# Remote Renewable Hubs For Carbon-Neutral Synthetic Fuel Production

Mathias Berger 1,∗, David Radu 1, Ghislain Detienne 2, Thierry Deschuyteneer
2, Aurore Richel 3 and Damien Ernst 1

###### Abstract

This paper studies the economics of carbon-neutral synthetic fuel production
from renewable electricity in remote areas where high-quality renewable
resources are abundant. To this end, a graph-based optimisation modelling
framework directly applicable to the strategic planning of remote renewable
energy supply chains is proposed. More precisely, a (hyper)graph abstraction
of planning problems is introduced, wherein nodes can be viewed as
optimisation subproblems with their own parameters, variables, constraints and
local objective. Nodes typically represent a subsystem such as a technology, a
plant or a process. Hyperedges, on the other hand, express the connectivity
between subsystems. The framework is leveraged to study the economics of
carbon-neutral synthetic methane production from solar and wind energy in
North Africa and its delivery to Northwestern European markets. The full
supply chain is modelled in an integrated fashion, which makes it possible to
accurately capture the interaction between various technologies on hourly time
scales. Results suggest that the cost of synthetic methane production and
delivery would be slightly under 150 €/MWh (higher heating value) by 2030 for
a system supplying 10 TWh annually and relying on a combination of solar
photovoltaic and wind power plants, assuming a uniform weighted average cost
of capital of 7%. A comprehensive sensitivity analysis is also carried out in
order to assess the impact of various techno-economic parameters and
assumptions on synthetic methane cost, including the availability of wind
power plants, the investment costs of electrolysis, methanation and direct air
capture plants, their operational flexibility, the energy consumption of
direct air capture plants, and financing costs. The most expensive
configuration (around 200 €/MWh) relies on solar photovoltaic power plants
alone, while the cheapest configuration (around 88 €/MWh) makes use of a
combination of solar PV and wind power plants and is obtained when financing
costs are set to zero.

##  1 Keywords:

optimisation, renewable energy, carbon neutral, synthetic fuels, remote supply
chain, linear programming, structured models, graph

##  2 Introduction

Electricity generation from renewable resources combined with wide-ranging
electrification has been a mainstay of European climate and energy policies,
with the primary goal of decarbonising the power sector as well as other
carbon-intensive sectors.

Major obstacles to such endeavours have nevertheless surfaced in recent years.
Firstly, sectors like aviation, shipping, heating or industry have proved
difficult to fully electrify. Indeed, feedstocks and energy carriers with
specific properties such as a high energy density are typically required
(Eveloy et al., 2021). Hence, the production of carbon-neutral synthetic fuels
and feedstocks from renewable electricity has been the focus of a growing body
of literature. For example, the synthesis of carbon-neutral hydrogen
(Borgschulte, 2016), methane (Biswas et al., 2020), methanol (Centi et al.,
2020) and ammonia (Ghavam et al., 2021) have all been considered. A number of
demonstration projects have been carried out as well (Wulf et al., 2020).
Secondly, it has become clear that the technical renewable potential of some
European countries (i.e., the maximum amount of renewable electricity that may
produced within a country’s borders and exclusive economic zone, while
accounting for a variety of land eligibility constraints (Ryberg et al.,
2018)) is insufficient to supply current energy demand levels (e.g., in
densely-populated countries like Belgium (Berger et al., 2020; Limpens et al.,
2020) or the United Kingdom (MacKay, 2008)). It is still unclear whether
pooling renewable resources at the European level would alleviate the problem.
On the other hand, it is well-documented that social acceptance issues tend to
compound it (Segreto et al., 2020).

A simple solution consists in harvesting renewable resources in remote areas
where they are abundant, synthesising carbon-neutral fuels or feedstocks using
renewable electricity and transporting them back to demand centres (Fasihi et
al., 2015; Chapman et al., 2017; Heuser et al., 2019). However, two conditions
must be satisfied for such an approach to be worth pursuing. Firstly,
transport should be energy-efficient and cost-effective. This will often
depend on the physics of the commodity considered and the maturity of
technologies available to handle it. Secondly, very-high-quality renewable
resources should be tapped. The quality of such resources is typically
estimated via the annual capacity factor of a given technology harnessing
them, which directly reflects the amount of electricity that may be produced
per unit capacity. Since renewable power generation technologies usually have
very low operating costs, the higher the capacity factor, the lower the
electricity cost. Regions with outstanding resources and vast technical
potential include Patagonia (wind) (Heuser et al., 2019), North Africa (sun
and wind) (Fasihi et al., 2015) and Greenland (wind) (Radu et al., 2019).
Providing an accurate quantitative assessment of the economics and efficiency
of such remote renewable energy supply chains and pathways is critical to
evaluate future sustainable energy supply options available to policy makers
and society at large as well as to identify where to direct future research
and innovation efforts.

From a conceptual standpoint, a supply chain can be viewed as a networked
system composed of dynamical subsystems interacting with each other. In order
to tackle the problem formulated above, the collection of processes and
technologies forming a remote renewable energy supply chain must be analysed
in an integrated fashion, which makes it possible to properly capture the
interactions between subsystems in space and time. In addition, a sufficient
level of technical detail and temporal resolution should be used to properly
model their operation (Poncelet et al., 2016). This paper formalises these
considerations and proposes a graph-based optimisation modelling framework
directly applicable to the strategic planning and analysis of remote renewable
energy supply chains. More precisely, a graph abstraction of planning problems
is introduced, wherein nodes can be viewed as optimisation subproblems with
their own parameters, variables, constraints and local objective, and
typically represent a subsystem such as a technology, a plant or a process.
Edges, on the other hand, express the connectivity between subsystems. The
framework is then leveraged to study the economics of carbon-neutral synthetic
methane production from renewable electricity and atmospheric carbon dioxide
in North Africa and its delivery to Northwestern European markets. Synthetic
methane is an appealing carbon-neutral energy carrier, as some downstream
transport infrastructure is readily available in Northwestern European
countries, and the liquefied methane chain is mature and cost-effective
(Timera Energy, 2018). It would also obviate the need for replacing or
upgrading appliances and processes presently used for residential heating and
in industry that a switch to other fuels would entail. In this paper, the
carbon-neutral synthetic methane supply chain is modelled end-to-end, from
power generation in North Africa to methane regasification in Northwestern
Europe. A detailed description of each process and technology is provided,
along with comprehensive data resources. The modelling framework also served
as a basis for the development of an open source optimisation modelling
language (Berger et al., 2021) and tool (Miftari et al., 2021). In the
interest of transparency, the input files and full data enabling others to
reproduce the analyses presented in this paper are also available in the
associated repository (Miftari et al., 2021).

This paper is structured as follows. Section 3 reviews the relevant
literature. Section 4 details the proposed modelling framework, while Sections
5 and 6 describe the case study and discuss results, respectively. Finally,
Section 7 concludes the paper and discusses future work directions.

##  3 Literature Review

To the best of the authors’ knowledge, Hashimoto et al. (1999) were the first
to suggest the production of hydrogen from renewable electricity in remote
areas followed by the synthesis of hydrocarbons using captured carbon dioxide
as a means of producing carbon-neutral fuels. The paper, however, did not
provide a quantitative techno-economic analysis of the proposed supply chains.
By contrast, Zeman and Keith (2008) performed one of the first quantitative
economic analyses of carbon-neutral synthetic fuel production using carbon-
neutral hydrogen and atmospheric carbon dioxide. Production cost estimates for
this route were found to be between 23.5 and 30.0 US$/GJ (which would roughly
correspond to 74.1 and 94.6 €/MWh, using the 2020 average exchange rate of
$1.142 for 1.0€). The production of carbon-neutral synthetic methane and
liquid fuels in remote areas with abundant renewable resources has been
considered in Fasihi et al. (2015) and Fasihi et al. (2017), respectively. In
the first study, the authors estimate that the cost of producing synthetic
methane from renewable electricity in the Maghreb and North Africa
(specifically in central and southern Algeria) and delivering it to Japan
could be around 65-75 €/MWh by 2030 for a hybrid solar-wind system, assuming a
uniform weighted average cost of capital (WACC) of 7%. It is not specified
whether the higher heating value (HHV) or the lower heating value (LHV) of
methane was used to compute these costs. In the second study, the cost of
producing synthetic methane in the same region and delivering it to Finland is
found to be between 100-110 €/MWh (HHV) by 2030 and between 90-100 €/MWh (HHV)
by 2040, respectively, using a WACC of 7%. Finally, the economics of carbon-
neutral fuel production is also analysed in Agora Verkehrswende, Agora
Energiewende and Frontier Economics (2018). Cost estimates close to 140-150
€/MWh (LHV) by 2030 and 110 €/MWh (LHV) by 2050 (using a WACC of 6% in both
cases) are found for synthetic methane production in North Africa
(specifically in central and southern Algeria) and delivery to Germany based
on both solar energy alone and hybrid systems combining solar and wind power
plants.

It is also informative to review the modelling approaches followed in these
studies. Firstly, Zeman and Keith (2008) do not specify the technologies used
to implement the various conversion processes, and instead rely on a set of
assumptions about conversion efficiencies and the cost of producing input
commodities (in stoichiometric proportions) to come up with a cost estimate
for the final product. Then, Fasihi et al. (2015) resort to a so-called
annual-basis model estimating the annual number of equivalent full load hours
of renewable power production in order to calculate electricity and synthetic
methane costs based on a set of techno-economic assumptions. This method is
equivalent to estimating annual power production and costs using an average
capacity factor value, and the model is therefore not temporally-resolved. A
so-called hourly-basis model enabling the sizing of solar photovoltaic (PV)
and wind power plants is mentioned in Fasihi et al. (2015, 2017), but no
mathematical model is explicitly described and no computer code implementing
it is made available, which makes the approach difficult to interpret and
scrutinise. Somewhat surprisingly, very minor differences in cost estimates
are observed between the annual-basis and hourly-basis models in Fasihi et al.
(2015). In Agora Verkehrswende, Agora Energiewende and Frontier Economics
(2018), an annual full load hour model similar to that of Fasihi et al. (2015)
is used. For systems driven by variable renewable energy resources, it has
been shown that using a high temporal resolution (e.g., hourly) and adopting a
proper level of technical detail (i.e., representing the flexibility of
technologies, or lack thereof) is key for accurately sizing plants and
estimating both investment and operating costs properly (Poncelet et al.,
2016). It is worth noting that the aforementioned papers rely on models that
have both a very low level of technical detail and a very low temporal
resolution. Furthermore, none of these models makes it possible to design the
supply chain in an integrated fashion while properly accounting for
interactions between subsystems. Similar shortcomings can be found in studies
focussing on other energy carriers such as hydrogen (Dagdougui, 2012; Heuser
et al., 2019).

