# Graph-Based Optimization for Technology Pathway Analysis:
A Case Study in Decarbonization of University Campuses

Blake Lopez†, Jiaze Ma†, Victor M. Zavala†

†Department of Chemical and Biological Engineering
University of Wisconsin - Madison, 1415 Engineering Dr, Madison, WI 53706, USA
Corresponding Author<EMAIL_ADDRESS>
###### Abstract

Industrial sectors such as urban centers, chemical companies, manufacturing
facilities, and microgrids are actively exploring strategies to help reduce
their carbon footprint. For instance, university campuses are complex urban
districts (involving collections of buildings and utility systems) that are
seeking to reduce carbon footprints that originate from diverse activities
(e.g., transportation operations and production of heating, cooling, and power
utilities). This work presents an optimization framework to identify
technology pathways that enable decarbonization of complex industrial sectors.
The framework uses a graph abstraction that compactly captures
interdependencies between diverse products and technologies as well as diverse
externalities (e.g., market, policy, and carbon prices). Duality analysis
reveals that the formulation can be interpreted as an economy, market, or
value chain that uses technologies to generate economic value (wealth) by
transforming basic products into higher value products. This interpretation
also reveals that the formulation identifies pathways that maximize the profit
of stakeholders, helps reveal the inherent value (prices) of intermediate
products, and helps analyze the impact of externalities and technology
specifications on product values. Our developments are illustrated via a case
study involving a prototypical university campus that seeks to identify
pathways that reduce its carbon footprint (e.g., via electrification and
deployment of hydrogen technologies). We use the framework to determine carbon
tax values, technology specifications, and investment budgets that activate
different technology pathways and that achieve different levels of
decarbonization.

Keywords: optimization; technology pathways; graph theory; decarbonization

##  1 Introduction

The global mean surface temperature reached 1℃ above pre-industrial times in
2017 and is projected to reach 1.5℃ as early as 2030 [31]. Significant efforts
need to be made to decrease greenhouse gas emissions to limit warming. The
most notable effort to accomplish this is the Paris agreement, which seeks to
limit warming to at most 2℃ by providing technological and financial support
as well as capacity building between the 192 countries in the agreement [34].
To satisfy the goals of global agreements, industrial sectors such as chemical
companies, manufacturing facilities, farms, microgrids, and urban centers need
to shift their operations so as to minimize CO2 emissions. For instance,
ammonia production facilities are exploring the implementation of electrolysis
technologies that harness renewable power to replace hydrogen sourced from
natural gas, and utility companies are relying on falling renewable costs and
energy storage technologies to mitigate carbon emissions [29, 43, 42].

An important and representative set of industrial systems that is actively
seeking to decarbonize operations are university campuses. To give some
perspective into the magnitude of these systems, we note that there are over
16.5 million students in the US reside on university campuses [2]. Campuses
involve large collections of buildings that consume heating, cooling, power,
and transportation services. Many university campuses own and operate utility
systems that provide steam, hot water, and chilled water to buildings and
research facilities. Currently, most of these systems are powered using
combined heat and power (CHP) systems that run on natural gas [39]. Electrical
power is also used for diverse essential services (e.g., ventilation,
lighting, circulation, lectures) and for resource-intensive equipment such as
computational facilities (e.g., computing clusters and data centers). The
generation and distribution of all these utilities generate a massive carbon
footprint. Moreover, universities also typically operate transportation
services, which further contribute to the carbon footprint. In essence,
university campuses are representative systems in that they combine elements
of urban centers and manufacturing facilities. It has been estimated that
university campuses in the US emit 7.7 MTCO2e per student and are responsible
for 2% of the 5,222 million MTCO2e of the US greenhouse gas emissions [39,
44]. Notably, these emissions are higher than those of ammonia production.

Technologies such as renewable power generation, heat pumps, electric and
hydrogen-based transportation vehicles, and energy storage systems are
actively being studied as means to help decrease carbon footprints [19, 28,
45, 37, 27]. The number of new technologies that can be used is steadily
increasing and complicates decision-making processes, because it is important
to understand how new technologies will interact with existing infrastructure
and operations. In the context of university campuses, cost-minimization
models have been proposed to determine the energy mix, capacities, and storage
requirements for a university campus (model is cast as a mixed integer
nonlinear program) [40]. This modeling approach requires detailed data for
investment costs, operating costs, operating lifetimes, and scaling factors
for the technologies proposed and time-dependent demand data to determine a
minimized cost scenario with no emissions. Other modeling approaches have used
multiobjective optimization to determine technology operations that can help
reach emissions reduction targets, determine lowest cost for maximum renewable
use, and determine cost of reducing emissions [35, 33, 25]. Existing
optimization models are typically intended to provide long-term investment
plans to achieve decarbonization. Detailed models are ought to be used to make
important investment decisions but these often hindered by the fact that there
might not be sufficient data to evaluate emerging technologies; in addition,
the use of detailed models might require significant implementation and
computational effort. As such, there is also a need for simple models that can
help screen options prior to investing resources in model detailed studies.

In this work, we provide an optimization framework for analyzing technology
pathways that can provide desired product demand targets while minimizing
supply and technology costs. The framework uses a compact, graph/network
representation that aims to facilitate the analysis of complex
interdependencies that arise between products and technologies. In the
proposed representation, we consider the supply of basic products to the
system (e.g., natural gas and electricity) and technologies that transform
these basic products to higher-value intermediate and final products (e.g.,
hydrogen and cooling water) that are delivered to consumers. The formulation
operates at a high level of abstraction, which enables capturing diverse types
of products (e.g., material, energy, labor, services) and byproducts (e.g.,
carbon emissions and waste) in a unified manner. The proposed formulation can
be viewed as a superstructure network that interconnects different
stakeholders in a system and helps determine how externalities (e.g., policy
and technology specifications) impact the system. Moreover, duality analysis
reveals that the formulation has a natural economic/market interpretation;
this reveals that the formulation aims to identify technology pathways that
maximize total economic value (by using technologies for transforming products
into higher-value products) and explains how externalities can help
activate/deactive pathways. The market interpretation also allows us to
discover inherent values (prices) for key intermediate products and helps
evaluate the impact of externalities and technology specifications on such
values. Moreover, the framework provides insights into how expanding the
domain of the system (e.g., by considering alternative products) can activate
technology pathways. The modeling framework uses minimal data on products and
technologies, which facilitates analysis and enables fast screening of diverse
technologies and externalities. We provide a case study that analyzes
decarbonization pathways for a prototypical university campus to illustrate
the developments.

##  2 Graph-Based Optimization Framework

We consider a system comprised of a set of basic, intermediate, and final
products as well as a set of technologies that transform basic products into
intermediate and final products. The goal is to determine technology pathways
that can maximize the value of served product demands, while minimizing supply
and technology costs.

A couple of optimization models are introduced to determine optimal technology
pathways that conduct desired goals. A management model will be used to
understand how suppliers, technologies, and demands interact and to determine
optimal allocations for all these stakeholders. The model uses a graph
abstraction that captures the topology/connectivity that arises from
interactions between products and technologies. The model can be interpreted
as a value chain that aims to identify pathways that maximize profit for all
stakeholders involved; this is done by using technologies to create wealth
(transforming lower-value products into higher-value products). This
interpretation will also allows us to determine the inherent value (prices) of
products, which is particularly useful in attributing value for intermediate
products and to understand how externalities (e.g., carbon prices or disposal
costs for waste products) and technology costs/efficiencies propagate through
the technology pathways and influence product prices. An investment model will
be used to prioritize the selection of pathways under constrained investment
budgets and to trade-off profits with investment costs.

Our models aim to use minimum specification data for technologies (e.g.,
efficiencies, capacities, operating costs, investment costs), so as to provide
$``$high-level$"$ picture of the technology landscape and on potential factors
that influence their economic viability. This is also, in part, motivated by
the fact that there is often limited data available for existing technologies.
As expected, any simplification to the representation of technologies will
come with a downside of inaccuracy. Our high-level abstraction aims to provide
an intuitive approach that helps navigate complex interdependencies between
products, technologies, and externalities.

The proposed model can be interpreted as a value chain in which there is no
transportation and spatial/geographical context; specifically, we note that
our model is a simplification of the multi-product supply chain model
presented in [38, 41]. This observation is important, as the model proposed
here can provide valuable preliminary insights that can inform the development
of more sophisticated models; for instance, it can help determine technology
pathways that will or will not be selected in supply chain designs.

###  2.1 Management Model

We define a system comprised of a set of products $\mathcal{P}$, set of
suppliers $\mathcal{S}$ that offer products, technologies $\mathcal{T}$ that
transform products, and consumers $\mathcal{D}$ that request products. These
elements are interconnected via a graph that captures product-technology
dependencies. We can think of this system as an economy, market, or value
chain that aims to generate economic value (wealth) by transforming basic
products into higher value products. In other words, we think about this
system as an economy in which elements (suppliers, consumers, technologies)
are stakeholders that provide/request services and that aim to generate
wealth. Factors that affect the ability of this economy from generating wealth
include costs of basic products, technology factors (e.g., costs, capacities,
and efficiencies), and externalities (e.g., policy, external markets, and
taxes).

