as the finite element mesh solver. The initial stage of the simulation consisted of initializing all parameters and variables used in the model. In this step, the applied voltage waveform was defined, either purely sinusoidal or with harmonic distortion. Then the model geometry was created and the finite element mesh was defined. At first, the model was simulated without the occurrence of PDs. This was necessary to determine the term $E_{c0}\left(t\right)$, which was later used to calculate $E_c\left(t\right)$ and then to model the surface charge decay through ${\sigma }_{cs}\left(t\right)$. Once this was undertaken, the main simulation was started by setting $t=$ 0 s.

Two different integration time steps were used in the developed algorithm, one for when the PD was confirmed ($\Delta t_L$) and another for when there was no PD ($\Delta t_H$). As PD occurred in a very short period (in the order of ns), a smaller integration time step was required. On the other hand, when there was no PD confirmation, a larger integration time step was used. This strategy was used to optimize the simulation time without lessening the quality of the results. The MATLAB^® (Natick, MA, USA) code controlled the integration time steps and the PDs’ occurrence. The PD was only confirmed when the two conditions necessary to start the electron avalanche were satisfied, that is, when $E_c\left(t\right)>E_{inc}$ and $P\left(t\right)>R\left(t\right)$. When the PD was confirmed, the cavity conductivity was increased to its maximum value, making the air cavity conductive, which quickly reduced the intensity of $E_c\left(t\right)$. When $E_c\left(t\right)$ became equal to the $E_{ext}$, the PD was interrupted, and the true and apparent charges generated by the PDs were calculated from the values of the current densities ${\overrightarrow{J}}_c\left(t\right)$ and ${\overrightarrow{J}}_{elec}\left(t\right)$, which were stored during the PD process. Afterward, the cavity conductivity was changed to its initial value, and if the total simulation time ($t_{end}$) had not yet been reached, then $t$ was increased with $\Delta t_H$ and the process was repeated. Otherwise, the PRPD pattern was determined for all PDs charges produced during the simulation, all results were stored, the respective graphs were plotted and the simulation ended.

4. Results and Discussion

To evaluate the impact of harmonic voltage components on the production of partial discharges, simulations were carried out for fifty cycles for sinusoidal voltage with a 60 Hz frequency and voltages distorted by third, fifth, and seventh order harmonics, with different magnitudes of THD (1%, 3%, 5%, 10%, 15%, and 20%). In addition, the model was also simulated for different phase angles of the third harmonic component. The results obtained in the simulations are presented and discussed in this section.

4.1. Impacts of Harmonic Voltage Distortions on the PDs Dynamic Behavior

Figure 4 shows the typical behaviors of the $E_{c0}\left(t\right)$, $E_c\left(t\right)$ and $E_{cs}\left(t\right)$ fields and of the current pulses $I_c\left(t\right)$ generated by the PDs in the cavity for the initial case study, when the voltage applied to the high-voltage electrode was purely sinusoidal. The $E_{c0}\left(t\right)$ waveform had the same behavior as the applied voltage waveform ($V_{appl}\left(t\right)$), while the $E_c\left(t\right)$ depended on the discharges that occurred in the cavity over time, and $E_{cs}\left(t\right)$ was obtained through Equation (12). It can be seen in Figure 4 that there are situations in which even when $E_c\left(t\right)$ exceeds $E_{inc}$ there is no PD occurrence. This fact was physically observed and implemented in the simulation model. Such effects occurred due to the stochastic nature of the PD phenomenon and led to delays in the occurrence of PD. When the delay was significant, the PD occurred at an instant when the electric field was strongest in the cavity, which resulted in current pulses with higher magnitudes.

In comparison with the initial case study shown in Figure 4, other simulated results indicated that harmonic voltage distortions produced effects on the dynamic behavior of PDs. To visualize and analyze such effects, distorted voltages with different THD values were applied. Figure 5, Figure 6 and Figure 7 present the waveforms of electric fields and current pulses for third, fifth, and seventh harmonics in the cavity during the PDs’ occurrences. The plotted graphs show the results for the first eight voltage cycles for cases where the applied voltages had THDs equal to 20%. It is also worth mentioning that in these simulations the phase angles of all harmonic components were considered equal to 0°.

With the addition of the third harmonic component, it is observed, in Figure 5a, the appearance of local minima in the waveform of $E_{c0}\left(t\right)$, in the regions where the phase angles of the fields (${\phi }_c$) were 90° and 270°. These minima were capable of stopping the occurrence of PDs, causing large time lags between consecutive discharges, as can be seen in Figure 5b. For this reason, it can be seen in Figure 5b that the number of PDs generated when the third harmonic was present was smaller compared with the initial case study in which the applied voltage was purely sinusoidal.

