# A Hetero-functional Graph Structural Analysis of the American Multi-modal
Energy System

Dakota J. Thompson  Thayer School of Engineering at Dartmouth
Hanover NH, United States
<EMAIL_ADDRESS>Amro M. Farid  Thayer School of
Engineering at Dartmouth
Hanover NH, United States
MIT Mechanical Engineering
Cambridge MA, United States
<EMAIL_ADDRESS>
###### Abstract

As one of the most pressing challenges of the 21st century, global climate
change demands a host of changes across four critical energy infrastructures:
the electric grid, the natural gas system, the oil system, and the coal
system. Unfortunately, these four systems are often studied individually, and
rarely together as integrated systems. Instead, holistic multi-energy system
models can serve to improve the understanding of these interdependent systems
and guide policies that shape the systems as they evolve into the future. The
NSF project entitled “American Multi-Modal Energy System Synthetic & Simulated
Data (AMES-3D)” seeks to fill this void with an open-source, physically-
informed, structural and behavioral machine-learning model of the AMES. To
that end, this paper uses a GIS-data-driven, model-based system engineering
approach to develop structural models of the American Multi-Modal Energy
System (AMES). This paper produces and reports the hetero-functional incidence
tensor, hetero-functional adjacency matrix, and the formal graph adjacency
matrix in terms of their statistics. This work compares these four hetero-
functional graph models across the states of New York (NY), California (CA),
Texas (TX), and the United States of America (USA) as a whole. From the
reported statistics, the paper finds that the geography and the sustainable
energy policies of these states are deeply reflected in the structure of their
multi-energy infrastructure systems and impact the full USA’s structure.

###### Index Terms:

Hetero-Functional Graph Theory, Model Based Systems Engineering, sustainable
energy transition, American Multi-modal Energy System, Sustainability

##  I Introduction

As one of the most pressing challenges of the 21st century, global climate
change demands a host of changes across at least four critical energy
infrastructures: the electric grid, the natural gas system, the oil system,
and the coal system. In the context of the United States, this paper refers to
this system-of-systems as “the American Multi-Modal Energy System (AMES)”. The
needs of climate change demand mitigation and adaptation strategies which are
far more demanding than the needs of just mitigation alone. Therefore, as
policies are developed to drive the sustainable energy transition forward,
they must not just aim to _mitigate_ climate change but must also _adapt_ to
its effects with resilient architectures. In effect, the need for
decarbonization must be harmonized with the need for economic development,
national energy security, and equitable energy access[1, 2, 3]. These combined
requirements to develop effective policies necessitate an understanding of the
AMES interdependence’s and how they vary geographically and temporally[4].
Furthermore, this cross-sectoral interdependency can introduce architectural
fragility[5] that must be managed as an integral part of the sustainable
energy transition.

Holistic multi-energy system models should serve to improve the understanding
of these interdependent systems as they evolve into the future. Unfortunately,
while there have been many attempts at modeling multi-energy systems and large
scale flows of energy, the field remains relatively nascent[6, 7, 8, 9, 10,
11, 12, 13, 14]. While initial models were developed in response to the oil
crisis and then to mitigate climate change with decarbonization they have also
been developed to introduce new energy streams such as hydrogen, synthetic
fuels, bio-fuels, and other renewable energy sources. These works introduce
their own set of weaknesses including the lack of asset level granularity,
difficulty of use, and specific one-off geographically-specific use case
models. Additionally, a majority of the works investigating these energy
systems in the past have been performed on individual energy networks [15, 16,
17, 18, 19]. More recently, work has been published analyzing only a couple
systems together such as pairing the electric grid with one of the others
fossil fuel systems that compose the AMES[6, 7, 8, 9, 10, 11, 12, 13]. These
works however, do not include all four critical energy infrastructures and do
not extend to the entire American geography. As an exception, the EIA
developed a comprehensive model called the National Energy Modeling System
(NEMS) which it uses to produce the (American) Annual Energy Outlook[20].
Despite serving this important function and being publicly available, this
software tool remains opaque and difficult to use. The EIA website itself
recognizes: ”[The] NEMS is only used by a few organizations outside of the
EIA. Most people who have requested NEMS in the past have found out that it
was too difficult or rigid to use [21]”. Consequently, holistic multi-energy
system models of the AMES remain a present need for open-source, citizen-
science to inform policies.

With a deficit of spatially and functionally resolved data, and with the
current methods for modeling multi-energy systems having their limitations,
the (American) National Science Foundation (NSF) put forth a call for
“research to develop and make available simulated and synthetic data on
interdependent critical infrastructures (ICIs), and thus to improve
understanding and performance of these systems”[22]. The NSF project entitled
”American Multi-Modal Energy System Synthetic & Simulated Data (AMES-3D)”
seeks to fill this void with an open-source structural and behavioral model of
the AMES. Adhering to a Model Based Systems Engineering (MBSE) approach [23,
24], this project develops an interdependent system data set and its
associated models on top of a strong theoretical foundation in systems
engineering. As a result, it can be used for practical applications in the
energy systems field to address not just mitigation of climate change but
adaptation and resilience as well. This is made possible by using asset-level,
openly-available datasets to infer the AMES’ reference architecture[25] to
organize and define the interconnections between the four subsystems. The
reference architecture uses SysML[24] to model the four interdependent energy
systems, and the flows of mass and energy within and between them. This
reference architecture provides a more detailed and self-consistent MBSE
foundation for energy models moving forward relatively to the “reference
energy systems” that have been used in some national energy system
optimization models[26, 27, 28, 29]. The datasets used to infer reference
architecture are also used to instantiate the AMES into an instantiated
architecture[25]. While the NSF project seeks to develop both a structural and
behavioral model, this paper focuses its scope on the former.

The development of the AMES reference architecture provides several immediate
benefits. The first is that a SysML based reference architecture describes the
system’s form, function and the allocation of the latter onto the former[23,
24]. Therefore, the reference architecture describes not just what the system
is made of, but also what it does. Second, a SysML based reference
architecture can be readily translated into mathematical models including both
the form and function. Standard structural models include formal graphs that
describe energy facilities and how they are connected. In the meantime,
hetero-functional graphs (HFG) describe how the wide variety of capabilities
in a system are interconnected and the flows of operands between them. HFGs
have been shown to provide more information than formal graphs when analyzing
an evolving instantiated architecture[30, 31]. In effect, HFGT provides a
means to quantitatively interpret the graphical SysML-based models from both a
formal and functional lens. Such an analysis has already been conducted on a
small scale electric power distribution systems [30] as well as on a large
scale for the entirety of the American electric power system[31]. This paper
now builds on these electricity-only analyses to study the structure of the
whole AMES.

###  I-A Original Contributions

This paper uses a data-driven, MBSE-guided approach to develop open-source
structural models of the American Multi-Modal Energy System. More
specifically, the AMES reference architecture[25] is applied to an asset-level
GIS data called Platts Map Data Pro [32] to create models of several regions.
The instantiated structural models include for the first time the electric
grid, the natural gas system, the oil system, the coal system and the
interconnections between them as defined by the AMES reference architecture,
for the full contiguous United States of America (USA). Initial results are
organized into two categories; a formal and hetero-functional graph for each
of the regions being studied, New York (NY), California (CA), Texas (TX), and
the full contiguous (USA). The states are chosen for their large size and the
diversity of their energy policies. Consequently, the chosen regions have also
taken distinct directions to advance the sustainable energy transition. In
2019, CA had the most renewable energy generation out of all the states [33].
In the meantime, NY’s efforts to expand renewable energy capacity are balanced
by its reliance on natural gas and oil to meet space heating energy
demands[34]. Alternatively, Texas, while being the nations leading crude oil
and natural gas producing state, is also the nations leading producer of wind
powered electric generation[35]. By using MBSE and HFGT, new open source data
models are presented for these three states and the full USA to aid in
advancing and guiding the sustainable energy transition and energy policies.

###  I-B Paper Outline

The remainder of the paper proceeds as follows. Section II is a description of
the background literature and the lack of open data models used to develop the
instantiated architecture models and guide policy. Section III then presents
the data source that drove the instantiated models in subsection III-A,
followed by defining the AMES Reference Architecture in subsection III-B. The
data processing in subsection III-B in then presented followed by a subsection
on hetero-functional graph theoryIII-D. The paper then presents a comparison
of the formal graphs and hetero-functional graphs network statistics for each
state in the Results section IV. The Results first starts with an analysis of
the computational intensity in section IV-A, it then presents the formal and
hetero-functional graph statistics in sections IV-B and IV-C respectfully.
Section IV continues to then present the formal graph resource distributions
in section IV-D and the HFG capability distributions in section IV-E. A
discussion of the HFG process degree distributions in section IV-F brings the
methodology section to close. Finally the paper is brought to a conclusion in
section V.

##  II Background

This section serves to situate the paper with respect to existing literature
and the ongoing trends in the field. Section II-A describes existing multi-
energy system models. Section II-B discusses the general lack of
interdependent infrastructure system data and models. Section II-C discusses
the emergy trend towards open data and source code in the multi-energy system
field. Finally, Section II-D introduces some essential concepts in HFGT that
are used in the remainder of the paper.

###  II-A Existing Multi-Energy System Models

Many existing multi-energy system models are in effect national models used to
inform national energy policy. Many such national models have an often
implicitly stated “reference energy system” which serves as the first step for
defining energy flow relationships[26, 27, 28, 29].

######  Definition 1:

\- Reference Energy System[26] As a tool to begin modeling energy systems, “it
represents a simplified and aggregated graphical representation of the entire
energy system under analysis which shows all existing and potential new energy
supply chains from primary energy to final demand.”

