# Identifying Distribution Network Faults Using Adaptive Transition
Probability

Xinliang Ma  Luohe Power Supply Company
Luohe, China  Weihua Liu  Luohe Power Supply Company
Luohe, China  Bingying Jin  Academy of Electrical Engineering
Shanghai Jiaotong University
Shanghai, China

###### Abstract

A novel approach is suggested for improving the accuracy of fault detection in
distribution networks. This technique combines adaptive probability learning
and waveform decomposition to optimize the similarity of features. Its
objective is to discover the most appropriate linear mapping between simulated
and real data to minimize distribution differences. By aligning the data in
the same feature space, the proposed method effectively overcomes the
challenge posed by limited sample size when identifying faults and classifying
real data in distribution networks. Experimental results utilizing simulated
system data and real field data demonstrate that this approach outperforms
commonly used classification models such as convolutional neural networks,
support vector machines, and k-nearest neighbors, especially under adaptive
learning conditions. Consequently, this research provides a fresh perspective
on fault detection in distribution networks, particularly when adaptive
learning conditions are employed.

###### Index Terms:

Distribution network; early-stage fault; fault identification; feature
extraction; adaptive probability learning

##  I Introduction

Distribution networks play a critical role in the electricity supply system by
serving end-users and ensuring power quality, operational efficiency, and
innovative customer services [1] [2]. Despite their extensive coverage, varied
equipment, and relatively low replacement costs, distribution networks are
often neglected by power utilities when it comes to ensuring reliable supply
[3]. Consequently, the development of fault identification technologies in
these networks has been slow [3]. Currently, most faults in distribution
networks are only addressed through repairs after they occur, leading to
significant service disruptions for users. However, as the power grid evolves
and expectations for supply reliability increase, power utilities are now
placing more importance on predicting and diagnosing equipment faults in
distribution networks. This shift aims to promptly address safety risks and
minimize the frequency of power outages.

Faults in equipment within distribution networks can occur suddenly or
gradually over time [4]. One common fault is a ground fault, where protective
devices isolate the faulty portion and restore normal operation once the fault
is resolved [5]. However, the electrical arcing during the fault can cause
irreparable damage to the insulation. If this process repeats multiple times,
it can lead to insulation degradation and eventual breakdown [6]. These
initial phase faults, referred to as ”early-stage faults” in this article, are
often disregarded by power utilities but contain valuable information about
the insulation [7]. If effectively utilized, the waveforms associated with
these faults can provide early warnings of faults in distribution networks and
improve supply reliability [8] [9].

Due to the complex nature of distribution network structures, traditional
waveform analysis methods based on mechanisms are not efficient [10]. However,
the integration of multiple sensors has allowed for the adoption of data-
driven models in this field. The identification of faults in distribution
networks is a challenging task, particularly when it involves electrical
arcing in ground faults, which are characterized by randomness and uncertainty
[11]. Obtaining sufficient data for training models is difficult due to the
rarity of self-recoverable faults [12]. As a result, many fault identification
algorithms rely on simulated or experimental data for developing and testing
models. Only a small number of algorithms incorporate real field data, which
is more complex and influenced by multiple interfering factors [13] [14].
Therefore, it is crucial to assess model performance using real field data
[15][16]. Furthermore, modern artificial intelligence techniques such as
diverse convolutional neural networks often lack interpretability. The
features extracted from these models are not easily understandable to humans,
making it challenging to evaluate the quality of features and incorporate
prior knowledge.

We present a proposed solution to tackle the previously mentioned challenges
associated with identifying faults in distribution networks. Our approach
incorporates adaptive probability learning, which entails training the model
using simulated data and testing it with actual field data. By evaluating
feature similarity and extracting universal features, our adaptive probability
learning algorithm overcomes the difficulties posed by varying network
structures, line parameters, and operating conditions. This is crucial as the
distributions of simulated and real data often diverge in these aspects.

