# Evaluating Distribution System Reliability with Hyperstructures Graph
Convolutional Nets

Yuzhou Chen1,2, Tian Jiang3, Miguel Heleno2, Alexandre Moreira2, Yulia R.
Gel3,2
1Department of Computer and Information Sciences, Temple University,
Philadelphia, USA
2Energy Storage and Distributed Resources Division, Lawrence Berkeley National
Laboratory, Berkeley, USA
3Department of Mathematical Sciences, The University of Texas at Dallas,
Dallas, USA

###### Abstract

Nowadays, it is broadly recognized in the power system community that to meet
the ever expanding energy sector’s needs, it is no longer possible to rely
solely on physics-based models and that reliable, timely and sustainable
operation of energy systems is impossible without systematic integration of
artificial intelligence (AI) tools. Nevertheless, the adoption of AI in power
systems is still limited, while integration of AI particularly into
distribution grid investment planning is still an uncharted territory. We make
the first step forward to bridge this gap by showing how graph convolutional
networks coupled with the hyperstructures representation learning framework
can be employed for accurate, reliable, and computationally efficient
distribution grid planning with resilience objectives. We further propose a
Hyperstructures Graph Convolutional Neural Networks (Hyper-GCNNs) to capture
hidden higher order representations of distribution networks with attention
mechanism. Our numerical experiments show that the proposed Hyper-GCNNs
approach yields substantial gains in computational efficiency compared to the
prevailing methodology in distribution grid planning and also noticeably
outperforms seven state-of-the-art models from deep learning (DL) community.

###### Index Terms:

distribution networks, hyperstructures, representation learning, graph neural
networks, resilience analysis

##  I Introduction

The most recent cases of cyber-physical failures and natural disasters, such
as 2021 Texas power crisis, have caused billions of dollars of irreparable
damage in grid assets and long-term interruption of service. As a result,
there is raising governmental awareness regarding resilience of energy
infrastructures [1] to face such High Impact Low Probability (HILP) events.
From a power grid perspective, these events are different from normal routine
failures and are often focused on the tails of outage distributions,
describing resilience as a risk metric, such as Value at Risk (VaR) or
Conditional Value at Risk (CVaR), associated with the loss of load [2].

Stochastic optimization is arguably the most popular approach to solve power
distribution grid expansion and planning problems that explicitly models
extreme events [3, 4]. Although these stochastic optimization approaches allow
for assessing resilience while determining investment in expansion plans, they
entail a significant computational burden. Particularly in the case of
distribution networks, composed by a large number of nodes and multiple small
candidate assets, the number of variables and constraints of stochastic
programs dramatically increase, creating a barrier for the application of
accurate resilience planning to real systems. As a result, it is of critical
importance to develop new techniques enabling us to produce accurate and
feasible expansion plans in the context of real-scale utility networks. This
opens a new opportunity for AI and, especially, automatic classification
algorithms. Indeed, such AI approaches might be able to significantly improve
the computational time of the distribution grid planning process, just by
deriving relevant topological information from pre-computed plans, i.e.,
without requiring a detailed modeling of the intra-hour stochastic events.
These classification methods, particularly, based on the graph convolutional
network (GCN) architectures, already proved to be efficient in bioinformatics,
social sciences, and transportation systems [5, 6, 7]. Furthermore, most
recently GCNs have emerged into the analysis of power transmission systems [8,
9, 10]. However, the capability of these classification approaches to
reproduce distribution grid resilience metrics (such as the CVaR of the loss
of load) is still an open question and to the best of our knowledge, GCNs have
never been used for assessment of distribution grids and the associated
distribution expansion plans.

We propose a novel resilience classification approach for accurate, reliable,
and computationally efficient distribution grid planning called
Hyperstructures Graph Convolutional Neural Networks (Hyper-GCNNs). Hyper-GCNNs
is based on the convolutional architecture with a hyperstructure learning
mechanism and innovative graph-theoretic summaries of distribution grid
vulnerability. We then evaluate how well Hyper-GCNNs captures the overall
resilience of the expansion plans and mirror conclusions of the substantially
more expensive CVaR metric. (Our expansion plans here are constructed using
stochastic simulation with the optimal distribution network operation as
adopted in operational planning.) Hyper-GCNNs ranks the potential power
expansion plans in terms of their resiliency while using only the higher-order
network information and bypassing costly optimization. Such classification
results can be then employed by utilities and regulators to enhance their
understanding how each expansion plan changes the operational risk of the
overall system, to pre-select the most promising plan candidates, and more
generally, to develop pro-active risk mitigation policies.

Beyond Distribution Grids While here we primarily focus on data mining for
distribution grid investment planning, the proposed Hyper-GCNNs methodology is
broadly applicable to many other diverse networks which exhibit multimodality
of nodes and intrinsic heterogeneity in higher-order interactions, such as
crypto-exchanges and individual addresses or transportation hubs, urban and
rural stations.

The key novelty of this paper are summarized as follows:

  * •

This is the first attempt to introduce the concepts of GDL, hypergraph
learning, and data mining to the yet uncharted territory of distribution grid
investment planning.

  * •

We introduce a new graph-theoretic summary, uniqueness scores (U-scores) which
allows for exploiting the efficiency of information flow among different types
of nodes such as loads and substations in distribution grids or hubs and rural
areas in transportation systems.

  * •

We develop a new Hyperstructures Graph Convolutional Neural Networks (Hyper-
GCNNs) which enable for systematic characterization of the latent higher order
network representations with an attention mechanism. Our experiments on the
validation of Hyper-GCNNs in conjunction with distribution grid planning
indicates that Hyper-GCNNs delivers reliable classification of power expansion
plans while exhibiting highly competitive computational efficiency.

##  II Related Work

Artificial Intelligence Tools for Power System Analysis AI tools and,
particularly, various machine learning (ML) and deep learning (DL) approaches
start to propagate into a broad range of power system studies [11, 12, 13].
Some of the most prominent recent examples are L2RPN: Learning to Run a Power
Network in a Sustainable World [14] under NeurIPS2020 and The Learning to run
a power network challenge under at NeurIPS2018 [15], both largely focusing on
the utility of reinforcement learning for smart-grid operations. Other recent
examples are Bayes networks and meta-action for cascading failure propagation
[16, 17], node classification in transmission power grids using graph neural
networks (GNNs) [18], semi-supervised learning for load monitoring and
scheduling [19], fault detection using Support Vector Machine (SVM) and other
ML models [20, 21], as well as topology-based systems as a part of neural
network architectures for impact metrics classification and prediction in
power systems [22]. However, comparing to other knowledge domains, the AI
methodology still remains severely underused in the power system analysis [23,
13, 24] and most existing DL approaches primarily focus on power transmission
system [25, 26, 27, 8, 9, 10]. To the best of our knowledge, AI and DL, in
particular, have neither been used before to assess resilience of potential
distribution grid investment plans, nor with distribution grid planning.