A reference energy system defines how the primary energy flows are processed
and converted into different energy carriers and where the energy is
ultimately utilized in end-use sectors. While the RES does identify all the
modes of energy being tracked through the model, it is driven purely by system
behavior. This means it only incorporates the system function but does not
explicitly describe system form. In contrast, the practice of MBSE requires
the definition of system function and system form[23, 24] in a graphical
modeling language like SySML. When a “reference energy system” omits the
system form, asset-level, and technology-specific interactions are ultimately
neglected. Additionally, by ignoring the form at the starting point for these
energy system models, simulation model outputs and results become less
transparent. Consequently, the absence of a description of system form further
impedes the comparability of different models because they do not explicitly
state which technologies exist and how they interact. This lack of model
comparability further impedes the comparability of differing strategies to
global climate change mitigation and adaptation. As an alternative to RES
models, MBSE and SysML utilize reference architectures as the basis of all
downstream modeling activities.

######  Definition 2:

\- Reference Architecture[36] “The reference architecture captures the essence
of existing architectures, and the vision of future needs and evolution to
provide guidance to assist in developing new instantiated system
architectures. …Such reference architecture facilitates a shared understanding
across multiple products, organizations, or disciplines about the current
architecture and the vision on the future direction. A reference architecture
is based on concepts proven in practice. Most often preceding architectures
are mined for these proven concepts. For architecture renovation and
innovation validation and proof can be based on reference implementations and
prototyping. In conclusion, the reference architecture generalizes
instantiated system architectures to define an architecture that is generally
applicable in a discipline. The reference architecture does however not
generalize beyond its discipline.”

A reference architecture specifically includes a description of system form
and function. Consequently, it makes transparent all flows of matter and
energy and which energy assets are used for transforming and transporting
these flows. Furthermore, It becomes much easier to compare energy system
models and their underlying assumptions. Finally, it is straightforward to
determine how much of a reference architecture appears in the instantiated
architecture that pertains to the geography of a specific region or case
study.

######  Definition 3:

\- Instantiated Architecture[36, 24] A case specific architecture, which
represents a real-world scenario, or an example test case. At this level, the
physical architecture consists of a set of instantiated resources, and the
functional architecture consists of a set of instantiated system processes.
The mapping defines which resources perform what processes.

The transparency and comparability of a reference architecture becomes even
more valuable in light of the open energy modeling initiative. It began in
2014 as a result of the open access movement which had begun in 2010[37, 26].
These movements originated from a recognition of the limited transparency and
reproducibility of energy systems models[38]. The open energy modeling
initiative seeks to encourage the use of open access models in research to
guide energy policies and the sustainable energy transition. While this
initiative has resulted in the production of many open access models, much
work remains to compare and converge these models towards the ultimate goals
of sustainable energy transition.

Several powerful energy system models have come out of the energy modeling
initiative including the OSeMOSYS, NEMS, and PRIMES[26, 39, 40, 41]. However,
these models each had their own weaknesses. For example, as previously
mentioned, the NEMS is extremely difficult to use making it unusable for many
open access projects. While the NEMS was a large model incorporating many
different modes of energy to inform the annual energy outlook for policy
planning, there have also been a plethora of models developed for specific
systems rather than focusing on the interconnection of multiple critical
infrastructures[15, 16, 17, 18, 19]. Alternatively, other energy system
optimization models (ESOMs) such as the EnergyScope TD have been designed for
the analysis of intermittent integration of renewable on an hourly scale.
Similarly, the Electric Power Enterprise Control Simulator addresses the
integration of variable renewable energy on multiple time-scales[42, 43].
These operational time-scale models, however, are not designed to explore the
annual transformation pathways of the sustainable energy transition[26, 44,
45]. Finally, the development of multi-energy system models did not truly gain
concerted attention until 2016[46].

###  II-B Interdependent Infrastructure System Data & Models

As multi-energy system model developed, it created a greater need for
interdependent infrastructure system data and models. In 2016, the NSF
released a call for open interdependent critical infrastructure system
data[22] that could be specifically used in system resilience and climate
change adaptation research. This call directly addresses the lack of existing
open data sources on energy systems.

In the meantime, the existing methods for organizing such data into multi-
modal energy system models had its own limitations. Multi-layer networks, for
example, was often looked at as the leading candidate to structurally model
these interdependent infrastructures. Unfortunately, multi-layer networks have
often been unable to address the explicit heterogeneity often encountered in
engineering systems[47, 48]. In a recent comprehensive review, Kivela et.al
[48] wrote: “The study of multi-layer networks $\ldots$ has become extremely
popular. Most real and engineered systems include multiple subsystems and
layers of connectivity and developing a deep understanding of multi-layer
systems necessitates generalizing ‘traditional’ graph theory. Ignoring such
information can yield misleading results, so new tools need to be developed.
One can have a lot of fun studying ‘bigger and better’ versions of the
diagnostics, models and dynamical processes that we know and presumably love –
and it is very important to do so but the new ‘degrees of freedom’ in multi-
layer systems also yield new phenomena that cannot occur in single-layer
systems. Moreover, the increasing availability of empirical data for
fundamentally multi-layer systems amidst the current data deluge also makes it
possible to develop and validate increasingly general frameworks for the study
of networks.

$\ldots$ Numerous similar ideas have been developed in parallel, and the
literature on multi-layer networks has rapidly become extremely messy. Despite
a wealth of antecedent ideas in subjects like sociology and engineering, many
aspects of the theory of multi-layer networks remain immature, and the rapid
onslaught of papers on various types of multilayer networks necessitates an
attempt to unify the various disparate threads and to discern their
similarities and differences in as precise a manner as possible.

$\ldots$ [The multi-layer network community] has produced an equally immense
explosion of disparate terminology, and the lack of consensus (or even
generally accepted) set of terminology and mathematical framework for studying
is extremely problematic.”

In this context, the NSF funded the ”American Multi-Modal Energy System
Synthetic & Simulated Data (AMES-3D)” to develop an open-source energy system
data set using modeling methods, and more specifically hetero-functional graph
theory (HFGT) that do not exhibit the limitations of multi-layer networks[47].
Consequently, this paper uses HFGT to present a hetero-functional graph
structural model (i.e. instantiated architecture) and analysis of the American
multi-modal energy system as part of the NSF-funded AMES-3D project.

###  II-C Transparency through Open Data and Source Code

In addition for the need for reference architectures (Sec. II-A), and the need
for interdependent infrastructure system data and models (Sec. II-B), the
multi-energy systems literature also recognizes a pressing need for open data
and source code. To that effect, the open modeling initiative (OMI) was
founded in 2014[37]. It was founded with the intent to promote open access
models research to guide energy policy[26]. Similarly, the open modeling
foundation (OMF) was founded in 2021[26, 49] as an “international open science
community that works to enable the next generation modeling of human and
natural systems” by making models more easily discoverable and globally
accessible[49]. The OMF seeks to create a collection of reusable,
interoperable models to study complex interactions between people and the
environment at multiple scales.

In direct alignment with the missions of the OMI and OMF is the NSF’s call for
open interdependent critical infrastructure system data as previously
mentioned. The work presented in this paper addresses these needs with
transparent open-source code and reproducible mathematics. First, the work
relies on the previously developed AMES reference architecture[25] that serves
to visualize all of the AMES components, functions, and interactions. In many
fields, “mature” reference architectures often develop into international
standards that converge and reconcile the otherwise inevitable proliferation
of models. Second, the mathematics of hetero-functional graph theory is used
to instantiate the AMES reference architecture for NY, CA, TX, and the
contiguous United States. The explicit statement of the HFGT mathematics not
only validates the work and makes it entirely reproducible, but it also makes
transparent the open-source HFGT toolbox used to produce the computational
results of this work. In all, the open data and open source commitments made
in the AMES-3D project has necessitated a research methodology that is
enhances usability, re-usability, transparency, and comparability of the
models, tools, and results.

###  II-D Hetero-functional Graph Theory

In the context of this work, hetero-functional graph theory serves to
translate and instantiate a graphical, SysML-based reference architecture into
its mathematically equivalent hetero-functional graph. The reliance on a
reference architecture grounds the work in a strong MBSE foundation.
Additional, the AMES reference architecture itself provides a clear definition
of the assets, functionality, and modes of energy that are included in the
AMES. This lossless translation from an MBSE-SysML model to a HFG maintains
the allocation of function onto form as capabilities composed of subject +
verb + object sentences where the subjects are resources, the predicates are
the processes, and the operands are the objects of the verbs. This translation
form the AMES reference architecture to a HFG (using the HFGT toolbxo) has
been well demonstrated for the single-commodity American electric power
system[31]. Now this demonstrates the same process to create an multi-energy
instantiated model of the full continguous USA for a structural (network)
analysis.

The following HFGT definitions are presented to facilitate the remainder of
the paper.

######  Definition 4 (System Operand):

[50] An asset or object $l_{i}\in L$ that is operated on or consumed during
the execution of a process.

######  Definition 5 (System Process[51, 50]):

An activity $p\in P$ that transforms a predefined set of input operands into a
predefined set of outputs.

######  Definition 6 (System Resource):

[50] An asset or object $r_{v}\in R$ that is utilized during the execution of
a process.

######  Definition 7 (Buffer[47]):

A resource $r\in R$ is a buffer $b_{s}\in B_{S}$ iff it is capable of storing
one or more operands at a unique location in space. $B_{S}=M\cup B$.

######  Definition 8 (Capability[52, 53, 54, 55, 56, 57]):

An action $e_{wv}\in{\cal E}_{S}$ (in the SysML sense) defined by a system
process $p_{w}\in P$ being executed by a resource $r_{v}\in R$. It constitutes
a subject + verb + operand sentence of the form: “Resource $r_{v}$ does
process $p_{w}$”.

##  III Methodology

As mentioned in the introduction, this paper utilizes a data driven, MBSE-
guided methodology to infer a hetero-functional graph structural model of the
American Multi-modal Energy System. This section succinctly relays this method
in four subsections:

  1. 1.