In Figure 6a, the effect of the fifth harmonic component on increasing the peak value of the $E_{c0}\left(t\right)$ waveform can be observed, creating local maxima at ${\phi }_c=$ 90° and at ${\phi }_c=$ 270°. In addition, local minima were also created at ${\phi }_c=$ 45°, ${\phi }_c=$ 135°, ${\phi }_c=$ 225° and at ${\phi }_c=$ 315°. Note that in the time intervals between the local minima and maxima, the slope $d{E}_{c0}\left(t\right)/dt$ of the $E_{c0}\left(t\right)$ waveform is very high, which caused large amounts of PDs to occur in these regions. This fact, along with the fact that there was an increase in the electric field peak value, justified the increase in the number of PDs generated when the fifth harmonic component was present, as can be seen in Figure 6b. On the other hand, in regions where there were local minima, the PD process was interrupted, also causing delays between consecutive PDs, creating regions called ‘dead zones’, where PDs did not occur.

The impacts of the seventh harmonic component on the $E_{c0}\left(t\right)$ waveform are shown in Figure 7a. The appearance of local minima was observed at ${\phi }_c=$ 33.3°, ${\phi }_c=$ 90°, ${\phi }_c=$ 123.3°, ${\phi }_c=$ 213.3°, ${\phi }_c=$ 270° and ${\phi }_c=$ 303.3°. Again, these local minima caused interruptions of the PDs, also creating dead zones for this case. However, Figure 7a shows the appearance of higher peaks for the $E_{c0}\left(t\right)$ waveform, in comparison with the initial case without harmonic distortion. Furthermore, regions of high slope $d{E}_{c0}\left(t\right)/dt$ appear in the $E_{c0}\left(t\right)$ waveform. For these two reasons, there was an increase in the number of PDs generated for this case, compared with the first case without harmonic distortion, as shown in Figure 7b. Nevertheless, this increase was smaller than it was for the fifth harmonic component, as the seventh harmonic created more local minima, causing additional stops for PDs occurrences.

The mean number of PDs generated per cycle of applied voltage ($N_{PDs/cycle}$) is one of the main indicators of the severity of PDs’ occurrence for the insulation condition [25]. The results of the performed simulations showed that $N_{PDs/cycle}$ varied for different compositions of harmonic orders and THDs, as can be seen in Figure 8a. For the presence of an individual harmonic component, the changes of $N_{PDs/cycle}$ in relation to the undistorted case was greater for higher THD values. For THD = 1%, the changes were minimal (almost imperceptible). However, for THD values above 3%, the voltage waveform distortions by the harmonic components started to affect the PDs activities.

It is noticeable in Figure 8a, that when the applied voltage was purely sinusoidal, an average of 8.24 PDs were generated per cycle. This number will be used as a reference for comparing the average numbers of PDs per cycle obtained for the other simulations when harmonic components are present.

Figure 8a demonstrates that the addition of the third harmonic component leads to a reduction in the number of PDs per cycle, and the reduction is greater as the THD increases. For THD = 20%, the mean number of PDs per cycle was 5.64, that is, there was a reduction of 31.55% compared with the case in which the applied voltage was purely sinusoidal. When the fifth and seventh harmonics were added, the opposite behavior was observed. The addition of the fifth harmonic caused the largest increase in $N_{PDs/cycle}$. For example, according to Figure 8a, for THD = 20%, $N_{PDs/cycle}=$ 10.68, which corresponded to an increase of 29.61% over the reference number. As for the addition of the seventh harmonic, there were also increases in the number of PDs generated as the THD was increased, but these were less significant than in the previous case. In the extreme case where THD = 20%, $N_{PDs/cycle}=$ 9.22 was obtained, which corresponded to an increase of 11.89% compared with the case in which the voltage without distortion was applied.

Variations in the number of PDs per cycle caused by harmonic voltage distortions also affected the mean charges generated per cycle, as shown in Figure 8b. The mean apparent charge generated per cycle ($q_{app/cycle}$) was calculated by dividing the total sum of all PDs apparent charge magnitudes ($q_{app\_tot}$) by the number of simulated cycles ($n_{cycles}$). Thus, $q_{app/cycle}$ depends on the number of PDs per cycle and the magnitudes of the individual charges of each PD. Analyzing Figure 8b, behaviors very similar to Figure 8a can be observed, which shows that the number of PDs per cycle was predominant over the differences in the magnitudes of each PD individual charge.

