#  A Heterogeneous Graph-Based Multi-Task Learning for Fault Event Diagnosis
in Smart Grid

Dibaloke Chanda , ,
Nasim Yahya Soltani Part of this work will be presented at the Intl. Workshop
on Machine Learning for Signal Processing, Rome, Italy, September 2023.
0000-0001-5993-659X 0000-0002-4502-8715

###### Abstract

Precise and timely fault diagnosis is a prerequisite for a distribution system
to ensure minimum downtime and maintain reliable operation. This necessitates
access to a comprehensive procedure that can provide the grid operators with
insightful information in the case of a fault event. In this paper, we propose
a heterogeneous multi-task learning graph neural network (MTL-GNN) capable of
detecting, locating and classifying faults in addition to providing an
estimate of the fault resistance and current. Using a graph neural network
(GNN) allows for learning the topological representation of the distribution
system as well as feature learning through a message-passing scheme. We
investigate the robustness of our proposed model using the IEEE-123 test
feeder system. This work also proposes a novel GNN-based explainability method
to identify key nodes in the distribution system which then facilitates
informed sparse measurements. Numerical tests validate the performance of the
model across all tasks.

###### Index Terms:

Distribution system, explainability, fault event diagnosis, heterogeneous
multi-task learning, smart grid.

##  I Introduction

Fault diagnosis is a crucial task for the operation and maintenance of power
systems, particularly in distribution networks due to the nature of complex
interconnectivity and scale of the network. Failure to take proper action
during a fault event can result in a cascading outage of the distribution
system [1, 2]. For uninterrupted operation in the case of a fault occurrence
grid operators need to identify the precise location of the fault in addition
to the type of the fault. Furthermore, knowing the fault resistance and fault
current before taking action for fault isolation and fault clearance
guarantees the implementation of appropriate safety measures. This additional
information allows grid operators to make more informed decisions and also to
plan for necessary repairs or equipment replacements. Not only that, but it
also provides insight into post-fault analysis to identify if all the
protection systems performed as intended. Due to the advancement in deep
learning in recent years, most fault diagnosis systems are utilizing a data-
driven approach. However, there is a lack of unified methods that take into
account the challenges associated with real-world deployment and many research
provides analysis based on theoretical assumptions only.

In this work, we propose a unified heterogeneous MTL-GNN architecture that is
capable of performing fault detection, fault localization, fault type
classification, fault resistance estimation and fault current estimation. All
the tasks are performed in a simultaneous manner as opposed to a sequential
manner which ensures the decoupling of tasks.

We call it a heterogeneous MTL in contrast to a homogeneous MTL due to the
fact that the proposed model performs both classification and regression
tasks. We take into account all $5$ types of short circuit faults that can
occur in a distribution system [3]. This includes asymmetrical faults
consisting of line-to-ground faults (LG), line-to-line faults (LL), line-to-
line-to-ground faults (LLG) and symmetrical faults consisting of line-to-line-
to-line-to-ground faults (LLLG), line-to-line-to-line faults (LLL). To address
the challenges associated with real-world deployment, our analysis takes into
account measurement error, variable fault resistance, small dataset and sparse
measurements. To make our contribution clear in the following section, the
related literature in this domain and the drawbacks and scope for development
have been reviewed.

##  II Literature Review

The literature on fault event diagnosis is very extensive [4, 5, 6]. This
includes different kinds of faults such as over-voltage, insulator, voltage
sag, arc, and short-circuit faults. They can be broadly structured into two
categories. One category are data-driven approaches which utilize deep
learning models [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] and
another category includes traditional methods [22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37] using statistical measures, signal processing
and the physics of fault events. The latter category can be further broken
down into separate categories like traveling wave-based[22, 23, 24, 25],
impedance-based [26, 27, 28, 29], morphology-based [30, 31, 32, 33, 34], and
voltage sag [35, 36, 37] methods.

As regards the data-driven approaches, there is a multitude of methods and
architectures but we broadly divide them into two categories which are GNN-
based approaches [15, 16, 17, 18, 19, 20, 21], multi-layer perceptron (MLP)
and convolutional neural network (CNN) based approaches[7, 8, 9, 10, 11, 12,
13, 14].

###  II-A Traditional Methods

The traveling wave-based methods [22, 23, 24, 25] analyze the characteristics
of traveling waves generated when a fault event occurs. During a fault event,
the sudden change in voltage and current around the fault location initiates a
transient disturbance that propagates along the distribution line. This
transient wave and its reflected counterpart are picked up by fault recorders
installed at a substation or two substations depending on whether it is a
single-end method or a double-ended method. Based on the time of arrival of
the transient wave and the reflected wave it is possible to deduce the
location of the fault.

TABLE I: Summary of Technical Differences with Previous GNN-Based Literature

Tasks |  |  Chen et al. [15] |  |  Sun et al. [16] |  |  Freitas et al. [17] |  |  Li et al. [18] |  |  Mo et al. [19] |  |  Hu et al. [20] |  |  Nguyen et al. [21] |  |  Ours
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Fault Resistance |  |  Variable |  |  Variable |  |  Variable |  |  Variable |  |  Constant |  |  Constant |  |  Constant |  |  Variable
|  Fault Detection
---
|  ✗ |  |  ✗ |  |  ✗ |  |  ✗ |  |  ✗ |  |  ✗ |  |  ✓ |  |  ✓
|  Fault Localization
---
|  ✓ |  |  ✓ |  |  ✓ |  |  ✓ |  |  ✓ |  |  ✓ |  |  ✓ |  |  ✓
|  Fault Classification
---
|  ✗ |  |  ✗ |  |  ✗ |  |  ✗ |  |  ✓ |  |  ✓ |  |  ✓ |  |  ✓
|  Fault Resistance Estimation
---
|  ✗ |  |  ✗ |  |  ✗ |  |  ✗ |  |  ✗ |  |  ✗ |  |  ✗ |  |  ✓
|  Fault Current Estimation
---
|  ✗ |  |  ✗ |  |  ✗ |  |  ✗ |  |  ✗ |  |  ✗ |  |  ✗ |  |  ✓
|  Types of Fault Considered
---
|  LG, LL, LLG |  |  LG, LL, LLG LLL, LLLG |  |  LG |  |  LG, LL, LLG |  |  LG, LL, LLG |  |  LG, LL, LLG LLL, LLLG |  |  LG, LL, LLG LLL, LLLG |  |  LG, LL, LLG LLL, LLLG

In impedance-based methods [26, 27, 28, 29], current and voltage signals are
measured along different places on the distribution line. In the case of a
fault event, these measured signals are used to isolate the fundamental
frequency to estimate the apparent impedance. This apparent impedance is then
used to locate the fault event.