Input Data: Platts Map Data Pro

  2. 2.

Infer AMES Reference Architecture

  3. 3.

Convert GIS Shape Files to XML Data

  4. 4.

Calculate Hetero-functional Graph Structural Model

Figure 1: GIS Layers from the Platts Map Data Pro dataset for the electric
grid, natural gas system, oil system, and coal system for California (A), New
York (B), Texas (C), and the USA (D).

###  III-A Input Data: Platts Map Data Pro

Following a data driven approach, the Platts Map Data Pro [32] data set is
used to infer and instantiate the AMES Structural models. This input dataset
consists of Graphic Information System (GIS) layers for each of the four
subsystems in the AMES[32]. These geo-spacial layers include meta-data
attributes of the physical resources/facilities that compose the AMES
infrastructure in addition to their GPS coordinates. As the Platts Map Data
Pro is directed towards wholesale energy decisions, the data is limited to
transmission system resources and neglects distribution level assets. This
limitation in the dataset notably excludes retail distribution of oil and gas
(by truck). It also cuts the electric grid off at the substations excluding
distributed electric generation assets such as roof-top solar that are an
integral part of the sustainable energy transition. Nevertheless, the Platts
Map Data Pro is likely the best available dataset because it allows inferences
of not just the AMES’s form but its function too. Visualizations of the Platts
map data pro GIS layers for each state addressed in this paper and the full
USA are presented in Figure 1.

Figure 2: The top level internal block diagram of the AMES. The domestic
supply sources, the energy imports, natural environment, domestic consumption,
energy exports, and water treatment are external to the AMES four subsystems
of coal, natural gas, oil, and electric grid.

###  III-B Infer AMES Reference Architecture

The inference of a hetero-functional graph structural model of the American
Multi-modal Energy System first requires the inference of the underlying
reference architecture. In the data-driven steps that follow, the reference
architecture plays a critical role in organizing the cleaned Platts GIS data
into defined resources with proper allocated functionality. Figure 2 shows the
top-level context diagram of the AMES reference architecture and it is further
elaborated in prior work[25]. The AMES reference architecture provides a
consistent blue print from which to develop AMES models irrespective of the
choice of region or scale. It also defines all energy resources/facilities,
and the functions that they can perform. The set of operands used to track the
flows of mass and energy between its many functions are also defined.
Consequently, the AMES reference architecture supports downsteam analysis that
address both climate change mitigation and adaptation.

###  III-C Convert GIS Shape Files to XML Data

In this step, the Platts Map Data Pro GIS shape files are converted into an
associated XML file for each region (i.e. NY, CA, TX, USA) that serves as the
input for the openly available HFGT toolbox [58]. Although the Platts Map Data
Pro GIS shape files are a commercially-curated dataset, they still need
substantial cleaning and processing before being organized into an XML file.
As an immediate first step, all resources marked with a canceled status,
closed status, or illegible identifying attributes are removed. From there,
the primary obstacle is that the Platts GIS data does not guarantee physical
continuity between all of the point-type (e.g. refineries and electric power
generation facilities) and line-type (e.g. oil pipelines and electric power
lines) resources in the dataset.

A novel operand-guided geographical clustering algorithm is developed and then
applied to all four energy subsystems simultaneously. It ensures that point-
type resources with overlapping GIS coordinates are aggregated into a single
resource cluster. It also ensures that line-type resources whose endpoints are
not connected to any resource cluster (e.g. “power lines to nowhere”) are
connected to a nearby resource cluster making sure to maintain the operand
compatibility of the line-type resource to the point-type resource cluster.
For example, an oil pipeline must connect to a refinery rather than an
electric power substation. The clustering algorithm also removes any isolated
point-resources from the dataset. The geographical clustering algorithm
provides an automated means for cleaning the Platts GIS data at the formidable
scale presented by the AMES.

The clustering algorithm proceeds in several steps.  1. Using a primary
clustering distance of 0.1 miles, point-type resources and the endpoints of
line-type resources endpoints with like operand associations are sorted into
clusters and connected in a manner that respects the physical continuity of
operands.  2. Then with a secondary clustering distance of 1 mile, the
remaining isolated nodes are added to existing clusters with like operands.
3. Finally, with a tertiary clustering distance of 35 miles, the remaining
isolated nodes are connected to existing clusters with like operands via the
creation of new transportation resources.  After the execution of this
clustering algorithm, the data attributes of these resources are converted to
strings in a manner that adheres to the AMES reference architecture and the
XML file format required by the hetero-functional graph theory toolbox[58].

###  III-D Calculation of Hetero-functional Graph Structural Model

The next step is to run the HFGT toolbox[58] on this newly produced XML file
so as to produce the positive and negative hetero-functional incidence tensors
(HFITs).

######  Definition 9 (The Negative 3rd Order Hetero-functional Incidence
Tensor (HFIT) $\widetilde{\cal M}_{\rho}^{-}$):

[59, 60] The negative hetero-functional incidence tensor $\widetilde{\cal
M_{\rho}}^{-}\in\\{0,1\\}^{|L|\times|B_{S}|\times|{\cal E}_{S}|}$ is a third-
order tensor whose element $\widetilde{\cal M}_{\rho}^{-}(i,y,\psi)=1$ when
the system capability ${\epsilon}_{\psi}\in{\cal E}_{S}$ pulls operand
$l_{i}\in L$ from buffer $b_{s_{y}}\in B_{S}$.

######  Definition 10 (The Positive 3rd Order Hetero-functional Incidence
Tensor (HFIT)$\widetilde{\cal M}_{\rho}^{+}$):

[59, 60] The positive hetero-functional incidence tensor $\widetilde{\cal
M}_{\rho}^{+}\in\\{0,1\\}^{|L|\times|B_{S}|\times|{\cal E}_{S}|}$ is a third-
order tensor whose element $\widetilde{\cal M}_{\rho}^{+}(i,y,\psi)=1$ when
the system capability ${\epsilon}_{\psi}\in{\cal E}_{S}$ injects operand
$l_{i}\in L$ into buffer $b_{s_{y}}\in B_{S}$.

In the context of the AMES, the operands are flows of matter and energy like
coal, oil, natural gas, and electricity. The buffers are the point-facilities
like electric power plants and refineries. The capabilities are “subject +
verb + object” sentences such as “NG refinery refines raw natural gas”. The
HFITs are important because they include all of the information necessary to
produce:  1. a Formal Graph (FG) adjacency matrix $(A_{Bs})$ where point-
facilities are connected via edge-facilities, and  2. a Hetero-Functional
Graph (HFG) adjacency matrix $A_{\rho}$ where the system’s capabilities follow
one another.

######  Definition 11 (The Formal Graph Adjacency Matrix $A_{Bs}$):

The formal graph adjacency matrix
${A_{Bs}\in\\{0,1\\}^{|B_{s}|\times|B_{S}|}}$ is a matrix whose element
${A_{Bs}(y_{1},y_{2})=1}$ when there is a physical connection between buffer
$y_{1}$ and buffer $y_{2}$.

######  Definition 12 (Hetero-functional Adjacency Matrix[56]):

A square binary matrix $A_{\rho}$ of size $|{\cal E}_{S}|\times|{\cal E}_{S}|$
whose element $A_{\rho}(\psi_{1},\psi_{2})\in\\{0,1\\}$ is equal to one when
string $z_{\psi_{1},\psi_{2}}\in{\cal Z}$ is available and exists.

TABLE I: The computational complexity of processing GIS data and the HFGT Toolbox for the Multi-Modal Energy Systems in NY, CA, TX, and the USA. | New York | California | Texas | United States
---|---|---|---|---
Time to Assign Clusters (sec) | 2.44 | 9.9 | 166.49 | 3,963.55
Time to Process Data (sec) | 23.60 | 105.31 | 1,445.81 | 49,373.63
Time to Process Data & Write XML (sec) | 28.29 | 121.43 | 1,499.93 | 51,470.68
Time to Calculate $A_{\rho}$ (sec) | 14.49 | 20.25 | 1,252.30 | 6,171.14
Time to run HFGT Toolbox (sec) | 19.13 | 29.45 | 1,382.97 | 6,439.74
XML size (MB) | 5.6 | 13 | 192 | 420.7
HFG resulting Pickle size (MB) | 8.8 | 20.6 | 314.9 | 682.6

Each of these adjacency matrices are then calculated from the incidence
tensors. The formal graph adjacency matrix $A_{Bs}$ requires two steps. First,
the two HFITs are summed along the operand dimension to produce two incidence
matrices[59, 60]:

$\displaystyle M_{B}^{+}(y,\psi)=$ $\displaystyle\sum_{i}^{|L|}{\cal
M}^{+}_{\rho}(i,y,\psi)$ (1) $\displaystyle M_{B}^{-}(y,\psi)=$
$\displaystyle\sum_{i}^{|L|}{\cal M}^{-}_{\rho}(i,y,\psi)$ (2)

It is important to recognize that the operand heterogeneity information is
lost in this process. Then, these incidence matrices are multiplied. This
multiplication also results in a loss of information where the allocation of
function onto form in the form of capabilities is dropped to describe the
physical connection of buffers[59, 60].

$\displaystyle A_{Bs}=M_{B}^{+}M_{B}^{-T}$ (3)

In the meantime, the hetero-functional graph adjacency matrix $A_{\rho}$ is
calculated without loss of information after the HFITs have been matricized
(or flattened) into hetero-functional incidence matrices $M_{\rho}^{+}$ and
$M_{\rho}^{-}$ with dimension $|L||B_{S}|\times|{\cal E}_{S}|$[59, 60].

