The Role of Space Charge on Corona Inception Threshold Conditions in Rod-Plane Air Gaps at DC/AC Combined Voltages

This work investigates the corona inception voltage and the motion of space charge near a rod-plane electrode under combined DC-AC voltages. For a high AC component, the corona inception voltage of the equivalent negative corona exceeds that of the equivalent positive corona. For a small AC component, the corona inception voltage of the equivalent positive corona exceeds that of the equivalent negative corona. The corona inception voltage decreases with increasing AC component. The charge-motion model based on hydrodynamics is used to analyze the charge motion for different AC/DC voltage ratios. The charge remaining for equivalent negative coronas exceeds that for equivalent positive coronas with high AC components. In addition, the charge remaining for equivalent positive coronas exceeds that of equivalent negative coronas with small AC components. As the AC component decreases, the remaining space charge increases. The results show that space charge tends to “block” corona inception. The more space charge remains, the higher the corona inception voltage is. Therefore, the variation in space charge and the corona inception threshold conditions under combined DC-AC voltage differ from those under only DC or AC voltage.


I. INTRODUCTION
Nowadays, with the expansion of the transmission grid in China, the lack of space for new power transmission lines is becoming a serious problem. To conserve land resources and improve transmission capacity, the DC-AC hybrid transmission line has been widely studied [1]- [3]. When the electric field on the electrode is above the corona inception, corona discharge will occur. Corona discharge can release a large of space charge [4] and lead to energy loss, audible noise, radio interference and insulation degradation [5]- [9].
The associate editor coordinating the review of this manuscript and approving it for publication was Mohd Zainal Abidin Ab Kadir . Compared with a DC or AC transmission line, the electromagnetic environment of a DC-AC hybrid transmission is more complex [3], which makes it more challenging to ensure the safety of transmission lines and electrical components.
The corona inception voltage is a key factor in the design of transmission lines and electrical components and electromagnetic environment optimization [10]- [12]. Corona inception field is an important criterion when predicting the acoustic noise and radio interference of transmission lines [13]. In the study of corona under AC-DC combined voltage, scholars mostly use the reduced-scale transmission line model [14]. In the calculation of transmission line models, most of them use the peek's formula to calculate corona inception field of the wire [15]. For the electric field on the wire surface, in order to reflect the effect of voltage variation and space charge, some scholars discard the Deutsch hypothesis and use the improved numerical algorithm in the model to calculate the electric field on the wire surface [4], [16]. However, for the complex electrode structures under the influence of AC-DC combined voltage, such as commutation valves. For the corona occurring at complex electrode structures, the corona inception criteria is usually used to obtain the corona inception voltage and corona inception field [17]- [20]. The corona inception voltage can be affected by space charge and other environmental factors [21]- [24].
Corona discharge generates a large amount of space charge, which enhances the synthetic field near the ground beneath the transmission line [1]. The space charge released by a corona discharge weakens the electric field near the electrode [22], [25], which renders corona inception more difficulties. Accumulated positive ions in the positive corona inhibit the formation of further streamers [26]. In addition, space charge influences the generation of Trichel pulse. The falling edge of the Trichel pulse is dominated by negative ions [27], [28], which reduces the electric field and cuts off the electron avalanche in the negative corona [29]. In the case of superposition of AC voltage and DC voltage, it is inevitable to consider the reciprocal motion of space charge under the AC voltage. Previous studies show that in the transmission line model, the AC voltage has negligible effect on the charge generated by the DC conductor [2]. However, it is questionable whether this conclusion also applies to short-gap model such as rod-plane model.
The study of space charge under combined DC-AC voltage mainly focuses on the low-frequency corona current due to the motion of the ion cloud [3], [30], [31], partial discharge in oil-paper insulation [32], [33], combined field strength near the ground [14], and the ion flow field [34], [35]. The ion flow field model neglects the ionosphere layer [36], [37]. However, in corona inception models, the ionosphere layer is not generally ignored [38]- [40]. Therefore, hydrodynamic models are commonly used to study the space charge and electric field at corona inception [41], [42]. For short gap electrodes, Qi and Zhou studied the partial discharge (PD) and found the inception voltage of PD increases as the DC component increasing because of the reverse electric field generated by charge [32], [43]. However, there are few micromechanical explanations for the corona inception characteristics under DC-AC combined voltage at the rod-plane electrode. Some phenomena different from pure DC and AC corona have not been studied yet. Therefore, it is worthwhile to consider simultaneously the space charge and the corona inception voltage and determine how they are related.
This study discusses the threshold conditions for corona inception under combined DC-AC voltage. Defining a uniform corona inception voltage allows synchronous current and voltage measurements to be made and thereby to obtain the corona inception voltage for different ratios of AC to DC components. Based on experiments, we develop a fluid model of space charge motion. Considering the effects of space charge, the corona inception voltage is analyzed under different conditions and compared from the perspective of space charge.