Figure 1: (a) General distribution system, containing the substation (main
power supply), the network lines and the consumption nodes, generally
represented by multiple buildings and (b) base resilience evaluation
methodology, in which the expansion and planning solution is tested against
multiple scenarios of network outages caused by either routine failures or by
extreme weather events. Each outage scenario leads to a loss of load which
entails significant costs to the system. The conditional value at risk (CVaR)
of this cost is a widely-used resilience metric in the planning-stage.

Methods for Distribution Grid Planning Current approaches in distribution grid
planning heavily rely on mathematical programming frameworks. For instance,
[28] formulates a distribution grid planning as a mixed-integer optimization
problem that seeks to identify the optimal portfolio of investments in lines,
transformers and substations in order to minimize costs associated with
expected energy not supplied. The approach developed by [29] presents an
optimization model to determine generation and network expansion across a
given period of time while considering reliability. In turn, [3] proposes a
linear programming based approach for determining investments in storage
devices in order to improve the resilience of the distribution system against
earthquakes. Recently, [30] develops a mixed-integer stochastic model to
optimize investments in line hardening, distributed generation and line
switches so as to decrease costs related to expected loss of load and
operation of distribution grids. The majority of optimization problems
associated with distribution grid planning belong to the mixed-integer linear
programming (MILP) class, which is well-known as an NP-Hard class. There are
many ways to facilitate (e.g., decomposition techniques) the solution of MILP
problems, especially when there are too many variables and constraints
involved as in the case of planning problems. However, all of these ways (in
classic optimization) involve the utilization of the Branch and Bound
algorithm, whose computational complexity is exponential. Hence, depending on
the size of the distribution system and the number of candidate assets, the
expansion problem may take a prohibitive time to be solved. Adequate
formulation of the distribution planning problem depends on the definition of
a metric to be improved. Traditional distribution grid planning focuses on
reliability indices, which are created to define routine failures of the power
grid [31], and later included into existing optimization expansion and
planning methods for the distribution network [29]. Here, instead of
reliability, we are interested in assessing the resilience of the system by
estimating the CVaR of the loss of load given a defined expansion plan. To do
so, unlike the conventional manner, we propose to replace conventional
optimization by a DL-based method so as to compute our metric in a
computational efficient way.

Hypergraph Neural Networks Hypergraph deep learning is a newly emerging
direction in GNNs which allows us to address various complex higher order
interdependencies (i.e., beyond node pairwise interactions). For instance,
HGNN [32] is arguably the first attempt to generalize GNNs to hypergraph
learning. In turn, [33] proposes an alternative weighting mechanism to derive
hypergraph Laplacian via mediators. In spirit of GraphSAGE, HyperSAGE of [34]
uses a spatial approach to study message propagation in and between
hyperedges. Most recently, [35] compares existing graph and hypergraph neural
networks, and formulates a unified framework for hypergraph learning. Finally,
[36] investigates tensor-based propagation on $d$-uniform hypergraphs. To the
best of our knowledge, hypergraph learning has never been applied to power
systems and, distribution grids, in particular.

##  III Methodology

The broader objective of this paper is to investigate utility of DL-based
classification methods to evaluate power distribution grid expansion plans in
terms of their resilience performance. In particular, our analysis aims to
assess the ability to approximate explicit but computationally expensive risk-
based resilience metrics, such as the CVaR of loss of load, through the
topological descriptors of the distribution networks. That is, our ultimate
idea is to gather different distribution expansion plans and to compare their
resilience evaluation, based on the prevailing CVaR calculations, with the new
topology-based metrics. In this section, we present (i) an overview of the
traditional risk-based resilience evaluation methods used for comparison in
our analysis, (ii) the algorithm to obtain the new topological signature of
distribution grid networks, i.e., uniqueness scores, and (iii) our proposed
Hyper-GCNNs designed for graph learning.

For the purpose of modeling distribution grids (as in the stylized
representation of Fig. 1 (a)), we consider a graph
$\mathcal{G}=(\mathcal{V},\mathcal{E},\omega)$ with a node set $\mathcal{V}$,
an edge set $\mathcal{E}$, and (edge)-weighted function
$\omega:\mathcal{E}\mapsto\mathbb{R}_{\geq 0}$. Number of nodes
$|\mathcal{V}|=N$ and distance among two nodes in $\mathcal{G}$ is denoted by
$d_{uv}$, with $d_{uv}\equiv 0$ if there exists no path connecting nodes $u$
and $v$. Let $A\in\mathbb{R}^{N\times N}$ denote a symmetric adjacency matrix
with $N$ nodes, and $D\in\mathbb{R}^{N\times N}$ be a diagonal matrix with
$D_{uu}=\sum_{v}A_{uv}$. Here, nodes represent buses and substations; edges
reflect distribution lines, and the (edge)-weighted function is used to model
power flow information about the system. In contrast to currently existing
approaches based on graph-theoretical analysis of power grid networks that do
not distinguish among types of nodes, we make a clear distinction between
substations (nodes of the distribution system that serve as a connection to
the main transmission grid) and buses (nodes of the distribution system that
may have a power load).

Mathematical Formulation of the Problem Our targets are to (i) learn a mapping
function $\mathfrak{F}_{c}:\mathcal{G}\rightarrow\mathcal{Y}_{c}$ that
classify distribution grids into different classes (where $\mathcal{Y}_{c}$ is
the label of distribution network $\mathcal{G}$) and (ii) find a function
$\mathcal{F}_{r}$ to predict CVaR value of distribution grid based on graph
structural information and node features, i.e.,
$\mathfrak{F}_{r}:\mathcal{G}\rightarrow\mathcal{Y}_{r}$ (where
$\mathcal{Y}_{r}$ is CVaR value of distribution network $\mathcal{G}$).

