Statistical models applied to PHD
The PHD pattern can be modeled by various statistical distributions. These sta- tistical models contain parameters with values that are correlated to the type of the defect where possible. Thus the parameters indicating the magnitude and the shape of the distributions can be used as indicators. This section introduces sta- tistical models to be applied to PHD patterns. Although, all the presented models
4.4. STATISTICAL PARAMETERS FOR PD PATTERNS 45 can be used to roughly represent the data, some distributions appear to be more practical than others. Trends in the model parameters over time can be used to reveal the progressive degradation occurring in the speci c location.
Normal model - The probability density function for the normal distribution is given by: f q; µ, σ2 =√ 1 2π · σ2 · e − (q−µ)2 2 ·σ2 q∈R (4.5)
where q is the PD magnitude, µ is the average value and σ is the variance that represents the width of the distribution.
The PHD pattern for corona discharges can be well modeled by this distri- bution [72, 73]. Since it describes only symmetrical distributions, it is not used in this work as a general model to represent the PHD pattern.
Gamma model - The density function for Gamma distribution is given by:
f (q; k, θ) = qk−1· e
−q θ
θk· Γ (k) q∈R
+ (4.6)
where q is the PD magnitude, k is the shape factor and θ represents the scale factor.
This model has more exibility in modeling the PD data during dierent stages of the degradation. In [74] it is expressed that the amplitude of the PDs are gamma distributed. However, during the life span of the cable in- sulation, dierent PD amplitudes, in dierent occurrence numbers are mea- sured. In case that higher PD magnitudes appear with higher occurrence rate compared to lower PD magnitudes, then this model fails to provide a good t since Gamma distributions are right-skewed.
Weibull model - Probability density function for this distribution is described by: f (q; α, β) = β α· q α β−1 · e−(αq) β q∈R+ (4.7)
where q is the PD magnitude, α is the scale factor and β represents the shape factor.
Weibull distribution is a exible model, that can either be right-skewed or left-skewed. Such characteristic of this distribution makes it an appealing option to be used to represent the PHD pattern. For β . 2.6 this model has a longer tail meaning that it is right skewed, for 2.6 . β . 3.7, this model tends to be centered, and in fact approaches the Normal distribution, for β & 3.7 this model changes to be left-skewed. This property makes the Weibull an attractive option to represent the PD activity from dierent sources at dierent degradation stage of the insulation.
0 200 400 600 800 1000 0 0.5 1 1.5 2 2.5 3 x 10−3 PD magnitude [pC] Density PD magnitude Weibull Distribution Normal Distribution Gamma Distribution (a) 0 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 PD magnitude [pC] Cumulative probability PD magnitude Weibull Distribution Normal Distribution Gamma Distribution (b)
Figure 4.4: Models t to PD magnitude: a) probability function, b) cumulative density function -Circuit A
The models presented through equations 4.5 to 4.7, are applied to the PD data collected from monitoring a live cable circuit to graphically compare their feasibil- ity in providing proper description of the data. For statistical signi cance of the model parameters the distribution should contain su cient data. PDs captured over an hour of measurement are often not su cient for reliably assigning a sta- tistical distribution. Therefore, in our research work a minimum of 25 events is taken to obtain signi cant values for the distributions parameters. That means larger time blocks are needed than the ones for the PD charge density and occur- rence rate distributions. The presented models are applied to the measurements performed for the PILC cable with mapping diagram shown in Figure 4.1. PDs from a defective location at 140 m are considered for the modeling.
Figure 4.4 depicts the ts of the various models applied to the PDs from defect captured over 72 hours.
