aaaifss2022_18.txt

Graph Representation Learning for Energy Demand Data:
Application to Joint Energy System Planning under Emissions Constraints
Aron Brenner*1, Rahman Khorramfar*2, Dharik Mallapragada3, Saurabh Amin4
1,4Civil and Environmental Engineering (CEE) and Laboratory for Information &Decision Systems (LIDS)
2MIT Energy Initiative (MITEI) and Laboratory for Information &Decision Systems (LIDS)
3MIT Energy Initiative (MITEI)
{abrenner, khorram, dharik, amins }@mit.edu
Abstract
A rapid transformation of current electric power and natural
gas (NG) infrastructure is imperative to meet the mid-century
goal of CO 2emissions reduction requires. This necessitates
a long-term planning of the joint power-NG system under
representative demand and supply patterns, operational con-
straints, and policy considerations. Our work is motivated by
the computational and practical challenges associated with
solving the generation and transmission expansion problem
(GTEP) for joint planning of power-NG systems. Specifi-
cally, we focus on efficiently extracting a set of representa-
tive days from power and NG data in respective networks and
using this set to reduce the computational burden required
to solve the GTEP. We propose a Graph Autoencoder for
Multiple time resolution Energy Systems (GAMES) to cap-
ture the spatio-temporal demand patterns in interdependent
networks and account for differences in the temporal resolu-
tion of available data. The resulting embeddings are used in a
clustering algorithm to select representative days. We evalu-
ate the effectiveness of our approach in solving a GTEP for-
mulation calibrated for the joint power-NG system in New
England. This formulation accounts for the physical interde-
pendencies between power and NG systems, including the
joint emissions constraint. Our results show that the set of
representative days obtained from GAMES not only allows
us to tractably solve the GTEP formulation, but also achieves
a lower cost of implementing the joint planning decisions.
Introduction
One of the most significant societal challenges that we cur-
rently face is to transition to a reliable, low-carbon, and
sustainable energy system as soon as possible, and to meet
the mid-century goal of limiting global warming below 2◦C
(UN-FCCC 2015; Gielen et al. 2019). This requires a signif-
icant use of renewable energy resources and well-planned
integration of various energy vectors, including emerging
clean energy sources such as hydrogen and other renew-
able energy sources. Our work is motivated by the enor-
mous potential of machine learning (ML) models in pro-
moting sustainable energy systems. In particular, we focus
on ML modeling for extracting a set of representative days
from heterogeneous demand data associated with real-world
*The first two authors contributed equally to this work.
Copyright © 2022, Association for the Advancement of Artificial
Intelligence (www.aaai.org). All rights reserved.electric power and natural gas (NG) systems, and using this
set for joint power-NG network planning under emissions
constraints. In doing so, we leverage ML-extracted repre-
sentative days to tractably solve an optimization problem
that determines a capacity and network expansion plan for
regional-scale energy systems such as that of New England.
Broadly speaking, our work addresses several practi-
cal and computational challenges associated with capac-
ity expansion models (CEMs) for decarbonization of in-
terdependent power-NG infrastructures. Classical examples
of such models include the generation expansion problem
(GEP) and generation and transmission expansion problem
(GTEP), both of which are well-studied in the context of
power systems (Li et al. 2022a; He et al. 2018). Our opti-
mization model is a GTEP that determines the optimal lo-
cation and timing of generation units, transmission lines,
and pipelines to meet future energy demands under a range
of operational and policy constraints such as joint emission
constraints. In our work, we extend the model to include
two main interdependencies between power and NG sys-
tems. The first interdependency captures the increasing role
of gas-fired power plants in the generation mix of electricity
production ( EIA; He et al. 2018). The second interdepen-
dency reflects the joint emission of CO 2in both systems.
