Study on the Propagation Characteristics of Partial Discharge in Switchgear Based on Near-Field to Far-Field Transformation

Ultra-high frequency (UHF) electromagnetic (EM) signals generated by the partial discharge (PD) process of high-voltage equipment are now widely used in PD detection. The computation of EM propagation generated by a local discharge source using a uniformly hardwiring source can hardly reveal the discharge characteristics. In this paper, a method of near-field to far-field transformation is proposed to realize the study of the propagation characteristics of the PD signal. A short gap discharge model is established to get the near-field electromagnetics and the proposed method is validated by comparing the directly calculated results with the results of the near-field source. In the end, a model of switchgear is employed to study the propagation characteristics of the EM signal based on the proposed method. Via numerical calculation, the influence of the equipment in the switchgear on the propagation of the discharge EM is studied. It is found that the direction of the discharge source has a significant effect on the distribution of the electric field, which indicates that the discharge source cannot be simplified to a uniformly hardwiring source. In addition, it is also obtained that the amplitude of the electric field shows the same trend with the growth of the discharge channel, which gives a method for evaluating the development of the PD. Particularly, the near-field to far-field transformation can provide an effective method for studying the propagation of discharge EM waves in large-scale equipment.


Introduction
With the advantages of high integration and reliability, high-voltage (HV) switchgear is widely used in the power grid and has already become an essential equipment in power transmission and transformation system [1][2][3][4].As a potential threat to the insulation safety of HV switchgears, partial discharge (PD) directly affects the safety and reliability of power supply [5][6][7].When a partial discharge occurs, the generated electromagnetic signal contains abundant information of insulation states and ultra-high frequency (UHF) components is widely used to detect and diagnose insulation of HV equipment [8][9][10][11].
Currently, researchers have designed many external and built-in UHF antennas based on IEC-62487 for online PD detection in transformers, generators, gas-insulation switches (GIS) and so on [12][13][14][15][16].In order to improve the detection of partial discharge, some researchers, such as Albarracín, Energies 2018, 11, 1619 2 of 12 R. considers the influence of metal surroundings on UHF antenna [14].In parallel with UHF Antenna design, the propagation of UHF signals generated by PD is also the focus of research.To study the propagation characteristics of the UHF signal, researchers have adopted the finite-difference time-domain (FDTD) method to simulate the discharge signal.Considering that the size of the switchgear is too large and the computer memory cannot meet the requirement, researchers simplified the PD source to a uniformly hardwiring source, which leads to the minimum mesh size reaches one tenth of the wavelength λ [15,[17][18][19].This provides a good method for studying the propagation of discharge signal in a wide area of space.While for the occurrence of discharge signal is in a limited size of electrical equipment, UHF electromagnetic (EM) wavelength λ (ranges from 10 to 100 cm) is comparable to the size of the electrical equipment.In this case, the PD source cannot be regarded as a uniformly hardwiring source, which leads to an inaccurate calculation.
In recent years, with the development of high-performance computers and numerical algorithms, many scholars have devoted themselves to studying atmospheric pressure gas discharges.Currently, a number of researchers have proposed some fluid chemistry models, which are used to describe the phenomenon of PD [20][21][22][23].Peng Qingjun and Zhang Yun established a hydrodynamic drift-diffusion model to describe the spatial-temporal evolution of electrons and ions in the discharge process, in which they obtained the radiation of EM waves propagated into the space.At the same time, the length and radius of the discharge channel were obtained, which provides the possibility of establishing an equivalent discharge source to study the streamer propagation [24,25].
In order to obtain the propagation characteristics of PD in limited large size electrical equipment, the near-field to far-field transformation is adopted in our calculation.First of all, an equivalent PD source is established.Secondly, a Huygens' surface is surrounding the discharge signal to get the near-field distribution of electromagnetic field based on the near-field to far-field transformation.Furthermore, the directly calculated electric field and directivity diagram are compared with these radiated results of the near-field source to validate the proposed method.Finally, the near-field distribution of EM is used to replace a discharge signal in a real sized switchgear and the propagation characteristics are obtained.