Each supplier $i\in\mathcal{S}$ has an offered value
$\alpha_{i}\in\mathbb{R}$, associated allocation $s_{i}\in\mathbb{R}$ (flow of
product provided to the system), an available capacity
$\bar{s}_{i}\in\mathbb{R}$ (maximum product amount that it can provide), and a
supplied product $p(i)\in\mathcal{P}$. Each consumer $j\in\mathcal{D}$ has an
associated offered value $\alpha_{j}\in\mathbb{R}$, allocation
$d_{j}\in\mathbb{R}$ (flow of product extracted from the system), an available
capacity $\bar{d}_{j}\in\mathbb{R}$ (maximum product amount that it can take),
and a requested product $p(j)\in\mathcal{P}$. We denote the set of suppliers
or consumers that provide/request a given product $p\in\mathcal{P}$ as
$\mathcal{S}_{p}\subseteq\mathcal{S}$ and
$\mathcal{D}_{p}\subseteq\mathcal{D}$, respectively.

Each technology $k\in\mathcal{T}$ has an offered value
$\alpha_{k}\in\mathbb{R}$ (service cost), has an allocation flow
$t_{k}\in\mathbb{R}$ (flow processed by the technology); this flow corresponds
to a reference (input) product $p(k)\in\mathcal{P}$ and an available
processing capacity $\bar{t}_{k}\in\mathbb{R}$ (maximum amount of reference
product that it can process). Technologies are key assets, as they generate
wealth by conducting transformation of products; this is captured using the
transformation factors
$\gamma_{k,p}\in\mathbb{R},\;k\in\mathcal{T},p\in\mathcal{P}$. A positive
transformation factor ($\gamma_{k,p}>0$) indicates that product $p$ is
produced by technology $k$, a negative factor ($\gamma_{k,p}<0$) indicated
that the product is consumed by the technology, and a zero factor
$\gamma_{k,p}=0$ indicates that the product is neither produced nor consumed
(does not participate in the technology). The set of technologies that
generate or consume a specific product $p\in\mathcal{P}$ are denoted as
$\mathcal{T}_{p}\subseteq\mathcal{T}$. Transformation factors play a key role
in dictating the behavior of the system, as they provide measures of
efficiency for technologies; moreover, these factors capture product-
technology interdependence (connectivity) and capture the generation of
desirable (e.g., value-added) products and undesirable (e.g., waste)
byproducts. In other words, transformation of products that occur in
technologies define the topology of the system graph; this topology captures
diverse pathways that might exist to obtain specific products and encodes key
information that explains how diverse externalities propagate through the
system.

The data for suppliers, consumers, and technologies is summarized into a graph
(a superstructure) that captures all product-technology interdependencies. The
goal is to determine pathways in this superstructure that generate maximum
wealth, while filtering out those that do not generate wealth. Figure 1
provides a simple illustration on how elements are connected in the graph and
how competing technology pathways emerge. Moreover, this aims to illustrate
how externalities (emissions costs/taxes) can favor some pathways over others.
A more complex graph showing diverse pathways to decarbonize a university
campus is presented in Figure 2. This illustrates how complexity quickly
arises due to the presence of diverse pathways and product-technologies
connectivity; as such, it is necessary to use systematic techniques to
identify optimal pathways.

Figure 1: Illustration of a simple graph in which supplies of basic products
(gasoline and solar radiation) are used by technologies (gas-powered vehicles,
electricity vehicles, and solar panels) to satisfy demand of higher value
products (travel distance) and that generate undesired products (CO2
emissions). In this case, electrical power generated by the photovoltaic
panels is an intermediate product that is used by the electric vehicle to
generate a value-added and final product (distance). If the cost of CO2
emissions is not taken into account (and/or gasoline is inexpensive), the gas
vehicle pathway will be preferred. On the other hand, when emissions costs are
taken into account and/or renewable power is inexpensive, the electric vehicle
pathway will be preferred. Figure 2: Graph showing diverse suppliers,
technologies, and consumers that participate in a university campus. Complex
interdependencies/connectivity arises between products and technologies and it
is thus nonobvious what are the best pathways. See Table 1 for technology
descriptions.

We now propose an optimization formulation that helps identify optimal
technology pathways from the graph. The formulation is given by:

$\displaystyle\max_{(s,d,t)}$
$\displaystyle\quad\sum_{j\in\mathcal{D}}{\alpha_{j}}d_{j}-\sum_{i\in\mathcal{S}}{\alpha_{i}}s_{i}-\sum_{k\in\mathcal{T}}{\alpha_{k}t_{k}}$
(1a) s.t
$\displaystyle\quad\sum_{i\in\mathcal{S}_{p}}{s_{i}}+\sum_{k\in\mathcal{T}_{p}}{\gamma_{k,p}t_{k}}-\sum_{j\in\mathcal{D}_{p}}{d_{j}}=0,\quad
p\in\mathcal{P},\quad(\pi_{p})$ (1b) $\displaystyle\quad 0\leq
s_{i}\leq\bar{s}_{i},\quad i\in\mathcal{S}$ (1c) $\displaystyle\quad 0\leq
d_{j}\leq\bar{d}_{j},\quad j\in\mathcal{D}$ (1d) $\displaystyle\quad 0\leq
t_{k}\leq\bar{t}_{k},\quad k\in\mathcal{T}$ (1e)

This $``$management$"$ model aims to identify product allocations $(s,d,t)$
for all system elements that maximize the total surplus (1b). The total
surplus balances the value of demand served (to be maximized) and the costs of
all supplies and technologies (to be minimized). We will see (via duality
analysis) that the total surplus is equivalent to the total system profit
(revenue minus cost) and captures the profit of all system elements.

The constraints (1b) encode the graph that captures the connectivity between
products and technologies. In this graph, products are interpreted as nodes
and we enforce conservation of product at such nodes. Specifically, for each
product $p\in\mathcal{P}$, we must have that all input by suppliers,
generation/consumption by technologies, and extraction by consumers must be
balanced.

The dual variables $\pi_{p},p\in\mathcal{P}$ of the balance constraints (1b)
can be interpreted as inherent values (prices) for the products. Specifically,
the dual variables capture how valuable a given product is in the economy. For
instance, a high-value product can be critical because it enables diverse
technology pathways and generate wealth via sales of final products;
conversely, a low-value product can be of low priority (or even irrelevant)
for the generation of wealth in the system. In fact, we will see later that
prices can have negative values, indicating that certain products (e.g.,
waste) can be detrimental to the creation of wealth in the economy. However,
products with negative values might be necessary to achieve alternate goals in
the system, such as mitigation of social and environmental impacts by the
economy. Determining prices for intermediate products is particularly
important, as such products typically do not have external markets (they are
means to an end).

We can gain important insights into the economic properties of the
optimization problem by analyzing the (partial) Lagrangian dual function:

$\displaystyle\mathcal{L}(s,d,t,\pi)=\sum_{j\in\mathcal{D}}{\alpha_{j}}d_{j}-\sum_{i\in\mathcal{S}}{\alpha_{i}}s_{i}-\sum_{k\in\mathcal{T}}{\alpha_{k}t_{k}}+\sum_{p\in\mathcal{P}}\pi_{p}\left({\sum_{i\in\mathcal{S}_{p}}{s_{i}}+\sum_{k\in\mathcal{T}_{p}}{\gamma_{k,p}t_{k}}-\sum_{j\in\mathcal{D}_{p}}{d_{j}}}\right).$
(2)

If we now consider the following identities:

$\displaystyle\sum_{p\in\mathcal{P}}\pi_{p}\sum_{i\in\mathcal{S}_{p}}{s_{i}}=\sum_{i\in\mathcal{S}}{\pi_{i}s_{i}}$
(3a)
$\displaystyle\sum_{p\in\mathcal{P}}\pi_{p}\sum_{k\in\mathcal{T}_{p}}{\gamma_{k,p}t_{k}}=\sum_{k\in\mathcal{T}}{\pi_{k}t_{k}}$
(3b)
$\displaystyle\sum_{p\in\mathcal{P}}\pi_{p}\sum_{j\in\mathcal{D}_{p}}{d_{j}}=\sum_{j\in\mathcal{D}}{\pi_{j}d_{j}},$
(3c)

we can see that the Lagrange function can be rewritten as:

$\displaystyle\mathcal{L}(s,d,t,\pi)=\sum_{j\in\mathcal{D}}{(\alpha_{j}^{\mathcal{D}}-\pi_{j})d_{j}}+\sum_{i\in\mathcal{S}}{(\pi_{i}-\alpha_{i}^{\mathcal{S}})s_{i}}+\sum_{k\in\mathcal{T}}{(\pi_{k}-\alpha^{\mathcal{T}}_{k})t_{k}}.$
(4)