The design and analysis of energy systems and supply chains has often been
tackled using mathematical programming techniques in the literature (Garcia
and You, 2015; Conejo et al., 2016). Different classes of models may be used,
ranging from linear and mixed-integer linear programs (LPs and MILPs) to
nonlinear and mixed-integer (possibly nonconvex) nonlinear programs (NLPs and
MINLPs) (Biegler and Grossmann, 2004). Parameter uncertainty may also be taken
into account (Sahinidis, 2004). The type of model used typically depends on
the research scope, the available computational resources and the data at
hand. For example, the design of a single piece of equipment used in a process
may require NLP or MINLP models to accurately represent its physics and
operating modes (Grossmann, 2002). On the other hand, supply chains can be
viewed as collections of interconnected plants or processes, which themselves
rely on a variety of complex pieces of equipment. Representing each of them in
their full complexity would require vast amounts of data and result in
intractable models. Thus, for the purpose of strategic or high-level system
design analyses, aggregate plant models are typically employed (Chen and
Grossmann, 2017; Montastruc et al., 2019). In such models, mass and energy
conservation laws are enforced at plant level while accounting for basic
operational constraints. Mass and energy balances are also enforced between
interconnected plants in order to guarantee consistency of flows at system
level. Such approaches, which usually rely on LP or MILP models, have for
instance been applied to the design of integrated biorefineries (Kokossis et
al., 2015), the design of power-to-syngas processes (Maggi et al., 2020) and
power-to-chemicals networks (Schack et al., 2016; Liesche et al., 2019). Such
an approach is adopted in this paper, as discussed next.

##  4 Methodology

This section formally introduces the abstract graph-based optimisation
modelling framework and describes how it can be applied in the context of
strategic energy supply chain planning.

###  4.1 Graph-Based Optimisation Modelling Framework

In this paper, supply chain planning problems are formulated as structured
linear programs. These problems typically involve the optimisation of
discrete-time dynamical systems over a finite time horizon and exhibit a
natural block structure that may be encoded by a sparse graph or hypergraph
(Gallo et al., 1993). A graph abstraction is therefore employed to represent
them, wherein nodes model optimisation subproblems, while hyperedges express
the relationships between nodes. A global discretised time horizon and
associated set of time periods common to all nodes are also defined. Each node
is equipped with a set of so-called internal and external (or coupling)
variables. A set of constraints is also defined for each node, along with a
local objective function representing its contribution to a system-wide
objective. Finally, for each hyperedge, constraints involving the coupling
variables of the nodes to which it is incident are defined in order to express
the relationships between nodes. In the following paragraphs, we formally
define variables, constraints, objectives and formulate the abstract model
that encapsulates the class of problems considered.

Let $T$ be the time horizon considered, let $\mathcal{T}=\\{0,1,\dots,T-1\\}$
be the associated set of time periods, and let
$\mathcal{G}=(\mathcal{N},\mathcal{E})$ be a (possibly directed) hypergraph
encoding the block structure of the problem considered, with node set
$\mathcal{N}$ and hyperedge set $\mathcal{E}\subseteq 2^{\mathcal{N}}$ (i.e.,
each hyperedge corresponds to a subset of nodes). Let
$X^{n}\in\mathcal{X}^{n}$ and $Z^{n}\in\mathcal{Z}^{n}$ denote the collection
of internal and coupling variables defined at node $n\in\mathcal{N}$. Note
that all variables are assumed to take values in continuous sets (i.e.,
$\mathcal{X}^{n}$ and $\mathcal{Z}^{n}$ are continuous). In addition, for any
hyperedge $e\in\mathcal{E}$, let $Z^{e}=\\{Z^{n}|n\in e\\}$ denote the
collection of coupling variables associated with each node to which this
hyperedge is incident.

Let $F^{n}$ denote the function defining the local objective at node
$n\in\mathcal{N}$. In this paper, we consider scalar objectives of the form

$F^{n}(X^{n},Z^{n})=f_{0}^{n}(X^{n},Z^{n})+\sum_{t\in\mathcal{T}}f^{n}(X^{n},Z^{n},t),$
(1)

where $f_{0}^{n}$ and $f^{n}$ are (scalar) affine functions of $X^{n}$ and
$Z^{n}$.

Both equality and inequality constraints may be defined at each node
$n\in\mathcal{N}$. More precisely, an arbitrary number of constraints that can
each be expanded over a subset of time periods may be defined. Hence, we
consider equality constraints of the form

$h_{k}^{n}(X^{n},Z^{n},t)=0,\mbox{ }\forall
t\in\mathcal{T}_{k}^{n},\vspace{-5pt}$ (2)

with (scalar) affine functions $h_{k}^{n}$ and index sets
$\mathcal{T}_{k}^{n}\subseteq\mathcal{T}$, $k=1,\ldots,K^{n}$, as well as
inequality constraints

$g_{k}^{n}(X^{n},Z^{n},t)\leq 0,\mbox{ }\forall
t\in\bar{\mathcal{T}}_{k}^{n},\vspace{-5pt}$ (3)

with (scalar) affine functions $g_{k}^{n}$ and index sets
$\bar{\mathcal{T}}_{k}^{n}\subseteq\mathcal{T}$, $k=1,\ldots,\bar{K}^{n}$.

Likewise, both equality and inequality constraints may be defined over any
hyperedge $e\in\mathcal{E}$. These constraints, however, can only involve the
coupling variables of the nodes to which hyperedge $e\in\mathcal{E}$ is
incident (i.e., nodes such that $n\in e$). More precisely, let $H^{e}$ and
$G^{e}$ be affine functions of $Z^{e}$ used to define the equality and
inequality constraints associated with a given hyperedge $e\in\mathcal{E}$.

Using this notation, the class of problems that can be represented in this
framework reads

$\begin{array}[]{rl}\min&\sum_{n\in\mathcal{N}}F^{n}(X^{n},Z^{n})\vspace{3pt}\\\
\hbox{s.t.}&h_{k}^{n}(X^{n},Z^{n},t)=0,\mbox{ }\forall
t\in\mathcal{T}_{k}^{n},\hskip 2.0ptk=1,\ldots K^{n},\hskip 2.0pt\forall
n\in\mathcal{N}\vspace{3pt}\\\ &g_{k}^{n}(X^{n},Z^{n},t)\leq 0,\mbox{ }\forall
t\in\bar{\mathcal{T}}_{k}^{n},\hskip 2.0ptk=1,\ldots\bar{K}^{n},\hskip
2.0pt\forall n\in\mathcal{N}\vspace{3pt}\\\ &H^{e}(Z^{e})=0,\hskip
2.0pt\forall e\in\mathcal{E}\vspace{3pt}\\\ &G^{e}(Z^{e})\leq 0,\hskip
2.0pt\forall e\in\mathcal{E}\vspace{3pt}\\\
&X^{n}\in\mathcal{X}^{n},Z^{n}\in\mathcal{Z}^{n},\hskip 2.0pt\forall
n\in\mathcal{N}.\end{array}$ (4)

Figure 1 schematically illustrates the class of problems that can be modelled
in this framework.

Figure 1: Graph abstraction of a hypothetical problem whose block structure is
represented by four nodes (i.e., $\mathcal{N}=\\{n_{1},n_{2},n_{3},n_{4}\\}$)
and two hyperedges (i.e., $\mathcal{E}=\\{e_{1},e_{2}\\}$, with
$e_{1}=\\{n_{1},n_{2}\\}$ and $e_{2}=\\{n_{2},n_{3},n_{4}\\}$). Note that
$e_{1}$ has only equality constraints while $e_{2}$ has only inequality
constraints.

###  4.2 Application to Energy Supply Chains

The framework presented in Section 4.1 can be readily leveraged to model
energy systems and supply chains. In this case, nodes typically represent a
technology, a plant or a process, while hyperedges may be used to enforce some
coupling between plants. Introducing a few generic (parametrised) nodes and
hyperedges often suffices to model a broad range of system configurations. In
the following, some key modelling assumptions are introduced, along with two
generic nodes and one generic hyperedge, namely conversion and storage nodes,
as well as conservation hyperedges.

####  4.2.1 Modelling Assumptions

Central Planning & Operation Investment decisions are made by a single entity
that also operates the system, and whose goal is to minimise total system
costs.

Perfect Foresight & Knowledge The entity planning and operating the system has
perfect foresight and knowledge, that is, future weather events and demand
patterns, as well as all technical and economic parameters are assumed to be
known with certainty.

Investment & Operational Decisions A static investment model is used, whereby
investment decisions are made at the beginning of the time horizon and assets
are immediately available. Operational decisions are made at hourly time
steps. The investment and operational problems are solved simultaneously.

Technology & Process Models The sizing and operation of technologies are
modelled via a set of affine input-output relations that typically express
mass and energy balances at plant or process level. Input or output dynamics
are considered for some technologies, but only storage technologies have a
simple state space representation.

####  4.2.2 Nodes

Preliminaries In the following developments, Latin letters denote optimisation
variables and indices, while Greek letters indicate parameters.

Conversion Let $n\in\mathcal{N}$ be a node representing a so-called conversion
technology that processes a set of commodities (e.g., an electrolysis plant
splits water into hydrogen and oxygen using an electric current and therefore
processes four commodities). Commodity flows are modelled as external
variables. An index $i\in\mathcal{I}^{n}$ is thus assigned to each commodity
(i.e., each $i\in\mathcal{I}^{n}$ corresponds to a time-indexed external
variable). The processing of commodities by technology $n$ is modelled via a
set of linear equations linking the flow of a reference commodity
$r\in\mathcal{I}^{n}$ (e.g., hydrogen may be taken as the reference commodity
for electrolysis plants) to the flow of all other commodities
$i\in\mathcal{I}^{n}\setminus\\{r\\}$, which read

$q_{rt}^{n}-\phi_{i}^{n}q_{i(t+\tau_{i}^{n})}^{n}=0,\mbox{ }\forall
i\in\mathcal{I}^{n}\setminus\\{r\\},\mbox{ }\forall t\in\mathcal{T}^{n},$ (5)

where $q_{it}^{n}\in\mathbb{R}_{+}$ represents the flow of commodity $i$ at
time $t$, $\phi_{i}^{n}\in\mathbb{R}_{+}$ is the so-called conversion factor
between commodity $r$ and $i$ (which may be derived, e.g., from stoichiometric
coefficients or the enthalpy of the underlying reaction), while
$\tau_{i}^{n}\in\mathbb{N}$ is the amount of time that may be required for the
conversion process to take place and $\mathcal{T}^{n}\subseteq\mathcal{T}$ is
a suitable subset of time periods. The capacity of a technology is typically
modelled as an internal variable and defined as the maximum flow of a
reference commodity $r^{\prime}\in\mathcal{I}^{n}$ according to which the
technology is sized. Note that $r^{\prime}$ may be different from $r$ (e.g.,
the size of electrolysis plants is typically expressed in terms of their
electrical capacity, although hydrogen may be the reference commodity used in
Equation (5)). Since a static investment model is considered, capacity
deployments occur at the beginning of the time horizon and remain constant
throughout, i.e.,

$K_{0}^{n}-K_{t}^{n}=0,\mbox{ }\forall t\in\mathcal{T}\setminus\\{0\\},$ (6)

where $K_{t}^{n}\in\mathbb{R}_{+}$ denotes the new capacity of technology $n$.
In the following, $K^{n}$ will be used as shorthand for $K_{0}^{n}$. Thus, the
total capacity of technology $n$ is defined via