4.2. Impacts of Harmonic Voltage Distortions on the PRPD Patterns

Figure 9a shows all the apparent charges generated by the PDs and the times of occurrence of each for the initial simulation without harmonic distortion after fifty cycles of the applied voltage. Figure 9b displays the corresponding PRPD pattern. This plot shows the amplitude of the apparent charge generated by each PD in relation to the phase angle of the alternating voltage at the moment the PD occurred [26]. The PRPD pattern is the main way of representing the results of partial discharge measurements, being widely used as a diagnostic tool for the insulation conditions of medium and high voltage equipment. With a correct interpretation of the PRPD pattern, it is possible to identify the type of PD, its severity, and the location in the insulation system where it occurred [27,28,29].

Figure 10, Figure 11 and Figure 12 show the PRPD patterns obtained for the simulations with the presence of the third, fifth, and seventh harmonic components, respectively; and for different percentages of THD. The effects explained in the previous subsection can also be seen in the obtained PRPD patterns, which underwent significant changes as the THDs values became higher.

Regarding the addition of the third harmonic component, it can be noted in Figure 10 that as the THD increased, the PDs occurred in smaller phase angles ranges, as there was the creation of dead zones in the regions where the local minima appeared in ${\phi }_c=$ 90° and in ${\phi }_c=$ 270°. For example, for very high values of THD (Figure 10e,f), the shift in the charge concentration to the regions outside the minima can be clearly seen. As shown in Figure 8a, comparing it with Figure 10, it appears that the total number of PDs generated is smaller as the magnitude of the third harmonic increases.

Alternatively, in the graphs of Figure 11, substantial changes can be observed in the PRPD pattern as the distortion of the voltage waveform by the fifth harmonic becomes greater. It is noted that for higher values of THDs, dead zones were created in the regions of local minima and there were concentrations of PDs in the regions of higher slopes $d{E}_{c0}\left(t\right)/dt$. Nonetheless, unlike the previous case, the total number of PDs produced was greater as the fifth harmonic influence on the voltage waveform increased. Figure 12 also shows that the seventh harmonic component significantly changed the PRPD pattern as the THD increased. In this case, there was the appearance of more dead zones than in the previous cases, as there was a greater number of local minima. Likewise, as in the previous cases, the PDs were concentrated in the regions of high slopes $d{E}_{c0}\left(t\right)/dt$.

Changes in the PRPD patterns shown in Figure 10, Figure 11 and Figure 12, caused by harmonic voltage components, may impair the identification of the PDs’ sources and the evaluation of their severity. From scientific publications and international standards, it is known that each PD source produces a characteristic PRPD pattern [30,31,32,33,34,35], which normally allows a monitoring system to immediately recognize the type of defect and the intensity of the PDs, after comparing the characteristics of the PRPD pattern obtained with the previously registered PRPD patterns. As shown in Figure 10, Figure 11 and Figure 12, the PRPD pattern is completely altered due to changes in the numbers of PDs generated per cycle and the creation of dead zones caused by harmonic components. As a consequence, it becomes difficult to correlate the PRPD pattern obtained with the typical PRPD patterns of the different sources of PDs, which may hinder the condition monitoring of the insulation system.

4.3. Impacts of Changing the Third Harmonic Component Phase Angle on the PDs Characteristics

The results presented in Section 4.1 showed that the mean number of PDs per cycle and the mean PDs apparent charges per cycle were reduced when the applied voltage was distorted by the third harmonic component. These results were obtained considering the third harmonic phase angle equal to 0°. However, it is known that due to the presence of non-linear loads and other disturbances in the electrical system, different types of harmonic components can arise, including those with phase angles other than 0°. Therefore, in order to evaluate the effects of the third harmonic phase angle on the generation of PDs, the implemented model was simulated for three other different phase angles (90°, 180°, and 270°). For the three simulations, the total harmonic distortion was kept the same as in the case of Figure 5, that is, THD = 20%.

The plots of the electric fields and current pulses of the PDs in the cavity obtained in these simulations are shown in Figure 13, Figure 14 and Figure 15. These three figures show that, unlike the Figure 5 case with ${\phi }_h=0°$, the effects of the phase angles ${\phi }_h=90°$, ${\phi }_h=180°$ and ${\phi }_h=270°$ led to increases in the peak values of the $E_{c0}\left(t\right)$ waveforms, creating local maxima for the three cases. It was also verified that the distortions created by the third order components with the shifted phase angles created regions of high slope $d{E}_{c0}\left(t\right)/dt$ in the $E_{c0}\left(t\right)$ waveform. These two effects led to increases in the values of $N_{PDs/cycle}$ and $q_{app/cycle}$, as can be seen in

Table 2. PDs’ simulated results for different 3rd harmonic phase angles with THD = 20%.

Third Harmonic Phase Angle	Number of PDs per Cycle	Mean Apparent Charge per Cycle
0°	5.64	3.61 nC
90°	9.16	4.94 nC
180°	10.52	5.42 nC
270°	9.72	5.07 nC

. These increases were greater for the case where ${\phi }_h=180°$, as this configuration was the one that generated the highest peak value.