Near-Field to Far-Field Transformation Method
When PD occurs in HV equipment, the near-field to far-field transformation used for the study the signal propagation is based on Huygens' Principle (the equivalence theorem) [26][27][28].The radiating sources of PD are enclosed inside surface S, as shown in Figure 1.If the EM field outside the enclosed surface S is the only field of interest, one can substitute the discharge sources with equivalent electric and magnetic currents (J and J m ) placed on the surface of S. based on Love's equivalence principle [29], the field within the closed surface S is set to zero, and the equivalent sources become where H S and E S denote the external electric field and magnetic field of the closed surface S, respectively.The EM field generated by those equivalent electric and magnetic currents satisfies the following equation Here E is the electric field and H is the magnetic field in the passive space; the wavenumber k is defined as k = ω √ µε and its unit is rad/m; ω is the radian frequency, rad/s; µ is the permeability and ε is the permittivity.According to vector Green's theorem [30], Here the vector P and Q have successive second derivatives.Let assume P = E(r), Q = e n G(r, r ); e n is the unit normal vector; r' denotes the distance of source point and r denotes the distance of the field point.Then Considering the Green function Combined with Equation ( 1), the electric field distribution can be obtained For the passive area, the value of the first integral item on the right of the Equation ( 8) is zero.The electric field E(r) outside surface of S can be calculated by Similarly, the magnetic field H(r) outside surface of S can be derived as Energies 2018, 11, x FOR PEER REVIEW 3 of 11 Here the vector P and Q have successive second derivatives.Let assume ( ) ( , ') n G = Q e r r ; en is the unit normal vector; r' denotes the distance of source point and r denotes the distance of the field point.Then  (6) Considering the Green function Combined with Equation (1), the electric field distribution can be obtained e H e E e E r r (8) For the passive area, the value of the first integral item on the right of the Equation ( 8) is zero.The electric field E(r) outside surface of S can be calculated by Similarly, the magnetic field H(r) outside surface of S can be derived as It can be obtained from Equations ( 9) and (10) that EM field (E(r), H(r)) in the passive area generated by PD can be obtained by calculating the distributions of electric field ES and magnetic field HS on the closed surface of S. Thus the entire calculation process of the near-field to far-field transformation can be divided into two steps: first, the EM distributions on the closed surface of S are calculated as the near-field source; then the calculated near-field source is used as a secondary source to compute distributions of electric field E and magnetic field H on the passive area.It can be obtained from Equations ( 9) and ( 10) that EM field (E(r), H(r)) in the passive area generated by PD can be obtained by calculating the distributions of electric field E S and magnetic field H S on the closed surface of S. Thus the entire calculation process of the near-field to far-field transformation can be divided into two steps: first, the EM distributions on the closed surface of S are calculated as the near-field source; then the calculated near-field source is used as a secondary source to compute distributions of electric field E and magnetic field H on the passive area.

Discharge Model Setup
When PD occurs, these generated up to UHF EM waves provide a method to evaluate the PD severity states.Peng et al. found out that the short air gap discharge forms an obvious discharge channel and channel radius [24,25,31].Based on the above characteristics, an equivalent schematic diagram of a discharge model is shown in Figure 2.