$q_{r^{\prime}t}^{n}-\pi_{t}^{n}(\underline{\kappa}^{n}+K^{n})\leq 0,\mbox{
}\forall t\in\mathcal{T},$ (7)

where $\pi_{t}^{n}\in[0,1]$ indicates the availability of technology $n$ at
time $t$ and $\underline{$\kappa$}^{n}\in\mathbb{R}_{+}$ represents the
existing capacity. The so-called availability parameter $\pi_{t}^{n}$ may for
instance represent the instantaneous capacity factor of a renewable power
plant. The maximum capacity of a technology may be bounded, which leads to the
introduction of an additional constraint,

$(\underline{\kappa}^{n}+K^{n})-\bar{\kappa}^{n}\leq 0,$ (8)

with $\bar{\kappa}^{n}\in\mathbb{R}_{+}$ the maximum capacity of technology
$n$ that may be installed. A variety of operational constraints may also be
considered. For instance, some conversion technologies may have a limited
operating range, and may only work if a minimum flow of commodity
$i\in\mathcal{I}^{n}$ is maintained, which can be expressed as

$\mu^{n}(\underline{\kappa}^{n}+K^{n})-\frac{\phi_{i}^{n}}{\phi_{r^{\prime}}^{n}}q_{it}^{n}\leq
0,\mbox{ }\forall t\in\mathcal{T},$ (9)

where $\mu^{n}\in[0,1]$ represents the minimum operating level (as a fraction
of the installed capacity). Since the technology is sized with respect to the
flow of commodity $r^{\prime}$, the flow of a commodity $i\neq r^{\prime}$
must be scaled by the ratio of conversion factors in Eq. (9). The rate at
which the flow of commodity $i\in\mathcal{I}^{n}$ can vary may also be
limited, leading to the introduction of so-called ramping constraints,

$\frac{\phi_{i}^{n}}{\phi_{r^{\prime}}^{n}}(q_{it}^{n}-q_{i(t-1)}^{n})-\Delta_{i,+}^{n}(\underline{\kappa}^{n}+K^{n})\leq
0,\mbox{ }\forall t\in\mathcal{T}\setminus\\{0\\},$ (10)

and

$\frac{\phi_{i}^{n}}{\phi_{r^{\prime}}^{n}}(q_{i(t-1)}^{n}-q_{it}^{n})-\Delta_{i,-}^{n}(\underline{\kappa}^{n}+K^{n})\leq
0,\mbox{ }\forall t\in\mathcal{T}\setminus\\{0\\},$ (11)

with $\Delta_{i,+}^{n}\in[0,1]$ and $\Delta_{i,-}^{n}\in[0,1]$ the maximum
rates at which flows can be ramped up and down (as a fraction of the installed
capacity per unit time), respectively. Finally, the local objective function
associated with this node reads

$F_{n}=\nu(\zeta^{n}+\theta_{f}^{n})K^{n}+\sum_{t\in\mathcal{T}}\theta_{t,v}^{n}q_{r^{\prime}t}^{n}\delta
t,$ (12)

where $\nu\in\mathbb{N}$ is the number of years spanned by the optimisation
horizon, $\zeta^{n}\in\mathbb{R}_{+}$ represents the (annualised) investment
cost (also known as capital expenditure, CAPEX),
$\theta_{f}^{n}\in\mathbb{R}_{+}$ models fixed operation and maintenance (FOM)
costs and $\theta_{t,v}^{n}\in\mathbb{R}_{+}$ represents variable operation
and maintenance (VOM) costs, which may be time-dependent.

Storage Let $n\in\mathcal{N}$ be a node representing a storage technology. A
storage technology is assumed to hold one commodity, although its operation
may involve other commodities (e.g., a compressed hydrogen storage system
stores hydrogen but requires electricity to drive compressors). The inventory
level of the storage system is defined as an internal variable, while the
charge and discharge flows are defined as external variables, respectively.
Let $i\in\mathcal{I}^{n}$ and $j\in\mathcal{I}^{n}$ be the indices of the
in/outflows of the commodity stored in technology $n$, respectively. Then, the
basic equation governing the operation of storage systems describes the
inventory level dynamics and reads

$e_{t+1}^{n}-(1-\eta_{S}^{n})e_{t}^{n}-\eta_{+}^{n}q_{it}^{n}+\frac{1}{\eta_{-}^{n}}q_{jt}^{n}=0,\mbox{
}\forall t\in\mathcal{T}\setminus\\{T-1\\},$ (13)

where $e_{t}^{n}\in\mathbb{R}_{+}$ is the inventory level at time $t$,
$q_{i_{u}t}^{n}\in\mathbb{R}_{+}$ and $q_{i_{y}t}^{n}\in\mathbb{R}_{+}$
represent commodity in- and outflows at time $t$, respectively,
$\eta_{S}^{n}\in[0,1]$ is the self-discharge rate, $\eta_{+}^{n}\in[0,1]$ is
the charge efficiency and $\eta_{-}^{n}\in[0,1]$ is the discharge efficiency.
Charging a storage system may also require the consumption of another
commodity $l\in\mathcal{I}^{n},l\neq i,j$ (e.g., electricity consumed by
compressors), which is typically modelled via an additional external variable
$q_{lt}^{n}\in\mathbb{R}_{+}$ and equations

$q_{lt}^{n}-\phi_{i}^{n}q_{it}^{n}=0,\mbox{ }\forall t\in\mathcal{T}.$ (14)

In order to avoid spurious transient effects in storage operation, inventory
levels are typically required to be equal at the beginning and at the end of
the optimisation horizon,

$e_{0}^{n}-e_{T-1}^{n}=0.$ (15)

The stock capacity of the storage technology is modelled as an internal
variable and it is defined by the maximum inventory level. Since a static
investment model is used, the stock capacity is constant throughout the entire
time horizon, i.e.,

$E_{0}^{n}-E_{t}^{n}=0,\mbox{ }\forall t\in\mathcal{T}\setminus\\{0\\},$ (16)

where $E_{t}^{n}\in\mathbb{R}_{+}$ is the new capacity. In the following,
$E^{n}$ will be used as shorthand for $E_{0}^{n}$. The total storage capacity
is therefore defined via

$e_{t}^{n}-(\underline{\epsilon}^{n}+E^{n})\leq 0,\mbox{ }\forall
t\in\mathcal{T},$ (17)

where $\underline{$\epsilon$}^{n}\in\mathbb{R}_{+}$ denotes the existing stock
capacity. Note that the total stock capacity itself may be constrained,

$(\underline{\epsilon}^{n}+E^{n})-\bar{\epsilon}^{n}\leq 0,$ (18)

with $\bar{\epsilon}^{n}\in\mathbb{R}_{+}$ the maximum stock capacity that may
be deployed. In addition, some storage technologies may require a minimum
inventory level to be maintained, which can be expressed as

$\sigma^{n}(\underline{\epsilon}^{n}+E^{n})-e_{t}^{n}\leq 0,\mbox{
}t\in\mathcal{T},$ (19)

where $\sigma^{n}\in[0,1]$ represents the minimum inventory level (as a
fraction of the stock capacity). The maximum inflow capacity is sized
independently of the stock capacity. For example, in the case of battery
storage systems, this implies that the energy-to-power ratio is not fixed a
priori. The maximum inflow is modelled using an internal variable that is also
constant throughout the time horizon considered, as in Eq. (6). It is defined
as follows

$q_{it}^{n}-(\underline{\kappa}^{n}+K^{n})\leq 0,\mbox{ }\forall
t\in\mathcal{T},$ (20)

where $\underline{$\kappa$}^{n}\in\mathbb{R}_{+}$ denotes the existing flow
capacity and $K^{n}\in\mathbb{R}_{+}$ is used as shorthand for the new
capacity. The maximum in- and outflows may be asymmetric, depending on the
properties of the underlying technology, which is modelled via

$q_{jt}^{n}-\rho^{n}(\underline{\kappa}^{n}+K^{n})\leq 0,\mbox{ }\forall
t\in\mathcal{T},$ (21)

where $\rho^{n}\in\mathbb{R}_{+}$ represents the maximum discharge-to-charge
ratio. Finally, the local objective function associated with this node reads

$F^{n}=\Big{[}\nu(\varsigma^{n}+\vartheta^{n}_{f})E^{n}+\sum_{t\in\mathcal{T}}\vartheta_{t,v}^{n}e_{t}^{n}\Big{]}+\Big{[}\nu(\zeta^{n}+\theta^{n}_{f})K^{n}+\sum_{t\in\mathcal{T}}\theta_{t,v}^{n}q_{it}^{n}\delta
t\Big{]}.$ (22)

where $\varsigma^{n}\in\mathbb{R}_{+}$ and $\zeta^{n}\in\mathbb{R}_{+}$
represent the stock and flow components of CAPEX,
$\vartheta_{f}^{n}\in\mathbb{R}_{+}$ and $\theta_{f}^{n}\in\mathbb{R}_{+}$
model the stock and flow components of FOM costs, while
$\vartheta_{t,v}^{n}\in\mathbb{R}_{+}$ and $\theta_{t,v}^{n}\in\mathbb{R}_{+}$
represent the stock and flow components of VOM costs, which may be time-
dependent.

####  4.2.3 Hyperedges

Conservation Let $e\in\mathcal{E}$ be a so-called conservation hyperedge that
enforces local flow conservation of some commodity. More precisely, one
commodity is associated with a given conservation hyperedge. Let
$i\in\cap_{n\in e}\mathcal{I}^{n}$ denote this commodity (i.e., each node
$n\in e$ has an external variable representing a flow of commodity $i$). In
addition, let us assume that hyperedge $e$ is directed (i.e., it can be
partitioned into two disjoint subsets $e_{T}$ and $e_{H}$ that are called its
tail and head, respectively). Roughly speaking, $e$ can be interpreted as
”going from nodes in $e_{T}$ to nodes in $e_{H}$”. Then, flow conservation of
commodity $i$ over hyperedge $e$ can simply be expressed as

$\sum_{n\in e_{T}}q_{it}^{n}-\sum_{n\in
e_{H}}q_{it}^{n}-\lambda_{t}^{e}=0,\mbox{ }\forall t\in\mathcal{T},$ (23)

where the first two sums on the left-hand side represent the aggregate flows
from the nodes in $e_{T}$ and $e_{H}$ (whose signs depend on the orientation
of $e$), while $\lambda_{t}^{e}\in\mathbb{R}$ represents exogenous withdrawals
or injections that may take place over $e$ at each time period $t$. Note that
Eq. (23) may sometimes be relaxed to a greater-than-or-equal-to inequality
constraint (in which case the net flow must exceed the injections/withdrawals
at each time period).