Changes in the phase angle of the third harmonic also caused changes in the PRPD patterns obtained, as can be seen in Figure 16. Comparing the three graphs in Figure 16 with the graph in Figure 10f, it can be seen that the number of PDs generated for the three cases was considerably increased. Furthermore, the changes in ${\phi }_h$ caused shifts in the peaks of the voltage and $E_{c0}\left(t\right)$ waveforms, so there were also charge concentrations at different phase angle intervals. For example, for ${\phi }_h=90°$, the peak value of $E_{c0}\left(t\right)$ was shifted to the right to a value greater than 90°, causing the widening of the PDs’ occurrence ranges. For ${\phi }_h=270°$, the peak value of $E_{c0}\left(t\right)$ was shifted to the left to a value smaller than 90°, which also affected the intervals of PDs’ occurrence. In this way, it appears that in addition to the harmonic order and the THD, the phase angle of the harmonic component also had a significant influence on the PDs activities and the PRPD patterns. Therefore, this variable must also be evaluated for the correct analysis of the PRPD patterns recorded in the PD measurement equipment.

4.4. Comparison between Simulated Results and Experimentally Measured Results Reported in the Literature

The results of the simulation model presented in this work showed similar trends when compared with the experimental measurements results found in the literature [11,12]. First, in [12], PD measurements performed on a test object containing a gaseous inclusion embedded in an electrical insulation revealed a greater number of PDs generated when the fifth harmonic component was added to the applied voltage waveform, whereas for the addition of the third harmonic component a reduction in the number of PDs generated was reported. These same trends were observed in the results of this article and were explained in detail previously in Section 4.1.

Regarding the effects of harmonic components on the PRPD patterns, Section 4.2 verified the creation of dead zones in regions where local minima were created and a high concentration of PDs in regions of high slope $d{E}_{c0}\left(t\right)/dt$ in the waveform of E_c0(t). These same effects were also observed on the PRPD patterns obtained experimentally in [11,12]. Furthermore, in [11,12] there was a trend to increase the number of PDs generated per cycle when the phase angle of the third harmonic component varied from 0° to 180°, which was also demonstrated in this work in Section 4.3. The similarities mentioned above between the trends of simulated results and experimental results found in the literature corroborate qualitatively to the results presented in this article.

5. Conclusions

This work analyzed the impacts of the third, fifth and seventh harmonic components for different levels of THD on the PDs generated in a cavity inside a cylinder filled with dielectric material. For this, a computer simulation model based on FEA was implemented, with which it was possible to obtain the dynamic behavior of the PDs in the time domain and the PRPD patterns for each applied voltage waveform.

The results obtained from the simulations showed that the three harmonic components analyzed were able to change the mean number of PDs generated per cycle, the mean PDs apparent charges per cycle, and the PRPD patterns. These changes were more significant as the THD was increased. The fifth harmonic was the component that caused the greatest increase in the mean number of PDs per cycle, while the third harmonic reduced this number. Despite this, when the third harmonic phase angle was modified, significant increases in the number of PDs generated and changes in the PRPD patterns were also noticed. This indicates that the phase angle of the harmonic component can also affect the PDs’ characteristics.

Furthermore, it was found that the local minima created in the waveforms of the applied voltage and the electric field in the cavity could interrupt the PD process, creating dead zones in the PRPD patterns, where there was no PD occurrence. On the other hand, the deformations caused by harmonics also created high slope intervals in the applied voltage waveform, causing higher concentrations of PDs charges in these specific regions. These effects indicate that the proper interpretation of PRPD patterns and the recognition of PD sources may be impaired due to the presence of harmonic components.

The simulations performed in this article were limited to the first fifty cycles of the applied voltage, when the dielectric material was at the beginning of its operating life with full performance. However, for a more accurate diagnosis of the insulation condition, the aging of the dielectric material must also be considered, which will be addressed in future work. The results presented in this paper are intended to serve as a groundwork for future experimental studies. For this, an experimental set up arrangement is being planned to submit an epoxy resin with an internal air cavity to electrical stresses caused by distorted voltages with different harmonic orders and THD levels, as was conducted in the simulations. The idea is to use the electrical method with coupling capacitor based on the IEC 60270 standard [2] for PD data acquisition. In addition, the authors intend to apply distorted voltages on the electrodes for different aging stages of the dielectric material, which will allow for more assertive conclusions regarding the individual impact of each harmonic component on the PDs’ characteristics during the entire operating life of the insulation system. Moreover, this research may also be extended to analyze the application of distorted voltages in insulation systems of more complex test objects, such as Roebel bars used in stators of large electrical machines.