Discharge Model Setup
When PD occurs, these generated up to UHF EM waves provide a method to evaluate the PD severity states.Peng et al. found out that the short air gap discharge forms an obvious discharge channel and channel radius [24,25,31].Based on the above characteristics, an equivalent schematic diagram of a discharge model is shown in Figure 2.Where Is denotes the applied pulse current, Rs is the equivalent input impedance, U denotes voltage drop in the streamer discharge channel.Since the length of the discharge channel is much smaller than the size of system, the discharge channel can be equivalent as a dipole antenna [32].Based on the characteristics of the discharge channel in reference [25], the maximum radius of the streamer discharge in a 5 mm air gap reaches 0.3 mm.Therefore, a typical length of the equivalent dipole antenna is set to 5 mm and its equivalent radius is set as 0.3 mm.A Gaussian pulse is added on the equivalent dipole antenna as the PD current pulse [33].
Here t0, τ and I0 denote the time at the peak value of the pulse current; the width of the pulse current, the amplitude of the pulse current, respectively [34].The capacity of the discharge pulse Q is calculated by integrating Equation ( 12) A typical Gaussian pulse with a bandwidth of 3 GHz is set as a PD pulse, which is shown in Figure 3.The amplitude I0 is set as 1 ampere and its pulse width τ is 560 ps and t0 is 450 ps, thus the total capacity of the discharge pulse Q is 280 pC.Where I s denotes the applied pulse current, R s is the equivalent input impedance, U denotes voltage drop in the streamer discharge channel.Since the length of the discharge channel is much smaller than the size of system, the discharge channel can be equivalent as a dipole antenna [32].Based on the characteristics of the discharge channel in reference [25], the maximum radius of the streamer discharge in a 5 mm air gap reaches 0.3 mm.Therefore, a typical length of the equivalent dipole antenna is set to 5 mm and its equivalent radius is set as 0.3 mm.A Gaussian pulse is added on the equivalent dipole antenna as the PD current pulse [33].
Here t 0 , τ and I 0 denote the time at the peak value of the pulse current; the width of the pulse current, the amplitude of the pulse current, respectively [34].The capacity of the discharge pulse Q is calculated by integrating Equation ( 12) A typical Gaussian pulse with a bandwidth of 3 GHz is set as a PD pulse, which is shown in Figure 3.The amplitude I 0 is set as 1 ampere and its pulse width τ is 560 ps and t 0 is 450 ps, thus the total capacity of the discharge pulse Q is 280 pC.

Discharge Model Setup
When PD occurs, these generated up to UHF EM waves provide a method to evaluate the PD severity states.Peng et al. found out that the short air gap discharge forms an obvious discharge channel and channel radius [24,25,31].Based on the above characteristics, an equivalent schematic diagram of a discharge model is shown in Figure 2.Where Is denotes the applied pulse current, Rs is the equivalent input impedance, U denotes voltage drop in the streamer discharge channel.Since the length of the discharge channel is much smaller than the size of system, the discharge channel can be equivalent as a dipole antenna [32].Based on the characteristics of the discharge channel in reference [25], the maximum radius of the streamer discharge in a 5 mm air gap reaches 0.3 mm.Therefore, a typical length of the equivalent dipole antenna is set to 5 mm and its equivalent radius is set as 0.3 mm.A Gaussian pulse is added on the equivalent dipole antenna as the PD current pulse [33].
Here t0, τ and I0 denote the time at the peak value of the pulse current; the width of the pulse current, the amplitude of the pulse current, respectively [34].The capacity of the discharge pulse Q is calculated by integrating Equation ( 12) A typical Gaussian pulse with a bandwidth of 3 GHz is set as a PD pulse, which is shown in Figure 3.The amplitude I0 is set as 1 ampere and its pulse width τ is 560 ps and t0 is 450 ps, thus the total capacity of the discharge pulse Q is 280 pC.

Near-Field Test
For simplicity, Figure 4 shows an equivalent dipole antenna placed along the z-direction.A Huygens' surface (20 × 20 × 20 mm) is surrounding the discharge model to carry out the near-field test.The model of equivalent dipole antenna is constructed using CST Microwave Studio and the entire calculation is carried out via this software.By numerical simulation, the EM field on the Huygens' surface is obtained as is shown on the right side of the figure, which is used as an equivalent radiating source.Then the electric field of the surface at a distance of 60 mm was calculated by the near-field to far-field transformation.Finally, these comparisons of electric field and far field directivity between Huygens' solution and direct calculation were analyzed.