###  III-A Base Resilience Evaluation Methodology

Reliability and resilience evaluation methods aim at assessing the ability of
the power grid to supply the load, considering multiple scenarios of grid
outages caused by routine failures of equipment or by High Impact Low
Probability (HILP) events, such as storms, earthquakes, wildfires, and cyber-
physical attacks. Here, there is a distinction between reliability and
resilience metrics: the reliability performance is associated with the routine
outages and, therefore, with the expected values of loss of load as well as
frequency and duration of the interruptions; in contrast, resilience is more
associated with the ability of the grid to withstand and mitigate long-
duration outages caused by HILP events, which means that the tail aspects
(risk) of the loss of load is the key for the resilience performance. Hence,
different power grid expansion plans with specific circuits connections,
cables and lines parameters, substation locations, etc. – lead to different
risks of loss of load, which means these resilience evaluations are essential
to support electric utilities planning decisions.

Considering an evaluation environment with a pre-defined set of scenarios of
routine failures and HILP events, the objective of the resilience evaluation
is to quantify the distribution of the loss of load on the different nodes of
the grid together with corresponding CVaR of the distribution. This is
typically achieved via Sequential Monte Carlo Simulation (SMCS) methods [37],
which perform simulations of loss of load over annual time horizons. More
specifically, to assess the annual CVaR of loss of load according to the
orange box in Fig. 1 (b), we simulate operation of the distribution system for
several scenarios (e.g., 2000 scenarios). Here each scenario corresponds to
365 days (one year) and the system is operated for each hour of each day. Due
to its respective failures, each annual scenario is associated with the annual
loss of load. Given the amounts of loss of load for all scenarios, we then
compute the annual CVaR of loss of load. Such a procedure can take up to 24
hours for a 54-bus system (see the case study in Section IV). Hence, one of
the primary motivations of this paper is to replace the costly simulations
related to the orange box in Fig. 1 (b) by a computationally efficient ML/DL
tool that can quickly classify distribution grids according to their
corresponding ranges of annual CVaR of loss of load.

###  III-B New Uniqueness Scores

Conventional resilience metrics for power grid networks based on graph-
theoretic foundations are average path length (APL), average betweenness
centrality (Avg. BC), diameter ($\mathfrak{D}$), small-worldness and giant
component properties [38]. That is, network summaries which first, do not
differentiate among types of nodes in the power grids (i.e., substations vs.
buses) and, second, largely address only global network characteristics and
disregard important information on local graph topology.

Our goal is to define a new node feature for load based on the simple paths
between load and substation. Inspired by the notion of graph APL, we propose a
new network summary, namely, uniqueness scores (U-scores), in order to exploit
the efficiency of information (i.e., flow) transport between different loads
and substations. We now summarize our U-scores algorithm in a framework for
different potential expansion plans of a distribution system. The U-scores
framework has three main components: (i) paths extraction, (ii) scores
calculation, and (iii) uniqueness scores finalization. We discuss each
component in details as follows.

Given the original graph structure $\mathcal{G}\textsubscript{Bus}$ and the
expansion plan $\mathcal{G}\textsubscript{Exp}$,
$\mathcal{G}\textsubscript{Comb}$ represents the combination of
$\mathcal{G}\textsubscript{Bus}$ and $\mathcal{G}\textsubscript{Exp}$. Let
$l_{j}$ and $s_{i}$ denote the load node $j$ and substation node $i$ in
$\mathcal{G}\textsubscript{Comb}$, respectively. We then define the new
U-scores through the steps below:

  1. 1.

Paths. We compute the list of simple weighted paths from $l_{j}$ and $s_{i}$,
i.e., $\textnormal{simplePaths}_{l_{j},s_{i}}$; then we sort
$\textnormal{simplePaths}_{l_{j},s_{i}}$ in increasing order by summarizing
the edge weights in path, and the sorted
$\textnormal{simplePaths}_{l_{j},s_{i}}$ is defined as
$\textnormal{simplePaths}^{\text{sorted}}_{l_{j},s_{i}}$.

  2. 2.

Scores. Suppose there are $\mathcal{K}$ (i.e.,
$\textnormal{len}(\textnormal{simplePaths}^{\text{sorted}}_{l_{j},s_{i}})$)
paths in $\textnormal{simplePaths}^{\text{sorted}}_{l_{j},s_{i}}$, we first
assign the uniqueness score of the shortest weighted path
$S^{(1)}_{l_{j},s_{i}}$ as 1, use a list $U_{l_{j},s_{i}}$ to store the
corresponding simple path by union operation, and calculate its length. For
clarity, we use $P_{l_{j},s_{i}}$ to record the total length of simple paths
in $\textnormal{simplePaths}^{\text{sorted}}_{l_{j},s_{i}}$. For $k$-th path
$\textnormal{simplePaths}^{\text{sorted}}_{l_{j},s_{i}}[k]$ (where $2\leq
k\leq\mathcal{K}$), the calculation of uniqueness score involves two extra
sub-steps: (i) generate the intersection $I_{l_{j},s_{i}}$ between
$\textnormal{simplePaths}^{\text{sorted}}_{l_{j},s_{i}}[k]$ and stored
$U_{l_{j},s_{i}}$, indicating that the repetition rate of $k$-th path; (ii)
calculate the normalized uniqueness score based on the ratio of the length of
the intersection to the length of $k$-th path.

  3. 3.

Uniqueness score. We finalize the uniqueness score of load $l_{j}$ on
substation $s_{i}$ as the ratio between the summation of scores and total
length of paths in $\textnormal{simplePaths}^{\text{sorted}}_{l_{j},s_{i}}$.

In our U-scores algorithm, we first encode path information of the
distribution system using scores for the load in the system, since traditional
topological metrics like [39] usually assess robustness of power system
network in graph-level. U-scores reduce computational costs of paths
extraction and only aggregate information between load and substation. The
basic idea is that, graph-level resiliency metrics are actually not useful to
capture the relationship between load and substation, since they only provide
very global and limited structural information.

What new do U-scores bring? In addition to accurately evaluating robustness of
distribution systems, the new U-scores metric sheds a valuable insight into
how each load affects the robustness of distribution system complex network.
Such quantitative evaluation yields multi-fold benefits. First, it allows
power system operators to enhance their understanding of the load dynamics
and, hence, to develop more efficient and pro-active risk mitigation
strategies. Second, armed with the U-scores metric, electrical engineers can
evaluate potential expansion plans of the existing distribution systems in a
more accurate and reliable manner via a comparison on U-scores summation
$\sum\sum{\boldsymbol{S}}$ (i.e., higher values indicate more resilient
distribution system).

