Modelling and Simulation of PD Characteristics in Non-Conductive Electrical Trees

Zepeng Lv, Simon M Rowland, Siyuan Chen and Hualong Zheng

The University of Manchester, School of Electrical and Electronic Engineering

Manchester, M13 9PL, UK

ABSTRACT

Electric tree growth is a key ageing mechanism leading to breakdown of high voltage electrical insulation. Partial discharges (PDs) are invariably associated with electrical tree inception and propagation. In turn, the physical structure of an electrical tree influences the characteristics of partial discharge activity. Interpretation of PD patterns is therefore central to developing an understanding of the tree propagation process, and also to the use of PD patterns as an asset management tool. Our previous research indicates that the phase resolved PD (PRPD) patterns and pulse sequence analysis (PSA) patterns evolve with tree propagation. A method was proposed to estimate the point-on-wave inception and extinction voltages of PDs in tree channels within each power cycle. It was shown that the evolution of PD patterns is a consequence of changes to PD inception and extinction voltages as a tree develops. This paper provides a deterministic model of partial discharge in tree channels. Simulations of PDs in a straight non-conductive tree channel are based on experimental PD inception, extinction and residual voltages. The quantitative simulations reproduce almost all the characteristics of observed PRPD and PSA patterns. It is concluded that PD events are determined by five key parameters: tree structure, applied voltage, PD inception voltage, PD extinction voltage and PD residual voltage. Key parameters estimated by the method, and the models proposed explain PD activity in non-conductive trees. It is suggested that the PSA and PRPD patterns should be discussed together to fully understand the PD events. This model forms a platform for generating robust information for asset managers using PD measurements from high voltage equipment in service.

Index Terms — Trees (insulation), epoxy resin insulation, partial discharge, modeling.

1 INTRODUCTION

ELECTRICAL treeing is an important electrical degradation process in high voltage insulation [1]. Tree growth under high electric fields is associated with partial discharge (PD) activity. A strong relationship exists between the visual aspect of tree growth and partial discharge activity as characterized by phase resolved PD (PRPD) patterns and pulse sequence analysis (PSA) [2, 3].

In practice, optical images of trees are not available in HV equipment in-situ, so partial discharge measurements have been employed to monitor insulation degradation [4, 5]. Most commonly it is understood that higher magnitude PD events reflect more severe discharging and a greater release of energy. This is considered to be the driver of material degradation and tree growth [3, 6, 7]. Measurement of PD magnitude may then be thought of as a simple asset management tool. However, in both epoxy and polyethylene it is found that sometimes the PD magnitudes are reduced to the point of becoming undetectable whilst the tree channel becomes conductive and continues to grow [8, 9]. Full understanding of how PDs influence tree growth and how, in turn, tree structure influences PD activity, is therefore required to reliably distinguish whether there is an electrical tree in, for example, a cable and how far the electrical tree has propagated. Knowledge of the nature of a tree is then also required to direct the life management strategy of on-line equipment.

Manuscript received on 26 March 2018, in final form 2 June 2018, accepted 3 June 2018. Corresponding author: Z. Lv.

Our previous work has discussed the evolution of PD characteristics during early tree propagation [3]. It was found that the PRPD patterns in a tree channel can be regarded as one of, or a combination of, wing-like and turtle-like patterns. The PD magnitudes in a wing-like pattern increase with the phase during each half-cycle forming a wing shape in the phase resolved analysis; the PD magnitudes in a turtle-like pattern are almost constant and have no phase dependence [3, 10, 11]. The wing-like patterns correspond to PD in long tree channels. Their maximum magnitudes increase with the tree length. The PDs reaching the tree tips result in tree length growth [10]. For deeper understanding of the PDs in tree channels, both PRPD patterns and PSA were employed [11]. It was found that the PSA shows regular patterns, and the voltage differences between adjacent PD events have characteristic values [11]. A model considering space charge accumulation inside a tree channel generated from a metallic needle was developed to explain the characteristic voltage differences between PDs. It was identified that in a non-conductive tree channel, the space charge and the corresponding Poisson electric field can be regarded as stable before a subsequent PD occurs; the Laplacian field (controlled by the applied voltage) is then totally responsible for changing the electric field at the needle tip from the discharge extinction field Ex to the inception field EI. As the electric field between the two adjacent PD events can be either in the same direction or reversed, the absolute values of voltage differences have one of two characteristic values. As a result the apparent PD inception and extinction point-on-wave voltages can be estimated from experimentally characteristic voltage differences [11].