Morphology-based methods [30, 31, 32, 33, 34] make use of mathematical
morphological operations like dilation, erosion, closing and opening on the
waveform generated during a fault event to extract features that correspond to
a fault event. After feature extraction is complete, the extracted features
are passed to different classifier algorithms like decision trees [31],
recursive least square stage [32], and random forest [34].

Voltage sag methods [35, 36, 37] use the characteristic of the reduction of
voltage magnitude in the case of a fault event to isolate the fault’s exact
location. When a fault event occurs there is a sudden dip in the voltage
magnitude. This sudden dip occurs only at the location of the fault event
which can isolated based on the characteristics of the voltage sag.

###  II-B Deep Learning Based Methods

####  II-B1 MLP and CNN Based Methods

These methods use historical or software-simulated data relating to fault
events in distribution systems and use them to train MLP and CNN architectures
to do prediction tasks like fault detection, fault localization and fault
classification.

The work in [7] is one of the early papers that use a MLP. The input to their
proposed model is the current measures of distributed generation units (DGs)
and substation and they perform fault localization as output. Another similar
work that uses MLP is [8] where the authors use the IEEE-13 bus system to
perform both fault classification and localization. In [12] the authors use
MLP but with an additional fuzzy layer and their analysis is on the IEEE-37
bus system.

In [9] the CNN architecture is adopted for fault localization. First, the
authors use a continuous wavelet transform (CWT) algorithm to convert current
phasors to images which are fed to a CNN model to localize the fault. The work
in [10] also uses a CNN to localize faults but considers partial observability
of the grid on IEEE-39 and IEEE-68 bus systems. In [11] the authors take a
slightly different approach by using 1-D convolutions with double-stage
architecture. The first stage extracts the features and the second performs
fault identification. Similar to [9], the work in [13] also uses CWT to
convert time-domain current signals to image domain and use the transformed
data for fault classification and localization on the IEEE-34 bus system. A
different approach by using a capsule-based CNN is proposed in [14] to do
fault detection, localization and classification.

####  II-B2 GNN Based Methods

The first prominent work to use GNN for fault localization is in [15]. In this
work, IEEE-123 and IEEE-37 are used as the test feeder system and the authors
consider a range of factors like metering error, changes in topology, etc. for
their analysis. The architecture employed is CayleyNets [38] which is a graph
convolutional neural network (GCN) based on spectral theory. The feature
vector considered as the input to their model consists of both voltage and
current phasors measured from the buses in the distribution system.

The subsequent notable work is [16] which utilizes not only node features but
also link features that include branch impedance, admittance and regulation
parameters of the distribution lines. The authors validate their approach on a
self-designed $6.6$ KV system with $12$ buses and $8$ loads.

Two other related research works are in [17] and [18]. For the first one, the
authors consider a gated GNN [39] architecture. However, they only consider
single-line-to-ground fault as opposed to [18] which considers all three types
of asymmetrical short circuit fault. Not only that, but the authors also
consider limited observability and limited labels in their implementation.
Their proposed method has a two-stage architecture with only voltage phasors
as input.

In [19] there is a more recent work that uses a different variant of GNN, a
graph attention neural network (GAT) [40] to do both fault localization and
classification in IEEE-37 feeder system. In their analysis, the authors
consider a constant fault resistance and the fault localization prediction is
dependent on the fault classification task.

These above-mentioned works consider instantaneous current and/or voltage
phasors meaning that they require measurement at the fault time, no pre-fault
or post-fault measurement is required. But the method proposed in [20]
requires a fault waveform sampled at $1$ KHz. Similar to [19] their analysis
considers constant fault resistance. One important contribution of this
research work is that they use MTL to do both fault classification and fault
localization at once in contrast to [19].

Another similar recent work employs spatial-temporal recurrent GNN to do three
tasks simultaneously which are fault detection, classification and
localization [21] . The authors report numerical results tested on a microgrid
and IEEE-123 bus system. Similar to the previous approach their analysis
considers constant resistance and due to the temporal nature of their proposed
method, it requires high time resolution of fault waveshape.

The summary of major technical differences between our proposed model with the
existing GNN-based models is outlined in Table I. We make the argument that
the models that are only trained for fault localization and/or classification
[15, 16, 17, 18, 19, 20] will require a separate method to first distinguish
between fault event and other events (non-fault) in distribution system like
load change.

Also, the models that consider constant resistance[19, 20, 21] will only
perform well as long as the actual fault resistance is similar to the fault
resistance considered during the generation of the training dataset. Even
though these research work report their model performance with different
resistance values, these reported values are only applicable to that specific
resistance value. In contrast, [15, 16, 17, 18] takes a more practical
approach and train their model considering a range of possible fault
resistance values. As long as the fault resistance is within that range, the
model is expected to hold its performance.

The drawback of [20, 21] is that their proposed model is dependent on temporal
characteristics. To perform well the dataset needs to have high temporal
resolution. As the scale of the distribution system grows data acquisition,
storage and training overhead for this approach becomes progressively more
demanding. In addition, there is a requirement for perfect time
synchronization which further complicates the system design.

Considering these drawbacks the key contributions of our work are outlined as:

  1. 1.

We propose a unified heterogeneous MTL-GNN architecture to perform $5$
different tasks simultaneously for a fault event which are fault detection,
fault localization, fault type classification, fault resistance estimation and
fault current estimation.

  2. 2.

Our proposed model works in the presence of measurement error, variable
resistance, small dataset and sparse measurements which are factors that
should be considered in real-world deployment.

  3. 3.

We utilize an explainability algorithm specific to GNN to identify key nodes
in the grid which provides the opportunity for informed sparse measurement.

  4. 4.

As all multi-task prediction/estimation shares a common backbone GNN for
feature extraction, it significantly reduces the parameter overhead and
computational complexity in addition to providing modularity for task-specific
layer.

The remaining parts of the paper are organized as follows. In section III, a
brief overview of the test feeder system and the process involving the dataset
generation is provided. The mathematical framework for the overall methodology
and architecture used is given in section IV. In the following section V, we
outline the details of the architecture, training procedure and
hyperparameters assumed during training. Numerical results and discussions on
them are presented in section VI. Finally, section VII concludes the paper.