II. EXPERIMENTAL SETUP
An experiment on the corona inception voltage of rod-plane electrode at combined DC-AC voltage was carried out.
A. EXPERIMENTAL PLATFORM Fig. 1 shows the experimental platform. The ambient temperature for experiments was around 14 ± 1 • C, and the relative humidity was 25% ± 2%. The rod-plane electrode was composed of a hemispherical copper rod and a disk. The rod radius was 1.5 mm, the disk radius was 20 cm, and the bottom end of the rod was 1-4 cm from the disk. The plane was connected to a DC voltage power through a 1.34 M protection resistor, and the rod was connected to the AC power through a 10.27 M protection resistor. DC power and AC power were added to the two ends of rod and plane. Physical isolation of DC and AC power was supplied to avoid interference between them. Circuit connection diagram of the experimental platform is shown in Fig. 2 and some physical views of experimental equipment are shown in Fig. 3.    When a corona occurs, the signal of the corona current in the plane was transmitted to the acquisition card through a coupling capacitor of 100 pF, and the DC voltage signal from the plane was transmitted to the acquisition card through a voltage divider with ratio of 1:1000. The corona current signal and AC voltage signal at the rod were then transmitted to the acquisition card through a 1050 pF coupling capacitor and a 1:1000 voltage divider. The current and voltage signals were simultaneously acquired by the PC via a four-channel acquisition card at a sampling frequency of 5 GS/s. And the current signal was detected by Person probes which can measure currents from 1.5 kHz to 200 MHz. The detailed experimental equipment parameters are shown in Table 1.
To reduce the effect of noise on the measurement, the bandpass filtering method was used to process the current single data and the lowpass filtering method was used to process the DC/AC voltage single. The range of bandpass filtering was from 10 kHz to 200 MHz, which can pass most of the corona current pulse component and filter out a significant portion of the noise component [44]. And the maximum frequency of lowpass filtering is 1kHz, which can filter out the most noise components.   The DC voltage was 18.60kV. The corona current pulse was occurring near the peak of AC voltage. Because corona discharge is a probabilistic event, the corona pulse was recorded 30 times for each voltage case, and the AC voltage corresponding to expected value of the normal distribution of corona pulses was taken as the actual AC component at the time of corona inception. The AC voltage was fixed at a certain value, and the DC voltage was gradually increased in increments of 0.1 kV. The voltage was held constant for 10 s after each voltage increment. The fixed value of AC voltage was 0 kV at the beginning, and the AC voltage was increased by 2 kV each time until the moment before AC corona occurring. When the corona pulse in Fig. 5 appears, the corona occurs. The corona current pulse is appearing in the both rod and plane at the same moment. The current probe can capture two types of pulses at the same time. The corresponding DC and AC components were recorded at this time. The advantage of using the corona current pulse method to detect the corona moment is that it is sensitive and accurate, and the phase information corresponding to the corona current can be detected. The experiments were done for rod-plane distances D = 1, 2, 3 and 4 cm. The DC voltage at the time of corona inception is taken as the DC component, denoted U DC , and the instantaneous value of AC voltage at the corona pulse occurring is taken as the AC component, denoted U AC or U sin(ωt + θ), which the U is the magnitude of AC voltage, ω is the Angular frequency of AC voltage, t is the time of corona pulse appearing and the θ is the initial phase of AC voltage. It should be noted that due to the uncertainty in the appearance of corona pulses, each θ is selected after 30 pulses captured, and the median of the normal distribution of the pulses is selected. The ratio U AC /U DC at the corona inception varies because AC voltage differs each time. For comparison, the corona that occurs when U r−p (i.e., the rod-plane potential difference) is positive is considered the equivalent positive corona (EPC)   Fig. 7(b) show that U DC decreases as U AC increases for both EPC and ENC. When U AC is small, the potential U DC required for EPC is higher than that required for ENC with the same U AC . Conversely, when U AC is high, the potential U DC required for EPC is lower than that required for ENC. For D = 2 cm, the curves of EPC and ENC intersect at point P 2 , while in the case of D = 4 cm, they intersect at point P 4 . For comparison, the U AC and U DC are called high U AC and U DC when they are higher than the value of intersection point, while the opposite is called low U AC and U DC . No accepted definition exists for corona-inception voltage for a combined DC-AC voltage. We therefore propose, given VOLUME 9, 2021 the small gap between the rod and plane, to define the combined DC-AC corona inception voltage U i between the rod and plane as: Fig. 8 gives the relationship among U AC , U DC , and U. The initial phase of AC voltage θ is set as 0 • . For D = 4 cm, Fig. 7(d) shows that, as U AC increases, U i decreases for both EPC and ENC. When U AC is small, the equivalent negative corona-inception voltage, denoted U n i , is smaller than the equivalent positive corona inception voltage, denoted U p i . U n i is 4.51% less than U p i when AC voltage is 2 kV as points a and b shown in Fig.7(d).