Several models have been developed to explain PD characteristics in tree channels [5, 8-12]. Patsch proposed a model which explained the PSA patterns of different types of defects by considering the PD inception voltage, reset voltage, decay time and other parameters [5]. However, the simulation parameters were chosen to fit the experimental results rather than extracted from experiments. Wu proposed a physical model describing PD in long narrow non-conductive tree channels, and explained that the PRPD patterns changed from turtle-like to wing-like with the increase of channel length [10]. Champion and Dodd proposed a two-dimensional deterministic model taking PD into account by adding dipolar charges to reduce the local electric field from the inception threshold to the extinction threshold [12]. The models of both Wu and Champion considered the PRPD pattern but not the PSA pattern, and key parameters used were estimated values fitted to the experimental results.

In this paper, the numerical model of PDs in tree channels proposed by Champion and Dodd is improved based on the authors’ discussion in [11]. With the resulting model, PD activity is simulated based on the parameters extracted from pulse sequence analysis. The PRPD and PSA patterns in non-conductive electrical tree are systematically discussed.

2 Experimental results

PD behavior reported in [11] shows combinations of typical wing-like and turtle-like patterns. Figure 1a is the pattern from 10 s of tree growth from a metallic needle at 10 kV in epoxy after 120 s of energization. The PRPD pattern is classic wing-like type. The PSA pattern is a typical result when the number of PDs per half cycle exceeds three. Similar but different characteristics from PDs in a tree channel in polyethylene are shown in Figure 2. The sample has a 2 mm needle to plane distance. The size of the sample is 3mm×3cm×3cm. The equipment and test procedure are identical to those in [11]. The PRPD and PSA patterns at 12 kV are shown in Figure 2a. It can be seen that the PRPD pattern is wing-like, but with an underlying lower amplitude PD pattern. This pattern is widely reported in polyethylene [13]. This kind of combination can also be found in epoxy resin.

(a) PRPD pattern (b) PSA pattern

(c) Frequency histogram

Figure 1. Experimental results of PD measurements in an electrical tree in epoxy resin at 10 kV [11].

Figure 2. Tree structure in polyethylene grown from a metallic needle at 12 kV.

(a) All PD data

(b) Wing-like PD only - extracted from (a)

Figure 3. PRPD and PSA pattern of PDs in polyethylene under 12kV peak 50Hz AC.

The PSA pattern shown in Figure 3a has similarities with Figure 1b. However, at its center, two triangles of PD events are seen rather than two concentrated ‘balls’. It should be recalled that each data point represented in a PSA plot is the voltage difference between consecutive PD events. Figure 3b shows the results excluding the bottom turtle-like PDs. It can be seen that the PSA pattern loses the triangle features and closely resembles Figure 1b. Figure 4 shows the histograms of the characteristic voltage differences. If only considering PDs in the wing-like clusters, two distinctive peaks are seen as in Figure 4b. It is also seen that PDs belonging to wing-like clusters are in a strict sequence, whilst PDs from the lower clusters are not.

(a)(b)

Figure 4. Histogram of voltage difference between consecutive PDs (dV); (a) all PDs, and (b) from the wing-like cluster only.

3 MODEL DESCRIPTION

To explain the characteristics of PD sequences and also verify the discussion in [11], the numerical model of PDs in tree channels proposed by Champion and Dodd is employed [12]. However, the calculation principles are modified according to the discussion in [11]. A simulation box with size of A×B nodes is set as shown in Figure 5. Lines between the nodes form segments of length h. A needle-plane electrode system is set inside the simulation box. The plane ground electrode is set as a row at x=0. The needle-plane distance is 2 mm, so that the needle tip is set as a sphere at the node in location [0.3 mm, 2.0 mm]. The needle tip and ground are considered to be metal and conductive, other nodes and segments are considered to be epoxy resin or gas-filled tree channels inside epoxy resin and non-conductive. All space charges are considered to be concentrated on the nodes as non-conductive spheres with a diameter of h, and the radius of the needle tip is set to be half of h.