The key computational challenge in solving the GTEP
arises from the fact that it links long-term investment de-
cisions (e.g. capacity and network expansion) to short-term
operational ones (e.g. unit commitment, power production,
and energy storage). The former decisions have a planning
horizon of 10-30 years with yearly granularity, while the
latter usually require hourly or sub-hourly resolution. Un-
der reasonable assumptions, we can express the GTEP as a
large-scale mixed-integer linear program (MILP), but cur-
rent literature has limited success in tractably solving these
problems to an adequate level of spatial and temporal resolu-
tion. In our case, the computational difficulty in solving the
GTEP increases further because we model both power and
NG networks. Thus, taking into account (projected) demand
information on a day-to-day basis becomes prohibitively
expensive from a computational viewpoint. In the classi-
cal GTEP problems for power systems, the computational
challenge is addressed by aggregating power system nodes
(buses) within a geographical neighborhood (power zone) to
a single node (Li et al. 2022a) and by solving the GTEP for a

set of representative days (Hoffmann et al. 2020). Crucially,
the set of representative days needs to capture demand and
supply patterns. To the best of our knowledge, the notion of
representative days has not been clearly defined and devel-
oped in the context of joint power-NG planning problem –
this is where we leverage our graph representation learning
approach.
Our work also addresses the practical issues arising from
coarse data availability from the NG network. Firstly, we do
not have access to the detailed connectivity and transmis-
sion information in the NG network while this information
is readily available for the power network. Secondly, power
systems typically collect demand and generation data at a
fine temporal resolution (hourly or less), but this data is usu-
ally not publicly accessible for NG systems. These issues
thus require us to (a) formulate network constraints based on
loosely specified information on power and NG node con-
nectivity and (b) develop an approach to leverage demand
and supply data from the power system with demand data of
NG system despite their different temporal resolutions.
We address the aforementioned challenges by develop-
ing a graph representation learning approach that captures
the physical interdependencies between power and NG net-
works, and also handles the different granularity of data
at each network. We consider demand data for both sys-
tems, and consider capacity factor (CF) data for solar and
wind plants to reflect the supply pattern in the renewable-
dominated future grid. We utilize graph convolutions to cap-
ture the network interactions both within and across power
and NG networks, and adopt an autoencoder architecture
with tuneable reconstruction losses for the respective de-
mand and CF data. We demonstrate that the resulting Graph
Autoencoder for Multiple time resolution Energy Sys-
tems (GAMES) model is ideally suited to handle embed-
ding the spatio-temporal patterns in power and NG demand
as well as wind and solar CF data into a lower-dimensional
representation, which can be readily clustered to extract the
set of representative days. Furthermore, our approach to
computing the set of representative days can also enable an
accurate estimation of the trade-off between costs (both in-
vestment and operational) and joint emissions from power
and NG systems.1
Previous studies for selecting representative days propose
variants of k-means (Mallapragada et al. 2018; Li et al.
2022b; Teichgraeber and Brandt 2019; Barbar and Mallapra-
gada 2022), k-medoids (Scott et al. 2019; Teichgraeber and
Brandt 2019), and hierarchical clustering (Liu, Sioshansi,
and Conejo 2017; Teichgraeber and Brandt 2019). The dis-
tance matrices used in clustering algorithms for most previ-
ous works are constructed based on a set of time series inputs
such as load data and variable renewable energies (VRE)
capacity factors (Li et al. 2022a; Hoffmann et al. 2020).
Notably, these approaches neither account for demand data
with multiple time resolutions nor account for network in-
1We believe this capability can have a significant societal im-
pact by lowering the barriers to investment in renewable energy re-
sources and alleviating reliability concerns in a low-carbon energy
system.terdependencies. Hence, they cannot be readily extended to
address the task of extracting representative days for joint
power-NG systems – an aspect that is crucial for realism and
tractability in joint planning optimization models for decar-
bonizing these systems. We believe that our GAMES model
addresses these challenges and provides a promising path
to better extract representative days in interdependent power
and NG systems.
Graph Convolutional Autoencoder Approach
In this section, we describe the Graph Autoencoder for Mul-
tiple time resolution Energy Systems (GAMES) model, a
simple graph autoencoder with linear graph convolutions.