Here, we use the short-hand notation $\pi_{i}:=\pi_{p(i)}$ to denote the price
of the product that supplier $i\in\mathcal{S}$ provides. We also define the
prices of the products for the consumers as $\pi_{j}:=\pi_{p(j)}$ and we
define $\pi_{k}:=\sum_{p\in\mathcal{P}}\gamma_{k,p}\pi_{p}$ as the technology
price/value (this is the weighted sum of its input and output products,
weighted by the transformation factors). If we now define the profit
functions:

$\displaystyle\phi_{i}$ $\displaystyle:=(\pi_{i}-\alpha_{i})s_{i},\quad
i\in\mathcal{S}$ (5a) $\displaystyle\phi_{k}$
$\displaystyle:=(\pi_{t}-\alpha_{k})t_{k},\quad k\in\mathcal{T}$ (5b)
$\displaystyle\phi_{j}$ $\displaystyle:=(\alpha_{j}-\pi_{j})d_{j},\quad
j\in\mathcal{D},$ (5c)

we can see that the Lagrange function can be rewritten as:

$\displaystyle\mathcal{L}(s,d,t,\pi)=\sum_{j\in\mathcal{D}}{\phi_{j}}+\sum_{i\in\mathcal{S}}{\phi_{i}}+\sum_{k\in\mathcal{T}}{\phi_{k}}.$
(6)

Because the optimization problem is a linear program, we can assume that
strong duality holds and thus its solution can be determined by solving the
Lagrangian dual problem:

$\displaystyle\max_{\pi}\max_{(s,d,t)\in\mathcal{C}}\mathcal{L}(s,d,t,\pi)$
(7)

Here, the set $\mathcal{C}$ captures the capacity constraints (1c),(1d), and
(1e). From these observations, we can see that the optimization model is
seeking to maximize the total profit of all system elements (suppliers,
consumers, and technologies). Moreover, because strong duality holds, we have
that the optimal allocations and prices are such that total surplus equals the
total profit:

$\displaystyle\sum_{j\in\mathcal{D}}{\alpha_{j}}d_{j}-\sum_{i\in\mathcal{S}}{\alpha_{i}}s_{i}-\sum_{k\in\mathcal{T}}{\alpha_{k}t_{k}}=\sum_{j\in\mathcal{D}}{\phi_{j}}+\sum_{i\in\mathcal{S}}{\phi_{i}}+\sum_{k\in\mathcal{T}}{\phi_{k}}.$
(8)

The total surplus can thus be interpreted as the system-wide profit; this
profit in turn can be interpreted as the total wealth created by the system
(i.e., the system can be interpreted as a market or an economy).

Lagrangian duality analysis also reveals the role of the dual variables in
remunerating elements. Specifically, the profit of the supplier is given by
difference between the actual payment ($\pi_{i}s_{i}$) and the expected cost
($\alpha_{i}s_{i}$); as such, the supplier desires that
$\pi_{i}\geq\alpha_{i}$ (this guarantees non-negative profit) and that
$\pi_{i}$ is as large as possible (to maximize its profit). For the consumers,
we have that the profit is given by the difference between the expected cost
($\alpha_{j}d_{j}$) and the actual payment ($\pi_{j}d_{j}$); as such, we the
consumer desired that $\pi_{j}\leq\alpha_{j}$ (this guarantees non-negative
profit) and that $\pi_{j}$ is as low as possible (to maximize its profit). For
the technologies we have a more interesting case; specifically, for the profit
to be non-negative, we require that $\pi_{k}\geq\alpha_{k}$; by definition, we
have that $\pi_{k}=\sum_{p\in\mathcal{P}}\gamma_{k,p}\pi_{p}$. As such, it is
important to observe that we need that the value created by a technology needs
to be higher than its operating cost $\alpha_{k}$. This can only be achieved
if the generated output products have a higher value than the consumed input
products (weighted by the transformation factors). This insight is important,
as it clearly shows how technologies are wealth creators (generate higher-
value products from lower-value products). Moreover, any technology that
cannot achieve wealth creation (due to mismatches in input/output costs or
technologies inefficiencies) will simply not participate in the economy.

Additional insight into the prices $\pi_{p},\;p\in\mathcal{P}$ generated by
the optimization problem can be determined by posing its dual problem:

$\displaystyle\min_{\pi,\lambda}\quad\sum_{i\in\mathcal{S}}{\bar{s}_{j}\lambda_{i}}+\sum_{j\in\mathcal{D}}{\bar{d}_{j}\lambda_{j}}+\sum_{k\in\mathcal{T}}{\bar{t}_{k}\lambda_{k}}$
(9a) $\displaystyle\text{s.t}\qquad\pi_{i}-\lambda_{i}\geq\alpha_{i},\quad
i\in\mathcal{S},\quad(s_{i})$ (9b)
$\displaystyle\qquad\qquad\pi_{j}-\lambda_{j}\leq\alpha_{j},\quad
j\in\mathcal{D},\quad(d_{j})$ (9c)
$\displaystyle\qquad\qquad\pi_{k}-\lambda_{k}\geq\alpha_{k},\quad
k\in\mathcal{T},\quad(t_{k})$ (9d)

This formulation reveals that the prices must be bounded by the offered
values. Here, we have that $\lambda_{i},i\in\mathcal{S}$,
$\lambda_{j},j\in\mathcal{D}$, and $\lambda_{k},k\in\mathcal{T}$ are the dual
variables assigned for the upper bound constraints of the suppliers,
consumers, and technologies. These dual variables must be positive; as such,
they indicate that the supplier and technology prices must be higher than
their offering values, while the prices for the consumers must be lower than
their offering costs. This again indicates that prices generated by the model
are such that all system elements generate a profit (or at least break even).
This also indicates that the system-wide profit must be non-negative;
specifically, the entire system must generate wealth (or at least break even);
in other words, it makes no sense to have an economy that does not generate
value.

Note also that an outcome of the optimization model is that all allocations
are zero $(s,d,t)=(0,0,0)$. This can occur, for instance, if the supplied
products and technology services are too expensive compared to values of
requested products (and thus the system cannot generate wealth). As such, the
system-wide profit can be seen as a measure of economic efficiency (a total
profit of zero indicates the economy/system/value-chain is not economically
efficient). Another possible outcome of the model is that the allocations of
some specific technology pathways are zero; this would indicate that such
pathways do not generate wealth. This also indicates that the model has the
goal of identifying pathways that generate the most value (and eliminate the
ones that do not generate value); as such, the model can be used to quickly
identify promising pathways. From all these observations, we can see that the
proposed model makes economic sense; this is important, as it facilitates
analysis and interpretation of model outcomes. For instance, it is possible to
use the framework to explore what values of technology efficiencies can help
activate pathways and generate wealth.

The primal formulation (1) of the optimization provides the physical
allocations of the products, while the dual variables
$\pi_{p},\,p\in\mathcal{P}$ from the dual formulation (1) provides economic
allocations of the products. These formulations together make up the
optimization model which takes in information from system elements, maximizes
the total profit, and returns the physical and economic allocations to the
elements as described by Figure 3.

Figure 3: Schematic representation of optimization model, indicating elements
involved as well as model inputs/outputs.

The model proposed treats products $p\in\mathcal{P}$ as abstract objects; as
such, these can represent any type of asset that is exchanged in the system
(e.g., material, energy, labor, mileage, computing service). For instance,
vehicles could take in fuel ($gal~{}gas$) to produce distance ($km$) and
emissions ($kg~{}CO_{2}$). The abstract product objects can also be used to
represent a discrete spectrum of products (e.g., green, blue, and brown
hydrogen or water of different types).

The model also treats values for products as abstract objects and we can use
these to capture a range of behaviors. For instance, offered values for
suppliers ($\alpha_{i}$) and consumers ($\alpha_{j}$) are typically positive
or non-negative. The price bounds obtained from duality indicate that, in such
a case, the product prices will be positive. However, the offering values can
can also be allowed to be negative; this provides flexibility to capture
policy and taxes or negative impact of byproducts from the system (e.g., waste
generation). For instance, one of the pathways shown in Figure 1 produces a
waste (CO2) as a byproduct of transportation. This waste CO2 has to be taken
by the environment; as such, we can interpret the environment as a consumer
that is willing to take the waste product but at a negative cost (the negative
cost can be interpreted as a carbon tax). This is analogous to how landfills
and wastewater treatment facilities operate (they charge a tipping fee to take
on waste). We can also envision a supplier that has a product that they want
to get rid of; as such, they can offer the product at a negative cost (this is
common when dealing with undesired waste). For instance, we are willing to pay
a fee for garbage to be taken away form our homes (as garbage is an undesired
product). Accounting for undesired outcomes of the economy (via negative
prices) is important in capturing social or environmental impacts of the
system.