##  III Dataset Generation

This section briefly covers the details of the IEEE-123 node feeder system and
the dataset generation process as well as the underlying assumptions
considered in the generation process.

###  III-A IEEE-123 Node Feeder System

Figure 1: Diagram of IEEE-123 node feeder system. The highlighted blue blocks
represent the node pairs that are considered connected. The number of voltage
regulators, transformers, and switches is mentioned in ($\cdot$) and the
number of buses, their phases and load connectivity are mentioned in the
table. The active substation (source bus) is connected to the node $150r$.

The IEEE-123 node feeder system [41] shown in Fig. 1 operates at a nominal
voltage of $4.16$ KV and consists of both overhead and underground lines. It
has three-phase, two-phase and single-phase lines and a couple of open and
closed switches, voltage regulators and a transformer. There are a total of
$85$ nodes that have loads connected to them and most of them are connected to
single-phase buses and the rest are connected to three-phase buses. For
simulating all $5$ types of short circuit faults including asymmetrical and
symmetrical faults all three phases need to be considered. Therefore, in our
analysis, similar to [15, 18] we only consider three-phase nodes which tally
up to $68$ nodes including the $2$ three-phase regulators $150$r and $160$r.
Also, similar to [15, 18] we also make the assumption that some specific pairs
of nodes are connected which are ($149$, $150$r), ($18$, $135$), ($13$,
$152$), ($60$, $160$, $160$r,), ($61$, $61$s), ($97$, $197$), ($9$, $9$r),
($25$, $25$r). The reason behind this assumption is the pairs ($18$, $135$),
($13$, $152$), ($61$, $61$s), ($97$, $197$), ($60$, $160$) are connected via
closed switches and the pairs ($149$, $150$r), ($9$, $9$r), ($25$, $25$r)
consists of buses and their corresponding regulators. This is shown in Fig. 1
via highlighted blue sections. One important thing to note here, in actuality,
considering all the $4$ regulators as separate nodes the total number of nodes
in the feeder system results in $128$ nodes.

###  III-B Dataset Description and Generation Procedure

For dataset generation, we use OpenDSS[42], open-sourced by the electric power
research institute (ERPI) as a power flow equation solver engine and use
py_dss_interface module in Python to interface with it.

Figure 2:  t-Distributed Stochastic Neighbor Embedding (t-SNE) visualization
of all the data points. (Left) shows the dataset generated with a variable
range of fault resistance sampled from a uniform distribution
$\mathcal{U}(0.05\Omega,20\Omega$). (Right) shows the dataset g with a
constant fault resistance $20\Omega$.

We opt to utilize only voltage phasors as measurements based on [15], where
the authors showed that the performance of their model is almost identical
with or without current phasors. Therefore, there is no incentive to use
current phasors as features because that would just double the amount of
computation needed at the expense of no performance increase. For the three-
phase buses shown in Fig. 1 for all three phases ( Phase A, Phase B, Phase C)
voltage amplitude and angle (in radian) can be measured. For single-phase and
two-phase buses values for the missing buses are padded with zero.

As our proposed model does $5$ tasks simultaneously, for each data point we
generate $5$ labels as fault detection, fault location, fault classification,
fault resistance, and fault current labels. The first three labels are
discrete values whereas the last two labels are continuous values. As
mentioned before for practical consideration we assume a range of fault
resistance instead of a single fault resistance value. Fault resistance values
are sampled from a uniform distribution (rounded up to $2$ decimal point)
which is $f_{res}\sim\mathcal{U}(\min_{res},\max_{res})$, where $f_{res}$ is
the sampled fault resistance value and $\min_{res}$ and $\max_{res}$ are the
lower and upper bound of the uniform distribution. For our initial analysis,
we assume a lower bound of $0.05~{}\Omega$ and an upper bound of
$20~{}\Omega$. The practical consideration claim we make is justified by Fig.
2 which clearly shows that for constant resistance analysis[19, 20, 21] the
fault diagnosis becomes trivial. For each fault type, we generate $20400$ data
samples. This results in a total of $20400\times 5=102000$ data points for the
fault events where there are $300$ samples per bus. For non-fault event data
generation, we vary all $91$ loads connected to the $85$ buses according to
another uniform distribution given by
$f_{load}\sim\mathcal{U}(\min_{load},\max_{load})$. We assume
$\min_{load}=20~{}\text{KW}$ and $\max_{load}=80~{}\text{KW}$ based on the
typical load range associated with the IEEE-123 node feeder system. For non-
fault events, the number of data points is also $20400$. All the values in the
feature vector are standardized by subtracting the mean value and dividing by
the standard deviation.

##  IV Mathematical Framework

Figure 3: Architecture of the proposed heterogenous MTL-GNN. The input
features go through the common backbone GNN to generate graph embeddings. For
visual clarity message passing across the layers for a couple of nodes
(highlighted in green) is shown, where the blue highlighted sections signify
nodes included in the message passing process (Left). The embeddings generated
by $128$ nodes are flattened and concatenated together to convert to a one-
dimensional vector (Middle). The concatenated feature vector is passed to $5$
heads, three classification heads and two regression heads. The corresponding
loss is computed based on the predicted output ($\hat{y}_{t}^{k}$) and ground
truth label ($y_{t}^{k}$). The computed loss for each task is weighted and
summed together (Right).

GNN uses message passing between nodes in the graph for the propagation of
features which allows feature representation learning in addition to
topological representation learning. GCN is a specific variant of GNN, first
introduced by [43]. The proposed model consists of GCN layers as a common
backbone followed by dense layers as prediction heads. GCN perform message
passing between a given target node $u$ in a particular layer $l$ and the
neighbors of the target node $v\in\mathcal{N}(u)$ according to the following
equation:-

$\mathbf{h}_{u}^{(l)}=\sigma\left(W^{(l)}\sum_{v\in\mathcal{N}(u)\cup\\{u\\}}\frac{\mathbf{h}_{v}}{\sqrt{\lvert\mathcal{N}(u)||\mathcal{N}(v)\rvert}}\right)$
(1)

where, $\frac{1}{\sqrt{\lvert\mathcal{N}(u)\mid\mid\mathcal{N}(v)\rvert}}$ is
the normalization factor, $\mathbf{h}_{v}$ is the hidden representation of the
neighboring nodes, $W^{(l)}$ is the weight matrix consisting trainable
parameters, $\sigma$ is the non-linear activation function and
$\mathbf{h}^{(l)}_{u}$ is the hidden representation of the target node. This
equation gives an intuitive understanding of how the message passing algorithm
works, but for implementation (2) is more common in literature.