Note that the currently existing graph-theoretical summaries do not allow for
such a comparison of expansion plans. For instance, Table I shows the
conventional graph-theoretic summaries of power grid vulnerability (i.e., APL,
$\mathfrak{D}$, and Avg. BC), the proposed new U-scores, and CVaR. Note that
lower APL, $\mathfrak{D}$, Avg. BC, CVaR [38] and higher U-scores are
preferred and considered as indicating higher system resilience. Here, APL,
$\mathfrak{D}$, Avg. BC and U-scores are based only on the topology of the
considered distribution networks, while CVaR is based on a stochastic
optimization approach and explicitly accounts for both network topology and
power flow information. As such, CVaR conclusions on resiliency of each plan
are viewed by power system analysts as the most reliable results, that is,
“ground truth” in our context. Remarkably, as Table I suggests conclusions of
APL, $\mathfrak{D}$ and Avg. BC are in contradiction with CVaR, while the new
U-scores, which uses substantially less information than CVaR, and is
noticeably more computationally efficient and has highest correlation with
CVaR. More specifically, as shown in Table I, we compute correlation
coefficients ($r$) between graph-based metrics and the inverse of CVaR values.
We find that (i) Avg. Degree vs. 1/CVaR: $r=0.869$, (ii) Avg. BC vs. 1/CVaR:
$r=0.282$, (iii) APL vs. 1/CVaR: $r=-0.133$, (iv) $\mathfrak{D}$ vs. 1/CVaR:
$r=-0.205$, and (v) U-scores vs. 1/CVaR: $r=0.893$. This phenomenon can be
explained by the fact that in contrast to conventional APL, $\mathfrak{D}$ and
Avg. BC metrics for power system resilience, U-scores explicitly
differentiates among substations and load buses and accounts for the local
topology of the distribution power grid.

TABLE I: Reliability metrics for systems # 1 and # 65. Metric | Expanded system # 1 | Expanded system # 65
---|---|---
APL | 5.79 | 7.00
$\mathfrak{D}$ | 19.96 | 24.41
Avg. BC | 45.37 | 103.70
CVaR | 28679.76 | 16736.37
U-scores | 25.32 | 43.01

###  III-C Hyperstructures

A number of recent results indicate that higher order substructures may play
important roles in resiliency and functionality of cyber-physical systems and,
in particular, power grids [40, 41, 42]. This phenomenon can be explained by
the intrinsic higher order relations in power distribution networks, where
buses interact with each another, and the power flow through branches may be
viewed as a result of such interactions. One of the emerging approaches to
address such higher order properties is via hypergraph representation
learning. Inspired by these recent results on hypergraph modeling, we propose
to represent high-order interactions on distribution grids via
hyperstructures. That is, we use similarity information about groups of nodes
(or edges) instead of only graph connectivity information such as network
motifs. We introduce the details of proposed hyperstructures for distribution
grid representation learning as follows. In the graph
$\mathcal{G}=(\mathcal{V},\mathcal{E},\omega)$, each edge is incident to two
nodes. Hypergraph can be viewed as a generalized graph that describes high-
order interactions in the data. It introduces a structure – hyperedge, which
can contain an arbitrary number of nodes. Let
$\mathcal{H}_{\mathcal{G}}=(\mathcal{V},\mathcal{H}_{\mathcal{E}},\omega_{\mathcal{E}})$
be a hypergraph, where $\omega_{\mathcal{E}}$ represents the hyperedge weight.
Then $\mathcal{G}$ is a special case when each hyperedge consists of exactly
two nodes, a 2-uniform hypergraph (hyperedge degree
$\delta(h_{\mathcal{E}})=|h_{\mathcal{E}}|=2$ for any
$h_{\mathcal{E}}\in\mathcal{H}_{\mathcal{E}}$). Similarly, we define incidence
matrix $Q_{\mathcal{E}}$ for connectivity between node and hyperedge, i.e.,
$q_{\mathcal{E}}(v,h_{\mathcal{E}})=1$ if node $v$ is incident to hyperedge
$h_{\mathcal{E}}$ and 0 otherwise. Then the normalized hypergraph Laplacian is
$L_{\mathcal{E}}=I-\frac{1}{2}D^{-1/2}Q_{\mathcal{E}}W_{\mathcal{E}}D_{\mathcal{E}}^{-1}Q_{\mathcal{E}}^{\top}D^{-1/2}$,
where $D$, $D_{\mathcal{E}}$, and $W_{\mathcal{E}}$ are diagonal matrices of
node degree, hyperedge degree and hyperedge weight, respectively.

Inspired by hypergraph, we further consider a generalization of line graph
(the dual graph represents the adjacencies between edges of a graph). We
denote hypernode as a set of arbitrary number of edges of graph
$\mathcal{G}=(\mathcal{V},\mathcal{E},\omega)$,
$h_{\mathcal{V}}\subset\mathcal{E}$ and its hypernode degree is
$\delta_{\mathcal{V}}(h_{\mathcal{V}})=|h_{\mathcal{V}}|$. Then the augmented
graph with hyperstructure becomes
$\mathcal{H}_{\mathcal{G}}=(\mathcal{V},\mathcal{E},\mathcal{H}_{\mathcal{V}},\mathcal{H}_{\mathcal{E}},\omega,\omega_{\mathcal{V}},\omega_{\mathcal{E}})$.
The connectivity between edge and hyperedge can be analogously expressed by
the incidence matrix $Q_{\mathcal{V}}$. Note that node degree matrix $D$ is
also updated by corresponding edge degree in the hyper-line-graph,
$d(e)=\sum_{h_{\mathcal{V}}}\omega_{\mathcal{V}}(h_{\mathcal{V}})q(e,h_{\mathcal{V}})$.
So we can utilize message propagation with high-order structures in the GNN
framework of [43] to capture high-order interdependencies in the data. The
detailed process of building hyperstructures for a distribution network can be
found in Algorithm 1.

Figure 2: An example of graph and hyperstructures: (a) graph , (b) line graph,
(c) hypergraph, and (d) hyper-line-graph.

Input: Graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$

Output: Hyperedge $H_{\mathcal{E}}$ and Hypernode $H_{\mathcal{V}}$

Function: $k$-nearest neighborhood $f_{knn}$

1 for  _$v$ in substation nodes_ do

2 $e_{h}$ = $f_{knn}$($v$, $\mathcal{V}$, $k$);

3 $H_{\mathcal{E}}$.insert($e_{h}$)

4for  _$e$ in edge set $\mathcal{E}$_ do

5 if  _$e$ is incident to a substation_ then

6 $v_{h}$ = $f_{knn}$($e$, $\mathcal{E}$, $k$);

7 $H_{\mathcal{V}}$.insert($v_{h}$)

8

9return _$H_{\mathcal{E}},H_{\mathcal{V}}$_ ;

Algorithm 1 Hyperstructures Construction

Fig. 2 presents a toy example of our augmented graph with hyperstructures. The
top left diagram gives a graph with five nodes and five edges. Its line graph
is shown on the right, for example, edge $e_{3}$ connects all other edges
since it shares node $v_{1}$ with $e_{1}$ and $e_{2}$, and shares node $v_{2}$
with $e_{4}$, $e_{5}$ in the original graph. The bottom diagrams illustrate
hyperedge about nodes in (c) and hypernode about edges in (d). For real data
analysis, we may predefine hyperstructures from contexts or build them
according to graph structure characteristics and some clustering algorithms.