Our method comprises two stages: waveform decomposition to obtain feature
vectors in the first stage, and linear mapping in the second stage for
dimension reduction and feature reconstruction. We determine the optimal
linear mapping by maximizing the likelihood of consistent reconstruction.
Additionally, we utilize clustering in the reduced-feature space to classify
events. Compared to other approaches, our model effectively addresses the
disparities between simulated and real data and offers strong
interpretability. This presents a fresh and efficient strategy for fault
identification in distribution networks.

##  II Adaptive Probability Learning

An objective of adaptive learning is to address the discrepancy in data
distributions between simulated and real-world data. One proposed strategy to
overcome this problem involves measuring a model’s performance across
different data contexts using feature similarity as a metric [16][17]. On the
other hand, adaptive probability learning utilizes probability to evaluate the
similarity between different features by calculating the reconstruction error,
providing valuable insights [18][19][20].

###  II-A Adaptive Learning

Adaptive learning relies on two kinds of data: simulation data and real-world
data [21]. Simulation data ($D_{s}$) includes waveforms and event classes from
simulated events, while real-world data ($D_{t}$) consists of waveforms of
events with unknown classes. The main idea behind adaptive learning is that
the event categories in the simulation data are similar to those in the real-
world data. This similarity allows us to make informed guesses about the event
categories in the real-world data by utilizing the knowledge gained from the
simulated data [22]. However, it is important to note that these two datasets
often have different distributions due to variations in grid structure, line
parameters, and operational conditions. In simpler terms, the probability
distribution of the simulation data ($P_{s}\left(x_{i}^{s},y_{i}^{s}\right)$)
may not be the same as that of the real-world data
($P_{t}\left(x_{i}^{t},y_{i}^{t}\right)$). Therefore, adapting knowledge from
one domain to another presents challenges and complexities.

Figure 1: Illustration of Waveform Decomposition

It is important to consider that the performance of a classification model, as
measured by the error $L_{s}$ on simulated data, may not accurately reflect
its performance on real-world data, as indicated by the error $L_{t}$. This
discrepancy arises because we need to determine if the features generated by
the model have meaning and can be transferred between simulated and actual
data. To address this concern, our proposed approach utilizes waveform
decomposition to enhance feature congruence and identify the optimal linear
mapping. This relationship can be expressed as $L_{t}=L_{s}+L_{sim}$,
illustrating that a reliable classification model should not only perform well
on simulated data but should also exhibit a high degree of similarity in the
extracted features from simulated and actual datasets.

###  II-B Feature Extraction

The research utilizes a wavelet decomposition-based approach for feature
extraction, which is advantageous compared to deep neural networks. One
advantage is that it necessitates less data, making it appropriate for
situations with limited samples such as distribution network fault diagnosis
[23]. Another advantage is that the extracted features are highly
interpretable, facilitating the incorporation of prior knowledge and enhancing
accuracy [24].

Figure 2: An illustration of principles of reconstruction error

The method steps are as follows: first, waveforms are decomposed into
approximate and detail components through wavelet transformation. The
approximate component reflects the overall shape of the waveform, while the
detail component reflects distortions. Based on this decomposition, the
fundamental component $Z_{o}$ and bias $Z_{\text{off }}$ are extracted from
the approximate component, and pulse $z_{p}$, harmonic $Z_{h}$, and distortion
$Z_{d}$ are extracted from the detail component. For different components,
corresponding features are extracted. For example, the fundamental component
$Z_{o}$ corresponds to features like amplitude $A_{o}$ and frequency $f_{o}$,
bias $z_{\text{off }}$ corresponds to amplitude $A_{\text{off }}$, pulse
$z_{p}$ corresponds to peak value $A_{p}$ and pulse width $t_{p}$, harmonic
$Z_{h}$ corresponds to amplitude $A_{h}$ and frequency $f_{h}$, and distortion
$z_{d}$ corresponds to distortion factor $w_{d}$. All features are normalized
to eliminate scale effects. In addition to the features of the components
themselves, the time intervals between components
$t\left(z_{i},z_{i+1}\right)$ are also considered, where $z_{i}$ represents
the $i$-th component. Figure 1 gives an illustration of waveform
decomposition, and after this decomposition, any waveform
$w=\left\\{I_{\mathrm{A}},I_{\mathrm{B}},I_{\mathrm{C}},U_{\mathrm{A}},U_{\mathrm{B}},U_{C}\right\\}$
can be uniquely determined by a feature vector
$\phi(w)=\left[A_{o},f_{o},A_{off},A_{p},t_{p},A_{h},f_{h},w_{d},t\left(z_{i},z_{i+1}\right)\right]$.