Normal model is not a proper model to t the data, as can be seen in 4.4a, the PD magnitude are not symmetrically distributed, therefore, the distribution tends to move to the negative domain and start from values below zero to better t the higher weighted PDs which are concentrated in the center of the pattern. This leads to signi cant deviation since it can not accurately t the low PD values. The Gamma and the Weibull distributions show better t for the data ranging from small to larger PD values. The two models are aligned for small PD values, however, for larger ones Weibull provides slightly closer estimate of the data. As discussed earlier, another advantage of the Weibull is its capability to skew to right or left. Therefore, it is capable to model the patterns where the distribution of the histogram is more concentrated on the right. Figure 4.5 shows a mapping diagram of an example (referred as Circuit B) where larger PDs are appearing in larger number as compared to the smaller ones. PHD pattern is made for the PD magnitude captured from the location of the defect and the pattern is modeled
4.4. STATISTICAL PARAMETERS FOR PD PATTERNS 47 0 50 100 150 200 0 200 400 600 800 1000 1200 1400 Location [m] Discharge magnitude [pC]
Figure 4.5: PD mapping for Circuit B
by the Weibull and Gamma distributions (Figure 4.6). By visually comparing the two distributions, one can deduce that the Weibull is a better choice to represent this pattern. Figure 4.6b shows the cumulative distribution model including the 95% con dence interval. The Weibull distribution including its con dence level represent the experimental PD magnitude curve better than the Gamma distri- bution. Therefore, the Weibull model is taken to represent the PD magnitude pattern. In [4, 41, 58, 75, 76, 77, 78] it is shown that the Weibull scale and shape parameters tend to be a good indicator of the defect type and, in fact, it is noted that the shape parameter of this model is correlated to the main defect types namely, internal, surface and corona discharges. It is shown in [58] that generally the value of the shape parameter is β . 2 for surface discharges, 2 . β . 8 for internal discharges and β & 8 provides indication of corona discharges. A trend in this parameter can be used as an indicator for change of the degradation status.
In addition to the Weibull parameters' values, their con dence bounds must be determined to indicate the reliability of the estimates. Basically, due to inherent statistical uctuations, but also due to the fact that it is only an assumption that the PD magnitude can be modeled by a Weibull distribution, there is an uncertainty in the estimated parameters. There is no underlying physical reason that PDs should follow precisely the Weibull or any other distribution. In this work, Con dence Interval (CI) with a 95% con dence level (δ) estimated by the Fisher matrix [79] is used. Details are discussed in Appendix A.
Figure 4.7 and Figure 4.8 show the parameter of the Weibull modeling over the complete length of the cable and the location of the defect versus time corre-
0 200 400 600 800 1000 1200 0 0.5 1 1.5 2 2.5 3 3.5 4x 10 −3 PD magnitude [pC] Density PD magnitude Weibull Distribution Gamma Distribution 0 200 400 600 800 1000 1200 0 0.2 0.4 0.6 0.8 1 PD magnitude [pC] Cumulative probability PD magnitude Weibull Distribution Weibull 95% confidence bounds Gamma Distribution Gamma 95%confidence bounds
Figure 4.6: Comparison of Weibull distribution and Gamma distribution -Circuit B
Figure 4.7: Weibull model applied to PD magnitude for Circuit A: a) normalized scale parameter (α), b) shape parameter (β)
sponding to Figure 4.1, respectively. In the patterns presented in Figures 4.1 and 4.7, three distinct discharging location are distinguishable: one around 140 m, and two distributed around 200 m. As it is mentioned in Figure 4.7, the joint located at around 140 m, was replaced at certain time due to the critical condition revealed by PDs captured via SCG. After this, the suspected joint was subjected to a DC test for further assessing of its condition, which indeed resulted in its failure, con rming its lowered insulation capability. The values of the scale and shape factors (Figure 4.8) indicated that the location around 140 m suered from
4.4. STATISTICAL PARAMETERS FOR PD PATTERNS 49
01−Sep−20070 01−Oct−2007
0.5 1
Date−Time
Scaled Alpha parameter [pC]
Scaled Alpha paramterer Trend line 01−Sep−20070 01−Oct−2007 1 2 3 4 Date−Time Beta parameter
Figure 4.8: Weibull parameters for PD magnitudes from defective joint located at about 140 m, a) normalized scale parameter, b) shape parameter - Circuit A.
an intense internal discharge source. The (normalized) scale parameter showed on average an increasing PD level.
Descriptive statistics
PD charge density and occurrence are indicative for the degradation process. The values can be calculated for PD pulses that are measured during each power cycle. Both mean and maximum values accumulated in a certain time block are used. In this work, all the derived quantities are scaled to values between 0 and 1, i.e. 0 symbolizes zero and 1 symbolizes the maximum value in the related pattern.
Equations 4.1 and 4.2, denote the average PD charge density and occurrence rate. Alternatively, also the maximum PD charge density and maximum PD charge occurrence rate seen in one of the power frequency cycles within Tef f can be
taken to characterize the insulation state. These quantities retain information on occasional high activity in contrast to the averaged values over many cycles. Equations 4.8 and 4.9 show the related maximum values on position l and at time t. P Dmax.dens. (lm, tn) = maxj P i qi, j Tef f, j/Tcycle (4.8) P Dmax.occ. (lm, tn) = maxj P i ni, j Tef f, j/Tcycle (4.9)
The function "max" takes the highest value with respect to index j, i.e. the highest value that has occurred during the chosen time block (for instance one hour) for each 1 cable length.