###  III-D Hyperstructures Graph Convolutional Neural Networks

Although graph-based semi-supervised learning (G-SSL) yields promising
performance on many datasets, G-SSL uses only the given adjacency matrix $A$
and the label matrix $Y$, but not the feature matrix $X_{\mathcal{V}}$. This
limitation is important, particularly when classifying data that not only
exhibit a sophisticated topological structure but also provide valuable
information on node features. To address this limitation, there have been
recently proposed many graph-based neural networks methods, e.g., GCN, which
use the feature matrix $X_{\mathcal{V}}$ instead of the label matrix $Y$ and
encode the graph structure by using neural network framework. Here, we
interpret GCN as the learnable graph filter in the spectral domain.

####  III-D1 Graph Convolutional Networks

GCN conducts graph filtering operation in each convolutional layer with the
filter $g$ and the signal matrix $Z_{\textnormal{GC}}^{(\ell-1)}$ (i.e., the
matrix of activations fed to the $\ell$-th layer), where $g=I-L$ and the
initial input signal matrix $Z_{\textnormal{GC}}^{(0)}=X_{\mathcal{V}}$.
Hence, we can obtain the following expression

$g_{\theta^{\prime}}\star
x_{\mathcal{V}}\approx\theta_{0}^{\prime}x_{\mathcal{V}}+\theta_{1}^{\prime}\left(I-L\right)x_{\mathcal{V}}=\theta_{0}^{\prime}x_{\mathcal{V}}-\theta_{1}^{\prime}D^{-{1}/{2}}AD^{-{1}/{2}}x_{\mathcal{V}}.$

When constraining a single parameter in this convolutional filter, i.e.,
$\theta=\theta^{\prime}_{0}=-\theta^{\prime}_{1}$, the filter operation
becomes

$g_{\theta}\star
x_{\mathcal{V}}\approx\theta(I+D^{-{1}/{2}}AD^{-{1}/{2}})x_{\mathcal{V}}.$

To avoid the exploding and vanishing gradient, GCN transforms
$I+D^{-{1}/{2}}AD^{-{1}/{2}}$ to
$\tilde{D}^{-{1}/{2}}\tilde{A}\tilde{D}^{-{1}/{2}}$ using the re-normalization
trick, where $\tilde{A}=I+A$ and $\tilde{D}_{ii}=\sum_{j}\tilde{A}_{ij}$.
Finally, the graph convolution operation on node-level representation learning
can be formulated as

$\displaystyle
Z_{\textnormal{GC}}^{(\ell)}=f_{\textnormal{GMP}}(\sigma(\tilde{D}^{-{1}/{2}}\tilde{A}\tilde{D}^{-{1}/{2}}Z_{\textnormal{GC}}^{(\ell-1)}\Theta^{(\ell-1)})),$
(1)

where $f_{\text{GMP}}$ is the global max pooling, $\sigma(\cdot)$ is the
activation function, e.g., ReLU, and $\Theta^{(\ell-1)}$ is the trainable
weight matrix in the layer.

Figure 3: The Hyper-GCNNs framework consists of three components: (I) hyperedge representation learning scheme, (II) hypernode representation learning scheme, and (III) graph convolution operation.  TABLE II: Average accuracy (%) ($\pm$standard error) comparison with different methods on 54-bus system i@ and 54-bus system<EMAIL_ADDRESS>Model | 3 classes | 4 classes | 5 classes
---|---|---|---
54-bus system i@ | 54-bus system ii@ | 54-bus system i@ | 54-bus system ii@ | 54-bus system i@ | 54-bus system ii@
GCN [44] | 89.10$\pm$1.98 | 72.00$\pm$2.39 | 84.90$\pm$2.61 | 69.14$\pm$2.17 | 73.00$\pm$2.32 | 65.43$\pm$2.56
GraphSAGE [45] | 90.60$\pm$2.46 | 82.38$\pm$2.33 | 85.00$\pm$2.39 | 80.07$\pm$3.89 | 72.90$\pm$3.63 | 67.45$\pm$4.56
GAT [46] | 88.70$\pm$2.64 | 80.00$\pm$2.80 | 84.10$\pm$2.77 | 79.96$\pm$3.98 | 71.10$\pm$1.71 | 69.89$\pm$3.78
GIN [47] | 89.10$\pm$2.12 | 84.61$\pm$3.28 | 85.20$\pm$2.70 | 76.60$\pm$2.99 | 73.00$\pm$2.63 | 66.51$\pm$3.39
DiffPool [48] | 90.90$\pm$3.39 | 85.71$\pm$3.30 | 84.33$\pm$2.47 | 78.68$\pm$4.00 | 75.50$\pm$3.61 | 73.02$\pm$5.02
SNNs [49] | 83.80$\pm$2.55 | 81.39$\pm$3.76 | 79.50$\pm$2.42 | 73.95$\pm$3.33 | 70.20$\pm$2.95 | 63.00$\pm$3.40
FC-V [50] | 86.80$\pm$2.71 | 82.85$\pm$2.15 | 81.30$\pm$2.66 | 75.56$\pm$2.47 | 71.66$\pm$3.55 | 64.75$\pm$2.96
Hyper-GCNNs (ours) | 91.80$\pm$2.34 | 88.28$\pm$3.17 | 86.37$\pm$1.65 | 84.29$\pm$2.26 | 78.00$\pm$2.11 | 73.35$\pm$3.29

####  III-D2 Hyperstructures Representations Learning

In our experiments, we use Multilayer Perceptron (MLP) to learn the
hyperstructures (i.e., hyperedge and hypernode) of a given distribution
system. Given the hyperedge $\mathcal{H}_{\mathcal{E}}$, we employ MLPs and
global max pooling (GMP) to obtain the graph-level representation