###  II-C Adaptive Probabilistic Learning

The problem involves analyzing both simulated data ($x_{i}^{s}$) and real data
($x_{j}^{t}$). We know the category of the simulated data ($y_{i}^{s}$), but
the category of the real data ($y_{j}^{t}$) is unknown. We extract features
from the data, resulting in feature vectors denoted as
$A_{i}:=\phi\left(x_{i}^{s}\right)$ and $B_{j}:=\phi\left(x_{j}^{t}\right)$.
Since the feature vectors $A_{i}$ and $B_{j}$ have high dimensionality, we use
a linear mapping $\varphi$ to reduce their dimensions, resulting in reduced
feature vectors $A_{i}^{\prime}:=\varphi\left(A_{i}\right)$ and
$B_{j}^{\prime}:=\varphi\left(B_{j}\right)$. We then assess the similarity
between $A_{i}^{\prime}$ and $B_{k}^{\prime}$ by calculating the
reconstruction error. This allows us to determine the probability of
transforming between $A_{i}^{\prime}$ and $B_{k}^{\prime}$ and vice versa. It
is important to note that although $A_{i}^{\prime}$ and $A_{j}^{\prime}$ may
be different, their respective categories ($y_{i}^{s}$ and $y_{j}^{s}$) must
be the same. Figure 2 illustrates the concept of the reconstruction error.

First, to determine the likelihood of transforming $A_{i}^{\prime}$ into
$B_{k}^{\prime}$, we compute the inner product of the two vectors, denoted as
$M_{ik}=\left\langle A_{i}^{\prime},B_{k}^{\prime}\right\rangle$. Next, we
evaluate the correlation probability between the two vectors using a
undisclosed formula.

$P_{ik}^{ab}=P(B_{k}^{\prime}|A_{i}^{\prime}):=\frac{\exp(M_{ik})}{\sum_{k^{\prime}}\exp(M_{ik^{\prime}})}$

Likewise, the likelihood of $B_{k}^{\prime}$ changing into $A_{i}^{\prime}$
can be represented as follows:

$P_{kj}^{ba}=P(A_{j}^{\prime}|B_{k}^{\prime}):=\frac{\exp(M_{kj})}{\sum_{k^{\prime}}\exp(M_{k^{\prime}j})}$

Therefore, converting from $A_{i}^{\prime}$ to $B_{k}^{\prime}$ and then to
$A_{j}^{\prime}$ in this cycle has a probability of:

$P_{ij}^{aba}=\left(P^{ab}P^{ba}\right)_{ij}=\sum_{k}P_{ik}^{ab}P_{kj}^{ba}$

If the values $y_{i}^{s}$ and $y_{j}^{s}$ are in agreement, then the
anticipated distribution of $P_{ij}^{aba}$ can be expressed as:

$T_{ij}^{aba}=\frac{1}{2}\left(P_{ij}^{aba}+P_{ij}^{aba}\right)$

$T_{ij}:=\begin{cases}1/N_{c}(y_{i}^{s})&A_{i}^{\prime},A_{j}^{\prime}\text{
have the same category }\\\ 0&\text{ otherwise }\end{cases}$

The measure $H$ quantifies the difference between $P_{ij}^{aba}$ and $T$. In
order to calculate $P_{ij}^{aba}$, we use the count of instances that belong
to category $y_{i}^{s}$ in dataset $D_{s}$, denoted as $N_{c}(y_{i}^{s})$.