###  4.3 Implementation

The graph-based modelling framework discussed in Section 4.1 has been used as
a basis for developing an optimisation modelling language for structured
linear and mixed-integer linear programs called the graph-based optimisation
modelling language (GBOML) (Berger et al., 2021). The language blends elements
of both algebraic (Kallrath, 2012) and object-oriented (Schichl, 2004)
modelling languages in order to facilitate problem encoding and post-
processing, promote model re-use and improve portability. The full description
of GBOML, which is beyond the scope of this paper, is detailed in a separate
tutorial paper (Berger et al., 2021). A parser for GBOML, called the GBOML
compiler, has also been implemented in Python 3.8 (using the PLY library), and
is released as open source software (Miftari et al., 2021). The GBOML compiler
directly interfaces with both commercial and open source LP and MILP solvers
(namely Gurobi, CPLEX and Clp/Cbc), enabling users to model problems, interact
with solver APIs, query solutions and retrieve post-processed results in an
integrated fashion. For the sake of transparency and reproducibility, the
input file that encodes the model and full data allowing one to reproduce the
case study and results discussed in Sections 5 and 6 are also provided in the
GBOML repository (Miftari et al., 2021). One instance of the resulting linear
programming model can be solved in about 10 minutes with the homogenous
barrier algorithm (cross-over disabled) of Gurobi 9.1.1 on a laptop with 16 GB
of RAM and a Quad-Core Intel i7 processor clocking at 2.6 GHz.

##  5 Case Study

This case study aims to analyse the economics of producing carbon-neutral
methane from renewable electricity in areas of North Africa where abundant and
high-quality renewable resources are readily available, and exporting it to
Northwestern European markets. More specifically, the entire supply chain is
modelled and optimised in an integrated fashion over a time horizon of five
years with hourly resolution (i.e., $T=43824$, since 2016 is a leap year),
from the remote generation of electricity to the synthesis and liquefaction of
carbon-neutral methane in North Africa, to its eventual delivery and
regasification at a Northwestern European gas terminal. Figure 2 schematically
displays the supply chain considered.

Figure 2: Remote carbon-neutral methane supply chain: electricity is produced
in a remote inland cluster in central Algeria and transported to a coastal
cluster where carbon-neutral methane is synthesised and liquefied for export
to Northwestern European markets. Figure 3: Remote hub system configuration.
Icons represent conversion or storage nodes, while bullets and arrows
schematically represent conservation hyperedges.

###  5.1 System Configuration

A more detailed representation of the system configuration considered in this
study is shown in Figure 3, where icons correspond to conversion or storage
nodes, while bullets and arrows schematically represent conservation
hyperedges. For the sake of readability, the set of nodes is split into three
clusters, which also correspond to the different geographical areas displayed
in Figure 2. The nodes and hyperedges used to model this system are described
in the following subsections.

####  5.1.1 Conversion Nodes

Conversion nodes are discussed in this subsection. Tables 1 and 2 gather the
techno-economic data (2030 estimates) used to model conversion nodes along
with the original data sources and complement the descriptions below. In the
model, power flows are measured in GW (GWh/h), energy is measured in GWh, mass
flows are measured in kt/h, mass is measured in kt, and money is measured in
M€.

Table 1: Technical parameters used to model conversion nodes.

| $\phi_{1}$ | $\phi_{2}$ | $\phi_{3}$ | $\mu$ | $\Delta_{+,-}$
---|---|---|---|---|---
HVDC Interconnection | 0.9499 |  |  |  |
(Xiang et al., 2016; IEA ETSAP, 2014) | - |  |  |  |
Electrolysis | 50.6 | 9.0 | 8.0 | 0.05 | 1.0
(Götz et al., 2016) | GWhel/kt${}_{H_{2}}$ | kt${}_{H_{2}O}$/kt${}_{H_{2}}$ | kt${}_{O_{2}}$/kt${}_{H_{2}}$ | - | -/h
Methanation | 0.5 | 2.75 | 2.25 | 1.0 | 0.0
(Götz et al., 2016; Roensch et al., 2016) | kt${}_{H_{2}}$/kt${}_{CH_{4}}$ | kt${}_{CO_{2}}$/kt${}_{CH_{4}}$ | kt${}_{H_{2}O}$/kt${}_{CH_{4}}$ | - | -/h
Desalination | 0.004 |  |  | 1.0 | 0.0
(International Renewable Energy Agency (2012), IRENA) | GWhel/kt${}_{H_{2}O}$ |  |  | - | -/h
Direct Air Capture | 0.1091 | 0.0438 | 5.0 | 1.0 | 0.0
(Keith et al., 2018) | GWhel/kt${}_{CO_{2}}$ | kt${}_{H_{2}}$/kt${}_{CO_{2}}$ | kt${}_{H_{2}O}$/kt${}_{CO_{2}}$ | - | -/h
CH4 Liquefaction | 0.616 |  |  | 0.0 | 1.0
(Pospisil et al., 2019) | GWhel/kt${}_{LCH_{4}}$ |  |  | - | -/h
LCH4 Carriers | 0.994 |  |  |  |
(Howard Rogers (2018), for the Oxford Institute for Energy Studies) | - |  |  |  |
LCH4 Regasification | 0.98 |  |  |  |
(Pospisil et al., 2019) | - |  |  |  |

Table 2: Economic parameters used to model conversion nodes (2030 estimates).

| CAPEX | FOM ($\theta_{f}$) | VOM ($\theta_{v}$) | Lifetime
---|---|---|---|---
Solar Photovoltaic Panels | 380.0 | 7.25 | 0.0 | 25.0
(Agora Verkehrswende, Agora Energiewende and Frontier Economics, 2018) | M€/GWel | M€/GWel-yr | M€/GWhel | yr
Wind Turbines | 1040.0 | 12.6 | 0.00135 | 30.0
(Danish Energy Agency, 2020b) | M€/GWel | M€/GWel-yr | M€/GWhel | yr
HVDC Interconnection | 480.0 | 7.1 | 0.0 | 40.0
(CIGRE C1.35 Working Group, 2019; EIA, 2018) | M€/GWel | M€/GWel-yr | M€/GWhel | yr
Electrolysis | 600.0 | 30.0 | 0.0 | 15.0
(Danish Energy Agency, 2020c) | M€/GWel | M€/GWel-yr | M€/GWhel | yr
Methanation | 735.0 | 29.4 | 0.0 | 20.0
(International Energy Agency (2019), IEA) | M€/GW${}_{CH_{4}}$ (HHV) | M€/GW${}_{CH_{4}}$-yr (HHV) | M€/GWh${}_{CH_{4}}$ (HHV) | yr
Desalination | 28.08 | 0.0 | 0.000315 | 20.0
(CMI Marseille, 2016) | M€/(kt${}_{H_{2}O}$/h) | M€/(kt${}_{H_{2}O}$/h)-yr | M€/kt${}_{H_{2}O}$ | yr
Direct Air Capture | 4801.4 | 0.0 | 0.0207 | 30.0
(Keith et al., 2018) | M€/(kt${}_{CO_{2}}$/h) | M€/(kt${}_{CO_{2}}$/h)-yr | M€/kt${}_{CO_{2}}$ | yr
CH4 Liquefaction | 5913.0 | 147.825 | 0.0 | 30.0
(Brian Songhurst, 2018) | M€/(kt${}_{LCH_{4}}$/h) | M€/(kt${}_{LCH_{4}}$/h)-yr | M€/kt${}_{LCH_{4}}$ | yr
LCH4 Carriers | 2.537 | 0.12685 | 0.0 | 30.0
(Economic Research Institute for ASEAN and East Asia (2018), ERIA) | M€/kt${}_{LCH_{4}}$ | M€/kt${}_{LCH_{4}}$-yr | M€/kt${}_{LCH_{4}}$ | yr
LCH4 Regasification | 1248.3 | 24.97 | 0.0 | 30.0
(Dongsha et al., 2017) | M€/(kt${}_{CH_{4}}$/h) | M€/(kt${}_{CH_{4}}$/h)-yr | M€/kt${}_{CH_{4}}$ | yr

Solar PV Solar photovoltaic panels are used for power generation. The plants
are modelled with one external variable representing the output power and one
internal variable representing the plant capacity, respectively. Constraints
(7) and (8) are used along with the local objective function (12). In order to
construct the capacity factor time series $\pi_{t}^{n}$, five years
(2015-2019) of irradiance data at hourly resolution are retrieved from the
ERA5 database (European Center for Medium Range Weather Forecasts (2020),
ECMWF) for each grey point in Figure 2 and converted into capacity factors
using a generic transfer function (HOMER, 2020) and TrinaSolar Tallmax M
tilted module data (TrinaSolar, 2017). Sites with a five-year average capacity
factor value exceeding $24.5$% are retained (11 in total, shown by red crosses
in Figure 2) and the associated time series are then aggregated (spatially
averaged) into a single time series $\pi_{t}^{n}$, which is illustrated in
Figure 4 for a set of weekly periods in 2016.

Wind Turbines Wind turbines are used for generating power as well. Wind power
plants are modelled in a similar fashion to solar PV plants, that is, with one
external variable representing the power output and one internal variable
representing the plant capacity, respectively. Constraints (7) and (8) are
used along with the typical local objective function (12). In order to
construct the capacity factor time series $\pi_{t}^{n}$, five years
(2015-2019) of wind speed data at hourly resolution are retrieved from the
ERA5 database (European Center for Medium Range Weather Forecasts (2020),
ECMWF) for each grey point in Figure 2 and converted into capacity factors
using the transfer function of the Vestas V90 turbine available in the
windpowerlib library (Haas et al., 2019). Sites with a five-year average
capacity factor value exceeding $50$% are retained (5 in total, shown by blue
crosses in Figure 2) and the associated time series are then aggregated
(spatially averaged) into a single time series $\pi_{t}^{n}$, which is also
displayed in Figure 4.

Figure 4: Capacity factor time series $\pi_{t}^{n}$ used for solar PV and wind
power plants in 2016. For all other nodes except liquefied methane carriers,
$\pi_{t}^{n}=1$ over the entire time horizon.

HVDC Interconnection Ultra high voltage direct current (HVDC) overhead lines
($800$ or $1100$ kV) are assumed to be used for bulk power transmission from
the first cluster (inland) to the second one (coastal hub) (CIGRE C1.35
Working Group, 2019). Note that the yellow area containing the solar and wind
sites in Figure 2 is assumed to be a copper plate for the purpose of this
study, which implies that solar PV and wind power plants feed directly into
the electricity interconnection and the cost of the infrastructure connecting
power plants to the HVDC interconnection is neglected. Voltage source
converters (VSC) are well-suited for remote applications, as they are self-
commutated and are much more controllable than typical line-commutated (LC)
alternatives, although they are more expensive and have higher conversion
losses (Xiang et al., 2016). In this case, two VSC stations are placed on each
side of an overhead HVDC cable whose length is assumed to be $1000$ km. Losses
in each converter station roughly amount to $1.8$% of the power flowing
through it, while approximately $1.5$% of the power transiting through the
HVDC cable is lost. Combining these figures yields the overall efficiency
reported in Table 1. Economic data shown in Table 2 include the costs of both
converter stations and the cable. The interconnection is modelled using two
external variables and one internal variable. The external variables represent
the power flows into (i.e., leaving the first cluster) and out of (i.e.,
reaching the second cluster) the HVDC interconnection, respectively. The
internal variable is the capacity of the converter-line pair. Investment costs
in lines and converter stations are accounted for in the local objective (12),
along with operating costs.