Near-Field Test
For simplicity, Figure 4 shows an equivalent dipole antenna placed along the z-direction.A Huygens' surface (20 × 20 × 20 mm) is surrounding the discharge model to carry out the near-field test.The model of equivalent dipole antenna is constructed using CST Microwave Studio and the entire calculation is carried out via this software.By numerical simulation, the EM field on the Huygens' surface is obtained as is shown on the right side of the figure, which is used as an equivalent radiating source.Then the electric field of the surface at a distance of 60 mm was calculated by the near-field to far-field transformation.Finally, these comparisons of electric field and far field directivity between Huygens' solution and direct calculation were analyzed.Since there are six faces in Huygens' box, for clarity, the electric field on the face of x-z plane was used as the test analysis.1500 MHz was selected as the test frequency.The directly calculated tangential electric field components of Ex and Ez were compared with the results obtained from the near-field to far-field transformation.It can be seen from Figure 5 that the comparison of electric fields at 60 mm reveals a similar behavior.Although some numerical noise is present in the calculated results, these noises have little effect on the accuracy of the far-field calculation, which can be verified by the far-field pattern.Figure 6 shows the far-field directivity results for 1500 MHz on the x-z plane and angular widths for Huygens' solution and direct calculation are approximately the same, which also can be obtained that there is a significant difference between the uniformly hardwiring source and dipole antenna in far field directivity.Results of the direct simulation and the Huygens' simulation using the equivalent near-field source are nearly identical, providing evidence that the proposed method was correctly implemented.These comparisons of two methods in computer consumption are listed in Table 1.Comparing the run time spent in Huygens' solution accounts for 48.78% of that in the direct calculation.The consumed maximum memory in Huygens' solution is 0.62% of that spent in the direct calculation, which is mainly because the model itself is relatively small and the advantage is not particularly obvious.In summary, the method of Huygens' calculation obviously improves the calculation efficiency.Since there are six faces in Huygens' box, for clarity, the electric field on the face of x-z plane was used as the test analysis.1500 MHz was selected as the test frequency.The directly calculated tangential electric field components of E x and E z were compared with the results obtained from the near-field to far-field transformation.It can be seen from Figure 5 that the comparison of electric fields at 60 mm reveals a similar behavior.Although some numerical noise is present in the calculated results, these noises have little effect on the accuracy of the far-field calculation, which can be verified by the far-field pattern.Figure 6 shows the far-field directivity results for 1500 MHz on the x-z plane and angular widths for Huygens' solution and direct calculation are approximately the same, which also can be obtained that there is a significant difference between the uniformly hardwiring source and dipole antenna in far field directivity.Results of the direct simulation and the Huygens' simulation using the equivalent near-field source are nearly identical, providing evidence that the proposed method was correctly implemented.These comparisons of two methods in computer consumption are listed in Table 1.Comparing the run time spent in Huygens' solution accounts for 48.78% of that in the direct calculation.The consumed maximum memory in Huygens' solution is 0.62% of that spent in the direct calculation, which is mainly because the model itself is relatively small and the advantage is not particularly obvious.In summary, the method of Huygens' calculation obviously improves the calculation efficiency.

Propagation Characteristics of Discharge Signal in Switchgear
Based on the above verification, the near-far field to far-field transform method is adopted for the study of the propagation characteristics of the PD signal in large-scale HV equipment.To achieve this, a typical model of switchgear with the dimensions of 1110 × 1250 × 2620 mm is established in CST Microwave Studio.Figure 7 shows the model which consists of a cabinet, bus-bar, insulators, breaker, lightning arrester and bushings.S1~S4 represent the near-field sources and P1~P5 the electric field probes in Figure 7. Table 2 lists these materials parameters of the main component in switchgear.Since the size of the PD source is much smaller than the large-scale switchgear, adopting the method of direct calculations will cause simulation difficulties.Hence, in this section, the EM field on the Huygens' surface (20 × 20 × 20 mm) obtained in Section 3.2 is used as the nearfield source to study the propagation of PD signal in switchgear.