Figure 5. The simulation box. In this case a single straight tree channel is shown four segments long.

3.1 Boundary conditions

AC voltage is applied between the needle-plane electrode system. The potential of the needle tip Vap changes with time t as a 50 Hz sine function:

(1)

where Vp is the peak value of the AC voltage. The potential of ground is always 0.

3.2 Calculation of potential distribution

The distributed space charge Q(r) inside a tree channel will distort the electric field generated by the voltage on the electrodes and so influence subsequent PD events. To simplify the calculation of the potential distribution in a needle-plane electrode, mirror image charges Q′(r) were employed in [12] in place of calculating the induced image charge on plane electrode and the potential caused by it. The potential V(r,t) at position r inside the needle-plane electrode system is constituted by the Laplacian field potential due to applied voltage Vap(t), the Poisson potential VQ(r,t) due to the local space charge distribution and the field potential (Vim(t)∙Vuapp(r)) due to the induced image charge (Qim(t)) on the needle electrode. Vim(t) is the corresponding potential caused by the induced image charge Qim(t) on needle tip, and Vuapp(r) is the potential at position r within the tree due to an applied unit potential at the needle tip. The details of the calculation method can be found in [12]. Both Vuapp(r) and VQ(r,t) consider the image charges due to the plane electrode,

(2)3.3 Partial discharge event

The inception and extinction of partial discharges are determined by the local electric field. It is considered that once the local electric field exceeds the inception field EI, a PD occurs; and once the electric field drops below the extinction field Ex, the PD will cease [12]. In the simulation, the potential distribution is more easily calculated than the electric field, so the threshold values considered are PD inception and extinction voltages. Here two things should be noticed: Firstly, the value of inception and extinction voltages, VI and VX, can be obtained from PD results. However, the values cannot be directly used in the simulation. This is because the potential difference across a segment is used as the simulation building block. But the estimated PD inception and extinction voltages are the voltage difference on the needle tip rather than across a segment. As the length of the tree segment is two times the needle radius, distances from the two sides of the first segment to the center of the needle tip are separately h/2 and 3h/2. When the potential of needle tip increases ∆V (the nearer side of the first segment), the potential increase of the further side is one third of the nearer side. So the change of potential difference across the first segment is 2/3×∆V. Therefore in the simulation, the threshold value determining the PD inception Von and the threshold value determining the PD extinction Voff for individual segments are respectively 2/3 of the estimated PD inception voltage VI and extinction voltages VX. Secondly, from our experimental results it is found that the residual electric field (Er) just after PD event extinction is much lower than the calculated PD extinction electric field (EX); if the residual electric field is the PD extinction voltage, the PD can only propagate tiny distance [11]. This indicates that PD extinction inside the tree channel should be determined by the residual electric field rather than the PD extinction electric field. The calculated PD extinction field/voltage should be only valid at the needle tip. Thus in this work, the inception of PD is determined by the estimated PD inception voltage of all segments; the extinction of PD in segments adjacent to the needle tip is determined by the estimated PD extinction voltage; and the extinction of PD in other tree segments is determined by the corresponding PD residual voltage drop on a segment Vr (the product of the residual field Er and the length of the segment h).

A partial discharge event is controlled by adding the dipolar charge at the two nodes of the discharging segments [12]. If the potential difference across a segment Vseg is higher than the PD inception voltage Von, a local electron avalanche is generated. Dipolar charges are added at the two nodes of that segment. The dipolar charges have the same quantity and opposite polarity, positive charge is added downstream of the electric field along the segment, negative charge is added upstream of the electric field. The dipolar charge reduces local electric field (the potential gradient), until the potential difference across the segment Vseg is not able to sustain the discharge. This latter state is when Vseg is lower than the threshold value Vt: at the needle tip the threshold value is the PD extinction voltage Voff; within a tree channel the threshold value is the residual voltage drop Vr. At this moment the voltage difference across the segment is lower than PD extinction voltage, VX. In other words, the potential difference across that segment decreases from Vseg to Vt-Verr. Here we introduce Verr, a small value to make the potential difference lower than Vt to prevent endless calculation in the software; it is not a part of the physical model. The amount of dipolar charge Qea can then be calculated by the local potential change. A partial discharge event is a series of local discharges across the tree segments after which no more local discharges can happen in any tree segments at that moment.