$\displaystyle A_{\rho}=M_{\rho}^{+T}M_{\rho}^{-}$ (4)

While the FG adjacency matrix shows the physical connections from one point-
resource (i.e. buffer) to another, the HFG adjacency matrix shows the logical
sequence of capabilities one after the other. This results in the HFG
adjacency matrix not just describing the physical connections but also the
flow of functionality on matter and energy. As previous works have shown, the
HFG allows for more comprehensive resilience analyses; be it for small
electric power distributions systems [30] or for full scale analysis of the
American electric power system [31]. The open-source HFGT toolbox [58]
provides an automated means for processing the input XML files at the
formidable scale presented by the AMES.

##  IV Results: Structural Model Statistics

Once created, the hetero-functional incidence tensor, the formal graph, and
the hetero-functional graph for the four regions of NY, CA, TX, and the full
USA can be compared. The following section presents these comparisons in the
following subsections:

  1. 1.

Computational Performance of the HFGT Toolbox

  2. 2.

Formal Graph Statistics

  3. 3.

Hetero-Functional Graph Statistics

  4. 4.

Formal Graph Resource Distribution

  5. 5.

Hetero-Functional Capability Distribution

  6. 6.

Hetero-Functional Graph Process and Operand Degree Distribution

###  IV-A Computational Performance of HFGT Toolbox

The hetero-functional graph theory toolbox[30] is used as a computational tool
that supports hetero-functional graph theory computations[47, 59, 52]. For
this work, the exceptional scale of the AMES required extensive computational
improvements to the HFGT toolbox so as to reduce computation times and memory
used to produce the XML files and HFGs. For this work, all the models and data
processing to produce the XML files was done on an iMac desktop with a 4 GHz
Quad-Core Intel Core i7 processor and 32GBs of RAM. The computation times of
major code milestones are presented in Table I. The resulting file sizes are
also presented in Table I. Note that the overwhelming majority of the
computation is devoted to converting the GIS shape files into XML files. More
specifically, automatic data cleaning and processing activities dominated the
computation. In comparison,

Interestingly, the majority of the computation time came from processing and
cleaning the GIS data. Specifically, the processing and cleaning post
assigning clusters to points takes the largest amount of time when producing
the XML. The actual assignment of clusters takes about $1/10^{th}$ of the
total time to produce the XMLs for all regions. When looking at the
computation time of the HFGT Toolbox nearly all of the required time is spent
calculating $A_{\rho}$. This calculation is ported over to Julia from Python
through the use of CSV files, $A_{\rho}$ is calculated, then ported back to
Python via another CSV. The hetero-functional adjacency matrix is at the heart
of the HFG as it does represent all flows of capabilities with the system. In
all, the HFGT toolbox computations themselves are highly optimized and are
produced in under two hours on a moderately sized computer even for the full
AMES dataset.

In line with the scale of each region, the smaller the region is, the faster
both the XML file and HFG is produced; with NY being the fastest followed by
CA, TX, and then the full USA. While CA is about twice the size of NY it took
4 times as long to produce the XML than NY. When comparing TX to NY it can be
seen that TX is 26 times as large and takes 60 times as long to produce an
XML. This suggests the computation efficiency of creating an XML is that of
$2N$. When comparing the computation times for the HFGT Toolbox, the same
trend emerges suggesting the toolbox also has a computational efficiency of
$2N$. Additionally, seeing that TX takes about a quarter of the time to
complete the HFGT toolbox as the USA, it suggests that TX composes about half
of the American energy infrastructure. The significant portion of energy
resources that TX commands is confirmed by the later sections.

###  IV-B Formal Graph Statistics

From the assessment of the computational intensity, the basic statistics of
the formal graph models for the four regions can be compared relative to the
size of their populations and land areas (Table II).

TABLE II: Formal Graph Statistics for the Multi-Modal Energy Systems in NY, CA, TX, and the USA. | New York | California | Texas | United States
---|---|---|---|---
# of Operands | 13 | 13 | 14 | 14
# of Buffers | 7,686 | 16,754 | 197,108 | 473,321
# of Edges | 9,115 | 20,674 | 179,895 | 511,802
Formal Graph Sparsity | 1.5438e-4 | 7.3661e-5 | 4.6306e-6 | 2.2848e-6
Population (millions) | 19.8 | 39.2 | 29.5 | 306.7
Land Area (sqr miles) | 54,556 | 163,696 | 268,597 | 3,119,884
Population Density(ppl/sqr mile) | 362.9 | 239.5 | 109.8 | 98.3
Buffers/Land Density (buffers/sqr mile) | 0.1409 | 0.1023 | 0.7338 | 0.1518
Buffers/Population Density (buffers/ppl) | 3.8818e-4 | 4.2740e-4 | 6.6816e-3 | 1.5432e-3
Edges/Land Density (edges/sqr mile) | 0.1671 | 0.1263 | 0.6698 | 0.1641
Edges/Population Density (edges/ppl) | 4.6035e-4 | 5.2740e-4 | 6.0981e-3 | 1.6687e-3
TABLE III: Hetero-functional Incidence Tensor Statistics for the Multi-Modal Energy Systems in NY, CA, TX, and the USA. | New York | California | Texas | United States
---|---|---|---|---
# of Operands | 13 | 13 | 14 | 14
# of Buffers | 7,686 | 16,754 | 19,7108 | 473,321
# of Capabilities | 43,766 | 100,349 | 1,430,588 | 3,130,235
# of Elements in $M_{\rho}^{+}$. | 40,921 | 94,377 | 1,422,463 | 3,028,855
# of Elements in $M_{\rho}^{-}$. | 42,778 | 99,796 | 1,425,699 | 3,070,083
HFIT Sparsity | 4.6484e-5 | 2.2221e-5 | 1.4848e-6 | 6.7354e-7
HFAM Sparsity | 1.0618e-4 | 4.8221e-5 | 2.8641e-6 | 1.4259e-6
Capability Land Density (capabilities/sqr mile) | 0.8022 | 0.6130 | 5.3262 | 1.0036
Capability Population Density (capabilities/ppl) | 2.2104e-3 | 2.5599e-3 | 4.8495e-2 | 1.0206e-2

As expected by the relative size of population and land area, Texas’s energy
infrastructure is the largest of the three states with California presenting
the second largest and New York being the smallest in terms of the number of
point-resources (buffers). The same ordering holds true for the number of
edges connecting the buffers in each of the states with TX presenting the most
edges and NY presenting the least. All three states are of course included in
the full USA model including all of their buffers and edges. There are 9,115
edges in NY, 20,674 edges in CA, 179,895 edges in TX, and 511,802 edges in the
USA. This corresponds to an adjacency matrix sparsity of 1.5438e-4, 7.3661e-5,
4.6306e-6, and 2.2848e-6 for NY, CA, TX, and the USA respectively. As
expected, the larger the system the more sparse the adjacency matrix becomes.
While the three states display energy infrastructures of differing scales,
they all show a diverse energy mixture that utilize the same 13 operands with
Texas and the USA additionally utilizing solid biomass feedstock. Also,
despite these absolute measures, NY’s population density is significantly
higher than that of Texas, California, or the USA with TX and the USA having
the lowest population densities. This means that, if all else is held equal,
New Yorkers receive more energy infrastructure benefits where Californians and
Texans and the USA must spend more in energy infrastructure costs.

Table II also presents the buffers per land and population densities. When
looking at the buffers per square mile density, CA had the smallest followed
by NY, USA, then TX reporting 0.1023, 0.1409, 0.1518, and 0.7338 respectfully.
This interesting result shows the impact that TX has on influencing the USA’s
energy infrastructure as a whole. TX being such a large percentage of the
USA’s infrastructure, it brings up the buffer per square mile density above
other states like NY or CA. This is also seen in the buffers per population
density with NY having the lowest followed by CA, USA, and TX respectfully.
These trends can also be tracked through the edges per square mile or
population density. Just as with buffers, CA has the lowest edges per square
mile density followed by NY, USA, and TX respectfully. Then when comparing the
edges per population NY and CA switch, presenting NY with the lowest density
followed by CA, USA, and TX respectfully. These density measures can be used
as a course indicator for the energy investment costs in each region. The
lower the resource per population density, the greater the proxy cost per
individual. Similarly, the lower the resource per square mile density, the
more spread out the cost is across the region’s landscape.

###  IV-C Hetero-Functional Graph Statistics

The statistics of the hetero-functional graphs for NY, CA, TX, and the USA are
presented in Table III similarly to those of the formal graphs. Trends of
sparsity in the HFITs match those of the formal graph adjacency matrices. When
comparing the number of capabilities in each model, NY has the least, followed
by CA, TX, and the USA with $(43,766)$, $(100,349)$, $(1,430,588)$, and
$(3,130,235)$ capabilities respectively. Unsurprisingly, the trend in sparsity
of the HFITs follow that of the formal adjacency matrix. NY’s HFIT is twice as
dense as CA’s, which 31 times as dense as TX’s, and which is 69 times as dense
as the USA’s HFIT. Additionally, across all regions, the negative HFIT has
more filled elements than the positive indicating a greater amount of
capabilities that pull operands from capabilities rather than injecting them.
A network topology with more capabilities that pull operands than inject
operands is synonymous with a system that gathers and and refines operands
than distributes and decomposes them. In the largest sense, this mathematical
result is consistent with our understanding of the AMES which collects raw
energy commodities (especially coal, oil, and natural gas) and delivers them
as a relatively few “high- grade” energy commodities like electricity. The
heterofunctional adjacency matrix can be calculated directly from the HFITs so
as to link capabilities togther. Again, as expected, these sparse matrices
follow the same density trend as the HFIT with NY having the most dense matrix
followed by CA, TX, and the USA with sparsity values of 1.0618e-4, 4.8221e-5,
2.8641e-6, and 1.4259e-6 respectfully.