$\mathcal{H}^{(l+1)}=\sigma\left(\tilde{D}^{-\frac{1}{2}}\tilde{\mathcal{A}}\tilde{D}^{-\frac{1}{2}}\mathcal{H}^{(l)}W^{(l)}\right)$
(2)

where, $\tilde{\mathcal{A}}=\mathcal{A}+I$ is the adjacency matrix with self-
loops and $\mathcal{A}$ is the original adjacency matrix without self-loops,
$\tilde{D}$ is the degree matrix considering the modified adjacency matrix
$\tilde{\mathcal{A}}$ and $\mathcal{H}^{(l)}$ is the $l^{th}$ GCN layer
containing all the feature representation of all the nodes for that particular
layer.

The node feeder system is represented by a graph
$\mathcal{G}:=(\mathcal{V},\mathcal{E},\mathcal{A})$ where $\mathcal{V}$
represents the set of nodes in the feeder system which can be represented as
the union of three disjoint sets as shown in the following equation:

$\mathcal{V}:=\mathcal{V}_{1p}\cup\mathcal{V}_{2p}\cup\mathcal{V}_{3p}$ (3)

where $\mathcal{V}_{1p}$, $\mathcal{V}_{2p}$ and $\mathcal{V}_{3p}$
respectively represent the nodes associated with three-phase, two-phase and
single-phase buses and regulators. With the inclusion of the voltage
regulators as nodes, $\lvert\mathcal{V}_{3p}\rvert=68$,
$\lvert\mathcal{V}_{2p}\rvert=4$, $\lvert\mathcal{V}_{3p}\rvert=56$ which
equate to $\lvert\mathcal{V}\rvert=128$. The set of edges is denoted by
$\mathcal{E}$ where $\lvert\mathcal{E}\rvert=127$ and
$\mathcal{A}\in\mathbb{R}^{\lvert\mathcal{V}\rvert\times\lvert\mathcal{V}\mid}$
is the symmetric adjacency matrix. As $\mathcal{G}$ is a sparse graph, the
adjacency matrix is replaced by a coordinate list format (COO) representation
denoted by $E_{c}\in\mathbb{R}^{2\times 2\lvert\mathcal{E}\rvert}$. $E_{c}$
holds the node pair that is connected by an edge. Owing to the fact that the
graph under consideration is undirected in nature, if there is an edge between
node pair $(p,q)$ then both $(p,q)$ and $(q,p)$ are included in $E_{c}$.

From this point, we use superscript $k$ to denote the index associated with a
data point, subscript $i$ to denote the index of a specific node and subscript
$t$ to denote the task index.

The generated dataset contains a feature vector associated with each node in
the graph. That is represented by $\mathbf{z}^{k}_{i}\in\mathbb{R}^{6}$ which
is the feature vector for $i^{th}$ node of the $k^{th}$ data point.

Each feature vector holds the value of voltage phasor meaning voltage
amplitude ($V_{i}$) and angle ($\theta_{i}$) for three phases. This can be
mathematically represented by (4).

$\mathbf{z}_{i}^{k}:=[\left.V_{i}^{A},\theta_{i}^{A},V_{i}^{B},\theta_{i}^{B},V_{i}^{B},\theta_{i}^{C}\right]^{k}$
(4)

where, the superscript $A,B,C$ represent respectively the value associated
with Phase-A, Phase-B and Phase-C. It is worth noting that since for the nodes
in $\mathcal{V}_{2p}$ and $\ \mathcal{V}_{1p}$, values for some specific
phases do not exist, they are replaced with zeros.

Stacking all the feature vectors for all $\lvert\mathcal{V}\rvert$ nodes in a
graph $\mathcal{G}^{k}$ results in a feature matrix
$\mathcal{X}^{k}\in\mathbb{R}^{\lvert\mathcal{V}\rvert\times\lvert\mathcal{V}\rvert}$
associated with that graph. A particular row $i$ in the feature matrix
$\mathcal{X}^{k}$ corresponds to the feature vector for the $i^{th}$ node. It
should be noted the COO representation of the adjacency matrix is the same for
all the data points i.e. $E_{c}^{k}=E_{c},\forall k$. Therefore, the $k^{th}$
input data point of our dataset can be represented by
$X^{k}:=(\mathcal{X}^{k},E_{c})$.

For each input data point, there are $5$ labels which are represented
respectively by $y_{\text{detect}}$, $y_{\text{loc}}$, $y_{\text{type}}$,
$y_{\text{res}}$, $y_{\text{current}}$. These signify the fault detection
label, fault classification label, fault type label, fault resistance label
and fault current label. For ease of representation, the subscripts are
replaced by the corresponding task index. The first three labels are for
classification type prediction and the last two labels are for regression type
prediction. Now we define $Y$ as an ordered list of all the labels which can
be represented as the following:-

$Y^{k}:=(y^{k}_{1},y^{k}_{{2}},y^{k}_{3},y^{k}_{4},y^{k}_{5})$ (5)

Therefore, our dataset can be succinctly written as

$\mathcal{D}^{k}:=\\{(X,Y)^{k}\\}$ (6)

where a single data point and the corresponding labels are indexed by $k$. The
size of the dataset is denoted by $\lvert\mathcal{D}^{k}\rvert=N$ and the
training and testing part of the dataset is represented by
$\mathcal{D}^{k}_{tr}$ and $\mathcal{D}^{k}_{test}$ respectively, each of
which has the size $\lvert\mathcal{D}^{k}_{tr}\rvert=N_{tr}$ and
$\lvert\mathcal{D}^{k}_{test}\rvert=N_{test}$. Similarly, the inputs
associated with the training and testing dataset are expressed as $X_{tr}$ and
$X_{test}$ and labels associated are expressed by $Y_{tr}$ and $Y_{test}$.

Now in implementation, we modify the labels of fault current because they can
have a varying range of magnitude up to order of $10^{3}$ or more. This can
make the entire optimization process unstable and result in exploding
gradients during the training phase. Hence we first normalize the labels to
get $\tilde{y}_{t=5}$ followed by taking the negative log which can be
expressed as

$y^{k}_{5}:=-\ln(\frac{y^{k}_{5}}{\sum^{N}_{k=1}y^{k}_{5}})$ (7)

This makes the fault current labels in the same range as fault resistance and
hence results in much more effective training.