3.4 Conductivity of tree channel walls and gas channels

Previous work shows that the conductive trees tend to form in glassy epoxy resin, and non-conductive trees form in epoxy resins above their glass transition temperature [8]. The conductivity of a tree channel greatly influences the PD pattern and tree growth character [8, 9]. In the numerical model of this paper, the conductivity of a tree channel surface is considered, and follows the method in [12]. Ohmic contact is considered at the needle/tree channel interface. Constant resistances Rseg are set for tree segments; the resistances of non-tree segments are set to be 105 times higher than tree segments. So the charge transport from the surface of a tree channel into bulk epoxy resin is relatively slow. The change of charge distribution across the tree segments ΔQseg can then be calculated as:

(3)

where Vseg is the potential difference across the segment. Δts is the time step used to calculate the charge movement, and is 1/20 of the time step for calculating the PD events to prevent non-convergent calculations. ΔQseg is considered as existing on the surface of a tree channel wall, while the space charge created by a partial discharge Q(r) is considered only to exist in the air channel constituting the tree. In the simulation, they are recorded at the same node position, but are not allowed to recombine with each other.

3.5 Calculation of Partial discharge magnitude

When a partial discharge event occurs, new space charge is created in the tree channel. Due to the conductivity of the tree channel wall and the gas channel, space charge may move inside or along the tree channel. The addition and movement of space charge would cause a change of image charge on the electrodes. As the potential of the needle tip is fixed by the applied voltage, an extra impulse of current through the external circuit is induced and superposed on the sinusoidal current. The sinusoidal current is responsible for a change of capacitance charge on electrodes over a relatively long time scale (ms); the extra impulse current compensates for the loss or gain of charge on electrodes due to PDs. Then the quantity of charge transferred by the impulse current (that is the PD charge magnitude) should be equal to the change of image charge on electrode, but with opposite polarity. In the former model [12], the change of image charge was calculated by:

(4)3.6 Calculation Procedure

Figure 6. The diagram of simulation procedure.

When voltage is applied between the needle and plane electrodes, an electric field distribution will form inside the sample. Then the PD activity is simulated by the process shown in Figure 6. At the beginning of the simulation, a specific tree structure is set. Voltage thresholds Vt used to determine the PD inception and extinction are set for all the segments: Vt of all the tree segments are set to be Von; Vt of all the other segments are set to be 105 times of Von to prevent PDs at non tree segments. The potential of all nodes and the potential difference of all segments that are parts of the tree channel are calculated according to the applied voltage and space charge distribution. The segment with highest value of (|Vseg|-Vt) is determined. If the value of (|Vseg|-Vt) is greater than 0, discharge can happen at that segment. The partial discharge would create a negative discharge head upstream of the electric field and positive tail downstream, due to the electron avalanche. The threshold value in the discharged segment is then reduced from Vt to Vr , which corresponds to the residual electric field. Then the potential distribution is calculated again and the values of (|Vseg|-Vt) checked to see whether the original segment continues to discharge or other segments can discharge. If there are no more discharging segments, then the PD event ends for this time step, the PD magnitude is then calculated with Equation (4), and the voltage thresholds Vt for all segments are reset back to Von. The space charge and potential distribution are then updated, and calculation made of the charge movements on the surface of tree channel and air channel with a sub time step (1/20 of the time step). The calculation of the charge movement is repeated 20 times. With these 20 sub-steps completed, the simulation turns to the next time step. As the experimental equipment has noise level below 0.4 pC, only the PDs with magnitudes higher than 0.1 pC are recorded.