Table III also presents the capabilities per land and population densities.
When looking at the capabilities per square mile density, CA had the smallest
followed by NY, USA, then TX reporting 0.6130, 0.8022, 1.0036, and 5.3262
respectively. The ordering of capabilities per square miles density is the
same trend as presented with buffers per square miles in the formal graph
statistics. Similarly, it shows the impact that states like TX have on
influencing the full USA’s energy infrastructure trends. Additionally, due to
the nature of allocating function onto form, there are many more capabilities
than buffers which brings the value of the density over land statistics up.
The increase in density values is also seen in the capabilities per population
density with NY having the lowest followed by CA, USA, and TX respectfully.
These density measures can be used as an indicator for the benefit of each
regions energy system. The higher the capabilities per population density, the
greater the amount of functionality (or benefits) from the infrastructure the
population experiences. Similarly, the higher the capability per square mile
density, the more infrastructure benefits can be accessed across the regions
landscape.

###  IV-D Formal Graph Resource Distribution

Figure 3: Normalized distribution of buffer types for NY, CA, TX, and the full
USA.

While it is important to assess the number of buffers (e.g. point energy
facilities) in the multi-energy infrastructure of the three American states,
it is also important to differentiate them by type. Fig. 3 shows that 74.6%,
76.8%, 69.3%, and 66% of the buffers in the formal graph are electric power
substations in the states of NY, CA, TX, and the USA respectively. The high
percentage of substations reflects the highly ubiquitous nature of the
electric power system in all four regions. Furthermore, another 13.5%, 16.8%,
4.6%, and 6.4% of buffers are devoted towards electric power generation
facilities (of various types) for NY, CA, TX, and the USA respectfully.
Because coal, oil, and natural gas are very dense approximate forms of energy
their processing facilities for these types of energy have very strong
economies of scale. Therefore, there is a trend towards centralization and a
small number of point facilities for energy conversion. California, notably,
has a greater shift toward the electric grid with a greater presence of
substations and power plants than NY and TX. Meanwhile, Texas is notably a
large producer and trader of fossil fuels in the USA, and thus has the
infrastructure to match. That NY has a greater reliance on oil and gas
facilities is likely a byproduct of it being located in a colder climate. In
opposition to the need for fossil fuels for space heating in cold
environments, both TX and CA depend more on electricity for cooling spaces in
their warmer climates and can make a greater use of renewable energy resources
for such a cause.

Figure 4: Normalized capability distribution of NY, CA, TX, and the USA.

The first massive peak around substations reflects that there are more
substations than any other type of node by far. In addition to identifying the
prominent dependence on electric power in American life, the peak also shoes
that the transmission system comes a lot closer to the grid periphery of
electric power systems than the corresponding distribution systems for coal,
oil, or natural gas. By taking a data driven approach on the Platts data, the
discrepancies of how close each systems transmission level assets reach
towards the system periphery becomes much more apparent. For the electric
grid, the system boundary ends at substations; ignoring the distributions
system assets and consumption and generation at homes and busineses. When
looking at oil and natural gas, the system boundary stops at the terminals and
ports where they are ultimately distributed outward. By taking a data-driven
approach on the Platts data, tankers and smaller gas lines delivering fuel for
residential and industrial use are also ultimately omitted. As a result, gas
stations, homes, and other retail aspects of the oil and gas industries are
not included. Additionally, with coal being sold between commercial entities
and not going out to individual consumers, the number of coal facilities
needed to distribute to the demand is quite limited. Despite the system
boundary limitations of the Platts data, the analysis speaks to the state of
the existing infrastructure and how much easier the electric power mode is to
distribute than any other. The ease of electric power transportation, along
with the potential reduction in carbon emissions, is one of the advantages to
electrifying energy demands as a part of the sustainable energy transition.

Figure 5: Normalized formal graph degree distribution of NY, CA, TX, and the
USA.

Beyond the number and type of point-energy-facilities, the formal graph also
measures their interconnectedness. Despite the heterogeneity of point-energy-
facilities and the sparsity of the four formal graphs of each state and the
full USA, Figure 5 shows that the formal graph degree distributions for the
all four regions are remarkably similar. In contrast to the well-known power-
law degree distribution for electric power systems [61, 6, 62, 63, 64], these
multi-energy systems have degree distributions with a rather unusual shape.
Notability, all four regions peak at a single degree, then have a major lack
of nodes with a degree 2. After the initial dip, the degree distribution jumps
back up to nodes more prevalent with a degree 3 before tailing off into an
exponential decay. Each energy subsystem likely contributes its own power law
degree distribution so that the degree distributions depicted in Figure Figure
5 are actually a composition of phenomena associated with the delivery of each
energy commodity. These differing contributing factors are further
investigated through the hetero-functional graph distributions reported in the
following sections.

###  IV-E Hetero-functional Capability Distributions

In addition to looking at the type of nodes present in each region, the
presence of different capabilities can also be compared. As such, the
normalized counts of the different capabilities are presented in Figure 4 for
NY, CA, TX, and the full USA. Figure 4 visualizes the prevalence of differing
energy mixtures for each region as determined by their existing capabilities.
Figure 4 shows a large spike in electric power capabilities similar to the
formal graph node distributions shown in Figure 3. Another notable trend is
that TX and the full USA have very similar capability distributions;
presumably by virtue of the large size of the TX energy system relative to the
full USA. The striking difference between TX and the USA is that the USA has
an abundance of coal resources while TX has a very minimal amount. The lack of
coal in TX is likely a result of the coal sources residing most prominently in
the Appalachian region. Additionally, NY, TX, and the USA have a heavy
reliance on natural gas functionality. All three regions have heavy spikes in
functionality related to importing and exporting gas resources. NY, as
relatively cold northeastern state, relies heavily on natural gas for space
heating during the cold winters. In contrast, TX’s natural gas functionality
comes as a result of its large oil and gas economy. The USA’s capability
distribution mixes these two dependencies. Alternatively, CA is much more
reliant on electric power generation as it has space conditioning requirements
are largely dirven by electrified cooling rather natural-gas based heating.
Notably, the functionality of solar, hydro and natural gas electric generation
are most prominent as a result of CA’s sustainable energy transition and the
flexibility of natural gas electric generation to balance variable energy
resources (VERs). These natural gas electric generation capabilities provide
valuable ramping functionality to balance the electric grid through sharp
upward and downward ramps in response to VER generation and in demand levels.
These results are further confirmed and supported by the results in Figure 6
which presents the normalized capacities of each electric generation
capability normalized by region.

Figure 6: Normalized Electric Generation Capacity by fuel source for NY, CA,
TX, and the full USA.

As the sustainable energy transition requires an electrification of the
demands placed on the AMES, it becomes important to investigate the electric
power generation mix. In accordance with single energy mode analysis, when the
electric power generation capabilities in Figure 4 are weighted by generation
capacity, it produces Figure 6 which shows the electric generation mix for the
three states and the USA. The main source of generation is found to be natural
gas for all four presented regions. The high presence of natural gas
generation indicates that the need for quick response generation sources to
support VERs for NY, CA, TX, and the USA in addition to low fuel costs has
lead to a large dependency on natural gas for electric power generation.
Additionally, NY sees a larger generation capacity from processed oil and
nuclear power facilities than the other regions. Again, the reliance on oil-
fired generation stems from the need for “peaker” units that support
electrified heating in the coldest winter periods.CA’s commitment to renewable
energy generation is demonstrated in its solar and hydro capacity. While Texas
has a large investment in wind power. Both TX and the USA also have
significant capacity in coal-fired electric power generation. In contrast,
Figure 6 shows CA’s and NY’s energy transition to cleaner fuel sources and
away from coal. Finally, the electric power generation mix in Figure 6
demonstrates that when HFGT is restricted to a single commodity network (e.g
electricity), it is able to reproduce the familiar analytics associated with
individual (and often siloed) infrastructures.

Just as the degree distributions of the AMES FGs can be presented so can the
degree distributions for the HFGs. Figures 7(a) and 7(b) show the AMES’ HFG
in-degree and out-degree distributions respectively. As HFGs describe the
logical sequence of capabilities, they are fundamentally directed in nature.
As such, both the in-degree and out-degree must be presented. These plots
present similar trends to those seen in the FG degree distributions shown in
Figure 5. Neither the FG degree distribution nor the HFG degree distributions
follow a traditional power law decay, but the long tails in each plot suggest
that there is some underlying power law behaviors. From the HFG degree
distributions, it is clear that there is some combination of exponentially
decaying degree distributions from each of the four subsystems composing the
AMES. Interestingly, the general shape of both the in and out degree plots are
very similar. Both plots have normalized distributions that bottom out by
degree 20 and have peaks at degrees 2 and 4. Additionally, all four regions
also have a major dip in their degree distributions at degree 3. While the
four regions all seem to have some underlying power law behaviors, each region
presents their own combination of decays. The differing degree distribution
combinations suggest that there are geographical dependencies that influence
the degree distributions. Ultimately, while the results in Figure 7 confirm
those of Figure 5, the HFG model invites further investigation (in the
following section) by classifying the HFG’s capabilities by their underling
process.

(a) In-degree

(b) Out-degree

Figure 7: Normalized hetero-functional graph degree distribution of NY, CA,
TX, and the USA.

###  IV-F Hetero-functional Graph Process Degree Distribution

Figures 8 and 9 present three dimensional (3D) degree distributions of the
AMES’ HFG where the third dimension classifies the node-capabilities by their
underlying process. This choice of classification is of critical analytical
importance. As discussed extensively in prior works[47, 59], capabilities can
be associated with either point-type buffers or line-type transportation
resources. Therefore, a classification scheme based on buffers is both
incomplete and logically inconsistent with the nature of a HFG. In the
meantime, a classification schmee based on the capabilities’ (input or output)
operands runs the risk of double counting the capabilities because each
capability can have more than one type of operand (e.g. a natural gas electric
power plant). Therefore, the most straightforward way of classifying
capabilities into mutually-exclusive and totally exhaustive sets while
preserving the objectivity of the underlying statistics is based upon the
capabilities’ underlying process.