We argue that this architecture efficiently captures spatio-
temporal demand patterns in power and NG systems.
Autoencoders
To begin with, we note that direct use of clustering algo-
rithms to identify representative days for any large-scale en-
ergy system is prone to the “curse of dimensionality” due
to the high dimensionality of time series data. In such set-
tings, it is desirable to first extract low-dimensional and de-
noised representations of the data before clustering (Par-
sons, Haque, and Liu 2004). To identify a set of represen-
tative days, we choose to utilize a state-of-the-art autoen-
coder architecture for learning low-dimensional embeddings
for power-NG systems (that have different time resolutions)
prior to clustering.
Given a high-dimensional input such as a time series of
graph signals, X∈Rp, an autencoder can be trained to
jointly learn an encoder, g:Rp→Rk, and a decoder,
f:Rk→Rpthat minimize the reconstruction loss func-
tion∥X−ˆX∥2
2, where ˆX=f(g(X))is the reconstructed
signal. Here, k≪pdenotes the dimension of the learned
latent space.
Variable Interpretation Granularity Nodes
XE Electricity Hourly 188
XW Wind Hourly 188
XS Solar Hourly 188
XG Natural Gas Daily 18
Table 1: Notation for input variables.
We denote by XE∈Rd×nE×tEthe data tensor of elec-
tricity demands over all days d, nodes nE, and times tE.
Similarly, we denote the natural gas data tensor by XG∈
Rd×nG×tG, the wind capacity factor tensor by XW∈
Rd×nW×tW, and the solar capacity factor data tensor by
XS∈Rd×nS×tS(see Table 1). Because the GTEP considers
different associated costs for investment and operational de-
cisions related to power, NG, wind, and solar, we introduce
hyperparameters αG, αW, αSin the autoencoder objective
function to tune the trade-off between the multiple recon-
struction losses. This parameter reflects the contribution of
each system towards the total cost. For example, if the NG
system cost is twice the power system cost, then higher val-
ues of αGensure that the reconstruction cost is penalized

05101520Hour°2.0°1.5°1.0°0.50.0Electricity Demand (std. dev.)Node 8Node 205101520Hour°2.0°1.5°1.0°0.50.0Electricity Demand (std. dev.)Node 72Node 92
05101520Hour°2.0°1.5°1.0°0.50.0Electricity Demand (std. dev.)Node 142Node 13005101520Hour°2.0°1.5°1.0°0.50.0Electricity Demand (std. dev.)Node 170Node 155Figure 1: Adjacent nodes in the power network demonstrate similar variations in demand over the course of the day. These
spatial dependencies are modeled explicitly by graph convolutional layers in the GAMES architecture.
more when deviating from the data of the NG system. This
gives us the following loss function:
dX
i=11
dnEtE∥X(i)
E−ˆX(i)
E∥2
F+αG
dnGtG∥X(i)
G−ˆX(i)
G∥2
F
+αW
dnWtW∥X(i)
W−ˆX(i)
W∥2
F+αS
dnStS∥X(i)
S−ˆX(i)
S∥2
F
,
where ∥ · ∥Fdenotes the Frobenius norm.
In our case study, we set αG= 2, αS= 0.5, αW= 0.5.
However, we note that it is possible to choose the hyperpa-
rameters by evaluating the downstream GTEP objective for
different values. Specifically, this can be performed using a
grid search in which the quality of a combination of hyper-
parameters {αG, αW, αS}is measured by GTEP objective
costs given by solving the optimization model rather than
the GAMES validation loss directly.
Graph Representation Learning
Next, we provide a brief introduction to modeling with graph
convolutional networks (GCNs).Preliminaries We encode the network topology with the
binary adjacency2matrix A, which we construct such that
Aij=0 (i, j)/∈ E
1 (i, j)∈ E.
We also construct the diagonal degree matrix Dsuch that
Dii=P
jAij.