4 SIMULATION in Non-conductive tree channel

In [11], the PRPD and PSA pattern of PD in an early branch tree have been reported. The PD inception voltage, and extinction voltage were calculated according to the newly proposed methods; the residual electric field is also estimated. The PDs in a single straight tree channel (as shown in Figure 5) are now simulated with the estimated values to test the estimated parameters and the proposed model, and to understand PD propagation in the tree channel.

Table 1. Parameters used for the simulation.

Parameter

Value

Unit

Number of nodes at X axis (A)

3

Number of nodes at Y axis (B)

334

Segment length (h)

6

μm

Time step (Δt)

0.02/360

s

Simulation time (t)

4

s

Sub time step (Δts)

2.78

μs

Sub step numbers

20

Relative permittivity of the polymer (εr)

3.7

Tree channel length

120

μm

Applied peak voltage (Vp)

6-10

kV

PD inception voltage (Von)

5.126

kV

PD extinction voltage (Voff)

2.826

kV

Residual voltage (Vr)

0.3

kV

Charge mobility in tree channel (μ)

10-10

m2V-1s-1

Charge resistance of each segment at the tree channel wall (Rseg)

1×1020

Ω

As the structure is centrosymmetric, the calculation can be regarded as one-dimensional, and the dimension of X axis can be small. Here, the length of X axis is 2 segments. In our sample, the radius of needle tip is 3 μm. So the segment length h is set to 6 μm. The dimension of Y axis is set to include 334 nodes. The plane electrode is at the position Y = 1. The needle tip is at the position of [2, 334]. The needle-plane distance is 334x0.006 ≈ 2 mm.

To fit the experiment results shown in [11], the tree length is set to 120 μm. The PDs at different applied voltage are simulated and discussed in Section 4.1. The PD inception and extinction voltage obtained after 10 s at 10 kV are separately 7690 V and 4240 V. So the Von and Voff in the simulation are set to be 5126 V and 2826 V. The parameters for the simulation in Section 4.1 are listed in Table 1. It was previously identified that the residual electric field Er in a PD event is less than 0.07 kV/μm [11]. Here Er is set to be 0.05 kV/μm. As the segment length is 6 μm, the residual voltage drop should be 0.3 kV. It should be noted that the characteristic voltage differences obtained experimentally are not a specific value. They show a normal distribution as shown in Figures 1 and 4. In the simulation the value of PD extinction voltage is deterministic, and has a fixed value, while the actual PD inception voltage von is set by the following equation to introduce stochastic characteristics:

(5)

The values of Von and Voff are experimentally calculated. The function of ‘normrnd’ generates random numbers from normal distribution with mean parameter as 1, and standard deviation as 0.2.

4.1 PD patterns at different applied voltages

The simulated results presented in Figure 7 reproduce most of the key characteristics of the experimental results. With the increase of the applied voltage, the maximum PD magnitude increases, the phase range of the PDs increases (extending from the voltage peak to earlier in the cycle), the PRPD pattern becomes more wing-like, and the number of PDs per cycle is similar to the experimental results and also increases with the applied voltage (also shown in table 2). The PSA pattern is a straight line at 5 kV; and nodes forming triangle, rhombus and two-sided hook geometries appear one after another with the increase of the applied voltage. This is due to the increase of number of PDs per cycle. The voltage difference sequence changes from [dV2, -dV2, dV2, -dV2, …] to [dV2, dV1, -dV2, -dV1, dV2, dV1, -dV2, -dV1, … ] and [dV2, dV1, dV1, -dV2, -dV1, -dV1, dV2, dV1, dV1, -dV2, -dV1, …] with the increase of number of PDs per cycle. So the nodes of the PSA pattern change from two forming the straight line to the more complex forms. Most of the PSA patterns in Figure 7 are a combination of two or three patterns. This is because the value of von is set to be statistical within a range, shown in Equation (5), and so the number of PDs per cycle also changes as a result.

(a) 5 kV

(b) 6 kV

(c) 7 kV

(d) 8 kV

(e) 9 kV

(f) 10 kV

Figure 7. The PRPD and PSA patterns of simulated PDs at different applied voltages.