Fig. 7(a) and
i is 4.16% less than U n i when the U ≈ 16 kV as points c and d shown in Fig.7(d). This is different from ether DC corona or AC corona. When U AC ≈ 9.50 kV, the curves of EPC and ENC intersect at point P 8 . When D = 2 cm, the relationship between the U i and the U AC is similar to that when D = 4 cm, but U i is smaller and the intersection of EPC and EAC is at point P 6 . The variation in potential difference between the rod-plane gap under a combined DC-AC voltage with varying AC/DC ratios can be divided into the following four cases: Fig. 6(a) shows the EPC when U DC is high, Fig. 6(b) shows the ENC when U DC is high, Fig. 6(c) shows the EPC when U AC is high, and Fig. 6(d) shows the ENC when U AC is high. Each of these four cases corresponds to the four cases a-d in Fig. 7(d). The potential difference between rod and plane is not reversed in cases a and b. In cases c and d, the potential difference is reversed in polarity, which, over a cycle, first goes through the positive (negative) phase and then through the negative (positive) phase. The first phase is defined as stage 1 and the second phase is defined as stage 2, as shown in Fig. 6(c) and Fig. 6(d). The corona occurs in stage 1. The typical moments T 1 -T 4 in Fig. 6(a) and Fig. 6(b) and T 1 -T 5 in Fig. 6(c) and Fig. 6(d) are selected for analysis. The movement of charge differs in the four cases, which leads to different values of U i in the four cases.

III. NUMERICAL STUDIES A. CORONA FLUID MODEL
To find the connection between space charge and U i , we developed a model to compute the space charge density and distribution at combined DC-AC voltage. The model is based on the continuity equations and Poisson's equation: where ϕ indicates the electric potential; E is the electric field; N e , N p , and N n are the electron, positive ion, and negative ion densities, respectively; e is the elementary charge; S t is the photoionization term; ε 0 is the dielectric constant of air; α is the collision-ionization coefficient; η is the adsorption coefficient; W e , W p , and W n are the velocity vectors for electrons and positive and negative ions, respectively; β ep is the electron-positive-ion complex rate; β np is the positiveion-negative-ion composite rate; and D e , D p , and D n are the diffusion coefficients for electrons, positive ions, and negative ions, respectively. Fig. 9 shows the calculation domain and boundary of the model. U r−p can be reversed at combined DC-AC voltage. Therefore, the EPC and ENC are modelled separately,   Table. 2 and Table. 3, where n is unit normal vector.  The parameters of the model are shown in appendix [45].