$L_{w}:=H(P_{ij}^{aba},T)$

To enhance the integration of real data in the reconstruction process, we
consider traversal error. This error evaluates the likelihood of each genuine
data element being included in the reconstruction procedure.

$L_{v}=H(P^{visit},V)$

$P_{k}^{v}=\sum_{x_{i}^{s}}P_{ik}^{ab},V_{k}=1/|D_{t}|$

During the training process of the model, the overall error, represented as
$L_{sim}$, is calculated by combining the errors that arise from measuring
feature similarity and analyzing simulation data. These calculations entail
evaluating and integrating errors from both categories.

##  III Experimental Data and Procedure

In order to verify and assess the reliability of the proposed model, a series
of experiments are conducted, utilizing both simulated data and real-world
data gathered in real-life situations. The key objective of adaptive learning
is to initially train the model using simulated data and subsequently enable
it to autonomously identify and categorize real data.

###  III-A Simulation system

Figure 3: Configuration of the simulation system

The simulation system shown in Figure 3 is based on the IEEE 13 node model and
operates at a voltage level of 10 kV and frequency of 50 Hz. A sampling
frequency of 4 kHz is used in the simulation. The figure indicates the fault
location and load conditions. The simulation system is created using PSCAD
software [25]. To simulate faults, the Kizilcay arc model is employed, which
captures the dynamic behavior of the arc using control theory principles,
specifically focusing on the energy balance within the arc column. The
mathematical expression for this arc model is provided as follows:

$\begin{gathered}\frac{dg(t)}{dt}=\frac{1}{\tau}\left(\frac{\left|i_{f}(t)\right|}{u_{o}+r_{o}\left|i_{f}(t)\right|}-g(t)\right)\\\
g(t)=\frac{u_{f}(t)}{i_{f}(t)}\end{gathered}$

Different variables such as fault impedance, fault starting angle, fault
distance, line parameters, load parameters, and noise levels were modified to
simulate network fault data under different conditions. These modifications
allowed for the creation of simulation data for four types of events: single-
frequency early fault, multi-frequency early fault, permanent fault, and
transient interference [26]. Figure 3 indicates the potential locations of
early faults, permanent faults, capacitors, and loads. A total of 10 sets of
simulation data were randomly generated for all possible scenarios. The arc
conductance $g(t)$, arc current $i_{f}(t)$, and arc voltage $u_{f}(t)$ are
measured in S/m, A, and V, respectively. The arc time constant $\tau$, arc
characteristic resistance $r_{o}$, and arc characteristic voltage $u_{o}$ are
measured in seconds, ohms, and volts, respectively. The parameter ranges for
$\tau$, $u_{o}$, and $r_{o}$ are 0.2-0.4 ms, 300-4000 V, and 0.01-0.015
$\Omega$, respectively.

Figure 4: Typical waveform for different types of events

###  III-B Actual Data

Between February and May 2021, data was gathered in Guangdong Province from
fault detection devices installed on a 10 kV overhead line. The system
utilizes low-current grounding and the fault detection device samples voltage
and current signals at a frequency of 4096 Hz. Recording commences when the
voltage or current signal surpasses a predefined threshold. The device
captures the three-phase voltage and current waveforms before and after a
fault occurrence, with a recording duration of sixteen cycles.

TABLE I: Distribution of event type Data Type | SIF | MIF | PF | TD
---|---|---|---|---
Simulation Data | 80 | 80 | 80 | 80
Actual Data | 71 | 64 | 93 | 88
Total | 151 | 144 | 173 | 168

Events in this classification are categorized based on waveform analysis and
onsite confirmation of the fault’s cause. There are three types of events:
Early faults, Permanent faults, and Transient interferences. Early faults are
transient faults that can be recovered and are further divided into single-
cycle early faults and multi-cycle early faults, indicating varying levels of
severity. Permanent faults cannot be self-recovered and require intervention
from protective devices [24]. While fault detection devices can also be
triggered by overvoltages caused by operations and lightning, these
overvoltages are not considered faults but are categorized as transient
interferences [27]. Figure 4 illustrates typical waveforms associated with
each event type. In addition, Table 1 provides statistical data on the
quantities of each event type.