We define our heterogeneous MTL model as $f_{\theta}$, parameterized by
$\theta$. The model can be sectioned into two parts. The first part is the
common backbone GNN represented by $g_{\theta^{sh}}$ where $\theta^{sh}$ are
the parameters of the common backbone $g$. Now for each task $t$, there is a
sequence of separate dense layers which can be represented by $h_{\theta^{t}}$
where $\theta^{t}$ represent parameters associated with a specific task $t$.
As there are a total of $T=5$ tasks, we can represent $t$ as
$t\in\\{1,2,\cdots,T\\}$ which means the network parameters can be expressed
as $\theta:=\\{\theta^{sh},\theta^{1},\theta^{2},\cdots,\theta^{T}\\}$. Now
the predicted output $\hat{y}^{k}_{t}$ of a task $t$ for $k^{th}$ data point
can be expressed as (8)

$\hat{y}^{k}_{t}:=h_{\theta^{t}}\circ g_{\theta^{sh}}(X^{k}_{tr})$ (8)

TABLE II: Number of parameters for a single batch of size $32$

Layers |  $\\#$ Parameters | Output Shape
---|---|---
Input Layer ($\mathcal{H}^{0}$) | — | $(32,128,6)$
GNN Backbone($g_{\theta^{sh}}$): |  |
GCNConv ($\mathcal{H}^{1}$) | $224$ | $(32,128,32)$
Layer Normalization | $64$ | $(32,128,32)$
GCNConv ($\mathcal{H}^{2}$) | $1056$ | $(32,128,32)$
Layer Normalization | $64$ | $(32,128,32)$
GCNConv ($\mathcal{H}^{3}$) | $1056$ | $(32,128,32)$
Concatenation Layer | — | $(32,4096)$
Classification Heads($h_{\theta^{cls}}$): |  |
Fault Detection Head ($h_{\theta^{1}}$) | $172,974$ | $(32,2)$
Fault Localization Head ($h_{\theta^{2}}$) | $135,328$ | $(32,128)$
Fault Type Classification Head ($h_{\theta^{3}}$) | $173,222$ | $(32,6)$
Regression Heads($h_{\theta^{reg}}$): |  |
Fault Resistance Estimation Head ($h_{\theta^{4}}$) | $172,969$ | $(32,1)$
Fault Current Estimation Head ($h_{\theta^{5}}$) | $172,969$ | $(32,1)$
Total | $829,926$ |

For each task, we define a loss function $\mathcal{L}_{t}$ which takes in
$X^{k}_{tr}$ and the parameters of the proposed model and computes the loss
for that particular task. Each loss is weighted by a weighting factor $w_{t}$.
The overall objective function including L2 regularization can be expressed as
(9) where $\lambda\in\mathbb{R}$ is the regularization hyperparameter.

$\min_{\theta}\sum_{k=1}^{N_{tr}}\sum_{t=1}^{T}w_{t}\mathcal{L}_{t}({\theta},X^{k}_{tr})+\lambda\|\theta\|_{2}$
(9)

For the classification layers jointly expressed as
$h_{\theta^{cls}}=\\{h_{\theta^{1}},h_{\theta^{2}},h_{\theta^{3}}\\}$ the
negative log-likelihood (NLL) loss function is used and for the regression
layers jointly expressed as
$h_{\theta^{reg}}=\\{h_{\theta^{4}},h_{\theta^{5}}\\}$ the mean squared error
(MSE) loss is used. For generality, we express the classification losses as
$\mathcal{L}_{cls}$ and regression losses as $\mathcal{L}_{reg}$ and they are
defined as following over entire training data:-

$\mathcal{L}_{cls}=-\sum_{k=1}^{N_{tr}}y_{t}^{k}\log\frac{\exp(h_{\theta^{t}}\circ
g_{\theta^{sh}}(X^{k}_{tr}))}{\sum_{j=1}^{m}\exp(h_{\theta^{t}}\circ
g_{\theta^{sh}}(X^{k}_{tr}))}$ (10)

where $t\in\\{1,2,3\\}$ which are the task index for classification task and
$m$ is the number of classes for task $t$.

$\mathcal{L}_{reg}=\frac{1}{N_{tr}}\sum_{k=1}^{N_{tr}}(\hat{y}^{k}_{t}-y_{t}^{k})^{2}$
(11)

in this case $t\in\\{4,5\\}$ which are the indices for regression tasks.

##  V Architecture Detail and Model Training

The GNN backbone of the proposed model consists of $3$ GCN layers which are
$\mathcal{H}^{1},\mathcal{H}^{2},\mathcal{H}^{3}$ and $\mathcal{H}^{0}$ is the
input layer.

TABLE III: Hyperparameters and their values

Hyperparameters |  Value
---|---
Batch Size |  $32$
Epochs |  $500$
Hidden Layer Activation ($g_{\theta^{sh}}$) |  ReLU
Output Layer Activation ($h_{\theta^{cls}}$) |  Log Softmax
Output Layer Activation ($h_{\theta^{reg}}$) |  –
Optimizer |  AdamW
Initial Learning Rate ($\alpha$) |  $0.001$
Gradient Clipping Threshold ($\mathcal{C}$) |  $5$
Weight Decay ($\lambda$) |  $10^{-3}$
Dropout Rate ($D_{r}$) |  $0.2$
Train-Test Split |  $80-20$
$w_{t}\rightarrow(w_{1},w_{2},w_{3},w_{4},w_{5})$ |  ($0.01,0.8,0.9,0.1,0.04,0.05$)
Random Seed (for reproducibility) |  66

Algorithm 1 Training Algorithm of MTL-GNN

1$\mathcal{D}_{tr},\alpha,\lambda,\mathcal{C},w_{t}$ $\triangleright$ Training
set and hyperparameters