We highlight that the optimization model does not enforces satisfaction of
demands explicitly ($d_{j}=\bar{d}_{j}$); instead, the model will only decide
to satisfy demands if this makes economic sense (i.e., if it maximizes the
total profit). For instance, it might be possible that forcing the
satisfaction of a demand of a non-valuable product will incur significant
costs that render the system non-profitable. It is also worth highlighting
that the model will select pathways between basic products and final products
that make the most economic sense; as such, the model is fully driven by
economic incentives. It is possible to modify the model to enforce demand
satisfaction in a couple of ways. For instance, we can set a large offering
value $\alpha_{j}^{\mathcal{D}}$ to demands that need to be satisfied; this is
analogous to how electricity markets operate (a high value to electricity
demands is set to ensure delivery). It is also possible to enforce strict
constraints on satisfaction of demands, but such constraints can be shown to
introduce economic inefficiencies [41], because this can force the selection
of pathways that are non-profitable.

The proposed framework is useful as a screening tool because typical
applications contain a large number of technology pathways. For instance, a
superstructure of all possible pathways for a specific application is shown in
Figure 2. This graph contains the current infrastructure and potential new
pathways to reach decarbonization for a university campus (as we will discuss
in our case study). For this example, electrolysis, fuel cells, electric
vehicles, hydrogen vehicles, hydrogen generators, and combined heat and power
systems can be added as potential technologies. This could become even more
complex as new technologies are developed or new supplies are discovered or
demands shift towards new products (thus expanding the boundary of the
system). The bottom right demand is for CO2 which has to be accounted for
analysis, as this is an undesired byproduct of the economy/system. The
proposed optimization model can be used to screen technology pathways that
make most sense under different market conditions (e.g., under different
carbon tax scenarios).

###  2.2 Investment Model

It is often of interest to restrict the selection of technology pathways based
on investment costs or budgets that might be available. We can easily extend
the proposed model to account for this; we will use this formulation to study
how available investment budgets can constraint certain technology pathways,
as well as to understand how we should prioritize technologies. We define a
binary variable $y_{k}\in\\{0,1\\},~{}k\in\mathcal{T}$ to represent if a
technology is selected with an investment cost of $\beta_{k}\in\mathbb{R}$.
This gives the formulation:

$\displaystyle\max_{s,d,t,y}$
$\displaystyle\quad\sum_{j\in\mathcal{D}}{\alpha_{j}}d_{j}-\sum_{i\in\mathcal{S}}{\alpha_{i}}s_{i}-\sum_{k\in\mathcal{T}}{\alpha}_{k}t_{k}$
(10a) s.t
$\displaystyle\quad\sum_{i\in\mathcal{S}_{p}}{s_{i}}+\sum_{k\in\mathcal{T}_{p}}{\gamma_{k,p}t_{k}}-\sum_{j\in\mathcal{D}_{p}}{d_{j}}=0,\quad
p\in\mathcal{P}$ (10b) $\displaystyle\quad 0\leq s_{i}\leq\bar{s}_{i},\quad
i\in\mathcal{S}$ (10c) $\displaystyle\quad 0\leq d_{j}\leq\bar{d}_{j},\quad
j\in\mathcal{D}$ (10d) $\displaystyle\quad 0\leq t_{k}\leq\bar{t}_{k}\cdot
y_{k},\quad k\in\mathcal{T}$ (10e)
$\displaystyle\quad\sum_{k\in\mathcal{T}}{\beta_{k}\cdot
y_{k}}\leq\bar{\beta}$ (10f)

The constraint (10f) enforces that the total investment cost is less than or
equal to a given budget $\mathcal{\beta}\in\mathbb{R_{+}}$. The maximum
technology input constraint (10e) is augmented such that if the technology is
not built ($y_{k}=0$) then the corresponding input value will be zero
($t_{k}=0$). For technologies that already in place, we simply set the
corresponding binary values to one. In our modeling approach, we can capture
economies of scale by defining technologies of different capacities. The
budget constraint will enable studies to determine what budget is required to
transition to different pathways. We note that it is also possible to
incorporate the investment costs into the objective and let the model decide
what investments are needed to maximize the total surplus (without any
constraints in budget). In this context, it is important to ensure that
operating costs and investment costs are on the same time basis (e.g.,
annualized costs). The investment model reduces to a management model if the
binary variables are fixed. We can thus think of the investment model as a
framework that can be used to study how strategic deployment of technologies
can be used to activate an economy and generate wealth.

##  3 Case Study: Decarbonizing a University Campus

We consider a case study for a simulated university campus to illustrate the
modeling capabilities of the framework and the types of insights that it can
provide. The model was developed using basic data and assumptions that aim to
capture general features of a prototypical, large campus (inspired by the UW-
Madison campus); however, the model is not a real and validated representation
of the actual system. As such, results and conclusions reached in this study
are only intended to illustrate the capabilities of the model, and should not
be used as actual recommendations.

We consider a campus that serves 48,000 students and 24,000 staff members on a
900-acre campus with 25 million square feet of buildings [4, 22]. Under
existing operations (and associated technology pathways), the university
outputs 500,000 tonnes of CO2 annually to meet demands of four key products
(see Figure 4) [22]. The carbon footprint is equivalent to the annual
emissions of 34,000 average citizens in the US [10] (thus comparable in
magnitude to the campus population). To decrease the CO2 footprint, we must
identify new technology pathways that can satisfy the product demands (campus
services) while reducing CO2 emissions.

Figure 4: Sankey diagram summarizing the annual CO2 emissions and their
sources for a university campus. Generation of electrical power and heating
are the dominant sources.

###  3.1 Existing Technologies

We assume that the campus purchases electricity from the local utility with a
special rate scheme due to the large demand of the university of 400,000 MWh
per year [22]. The university has some small renewable energy projects to
generate electricity, but these total to only 30 MWh per year. These are
primarily rooftop solar photo-voltaic solar panels. The rest of the power
comes from the power grid. The largest power sources for electricity in the
state are coal and natural gas at 42% and 34%, respectively [16]. There is one
large nuclear plant supplying 15% of electricity and hydroelectric supplies
4%; the remaining 5% is composed of non-hydroelectric renewables. This leaves
the grid electricity with an estimated CO2 output of 560 kg CO2/MWh [16]. Grid
electricity is represented by GP in Table 4 and 5, which takes in MWh grid and
converts it into MWh and kg CO2. The grid is represented as a technology, to
account for generated CO2 emissions associated with the use of grid
electricity. Operating costs for the technologies are estimated and are
represented relative to the reference product.

The campus has an annual average highest outdoor temperature of 35℃ and lowest
temperature of -27℃, so both heating and cooling is required to keep buildings
comfortable for teaching and research [17]. To meet these heating and cooling
demands, the campus uses utility network that delivers steam and chilled water
to buildings [26]. These services are supplied by a couple of heating and
cooling plants and one combined heat and power (CHP) plant. These plants burn
natural gas to produce steam, which is used to heat the buildings and are
represented by SGNG. The plants also use refrigeration systems to chill water
to cool the buildings, represented by WC. These plants, along with the CHP
plant, meet the annual demand for the university of 1 million tonnes of steam
and 24 million tonnes of chilled water to maintain buildings at a comfortable
temperature as shown in Table 2.

The natural gas-fired CHP plant has a rated electricity output of 150 MW, can
produce 220 tonne/hr of steam, and 27,000 tonne/hr of chilled water [1]. For
this plant, the water chillers are powered by electricity and will be
represented separately by WC. The steam and power output will be represented
by SGNG.

The campus has a fleet of 912 vehicles to maintain the campus and meet travel
demands (measured as distance) [22]. A total of 865 of the vehicles are
exclusively gasoline powered with the rest being hybrid vehicles and just 2
fully electric vehicles. To compare different types of vehicles, there is a
demand of distance (km) to meet demands rather than a demand for fuel for
transportation purposes. These vehicles satisfy the 6 million km the
university requires each year at a rate estimated by the university fleet rate
[9]. These vehicles are represented by GV with operating costs based on
average values for internal combustion vehicles.

These technologies make up the current pathways that meet the campus demands
and are displayed in Figure 5. Supplies of natural gas, gasoline, and grid
electricity are added with pricing details found in Table 3. When implemented
in the model, the quantities for these supplies are set to a sufficiently
large value such that the technologies are not limited by the available
supply. Demand data can be found in Table 2.

Figure 5: Superstructure graph showing currently-available suppliers,
technologies, and demands for the university campus. See Table 1 for
technology descriptions.