$\displaystyle
Z_{\mathcal{E}}=f_{\textnormal{GMP}}(\textnormal{MLP}(\mathcal{H}_{\mathcal{E}}\cdot\textnormal{MLP}(\mathcal{H}^{\top}_{\mathcal{E}}X_{\mathcal{V}}))),$
(2)

where MLP is a MLP model, $X_{\mathcal{V}}\in\mathbb{R}^{N\times
c_{\mathcal{V}}}$ represents the node feature matrix, and
$Z_{\mathcal{E}}\in\mathbb{R}^{d^{\textnormal{out}}_{\mathcal{E}}}$ denotes
the output embedding for a bus system based on hyperedge information.
Similarly, the graph-level representation based on hypernode
$\mathcal{H}_{\mathcal{V}}$ can be formulated as

$\displaystyle
Z_{\mathcal{V}}=f_{\textnormal{GMP}}(\textnormal{MLP}(\mathcal{H}_{\mathcal{V}}\cdot\textnormal{MLP}(\mathcal{H}^{\top}_{\mathcal{V}}X_{\mathcal{E}}))),$
(3)

where $X_{\mathcal{E}}$ is the edge feature matrix and
$Z_{\mathcal{V}}\in\mathbb{R}^{d^{\textnormal{out}}_{\mathcal{V}}}$ represents
the output embedding for a distribution system based on hypernode information.

To adaptively learn the intrinsic dependencies among different representations
from graph structure and hyperstructures, we utilize the attention mechanism
to focus on the importance of task relevant parts of the learned
representations for decision making, i.e.,
$(\alpha_{\textnormal{GC}},\alpha_{\mathcal{E}},\alpha_{\mathcal{V}})=Att(Z_{\textnormal{GC}},Z_{\mathcal{E}},Z_{\mathcal{V}})$.
In practice, we compute the attention coefficient as follows

$\displaystyle\begin{split}\alpha_{i}&=\text{softmax}_{i}(\Upsilon_{\text{Att}}\tanh{(\Xi
Z_{i})})\\\ &=\frac{\exp{(\Upsilon_{\text{Att}}\tanh{(\Xi
Z_{i})})}}{\sum_{j\in\\{\textnormal{GC},\mathcal{E},\mathcal{V}\\}}\exp{(\Upsilon_{\text{Att}}\tanh{(\Xi
Z_{j})})}},\end{split}$ (4)

where $\Upsilon_{\text{Att}}\in\mathbb{R}^{1\times d_{\text{out}}}$ is a
linear transformation, $\Xi$ is the trainable weight matrix, and the softmax
function is utilized to normalize the attention vector. Then we obtain the
final embedding $Z$ by combining all embeddings

$\displaystyle Z=\alpha_{\textnormal{GC}}\times
Z_{\textnormal{GC}}+\alpha_{\mathcal{E}}\times
Z_{\mathcal{E}}+\alpha_{\mathcal{V}}\times Z_{\mathcal{V}}.$ (5)

Lastly, we feed the final embedding $Z$ into an MLP layer and use a
differentiable classifier for graph learning tasks. The overall framework of
Hyper-GCNNs is shown in Fig. 3.

##  IV Experimental Studies

Datasets In our experiments, we consider learning CVaR of Annual Load Shedding
through multi-class classification and graph regression. For the graph
classification tasks, to better study the challenges of CVaR analysis under
different expansion plans, we work on three different number of classes based
on ranges of annual CVaR of loss of load (kWh), i.e., 3 classes, 4 classes,
and 5 classes. Table III shows the details of these classes and their
respective ranges of annual CVaR of loss of load. Here we follow a standard
statistical practice [51] and bin expansion plans into classes based on
quantile ranges of annual CVaR of loss of load (kWh), that is, 3 classes
(i.e., low-, moderate-, and high-risk), 4 classes (i.e., low-, moderate-,
middle-, and high-risk), and 5 classes (i.e., low-, moderate-, high-, very
high-, and extreme high-robustness). Such binning can be viewed, for example,
as an analogue of quantitative requirements under Basel III for banking and
the regulatory framework of Solvency II for the European insurance industry,
where CVaR is referred to as expected shortfall (ES) in financial management
and supervision. Optimal binning practices and their connection to planning
standards are topics of standalone interest which we leave for future
research. For example, when using 3 classes for the 54-bus system I, we
consider the 1st class as the range between 0 kWh and 10,000 kWh, the 2nd
class associated with the range between 10,000 kWh and 20,000 kWh, and the 3rd
class related to more than 20,000 kWh of annual CVaR of loss of load.

TABLE III: Ranges of annual CVaR of loss of load. # Classes | 54-bus system i@ | 54-bus system ii@
---|---|---
Label | Range (kWh) | Label | Range(kWh)
# 3 classes | 0 | [0, 1.0e4] | 0 | [0, 3.0e4]
1 | (1.0e4, 2.0e4] | 1 | (3.0e4, 4.0e4]
2 | (2.0e4, $\infty$) | 2 | (4.0e4, $\infty$]
# 4 classes | 0 | [0, 1.0e4] | 0 | [0, 3.0e4]
1 | (1.0e4, 2.0e4] | 1 | (3.0e4, 3.5e4]
2 | (2.0e4, 3.0e4] | 2 | (3.5e4, 4.0e4]
3 | (3.0e4, $\infty$] | 3 | (4.0e4, $\infty$)
# 5 classes | 0 | [0, 1.0e4] | 0 | [0, 3.0e4]
1 | (1.0e4, 1.5e4] | 1 | (3.0e4, 3.5e4]
2 | (1.5e4, 2.0e4] | 2 | (3.5e4, 4.0e4]
3 | (2.0e4, 2.5e4] | 3 | (4.0e4, 4.5e4]
4 | (2.5e4, $\infty$) | 4 | (4.5e4, $\infty$)

The proposed methodology has been applied to two distribution systems, namely
54-bus system I and 54-bus system II, which are modified versions of the
54-bus system described by [29]. In the 54-bus system I, we have 72 lines (50
existing and 22 candidate lines respectively), 4 substation nodes and 50 load
nodes. In the 54-bus system II, we have 72 lines (52 existing and 20 candidate
lines respectively), 2 substations, 50 load nodes and 2 non-load nodes. Node
features consists of (i) U-scores (where the dimension of U-scores depends on
the number of substations), (ii) node degree, and (iii) node betweenness
centrality; thus, the feature matrix of 54-bus system is
$X_{\mathcal{V}}\in\mathbb{R}^{N\times(\mathcal{N}_{\textnormal{substation}}+2)}$.
Moreover, we use pairwise cosine similarity of node information as edge
features $X_{\mathcal{E}}\in\mathbb{R}^{M\times 1}$ (where $M$ denotes the
number of edges).