Graph Convolutions Our graph autoencoder approach
follows (Kipf and Welling 2017) in utilizing Chebyshev con-
volutional filters , which approximate spectral convolutions
to learn node embeddings as weighted local averages of em-
beddings of adjacent nodes. This is ideal for learning low-
dimensional embeddings of energy networks as neighbor-
hoods of nodes typically exhibit similar energy demands
patterns and can thus be represented jointly. Chebyshev fil-
ters operate on the “renormalized” graph Laplacian ˜L=
˜D−1
2˜A˜D−1
2, where ˜D=I+Dand˜A=I+A, and
perform a form of Laplacian smoothing (Li, Han, and Wu
2018; Taubin 1995). We initialize H(0)=Xand apply con-
volutional filters to learn subsequent node embeddings as
2Ideally, one should construct an affinity matrix Awith a Gaus-
sian kernel such that Aij= exp
−dist(i,j)2
σ2
for all edges (i, j),
where dist(i, j)denotes the distance of edge (i, j)andσdenotes
the standard deviation of distances in the network (Shuman et al.
2012). Since we do not have access to edge distance data in our
case study, we proceed with the binary adjacency matrix.

follows:
H(l+1)=σ(˜LH(l)Θ(l)),
where Θ(l)is a trainable weight matrix and H(l)is a matrix
of node embeddings in layer l.σ(·)is typically a nonlinear
activation function, such as ReLU ortanh .
In each layer, GCNs aggregate features from the imme-
diate neighborhood of each node. Deep GCNs stack multi-
ple layers with nonlinear activations to learn node embed-
dings as nonlinear functions of both local and global node
features. In contrast, (Salha, Hennequin, and Vazirgiannis
2019) propose a simpler graph autoencoder model, which
they demonstrate to have competitive performances with
multilayer GCNs on standard benchmark datasets despite
being limited to linear first-order interactions. Shallow neu-
ral architectures are also better suited for settings where data
is scarce. This is particularly significant in modeling energy
systems whose data may only be available for a few his-
torical years. Indeed, we find this simpler GCN approach to
perform well for our case study. We now introduce GAMES,
an augmented version of the linear GCN autoencoder for en-
ergy systems with multiple time resolutions.
GAMES
Our proposed GAMES architecture is designed as follows
and illustrated in Fig. 2.
Encoder Consider the power, wind CF, solar CF, and NG
time series corresponding to day i,X(i)
E,X(i)
W,X(i)
S,X(i)
G.
We begin by constructing the data matrix X(i)as
X(i)= 
X(i)
EX(i)
WX(i)
W 0
0 0 0 X(i)
G!
.
Note that X(i)∈Rn×t, where n:=nE+nGandt:=
tE+tW+tW+tG. This is because capacity factor data
exists for all nodes in the power network and utilizes the
same network topology. X(i)is then passed through a sin-
gle convolutional layer to produce the low-dimensional em-
bedding Z(i)∈Rn×k. The hyperparameter kdefines the
bottleneck of the autoencoder architecture (i.e. the dimen-
sion of each node embedding) and consequently the tradeoff
between compression and reconstruction loss. In our case
study, we find k= 3 to show a sufficient performance for
our application of identifying representative days.
Decoder Z(i)is passed through a convolutional layer to
produce the embedding H(i)∈Rn×t. This reconstructed
matrix is then split along the second dimension into two
blocks: H(i)
E,W,S∈R(nE+nW+nS)×tandH(i)
G∈RnG×t.
Each block is then passed to a separate series of fully con-
nected layers with tanh activations that map the node em-
beddings in H(i)
E,W,SandH(i)
Grespectively to the reconstruc-
tions ˆX(i)
E,W,SandˆX(i)
G. Finally, the tensor ˆX(i)
E,W,Sis split
into the reconstructions ˆX(i)
E,ˆX(i)
W,ˆX(i)
S.Clustering
After the model is trained, the power
and NG time series from each day i, i.e.