###  III-C Experimental Procedure

Two experiments were carried out to assess the adaptability of the model in
this study. In the first experiment, the model was trained using all simulated
data, while a random sample of actual data was used for validation. The
remaining actual data was then used as the test set, with known labels
obtained from the validation set. The second experiment followed a similar
setup, with all simulated data used for training, a random sample of actual
data used for validation, and the remaining actual data used for testing.
However, in this case, the labels of the validation set were unknown. It is
important to note that each of these three sets serves a specific purpose: the
training set is used to train the model, the validation set assesses
performance and tunes hyperparameters, and the test set evaluates the final
model’s performance.

The experimental data is divided into three groups [28]: the training set
comprising 320 samples, the validation set consisting of 160 samples, and the
test set containing 156 samples. To minimize the impact of event type
distribution, each experiment is replicated 10 times, and the average
performance across these 10 iterations is measured. The F1 score is utilized
as the evaluation metric to assess the model’s performance.

##  IV Experimental Results and Analysis

In this section, we will compare the adaptive probabilistic learning model
introduced in this study with three commonly employed classifiers:
Convolutional Neural Network (CNN), Support Vector Machine (SVM), and
K-Nearest Neighbors (KNN) algorithm. The objective is to illustrate the
superiority of our proposed method. Unlike traditional classifiers, our
adaptive probabilistic learning model considers the variances between
simulated and real data distributions and integrates the notion of feature
similarity to tackle this problem. As a result, it achieves substantially
improved performance.

###  IV-A Adaptive Probabilistic Learning

The adaptive probabilistic concept learning model undergoes a training process
that involves breaking down waveforms into different components, such as
fundamental wave, bias, pulse, harmonic, and distortion. For each component,
feature values and time intervals are calculated, and these feature vectors
are then reduced in dimensionality through a linear mapping step. The
similarity of features is measured by the reconstruction error. The training
of the model involves estimating the actual data error, which combines the
feature similarity error and the classification error of simulated data. This
estimation helps determine the optimal parameters of the linear mapping.
During testing, test waveforms are decomposed and mapped onto the feature
space using linear mapping. Once in this space, they are clustered to make
predictions about the corresponding event types. In Experiment 1, the
validation set contains the actual data labels. This allows the model trained
using the aforementioned method to directly predict the validation set. Using
these predicted results, the best model can be identified by comparing them
with the true labels. Subsequently, this best model is applied to the test set
to generate the final test results.

The validation set used in Experiment 2 includes unlabeled data. As a result,
the model can be directly used on the test set to obtain the final test
results.

Table 2 presents evidence of how the adaptive probabilistic learning model
effectively handles diverse faults. The model achieves this by taking into
account the dissimilarities in distribution between simulated and real data
and utilizing the similarities in characteristics to establish a correlation
between the model’s errors on both types of data. The central concept involves
identifying a suitable conversion technique that transforms the original
waveform into a feature vector space. This enables precise categorization of
both simulated and real data within this space, ensuring that similar data is
clustered together while dissimilar data is dispersed.

TABLE II: F1 score of different models Model | Exp. | SIF | MIF | PF | TD | Ave.
---|---|---|---|---|---|---
APL | 1 | 0.910 | 0.945 | 0.972 | 0.897 | 0.931
| 2 | 0.874 | 0.904 | 0.951 | 0.847 | 0.894
CNN | 1 | 0.742 | 0.784 | 0.801 | 0.714 | 0.760
| 2 | 0.701 | 0.722 | 0.741 | 0.684 | 0.712
SVM | 1 | 0.804 | 0.791 | 0.831 | 0.763 | 0.797
| 2 | 0.714 | 0.721 | 0.730 | 0.691 | 0.714
KNN | 1 | 0.725 | 0.740 | 0.734 | 0.667 | 0.717
| 2 | 0.684 | 0.707 | 0.700 | 0.624 | 0.679