Electrolysis Plants Proton exchange membrane (also called polymer electrolyte
membrane, PEM) electrolysis plants (Carmo et al., 2013) are used for producing
hydrogen. This technology makes it possible to split water into hydrogen and
oxygen by the passage of an electric current. Hence, the plants are modelled
with four external and one internal variables. The external variables
represent the power and water inflows as well as the hydrogen and oxygen
outflows, while the internal variable is the plant capacity. The reference
commodity $r$ used in Equation (5) is hydrogen, while the commodity
$r^{\prime}$ according to which the technology is sized in Equation (7) is the
power input. Electrolysis plants are also assumed to operate at $20$ bar and
$40$°C. This technology is flexible and can ramp up and down very quickly
(usually within seconds). Hence, no ramping constraints are used. However, a
minimum hydrogen production level around $5-10$% of the nominal capacity must
be maintained when the plant is switched on. Constraints (9) are therefore
used to model plant operation. The usual objective function (12) is also used.

Methanation Plants The carbon dioxide methanation (Sabatier) reaction enables
the conversion of carbon dioxide and hydrogen into methane and water (steam)
and is highly exothermic (i.e., the production of $1$ kg of methane releases
approximately $2.867$ kWh of high-temperature heat (Roensch et al., 2016)). In
this paper, cooled fixed-bed (catalytic) reactors operating at $300$°C and
$20$ bar are assumed to be used to produce synthetic methane via the Sabatier
reaction. Furthermore, carbon dioxide and hydrogen are assumed to be fed in
stoichiometric proportions, and the conversion of reactants is assumed to be
complete (which is facilitated by the use of, e.g., alumina-supported nickel
catalysts (Mills and Steffgen, 1974) that offer good selectivity and are
relatively cheap). Plants are modelled using four external variables and one
internal variable. The external variables represent the hydrogen and carbon
dioxide inflows as well as the methane and water (steam) outflows, while the
internal variable is the plant capacity. Methane is taken as the reference
commodity $r$ used to describe the process in Equation (5) as well as the
reference commodity $r^{\prime}$ used for sizing the plant in Equation (7).
Owing to the exothermicity of the reaction, cooled fixed-bed reactors are very
sensitive to changes in operating parameters such as the feed and coolant
temperatures (Schlereth and Hinrichsen, 2014) or the feed flow rate (Theurich,
2019), and can suffer from pronounced hot-spots or thermal runaway (Schlereth
and Hinrichsen, 2014), which can in turn lead to catalyst sintering and
deactivation. Hence, reactors usually have a (very) limited operating range,
although promising ways of improving this have been proposed (Bremer and
Sundmacher, 2019). Dynamic operation may also involve shutting down the
reactor, keeping it in hot standby and starting it up again (Gorre et al.,
2020), which nevertheless leads to inefficiencies (e.g., the reactor must be
flushed with hydrogen (Gorre et al., 2020)). Finally, note that maintaining
product quality is typically more difficult in unsteady operation. In the
light of these observations, in this paper, it is assumed that methanation
reactors operate in steady state. Constraints (9), (10), (11) are therefore
used to model plant operation, while investment and operating costs are
modelled via (12).

Water Desalination Plants Reverse osmosis (RO) plants are employed to
desalinate seawater and produce freshwater (Caldera et al., 2016). This
technology essentially pumps seawater into a chamber featuring a porous
membrane and produces a pressure differential across the membrane, enabling
dead-end filtration and the recovery of freshwater on the other side of the
membrane. The plants are modelled with two external variables and one internal
variable. The external variables are the power required to drive pumps and the
freshwater outflow, whereas the internal variable represents the plant
capacity. The reference commodity $r^{\prime}$ according to which the plant is
sized is the freshwater flow out of the system. For mechanical reasons,
membranes are usually designed to operate under constant pressure and plants
therefore operate more or less continuously. Hence, constraints (5),(7),(9),
(10), (11) are used to model plant sizing and operation, while investment and
operating costs are modelled via (12). Note that the seawater inflow and brine
discharge are not modelled. The implicit assumptions are that seawater is
freely available and that the brine by-product can be disposed of at no cost,
without any restriction on pumped volumes.

Direct Air Capture Units Direct air capture units extract carbon dioxide from
the atmosphere (Kiani et al., 2020). The process used in this paper is the one
proposed by Keith et al. (2018). Roughly speaking, this process relies on four
main chemical reactions, which are combined to form two chemical loops. In the
first loop, aqueous sorbents are used in an air contactor to chemically bind
carbon dioxide and form dissolved compounds. These compounds then react with
pellets in a fluidised-bed reactor, making it possible to recover the
aforementioned sorbents and trap carbon in solid compounds. The second loop
essentially recovers carbon dioxide by calcining the solid compounds and
replenishes the pellet stock by hydrating (slaking) the solid product of the
calcination reaction. The process requires electricity to power fans driving
air through the contactors, pumps maintaining the flow of aqueous solutions as
well as compressors compressing the output carbon dioxide stream from
atmospheric pressure to $20$ bar (the associated energy expense is
approximated via the polytropic compression work, assuming a polytropic
efficiency of $80$%). The net power consumption is obtained as the difference
between the total consumption of these subsystems and the power produced by a
steam turbine recovering slaking heat. A sustained water supply is also
necessary to form aqueous solutions, counter natural evaporation in the air
contactors and produce steam used in the slaker. Furthermore, a source of heat
at around 900°C is required for the calcination reaction. In the original
design, natural gas is burnt via an oxy-fuel combustion process at the bottom
of the calciner to provide this heat, and the off-gases fluidise the reactor.
In this paper, it is assumed that the high temperature heat (approximately
$1.46$ MWh per ton of carbon dioxide) is provided by burning hydrogen
(assuming a lower heating value of $33.3$ MWh per ton of hydrogen burnt).
Hence, the process is modelled using four external variables and one internal
variable. The external variables represent the power, water and hydrogen
inflows as well as the carbon dioxide outflow, while the internal variable is
the plant capacity. The reference flow according to which the plant is sized
is the carbon dioxide outflow. None of the technologies implementing the
various reactions really lend themselves to highly variable operation.
Constraints (5),(7),(9), (10), (11) are therefore used to model plant sizing
and operation, while investment and operating costs are modelled via (12).

Methane Liquefaction Units Liquefaction units turn gaseous methane into
liquefied methane (Pospisil et al., 2019). This technology typically relies on
compressors and pumps in order to progressively compress and cool the methane
inflow, which is eventually throttled and liquefied via the Joule-Thomson
effect. In this case, three external variables and one internal variable are
used. The external variables represent the methane inflow, the power
consumption of compressors and pumps as well as the liquefied methane outflow
(which is the reference commodity), while the internal variable represents the
plant capacity. Constraints (5),(7),(9), (10), (11) are used to model plant
sizing and operation, while investment and operating costs are modelled via
(12).

Liquefied Methane Carrier Vessels Liquefied methane is transported to market
with large ocean-going vessels powered by dual fuel diesel electric (DFDE)
engines (Howard Rogers (2018), for the Oxford Institute for Energy Studies).
These engines are particularly efficient and can run solely on natural boil-
off gas (i.e., gaseous methane resulting from the natural evaporation of
liquefied methane stored on board in insulated cargo tanks). This allows
vessels to sail at a speed of 19 knots, with approximately 0.1% of their cargo
evaporating due to natural boil-off per day spent at sea, which is used for
propulsion (i.e., no other fuel is needed). The liquefied methane heel that
must usually be maintained for the return journey to guarantee that the
onboard tanks remain cool (roughly 4-5% of the total cargo) is neglected in
this paper. Two external variables and one internal variable are used to
describe a stylised carrier vessel. The external variables represent the flow
of liquefied methane loaded at the coastal hub and the flow of liquefied
methane unloaded at the destination, respectively. The internal variable is
the vessel capacity. Eq. (5) is used to model the transport of liquefied
methane, with $\tau=116$ hours, as the berthing and travel time between the
coastal hub and the destination is assumed to take slightly less than 5 days.
The conversion factor $\phi\approx 0.994$ represents the transport efficiency,
computed from the boil-off consumption (0.125% of cargo per day) and trip
duration (116 hours). In addition, loading and unloading may only be possible
when the vessel is moored at the coastal hub and destination, respectively.
This is enforced via Eq. (7) and time series $\pi_{t}^{n}$ (with values equal
to 0 or 1), which defines a berthing, mooring, loading and unloading schedule
(loading or unloading take place when $\pi_{t}^{n}=1$). For the sake of
simplicity, $\pi_{t}^{n}$ represents an aggregate schedule constructed from 7
different, non-overlapping schedules corresponding to individual carrier
vessels. Some of these schedules are shown in Figure 5 (loading and unloading
is assumed to take 24 hours). The standard local objective (12) is used for
the stylised carrier.

Figure 5: Subset of non-overlapping schedules used to construct the aggregate
schedule $\pi_{t}^{n}$ of stylised liquefied methane carriers. These time
series are summed to obtain the aggregate schedule $\pi_{t}^{n}$. For all
other nodes except wind and solar PV power plants, $\pi_{t}^{n}=1$ over the
entire time horizon.

Liquefied Methane Regasification Units Regasification units are used to
transform liquefied methane into gaseous methane at the destination (Dongsha
et al., 2017). The heat required to do so can come from a variety of sources.
In this case, it is assumed to come from the combustion of a fraction of the
methane (around 2%). Thus, two external variables and one internal variable
are used. The external variables represent the liquefied methane inflow as
well as the gaseous methane outflow, and the internal variable is the plant
capacity. Constraints (5),(7) are used to model plant sizing and operation,
while investment and operating costs are modelled via (12).

####  5.1.2 Storage Nodes

Storage nodes are discussed in this subsection. Tables 3, 4 and 5 gather the
techno-economic data (2030 estimates) used to model storage nodes along with
the original data sources and complement the descriptions below. In the model,
power flows are measured in GW (GWh/h), energy is measured in GWh, mass flows
are measured in kt/h, mass is measured in kt, and money is measured in M€.