2$\theta$ $\triangleright$ Learned parameters of the model

3$X_{tr},Y_{tr}\leftarrow\mathcal{D}_{tr}$

4while $s<\text{number of epochs}$ do

5 for $k=1$ to $N_{tr}$ do

6 for $t=1$ to $T$ do

7 $\hat{y}^{k}_{t}\leftarrow h_{\theta^{t}}\circ g_{\theta^{sh}}(X^{k}_{tr})$

8 if $t=1$ and $\hat{y}^{k}_{1}==1$ then $\triangleright$ Non-fault event

9
$\mathcal{L}_{s}\leftarrow\sum_{k}\sum_{t\in\\{1,3\\}}w_{t}\mathcal{L}_{cls}$

10 else

11 $\mathcal{L}_{s}\leftarrow\sum_{k}\sum_{t}w_{t}\mathcal{L}_{t}$

12
$\nabla_{\theta}\mathcal{L}_{s}\leftarrow\begin{cases}\nabla_{\theta}\mathcal{L}_{s},&\text{if
}\|\nabla_{\theta}\mathcal{L}_{s}\|\leq\mathcal{C}\\\
\frac{\mathcal{C}}{\|\nabla_{\theta}\mathcal{L}_{s}\|}\cdot\nabla_{\theta}\mathcal{L}_{s},&\text{otherwise}\end{cases}$

13
$\theta_{s}\leftarrow\textsc{AdamW}(\theta_{s},\alpha,\lambda,\nabla_{\theta}\mathcal{L}_{s})$

14$\theta\leftarrow\theta_{s}$

The number of layers is restricted to $3$ to avoid over smoothing issue [44]
which is a common problem for deep GNNs. For normalization of the features
flowing through the layers, layer normalization is used [45]. In Fig. 3 the
entire forward propagation through the network is shown. The forward
propagation through GNN allows feature representation learning through message
passing. In addition, topological information is captured in the learned
representations. After that, learned embedding is extracted from all $128$
nodes which are then flattened and concatenated together to generate a feature
vector of dimension $128\times 32=4096$. This feature vector is passed through
the different heads for different prediction tasks. It is crucial to note that
for non-fault events we restrict all the losses except for $\mathcal{L}_{1}$
and $\mathcal{L}_{3}$, as all the other losses correspond to a fault event.
This allows gradient flow (during backpropagation) through the network only
for $\mathcal{L}_{1}$ and $\mathcal{L}_{3}$ loss for non-fault data samples.
Without this mechanism, the parameters associated with fault events
($\theta_{2},\theta_{4},\theta_{5}$) would get updated also.

The total number of parameters per layer and output shapes are mentioned in
Table II. The model was trained for $500$ epochs with a batch size of 32 with
AdamW [46] optimizer on an NVIDIA A100 80GB GPU. For stability of training
gradient clipping and L2 regularizer are used as well as dropout is used
($D_{r}=0.2$) to prevent overfitting. The hyperparameters are outlined in
Table III.

The training procedure is shown in Algorithm 1. For training the model the
training dataset $\mathcal{D}_{tr}$ and the hyperparameters : initial learning
rate ($\alpha$), regularizer hyperparameter/weight decay ($\lambda$), gradient
clipping threshold ($\mathcal{C}$), loss weighting factors ($w_{t}$) need to
be specified.

To determine the $w_{t}$ hyperparameters a trial run with the same weight was
conducted to get a gauge about which task is easier to learn for the model and
after that, these values are set taking into account the importance of the
task. We specify the epoch index as $s$ and the corresponding overall weighted
loss associated with that epoch is $\mathcal{L}_{s}$. After a forward pass
through the network, $\mathcal{L}_{s}$ is calculated followed by the gradient
of the loss with respect to model parameters which is
$\nabla_{\theta}\mathcal{L}_{s}$. Gradient clipping is applied if necessary
and finally, the AdamW optimizer updates the model parameters based on the
defined hyperparameters.

TABLE IV: Performance of proposed model on all $5$ tasks considering
measurement error, resistance change and dataset size

Criteria |  Fault Detection |  Fault Localization |  Fault Classification |  Fault Resistance Estimation |  Fault Current Estimation
---|---|---|---|---|---
| —————————————– | —————————————— |  ————————————— |  —————————– |  —————————–
| Balanced Accuracy | F1-Score | $\text{LAR}^{0}$ | $\text{LAR}^{1}$ | $\text{LAR}^{2}$ | F1-Score |  Accuracy | F1-Score |  MSE | MAPE |  MSE | MAPE
Measurement Error |  |  |  |  |  |  |  |  |  |  |  |
${}^{\dagger}\text{Low Noise}$: $n\sim\mathcal{N}(0,0.0001)$ | $1.0$ | $1.0$ | $0.982$ | $0.999$ | $0.999$ | $0.983$ |  $0.991$ |  $0.991$ |  $0.108$ |  $0.094$ |  $0.022$ |  $0.008$
${}^{\dagger}\text{Moderate Noise}$: $n\sim\mathcal{N}(0,0.001)$ | $1.0$ | $1.0$ | $0.980$ | $0.999$ | $0.999$ | $0.980$ |  $0.992$ |  $0.992$ |  $0.111$ |  $0.102$ |  $0.022$ |  $0.008$
${}^{\dagger}\text{High Noise}:$ $n\sim\mathcal{N}(0,0.01)$ | $1.0$ | $1.0$ | $0.715$ | $0.910$ | $0.963$ | $0.714$ |  $0.989$ |  $0.824$ |  $0.208$ |  $0.111$ |  $0.031$ |  $0.010$
$\text{High Noise}:$ $n\sim\mathcal{N}(0,0.01)$ | $1.0$ | $1.0$ | $0.954$ | $0.999$ | $0.999$ | $0.952$ |  $0.992$ |  $0.992$ |  $0.136$ |  $0.116$ |  $0.040$ |  $0.011$
Resistance Range Change |  |  |  |  |  |  |  |  |  |  |  |
$0.05\Omega-20\Omega$ | $1.0$ | $1.0$ | $0.984$ | $0.999$ | $0.999$ | $0.984$ |  $0.993$ |  $0.993$ |  $0.133$ |  $0.112$ |  $0.026$ |  $0.008$
$20\Omega-100\Omega$ | $1.0$ | $1.0$ | $0.983$ | $0.999$ | $0.999$ | $0.982$ |  $0.991$ |  $0.991$ |  $1.468$ |  $0.020$ |  $0.008$ |  $0.006$
$100\Omega-500\Omega$ | $1.0$ | $1.0$ | $0.917$ | $0.990$ | $0.999$ | $0.914$ |  $0.949$ |  $0.948$ |  $68.92$ |  $0.021$ |  $0.028$ |  $0.009$
% Samples /Fault Type |  |  |  |  |  |  |  |  |  |  |  |
$15\%$ | $1.0$ | $1.0$ | $0.934$ | $0.997$ | $0.999$ | $0.931$ |  $0.991$ |  $0.991$ |  $0.206$ |  $0.302$ |  $0.040$ |  $0.014$
$25\%$ | $1.0$ | $1.0$ | $0.957$ | $0.995$ | $0.999$ | $0.957$ |  $0.992$ |  $0.992$ |  $0.146$ |  $0.174$ |  $0.030$ |  $0.011$
$50\%$ | $1.0$ | $1.0$ | $0.975$ | $0.999$ | $0.999$ | $0.975$ |  $0.991$ |  $0.991$ |  $0.211$ |  $0.181$ |  $0.040$ |  $0.011$
†Out-of-Distribution (OOD) data |

##  VI Numerical Tests

In this section, we first introduce the evaluation metrics used to assess the
model performance, followed by the regression and classification performance
of the model. Then we evaluate the model performance on sparse measurements
where the sparse node-set is strategically chosen based on an explainability
algorithm.