The maximum magnitudes of simulated PDs are higher than the experimental results at each applied voltage, by a constant factor of about 1.3-1.4. The characteristic voltage differences are also very close to the experimental results. It can be seen that the simulation not only reproduces the key patterns but also the key quantifying characteristics. The model used in this simulation is therefore able to explain PD behavior in a non-conductive tree channel. The fundamental characteristics of the estimated PD inception and extinction voltages are therefore quantitatively close to the measured values. Analysis could, if desired, tune these parameters to provide optimal fitting to the data.

Table 2. Number of PDs per cycle (npc) and maximum magnitudes (Qmax)at different voltages. ‘Sim’ represents the simulation results; ‘Exp’ represents the experimental results.

5kV

6kV

7kV

8kV

9kV

10kV

npc Sim

0.5

1.5

2.9

4.2

5.4

6.2

npc

Exp

0.2

1.2

2.6

3.3

4.4

5.9

Qmax

Sim

5.2

7.3

9.2

11.5

14.2

16.9

Qmax

Exp

5.1

5.7

7.1

8.4

9.7

12

Comparing the modelled and experimental results there are two issues to be discussed further. Firstly, the ratio between the simulated and experimental maximum PD magnitude may be caused by several factors. The simulated tree channel is straight and aligned on the needle axis, whereas the experimental tree channel is not. The total voltage difference from the needle tip to the tree tip may then differ between the simulated tree and experimental tree. If the experimental tree thus has a smaller total voltage difference over its length, a smaller maximum PD magnitude will result. It is also possible that the PD measurement system may have some systematic error that may cause a ratio (< 1) between the measured apparent value and the true value of magnitude. The second consideration is that the experimental PSA patterns are not as asymmetric as the simulated results. The negative component patterns occur earlier than the positive component. This arises because the inception voltage of negative PDs (7.34 kV when the needle tip is negative) is smaller than that of positive PDs (8.23 kV when the needle tip is positive). The estimation method proposed by [11] considered the PD inception voltage and extinction voltage to be nearly the same value and opposite sign, especially for the results at 10 kV. Polarity dependence needs to be considered when the PSA pattern shows significant asymmetry.

4.2 PD characteristics with different tree channel lengths

It has been reported that the tree channel length greatly influences the PRPD pattern [10]. In this section, the parameters in table 1 are kept the same, and only the tree channel length is changed. The simulation results are shown in Figure 8. It can be seen that as a short tree channel increases in length, the turtle-like pattern turns to wing-like, and the maximum PD magnitude increases. The simulation results are consistent with the discussion in [10]. A short tree length limits the physical PD propagation, and all the PDs can reach the tree tip, so all the PDs have the same magnitude and the pattern is turtle-like. When the tree channel is long, it does not limit the PD propagation, and the physical length of each discharge event is determined by the applied voltage: PDs at lower voltages can only extend a short distance; PDs at higher voltage extend further. Then the PD maximum magnitude increases with the phase, and the pattern becomes wing-like. As shown in Figure 8, the tree channel length change does not influence the PSA patterns. This is because that the change of tree channel length does not influence the PD inception and extinction voltages which determine the voltage differences between PDs and thus the PSA pattern.

(a) 30 µm

(b) 60 µm

(c) 180 µm

Figure 8. The PRPD and PSA pattern of simulated PDs with different tree channel lengths at 10 kV.

(a) Experimental results

(b) Simulation with reduced Von and Voff

(c) Simulation with reduced Von, Voff and Vr

Figure 9. The experimental and simulation results of PRPD and PSA pattern of PDs at 480s at 10 kV. The simulations are with reduced inception (3.237 kV), extinction (2.353 kV) and residual voltage (0.1 kV).