Table 3: Technical parameters used to model storage nodes.

| $\eta^{S}$ | $\eta^{+}$ | $\eta^{-}$ | $\sigma$ | $\rho$ | $\phi$
---|---|---|---|---|---|---
Battery Storage | 0.00004 | 0.959 | 0.959 | 0.0 | 1.0 |
(Danish Energy Agency, 2020a) | - | - | - | - | - |
Compressed H2 Storage | 1.0 | 1.0 | 1.0 | 0.05 | 1.0 | 1.3
(Danish Energy Agency, 2020a) |  |  |  |  |  | GWhel/kt${}_{H_{2}}$
Liquefied CO2 Storage | 1.0 | 1.0 | 1.0 | 0.0 | 1.0 | 0.105
(Mitsubishi Heavy Industries (2004), for IEA Greenhouse Gas R&D Programme) |  |  |  |  |  | GWhel/kt${}_{CO_{2}}$
Liquefied CH4 Storage | 1.0 | 1.0 | 1.0 | 0.0 | 1.0 |
| - | - | - | - | - |
H2O Storage | 1.0 | 1.0 | 1.0 | 0.0 | 1.0 | 0.00036
(Caldera et al., 2016) |  |  |  |  |  | GWhel/kt${}_{H_{2}O}$

Table 4: Economic parameters used to model storage nodes (stock component,
2030 estimates).

| CAPEX | FOM ($\vartheta_{f}$) | VOM ($\vartheta_{v}$) | Lifetime
---|---|---|---|---
Battery Storage | 142.0 | 0.0 | 0.0018 | 10.0
(Danish Energy Agency, 2020a) | M€/GWh | M€/GWh-yr | M€/GWh | yr
Compressed H2 Storage | 45.0 | 2.25 | 0.0 | 30.0
(Danish Energy Agency, 2020a) | M€/kt | M€/kt-yr | M€/kt | yr
Liquefied CO2 Storage | 1.35 | 0.0675 | 0.0 | 30.0
(Mitsubishi Heavy Industries (2004), for IEA Greenhouse Gas R&D Programme) | M€/kt | M€/kt-yr | M€/kt | yr
Liquefied CH4 Storage | 2.641 | 0.05282 | 0.0 | 30.0
(Interior Gas Utility, 2013) | M€/kt | M€/kt-yr | M€/kt | yr
H2O Storage | 0.065 | 0.0013 | 0.0 | 30.0
(Caldera et al., 2016) | M€/kt | M€/kt-yr | M€/kt | yr

Table 5: Economic parameters used to model storage nodes (flow component, 2030
estimates).

| CAPEX | FOM ($\theta_{f}$) | VOM ($\theta_{v}$) | Lifetime
---|---|---|---|---
Battery Storage | 160.0 | 0.5 | 0.0 | 10.0
(Danish Energy Agency, 2020a) | M€/GW | M€/GW-yr | M€/GWh | yr
Liquefied CO2 Storage | 48.6 | 2.43 | 0.0 | 30.0
(Mitsubishi Heavy Industries (2004), for IEA Greenhouse Gas R&D Programme) | M€/(kt/h) | M€/(kt/h) | M€/kt | yr
H2O Storage | 1.55923 | 0.0312 | 0.0 | 30.0
(Caldera et al., 2016) | M€/(kt/h) | M€/(kt/h) | M€/kt | yr

Stationary Battery Storage Nickel manganese cobalt (NMC) oxide lithium-ion
batteries are used for short-term electricity storage (Danish Energy Agency,
2020a). Power in- and outflows are modelled using external variables. The
state of charge, power capacity and energy capacity, on the other hand, are
modelled as internal variables. Constraints (13), (15), (17), (18), (20), (21)
are used, while the local objective function is given in (22).

Hydrogen Storage Tanks Compressed hydrogen storage tanks are considered in
this paper. More precisely, overground, man-made steel storage vessels (type
I) withstanding pressure levels around 200 bar and suitable for stationary
applications are used (Danish Energy Agency, 2020a). A minimum inventory level
of 5% is assumed to represent the cushion gas (effectively reducing the
working volume). Since hydrogen at 20 bar and 40°C is produced by electrolysis
plants, the hydrogen inflow must be compressed to 200 bar using electric
compressors for storage purposes. The associated energy expense is
approximated via the polytropic compression work (assuming a polytropic
efficiency of 80%) ((2003), JRC). Thus, three external variables and three
internal variables are used. The external variables represent the hydrogen
inflow, the electricity consumption and the hydrogen outflow, while the state
of charge, the power capacity and the energy capacity are modelled as internal
variables. Constraints (13), (14), (15), (17), (18), (20), (21) are used,
while the local objective function is (22).

Liquefied Carbon Dioxide Storage Tanks Liquefied carbon storage tanks are used
to store carbon dioxide. Liquefaction and regasification units are also
required (Mitsubishi Heavy Industries (2004), for IEA Greenhouse Gas R&D
Programme). Liquefaction units consume electricity, while regasification units
are assumed to use ambient heat to recover gaseous carbon dioxide. Hence, in
this case, three external variables and five internal variables are used. The
external variables are the gaseous carbon dioxide inflow, the power
consumption of the liquefaction units and the gaseous carbon dioxide outflow,
while the internal variables represent the state of charge, the tank capacity,
capacities of liquefaction and regasification units, and the flows of
liquefied carbon dioxide in and out of the tanks. Constraints (13), (14),
(15), (17), (18), (20), (21) are used, while the local objective function is
(22).

Liquefied Methane Storage Tanks Liquefied methane is stored in full
containment tanks (i.e., tanks with both inner and outer containment walls and
such that the annular gap between both walls is sealed to prevent any gaseous
leaks (Interior Gas Utility, 2013)). It is assumed that the boil-off gas
keeping the content of the storage tanks cold is re-liquefied and pumped back
into the tanks but the electricity consumption required to do so is neglected.
Two external variables and two internal variables are used. The external
variables are the liquefied methane in- and outflow, while internal variables
represent the state of charge and the storage capacity. Constraints (13),
(15), (17), (18), (20), (21) are used, while the local objective function is
(22).

Water Storage Tanks Water is stored in tanks equipped with electric pumps
(Caldera et al., 2016). Three external variables and three internal variables
are used. The external variables correspond to the water inflow, the power
consumed by pumps and the the water outflow. The internal variables represent
the state of charge, the tank capacity and the flow capacity of pipes feeding
into the tank. Constraints (13), (15), (17), (18), (20), (21) are used, while
the local objective function is (22).

####  5.1.3 Conservation Hyperedges

Inland Power Balance This hyperedge enforces active power flow conservation
(which derives from Kirchhoff’s current law) in the inland cluster. It
therefore guarantees that the sum of power flows from the solar PV plant, the
wind power plant and the battery is equal to the sum of power flows to the
HVDC interconnection and the battery. Note that both in and outflows are used
for the battery, which correspond to discharge and charge flows, respectively.
No exogenous power injections and withdrawals take place over this hyperedge,
hence $\lambda_{t}^{e}=0,\forall t\in\mathcal{T}$. This is assumed to be the
case for all other hyperedges as well, unless otherwise stated.

Coastal Power Balance This hyperedge enforces active power flow conservation
in the coastal cluster. It therefore guarantees that the power flow from the
HVDC interconnection is equal to the sum of power flows to the direct air
capture plant, the electrolysis plant, the hydrogen storage system, the
methane liquefaction units, the desalination plant and the liquefied carbon
dioxide storage system.

Coastal Hydrogen Balance This hyperedge enforces conservation of hydrogen mass
flows in the coastal cluster. It therefore guarantees that the sum of flows
from the electrolysis plants and the storage system is equal to the sum of
flows to the direct air capture plants, the methanation plants and the storage
system.

Coastal Carbon Dioxide Balance This hyperedge enforces conservation of
(gaseous) carbon dioxide mass flows in the coastal cluster. It therefore
guarantees that the sum of flows from the direct air capture units and the
storage system is equal to the sum of flows to methanation plants and the
storage system.

Coastal Water Balance This hyperedge guarantees that the aggregate flow of
freshwater generated by desalination and methanation plants in the coastal
cluster exceeds the aggregate flow consumed by electrolysis and direct air
capture plants. Hence, it is assumed that any freshwater surplus may be
released into the environment without harm or used in other applications
(e.g., cooling), and the equality constraint in Eq. (23) is relaxed to a
$\geq$ inequality.

Coastal Methane Balance This hyperedge enforces conservation of (gaseous)
methane mass flows in the coastal cluster, and guarantees that flows from the
methanation plants and to the liquefaction units are equal.

Coastal Liquefied Methane Balance This hyperedge enforces conservation of
liquefied methane mass flows in the coastal cluster. Thus, it guarantees that
the sum of flows from the liquefaction units and the storage system is equal
to the sum of flows to the storage system and the liquefied methane carriers.

Destination Liquefied Methane Balance This hyperedge enforces conservation of
liquefied methane mass flows at the destination. Hence, it guarantees that the
sum of flows from the liquefied methane carriers and the storage system is
equal to the sum of flows to the regasification units and storage system.

Destination Methane Balance This hyperedge guarantees that the exogenous
demand for methane at the destination is satisfied by the flows from the
regasification units. The gas demand is set to $10$ TWh (HHV) per annum, but
no specific assumptions about the end-uses or sectors relying on synthetic
methane are made. Furthermore, the liquefied methane terminal located in
Northwestern Europe is assumed to be connected to some existing gas network
infrastructure. Although the latter is not explicitly modelled, it is assumed
to be able to absorb both short-term and seasonal demand variability (e.g.,
via its line pack and seasonal storage facilities (Correa-Posada and Sánchez-
Martín, 2015), as is the case in most systems). The demand profile, which
essentially represents injections from the terminal into the gas network, is
therefore assumed to be flat. Hence, assuming that synthetic methane has a HHV
of $15.441$ kWh/kg, the demand profile is obtained as
$\lambda_{t}^{e}=(10\times 10^{3}/8760)\times(1/15.441)\approx 0.07393$ kt/h,
$\mbox{ }\forall t\in\mathcal{T}$. Note that using the LHV would have resulted
in a higher mass flow rate.

###  5.2 Scenarios

One reference scenario and a comprehensive sensitivity analysis are presented.
The reference scenario studies a system configuration that relies on a
combination of solar PV and wind power plants for electricity generation. A
uniform weighted average cost of capital (WACC) of 7% is assumed for all
technologies, which represents the case where the funds required to finance
the system are borrowed on capital markets. Under these assumptions, for a
conversion or storage technology $n$, the CAPEX values in Tables 2, 4 and 5
are used to compute

$\zeta^{n}=\mbox{CAPEX}_{n}\times\frac{\mbox{w}}{(1-(1+\mbox{w})^{-\mbox{L}_{n}})},$
(24)

with $\mbox{L}_{n}$ the lifetime of technology $n$ and w the WACC. Hence,
$\zeta^{n}$ represents the annualised cost of investing in technology $n$.

The sensitivity analysis investigates the impact of a number of techno-
economic parameters and assumptions on synthetic methane cost, which include
the availability of wind power plants, the investment costs of electrolysis,
direct air capture and methanation plants, the operational flexibility of the
latter two technologies as well as the energy consumption of direct air
capture plants and the financing costs. More specifically, a hypothetical
situation where the cost of financing the system is zero is studied, such that
the cost of synthetic methane production and delivery solely reflects the cost
and efficiency of technologies in the supply chain. In this set-up, Eq. (24)
cannot be used and annualised investment costs are instead computed via

$\zeta^{n}=\frac{\mbox{CAPEX}_{n}}{\mbox{L}_{n}}.$ (25)

##  6 Results

###  6.1 Reference Scenario

In the reference scenario, a system configuration relying on solar and wind
power plants is studied assuming a WACC of $7$%. In this set-up, synthetic
methane is delivered to market in gaseous form at $149.7$ €/MWh (HHV), which
is computed as the ratio of total (annualised) system cost to methane volume
delivered ($10$ TWh HHV per year). It is worth noting that using the LHV would
have increased the cost per MWh, as this would have effectively reduced the
amount of energy that could have been retrieved per unit mass of methane
delivered.

Figure 6: Breakdown of synthetic methane cost at destination for reference
scenario. All contributions roughly sum to $149.7$ €/MWh (HHV).