(X(1)
E, X(1)
W, X(1)
S, X(1)
G), . . . , (X(N)
E, X(N)
W, X(N)
S, X(N)
G),
is passed through the encoder to generate the embedding
matrices Z(1), . . . , Z(N). Then, k-medoids clustering is
applied to select a set of Kcluster medians, denoted by
S ⊂ { 1, . . . , N }, and assign each day ito a corresponding
cluster j∈ S. We denote the set of days assigned to the
cluster defined by day jasCj. Given the number of clusters
K, the k-medoids algorithm aims to minimize the objective
function
minX
j∈SX
i∈Cj∥Z(i)−Z(j)∥2
F (1)
(Hastie, Tibshirani, and Friedman 2001). Note that every day
in the dataset must be assigned to exactly one cluster. Se-
mantically, (1) can be understood as aiming to ensure that
the set of representative days Sproportionately partitions
the full set of days in the dataset by minimizing squared Eu-
clidean distances in the latent space as constructed by the
autoencoder.
Capacity Expansion Model
The result of the clustering algorithm is used to solve the
CEM for joint power and NG planning, which is formulated
as a GTEP. The problem determines the minimum invest-
ment cost and operational decisions for the year 2050 un-
der various investment, operational, and policy constraints.
The investment decisions include establishing new power
plants, transmission lines, and pipelines as well as decom-
missioning existing plants. The operational constraints in-
clude minimum production, ramping, energy balance, trans-
mission, and storage. We consider emission limits and min-
imum share of VREs as policy constraints. Importantly, in
our formulation, the emissions constraint limits CO 2emis-
sions incurred by the consumption of NG in both networks.
We introduce our model with simplified notation in this
section and provide a detailed formulation in the supple-
mentary material (SI 2022). Let ze= (xe,ye,p)represent
the set of variables for the power system. The integer vari-
ablexeis the variable establishing plants, decommissioning
plants, and establishing new transmission lines. The contin-
uous variable pcaptures the power generation in NG-fired
plants while yeis a continuous variable that captures all the
remaining variables including power generation from non
NG-fired plants and power flow between nodes, storage, and
load shedding variables. We use zg= (xg,yg,f)to de-
note the set of variables associated with the NG system. The
mixed-integer variable xgis the set of all investment, stor-
age, and load shedding decisions. The continuous variable
ygrepresents the intra-network flow, i.e. the flow between
NG nodes or the flow between NG nodes and NG storage
facilities. The flow between NG and electricity systems is
denoted by f. We formulate the joint power-NG system as

[XE∥XW∥XS]
XGX ConvEncoder
Z ConvDecoder
HHE,W,S
HGFC tanh FC [ˆXE∥ˆXW∥ˆXS]
FC tanh FC ˆXG
Extract Node EmbeddingsFigure 2: The GAMES Architecture. The electric power, wind CF, solar CF, and NG time series are combined into the block
matrix Xwithnrows and tE+tW+tS+tGchannels. A single linear graph convolutional layer constructs matrix Zby
embedding each row of Xintokdimensions. Another graph convolutional layer scales each row of Zback to tE+tW+tS+tG
dimensions, which are then separated and fed through fully connected layers to reconstruct the two time series. After the model
is trained, the embeddings are extracted by feeding the daily time series inputs through the encoder, at which point clustering is
applied.
follows:
min (ce
1xe+ce
2ye+ce
3p) + (cg
1xg+cg
2yg+cg
3f)
(2a)
s.t.Aexe+Beye+Dep≤be
1 (2b)
Heye≥be
2 (2c)
Agxg+Bgyg+Dgf≤bg
1 (2d)
f=E1p (2e)
G2yg+E2p≤η (2f)
xe∈Z+,ye,xg∈Z+×R+,p,yg,f∈R+(2g)
The objective function (2a) minimizes the investment and
operational costs for the power system (first term) and NG
system (second term). The constraint (2b) represents all in-
vestment, commitment, and operational constraints for the
power system including the production limit, ramping, stor-
age, and energy balance constraints. The constraint (2c) en-
forces policy considerations such as the minimum require-
ment for renewable portfolio standard (RPS). The NG con-
straints are reflected in constraint (2d), which includes tech-
nological and operational constraints such as the supply
limit at each node, flow between NG nodes, and storage.