In our experiments, we have generated several possible expansion plans for the
two aforementioned systems (200 for the 54-bus system I and 74 for 54-bus
system II). These expansion plans have been generated by selecting different
subsets of the available candidate lines. Then, for each expansion plan of
each system, we have simulated 2000 scenarios of annual operation, with hourly
resolution. We simulate independent Bernoulli trials for the availability of
line segments of the distribution grid for each hour of each scenario, setting
the rate of routine failures (single-line failures) as 0.4 times per year and
the rate of HILP failures (failures involving more than one line segment) as
0.01 times per year. As a result, we attain the CVaR of annual loss of load
for each expansion plan as described in Section III-A. The procedure to obtain
the dataset has been implemented using Julia 1.1 via the JuMP package combined
with CPLEX 12.9, which is run on Intel Core i5-7200U CPU @2.50GHz. For 54-bus
system I dataset, we build hyperedge for each of the 4 substations by 10
nearest neighbors in its feature space; for hypernodes, edges incident to
substation nodes are selected as centroids and clustering is based on
concatenated feature space of incident nodes. The number of edges vary from 50
to 67 over the 200 grids and the number of hypernodes vary from 8 to 12 based
on connectivity of grids. A similar preprocessing is performed on the 54-bus
system II dataset with 2 substations. There are 52 to 56 edges and resulted 4
to 5 hypernodes over the 74 distribution grids.

Baselines For the graph classification tasks, we compare our Hyper-GCNNs with
7 baselines representative of three different categories (i.e., GNN-based
models, simplicial complex-based neural networks, and topological machine
learning method), including (i) Graph Convolutional Networks (GCN) [44], (ii)
GraphSAGE [45], (iii) Graph Attention Networks (GAT) [46], (iv) Diffusion-
convolutional neural networks (GIN) [52], (v) DiffPool [48], (vi) Simplicial
Neural Networks (SNNs) [49], and (vii) Filtration Curves with Random Forest
(FC-V) [50]. For the graph regression tasks, we implement the Random Forest,
GCN [44], and GraphSAGE [45] as baselines. We use the generalization accuracy
and Root Mean Squared Errors (RMSE) to measure the performances on 54-bus
system i@ and 54-bus system ii@ for graph classification and graph regression
respectively. Note that, we split the dataset (i.e., 54-bus system i@ and
54-bus system ii@) to different partitions as training and test sets splits of
80% and 20%, respectively.

Experimental Settings We use the Adam optimizer for 100 epochs to train Hyper-
GCNNs. For the graph classification tasks, (i) 54-bus system i@: Hyper-GCNNs
consists of 3 layers whose hidden feature dimension is 64, and each layer
consists of 2 MLP blocks; the learning rate is 0.01 the dropout is set as 0.5,
and the batch size is set as 16; and (ii) 54-bus system ii@: Hyper-GCNNs
consists of 3 layers whose hidden feature dimension is 16, and each layer
consists of 2 MLP blocks; the learning rate is 0.05 the dropout is set as 0.0,
and the batch size is set as 8. For the graph regression tasks, (i) 54-bus
system i@: Hyper-GCNNs consists of 3 layers whose hidden feature dimension is
64, and each layer consists of 2 MLP blocks; the learning rate is 0.01 the
dropout is set as 0.5, and the batch size is set as 8; and (ii) 54-bus system
ii@: Hyper-GCNNs consists of 2 layers whose hidden feature dimension is 8, and
each layer consists of 2 MLP blocks; the learning rate is 0.0001 the dropout
is set as 0.8, and the batch size is set as 8. Moreover, for both datasets,
the filter size, kernel size, stride, and the size of global max pooling is
set to be 8, 2, 2, and $3\times 3$ respectively. In our work, hyperedges and
hypernodes are based on 10-nn (i.e., $k=10$ in $k$-nearest neighbor algorithm)
about substation nodes and edges incident to substation nodes, respectively.
For the 54-Bus system I, we have 4 hyperedges and around 10 hypernodes for
each plan; for the 54-Bus system II, we have 2 hyperedges and around 4
hypernodes for each plan. We select the best hyperparameter setting of our
Hyper-GCNNs through cross-validation. For baselines, we use the original
papers’ codes in related code repositories. Our data and codes are publicly
available at https://github.com/hypergcnns/hypergcnns.git.

Findings Graph classification results on two 54 bus systems, averaged over 10
cross-validation runs, are summarized in Table II. It suggests the following
key observations: (i) Hyper-GCNNs surpasses all state-of-the-art baselines in
terms of classification performance over all considered scenarios across all
datasets; (ii) Hyper-GCNNs brings relative gains with respect to the next best
approach from 0.98% (for 3 classes of the 54-bus system II and DiffPool as
competitor) to 3.20% (for 5 classes of the 54-bus system I and DiffPool as
competitor); (iii) Hyper-GCNNs yields substantial reductions in computational
costs with respect to the existing stochastic optimization methods adopted by
the power system community and the second lowest running time among DL models,
i.e., the running time (seconds: s) of the methods tested by us on 54-bus
system (i.e., 54-bus system i@) are: GCN (1.00 s); GAT (1.31 s); GraphSAGE
(1.55 s); GIN (1.80 s); DiffPool (2.03 s); SNNs (2.50 s); and FC-V (0.80 s);
and (iv) as expected, performance of all models deteriorates with an increase
of a number of classes. The graph regression evaluation results are summarized
in Table IV. Compared with 3 baseline methods, our Hyper-GCNNs has achieved
the best performance on both 54-bus system i@ and 54-bus system ii@ datasets.
On average, Hyper-GCNNs outperforms the best baseline by 2.5% on RMSE. These
results prove that Hyper-GCNNs to be the most competitive approach for both
graph classification and graph regression of the distribution expansion plans
both in terms of computational costs and accuracy.