Figure 8: In-Degree distribution of all Processes Figure 9: Out-Degree
distribution of all Processes

Returning to Figures 8 and 9, the former presents the 3D in-degree
distribution while the latter presents the 3D out-degree distribution. Both
figures reveal, for the first time, the underlying nature of the AMES’ hetero-
functional structure. With rare exception, the AMES exhibits a power-law
degree distribution for each set of capabilities classified by process. These
figures provide a fascinating empirical result. Previous works demonstrate
that HFGs of single operand networks have power law degree distributions just
like formal graphs of single operand networks[30, 31]. Now, this paper shows
that a HFG of a multi-operand network exhibits a power law degree distribution
for each type of process so that the final HFG degree distribution in Figure 7
shows a superposition of the degree distributions associated with each type of
process. In contrast, such an empirical result can not be reached using a
formal graph with buffer-nodes because of the information loss caused by the
sums in Eq. 1 and 2. In other words, HFGT has confirmed well-known results in
the network science literature and successfully generalized them for systems
that are fundamentally hetero-functional in nature.

This theoretical insight has direct practical relevance to the fundamental
systems science principles underlying the sustainable energy transition. The
3D degree distributions in Figures 8 and 9 show that the AMES’ structure is
evolving at different rates depending on the underlying process of each
capability. In other words, from a graph theory perspective, the sustainable
energy transition can be understood as _network phenomena_ where the
decommissioning of carbon-intensive processes (e.g. oil and gas refining and
coal-fired electric power generation) is occurring at a certain rate, and
carbon-light processes (e.g. wind and solar power generation) is occurring at
a different rate. Similarly, broad trends toward electrification (or fuel
switching) is a superposition of the network phenomena driving the consumption
of carbon-intensive fuels relative to consuming electricity. Said differently,
HFGT points to the network phenomena underpinning “infrastructure lock-in”
like homeowners who wish switch to heat-pump technology but find themselves
“locked in” to furnace-based heating. Similarly, HFGT points to the network
phenomena underpinning “infrastructure platforms” like car owners who wish to
buy an electric vehicle but find themselves worrying about local charging
infrastructure. Ultimately, a deeper study of the power laws in Figures 8 and
9 is warranted because the relative rates underpin the success of the
sustainable energy transition. From a policy perspective, the sustainable
energy transition requires that the rate of integrating certain types of
sustainable energy capabilities be “sped-up” relative to more carbon intensive
ones.

##  V Conclusions and Policy Implications

This paper uses a data-driven, MBSE-guided approach to develop open-source
software that produces open structural models of the American Multi-modal
Energy System. It is part of a larger NSF project entitled “American Multi-
Modal Energy System Synthetic & Simulated Data (AMES-3D)” which seeks to
produce open-source structural and behavioral models of the American Multi-
modal Energy System. The creation of open-source software and open-data models
of the AMES fills an important need in open, citizen-based science in
America’s sustainable energy transition. It also provides one of the few
multi-energy system datasets in which to advance fundamental methods. The
AMES’ structural models are inferred from the Platts Map Data Pro GIS dataset
and is complemented by the previously developed American Multi-modal Energy
System Reference Architecture [25]. Together, these two data sources serve as
the basis for an XML-based input data file for the open-source hetero-
functional graph theory toolbox.

This paper specifically reports the hetero-functional incidence tensor, the
formal graph adjacency matrix and hetero-functional graph adjacency matrix
statistics for the multi-energy infrastructure systems for the states of NY,
CA, TX, and the full USA. Here, the application of hetero-functional graph
theory facilitates a nuanced analysis that respects the heterogeneity in this
highly interdependent system-of-systems. The paper finds that the geography
and sustainable energy policies of the states are deeply reflected in the
structure of their multi-energy infrastructure. Because New York’s cold north
eastern climate drives heating demand, it has a multi-energy system with a
greater emphasis on oil and gas. In the meantime, California’s warm climate is
reflected in a multi-energy system with a greater emphasis on electric power
systems. Additionally, Texas has a large oil and gas economy and thus has a
large percentage of energy infrastructure pertaining to these fossil fuels but
also has build out of a tremendous amount of wind energy for electric
generation. Along these lines, California and Texas have also geared their
natural gas resources infrastructure towards electric power generation to
support their growing reliance on variable energy resources. These trends
appear as components of the AMES as a whole. Additionally, states with a large
energy infrastructure like Texas have a greater impact on shaping the USA’s
energy infrastructure. It is also important to note the import and export
functionality of these fuel sources are also very prominent. Identifying the
abundance of import and export functionality is important for the evolution of
a multi-modal energy system for two reasons. Import and export functionality
provides open interfaces to new modes of energy delivery such as hydrogen.
Important export functionality in a HFG also creates a theoretical foundation
from which to investigate energy systems with relatively large import/export
economies (e.g. Australia). Finally, through the analysis of HFG degree
distributions and their associated processes, it becomes clear that power-law
degree distributions are fundamentally tied to the processes rather than
point-resources (i.e. buffers).

From the perspective of understanding the dynamics of the sustainable energy
transition, the power laws imply that for each type of process, the ”popular”
(e.g. most well-connected) become even more so. The degree distribution power
law for each process reinforces the connections of the capabilities with more
connected capabilities. The AMES’ HFG has a power law associated with every
type of process and each power law reflects a rate of evolution or adoption.

Consequently, many new sustainable energy process technologies may require
policies, at least initially, that support them relative to incumbent process
technologies. These initial policies are likely to instigate network-driven
positive feedback loops that accelerate adoption rates. For example, as
renewable energy sources are adopted, it expands the electric grid itself,
which in turn supports electrification on the AMES’ demand side. These
implications are also applicable to hydrogen as it becomes a more viable and
prominent technology. As this work was driven by the Platts GIS data, it was
ultimately confined to incumbent conventional energy sources, rather than new
energy pathways tied to hydrogen, synthetic fuels and bio-energy. Naturally,
the transparent natures of the AMES reference architecture lends itself to
revision so as to include these new energy carriers based upon first-principle
process physics. Finally, the power law results found in Figures 8 and 9
present a ripe opportunity for further investigation.

Beyond the structural analysis presented here, the AMES’ structural model
presents multiple avenues for future open-science research. First, behavioral
data can be incorporated so as to develop a physically-informed machine-
learning behavioral model of the AMES. Secondly, the AMES can be studied
rigorously for its sustainability and resilience properties using novel
methods rooted in hetero-functional graph theory. Finally, scenarios varying
the energy mixtures seen in these regions and across the full USA can be
investigated to explore potential sustainable energy transitions.

##  VI Acknowledgement

This work is supported by funding from the National Science Foundation.