The coupling constraint (2e) ensures that NG-fired plants
operate based on the gas flow they receive from the NG
network. The second coupling constraint (2f) is the decar-
bonization constraint that limits emissions resulting from
NG consumption to serve both electricity (via NG power
plants) and non-power related NG loads to η. The coeffi-
cient matrices E1,G2, andE2represent the heat rate, emis-
sion factors for NG usage, and emission factor for NG-fired
plants, respectively. Indeed, emissions from coal-fired plants
is a major driver for decarbonization efforts and NG remains
as primary fuel for which emissions need to be regulated.
Therefore, given the declining role of coal in the US energy
system, the constraint (2f) reflects a futuristic setting where
such plants are already decommissioned.Input Data
Using publicly available data, we consider the New England
region and construct its corresponding power and NG net-
work. We then calibrate the resulting networks using his-
torical data. The power network consists of 188 nodes with
338 existing and candidate transmission lines. The NG net-
work consists of 18 NG nodes and 7 storage nodes. We as-
sume that each NG node is connected to two other storage
nodes. We also assume that each power node is connected
to three of its closest NG nodes. The Supplementary Infor-
mation provides the details of the input data for the joint
power-NG planning model (SI 2022).
Computational Experiments
GAMES Performance
We train GAMES on a dataset of 292 days using the Adam
optimizer with a learning rate of 0.001. We use the full batch
of 292 data points for each update step and perform early
stopping to end training when the validation loss no longer
decreases. We report the validation reconstruction loss on
a set of 73 days for various node embedding dimensions k
in Table 2. We observe slightly diminishing returns for the
Embed. Dim. k= 1 k= 2 k= 3 k= 4
MSE Loss 0.727 0.398 0.244 0.160
Table 2: The reconstruction loss shows diminishing returns
fork >3node embedding dimensions.
validation reconstruction loss for k >3. Consequently, we
proceed with our representative day selection using embed-
dings generated by the model corresponding to k= 3.
Representative Days Comparison
Setup We use the k-medoids clustering algorithm to ob-
tain different sets of representative days. We apply the clus-
tering algorithm to both raw data and the embeddings ob-

2 6 10 14 18 22 26 30 34 38 421.52.01e10
Total Cost
Raw Data GAMES
2 6 10 14 18 22 26 30 34 38 421.01.51e10
Power System Cost
2 6 10 14 18 22 26 30 34 38 424.55.01e9
NG System Cost
2 6 10 14 18 22 26 30 34 38 421.01.21e10
Investment and FOM for Geneneration and Storage (Pow. Sys)
2 6 10 14 18 22 26 30 34 38 420241e9
Power System Load Shedding Cost
2 6 10 14 18 22 26 30 34 38 42
Number of Representative Days051e6
Emission from Power System(a) GAMES vs. raw data clustering comparison under an 80%
carbon reduction goal.
2 6 10 14 18 22 26 30 34 38 421.752.002.251e10
Total Cost
Raw Data GAMES
2 6 10 14 18 22 26 30 34 38 421.01.51e10
Power System Cost
2 6 10 14 18 22 26 30 34 38 42561e9
NG System Cost
2 6 10 14 18 22 26 30 34 38 421.01.21.41e10
Investment and FOM for Geneneration and Storage (Pow. Sys)
2 6 10 14 18 22 26 30 34 38 420241e9
Power System Load Shedding Cost
2 6 10 14 18 22 26 30 34 38 42
Number of Representative Days051e6
Emission from Power System(b) GAMES vs. raw data clustering comparison under a 95% car-
bon reduction goal.