###  3.2 Potential New Technologies

Electricity generation on campus produces the most CO2; to decarbonize
generation, renewable electricity generation is required. To do this, the
university would have to either wait for the power grid to use more renewable
power or install their own renewable generators. However, renewable capacity
depends on location and cannot be placed wherever is convenient. Further, the
times that renewable power generation occurs does always match the demand,
meaning some energy would have to be stored for later use. Based on this
constraint, the power produced from renewable power will be treated as a
separate product ($MWh~{}renew$) that cannot be used directly to satisfy
electricity demands. For the purposes of this case study, solar photovoltaic
and wind power will be considered. Solar power will be represented by SP and
wind power will be represented by WP to enable the use in the investment model
and capture the operational costs of the two technologies [13]. These
technologies are modular, meaning that additional capacity can be added in
small increments for the same cost. As such, the investment cost and capacity
reflect these aspects and, when implemented in the model, will have a set
number of units available. These technologies take into consideration the
required land usage, and an additional supply of land is added to account for
this. This means that the technologies will be competing for a supply of land
(in m2). In other words, our model captures land as a resource that needs to
be strategically used to various uses. This illustrates how the abstract
products can be used to capture diverse assets.

Hydrogen-based technologies have been proposed as an energy carrier with
similar applications as natural gas. As such, hydrogen-based technologies
could be critical to continue to meet the demands of the university. To be
decarbonized, hydrogen can be produced through electrolysis using renewable
power to create a CO2 free energy carrier. As such, electrolysis will be
represented by WS which takes in renewable power ($MWh~{}renew$) and produces
hydrogen. The hydrogen could then be used by a fuel cell for electricity by
FC, which is a modular technology as well [30, 32].

The produced hydrogen can be used by the existing CHP plant by blending
hydrogen and natural gas as represented by CHPB. This option has an investment
cost of just 1% of the initial investment cost of the turbines, because many
natural gas powered turbines can handle hydrogen blends up to a volume
fraction [49]. This study will use a blend of 20% hydrogen by volume.
Alternatively, the university could replace the existing turbines to run
entirely off hydrogen with an associated investment cost of 25% of the initial
cost of the turbines of the CHP plant as represented by CHPH2 [49].
Alternatively, hydrogen could be used by a fuel cell CHP plant which uses the
high temperatures a fuel cell operates at to generate heat that could be used
by the university as well as electricity. The fuel cell CHP will be
represented by CHPFC. Hydrogen could also be used to decarbonize the heating
demand of the university by implementing hydrogen steam generators as
described by SGH2. This can be implemented without having to make any
additional upgrades, unlike the CHP plant [47].

To reduce the CO2 emissions of the university travel demand, alternative
vehicles must be considered. Hybrid vehicles have 50% higher fuel efficiency
than traditional gasoline vehicles and will be represented by HV [3]. However,
even the most fuel efficient gasoline vehicle still emits CO2. Electric
vehicles could be a CO2 free alternative if the electricity is sourced from
renewable power such as wind and solar and will be represented by EV. The
shift to electric vehicles has already been observed as more companies release
new vehicles [5]. Hydrogen fuel-cell vehicles are another alternative,
although they are less commercially available than electric vehicles. However,
they have faster refueling times compared to the charging times of electric
vehicles [6, 7]. The hydrogen vehicles in this study will be described by H2V.
For all the alternative vehicle options, they will also be represented in a
modular way meaning the purchase of one of these vehicles provides the
capacity for 8000 km and the purchase of multiple vehicles will be required.

Modular technologies, such as vehicles and electrolysis, will have an
associated number of units that are available in Table 7. While the single
items like switching to a blend of hydrogen and natural gas feed for CHP are a
single unit. This will allow the model to see what technology pathways are
chosen at budgets that would not allow for a complete transition from, for
example, gasoline vehicles to electric vehicles. All of the new technologies
are shown in Figure 6.

All of the possible pathways from supply through technologies to demands
associated with potential technologies are displayed in Figure 2. While all of
the pathways are displayed here, a subset of them will be chosen by the model
as being the optimal pathway. By varying some key attributes such as a carbon
tax or the budget, we can determine what technologies and products will have
the most impact under those conditions.

Figure 6: Superstructure graph of potential technology pathways that could be
chosen by the university campus to decarbonize the system (current
technologies have been blurred out to facilitate comparison).

###  3.3 Results

####  3.3.1 Management Model

The purpose of the management model is to determine optimal technology
pathways for the campus to meet its needs, without any restrictions on the
budget needed to build such technologies. These optimal technologies will vary
based on external conditions of the system (e.g., policy, markets, technology
efficiencies). For instance, a key external condition is the cost of CO2.
There is work to determine the externality of releasing CO2 that places a cost
of 113 USD per tonne CO2, although estimates vary [46]. However, our work will
aim to ask the question: What CO2 cost/tax would incentivize the university to
reduce or stop emitting CO2? The tax does not necessarily have to be
interpreted as an externality (e.g., government-imposed), but can also be
interpreted as an implicit value that the university is placing to its carbon
footprint (which can potentially trigger negative public perception).

The carbon emissions are modeled as a demand for CO2 that is charged at a
negative bid value (it is a waste product). This can be interpreted in
different ways; for instance, this can be interpreted as a tax that the
government is introducing for any emissions generator by the system.
Alternatively, this can be interpreted as a “tipping fee” that the environment
is charging the system to take its CO2 waste. The CO2 cost can also be
interpreted as an internal (inherent) value that campus is placing on
emissions; this is analogous to how companies that are currently seeking to
decarbonize think about this waste (e.g., they are internally placing a
negative value to CO2 that might be implicitly connected to branding or public
perception). This illustrates how the proposed modeling framework can help
capture diverse scenarios of interest.

To establish a benchmark, the pathway model with all potential and current
technologies will first be optimized with no carbon tax. This leads to the
technology pathway as shown in Figure 7. This pathway, however, still emits
CO2 because it is using the natural gas-powered CHP plant as much as possible.
Meaning that without an incentive, the university has no need to fully
decarbonize. The pathway, however, makes use of solar power to produce
hydrogen to be used to supplement the CHP electricity for use in electric
vehicles, to operate the water chiller, and meet the electricity demands of
the university.

Figure 7: Optimal technology pathways under no CO2 tax showing the
technologies used satisfy the demands of university campus; pathways that are
not selected are blurred out. Product prices are displayed in the legend,
flows going in and/or out of each stakeholder are reported in the units of the
legend, and the number of technology units required are reported above each
technology.

To study the impact of the CO2 tax ($\alpha_{kg~{}CO_{2}}$), it will involve
discretizing a range of values and solving the management model for each
value. This is computationally cheap as the optimization problem is a linear
program and therefore quick to solve. The demand bid for CO2 will be negative
because it is being modeled as a tax; meaning the university will have to pay
the environment/government to take the CO2 it emits. The impact of this tax on
the utility cost of the university was also determined; utility cost is the
cost to operate the technologies, cost of supplies, and disposal cost of waste
products (CO2), and is given by:

$\displaystyle\sum_{k\in\mathcal{T}}{\alpha}_{k}t_{k}+\sum_{i\in\mathcal{S}}{\alpha_{i}}s_{i}-\alpha_{kg~{}CO_{2}}d_{kg~{}CO_{2}}$
(11)

Figure 8 shows the impact of varying the CO2 tax on the CO2 emissions and
utility cost of the university. This shows three different technology pathways
depending on the CO2 tax. Below 45 USD per tonne, the pathway chosen is the
same as in Figure 7. While the CO2 output stays constant for each pathway, the
utility cost increases when there is still CO2 emissions because the
university has to pay for the environment to take the emissions.

Figure 8: Dependence of CO2 output and utility cost of the university on the
CO2 tax (when all possible technologies can be used). Results indicate that
there are three different pathways that are activated at different tax levels.

Between 45 USD and 130 USD per tonne is the pathway shown in Figure 9. This
pathway is the one that would be chosen if the estimated externality of CO2
was implemented as a tax. The pathway has a 77% lower CO2 output than in
Figure 7 but also has a higher utility cost. This pathway has fully
decarbonized the demand for electricity, cooling, and transportation but still
relies on natural gas for steam.

Figure 9: Optimal technology pathways for a CO2 tax between 45 USD and 130 USD
per tonne showing the technologies used satisfy the demands of university
campus; pathways that are not selected are blurred out. Product prices are
displayed in the legend for a CO2 0.075 USD per tonne, flows going in and/or
out of each stakeholder are reported in the units of the legend, and the
number of technology units required are reported above each technology. This
pathway has decarbonized all of the university’s demands besides the one for
steam which still relies on natural gas fired steam generation.

For CO2 taxes above 130 USD per tonne, the optimal pathway is shown in Figure
10. This pathway has no CO2 emissions because it has switched to using
hydrogen fuel cell CHP to meet the demands for steam and some of the
electricity. Fuel cells are used to supplement the electricity for use by the
electric vehicles for distance, the water chiller for cooling, and the
universities demand for electricity. This has the highest utility cost at 55%
higher than when there is no CO2 tax. This pathway also has the highest steam
price as the fuel cell CHP plant is the most expensive to operate of the
options for CHP, while all other prices remained constant between the three
different pathways.