4.3 PD inception voltage, extinction voltage and residual voltage: Von, Voff and Vr

The model explains how the increase of tree channel length turns turtle-like PRPD patterns to wing-like patterns. However, sometimes wing-like PD patterns evolve into turtle-like patterns as experimental results at 10 kV illustrate in [11]. It is also found that with the tree growth, the PD magnitude decreases, the PD phase range increases, and the number of PDs per cycle increases. The calculated PD inception voltage and extinction voltage decrease during this transition. The calculated average PD inception and extinction voltage after 10 s are 7.69 kV and 4.24 kV respectively; the calculated average PD inception and extinction voltage after 480 s are 4.86 kV and 3.53 kV. The transition is modelled accordingly in [11]. Here, simulation is carried out with reduced von and voff. The values of von and voff are set to be 2/3 of the calculated values, as discussed in Section 3.3.

Figure 9a shows the experimental results of PRPD and PSA patterns of PDs at 480 s at 10 kV. The PRPD pattern is turtle-like; the PSA pattern is of smaller size. From Figure 9b it can be seen that with the reduced PD inception voltage and extinction voltage, the two sided hook-like PSA pattern shrinks in size showing that the voltage differences between PD events decrease. The average number of PDs per cycle increases to 18.3. The maximum PD magnitude decreases. These characteristics match the equivalent experimental results in [11]. However, the PDs still form a wing-like rather than turtle-like pattern. It seems that the reduction of only the PD inception and extinction voltage cannot fully explain the change of PRPD pattern.

Figure 9c shows the simulation results with reduced residual voltage, the other settings being identical with those which generated Figure 9b. It can be seen that the PD pattern becomes turtle-like, and the PD magnitude is also similar to the experimental results. The PSA pattern of this simulation is the same as with the previous model and experimental results.

It can be concluded that the change of a wing-like pattern to a turtle-like pattern is due to the reduction of the residual voltage which enables the PDs to propagate further under the same voltage. The reduction of the PD magnitude, the increase in the number of PDs per cycle and shrinking of the PSA pattern are all due to the reduction of PD inception and extinction voltages.

The reduction of PD inception and extinction voltages is important in the ageing process. As discussed in Section 4.2, under a fixed voltage and with constant PD inception and extinction voltages, the PDs can only propagate a certain length. If the tree channel is longer than the PD propagation length, the PDs cannot cause further degradation on the tree tip. Then the tree cannot extend further. If the PD inception, extinction and residual voltages reduce with the tree growth, as the experiments indicate, the PDs can propagate further and result in tree growth. However presently it is not clear what might cause the reduction of PD inception, extinction and residual voltages.

5 Conclusions

This paper provides an improved simulation model of PD propagation in a non-conducting tree channel. PD behaviour in a straight non-conductive channel has been simulated based on a point-on-wave PD inception voltage, extinction voltage and residual voltage extracted from experimental data. Along with the tree length, applied voltage, PD inception voltage, extinction voltage and residual voltage, the simulations reproduce almost all the characteristics of both phase-resolved PD (PRPD) patterns and pulse sequence analysis (PSA) patterns. Results provide a good quantitative and qualitative fit. For example, the simulations validate the model of PD propagation in non-conductive tree channels and the accuracy of the estimated PD inception and extinction voltages. The method proposed by [11] is shown to be efficient and accurate for non-conductive trees.

Tree structure can be obtained by the several methods, and in particular 3D XCT can provide detailed structure of tree channels, and the resolution of nano-size XCT can presently reach 50 nm [14]. The PD inception and extinction voltage can be obtained from the PSA pattern according to our method [11]. The residual voltage can be estimated according to the applied voltage, tree channel length and the PD pattern. Then it is possible to quantitatively reveal the PD evolution and non-conductive tree growth based on the time sequential PD signal and tree images. Based on the model proposed in this paper, the PDs in conductive tree channels can also be discussed by considering higher conductivity of the tree channel wall and higher mobility for the carriers generated by PDs.

The PRPD pattern and PSA patterns give complimentary information. The PRPD pattern reveals the PD magnitude and phase distribution. The information carried by PRPD pattern reflects how severe PDs events are and how much energy the PDs generate. The PSA pattern is formed by the voltage differences which determined by the PD inception voltage and extinction voltage. The calculated PD inception and extinction voltages are fundamental parameters determining the PD characteristics. It is enough to use PRPD pattern to study the influence of PDs on the treeing process. However, to fully understand PD events in non-conductive tree channels, the PSA pattern is of great importance. Analysis of PRPD together with the PSA pattern can give a clearer insight into the whole PD sequence and its development.