Propagation Characteristics of Discharge Signal in Switchgear
Based on the above verification, the near-far field to far-field transform method is adopted for the study of the propagation characteristics of the PD signal in large-scale HV equipment.To achieve this, a typical model of switchgear with the dimensions of 1110 × 1250 × 2620 mm is established in CST Microwave Studio.Figure 7 shows the model which consists of a cabinet, bus-bar, insulators, breaker, lightning arrester and bushings.S 1 ~S4 represent the near-field sources and P 1 ~P5 represent the electric field probes in Figure 7. Table 2 lists these materials parameters of the main component in switchgear.Since the size of the PD source is much smaller than the large-scale switchgear, adopting the method of direct calculations will cause simulation difficulties.Hence, in this section, the EM field on the Huygens' surface (20 × 20 × 20 mm) obtained in Section 3.2 is used as the near-field source to study the propagation of PD signal in switchgear.

Influence of Insulator on the Propagation of Discharge Signals
Insulators are mainly used as the HV insulation protection, which are largely and widely distributed in switchgear.The EM waves of discharge pulse are inevitably influenced by these insulators.To study the propagation characteristic of EM wave, the near-field source S1 (150 mm, 760 mm, 0 mm) is placed on the surface of an insulator and two probes are located in the internal space of the switchgear.One electric field probe P1 (150 mm, 900 mm, 0 mm) is placed on the same side of the equivalent source, of which position is near to the equivalent source.For another electric field probe P2 (150 mm, 520 mm, 0 mm) is placed on the opposite side of the near-field source and its position is far from the discharge source.
Figure 8 illustrates the EM waveforms detected by the electric field probe at P1 and P2.It can be obtained from Figure 8a that the amplitude of EM shows significant attenuation with the spread distance.By comparing the Fast Fourier Transform (FFT) waveforms in Figure 8b, it can be found that the low frequency components are obviously weakened by the insulator.Thus, the PD detection should avoid the shade from the insulator.

Influence of Insulator on the Propagation of Discharge Signals
Insulators are mainly used as the HV insulation protection, which are largely and widely distributed in switchgear.The EM waves of discharge pulse are inevitably influenced by these insulators.To study the propagation characteristic of EM wave, the near-field source S 1 (150 mm, 760 mm, 0 mm) is placed on the surface of an insulator and two probes are located in the internal space of the switchgear.One electric field probe P 1 (150 mm, 900 mm, 0 mm) is placed on the same side of the equivalent source, of which position is near to the equivalent source.For another electric field probe P 2 (150 mm, 520 mm, 0 mm) is placed on the opposite side of the near-field source and its position is far from the discharge source.
Figure 8 illustrates the EM waveforms detected by the electric field probe at P 1 and P 2 .It can be obtained from Figure 8a that the amplitude of EM shows significant attenuation with the spread distance.By comparing the Fast Fourier Transform (FFT) waveforms in Figure 8b, it can be found that the low frequency components are obviously weakened by the insulator.Thus, the PD detection should avoid the shade from the insulator.Comparing the time-domain waveforms, it can be obtained that the first-arrival EM at P 3 has an amplitude 0.0164 V/m while the amplitude at P 4 is 0.084 V/m, which is due to the electric field attenuated with the growth of distance.Meanwhile, it can be found from the FFT waveforms that bus-bar have a slight effect on the propagation of high-frequency EM waves.This is mainly due to the fact that the size of the bus-bar is much smaller than the wavelength of the EM.The EM produces a diffraction effect and the bus-bar has limited absorption on EM.

Propagation Characteristics of Discharge Signals in Different Direction
Due to the limited size and complex structure of HV switchgear, the discharge source on the insulation structure of switchgear are needed to consider its direction.Hence, two near-field sources S 1 /S 3 (150 mm, 760 mm, 0 mm) of discharge signal are placed at the same place on the surface of the supporting insulator apart from their directions are z-direction and x-direction, respectively.Two electric field probes are located at P 1 (150 mm, 900 mm, 0 mm) and P 2 (150 mm, 520 mm, 0 mm) to detect the electric field waveform.Figure 10 shows the comparison of the electric field waveforms detected by two probes.Through the time-domain waveforms, it is shown in Figure 10a that the electric field E z at the point P 1 generated by near-field source S 1 is evidently greater than the value generated by near-field source S 3 .However, at the detection point P 2 , the comparison of the electric fields is obviously opposite to that of in Figure 10b.Thus, the direction of the discharge signal has a significant effect on the distribution of electric field.Therefore, when studying the characteristics of the discharge signal, the discharge source cannot be simplified to a uniformly hardwiring source.