The synthetic methane cost breakdown is provided in Figure 6, where each bar
represents the contribution (in €/MWh) of the corresponding technology to
synthetic methane cost. Each bar can also be interpreted as representing the
contribution of the corresponding technology to total system cost. Wind
turbines account for the lion’s share of synthetic methane cost (roughly
$28.9\%$). Electrolysis ($19.6\%$) and solar PV plants ($11.3\%$) come in
second and third, respectively, although they each represent a much smaller
proportion of costs (taken together, they contribute slightly more than wind
turbines to total system cost). Overall, the technologies used to generate,
transport and store electricity (shown in gold in Figure 6) represent the
largest share of costs (around $56.6\%$). Hydrogen storage plants, which are
used as a buffer between flexible electrolysis and inflexible methanation
plants, make up approximately $5\%$ of total cost. Hence, the technologies
producing and storing hydrogen (shown in light blue in Figure 6) account for
roughly $25\%$ of total system cost. It is worth noting that the plants
upstream of the inflexible plants (i.e., methanation, direct air capture and
desalination plants) make up almost $80\%$ of total system cost. On the other
hand, methanation plants make up a minor share of total cost (approximately
$7.7\%$), and the full methane chain (i.e., production, liquefaction, storage,
transport and regasification, shown in light orange in Figure 6) accounts for
roughly $12.5\%$ of final product cost. Direct air capture plants also
represent a minor fraction of system cost (around $7.9\%$, shown in green in
Figure 6). Water desalination and storage technologies are deployed in
moderate quantities, resulting in a very small share of total costs (well
under 1%), while carbon dioxide storage is not deployed.

Figure 7: Material and energy balance diagram for the reference scenario,
along with technology capacities. All energy-equivalent flows of energy
carriers other than electricity have been computed using their HHV. Flow
values represent yearly averages (i.e., flows have been integrated over the
full time horizon and divided by the number of years) and all values have been
rounded up to keep significant digits only.

Analysing mass and energy balances provides some insight into system design
and operation. Figure 7 displays mass and energy balances (flow values are
integrated over the full optimisation horizon of one year) along with
technology capacities. Firstly, as can be seen in Figure 7, the average annual
electricity production of solar PV and wind power plants is equal to $8.90$
and $13.87$ TWh, respectively, which suggests that the full supply chain has a
conversion efficiency of roughly $43.9$%. However, the amount of curtailment
is substantial and stands at $5.26$ TWh, which represents slightly less than
one quarter of the useful power production. This effectively decreases the
capacity factors of the photovoltaic and wind power plants from their
theoretical maximum of $24.6$% and $50.0$% over 2015-2019 (i.e., corresponding
to the case where all electricity produced is used) to $23.8$% and $36.8\%$
(taking only useful power production into account). However, the high share of
curtailment attributed to wind power plants must be taken with caution.
Indeed, since curtailment is not penalised in the objective, some natural
symmetry exists in the model when solar PV and wind power plants jointly
produce more than the system can absorb (which typically occurs during hours
of peak solar PV production). More specifically, in such situations, solutions
curtailing a given share of solar PV output would yield the same objective
values as that of other solutions curtailing the same given share of wind
power output, and curtailment could therefore be attributed to one technology
or the other without any impact on the objective. The high overall rate of
curtailment can nevertheless be explained by the difficulty of effectively
absorbing the highly variable aggregate power input from renewable power
plants. This is a direct consequence of the fact that the operating regimes of
several key conversion technologies are inflexible, which has two further
implications. Firstly, battery and hydrogen storage systems are deployed at
great cost in order to smooth the variability of the power supply as much as
possible. Secondly, plants located upstream of the inflexible ones are
typically oversized (especially solar PV power plants), as the level of
smoothing required to guarantee steady power and hydrogen flows cannot be
economically provided by storage plants alone. Additional evidence supporting
this analysis is provided in the following section.

###  6.2 Sensitivity Analyses

The sensitivity analysis investigates the impact of a number of techno-
economic parameters and assumptions on synthetic methane cost. More precisely,
the impacts of i) being unable to deploy wind power plants ii) the operational
flexibility of direct air capture, methanation and desalination plants iii)
the investment costs of electrolysis, direct air capture and methanation
plants iv) the energy consumption of direct air capture plants v) financing
costs are assessed. The cost share, capacities and capacity factors of the
technologies that were found to contribute the most to total system cost in
Section 6.1 are computed and analysed in this section. Hence, Figure 8
displays the breakdowns of synthetic methane costs obtained under various
techno-economic assumptions, while Figure 9 gathers the capacities and average
capacity factors of key conversion and storage technologies. These results are
elaborated upon below.

Figure 8: Breakdown of synthetic methane costs obtained under various techno-
economic assumptions. Figure 9: Capacities and average capacity factors (shown
as greyed fractions of capacity bars) of key conversion and storage
technologies under various techno-economic assumptions.

Solar PV System This analysis assumes that wind power plants cannot be
deployed, and the full electricity supply must therefore come from solar PV
power plants. As can be seen in Figure 8, the cost of synthetic methane for
this configuration is around $202$ €/MWh, which is almost $35\%$ more
expensive than that found in the reference scenario. In this case,
electrolysis plants contribute the most to total system cost, followed by
solar PV power plants. In addition, it is clear from Figure 8 that the cost of
each technology located upstream of the inflexible plants (i.e., methanation,
direct air capture and desalination plants) increases in absolute terms
compared with the reference scenario. The only technologies whose cost remains
the same are direct air capture and methanation. This claim is supported by
the results shown in Figure 9. Indeed, the capacities of conversion and
storage technologies located upstream of the inflexible ones are larger in the
solar-only case, while the capacities of inflexible technologies are equal in
both cases. Furthermore, the average capacity factors of key conversion
technologies (e.g., electrolysis plants) are much lower in the solar-only
case. This observation suggests that these plants are oversized to help absorb
the highly variable input from solar PV power plants, as smoothing it with
storage systems alone would be uneconomical. Overall, relying on solar PV
power plants alone results in a system design that it is much less efficient
and much more expensive than the one identified in the reference scenario.

System Flexibility In the reference scenario, several technologies were
assumed to be inflexible, namely methanation (MT), direct air capture (DAC)
and desalination (DS) plants. Their inflexibility, combined with the fact that
the renewable power supply is highly variable, was found to have an impact on
synthetic methane cost. Although these assumptions are well-founded, the
minimum level and ramping constraints of the three aforementioned technologies
are now relaxed (i.e., $\mu=0.0$ and $\Delta_{+}=\Delta_{-}=1.0$) in order to
evaluate the sensitivity of our results. Figure 8 shows that shifting to a
system with fully flexible methanation, direct air capture and desalination
plants can lead to cost savings around $6$%, and has a substantial impact on
the capacities of several technologies. More specifically, the capacity of
solar PV power plants decreases by $10$%, while the capacities of battery and
hydrogen storage systems shrink by $40$% and $80$%, respectively. On the other
hand, the capacities of methanation and direct air capture plants increase by
$20$% and $11$%, respectively, which slightly offsets the cost savings made
elsewhere. Although not shown in Figure 9, the capacity of liquefied methane
storage tanks in the coastal hub more than doubles. This can be explained by
the fact that the transport of liquefied methane does not occur on a
continuous basis, which introduces some inflexibility in the supply chain.
Since liquefied methane storage is much cheaper than battery or hydrogen
storage, the buffer absorbing the variability of renewable power generation is
moved downstream in the supply chain. It is worth noting that even if the
transport of liquefied methane took place continuously, the mismatch between
the production and demand profiles would need to be balanced by some storage
capacity, which will always come at a cost.

Investment Costs A number of technologies used in the proposed remote carbon-
neutral synthetic methane supply chain have not yet reached full maturity. In
particular, electrolysis (EL), direct air capture and methanation plants are
still undergoing development, and their costs thus remain highly uncertain to
2030. The cost figures used for electrolysis plants in the reference scenario
lean towards the moderately optimistic side for 2030, and achieving them would
require both sustained research and development efforts and a commercial
scale-up (Schmidt et al., 2017). Likewise, the cost figures used for direct
air capture plants are on the fairly optimistic side for 2030. Indeed, the
cost of each ton of carbon dioxide captured from the atmosphere is between 65
and 70 € (energy costs excluded) in the reference scenario, which is close to
long-term cost targets of $100/ton (energy costs included) that Keith et al.
(2018) seek to reach. Methanation cost estimates, however, are rather
conservative (International Energy Agency (2019), IEA; Agora Verkehrswende,
Agora Energiewende and Frontier Economics, 2018). The uncertainty around these
costs and their impact on synthetic methane cost is resolved as follows.
Firstly, the CAPEX and FOM of electrolysis and direct air capture plants are
increased by 50%, which yields synthetic methane cost estimates around 168.1
€/MWh (which is approximately $12$% higher than that of the reference
scenario). Then, the CAPEX and FOM of electrolysis, direct air capture and
methanation plants are decreased by 50%, first on an individual basis and then
all at once. Although such drastic cost reductions seem very unlikely, they
nevertheless make it possible to estimate the sensitivity of synthetic methane
cost to these assumptions and provide a lower bound on costs that may
realistically be achieved. The least sensitive of these parameters is the cost
of direct air capture plants, followed by the cost of methanation plants.
Decreasing them individually only leads to methane cost reductions of $5$% or
less. Decreasing the cost of electrolysis plants by $50$%, however, has a much
greater impact and leads to cost savings around $10$%. Decreasing the costs of
all three technologies at once leads to cost savings of roughly $16$% and
yields a strong lower bound of $125.1$€/MWh on the reductions in synthetic
methane cost that may be achieved in this fashion. It is worth noting that
these cost savings are achieved with virtually no change in deployed
capacities (as shown in Figure 9), which suggests that the latter mostly
depend on renewable production profiles, the flexibility of technologies and
the demand that must be satisfied.

DAC Energy Consumption The direct air capture process used in this analysis
requires high-temperature heat to calcine calcium carbonate compounds and
release the carbon dioxide that they trap. In this paper, it is assumed that
this heat is provided by burning hydrogen, and slightly less than 20% of the
total hydrogen production is used to this end. Hence, the fact that this
hydrogen must be produced from renewable electricity leads to the deployment
of additional power generation, storage, transport, electrolysis and hydrogen
storage capacity, which directly translates into a lower overall efficiency of
the full supply chain and higher synthetic methane costs. Instead, a different
capture process that only uses low to medium-temperature heat and electricity
could be used (Wurzbacher et al., 2011). It has been estimated that this
process consumes approximately $0.5$ MWh of electricity per ton of carbon
dioxide captured and requires roughly $2.5$ MWh of heat at 100°C. Since the
production of one ton of synthetic methane by the Sabatier reaction releases
$2.87$ MWh of high temperature heat (and requires $2.75$ tons of carbon
dioxide), some of the heat required by the direct air capture process could be
supplied by nearby methanation plants. The impact of such a change is analysed
by increasing the electricity consumption of the direct air capture process
fivefold (i.e., setting it to roughly $0.5$ MWh/ton instead of $0.1$ MWh/ton
in the reference scenario) and setting the hydrogen consumption to zero. The
same cost figures those of the reference scenario are used. Results shown in
Figures 8 and 9 confirm the intuition that using hydrogen to satisfy the heat
requirements of a high-temperature direct air capture process has a
substantial impact on synthetic methane cost and the capacities of various key
technologies. More precisely, the cost of synthetic methane in this
configuration is approximately 10% lower than that found in the reference
scenario, while the capacities of power generation and electrolysis plants are
$10$% and $20$% smaller, respectively. Although promising, a detailed analysis
of the heat integration potential and the cost of this process should be
performed in order to confirm these findings.