Acknowledgement

The authors are grateful to the EPSRC for support of this work through the project 'Novel Composite Dielectric Structures with Enhanced Lifetimes’ EP/M016234/1.

This paper contains data which is openly available from www.manchestertrees.com.

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[8] J V Champion and S J Dodd, “Simulation of partial discharges in conducting and non-conducting electrical tree structures, J. Phys. D: Appl. Phys. 34, 1235-1242, 2001.

[9] Kai Wu, Yasuo Suzuoki, Teruyoshi Mizutani and Hengkun Xie, Model for partial discharge associated with treeing breakdown: III. PD extinction and regrowth of the tree”, J. Phys. D: Appl.Phys. 33, pp.1209-1218, 2000.

[10] Kai Wu, Yasuo Suzuoki, Teruyoshi Mizutani and Hengkun Xie, “A Novel Physical Model for Partial Discharge in Narrow Channels”, IEEE Trans. Dielectr. Electr. Insul., vol. 6, pp. 181-190, 1999.

[11] Zepeng Lv, Simon M Rowland, Siyuan Chen, Hualong Zheng and Kai Wu, “Modelling of partial discharge characteristics in electrical tree channels: estimating the PD inception and extinction voltages”, accepted IEEE Trans. Dielectr. Electr. Insul.

[12] J V Champion and S J Dodd, “An approach to the modelling of partial discharges in electrical trees”, J. Phys. D: Appl. Phys. 31, pp. 2305-2314, 1998.

[13] H. Kaneiwa, Y. Suzuoki and T. Mizutani, “Partial Discharge Characteristics and Tree Inception in Artificial Simulated Tree Channels”, IEEE Trans. Dielectr. Electr. Insul., vol. 7, pp. 843-848, 2000.

[14] Roger Schurch, Simon M. Rowland, Robert S. Bradley and Philip J. Withers, “Imaging and analysis techniques for electrical trees using X-ray computed tomography”, IEEE Trans. Dielectr. Electr. Insul., vol. 21, pp. 53-63, 2014.

Zepeng Lv received the B.S. degree in electrical engineering from Xi'an Jiaotong University, Xi’an, China, in 2009; and then he received his doctoral degree from the same university in 2015. Now he works as a post-doctoral research associate in School of Electrical and Electronic Engineering, The University of Manchester. His research interests are in charge transport and aging processes in dielectrics.

Simon M. Rowland (F ‘14) was born in London, England. He completed the B.Sc. degree in physics at The University of East Anglia, and the PhD degree at London University, UK. He has worked for many years on dielectrics and their applications and has also been Technical Director within multinational companies. He joined The School of Electrical and Electronic Engineering in The University of Manchester in 2003, and was appointed Professor of Electrical Materials in 2009, and Head of School in 2015. He was elected President of the IEEE Dielectric and Electrical Insulation Society in 2011 and again in 2012.

Plane

x

y

Needle tip

Nodes in tree

segment

Tree segment

(0,0)

h

h

r=h/2

�

Plane

x

y

Needle tip

app

()sin(250)

VtVt

p

=××

imQ

appapp

(,)=()()+()()+(,)

uu

VtVtVVtVVt

××

rrrr

segsegsegs

(/)

QVRt

D=D

pdnimnimnimn-1

()()(()())

QtQtQtQt

=-D=--

t = t+Δt

Update the applied voltage

Calculate the potential

distribution

Find the segment with

highest (|V

seg

|-V

t

)

Set dipolar charge on

the two nodes of

discharge segment

(|V

seg

|-V

t

)> 0

Yes

PD event of this

time step ends

No

Calculate the charge movements with

smaller time step

Set V

t

of discharge and

nearby segments as V

r

Calculate the

PD magnitude

Reset V

t

of all

segments as V

on

Start

Set initial tree segments and

threshold V

t

of all segments

�

�

�

t = t+ΔtUpdate the applied voltage

ononoffoff

normrnd(1,0.2,1,1)(-)+

vVVV

=´