###  IV-B Comparing with other models

The experiment utilized a convolutional neural network model based on the
architecture of AlexNet. The model consisted of 5 convolutional layers and 3
fully connected layers. The input layer had dimensions of 1 x N x 6, where N
represented the length of the sample including 6 groups of waveforms. The
first convolutional layer had a kernel size of 1 x 41, a stride of 20, and 40
convolutional kernels. The second convolutional layer had a kernel size of 1 x
20, a stride of 10, and 20 convolutional kernels. The third, fourth, and fifth
convolutional layers had kernel sizes of 1 x 10, strides of 5, and 10
convolutional kernels each. The three fully connected layers had sizes of 512,
512, and 4, respectively, with the last layer representing the number of
output classes. The process involved extracting and linearly transforming the
input waveform, with the final output indicating the probability of the event
belonging to each class. Choosing the appropriate kernel function was crucial
for the support vector machine model, and in Experiment 2, the kernel function
category with the highest accuracy in the training set was selected. Both
Experiment 1 and 2 confirmed the selection of the polynomial kernel function
type.

In the K-nearest neighbor algorithm, the selection of the hyperparameter K
involves testing different values and comparing their classification accuracy
on the validation set. The value of K that yields the highest accuracy on the
training set is chosen since the labels of the validation set are unknown in
Experiment 2. The optimal K value was found to be 5 in Experiment 1 and 10 in
Experiment 2. Table 2 presents the F1 scores of different models, with the
adaptive probability learning model showing significantly higher
classification accuracy compared to the other three models. This is because
the model takes into account the dissimilarities between the training and test
sets, and the extracted features exhibit high similarity between the two sets,
indicating that the model captures general features well. On the other hand,
the other three models perform well on the training set but poorly on the test
set due to differences in data distribution. Additionally, Experiment 1
demonstrates significantly higher accuracy than Experiment 2, suggesting that
the partially known labels in Experiment 1 help the model overcome
distribution differences and achieve better classification accuracy. The
difference in accuracy between Experiment 1 and Experiment 2 highlights the
adaptive learning capability of the model. It is worth noting that the
adaptive probability learning model outperforms the other three models in
terms of this ability, indicating reduced reliance on actual data labels.

Figure 5: Distribution of Average F1 Score over classes for different models

Figure 5 illustrates the stability of the average class F1 scores [29] for the
four models, indicating whether their classification accuracy varies with
changes in event distribution. The adaptive probability learning model
demonstrates highly consistent classification accuracy. In experiment 1, its
accuracy consistently falls within the range of 0.90 to 0.95, while in
experiment 2, it remains concentrated around 0.90. In contrast, the
convolutional neural network exhibits more scattered accuracy, suggesting a
greater vulnerability to variations in data type distribution. This can be
attributed to its numerous parameters and reliance on substantial amounts of
data for determining network weights. Consequently, in scenarios with limited
samples and significant variations in event types, training results often
differ, leading to fluctuating accuracy. While the support vector machine and
K-nearest neighbor algorithm also display relatively stable classification
accuracy, their overall performance level is low and not suitable for
practical real-world scenarios. In conclusion, the proposed method is best
suited for identifying faults in distribution networks.

##  V Conclusion

The scarcity of training samples is a major hurdle when it comes to
identifying faults in distribution networks. To address this problem, the
adaptive probability learning method utilizes a two-step approach. First, it
extracts feature vectors by breaking down waveforms, and then it reduces the
dimensionality through linear mapping. The model then solves an optimization
problem to establish a relationship between errors in simulated and real data,
with the goal of maximizing consistency probability during the reconstruction
process. This ultimately results in the achievement of an optimal linear
mapping.

There are several advantages to using the adaptive probability learning method
instead of other methods. One advantage is that it produces extracted features
that are easy to interpret, which makes it easier to incorporate prior
knowledge. Additionally, this method is able to make good use of simulated
data during training, which helps overcome the issue of having limited samples
when identifying faults in distribution networks. Finally, by incorporating
field actual data, the model’s performance can be further improved, allowing
maintenance personnel to create a sample library right from the start.

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