###  VI-A Metrics Used for Evaluation of the Model Performance

For fault detection which is a binary classification task, we report balanced
accuracy and f1-score. The reason for reporting balanced accuracy as opposed
to accuracy is the class imbalance for fault detection. For fault
localization, we report the location accuracy rate (LAR) and f1-score. The
$\text{LAR}^{h}$ ($h$ indicate the number of hops) is a common metric to
evaluate the performance for fault localization. It quantifies the percentage
of correctly identified fault locations within a certain $h$-hop distance from
the actual fault location. $\text{LAR}^{0}$ capture the accuracy for exact
fault location. In contrast, $\text{LAR}^{1}$ measures the performance of the
identified fault locations within $1$-hop distance from the actual fault
location. $\text{LAR}^{2}$ computes the same thing but for $2$-hop distance.
The reason for computing this metric is to evaluate the model’s ability to
provide an estimation of the fault’s approximate position, even if the model
cannot precisely identify the exact fault location.

For fault type classification we report accuracy and f1-score. In addition, we
also use the confusion matrix which provides insight into the model’s
performance specific to a fault type.

For the regression tasks, fault resistance and fault classification estimation
we report MSE and mean absolute percentage error (MAPE) for the test data set.
These two metrics are given by the following equation.

$\mathrm{MSE}_{test}=\frac{1}{N_{test}}\sum_{k=1}^{N_{test}}(\hat{y}^{k}_{t}-y_{t}^{k})^{2}$
(12)

$\mathrm{MAPE}_{test}=\frac{1}{N_{test}}\sum_{k=1}^{N_{test}}\left|\frac{\hat{y}^{k}_{t}-{y}^{k}_{t}}{{y}^{k}_{t}}\right|\times
100$ (13)

where $t=4$ indicates the metrics for fault resistance and $t=5$ indicates the
metrics for fault current. The reason for reporting MAPE in addition to MSE is
the sensitivity to scale for MSE. As we report performance for different
ranges of fault current and resistance, this varying range needs to be taken
into account. All results are summarized in Table IV. For simulating possible
measurement errors, we introduce noise $n$ sampled from a zero mean gaussian
distribution $\mathcal{N}(0,\sigma_{noise})$ with a variance value set to
$\sigma_{noise}$. The variance $\sigma_{noise}$ is respectively set to
$0.0001$, $0.001$ and $0.01$ for different levels of noise. We also consider
the model performance under varying ranges of fault resistance values which
are $0.05\Omega-20\Omega$, $20\Omega-100\Omega$ and $100\Omega-500\Omega$.
Furthermore, we consider the fact there might be a lack of historical data for
fault events. To imitate this, we decrease the size of the dataset and
evaluate the model performance on the decreased dataset. The smallest size we
consider is $15\%$ of the original which results in only $45$ samples per
node.

###  VI-B Performance of the Model for Classification Tasks

From Table IV it is apparent that the model can easily distinguish between a
load change event and a fault event which is intuitive given that there are
only two classes.

Figure 4: Confusion matrix for fault type classification task. The first three
classes are asymmetric faults, the next two classes are symmetric faults and
the final class corresponds to non-fault events.

In addition, the voltage fluctuation in the case of a fault event differs
significantly from that of a load change event. In the case of fault
localization, despite the different variations, the model is robust enough to
maintain relatively high $\text{LAR}^{1}$ and $\text{LAR}^{2}$. When we
consider measurement error, first we evaluate the performance with an out-of-
distribution (OOD) test set meaning the training dataset didn’t contain noisy
samples. For low and moderate noise, even with OOD samples the fault
localization performance holds. For high levels of noise, the performance goes
down but if the noisy samples are included in the training set the location
accuracy improves significantly. Also, the model manages to sustain relatively
high accuracy despite a broad range of resistance values. Similar conclusions
can be made for varying dataset sizes. One important thing to note here, even
when the model is not able to localize the exact fault point, it can
approximate the location with high accuracy.

For fault-type classification, it is important to outline per-class
performance as the probability of all fault types is not the same and varies
according to fault type; LG (70% - 80%), LLG (17% - 10%), LL (10% - 8%), LLL,
LLLG (3% - 2%) [3]. This means it is more important the model is able to
classify the asymmetrical faults compared to the symmetrical faults.

Figure 5: t-SNE visualization of last layer features of fault type
classification head $h_{\theta}^{3}$ for the test set $\mathcal{D}_{test}$.
Except for a few samples in the LLL fault type and LLLG fault type, most
features are clearly separable based on fault type.

The confusion matrix shown in fig. 4 summarizes the performance for each fault
type. The misclassified fault types belong to the LLL and LLLG fault class
which is further corroborated by the t-SNE visualization of the last layer
features of $h_{\theta}^{3}$ shown in Fig. 5. Samples from other fault types
are clearly separable even when they are projected into a 2-dimensional
feature space.

###  VI-C Performance of the Model for Regression Tasks

For both regression tasks, the performance of the model remains consistently
high across all the variations considered. Even though for resistance level
$100\Omega-500\Omega$ because of the scale sensitivity, MSE is high, MAPE
remains low similar to other resistance levels.

Fig. 6 shows the distribution plot for the ground truth and predicted value
for fault resistance on the test set. For the most part, the predicted
distribution closely resembles the ground truth distribution. Similarly, Fig.
7 shows the ground truth distribution and predicted distribution for the fault
current. One important thing to mention here, this plot is for the transformed
fault current label mentioned in (7) rather than the actual fault current.