B. CORONA INCEPTION CRITERION
U AC and U DC at corona inception are computed by using the corona inception criterion based on photon effect [17], which demands that the number of photons produced by the secondary electron collapse is greater than the number of photons produced by the primary electron collapse.
As shown in Fig. 10, the bottom of the rod is the origin, and the central axis is the z axis. The corona inception criterion is computed on the z axis. For the EPC, when an electron arrives at the surface of the rod electrode from position z on the central axis, it produces multiple impact ionization. The total number of electrons produced by this electron is: where z is the virtual integral variable. Considering the adsorption process of electrons in air, the total number of positive ions produced by the electron is not equal to (7), it should be: (8) where z is also the virtual integral variable. The (8) can also represent the number of impact ionizations.
Supposing that the electron is triggered at the ionization boundary z i , it produces a large number of excited particles by impact ionization as it moves towards the rod. When these excited-state particles jump back to the ground state, a lot of photons are released. If the probability of an impact collision producing a photon is proportional to the number of collisions, the number of photons produced by the primary electron avalanche is: where f 1 is the probability of producing a photon per collision; z i is the ionization boundary.
The following is the analysis of secondary electron avalanche. The gas molecules in space absorb the photons and produce photoelectrons. The photoelectrons continue moving towards the rod and repeat the process of primary electron avalanche. When the N 1 photons produced by the primary electron avalanche are emitted from the surface of the electrode, the total number of photons produced by all secondary electron avalanches is: (10) where µ is the photon-absorption coefficient of air; and g p (z) is the area factor of the rod-plane electrode when a positive voltage is applied [46]; f 2 is the probability that a gas molecule absorbs a photon to produce a photoelectron.
To form a corona discharge, the number of photons generated by the secondary electron avalanche is larger than the number of primary electron avalanche photons: Considering the collision process of electrons very close to the electrode and the collision ionization coefficient is much larger than the attachment coefficient, the total number of positive ions in an electron avalanche can be simplified as: Organizing the above equations, the corona inception criterion for EPC can be obtained as: In the case of ENC, the photons absorbed by air molecules to generate photoelectrons are no longer considered, but the process of positive ions bombarding the cathode to produce electrons is considered. The corona inception criterion for ENC can be obtained as: where the g n (z) is the area factor when the negative voltage is applied to the rod-plane electrode. At the end of stage 2, the corona inception moment of the next cycle is computed for EPC and ENC.

C. THE CASE OF HIGH U AC
When U AC is high in Fig. 6(c) and Fig. 7(d), five moments  Table. 4 and Table. 5 list the moments and the corresponding AC voltages (magnitudes) for the EPC and the ENC. The model first calculates the charge distribution in one cycle. Then the charge distribution of the first cycle is used as the initial value to calculate the corona inception voltage for the next cycle.    For the EPC, Fig. 12(a) and Fig. 12(b) show the charge of the electrons and of the positive and negative ions on the axis in stage 1. The space charge is dominated by positive ions and increases rapidly at T 2 , which move away from the rod head due to the action of the electric field. A similar charge distribution can be found in [26], where the maximum density of positive ions on the axis is 2.12×10 19 m −3 . When stage 1 ends (at T 3 ), the residual density of positive ions on the axis is 8.73×10 16 m −3 . Since the densities of the three particles are not in the same order of magnitude, the diagrams of the densities of non-dominant particles at T 3 , T 4 and T 5 in space are shown in Fig. 14 and Fig. 15.   . 12(c) shows the variation in charge on the axis of the EPC after entering stage 2. The electric field is reversed at T 4 . The positive charge of the EPC moves toward the rod and gradually decreases. In addition, the positive ions enter the rod electrode, triggering the cathode to emit photoelectrons, further depleting the positive ions. Fig. 12(d) shows the positive ion density for the EPC at T 5 . The maximum density of positive ion residues at T 5 is 8.8×10 15 m −3 . These positive ions suppress the corona discharge in the next cycle.    of the rod and disappear. Negative ions accumulate on the outside of the rod and move away from the rod, so the space charge gradually becomes dominated by negative ions. At T 3 , the maximum density of negative ions is 8.31×10 16 m −3 . Fig. 13(c) shows the variation of charge on the axis for the ENC after entering stage 2. At point T 4 , the electric field is reversed, and the negative charge dominates the space charge. The negative charge moves toward the rod head and gradually decreases. Fig. 13(d) shows when stage 2 ends (at T 5 ), the maximum density of residual negative ions for the ENC is 1.67×10 16 m −3 . These negative ions act as a barrier in the next cycle of corona. Fig. 14 shows the variation of electrons and negative ions at T 3 , T 4 , and T 5 for the EPC. Fig. 15 shows the variation of electrons and positive ions at T 3 , T 4 , and T 5 for the ENC. For electron and negative ion in EPC and electron and positive ion in ENC, their orders of magnitudes are smaller compared to the dominant charge (positive ion for EPC and negative for ENC). Due to the presence of cathode secondary emission in stage 2 for EPC, the density of electrons increases after the moment T 3 . For the same reason, the negative ion grows in density from T 3 to T 5 as it moves away from the rod head. In the case of ENC, the density of electron is close to 0 at T 4 and T 5 because there is no cathode secondary emission in stage 2 of the ENC. The positive ions in the ENC are moving away from the rod and decreasing from T 3 to T 5 .   the positive ions. In contrast, no such process exists for the ENC in stage 2. Therefore, the space charge produces more of an obstacle for the ENC in the next cycle. Fig. 16 shows the computed corona inception criterion for the next cycle, which gives U p i = 18.2 kV and U n i = 18.6 kV. This deviates from the experimentally obtained corona inception voltage by 2.47% for the EPC and by 1.22% for the ENC. Due to the existence of space charge, the same electrode at different ratios of DC/AC voltages has the similar corona inception field but the different corona inception voltage.
Figs. 17(a)-17(c) show the space charge residual at T 5 for the EPC when U AC = 16, 14, and 12 kV at D = 4 cm, and Figs. 17(d)-17(f) show the space charge residual at T 5 for the ENC. The maximum space charge density in the EPC at T 5 is always less than that in the ENC. As U AC decreases, the space charge in the residues of both the EPC and ENC at T 5 increases because the maximum voltage of stage 2 decreases as the AC component decreases, thereby hindering the dissipation of space charge. Eventually, the space charge becomes more obstructed. This scenario explains the decrease in U i for the positive and negative coronas as U AC increases.