Propagation Characteristics of Discharge Signals in Different Length
With the development of the discharge process, the length of the discharge channel gradually increases.To study the EM wave propagation characteristics for different length of discharge source, five discharge channels with 1 mm, 2 mm, 3 mm, 4 mm and 5 mm length are built to get these near field sources as is mentioned method in Section 3, respectively.Then these equivalent sources are respectively placed at S 4 (135 mm, 65 mm, 0 mm).An electric field probe P 5 (−450 mm, 360 mm, 200 mm) is placed near to the inner surface of the switchgear to detect the EM waves.The maximum electric field component in z direction (E z_max ) is plotted in Figure 11.According to Figure 11, the E z_max of discharge signal shows a significant growth with the increase of the discharge channel.This is mainly because the electromagnetic radiation capability of the discharge channel increases with its length, which can be used to judge the development status of the discharge channel in the long-term monitoring process.mm) is placed near to the inner surface of the switchgear to detect the EM waves.The maximum electric field component in z direction (Ez_max) is plotted in Figure 11.According to Figure 11, the Ez_max of discharge signal shows a significant growth with the increase of the discharge channel.This is mainly because the electromagnetic radiation capability of the discharge channel increases with its length, which can be used to judge the development status of the discharge channel in the long-term monitoring process.

Conclusions
In this paper, a PD model is established to replace the uniformly hardwiring source for the study of propagation characteristics in high voltage switchgear.To solve the difficulties in numerical calculation, as the fine meshed discharge model leads to insufficient computer memory and longer simulation time in large-scale equipment, the near-field to far-field transformation method is adopted in simulation.The proposed method is verified with direct calculation in surface electric field and far field directivity.Then, the near-field source obtained based on this method is used to study the propagation of electromagnetic waves in the large-scale HV switchgear.The simulation results indicate that insulators would affect the EM waves in low frequency components, and that bus-bars cause diffraction effects on EM waves.It is also found that the direction of the discharge has a significant effect on the distribution of the electric field and the maximum value of the EM field increases with the growth of the discharge channel.This research work provides a method for studying the EM propagation of a PD signal in large scale equipment.

Figure 4 .
Figure 4. Test model and its near-field source.

Figure 4 .
Figure 4. Test model and its near-field source.

Figure 5 .
Figure 5.Comparison of calculated electric field and Huygens' electric field on the surface at 60 mm.

Figure 6 .
Figure 6.Comparison of far field directivity between Huygens' solution and direct calculation.

Figure 5 .
Figure 5.Comparison of calculated electric field and Huygens' electric field on the surface at 60 mm.

Figure 6 .
Figure 6.Comparison of far field directivity between Huygens' solution and direct calculation.

Figure 6 .
Figure 6.Comparison of far field directivity between Huygens' solution and direct calculation.

Figure 10 .
Figure 10.Comparison of time-domain waveforms; (a) Waveforms detected at point P1; (b) Waveforms detected at point P2.Figure 10.Comparison of time-domain waveforms; (a) Waveforms detected at point P 1 ; (b) Waveforms detected at point P 2 .

Figure 11 .
Figure 11.Ez_max of EM wave for different length of discharge channel.

Figure 11 .
Figure 11.E z_max of EM wave for different length of discharge channel.

Table 1 .
Comparison of two methods in computer consumption.

Table 1 .
Comparison of two methods in computer consumption.

Table 2 .
Parameters of each component in switchgear.

Table 2 .
Parameters of each component in switchgear.