0 50 100 150 200 01−Sep−2007 09−Sep−2007 16−Sep−2007 23−Sep−2007 01−Oct−2007 Location [m] 1000 2000 3000 4000 5000 60001 0 (b)
Figure 4.9: PD charge density pattern (1-hourly basis) for Circuit A: a) Mean value, b) Maximum value.
Figure 4.9 shows the normalized density pattern based on mean and maximum PD charge density. The main defect of this cable circuit, located at around 140 m, shows intense PD activity in both patterns. In the mean density pattern, discharge activity at the very beginning is highest while the pattern with maximum values shows that the highest discharging level just before the replacement took place. This dierence can basically be attributed to the fact that the PD activity in terms of number of PDs in the early stage is higher, giving more weight to the density while in the last stages, the number of PDs decreased while their magnitudes increased. Such behavior itself is an indication of change in the status of the insulation. PDs distributed along the cable length (see Figure 4.1) are mostly averaged out. At both cable ends, PD spread over 10 m from the terminations are distinguishable, however, with lowered level. At 180 m some relatively large PDs occurred spread over a region of 20 m (see maximum pattern), these activities are lowered in the mean pattern, since these discharges do not repeat.
Figure 4.10 shows the normalized PD charge density patterns averaged over 8-hour and 24-hour time blocks respectively. As can be seen from the patterns the random PD activity as well as the distributed ones are partly averaged out. PDs spread around 0 m and 180 m are averaged out partly in 8-hourly pattern and almost completely in 24-hourly pattern due to averaging over a longer time. Therefore, more clear patterns are created for defect identi cation.
Longer averaging time may result in obtaining more clear pattern with more visible PD activity from real defects, especially when PDs are distributed along the cable length. However, in some cases, PD activity may rise for a very short period of time and disappear, especially when the defect has reached the last stage before the breakdown occurs. Some PD activity can only show very low level while still being crucial for the insulation. Too long averaging time may result in averaging out those activities as well as losing time in taking action for possible corrective
4.4. STATISTICAL PARAMETERS FOR PD PATTERNS 51
Figure 4.10: PD charge density pattern for Circuit A: a) 8-hourly pattern, b) 24-hourly pattern.
remedy. Therefore, as explained earlier in this chapter, 1-hourly time block is selected for analyzing PD patterns.
Figure 4.11 shows the normalized mean and maximum PD occurrence values. The defective joint is clearly visible in both patterns. By averaging the values over time, prominent PD activity stands out, whereas noise or spread activities merge with the background.
Mean patterns, rules out the discharge activities that are of low occurrence. This helps to distinguish locations with high PD activity from the background. Especially for PILC cables there is always a broad background of low magnitude PDs. These PDs, which might be generated between the cable cores in a belted cable, are usually considered to be harmless to the insulation.
Figure 4.11: PD occurrence rate pattern (1-hourly basis) for Circuit A: a) Mean value, b) Maximum value.
Chapter 5
Decision Support System for Smart
Cable Guard
In chapter 4, basic patterns that can help to reveal the status of the insulation media have been discussed. These patterns aim to visualize the potential mal- functioning within the system through projecting the suspected PD activities along cable length and time. However, analyzing the presence of the defects is still dependent on visually inspecting the patterns by specialists. The fast growth in the number of installed Smart Cable Guard (SCG) units in the eld, makes it virtually impossible to individually investigate all patterns for each circuit. This calls for developing an automated tool to analyze the circuits and in case of mal- functioning in any circuit, to perform estimations on the state of the insulation. Depending on the insulation state, this information is communicated in the form of a logged message or in the form of an acute alarm. A so-called decision support system is developed to enhance the capability of the SCG system by adding a status analyzer module to the existing system for assessing the cable insulation condition. Such a tool includes several sub-tools to process the data
to eliminate noise or non-PD data to identify potential defects
to calculate their failure probabilities and correlate them to a risk index Figure 5.1 depicts the process from data acquisition to assessing the risk level(s) of identi ed defects related to the insulation condition. The mentioned sub-tools are described in details in the following sections.