Figure 10: Optimal technology pathways for a CO2 taxes above 130 USD per tonne
showing the technologies used satisfy the demands of university campus;
pathways that are not selected are blurred out. Product prices are displayed
in the legend, flows going in and/or out of each stakeholder are reported in
the units of the legend, and the number of technology units required are
reported above each technology. This pathway has decarbonized all of the
university’s demands through the use of solar power and hydrogen fuel cell CHP
and hydrogen fuel cells.

The management model results identify that meeting the demand for steam to
heat the buildings seems to be a major factor that drives technology
selection. Additionally, the choice of using electric vehicles was consistent
for the three different pathways because they have the lowest cost to operate
per distance and the low cost of power from renewables. Solar power is chosen
over wind power for three pathways because it has a lower operating cost per
MWh produced than wind. With the use of solar power in each pathway, the
hydrogen price remained constant at 0.97 USD/kg H2 for the three different
pathways. The electricity price also remained constant at 62.3 USD/MWh as it
was set by the solar power to electrolysis to fuel cell pathway operating
cost. This made the price for distance (km) and chilled water to remain
constant.

Steam price did not remain constant across the three different pathways. Under
no CO2 tax (Figure 7) the steam price is -0.007 USD/kg. The negative value
means that the natural gas fired CHP would be willing to pay for the steam to
be taken because it already profits from electricity price being set by the
fuel cell pathway. When the steam is instead produced by the natural gas fired
steam generator (Figure 9) the steam price is no longer tied to the
electricity price. The operating cost of the steam generator and the tax on
the CO2 released must be covered by the steam demand and a positive price of
0.016 USD/kg steam is observed at a CO2 tax of 0.075 USD/kg CO2. When the
steam is produced by the hydrogen fuel cell CHP and there are no CO2 emissions
(Figure 9) the steam price is 0.022 USD/kg steam. For this pathway, the
positive price is a result of the revenue required from the steam to offset
the higher operating cost of the hydrogen fuel cell CHP than the hydrogen fuel
cell.

####  3.3.2 Investment Model

The investment model allows for the further consideration of a budget on
choosing the technology pathways under different conditions. The key condition
to incentives reducing CO2 emissions will be a CO2 tax, and analysis will
involve solving the model for each value in a range. However, the impact of
the budget has to also be considered. This means for each CO2 tax, the
investment model will have to optimize for each budget at each CO2 tax. This
is more computationally expensive than the management model because of the
binary variables, but it is still readily solvable for each CO2 tax and budget
combination.

Figure 11 shows the CO2 emissions (a) and utility cost of the university (b)
at varying CO2 taxes and at varying investment budgets. When the budget is
low, little can be done to shift away from the existing infrastructure so as
the CO2 tax increases, the emissions stay the same and the utility cost
increases. This can be observed for all CO2 taxes when the budget is less than
80 million USD. While at sufficiently large budgets ($\geq$2.7 billion USD),
the potential pathways match the results of the management model and the same
dependence on the CO2 tax is observed. This occurs because the investment
model has a high enough budget that it can choose any technology it desires,
essentially eliminating the budget constraint.

Figure 11: Dependence of the annual (a) CO2 output and (b) utility cost of the
university on the CO2 tax and budget when all possible technologies are
available for purchase and use.

As the budget increases above 80 million USD, technologies are chosen to
reduce the impact of the CO2 tax. Figure 12, shows the pathway at a budget of
100 million USD and a CO2 of 200 USD per tonne. This pathway is at a point
where the budget is insufficient to allow for a complete switch to technology
alternatives. Based on the management results, the optimal pathway at this CO2
tax would be the one observed in Figure 10; however, the budget is not high
enough for this pathway to be possible. Instead, the pathway chosen uses the
existing gasoline vehicles to meet the distance demand, the natural gas
powered CHP, and grid power which all produce CO2. There is some use of solar
power for production of hydrogen which is then used by a single fuel cell CHP
plant. The number of solar power units here are less than in the management
model. The inability to offset the CO2 tax results in a 2.4 times larger price
for the power in this pathway than what the price would be without any budget
constraint.

Figure 12: Investment model results at a CO2 tax of 200 USD per tonne and a
budget of 100 million USD showing the technologies used satisfy the demands of
university campus; pathways that are not selected are blurred out. Product
prices are displayed in the legend, flows going in and/or out of each
stakeholder are reported in the units of the legend, and the number of
technology units required are reported above each technology. This pathway
makes use of some solar power to hydrogen for a fuel cell CHP unit however not
demand is completely decarbonized.

Another point where the budget limits the pathway options is at a budget of 1
billion USD and a CO2 tax of 75 USD per tonne. At this CO2 tax, the optimal
pathway without any budget constraint would be the one displayed in Figure 9
which had decarbonized everything besides the demand for steam. However, the
budget limits the pathway to continue using the natural gas CHP plant and the
gasoline vehicles. There is more use of solar power than in Figure 12 due to
the higher budget and more fuel cell CHP units, but it is still insufficient
to meet the full demand for steam and electricity. Fuel cell units are used in
this pathway to supplement the electricity from the natural gas and fuel cell
CHP as well. This pathway has higher prices than in the budget unconstrained
case with a 35% higher power price. This is a result of the electricity price
being impacted by the operating cost of the natural gas fired CHP and the tax
on the CO2 it emits.

Figure 13: Investment model results at a CO2 tax of 75 USD per tonne and a
budget of 1 billion USD showing the technologies used satisfy the demands of
university campus; pathways that are not selected are blurred out. Product
prices are displayed in the legend, flows going in and/or out of each
stakeholder are reported in the units of the legend, and the number of
technology units required are reported above each technology. This pathway
uses no grid power but still uses natural gas fired CHP and steam generators.
Solar power is used to produce hydrogen to be used by the fuel cell CHP and
fuel cells.

The impact of the CO2 and budget constraint on the hydrogen production and
hydrogen price is reported in Figure 14. Hydrogen production is highest in the
pathway described in Figure 9 because it relies on fuel cells to meet the
entire electricity use and demand. Hydrogen production is lower when CHP is
used, because they are more efficient. The hydrogen price is 0.97 USD per kg;
however, it spikes to as high as 3.30 USD per kg when the budget is 810
million USD and the CO2 tax is 240 USD per tonne. This pathway is displayed in
Figure 15. The high hydrogen price is caused by the use of a hydrogen vehicle.
By participating in the same demand, the hydrogen price becomes tied to the
cost of using the most expensive technology used. For this pathway and
condition, gasoline vehicles are the most expensive because of the high CO2
tax and lower efficiency than the hybrid vehicles. This effect on hydrogen
price demonstrates the importance of understanding how intermediate products
are priced as a result of being used in competing pathways; specifically, the
inherent value of intermediate products is inherently tied to the final uses.

Figure 14: Dependence of (a) hydrogen production and (b) hydrogen prices on
the CO2 tax and budget when all possible technologies are available for
purchase and use. Figure 15: Investment model results at a CO2 tax of 240 USD
per tonne and a budget of 810 million showing the technologies used satisfy
the demands of the campus; pathways that are not selected are blurred out.
Product prices are displayed in the legend, flows going in and/or out of each
stakeholder are reported in the units of the legend, and the number of
technology units required are reported above each technology. This pathway
makes uses a mix of gasoline, hybrid, and a single hydrogen fuel cell vehicle
to meet the distance (km) demand of the university resulting in a high
hydrogen price.

The investment model results demonstrate the impact of the budget constraint
on technology selection under different budget and CO2 tax conditions. These
technology selections impact the utility cost for the university and the
prices of products. When the budget is low, little can be done to compensate
for the CO2 tax leading to high utility costs and prices. As the budget is
increased, technology selections favor the use of solar power production for
hydrogen production due to the low operating cost and greatest impact
offsetting the CO2 emissions associated with the use of grid power.
Additionally, the investment model results indicate that there are some
budgets where potential technologies (e.g., hybrid vehicles) are chosen
despite them not being used in the corresponding budget unconstrained case.
This suggests that, periodic investments of 100 million USD per year are made
for 10 years, may result in a different technology pathway compared to a
single investment of 1 billion USD.

##  4 Conclusions and Future Work

We have presented an optimization framework for conducting technology pathway
analysis. The proposed framework uses a graph abstraction that captures
interdependencies between products and technologies and diverse factors that
are of interest for high-level analysis such as techno-economic factors,
economies of scale, efficiencies, and externalities (e.g., policy). Duality
analysis also reveals that the optimization formulation has a natural economic
interpretation that reveals how stakeholders participating in the system can
be remunerated and that also reveals in inherent value of intermediate
products. We have illustrated the use of the framework for studying technology
pathways that enable decarbonization of a representative university campus.
Here, we show how the framework can model diverse types of products,
technologies, and externalities. This analysis identified key technologies and
products and demonstrated the importance of understanding product pricing
dependencies that can arise as decarbonization occurs. As part of future work,
we will explore the use of the proposed framework to decarbonize other systems
(such as plastics manufacturing) and to understand the impact of additional
technologies (such as ammonia production) and connections with other sectors
(such as agriculture).