Figure 6: Ground truth ($y_{4}$) and predicted ($\hat{y}_{4}$) fault
resistance distribution on the test set ($\mathcal{D}_{test}$). The grey
section is the correctly predicted part of the distribution. Figure 7: Ground
truth ($y_{5}$) and predicted ($\hat{y}_{5}$) redefined fault current
distribution on the test set ($\mathcal{D}_{test}$). The grey section is the
correctly predicted part of the distribution.

To get the actual fault current prediction and ground truth labels, the
inverse operation of (7) can be performed which is given by the following
equation:

$y^{k}_{5}:={\sum^{N}_{k=1}y^{k}_{5}}\times\exp(-y^{k}_{5})$ (14)

###  VI-D Using Explainability to Identify Key Nodes

Out of all the tasks, fault localization is the most important given that the
model performs really well for fault detection. Considering practical
implications, for the remaining three tasks, it is acceptable if the model can
approximate the prediction. So far in analysis, we assumed complete
observability which implies all node voltage phasors can be measured. As the
scale of the distribution system grows it becomes more and more harder to
maintain complete observability considering cost and system complexity.
Therefore, we propose a novel approach to locate key nodes using an
explainability algorithm GNNExplainer proposed by [47]. GNNExplainer uses an
optimization that maximizes the mutual information between a GNN’s prediction
and distribution of possible subgraph structures to identify important
subgraphs. The most notable characteristic of this algorithm is it does not
require ground truth labels. Using this algorithm we generate two sparse node
sets $\mathcal{V}_{50\%}$, $\mathcal{V}_{75\%}$ where the subscript denotes
the percentage of nodes out of $128$ which has data available. The feature
vector of the rest of the nodes is set to $0$.

Algorithm 2 Sparse Node-Set Generation Algorithm

1$X_{test},epoch_{GE},h_{\theta^{2}}\circ g_{\theta^{sh}}(trained),w_{th}$

2$\mathcal{V}_{x\%}$

3for $k=1$ to $N_{test}$ do

4
$E^{k}_{\mathcal{I}}\leftarrow\textsc{GNNExplainer}(X^{k}_{test},h_{\theta^{2}}\circ
g_{\theta^{sh}},epoch_{GE})$

5 for $p=1$ to $\mid\mathcal{E}\mid$ do

6 if $E^{k}_{\mathcal{I}}(p)>w_{th}$ then $\triangleright$ Thresholding with
$w_{th}$

7 $E_{w_{th}}^{k}(p)\leftarrow$ 1

8 else

9 $E_{w_{th}}^{k}(p)\leftarrow$ 0

10
$\\{\mathcal{V}_{sparse}\\}^{k}\leftarrow\textsc{GetConnectedNodePairs}(E^{k}_{w_{th}})$

11 $\mathcal{V}_{x\%}\leftarrow\bigcup_{k}\\{\mathcal{V}_{sparse}\\}^{k}$

To generate these sets we pass each test data sample $X_{test}^{k}$ and the
trained model with localization head $h_{\theta^{2}}\circ g_{\theta^{sh}}$ to
the GNNExplainer algorithm which generates an edge importance vector given by
$E_{\mathcal{I}}\in\mathbb{R}^{\mid\mathcal{E}\mid}$, indexed by $p$. For each
edge in $E_{c}$ a corresponding weight is generated which signifies the
importance of that edge. A weight value closer to $1$ means a more important
edge and a value closer to $0$ means a less important edge. We threshold the
values in $E_{\mathcal{I}}$ with a threshold $w_{th}$. After thresholding, the
transformed edge importance vector is expressed by
$E_{w_{th}}\in\mathbb{R}^{\mid\mathcal{E}\mid}$. Edges that have an importance
score of more than $w_{th}$ are kept and the rest are disregarded. The nodes
connected to the edges in $E_{w_{th}}$ after thresholding are regarded as the
important nodes for the $k^{th}$ data point. In this way, for each data point,
we get a sparse node set given by $\\{\mathcal{V}_{sparse}\\}^{k}$. Then the
union of these sparse important node sets generates the final sparse node set
which is given by the following equation:-

$\mathcal{V}_{x\%}:=\bigcup_{k=1}^{N}\\{\mathcal{V}_{sparse}\\}^{k}$ (15)

where the $x\%$ value depends on the threshold value. We set the value of
threshold $w_{th}$ to respectively $0.44$ and $0.487$ to generate
$\mathcal{V}_{75\%}$ and $\mathcal{V}_{50\%}$.

The entire process is described in the Algorithm 2. In addition to other
parameters mentioned above, an epoch number needs to be specified for the
internal optimization of the GNNExplainer.

TABLE V:  $\text{LAR}^{h}$ and F1-Score with Sparse Node-Set

Sparse Sets | $\text{LAR}^{0}$ | $\text{LAR}^{1}$ | $\text{LAR}^{2}$ | F1-Score
---|---|---|---|---
$\mathcal{V}_{75\%}$ | 0.975 | 0.999 | 0.999 | 0.975
$\mathcal{V}_{50\%}$ | 0.962 | 0.998 | 0.999 | 0.962
$\mathcal{V}^{random_{avg}}_{50\%}$ | 0.916 | 0.993 | 0.999 | 0.913
$\mathcal{V}^{random_{min}}_{50\%}$ | 0.886 | 0.989 | 0.998 | 0.881

To validate this approach we also randomly sample $50\%$ nodes several times
and train the model to generate fault localization metrics on the test set.
$\mathcal{V}_{50\%}^{random_{avg}}$ represent the average of these metrics and
$\mathcal{V}_{50\%}^{random_{min}}$ represent the minimum of these metrics
from the samples generated. The results of this analysis are summarized in
Table V. It is apparent that the model is robust enough to maintain the
$\text{LAR}^{1}$ and $\text{LAR}^{2}$ irrespective of which sparse node-set is
used. But for $\text{LAR}^{0}$ and F1-Score, the sparse set generated with
GNNExplainer is comparatively better. $\mathcal{V}_{50\%}$ has a $4.78\%$
increase in $\text{LAR}^{0}$ compared to $\mathcal{V}^{random_{avg}}_{50\%}$
and a $7.9\%$ increase compared to $\mathcal{V}^{random_{min}}_{50\%}$ which
means the sparse node-set generation algorithm was able to identify the
important node set for fault localization.

##  VII Conclusion

In this paper, we proposed an MTL-GNN capable of performing $5$ different
tasks simultaneously even with a sparse node-set. This sparse node-set is
generated with a novel algorithm and to the best of our knowledge this is the
first work that uses a GNN explainability algorithm for an informed node
selection.

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