Financing Costs A weighted average cost of capital of $7$% has so far been
used. In order to evaluate the impact of financing costs, a hypothetical
situation where the cost of financing the system is set to zero is now studied
(i.e., annualised CAPEX values are calculated using Eq. 25). Thus, in this
set-up, the cost of synthetic methane production and delivery solely reflects
the cost and efficiency of technologies in the supply chain, and provides an
absolute lower bound on costs that may be realistically achieved. Results in
Figure 8 suggest that neglecting financing costs leads to a synthetic methane
cost of $88.3$ €/MWh (which corresponds to a $40$% reduction compared with the
reference scenario), which is by far the lowest observed in this paper and
highlights the influence of weighted average cost of capital assumptions. It
is also worth noting that this cost decrease is achieved with little change in
the capacities of conversion and storage technologies compared with the
reference scenario.

###  6.3 Discussion

Discrepancies exist between the results presented in Sections 6.1 and 6.2 and
synthetic methane production cost estimates published elsewhere in the
literature. Indeed, recall that Zeman and Keith (2008) provide cost estimates
ranging from 74.1 to 94.6 €/MWh. Furthermore, in Fasihi et al. (2015), the
cost of producing synthetic methane from renewable electricity in central and
southern Algeria and delivering it to Japan is estimated to be around 65-75
€/MWh in 2030 for a hybrid solar-wind system using a WACC of 7%. In Fasihi et
al. (2017), the cost of producing synthetic methane in the same region and
delivering it to Finland is estimated to be between 100-110 €/MWh (HHV) by
2030 and between 90-100 €/MWh (HHV) by 2040, respectively, using a WACC of 7%.
Finally, in Agora Verkehrswende, Agora Energiewende and Frontier Economics
(2018), a uniform WACC of 6% is used, yielding cost estimates around 140 €/MWh
(LHV) for a solar PV configuration and around 150 €/MWh (LHV) for a hybrid
solar-wind configuration. It is worth noting that the hybrid solar-wind
configuration is slightly more expensive than the solar-powered system in
their reference cost scenario.

The methods used in the aforementioned papers, which are discussed in Section
3, suffer from several shortcomings. More precisely, they use a very low
temporal resolution (one time period per year in the so-called full load hour
model) that completely smoothes out the variability of power production
signals. Furthermore, their models have a very low level of technical detail.
The combination of these two features makes it very difficult to capture the
interaction between subsystems and accurately model the supply chain in an
integrated fashion. Hence, this typically removes the need for oversizing
renewable power generation technologies or deploying flexibility options such
as storage systems to balance the variable power supply and satisfy operating
constraints, in spite of the fact that the operation of some technologies
further down the chain is inflexible or discontinuous (e.g., methanation
plants or transport by liquefied methane carrier vessels). Oversizing plants
or deploying storage technologies is relatively expensive and both account for
a non-negligible share of the final methane cost, as discussed in Sections 6.1
and 6.2. For example, the fact that solar PV variability has been completely
smoothed out by the full load hour model used in Agora Verkehrswende, Agora
Energiewende and Frontier Economics (2018) explains the fact that solar-only
and hybrid solar-wind configurations yield very close methane cost estimates,
while the solar-only configuration is almost 35% more expensive than the
hybrid wind-solar configuration considered in this paper. Thus, the
aforementioned papers underestimate final product cost as a result of
inadequate modelling choices. In addition, some of the techno-economic
assumptions made in Fasihi et al. (2015, 2017) seem particularly optimistic.
For example, the CAPEX of electrolysis and methanation plants is approximately
two and three times lower than the values used in the reference case presented
in this paper, respectively. These assumptions clearly lead to low methane
cost estimates but are poorly supported. Indeed, to the authors’ best
knowledge, such assumptions do not appear elsewhere in the literature or in
publicly-accessible databases and are therefore difficult to cross-check.

##  7 Conclusion and Future Work

This paper studies the economics of carbon-neutral synthetic fuel production
in remote areas where high-quality renewable resources are abundant. With this
goal in mind, a (hyper)graph-based optimisation modelling framework directly
applicable to the strategic planning of remote renewable energy supply chains
is proposed. The method is leveraged to study the economics of carbon-neutral
synthetic methane production from solar and wind energy in North Africa.

The full supply chain is modelled and optimised in an integrated fashion over
five years (2015-2019) with hourly time resolution. Essential operational
constraints are taken into account, which is key for accurately capturing
interactions between subsystems. Results suggest that the cost of synthetic
methane delivery to northwestern European consumers would be around $149.7$
€/MWh (HHV) by 2030 for a system that relies on a combination of solar
photovoltaic and wind power plants, assuming a uniform weighted average cost
of capital of 7%. A comprehensive sensitivity analysis has also been carried
out in order to evaluate the impact of various techno-economic parameters and
assumptions on synthetic methane cost, including the availability of wind
power plants, the investment costs of electrolysis, methanation and direct air
capture plants, their operational flexibility, the energy consumption of
direct air capture plants, and financing costs. The most expensive
configuration (around 200 €/MWh) relies on solar photovoltaic power plants
alone, while the cheapest configuration (around 88 €/MWh) makes use of a
combination of solar PV and wind power plants and is obtained when financing
costs are set to zero. The cost estimates found for the reference scenario and
the configuration relying solely on solar PV power plants are much higher than
those previously published in the literature. This discrepancy can be partly
explained by the fact that the models used in previous studies had a very low
temporal resolution and failed to properly capture the interactions between
highly variable power generation plants (especially solar photovoltaic units)
and inflexible conversion technologies (such as methanation plants) and demand
profiles.

Several research directions can be pursued in future work. Firstly,
quantitatively analysing some of the options suggested for cost reductions
would provide more insight into the economic potential of an energy supply
pathway based on carbon-neutral methane synthesis in remote areas. Then,
leveraging the framework to study different pathways involving different
regions (and thus resource types and profiles) and energy carriers (e.g.,
hydrogen, methanol or ammonia), would allow one to draw a complete picture of
energy supply options and to compare their respective merits. Finally, the
graph-based modelling framework could be expanded in different ways. For
instance, the class of problems that can be represented could be broadened by
introducing nonlinear expressions. The graph representation could also be
exploited to facilitate preprocessing tasks and the analysis of model
properties, eventually opening the door to the deployment of more efficient
solution methods that better exploit problem structure (Jalving et al., 2019).

##  8 Nomenclature

| Sets and Indices
---|---
$\mathcal{E},e$ | set of hyperedges and hyperedge index
$e_{T},e_{H}$ | tail and head of hyperedge $e\in\mathcal{E}$
$\mathcal{G}$ | hypergraph with node set $\mathcal{N}$ and hyperedge set $\mathcal{E}$
$\mathcal{I}^{n},i$ | set of external variables at node $n$, and variable index
$\mathcal{N},n$ | set of nodes and node index
$\mathcal{T},t$ | set of time periods and time index
| Parameters
$\nu\in\mathbb{N}$ | number of years spanned by optimisation horizon
$\pi_{t}^{n}\in[0,1]$ | (operational) availability of conversion node $n$ at time $t$
$\underline{$\kappa$}^{n}\in\mathbb{R}_{+}$ | existing flow capacity of conversion or storage node $n$
$\bar{\kappa}^{n}\in\mathbb{R}_{+}$ | maximum flow capacity of conversion or storage node $n$
$\mu^{n}\in[0,1]$ | minimum operating level of conversion node $n$ (fraction of capacity)
$\delta t\in\mathbb{R}_{+}$ | duration of each time period
$\Delta_{i,+}^{n}\in[0,1]$ | maximum ramp-up rate for flow $i$ and conversion node $n$ (frac. of capacity per unit time)
$\Delta_{i,-}^{n}\in[0,1]$ | maximum ramp-down rate for flow $i$ and conversion node $n$ (frac. of capacity per unit time)
$\phi_{i}^{n}\in\mathbb{R}_{+}$ | conversion factor between reference flow $r$ and flow $i$ for conversion or storage node $n$
$\tau_{i}^{n}\in\mathbb{N}$ | conversion time delay for flow $i$ of conversion node $n$
$\eta_{S}^{n}\in[0,1]$ | self-discharge rate of storage node $n$
$\eta_{+}^{n}\in[0,1]$ | charge efficiency of storage node $n$
$\eta_{-}^{n}\in[0,1]$ | discharge efficiency of storage node $n$
$\sigma^{n}\in[0,1]$ | minimum inventory level of storage node $n$ (fraction of capacity)
$\bar{\epsilon}^{n}\in\mathbb{R}_{+}$ | maximum inventory capacity of storage node $n$
$\underline{$\epsilon$}^{n}\in\mathbb{R}_{+}$ | existing inventory capacity of storage node $n$
$\rho^{n}\in\mathbb{R}_{+}$ | maximum discharge-to-charge ratio of storage node $n$
$\lambda_{t}^{e}\in\mathbb{R}$ | withdrawal/injection at time $t$ and conservation hyperedge $e$
$\zeta^{n}\in\mathbb{R}_{+}$ | annualised CAPEX of node $n$ (flow component)
$\theta_{f}^{n}\in\mathbb{R}_{+}$ | FOM cost of of node $n$ (flow component)
$\theta_{t,v}^{n}\in\mathbb{R}_{+}$ | VOM cost of node $n$ (flow component)
$\varsigma^{n}\in\mathbb{R}_{+}$ | annualised CAPEX of storage node $n$ (stock component)
$\vartheta_{f}^{n}\in\mathbb{R}_{+}$ | FOM cost of storage node $n$ (stock component)
$\vartheta_{t,v}^{n}\in\mathbb{R}_{+}$ | VOM cost of storage node $n$ (stock component)
$\theta_{t,L}^{n}\in\mathbb{R}_{+}$ | cost of unserved demand at conservation node $n$
| Variables
$q_{it}^{n}\in\mathbb{R}_{+}$ | flow variable $i$ of node $n$ at time $t$
$K^{n}\in\mathbb{R}_{+}$ | new flow capacity of node $n$
$e_{t}^{n}\in\mathbb{R}_{+}$ | inventory level of storage node $n$ at time $t$
$E^{n}\in\mathbb{R}_{+}$ | new stock capacity of storage node $n$

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

Mathias Berger and Damien Ernst designed the research. Mathias Berger, David
Radu and Ghislain Detienne collected the data. Mathias Berger performed the
research and drafted the manuscript. David Radu, Ghislain Detienne, Thierry
Deschuyteneer, Aurore Richel and Damien Ernst provided feedback on the
research and manuscript.

## Acknowledgments

The authors would like to thank Adrien Bolland, Hatim Djelassi and Virginie
Pison for providing feedback on an earlier version of this manuscript. The
authors would also like to thank Julien Confetti for his precious help with
the design of figures and diagrams in this paper. Finally, the authors would
like to gratefully acknowledge the support of the Federal Government of
Belgium through its Energy Transition Fund and the INTEGRATION project.

## Data Availability Statement

The datasets and code used for this study can be found in the GBOML repository
(Miftari et al., 2021).

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