## Acknowledgments

We acknowledge funding from NSF CAREER award CBET-1748516.

## References

  * [1] MGE West Campus Cogeneration Facility. URL: www.mge.com.
  * [2] NCES Digest of Education Statistics. URL: nces.ed.gov.  Publisher: National Center for Education Statistics.
  * [3] NREL 2020 Transportation Annual Technology Baseline. URL: atb.nrel.gov.
  * [4] University of Wisconsin–Madison Facts. URL: www.wisc.edu.
  * [5] US DOE Alternative Fuels Data Center: Electricity Basics. URL: afdc.energy.gov.
  * [6] US DOE Alternative Fuels Data Center: Hydrogen Basics. URL: afdc.energy.gov.
  * [7] US DOE Alternative Fuels Data Center: Hydrogen Fuel Cell Electric Vehicle Availability. URL: afdc.energy.gov.
  * [8] US DOE Overview of CHP Technologies, Nov. 2017.
  * [9] UW-Madison Fleet Rates. URL: transportation.wisc.edu, July 2020.
  * [10] World Bank Group US CO2 Emissions. URL: data.worldbank.org, 2020.
  * [11] Madison Gas and Electric Company Electric Rates and Rules, Dec. 2021\.
  * [12] EIA Consttruction Cost For Electric Generators. URL: www.eia.gov, Aug. 2022.
  * [13] EIA Levelized Costs of New Generation Resources in the Annual Energy Outlook 2022. URL: www.eia.gov, Mar. 2022.
  * [14] EIA U.S. Natural Gas Industrial Price. URL: www.eia.gov, Aug. 2022\.
  * [15] EIA U.S. Retail Gasoline Prices. URL: www.eia.gov, Sept. 2022.
  * [16] EIA Wisconsin State Energy Profile. URL: www.eia.gov, July 2022.
  * [17] NOAA Annual Climatological Report. URL: forecast.weather.gov, Jan. 2022\.  Publisher: NOAA’s National Weather Service.
  * [18] USDA Land Values 2022 Summary, Aug. 2022.
  * [19] G. Coppez, S. Chowdhury, and S. Chowdhury.  The importance of energy storage in Renewable Power Generation: A review.  In 45th International Universities Power Engineering Conference UPEC2010, pages 1–5, Aug. 2010.
  * [20] K. Darrow, R. Tidball, J. Wang, and A. Hampson.  Catalog of CHP Technologies Section 7.pdf, Sept. 2017.
  * [21] P. Denholm, M. Hand, M. Jackson, and S. Ong.  Land Use Requirements of Modern Wind Power Plants in the United States.  Technical Report NREL/TP-6A2-45834, 964608, Aug. 2009.
  * [22] A. Frank.  University of Wisconsin-Madison STARS Report, July 2022.
  * [23] B. Grill.  The UW-Madison Heating Plants: Staving Off Wisconsin Winters, Oct. 2016.
  * [24] C. Harto.  Electric Vehicle Ownership Costs: Chapter 2—Maintenance.  page 4, Sept. 2020.
  * [25] Y.-F. Ho, C.-C. Chang, C.-C. Wei, and H.-L. Wang.  Multi-objective programming model for energy conservation and renewable energy structure of a low carbon campus.  Energy and Buildings, 80:461–468, Sept. 2014.
  * [26] N. Jandl.  A Lake Runs Through It: How the Charter Street Heating & Cooling Plant Helps UW-Madison Stay Comfortable Year-Round, Feb. 2019.
  * [27] J. Khan and M. H. Arsalan.  Solar power technologies for sustainable electricity generation – A review.  Renewable and Sustainable Energy Reviews, 55:414–425, Mar. 2016\.
  * [28] O. Lah.  Decarbonizing the transportation sector: policy options, synergies, and institutions to deliver on a low-carbon stabilization pathway.  WIREs Energy and Environment, 6(6):e257, 2017.  _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/wene.257.
  * [29] G. Luderer, S. Madeddu, L. Merfort, F. Ueckerdt, M. Pehl, R. Pietzcker, M. Rottoli, F. Schreyer, N. Bauer, L. Baumstark, C. Bertram, A. Dirnaichner, F. Humpenöder, A. Levesque, A. Popp, R. Rodrigues, J. Strefler, and E. Kriegler.  Impact of declining renewable energy costs on electrification in low-emission scenarios.  Nature Energy, 7(1):32–42, Jan. 2022.  Number: 1 Publisher: Nature Publishing Group.
  * [30] J. Marcinkoski, J. Spendelow, A. Wilson, and D. Papageorgopoulos.  DOE Hydrogen and Fuel Cells Program Record #15015.  Technical Report 15015, Sept. 2015.
  * [31] V. Masson-Delmotte, H.-O. Pörtner, J. Skea, P. Zhai, D. Roberts, P. R. Shukla, A. Pirani, R. Pidcock, Y. Chen, E. Lonnoy, W. Moufouma-Okia, C. Péan, S. Connors, J. B. R. Matthews, X. Zhou, M. I. Gomis, T. Maycock, M. Tignor, and T. Waterfield.  An IPCC Special Report on the impacts of global warming of 1.5°C above pre-industrial levels and related global greenhouse gas emission pathways, in the context of strengthening the global response to the threat of climate change, sustainable development, and efforts to eradicate poverty.  page 630.
  * [32] M. W. Menezes.  Department of Energy Hydrogen Program Plan.  Technical report, Nov. 2020.
  * [33] N. Mirzaei Alavijeh, D. Steen, Z. Norwood, L. Anh Tuan, and C. Agathokleous.  Cost-Effectiveness of Carbon Emission Abatement Strategies for a Local Multi-Energy System—A Case Study of Chalmers University of Technology Campus.  Energies, 13(7):1626, Jan. 2020.  Number: 7 Publisher: Multidisciplinary Digital Publishing Institute.
  * [34] C. of the Parties to the United Nations Framework Convention on Climate Change (21st sess. : 2015 : Paris).  Report of the Conference of the Parties on its 21st Session.  Technical report, Jan. 2016.  Publisher: UN,.
  * [35] D. J. Olsen, N. Zhang, C. Kang, M. A. Ortega-Vazquez, and D. S. Kirschen.  Planning Low-Carbon Campus Energy Hubs.  IEEE Transactions on Power Systems, 34(3):1895–1907, May 2019.  Conference Name: IEEE Transactions on Power Systems.
  * [36] S. Ong, C. Campbell, P. Denholm, R. Margolis, and G. Heath.  Land-Use Requirements for Solar Power Plants in the United States.  Technical Report NREL/TP-6A20-56290, 1086349, June 2013.
  * [37] A. Roberts, B. Thomas, P. Sewell, Z. Khan, S. Balmain, and J. Gillman.  Current tidal power technologies and their suitability for applications in coastal and marine areas.  Journal of Ocean Engineering and Marine Energy, 2(2):227–245, May 2016.
  * [38] A. M. Sampat, Y. Hu, M. Sharara, H. Aguirre-Villegas, G. Ruiz-Mercado, R. A. Larson, and V. M. Zavala.  Coordinated management of organic waste and derived products.  Computers & Chemical Engineering, 128:352–363, Sept. 2019.
  * [39] P. Sinha, W. A. Schew, A. Sawant, K. J. Kolwaite, and S. A. Strode.  Greenhouse Gas Emissions from U.S. Institutions of Higher Education.  Journal of the Air & Waste Management Association, 60(5):568–573, May 2010.  Publisher: Taylor & Francis _eprint: https://doi.org/10.3155/1047-3289.60.5.568.
  * [40] X. Tian, Y. Zhou, B. Morris, and F. You.  Sustainable design of Cornell University campus energy systems toward climate neutrality and 100% renewables.  Renewable and Sustainable Energy Reviews, 161:112383, June 2022\.
  * [41] P. A. Tominac and V. M. Zavala.  Economic properties of multi-product supply chains.  Computers & Chemical Engineering, 145:107157, Feb. 2021.
  * [42] A. Tullo.  CF plans green ammonia plant in Louisiana, Nov. 2020.
  * [43] A. Tullo.  Industry prepares ammonia for a second life as a fuel for the future, Mar. 2021.
  * [44] O. US EPA.  Inventory of U.S. Greenhouse Gas Emissions and Sinks, Feb. 2017.
  * [45] S. A. Vargas, G. R. T. Esteves, P. M. Maçaira, B. Q. Bastos, F. L. Cyrino Oliveira, and R. C. Souza.  Wind power generation: A review and a research agenda.  Journal of Cleaner Production, 218:850–870, May 2019.
  * [46] P. Wang, X. Deng, H. Zhou, and S. Yu.  Estimates of the social cost of carbon: A review based on meta-analysis.  Journal of Cleaner Production, 209:1494–1507, Feb. 2019.
  * [47] T. Wang, H. Zhang, Y. Zhang, H. Wang, J. Lyu, and G. Yue.  Efficiency and emissions of gas-fired industrial boiler fueled with hydrogen-enriched nature gas: A case study of 108 t/h steam boiler.  International Journal of Hydrogen Energy, 47(65):28188–28203, July 2022.
  * [48] M. M. Whiston, I. Azevedo, S. Litster, K. Whitefoot, C. Samaras, and J. Whitacre.  Total Cost of Ownership of Fuel Cell Electric Vehicles Using Expert Assessments.  ECS Meeting Abstracts, MA2018-02(42):1419, July 2018.  Publisher: IOP Publishing.
  * [49] S. Öberg, M. Odenberger, and F. Johnsson.  Exploring the competitiveness of hydrogen-fueled gas turbines in future energy systems.  International Journal of Hydrogen Energy, 47(1):624–644, Jan. 2022\.