Figure 3: Various costs and power emission for different number of representative days under different decarbonization goals.
tained from the GAMES model to compare the results of
the proposed model. Accordingly, two different sets are ob-
tained for each number of representative days. The optimiza-
tion model over the full power network is prohibitively chal-
lenging even for a very small number of days. Therefore, we
aggregate all buses in each state of the New England region
to obtain a 6-node power network. This aggregation allows
us to run the formulation for up to 42 representative days.
We obtain a feasible solution in two steps for each set of
representative days: (1) The optimization model is aggre-
gated to the set of representative days for tractability and
then solved. (2) Next, we consider the full planning horizon
(the entire year of 2050) and set the integer decision vari-
ables (i.e. investment decisions) to the values determined in
the first step. We note that the investment decision variables
in our formulation are (a) the only integer-valued decision
variables and (b) independent of planning periods. There-
fore, fixing them reduces the remaining operational problem
to a linear program (LP), which can be solved considerably
faster. The resulting solution from the second step is a fea-
sible solution to the full-year problem, with which we can
analyze resulting costs and decisions.In our computational experiments, we consider two de-
carbonization goals of 80 %and 95 %where the former is the
projected target for New England states (Weiss and Hagerty
2019), and the latter aims reflects a radical decarbonization
goal. Figures 3a and 3b show the results under 80 %and 95 %
emission reduction goals respectively. Both figures evalu-
ate the following quantities for the clusters obtained from
GAMES and raw data: i) “Total Cost” which is the objec-
tive function of model 2; ii) “Power System Cost” which is
the first term in the objective function (2a); iii) “NG Sys-
tem Cost” which is the second term in the objective func-
tion (2a); iv) “Investment and FOM for Generation and Stor-
age (Pow. Sysm)” (investment-FOM) which is part of the
power system cost and captures the capital investment and
fixed operating and maintenance (FOM) costs of installing
new power plants and storage systems; v) “Power System
Load Shedding Cost” which is part of the power system cost
and reflects the cost of unsatisfied electricity demand; and
v) “Emission from Power System” which is the tonnage of
emission as a result of operating NG-fired power plants in
the power system. We use “GAMES” to denote the feasi-
ble solution for the set of days obtained by GAMES. We do

Table 3: Average percentage change when using GAMES approach for for various costs and power emissions.
Reduction Goal Total Power NG Inv-FOM (Power) Shedding cost (Power) Emission from Power Sys
80% -5.14 -7.03 0.24 -4.64 -24.13 -9.87
95% -7.27 -10.50 1.5 -8.51 -27.80 -3.31
not report the wall-clock times, but all instances are solved
under 5 hours. As expected, run-times vary significantly de-
pending on the number of representative days utilized; in-
stances with 2 representative days typically run in fewer than
350 seconds, whereas 30-day instances may need to run for
2800 seconds. All instances are implemented in Python us-
ing Gurobi 9.5 and are run on the MIT Supercloud system
with an Intel Xeon Platinum 8260 processor containing up
to 96 cores and 192 GB of RAM (Reuther et al. 2018).
Results Table 3 presents the percentage change in various
quantities yielded by the GAMES representative days solu-
tion as compared to the solution using representative days
selected from clustering the raw data. The cost comparisons
are also plotted in Figures 3a and 3b. We observe, on aver-
age, a 5.14 %and 7.27 %improvement (decrease) in the total
cost when using GAMES under 80 %and 95 %decarboniza-
tion goals, respectively. This improvement may be attributed
to GAMES’ ability to model dependencies between power
and NG system data. Under more stringent decarbonization
targets, the share of VRE increases and the role of dispatch-
able power plants, such as NG-fired plants, diminishes. As
a result, modeling the influence of capacity factors and their
interactions with power and gas demands becomes more es-
sential. This phenomenon may underlie our observation for
the 22-day instance in which, while both approaches provide
similar results under the 80 %decarbonization goal, GAMES
significantly outperforms the raw data clustering as mea-
sured by total cost for the higher decarbonization goal. As
shown in Figures 3a and 3b, the total cost from GAMES out-
performs or matches the performance of the raw data clus-
tering in all instances (except the 30-day instance under an
80%reduction goal). Interestingly, this disparity in perfor-
mance is most drastic when 15 or fewer representative days
are utilized under both decarbonization goals. This is worth
noting as the optimization model instantiated on the full net-
work topology (i.e. without aggregating nodes by state) is
only tractable over a small set of representative days (i.e. af-
ter applying a very coarse temporal aggregation). It is espe-
cially important when the a model-year model only affords
to consider a handful of representative days for each year.