D. THE CASE OF LOW U AC
To investigate the variation in space charge when U AC is small in Fig. 6(a) and Fig. 6(b), we consider the point T1-T4 in these figures. T 1 is the moment when the AC voltage is half the amplitude. T 2 is the point at the maximum AC voltage, T 3 is the point at the half cycle of AC voltage, At T 4 , the potential difference between rod and plane is at its lowest point in the cycle.
The Fig. 18 shows the positive ion distribution of the EPC and the negative ion distribution of the ENC at T 1 -T 3 moments when U AC = 8kV. At T 1 moments, the voltage has not yet reached the corona inception voltage, the EPC and ENC gradually gather a portion of positive and negative ions near the rod head. At this time both the space charges are dominated by positive ions, so the negative ion order of magnitude in ENC is small. At T 2 moments, the corona begins to occur, and there is a sudden increase in the magnitude of both positive and negative ions in EPC and ENC. At this time, the space charge of both is still dominated by positive ions as shown in Fig. 12, Therefore, the magnitude of negative ions of ENC is small. After the T 2 moment, the negative ions of ENC gradually dominate the space charge. The positive ions of EPC and negative ions of ENC gradually drift outward under the action of electric field. During this process, the numbers of positive ions of EPC and negative ions of ENC decrease continuously. At T 3 , the positive ions of EPC and negative ions of ENC are already with the same level.
At T 4 , the potential difference between rod and plane is at its lowest point in the cycle. Fig. 19(a) and Fig. 19(b) show the distribution of the EPC space charge for U AC = 8 and 4 kV, and Fig. 19(c) and Fig. 19(d) show the distribution of the ENC space charge for U AC = 8 and 4 kV. As U AC decreases, the space charge density of the EPC and ENC residues increases. The number of charges in residue of the EPC is 4.51×10 16 m −3 for U AC = 8 kV and 4.88×10 16 m −3 for U AC = 4 kV. The number of charges in the residue of the ENC is 4.04×10 16 m −3 for U AC = 8 kV and 4.55×10 16 m −3 for U AC = 4 kV. More charge remains in the EPC than in the ENC and obstructs the corona inception, which makes U p i higher than U n i .

IV. CONCLUSION
To summarize, the corona inception voltage and the variation in space charge near the rod under the combined DC-AC voltage differ from those under either DC or AC voltage. The results indicate that U n i > U p i when U AC is high, which is mainly due to there is more space charge residue for the equivalent negative corona than for the equivalent positive corona when U AC is high. This space charge weakens the electric field near the rod electrode, which thereby hinders the equivalent negative corona inception. For high U DC , there is more space charge for the equivalent positive corona than the equivalent negative corona, where the space charge displaces similarly to the DC corona, and U p i > U n i . It should be noted that the laws obtained in this paper are verified in short gap electrodes like rod-plane. However, whether the same law could be applied to long gap electrodes such as wire-plane needs to be further verified. where P is the atmospheric pressure in Torr.