## Appendix

Table 1: Acronyms for technologies considered in University campus study. Acronym | Meaning
---|---
CHPNG | Combined Heat and Power Plant (using natural gas)
SGNG | Steam Generator (using natural gas)
WC | Water Chiller
GP | Grid Power
GV | Gasoline Vehicle
SP | Solar Power (Photovoltaic)
WP | Wind Power
WS | Water Splitting (Electrolysis)
CHPB | Combined Heat and Power Plant (using blend of natural gas and hydrogen)
CHPH2 | Combined Heat and Power Plant (using hydrogen)
CHPFC | Combined Heat and Power Plant (using Hydrogen Fuel Cell)
SGH2 | Steam Generator (using Hydrogen)
FC | Hydrogen Fuel Cell
H2V | Hydrogen Fuel Cell Vehicle
EV | Electric Vehicle
HV | Hybrid Gasoline Vehicle
Table 2: University Campus Annual Demands (dashed value(s) represent variable or not constrained) Product | Quantity | Current Price | Reference
---|---|---|---
Electricity | $4.03\times 10^{5}~{}MWh$ | $99.8~{}USD/MWh$ | [22, 11]
Steam | $1.04\times 10^{9}~{}kg~{}steam$ | $0.01~{}USD/kg~{}steam$ | [22]
Chilled Water | $2.43\times 10^{1}0~{}kg~{}chilled$ | $0.00023~{}USD/kg~{}chilled$ | [22]
Distance | $7.30\times 10^{6}~{}km$ | $0.10~{}USD/km$ | [9, 22]
CO2 | $-~{}kg~{}CO_{2}$ | $0~{}USD/kg~{}CO_{2}$ |
Table 3: University Campus Annual Available Supplies Product | Quantity | Price | Reference
---|---|---|---
Grid Power | $1\times 10^{10}~{}MWh~{}grid$ | $99.8~{}USD/MWh~{}grid$ | [11]
Natural Gas | $1\times 10^{10}~{}kg~{}CH_{4}$ | $0.14~{}USD/kg~{}CH_{4}$ | [14]
Gasoline | $1\times 10^{10}~{}gal~{}gas$ | $3.10~{}USD/gal~{}gas$ | [15]
Solar Power | $1\times 10^{10}~{}MWh~{}solar$ | $0~{}USD/MWh~{}solar$ |
Wind Power | $1\times 10^{10}~{}MWh~{}wind$ | $0~{}USD/MWh~{}wind$ |
Land | $1\times 10^{10}~{}m^{2}$ | $0~{}USD/m^{2}$ |
Table 4: Inputs and Outputs of the Current University Campus Technologies Supply | Technology | Equation
---|---|---
CHPNG | Natural Gas CHP | $253~{}kg~{}CH_{4}\rightarrow 1500~{}kg~{}steam+1~{}MWh+760~{}kg~{}CO_{2}$
SGNG | Natural Gas SG | $1~{}kg~{}CH_{4}\rightarrow 23~{}kg~{}steam+2.8~{}kg~{}CO_{2}$
WC | Water Chiller | $0.024~{}MWh\rightarrow 1000~{}kg~{}chilled$
GP | Grid Power | $1~{}MWh~{}grid\rightarrow 1~{}MWh+560~{}kg~{}CO_{2}$
GV | Gasoline Vehicle | $1~{}gal~{}gas\rightarrow 40~{}km+8~{}kg~{}CO_{2}$
Table 5: Economic and Capacity Properties of the Current University Campus Technologies (dashed values are treated as unconstrained) Acronym | Technology | Operating Costs | Annual Capacity | Reference
---|---|---|---|---
CHPNG | Natural Gas CHP | $0.063~{}USD/kg~{}CH_{4}$ | $3.3\times 10^{8}~{}kg~{}CH_{4}$ | [20, 1]
SGNG | Natural Gas SG | $0.03~{}USD/kg~{}CH_{4}$ | $2.9\times 10^{8}~{}kg~{}CH_{4}$ | [26, 23]
WC | Water Chiller | $1~{}USD/MWh$ | $-~{}MWh$ | [26]
GP | Grid Power | $0~{}USD/MWh~{}grid$ | $-~{}MWh~{}grid$ | [16]
GV | Gasoline Vehicle | $1.52~{}USD/gal~{}gas$ | $2.9\times 10^{5}~{}gal~{}gas$ | [3, 24]
Table 6: Inputs and Outputs of Potential Technologies Acronym | Technology | Equation
---|---|---
SP | Solar Power | $1~{}MWh~{}solar\rightarrow 1MWh~{}renew$
WP | Wind Power | $1~{}MWh~{}wind\rightarrow 1~{}MWh~{}renew$
WS | Electrolysis | $0.043~{}MWh~{}renew\rightarrow 1~{}kg~{}H_{2}$
CHPB | Blend CHP | $236~{}kg~{}CH_{4}+7.3~{}kg~{}H_{2}\rightarrow 1500~{}kg~{}steam+1~{}MWh+707~{}kg~{}CO_{2}$
CHPH2 | Hydrogen CHP | $107~{}kg~{}H_{2}\rightarrow 1500~{}kg~{}steam+1~{}MWh$
CHPFC | Hydrogen Fuel Cell CHP | $60~{}kg~{}H_{2}\rightarrow 1414~{}kg~{}steam+1~{}MWh$
SGH2 | Hydrogen SG | $1~{}kg~{}H_{2}\rightarrow 46~{}kg~{}steam$
FC | Fuel Cell | $1~{}kg~{}H_{2}\rightarrow 0.02~{}MWh$
H2V | Hydrogen Vehicle | $1~{}kg~{}H_{2}\rightarrow 100~{}km$
EV | Electric Vehicle | $1~{}MWh\rightarrow 100~{}km$
HV | Hybrid Vehicle | $1~{}gal~{}gas\rightarrow 60~{}km+8~{}kg~{}CO_{2}$
Table 7: Economic and Capacity Properties of Potential Technologies Acronym | Operating Costs | Annual Capacity | Investment Cost | Units | Reference
---|---|---|---|---|---
SP | $8.54~{}USD/MWh~{}solar$ | $5000~{}MWh~{}solar$ | $234,650~{}USD$ | $750$ | [13, 36, 18]
WP | $10.21~{}USD/MWh~{}wind$ | $5000~{}MWh~{}wind$ | $893,550~{}USD$ | $750$ | [13, 21, 18]
WS | $0.018~{}USD/MWh~{}renew$ | $5000~{}MWh~{}renew$ | $58,250~{}USD$ | $700$ | [32]
CHPB | $0.066~{}USD/kg~{}CH_{4}$ | $3.1\times 10^{8}~{}kg~{}CH_{4}$ | $763,000~{}USD$ | $1$ | [1, 49, 12]
CHPH2 | $0.15~{}USD/kg~{}H_{2}$ | $1.4\times 10^{8}~{}kg~{}H_{2}$ | $20,000,000~{}USD$ | $1$ | [1, 49, 12]
CHPFC | $0.60~{}USD/kg~{}H_{2}$ | $1.5\times 10^{6}~{}kg~{}H_{2}$ | $12,880,000~{}USD$ | $17$ | [8]
SGH2 | $0.07~{}USD/kg~{}H_{2}$ | $4.4\times 10^{7}~{}kg~{}H_{2}$ | $0~{}USD$ | $1$ | [47]
FC | $0.28~{}USD/kg~{}H_{2}$ | $2.5\times 10^{5}~{}kg~{}H_{2}$ | $12,500,000~{}USD$ | $100$ | [32]
H2V | $3.11~{}USD/kg~{}H_{2}$ | $80~{}kg~{}H_{2}$ | $50,000~{}USD$ | $913$ | [3, 24, 48]
EV | $66.4~{}USD/MWh$ | $2.3~{}MWh$ | $40,000~{}USD$ | $913$ | [3, 24]
HV | $2.05~{}USD/gal~{}gas$ | $133~{}gal~{}gas$ | $30,000~{}USD$ | $913$ | [3, 24]