TABLE IV: The CVaR prediction performance (RMSE$\pm$standard error) of Hyper-GCNNs and different methods on 54-bus system i@ and 54-bus system<EMAIL_ADDRESS>Model | 54-bus system i@ | 54-bus system ii@
---|---|---
Random Forest | 0.68$\pm$0.011 | 0.94$\pm$0.001
GCN [44] | 0.59$\pm$0.007 | 1.10$\pm$0.001
GraphSAGE [45] | 0.57$\pm$0.005 | 1.00$\pm$0.001
Hyper-GCNNs (ours) | 0.55$\pm$0.003 | 0.93$\pm$0.001
TABLE V: Computational costs for U-scores generation under each expansion plan and a single training epoch of Hyper-GCNNs. Dataset | Avg. # edges | Average Time Taken (sec)
---|---|---
U-scores | Hyper-GCNNs (epoch)
54-bus system i@ | 55.99 | 1.81 | 1.25
54-bus system ii@ | 54.38 | 0.03 | 0.89

Ablation Study we conduct an ablation study to examine the contributions of
different components in our proposed Hyper-GCNNs on 54-bus system I (3
classes) and 54-bus system II (3 classes) datasets. That is, we compare our
Hyper-GCNNs with four ablated variants, (i) Hyper-GCNNs without U-scores
(Hyper-GCNNs w/o U-scores), (ii) Hyper-GCNNs without hyperedge representation
learning (Hyper-GCNNs w/o Hyperedge), (iii) Hyper-GCNNs without hypernode
representation learning (Hyper-GCNNs w/o Hypernode), and (iv) Hyper-GCNNs
without attention mechanism (Hyper-GCNNs w/o Attention mechanism). Table VI
shows that, when ablating the above components, the accuracy of Hyper-GCNNs
drops significantly on graph classification tasks. These ablation results
indicate that integrating the information through U-scores and hyperstructures
representations learning with the attention mechanism is critical.

TABLE VI: Ablation study of the architecture of Hyper-GCNNs. Architecture | 54-bus system i@ | 54-bus system ii@
---|---|---
Hyper-GCNNs | 91.80$\pm$2.34 | 88.28$\pm$3.17
Hyper-GCNNs w/o U-scores | 90.40$\pm$2.37 | 86.52$\pm$3.40
Hyper-GCNNs w/o Hyperedge | 83.00$\pm$3.15 | 81.75$\pm$3.29
Hyper-GCNNs w/o Hypernode | 89.00$\pm$2.58 | 84.28$\pm$2.43
Hyper-GCNNs w/o Attention mechanism | 90.50$\pm$2.20 | 85.71$\pm$3.28

Computational Complexity The computational complexity for the computation of a
simple path between load and substation is $\mathcal{O}(N+M)$, where $N$ is
the number of nodes and $M$ is the number of edges. Since we consider the
U-scores for all nodes in the distribution network, the computational
complexity for computation of U-scores is $\mathcal{O}(N^{2}+NM)$. For $k$-nn
hyperstructure generation, the complexity is $\mathcal{O}(N\log(N)+M\log(M))$
since number of clusters and feature dimensions are usually very small with
respect to the numbers of nodes and edges.

##  V Utility and Limitations

The purpose of this paper is to investigate the ability of GNN-based model to
reproduce risk-based metrics used in electricity distribution grid resilience
planning, that is, the area being still an uncharted territory for AI. Our
findings indicate that the proposed GNN-based model can outperform the more
conventional methods in terms of both accuracy and computational efficiency in
the classification of CVaR for loss of load in electricity distribution grid
expansion plans. Indeed, the computational time to classify different grid
plans (in the order of seconds) is encouraging, especially when compared with
existing simulation based CVaR evaluation methodologies that may take several
hours to compute. This clearly indicates that classification methods, based on
GNN architectures, demonstrate a high potential to be successfully
incorporated into the traditional expansion and planning algorithms to improve
the computational times of the existing large-scale stochastic optimization
models. Additionally, the U-scores metric has proved to be an excellent
topology-based metric to express the risk of the system. This deserves to be
further investigated in future works, as the use of this metric can go beyond
planning applications. This is particularly relevant in distribution systems,
that are operated radially and where power and voltage stability elements
(substations, electric storage resources) are well identified in the network.
For example, in these systems, U-scores could be used to improve operations,
through the development of methods for fast topological reconfiguration based
on simple uniqueness score classifications.

Not Only Power Distribution Systems Our proposed methodology can be broadly
applied way beyond distribution systems to various types of networks
characterized by the prominent multi-node (i.e., higher-order) interactions
and heterogeneity of node roles in the underlying system organization such as
molecular, transportation, and blockchain transaction graphs. For instance, in
molecular networks, the relationships between different cell type markers
(e.g., microglia, astrocytes, oligodendrocytes and neurons) can be
quantitatively evaluated through U-scores, while hyperstructures
representation learning can help to capture hidden dependencies among them -
which will result in enhancing treatment outcomes. Similarly, to improve
traffic forecasting performance, hyperstructures and U-scores can be used to
capture the complex spatio-temporal dependencies in human mobility among
transportation hubs and rural areas. Our Hyper-GCNNs approach makes an
important step to address such intrinsic network properties in a systematic
graph-theoretic manner.

##  VI Discussion

Distributions networks are responsible for delivering power to individual
consumers and are composed by a large number of nodes with small distances
between them, resulting in relatively standardized electrical installations
under a substation, i.e., at the neighborhood/town level. Large electric
utilities have hundreds of such distribution substations, which means that
their grid is actually formed by multiple networks with similar properties.
This unique aspect of distribution grids opens up possibilities to apply
automatic classification algorithms based on GNNs to improve power system
computations, namely, stochastic planning methods (in the context of
resilience) that entail significant computational costs.

The first step to unlock these applications was achieved with this paper,
i.e., showing that it is possible to capture resilience via automatic
classification methodologies (which has been achieved with this paper). After
this first step, there are two possible ways of including classification
algorithms in power grid resilience planning. The first is as a replacement of
the traditional resilience and reliability evaluation methodologies, based on
SMCS described in Methods Section. In this case, the classification methods
based on GNNs are an alternative form to obtain a fast and standardized
resilience analysis of the network, i.e., exclusively based on the topology
and the grid assets. A tool with these characteristics could be used, for
example, by regulators to approve reliability/resilience investments and to
compare grid assets or benchmark resilience performance across different
territories and utilities. Second, classification methods based on graph
convolution operation and hyperstructures representation learning framework
can be used as a pre-solver of the stochastic expansion and planning
optimization methods. For example, U-scores can be applied in this pre-solver
to eliminate potential candidate plans or to identify specific nodes for
investments in batteries and substations. This will result in significant
decreases of the search space in optimization, reducing computational times
and allowing optimal expansion and planning methods to be applied in large-
scale distribution systems.

## Acknowledgment

This work was authored in part by the Lawrence Berkeley National Laboratory
(LBL), U.S. Department of Energy (DOE) National Laboratory operated by the
University of California and also was supported by the NSF grant # ECCS
2039701 and ONR grant # N00014-21-1-2530. The views expressed in the article
do not necessarily represent the views of the DOE, NSF, and ONR.

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