5.1 Noise Reduction Algorithm
Raw PD data (i.e. PD magnitude and number of PDs) are pre-processed and converted to characteristic quantities, namely PD charge density and PD charge occurrence rate. Patterns presenting these PD-related values have been introduced
Figure 5.1: Decision support module for SCG system
in the previous chapter. By visually examining the patterns, one may notice that in some circuits the measured data and the associated values are distributed over a wide region along the cable system. These signals occur either from distributed low PD activity which is harmless in case of paper-oil insulated cables, or arise from noise picked up by the cable connection. To maximize PD sensitivity, the SCG detection level can be remotely lowered until misinterpreted signals as PDs just start popping up. However, external conditions may vary, causing occasional high background of misinterpreted signals referred to as noise. Figure 5.2 shows examples of mapping diagrams of PD measurements for two live circuits addressed as "Circuit A" and "Circuit C". In both circuits intense PD activities as well as background noise are observed. The PD activity in Figure 5.2a at the location of a defect is clearly distinguishable from the background activity. The PD activity in Figure 5.2b, is spread over the whole length of the cable circuit. On top of the
0 50 100 150 200 0 500 1000 1500 Location [m] Discharge magnitude [pC] (a) 0 2000 4000 6000 0 1000 2000 3000 4000 5000 6000 Location [m] Discharge magnitude [pC] (b)
Figure 5.2: PD mappings diagram - a) Circuit A - 214 m PILC cable, b) Circuit C - 7 km XLPE cable
5.1. NOISE REDUCTION ALGORITHM 55 background, three concentrations near 500 m, 4500 m and 5000 m are present. Their observation is hampered by this broad noise background.
The e ciency of any algorithm that is utilized to automatically identify po- tential defects will be impeded by a high background. Therefore, it is necessary to perform noise reduction prior to any further data processing. However, noise reduction may not eliminate undesired data completely or may reject data from PDs.
The noise reduction in this research work is performed through applying a statistical approach. First step is to de ne a threshold based on the statistics of the measured data levels observed over the length of the cable in a certain time period. Knowing that defects that generate PDs are normally concentrated in localized regions only, then the next steps involve labeling data that does not reproduce themselves in the close spatial vicinity and/or over a time as "passive data" and exclude them from further processing. In the following sub sections, details of the noise reduction algorithms are presented.
Throughout this chapter the performance of the algorithms is exempli ed by two cases. "Circuit A" is a relatively short PILC cable, showing clearly con- centrated PD activity. "Circuit C" is a 7 km XLPE connection where a broad background conceals the local PD concentrations.
Threshold based on running average
A preliminary noise reduction is achieved by thresholding based on employing a moving average. In statistics a moving average is de ned as a nite impulse re- sponse lter used to analyze a set of data points by creating a series average of the full data set. The moving average is an approach to smoothen the data set. This section describes a practical tool developed based on moving average to de- ne a preliminary threshold for the data set to discard potential noise. Due to the fact that all data points distributed along the cable length are assumed to have similar weights, simple moving average (SMA) is preferred over other moving average techniques (cumulative, weighted and exponential moving averages). Al- gorithm 5.1 presents the preliminary noise reduction through thresholding based on computing an average of a stream of numbers by averaging over n elements from the stream. Next, the pre-processed measurement data is provided as input for thresholding based on levels assigned by the moving average algorithm. The moving average runs for data over the cable length for each single time block and sets the threshold value in accordance to the data level observed in that time block. The location averaging window (LAW), i.e. the length that the averaging is performed over, is an important factor in the calculation of moving average. Its value is estimated based on an experimental curve developed according to observed dispersion of defects. PDs are measured for each fractional time (FT). Fractional time for PD signals is de ned in relation to the signal propagation time of the whole cable circuit and it is expressed as an integer between 0 and 1000. That means that the length of the cable is scaled to a value between 0-1000. Therefore, defects from shorter cables are more distributed while discharges in longer cable
0 2000 4000 6000 8000 −10 −5 0 5 10 15 20 25 30 Cable length [m]
Defect dispersion (number of FTs)
FTs vs Cable length Fitted line 95% confidence bound
Figure 5.3: Empirical defect dispersion curve - Fractional time vs. cable length
are limited to a smaller region in terms of FTs. PDs from cables with various lengths have been studied, and the dispersion of their defects over the fractional time has been measured. The defect dispersion is plotted vs. the cable length in Figure 5.3 and a line is tted to the data. The 95% upper bound value is utilized further to calculate the averaging windows for dierent cable lengths.
Figure 5.4 illustrates pre-processed data in a single time block calculated for measured data from Circuit A and Circuit C and the calculated moving average curve for that particular time block. The moving average forms the threshold line and data below this line are to be excluded from the data set. Figure 5.5 de-