## References

  * [1] E. J. Moniz. (2022, July) A way forward on the climate crisis energy insecurity. [Online]. Available: https://www.bostonglobe.com/2022/07/18/opinion/way-forward-climate-crisis-energy-insecurity/
  * [2] C. Action, “Framework convention on climate change,” 2010.
  * [3] WEC, “World energy trilemma 2016: Defining measures to accelerate the energy transition,” 2016.
  * [4] G. Bridge, S. Bouzarovski, M. Bradshaw, and N. Eyre, “Geographies of energy transition: Space, place and the low-carbon economy,” _Energy policy_ , vol. 53, pp. 331–340, 2013.
  * [5] A. Vespignani, “The fragility of interdependency,” _Nature_ , vol. 464, no. 7291, pp. 984–985, 2010. [Online]. Available: https://doi.org/10.1038/464984a
  * [6] R. Albert, I. Albert, and G. L. Nakarado, “Structural vulnerability of the north american power grid,” _Physical review E_ , vol. 69, no. 2, p. 025103, 2004.
  * [7] G. Dong, R. Du, L. Tian, and R. Liu, “Robustness of network of networks with interdependent and interconnected links,” _Physica A: Statistical Mechanics and its Applications_ , vol. 424, pp. 11–18, 2015.
  * [8] P. Jean-Baptiste and R. Ducroux, “Energy policy and climate change,” _Energy Policy_ , vol. 31, no. 2, pp. 155–166, Jan 2003.
  * [9] E. Kriegler, G. Luderer, N. Bauer, L. Baumstark, S. Fujimori, A. Popp, J. Rogelj, J. Strefler, and D. P. Van Vuuren, “Pathways limiting warming to 1.5c: a tale of turning around in no time?” _Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences_ , vol. 376, no. 2119, p. 20160457, 2018.
  * [10] Robert Lempert, Benjamin L. Preston, Jae Edmonds, Leon Clarke, Tom Wild, Matthew Binsted, Elliot Diringer, Brad Townsend, “Pathways to 2050 alternative scenarios for decarbonizing the u.s. economy,” Center for Climate and Energy Solutions, Climate Innovation 2050, 2019.
  * [11] J. Rogers, K. Averyt, S. Clemmer, M. Davis, F. Flores-Lopez, D. Kenney, J. Macknick, N. Madden, J. Meldrum, S. Sattler, and E. Spanger-Siegfried, “Water-Smart Power: Strengthening the U.S. Electricity System in a Warming World,” Union for Concerned Scientists, Cambridge, MA, Tech. Rep., 2013.
  * [12] P. Lara, M. Sánchez, and J. Villalobos, “Enterprise modeling and operational technologies (ot) application in the oil and gas industry,” _Journal of Industrial Information Integration_ , vol. 19, p. 100160, 2020\. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S2452414X20300352
  * [13] M. Panteli, E. A. M. Ceseña, R. Moreno, and P. Mancarella, “Infrastructure planning under uncertainty: flexibility, resilience and multi-energy systems application tutorial,” 2019.
  * [14] D. Mejia-Giraldo, J. Villarreal-Marimon, Y. Gu, Y. He, Z. Duan, and L. Wang, “Sustainability and resiliency measures for long-term investment planning in integrated energy and transportation infrastructures,” _Journal of Energy Engineering_ , vol. 138, no. 2, pp. 87–94, Jun. 2012. [Online]. Available: http://dx.doi.org/10.1061/(ASCE)EY.1943-7897.0000067
  * [15] G. M. Masters, _Renewable and efficient electric power systems_. John Wiley & Sons, 2013.
  * [16] S. Mokhatab and W. A. Poe, _Handbook of natural gas transmission and processing_. Gulf professional publishing, 2012.
  * [17] M. V. Lurie, _Modeling of Oil Product and Gas Pipeline Transportation_. Wiley-VCH Verlag GmbH & Co. KGaA, 2009.
  * [18] EIA, “Coal market module of the national energy modeling system: Model documentation 2014,” _Independent Statistics and Analysis, U.S. Energy Information Administration, Department of Energy_ , 2014.
  * [19] E. Priyanka, S. Thangavel, X.-Z. Gao, and N. Sivakumar, “Digital twin for oil pipeline risk estimation using prognostic and machine learning techniques,” _Journal of Industrial Information Integration_ , p. 100272, 2021. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S2452414X21000704
  * [20] EIA, “Annual energy outlook 2020,” _Independent Statistics and Analysis, U.S. Energy Information Administration, Department of Energy_ , 2020.
  * [21] ——, “Availability of the national energy modeling system (nems) archive,” United States Energy Information Administration, Tech. Rep., 2017. [Online]. Available: https://www.eia.gov/outlooks/aeo/info_nems_archive.cfm
  * [22] J. K. Barry W. Johnson, “Dear colleague letter: Simulated andsynthetic data for infrustructure modeling,” March 2017.
  * [23] D. Dori, _Model-based systems engineering with OPM and SysML_. Springer, 2015.
  * [24] S. Friedenthal, A. Moore, and R. Steiner, _A Practical Guide to SysML: The Systems Modeling Language_ , 2nd ed. Burlington, MA: Morgan Kaufmann, 2011.
  * [25] D. J. Thompson and A. M. Farid, “A reference architecture for the american multi-modal energy system,” _arXiv preprint arXiv:2012.14486_ , 2020.
  * [26] F. Plazas-Niño, N. Ortiz-Pimiento, and E. Montes-Páez, “National energy system optimization modelling for decarbonization pathways analysis: A systematic literature review,” _Renewable and Sustainable Energy Reviews_ , vol. 162, p. 112406, 2022. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S1364032122003148
  * [27] A. Millot, A. Krook-Riekkola, and N. Maïzi, “Guiding the future energy transition to net-zero emissions: Lessons from exploring the differences between france and sweden,” _Energy Policy_ , vol. 139, p. 111358, 2020. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0301421520301154
  * [28] A. Dedinec, V. Taseska-Gjorgievska, N. Markovska, T. Obradovic Grncarovska, N. Duic, J. Pop-Jordanov, and R. Taleski, “Towards post-2020 climate change regime: Analyses of various mitigation scenarios and contributions for macedonia,” _Energy_ , vol. 94, pp. 124–137, 2016. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S036054421501453X
  * [29] D. Yangka and M. Diesendorf, “Modeling the benefits of electric cooking in bhutan: A long term perspective,” _Renewable and Sustainable Energy Reviews_ , vol. 59, pp. 494–503, 2016. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S1364032115016482
  * [30] D. J. Thompson, W. C. Schoonenberg, and A. M. Farid, “A hetero-functional graph analysis of electric power system structural resilience,” in _2020 IEEE Power Energy Society Innovative Smart Grid Technologies Conference (ISGT)_ , 2020, pp. 1–5.
  * [31] ——, “A hetero-functional graph resilience analysis of the future american electric power system,” _IEEE Access_ , vol. 9, pp. 68 837–68 848, 2021\.
  * [32] Platts, “Platts energy map data pro,” S&P Global Platts, Tech. Rep., 2017. [Online]. Available: https://www.spglobal.com/platts/en/products-services/oil/map-data-pro
  * [33] EIA, “California state energy profile,” _EIA U.S. States Overview_ , February 2021.
  * [34] ——, “New york state energy profile,” _EIA U.S. States Overview_ , October 2021.
  * [35] ——, “Texas state energy profile,” _EIA U.S. States Overview_ , May 2022\.
  * [36] R. Cloutier, G. Muller, D. Verma, R. Nilchiani, E. Hole, and M. Bone, “The concept of reference architectures,” _Systems Engineering_ , vol. 13, no. 1, pp. 14–27, 2010.
  * [37] S. Oberle and R. Elsland, “Are open access models able to assess today’s energy scenarios?” _Energy Strategy Reviews_ , vol. 26, p. 100396, 2019. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S2211467X19300896
  * [38] P. Lopion, P. Markewitz, M. Robinius, and D. Stolten, “A review of current challenges and trends in energy systems modeling,” _Renewable and Sustainable Energy Reviews_ , vol. 96, pp. 156–166, 2018. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S1364032118305537
  * [39] M. Howells, H. Rogner, N. Strachan, C. Heaps, H. Huntington, S. Kypreos, A. Hughes, S. Silveira, J. DeCarolis, M. Bazillian, and A. Roehrl, “Osemosys: The open source energy modeling system: An introduction to its ethos, structure and development,” _Energy Policy_ , vol. 39, no. 10, pp. 5850–5870, 2011, sustainability of biofuels. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0301421511004897
  * [40] EIA, “The national energy modeling system: An overview 2018,” _Independent Statistics and Analysis, U.S. Energy Information Administration, Department of Energy_ , April 2019.
  * [41] E3MLab, “Primes model version 2018 detailed model description,” _E3MLAB_ , 2018.
  * [42] A. Muzhikyan, S. O. Muhanji, G. Moynihan, D. Thompson, Z. Berzolla, and A. M. Farid, “ISO New England System Operational Analysis and Renewable Energy Integration Study: Final Project Report,” Laboratory for Intelligent Integrated Networks of Engineering Systems. Thayer School of Engineering at Dartmouth., Hanover, NH, USA, Tech. Rep., 2018. [Online]. Available: http://engineering.dartmouth.edu/liines/resources/TechReports/SPG-W07.pdf
  * [43] S. O. Muhanji, C. Barrows, J. Macknick, and A. M. Farid, “An Enterprise Control Assessment Case Study of the Energy-Water Nexus for the ISO New England System,” _Renewable and Sustainable Energy Reviews_ , vol. 141, pp. 110 766–110 785, 2021. [Online]. Available: https://doi.org/10.1016/j.rser.2021.110766
  * [44] J. N. P. van Stralen, F. Dalla Longa, B. W. Daniëls, K. E. L. Smekens, and B. van der Zwaan, “Opera: a new high-resolution energy system model for sector integration research,” _Environmental Modeling & Assessment_, vol. 26, no. 6, pp. 873–889, 2021. [Online]. Available: https://doi.org/10.1007/s10666-020-09741-7
  * [45] G. Limpens, S. Moret, H. Jeanmart, and F. Maréchal, “Energyscope td: A novel open-source model for regional energy systems,” _Applied Energy_ , vol. 255, p. 113729, 2019. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0306261919314163
  * [46] C. B. Heendeniya, A. Sumper, and U. Eicker, “The multi-energy system co-planning of nearly zero-energy districts – status-quo and future research potential,” _Applied Energy_ , vol. 267, p. 114953, 2020. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0306261920304657
  * [47] W. C. Schoonenberg, I. S. Khayal, and A. M. Farid, _A Hetero-functional Graph Theory for Modeling Interdependent Smart City Infrastructure_. Berlin, Heidelberg: Springer, 2019. [Online]. Available: http://dx.doi.org/10.1007/978-3-319-99301-0
  * [48] M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, and M. A. Porter, “Multilayer networks,” _Journal of complex networks_ , vol. 2, no. 3, pp. 203–271, 2014.
  * [49] (2022, July). [Online]. Available: https://openmodelingfoundation.github.io/about/
  * [50] SE Handbook Working Group, _Systems Engineering Handbook: A Guide for System Life Cycle Processes and Activities_. International Council on Systems Engineering (INCOSE), 2015.
  * [51] D. Hoyle, _ISO 9000 pocket guide_. Oxford ; Boston: Butterworth-Heinemann, 1998. [Online]. Available: http://www.loc.gov/catdir/toc/els033/99163006.html
  * [52] A. M. Farid and D. C. McFarlane, “A Development of Degrees of Freedom for Manufacturing Systems,” in _IMS’2006: 5th International Symposium on Intelligent Manufacturing Systems: Agents and Virtual Worlds_ , Sakarya, Turkey, 2006, pp. 1–6. [Online]. Available: http://engineering.dartmouth.edu/liines/resources/Conferences/IEM-C02.pdf
  * [53] A. M. Farid, “Reconfigurability Measurement in Automated Manufacturing Systems,” Ph.D. Dissertation, University of Cambridge Engineering Department Institute for Manufacturing, 2007. [Online]. Available: http://engineering.dartmouth.edu/liines/resources/Theses/IEM-TP00.pdf
  * [54] A. M. Farid and D. C. McFarlane, “Production degrees of freedom as manufacturing system reconfiguration potential measures,” _Proceedings of the Institution of Mechanical Engineers, Part B (Journal of Engineering Manufacture) – invited paper_ , vol. 222, no. B10, pp. 1301–1314, 2008. [Online]. Available: http://dx.doi.org/10.1243/09544054JEM1056
  * [55] A. M. Farid, “Product Degrees of Freedom as Manufacturing System Reconfiguration Potential Measures,” _International Transactions on Systems Science and Applications – invited paper_ , vol. 4, no. 3, pp. 227–242, 2008. [Online]. Available: http://engineering.dartmouth.edu/liines/resources/Journals/IEM-J04.pdf
  * [56] ——, “Static Resilience of Large Flexible Engineering Systems: Axiomatic Design Model and Measures,” _IEEE Systems Journal_ , vol. PP, no. 99, pp. 1–12, 2015. [Online]. Available: http://dx.doi.org/10.1109/JSYST.2015.2428284
  * [57] ——, “An engineering systems introduction to axiomatic design,” in _Axiomatic Design in Large Systems: Complex Products, Buildings & Manufacturing Systems_, A. M. Farid and N. P. Suh, Eds. Berlin, Heidelberg: Springer, 2016, ch. 1, pp. 1–47. [Online]. Available: http://dx.doi.org/10.1007/978-3-319-32388-6
  * [58] D. Thompson, P. Hegde, W. C. H. Schoonenberg, I. Khayal, and A. M. Farid, “The hetero-functional graph theory toolbox,” 2020.
  * [59] A. M. Farid, D. Thompson, P. Hegde, and W. Schoonenberg, “A tensor-based formulation of hetero-functional graph theory,” _arXiv preprint arXiv:2101.07220_ , 2021.
  * [60] D. Thompson and A. M. Farid, “Reconciling formal, multi-layer, and hetero-functional graphs with the hetero-functional incidence tensor,” in _IEEE Systems of Systems Engineering Conference_ , Rochester, NY, 2022, pp. 1–6.
  * [61] L. A. N. Amaral, A. Scala, M. Barthelemy, and H. E. Stanley, “Classes of small-world networks,” _Proceedings of the national academy of sciences_ , vol. 97, no. 21, pp. 11 149–11 152, 2000.
  * [62] R. Albert, H. Jeong, and A.-L. Barabási, “Error and attack tolerance of complex networks,” _Nature_ , vol. 406, no. 6794, pp. 378–382, Jul 2000\.
  * [63] M. Barthélemy, “Spatial networks,” _Physics Reports_ , vol. 499, no. 1, pp. 1–101, 2011.
  * [64] A.-L. Barabási and R. Albert, “Emergence of scaling in random networks,” _science_ , vol. 286, no. 5439, pp. 509–512, 1999.