The power system cost largely drives variation in the to-
tal cost under both decarbonization goals – the total cost is
lower for all solutions with a lower power system cost. Note
that the difference in performance is more pronounced in the
power system cost compared to the total cost as indicated
by the 7.03 %and 10.50 %power system cost improvement
for GAMES under the 80 %and 95 %decarbonization goals.
In Figure 3a, this trend aligns with load shedding costs ex-
cept for the 14-day instance. However, as the 24.13 %de-
crease shows, the GAMES approach results in significantly
lower load shedding on average. The 27.80 %improvementin the load shedding cost for GAMES under the 95 %goal
is plotted in detail in Figure 3b; GAMES outperforms the
raw data clustering for all instances. Moreover, the GAMES
approaches converges after 14 days with load shedding cost
significantly lower than those instances utilizing fewer rep-
resentative days.
In both figures the trends of investment-FOM cost and
power system cost are the same, indicating that the power
system cost is largely driven by investment-FOM cost, and
to a lesser extent, by load shedding cost. This is expected as
future energy systems will rely significantly on VREs such
as solar and wind power, which only incur investment and
FOM costs. Another interesting observation pertains to the
quantity of emissions in the power system caused by oper-
ating NG-fired plants. Emissions for the power system are
on average 9.87 %and 3.31 %lower for GAMES under the
two decarbonization goals. This indicates a greater share
of VREs in the GAMES approach, and correspondingly, a
higher share of gas-fired plants in the raw data clustering
approach. This is an interesting observation that may have
significant implications for energy policy-making. In partic-
ular, it suggests that the results from the raw data cluster-
ing approach may be misleading as they do not sufficiently
convey the radical changes required to transform the sys-
tem from the current gas-dominant generation portfolio to a
renewable-dominant power grid.
NG system cost is another essential component of the total
costs. Although NG costs are similar for GAMES and raw
data clustering for each instance, the NG cost increases with
the number of representative days. A possible explanation
might be that neither GAMES nor raw data clustering aim to
capture extreme days with separate clusters. Therefore, days
with loads similar to extreme days are more likely to be se-
lected as a cluster’s medoid as the number of representative
days increases, which inevitably raises the NG system cost.
This consideration is also consistent with the observed load
shedding cost for the power system, which is significantly
higher for instances with fewer than 15 representative days,
indicating that both approaches fail to account for extreme
days in cluster medoids.
Conclusion
In this work, we propose GAMES, a graph convolutional
autoencoder for modeling energy demand in interdependent
electric power and natural gas systems with heterogeneous
nodes and different time resolutions. GAMES is able to ex-
ploit spatio-temporal demand patterns to learn efficient em-
beddings of interdependent power and NG networks. We ap-
ply the k-medoids clustering algorithm to these embeddings
to identify a set of representative days with which we are
able to tractably solve an energy system infrastructure plan-

ning problem calibrated for the joint power-NG system in
New England. Our computational results show that the pro-
posed framework outperforms clustering methods applied to
the raw data and is effective in selecting a small number
of representative days to provide high-quality feasible so-
lutions for the optimization problem.
The current work can be extended in multiple directions.
The immediate extension of the GCN architecture is to ex-
plore alternative approaches to graph representation learning
such as Laplacian sharpening (Park et al. 2019). The extrac-
tion and inclusion of extreme days, or low-frequency days
with unusually low or high demand is another potential next
step which could prevent high load shedding costs and better
represents the NG system’s load patterns.
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