## Appendix

TABLE IV: This table presents the normalization factors for figures 3-7 for the regions NY, CA, TX, and the USA. | New York | California | Texas | United States
---|---|---|---|---
Normalization Factor Figure 3 | 3.15e-4 | 1.37e-4 | 1.14e-4 | 7.93e-6
Normalization Factor Figure 4 | 1.30e-4 | 5.97e-5 | 5.07e-6 | 2.11e-6
Normalization Factor Figure 5 | 2.27e-4 | 1.17e-4 | 6.88e-5 | 5.60e-6
Normalization Factor Figure 6 | 2.71e-5 | 1.52e-5 | 8.59e-6 | 9.25e-7
Normalization Factor Figure 7A | 2.29e-5 | 9.96e-6 | 6.99e-7 | 3.24e-7
Normalization Factor Figure 7B | 2.29e-5 | 9.96e-6 | 6.99e-7 | 3.24e-7
TABLE V: This table presents the normalization factors for figures 8 and 9 for the regions NY, CA, TX, and the USA. | NY In | NY Out | CA In | CA Out | TX In | TX Out | USA In | USA Out
---|---|---|---|---|---|---|---|---
Generate Electric Power from Water Energy | 5.62e-3 | 5.62e-3 | 3.60e-3 | 3.60e-3 | 4.76e-2 | 4.76e-2 | 7.70E-4 | 7.70E-4
Generate Electric Power from Processed Gas | 6.17e-3 | 6.17e-3 | 2.83e-3 | 2.83e-3 | 5.05e-3 | 5.05e-3 | 5.20E-4 | 5.20E-4
Generate Electric Power from Processed Oil | 4.17e-2 | 4.17e-2 | 0.125 | 0.125 | 0.33 | 0.33 | 1.63E-3 | 1.63E-3
Generate Electric Power from Uranium | 0.25 | 0.25 | 1 | 1 | 0.5 | 0.5 | 1.96E-2 | 1.96E-2
Generate Electric Power from Coal | 0.5 | 0.5 | 0.2 | 0.2 | 5.26e-2 | 5.26e-2 | 3.31E-3 | 3.31E-3
Generate Electric Power from Other | 5.26e-2 | 5.26e-2 | 1.47e-2 | 1.47e-2 | 5.88e-2 | 5.88e-2 | 2.97E-3 | 2.97E-3
Generate Electric Power from Wind Energy | 2.56e-2 | 2.56e-2 | 9.43e-3 | 9.43e-3 | 7.69e-3 | 7.69e-3 | 2.10E-3 | 2.10E-3
Generate Electric Power from Solar | 5.26e-2 | 5.26e-2 | 2.39e-3 | 2.39e-3 | 5.56e-2 | 5.56e-2 | 1.56E-3 | 1.56E-3
Consume Electric Power | 4.23e-4 | 4.23e-4 | 1.79e-4 | 1.79e-4 | 1.65e-4 | 1.65e-4 | 1.56E-4 | 1.56E-4
Compress processed gas | 2.38e-2 | 2.38e-2 | 4.22e-3 | 4.22e-3 | 3.81e-3 | 3.81e-3 | 1.19E-2 | 1.19E-2
Compress Syngas | 2.38e-2 | 2.38e-2 | 4.22e-3 | 4.22e-3 | 3.81e-3 | 3.81e-3 | 1.19E-2 | 1.19E-2
Compress Raw Gas | 2.38e-2 | 2.38e-2 | 4.22e-3 | 4.22e-3 | 3.81e-3 | 3.81e-3 | 1.19E-2 | 1.19E-2
Import Processed Gas | 3.94e-3 | 3.94e-3 | 1.16e-2 | 1.16e-2 | 7.11e-4 | 7.11e-4 | 2.45E-3 | 2.45E-3
Import Syngas | 3.94e-3 | 3.94e-3 | 1.16e-2 | 1.16e-2 | 7.11e-4 | 7.11e-4 | 2.45E-3 | 2.45E-3
Import Raw Gas | 3.94e-3 | 3.94e-3 | 1.16e-2 | 1.16e-2 | 7.11e-4 | 7.11e-4 | 2.45E-3 | 2.45E-3
Export Processed Gas | 3.94e-3 | 3.94e-3 | 1.16e-2 | 1.16e-2 | 7.11e-4 | 7.11e-4 | 2.45E-3 | 2.45E-3
Import Crude Oil | 1.35e-2 | 1.35e-2 | 1.02e-2 | 1.02e-2 | 4.63e-3 | 4.63e-3 | 2.13E-2 | 2.13E-2
Export Crude Oil | 1.35e-2 | 1.35e-2 | 1.02e-2 | 1.02e-2 | 4.63e-3 | 4.63e-3 | 2.13E-2 | 2.13E-2
Import Processed Oil | 1.35e-2 | 1.35e-2 | 1.02e-2 | 1.02e-2 | 4.63e-3 | 4.63e-3 | 2.13E-2 | 2.13E-2
Export Processed Oil | 1.35e-2 | 1.35e-2 | 1.02e-2 | 1.02e-2 | 4.63e-3 | 4.63e-3 | 2.17E-2 | 2.17E-2
Import Liquid Biomass feedstock | 1.35e-2 | 1.35e-2 | 1.10e-2 | 1.10e-2 | 4.83e-3 | 4.83e-3 | 2.22E-2 | 2.22E-2
Export Liquid Biomass feedstock | 1.35e-2 | 1.35e-2 | 1.10e-2 | 1.10e-2 | 4.83e-3 | 4.83e-3 | 2.22E-2 | 2.22E-2
Import Coal | 0.25 | 0.25 | 0.125 | 0.125 | 1.82e-2 | 1.82e-2 | 2.66E-3 | 2.66E-3
Export Coal | 0.33 | 0.33 | 0.17 | 0.17 | 6.67e-2 | 6.67e-2 | 0.05 | 0.05
Process Raw Gas | 0 | 0 | 5.26e-2 | 5.26e-2 | 3.10e-3 | 3.10e-3 | 0.04 | 0.04
Process Crude Oil | 0 | 0 | 5.26e-2 | 5.26e-2 | 3.85e-2 | 3.85e-2 | 0.2 | 0.2
Transport Electric Power | 9.95e-5 | 9.95e-5 | 3.44e-5 | 3.44e-5 | 1.31e-4 | 1.31e-4 | 7.73E-5 | 7.73E-5
Transport Processed Gas | 1.52e-4 | 1.52e-4 | 6.43e-5 | 6.43e-5 | 1.63e-5 | 1.63e-5 | 2.07E-4 | 2.07E-4
Transport Syngas | 1.65e-4 | 1.65e-4 | 6.77e-5 | 6.77e-5 | 1.63e-5 | 1.63e-5 | 2.09E-4 | 2.09E-4
Transport Raw Gas | 1.65e-4 | 1.65e-4 | 6.77e-5 | 6.77e-5 | 1.63e-5 | 1.63e-5 | 2.07E-4 | 2.07E-4
Transport Crude Oil | 8.47e-3 | 8.47e-3 | 4.82e-4 | 4.82e-4 | 2.73e-4 | 2.73e-4 | 2.11E-3 | 2.11E-3
Transport Processed Oil | 3.23e-3 | 3.23e-3 | 5.92e-4 | 5.92e-4 | 5.37e-3 | 5.37e-3 | 1.05E-2 | 1.05E-2
Transport Liquid Biomass Feedstock | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Transport Coal | 9.83e-5 | 9.83e-5 | 7.17e-5 | 7.17e-5 | 1.14e-3 | 1.14e-3 | 5.16E-4 | 5.16E-4
Transport Water Energy | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Transport Other | 0 | 0 | 0 | 0 | 0.01 | 0.01 | 4.76E-2 | 4.76E-2
Transport Solid Biomass Feedstock | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Total | 2.28e-5 | 2.28e-5 | 9.97e-6 | 9.97e-6 | 6.99e-7 | 6.99e-7 | 3.24E-7 | 3.24E-7